Engineering composites

ENGINEERING
COMPOSITE
MATERIALS

Bryan Harris

The Institute of Materials, London
1999

CONTENTS

1 THE NATURE OF COMPOSITE MATERIALS....................................................................... 5
1.1 WHAT ARE COMPOSITES? ..............................................................................................................................5
1.2 CONVENTIONAL MATERIALS AND THEIR LIMITATIONS ..................................................................................6
1.3 STRONG FIBRES..............................................................................................................................................7
1.3.1 Glass fibres ..........................................................................................................................................8
1.3.2 Carbon fibres.......................................................................................................................................8
1.3.3 Silicon carbide.....................................................................................................................................9
1.3.4 Alumina and alumina/silica compounds............................................................................................10
1.3.5 Organic Fibres ..................................................................................................................................10
1.3.6 Styles of reinforcement ......................................................................................................................11
1.4 THE SCOPE FOR REINFORCEMENT OF CONVENTIONAL MATERIALS................................................................12
1.4.1 Functions of the matrix......................................................................................................................12
1.4.2 Metals ................................................................................................................................................13
1.4.3 Polymeric materials...........................................................................................................................13
1.4.4 Glasses, ceramics and cement ...........................................................................................................14
1.4.5 Carbon...............................................................................................................................................15
1.5 REFERENCES TO CHAPTER 1 .........................................................................................................................15
1.5.1 General bibliography ........................................................................................................................16
1.5.2 Sources of data ..................................................................................................................................17
2 MAKING COMPOSITE MATERIALS .................................................................................... 18
2.1 THE COMBINING OF MATERIALS ...................................................................................................................18
2.2 THE INTERFACE ...........................................................................................................................................19
2.3 MANUFACTURING PROCESSES ......................................................................................................................20
2.3.1 Polymer-matrix composites ...............................................................................................................20
2.3.2 Metal-matrix composites ...................................................................................................................22
2.3.3 Ceramic-matrix composites ...............................................................................................................23
2.3.4 Hybrid composites .............................................................................................................................24
2.4 DEFECTS IN MANUFACTURED POLYMERIC COMPOSITES ...............................................................................25
2.5 METHODS OF NON-DESTRUCTIVE EVALUATION FOR POLYMER COMPOSITES.................................................28
2.5.1 Optical inspection..............................................................................................................................28
2.5.2 Radiographic methods.......................................................................................................................28
2.5.3 Thermal imaging................................................................................................................................28
2.5.4 Ultrasonic techniques ........................................................................................................................29
2.5.5 Optical fibre sensors..........................................................................................................................29
2.5.6 Microwave methods. ..........................................................................................................................30
2.5.7 Dynamic mechanical analysis. ..........................................................................................................30
2.5.8 Acoustic emission methods. ...............................................................................................................31
2.6 THE USE OF COMPOSITES ..............................................................................................................................31
3 ELASTIC PROPERTIES OF FIBRE COMPOSITES............................................................. 33
3.1 SIMPLE MICROMECHANICAL MODELS ...........................................................................................................33
3.2 THE HALPIN-TSAI EQUATIONS .....................................................................................................................36
3.3 THE COMPRESSION MODULUS.......................................................................................................................38
3.4 THE ANISOTROPIC PROPERTIES OF A THIN LAMINA .......................................................................................39
3.5 ORIENTATION-DEPENDENCE OF THE ELASTIC PROPERTIES OF A UNIDIRECTIONAL LAMINA...........................42
3.6 MULTI-PLY LAMINATES................................................................................................................................44
3.7 SHORT-FIBRE COMPOSITES ...........................................................................................................................48
3.8 HYBRID COMPOSITES ...................................................................................................................................51
3.9 RESIDUAL STRAINS ......................................................................................................................................54
3.10 TEXTILE COMPOSITES ..............................................................................................................................56
3.11 PLIANT COMPOSITES ................................................................................................................................58
3.12 REFERENCES TO CHAPTER 3 ....................................................................................................................59
4 THE STRENGTH OF FIBRE COMPOSITES................................................................................ 61

4.1 TENSILE STRENGTH ......................................................................................................................................61
4.1.1 Unidirectional continuous-fibre composites......................................................................................61
4.1.2 The problem of the strength of brittle fibres ......................................................................................63
4.1.3 Progressive damage and failure in unidirectional composites..........................................................65
4.1.4 Calculation of the tensile strength of a unidirectional composite .....................................................67
4.1.5 Transverse strength ...........................................................................................................................69
4.1.6 Orientation-dependence of composite strength .................................................................................71
4.1.7 The strengths of multi-ply laminates..................................................................................................72
4.1.8 Short-fibre composites .......................................................................................................................75
4.1.9 Hybrid composites .............................................................................................................................79
4.2 COMPRESSION STRENGTH.............................................................................................................................80
4.2.1 Compression strength after impact....................................................................................................84
4.3 SHEAR STRENGTH ........................................................................................................................................85
4.3.1 The shear strength of the interfacial bond.........................................................................................87
4.4 FLEXURAL STRENGTH ..................................................................................................................................92
4.5 FAILURE CRITERIA FOR COMPLEX STRESSES .................................................................................................93
4.6 THE STRENGTH OF CERAMIC-MATRIX COMPOSITES ......................................................................................94
5 FRACTURE AND TOUGHNESS OF COMPOSITES ............................................................ 99
5.1 STRUCTURAL ASPECTS OF COMPOSITES FAILURE ...........................................................................................99
5.1.1 Fracture formalisms ..........................................................................................................................99
5.2 FRACTURE PROCESSES IN COMPOSITES .......................................................................................................100
5.2.1 Matrix effects ...................................................................................................................................100
5.2.2 Fibre effects .....................................................................................................................................101
5.2.3 Simple fibre/matrix addition effects.................................................................................................101
5.3 TOUGHNESS DERIVING FROM COMPOSITE ACTION ......................................................................................102
5.3.1 Toughening mechanisms in fibre-reinforced plastics......................................................................102
5.3.2 Toughness of hybrid composites......................................................................................................111
5.3.3 Toughness of laminated structures ..................................................................................................112
5.4 TOUGHNESS OF PRACTICAL REINFORCED PLASTIC LAMINATES ...................................................................115
5.4.1 Short-fibre composites .....................................................................................................................117
5.4.2 Design of damage-tolerant composites ...........................................................................................118
5.5 TOUGHNESS OF CERAMIC-MATRIX COMPOSITES .........................................................................................118
5.5.1 Matrix cracking ...............................................................................................................................119
5.5.2 Fibre bridging..................................................................................................................................119
5.5.3 Fibre pull-out...................................................................................................................................120
5.6 APPLICATION OF FRACTURE MECHANICS TO COMPOSITES ..........................................................................121
5.6.1 Strength of notched composites and notch sensitivity .....................................................................121
5.6.2 The application of fracture mechanics .................................................................................121
5.6.3 Application to specific composites...................................................................................................122
5.7 CONCLUSION ..............................................................................................................................................125
5.8 REFERENCES TO CHAPTER 5 .......................................................................................................................127
6. FATIGUE BEHAVIOUR OF FIBRE COMPOSITES ........................................................... 130
6.1 INTRODUCTION ..........................................................................................................................................130
6.2 DAMAGE IN COMPOSITES ...........................................................................................................................131
6.2.1 Definition of failure and experimental scatter.................................................................................132
6.3 REVIEW OF SOME CLASSICAL FATIGUE PHENOMENA.................................................................................132
6.3.1 Stress/life curves ..............................................................................................................................132
6.3.2 Constant-life diagrams ....................................................................................................................133
6.3.3 Statistical aspects of the analysis of fatigue data ............................................................................133
6.3.4 Cumulative damage laws.................................................................................................................135
6.5 FATIGUE OF REINFORCED PLASTICS ...........................................................................................................136
6.4.1 Constant-stress fatigue curves.........................................................................................................136
6.4.2 Materials factors affecting fatigue behaviour of reinforced plastics...............................................139
6.4.3 Other factors affecting the fatigue of reinforced plastics ................................................................143
6.4.4 Damage accumulation and residual strength.....................................................................148
6.4.5 Fatigue under conditions of variable stress: block loading ............................................................151

6.4.6 Life prediction..................................................................................................................................152
6.5 FATIGUE OF METAL-MATRIX COMPOSITES ..................................................................................................153
6.6 FATIGUE OF CERAMIC-MATRIX COMPOSITES ..............................................................................................156
6.7 CONCLUSION ..................................................................................................................................................160
7 ENVIRONMENTAL EFFECTS............................................................................................... 164
7.1 INTRODUCTION ..........................................................................................................................................164
7.2 HYGROTHERMAL SENSITIVITY OF REINFORCED PLASTICS .........................................................................166
7.2.1 Fibre effects .....................................................................................................................................166
7.2.2 Resin effects .....................................................................................................................................167
7.2.3 Composite effects .............................................................................................................................168
7.3 TIME-DEPENDENT EFFECTS ........................................................................................................................170
7.3.1 GRP in Corrosive Environments .....................................................................................................171
7.4 HIGH-TEMPERATURE STRENGTH AND CREEP OF COMPOSITES ...................................................................171
7.4.1 Introduction .....................................................................................................................................171
7.4.2 Creep ...............................................................................................................................................173
APPENDIX 1......................................................................................................................................... 185
ORIENTATION-DEPENDENCE OF THE ELASTIC PROPERTIES OF A SINGLE UNIDIRECTIONAL LAMINA ......................185
APPENDIX 2......................................................................................................................................... 189
STATISTICAL ASPECTS OF THE STRENGTH OF BRITTLE SOLIDS ............................................................................189
The Weibull Model..........................................................................................................................................189
REFERENCES TO APPENDIX 2 ...............................................................................................................................192
APPENDIX 3......................................................................................................................................... 193
SOME APPLICATIONS OF COMPOSITES..................................................................................................................193
Aerospace .......................................................................................................................................................193
Automotive Engineering .................................................................................................................................193
BioEngineering ...............................................................................................................................................193
Chemical Engineering ....................................................................................................................................194
Civil/Structural Engineering...........................................................................................................................194
Domestic .........................................................................................................................................................194
Electrical Engineering....................................................................................................................................194
Marine Engineering........................................................................................................................................195
Sport................................................................................................................................................................195

1 THE NATURE OF COMPOSITE MATERIALS

1.1 WHAT ARE COMPOSITES?

C omposite materials are extending the horizons of designers in all branches of engineering, and yet
the degree to which this is happening can easily pass unperceived. The eye, after all, does not see
beyond the glossy exterior or the race performance of a GRP1 yacht, nor does it sense the
complexity of the structure of a composite helicopter rotor blade or of a modern CFRP2 tennis
racket. Nevertheless, this family of synthesized materials offers the possibility of exciting new solutions
to difficult engineering problems.

In composites, materials are combined in such a way as to enable us to make better use of their
virtues while minimising to some extent the effects of their deficiencies. This process of optimisation
can release a designer from the constraints associated with the selection and manufacture of conventional
materials. He can make use of tougher and lighter materials, with properties that can be tailored to suit
particular design requirements. And because of the ease with which complex shapes can be
manufactured, the complete rethinking of an established design in terms of composites can often lead to
both cheaper and better solutions.

The ‘composites’ concept is not a human invention. Wood is a natural composite material consisting
of one species of polymer — cellulose fibres with good strength and stiffness — in a resinous matrix of
another polymer, the polysaccharide lignin. Nature makes a much better job of design and manufacture
than we do, although Man was able to recognize that the way of overcoming two major disadvantages of
natural wood — that of size (a tree has a limited transverse dimension), and that of anisotropy
(properties are markedly different in the axial and radial directions) — was to make the composite
material that we call plywood. Bone, teeth and mollusc shells are other natural composites, combining
hard ceramic reinforcing phases in natural organic polymer matrices. Man was aware, even from the
earliest times, of the concept that combining materials could be advantageous, and the down-to-earth
procedures of wattle-and-daub (mud and straw) and ‘pide’ (heather incorporated in hard-rammed earth)
building construction, still in use today, pre-date the use of reinforced concrete by the Romans which
foreshadowed the pre-tensioned and post-tensioned reinforced concretes of our own era. But it is only in
the last half century that the science and technology of composite materials have developed to provide
the engineer with a novel class of materials and the necessary tools to enable him to use them
advantageously.

The simple term ‘composites’ gives little indication of the vast range of individual combinations that
are included in this class of materials. We have mentioned some of the more familiar ones, but the
diagram Figure 1.1 gives a clearer idea of the scope for ingenuity which is available to the Materials
Scientist and his customer, the Design Engineer. First, within each group of materials — metallic,
ceramic and polymeric — there are already certain familiar materials which can be described as
composites. Many members of the commonest and largest group of engineering materials, the family of
steels, consist of combinations of particles of hard ceramic compounds in a softer metallic matrix. These
particles are sometimes plate-like, sometimes needle-shaped, and sometimes spherical or polygonal.
Polymers, too, are often two-phased, consisting of a matrix of one polymer with distributions of harder
or softer particles contained within it; wood is a perfect example of this, as we have seen. And concrete
is a classic example of a ceramic/ceramic composite, with particles of sand and aggregate of graded sizes
in a matrix of hydrated Portland cement. These materials have been well known for many years, and
Materials Scientists have learned to control their properties by controlling their microstructures; that is to
say, the quantity, the form, and the distribution of what we might refer to as the ‘reinforcing phase’. The
idea of mixing components across the materials class boundaries is a natural extension of this idea.
Making additions of hard, or fire-resistant, or simply cheap, ceramic powders to plastics to make filled
__________________________________________________________________________________
1
GRP is the usual abbreviation for glass-fibre-reinforced plastics
2
CFRP is carbon-fibre-reinforced plastic

chapter one — The nature of composites 6

METALS AND
ALLOYS: metal-matrix composites,
metal-filled plastics ceramic-matrix composites,
steels,
(particulate and fibre fill) (including ordinary reinforced
aluminium alloys,
copper & brasses, concrete and steel-fibre rein-
titanium, etc. forced concrete)

PLASTICS:
CERAMICS &
resins (epoxies, etc.), GLASSES
thermoplastics,
rubbers, glass,
foams, fired ceramics,
textile fibres concrete,

fibre-reinforced plastics
(including GRP, CFRP,
glass/PTFE coated fabrics),
FRP-reinforced concrete

Figure 1.1. Relationships between classes of engineering materials,
showing the evolution of composites

polymers; and making additions of very hard, or abrasive, or thermally stable ceramic particles to metals
to make the class of materials known as ‘cermets’ to produce machine tool tips capable of cutting hard
metals at high speeds or high temperatures; are only two examples of important developments in our
exploitation of these materials. But even more significant is the extension of this principle to incorporate
filamentary metals, ceramics and polymers into the bulk forms of any of these three classes of materials
to make fibre composites — reinforced plastics, like CFRP and GRP, metal-matrix composites (MMCs)
like silicon-carbide-fibre-reinforced aluminium, and ceramic-matrix composites (CMCs) like carbon-
fibre-reinforced glass.

Ideally, the properties of engineering materials should be reproducible and accurately known. And
since satisfactory exploitation of the composite principle depends on the design flexibility that results
from tailoring the properties of a combination of materials to suit a particular requirement, we also need
to be able to predict those properties successfully. At the present time some of the more important
engineering properties of composites can be well predicted on the basis of mathematical models, but
many cannot.

1.2 CONVENTIONAL MATERIALS AND THEIR LIMITATIONS
It is difficult to draw up a table of materials characteristics in order to assess the relative strengths and
weaknesses of metals, plastics and ceramics because each of these terms covers whole families of
materials within which the range of properties is often as broad as the differences between the three
classes. A comparison in general terms, however, can identify some of the more obvious advantages and
disadvantages of the different types of material. At a simplistic level, then:

• Plastics are of low density. They have good short-term chemical resistance but they lack thermal
stability and have only moderate resistance to environmental degradation (especially that caused by
the photo-chemical effects of sunlight). They have poor mechanical properties, but are easily
fabricated and joined.

• Ceramics may be of low density (although some are very dense). They have great thermal stability
and are resistant to most forms of attack (abrasion, wear, corrosion). Although intrinsically very
rigid and strong because of their chemical bonding, they are all brittle and can be formed and shaped
only with difficulty.

• Metals are mostly of medium to high density — only magnesium, aluminium and beryllium can
compete with plastics in this respect. Many have good thermal stability and may be made corrosion-


resistant by alloying. They have useful mechanical properties and high toughness, and they are
moderately easy to shape and join. It is largely a consequence of their ductility and resistance to
cracking that metals, as a class, became (and remain) the preferred engineering materials.

On the basis of even so superficial a comparison it can be seen that each class has certain intrinsic
advantages and weaknesses, although metals pose fewer problems for the designer than either plastics or
ceramics.

1.3 STRONG FIBRES
The engineer who uses materials for structural or load-bearing purposes is quickly aware of an
important feature of engineering solids, which is that they are never as strong as we would expect them
to be from our knowledge of the strengths of the chemical bonds which hold them together. The reason
for this is that all materials contain defects of various kinds which can never be entirely eliminated in
practical manufacturing operations. For example, the strength of bulk glass and other ceramics is
determined not by their strong covalent or ionic bonds, but by the many tiny pores or sharp cracks that
exist either on the surface or in the interior. The most highly polished and dense bulk ceramic will rarely
have a strength that exceeds one thousandth of the theoretically predicted strength. Similarly, metals
contain faults in the stacking of atoms in their crystalline arrays, and the most damaging of these
imperfections — dislocations — cause most bulk metal samples to deform plastically at loads which are,
again, perhaps a thousandth or less of the theoretically calculated shear strength.

The strength of any sample of a glass or ceramic is actually determined by the size of the largest
defect, or crack, which it happens to contain. Roughly, the strength is proportional to the inverse square
root of the length of the largest flaw, a relationship developed in a thermodynamic argument by Griffith
in the 1920s. The relationship in its simplest form is:

2EγF
σmax = ..................................................................................................... [1.1]
πa

where σmax is the strength of the material, E is its elastic stiffness (Young's modulus), γF is the work
required to fracture the sample (a slight modification of Griffith's original idea that it represented the
surface free energy of the material), and a is the flaw size. Since materials of this kind inevitably
contain cracks and other defects with a wide range of sizes, one of the unfortunate consequences of the
Griffith model is that the measured strengths of a batch of supposedly identical samples of a brittle
material will be very variable simply because of the spread of flaw sizes.

If the flaw sizes can be reduced by careful control of the manufacturing process, the general level of
strength of the material will be raised and its variability reduced. Glass fibres, for example, are
manufactured by drawing molten glass very rapidly down to form fine filaments of the order of only ten
microns (10μm) in diameter. The resulting freshly formed surface is free of macroscopic defects and the
fibre itself is too fine to contain any defects of the size that are found in bulk glass. Careful
measurements of the strengths of freshly drawn glass fibres shows them to be up to 5GPa by comparison
with the very modest 100MPa or so of bulk glass. The fresh fibre needs to be protected from abrasion by
contact with hard objects and from corrosion by moisture in the atmosphere, both of which would result
in the reintroduction of surface flaws and loss of the high strength achieved by drawing.

Similarly, the strengths of polymeric filaments, such as the polyacrylonitrile fibres used to make
‘acrylic’ textiles, are limited by the weak chemical bonding between the molecular chains which make
up the filament and by the defects caused by the manufacturing process. By subjecting such filaments to
carefully controlled stretching, oxidation and carbonisation processes, however, the polymer can be
converted into fibres which are chemically almost entirely carbon, and with a crystal structure that is
approximately that of graphite, in which the proportion of strong C—C bonds that lie in the direction of
the fibre axis is very great. The load-bearing ability of these filaments, which are about 7μm in diameter,
is high, and the sizes of the residual defects are very small. As a result, each filament, finer than a human


hair, is capable of supporting a load of 10-20g without breaking, which is equivalent to strengths of up to
5GPa. And so it is with the many other types of fibres, organic and inorganic, that are used in composite
materials. The finer the filament that can be made from a given solid, the stronger it will be.

The theoretical strength of a given type of solid is determined by the strengths of the atomic or
molecular bonds that hold the solid together. And although the practical strengths of solids are
determined by the defects which they contain, it is nonetheless necessary to seek materials with the
strongest chemical bonds if we are to have the best chance of exploiting the principle of composite
materials construction. An appreciation of the nature and magnitude of chemical bonding leads us to
scan the Periodic Table for elements in which there is a high density of covalent or mixed covalent/ionic
bonds, and we naturally alight on the area containing light elements with directional bonds. The
elements most likely to provide what we seek are those like carbon, boron and silicon. In the 1950s,
when ideas about composite materials were first being formulated, the plain elements would not have
been considered to be particularly promising as high-performance structural materials, but the invention
of carbon fibres at the Royal Aircraft Establishment, Farnborough, in the UK, and the simultaneous
development of high-strength boron fibres at Texaco in the USA, resulted in a very rapid expansion of
research and development programs to exploit these new filamentary materials in engineering
composites.

Silicon also appears unpromising as a structural material, but in combination with another ‘light’
element, oxygen, as the compound silica (SiO2), it was long familiar as a high-strength filament in the
form of melt-drawn glass or quartz fibres. And the list of other combinations of light-weight elements
that have been successfully exploited in the search for high-performance reinforcing filaments includes
alumina (Al2O3), silicon carbide (SiC), and silicon nitride (Si3N4) to name but three of the more
important ceramic fibres. Polymeric fibres in which the strength and stiffness are due to the strong C—C
bond are also utilized in fibre composites, and perhaps the two most important are highly drawn forms of
the humble polyethylene and the more exotic aromatic polyamides (related to Nylon, but possessing
much greater thermal stability). Table 1.1 gives a rough comparison of the mechanical properties of
some of the more important commercial reinforcing fibres. Taking steel, in the form of piano wire, as a
basis for comparison, it is clear why some of the new fibrous materials are of such interest to engineers,
especially those concerned with light-weight structures. On a strength/density or stiffness/density basis,
the comparison is almost startling.

Since it is the reinforcing fibres that provide the means of creating composite materials of high
strength and stiffness, combined with low density, it is worth-while examining in a little more detail the
nature of these fibres and their origins. Information can be obtained on these and other fibres from the
publications edited by Bunsell (1988) and Mai (1994).

1.3.1 Glass fibres
Glass fibres are manufactured by drawing molten glass into very fine threads and then immediately
protecting them from contact with the atmosphere or with hard surfaces in order to preserve the defect-
free structure that is created by the drawing process. Glass fibres are as strong as any of the newer
inorganic fibres but they lack rigidity on account of their molecular structure. The properties of glasses
can be modified to a limited extent by changing the chemical composition of the glass, but the only glass
used to any great extent in composite materials is ordinary borosilicate glass, known as E-glass. The
largest volume usage of composite materials involves E-glass as the reinforcement. S-glass (called R-
glass in France) has somewhat better properties than E-glass, including higher thermal stability, but its
higher cost has limited the extent of its use. Wallenberger and Brown (1994) have recently described the
properties of experimental calcium aluminate glass fibres with stiffnesses as high as 180GPa.

1.3.2 Carbon fibres
By oxidising and pyrolysing a highly drawn textile fibre such as polyacrylonitrile (PAN), preventing
it from shrinking in the early stages of the degradation process, and subsequently hot-stretching it, it is
possible to convert it to a carbon filament with an elastic modulus that approaches the value we would
predict from a consideration of the crystal structure of graphite, although the final strength is usually


well below the theoretical strength of the carbon-carbon chain (Watt, 1970). The influence of strength-
limiting defects is considerable, and clean-room methods of production can result in substantial increases
in the tensile strength of commercial materials. Prior to sale, fibres are usually surface-treated by
chemical or electrolytic oxidation methods in order to improve the quality of adhesion between the fibre
and the matrix in a composite. Depending on processing conditions, a wide range of mechanical
properties (controlled by structural variation) can be obtained, and fibres can therefore be chosen from
this range so as to give the desired composite properties. Although the fibre is highly organized and
graphite-like, the structure is not identical with that of graphite and the fibres should not, strictly
speaking, be referred to by that name, although this is common in the US (and in UK advertising jargon
for sports equipment!). Recent developments in this field have led to the use of pitch as a precursor in
place of textile fibres, and these newer materials have extremely high stiffnesses, compared to PAN-
based fibres, but rather lower strengths (Fitzer and Heine, 1988).

Table 1.1. Typical properties of some familiar reinforcing fibres

Material Trade name Density, Fibre Young's Tensile
103 kg.m-3 diameter, Modulus, Strength,
µm GPa GPa
High-carbon steel wire eg. piano wire 7.8 250 210 2.8

Short fibres:
α-Al2O3 (whisker crystals) 3.96 1-10 450 20
δ-Al2O3 + SiO2 (discontinuous) Saffil (UK) 2.1 3 280 1.5
Continuous fibres:(inorganic)
α-Al2O3 FP (US) 3.9 20 385 1.8
Al2O3 + SiO2 + B2O3 (Mullite) Nextel480 (US) 3.05 11 224 2.3
Al2O3 + SiO2 Altex (Japan) 3.3 10-15 210 2.0
Boron (CVD on tungsten) VMC (Japan) 2.6 140 410 4.0
Carbon (PAN precursor) T300(Japan) 1.8 7 230 3.5
Carbon (PAN precursor) T800(Japan) 1.8 5.5 295 5.6
Carbon (pitch precursor) Thornel P755(US) 2.06 10 517 2.1
SiC (+O) Nicalon(Japan) 2.6 15 190 2.5-3.3
SiC (low O) Hi-Nicalon(Japan) 2.74 14 270 2.8
SiC (+O+Ti) Tyranno(Japan) 2.4 9 200 2.8
SiC (monofilament) Sigma 3.1 100 400 3.5
Silica (E glass) 2.5 10 70 1.5-2.0
Silica (S or R glass) 2.6 10 90 4.6
Silica (Quartz ) 2.2 3-15 80 3.5
Continuous fibres (organic)
Aromatic polyamide Kevlar 49 (US) 1.5 12 130 3.6
Polyethylene (UHMW) Spectra 1000 (US) 0.97 38 175 3.0

Carbon fibres are inherently expensive, since they are produced from relatively costly precursor
filaments. Since their introduction, however, their price has fallen from several hundred £/kg to tens of
£/kg, depending upon quality, and although carbon is still an expensive reinforcement, more so than
glass, for example, it is now much cheaper than most other common reinforcing fibres. The costs of the
fibres listed in Table 1.1 vary from about £30 per kg for PAN-based carbon to over £5,000 per kg for
boron.

1.3.3 Silicon carbide.
Continuous SiC monofilaments were first produced by pyrolytic decomposition of gaseous silanes
onto fine filaments of carbon or tungsten. These are thick fibres, of the order of 100μm in diameter,
which continue to be of major interest to manufacturers of metal- and ceramic-matrix composites. The
alternative production method, analogous to that for carbon described earlier, is based on the controlled
thermal degradation of a polymer precursor (Yajima, 1980). This process typically takes a precursor
such as some mixture of dimethyl dichlorosilane and diphenyl dichlorosilane and converts it in an
autoclave to a polycarbosilane from which a continuous fibre is made by melt-spinning. The fibre is then


converted by pyrolysis at 1300°C into a fibre consisting mainly of β-SiC of about 15µm diameter. The
characteristic commercial fibre of this type is that known as 'Nicalon' which is marketed by the Nippon
Carbon Company. It has a rough surface, making for good fibre/matrix adhesion, but is somewhat
reactive towards oxygen. It is well wetted by molten metals and is reasonably stable as a reinforcement
for MMCs based on aluminium and copper although it lacks long-term thermal stability. There have
been various attempts to improve this feature of the fibre, for example by reducing the oxygen content
(Hi-Nicalon; see Toreki et al, 1994) and by adding titanium (Tyranno fibre).

1.3.4 Alumina and alumina/silica compounds
The earliest work involving Al2O3 as a reinforcement for composite materials concerned the use of
tiny filamentary crystals, of the order of 50μm long and a few microns in diameter, which could be
grown from the vapour phase in a highly perfect state and which, in consequence, had great strength and
stiffness. These crystals, known as ‘whiskers’, proved to be very expensive and difficult to handle in
processing operations, and their potential could not be satisfactorily realized. The DuPont Company in
the USA subsequently developed an improved alumina fibre in the form of a polycrystalline yarn known
as FP fibre (Dhingra, 1980). These filaments, which were manufactured by a sol-gel process, had a
modulus comparable with those of boron and carbon, and a strength of the order of 1.8GPa. Like SiC,
this fibre also had a rough surface and was potentially well-suited for the reinforcement of metals such
as aluminium and magnesium on account of its chemical inertness, high-temperature stability, and its
ability to form a good bond with matrix alloys. As far as can be ascertained at present, however, FP has
never been an economic prospect for its manufacturer, and alternative forms of alumina, usually
containing other constituents, like Sumitomo’s Altex fibre, for example, are currently preferred. Many
of the oxide-based reinforcing filaments available at present are compounds of alumina and silica, and
Wallenberger and Brown (1994) have given a useful graphical plot, drawn in modified form in Figure
1.2, showing how the tensile stiffness of this particular family of materials varies with composition.
Some of the points in the diagram are labelled with their familiar trade names.

An alternative form of alumina-based reinforcement known as Saffil, a δ-Al2O3 containing certain
impurities which were advantageous for processing MMCs, was introduced in 1979 by ICI in response
to the need for a reinforcing filament that was cheaper than those currently available. This discontinuous
fibre was manufactured in the form of a continuous mat which could be used as layers or preforms, or as
a shredded material, with controlled fibre aspect ratios (the ratio of length to diameter) ranging from
50:1 to 500:1. The material combined ease of handling with low cost (up to one tenth the cost of
continuous reinforcing fibres), and unlike competing fibres was produced on a large-scale. Its intrinsic
properties were as good as those of
some competing fibres, but the fact that 500
it was produced as a random mat meant
monocrystalline
that the potential strength of composites Saphicon
Elastic modulus, E, GPa

containing it was more limited. It has 400
FP
been used for the selective
reinforcement of MMC components, but polycrystalline Safimax
300
not on any large scale. The
manufacturers have since relinquished Nextel 480
their interest in the material. 200 Altex
Nextel 312
1.3.5 Organic Fibres S glass
100
Bulk polymers have elastic moduli
no greater than 100MPa, but if the E glass amorphous
polymer is spun into fibres and cold 0
drawn so as to develop a high degree of 0 20 40 60 80 100
molecular orientation, substantial Alumina content, wt%
improvements in both strength and
rigidity can be achieved. In particular, Figure 1.2. Elastic modulus of SiO2/Al2O3 fibres as a function of
polyolefin fibres with an exceptionally composition (after Wallenberger & Brown, 1994)


highly oriented, extended molecular chain structure can be achieved by superdrawing in the solid state
(Ward, 1980). In this process, both the crystalline and non-crystalline phases of the initially isotropic
polymer are stretched out and aligned and there is an increase in crystal continuity. Such fibres have
high strengths, and their elastic moduli are similar to those of glass and aluminium. Apart from their
excellent mechanical properties such fibres have the important advantage over inorganic fibres that they
are not brittle.

The other major development in organic fibres over the last three decades has been the production by
DuPont of aromatic polyamide fibres, collectively known as aramids, of which the best known for many
years in the composites industry has been Kevlar-49 (Yang, 1993). These polymers are based on p-
oriented diamine and dibasic acid intermediates which yield liquid crystalline solutions in amide and
acid solvents. These solutions contain highly-ordered extended-chain domains which are randomly-
oriented in the absence of force, but which may be oriented by inducing shear forces in the liquid.
Highly-oriented fibres can therefore be produced by wet-spinning these solutions, and
poly(paraphenylene terephthalamide) fibres such as Kevlar and Twaron have strengths of the order of
2.6GPa and moduli up to 130GPa, depending on the degree of alignment of the polymer chains. Having
properties intermediate between those of carbon and glass, aramids offer an extra degree of flexibility in
composite design. One important characteristic of the fibre is that it is extremely difficult to cut on
account of its fibrillar structure. Laser and water-jet methods are essential for trimming bare fabrics and
composites containing them. Kevlar/resin composites are noted for their extremely high levels of
toughness and resistance to impact damage. One disadvantage of aramids, by comparison with carbon
fibres, for example, is that they are sensitive to moisture.

1.3.6 Styles of reinforcement
Many reinforcing fibres are marketed as wide, semi-continuous sheets of ‘prepreg’ consisting of
single layers of fibre tows impregnated with the required matrix resin and flattened between paper carrier
sheets. These are then stacked, as discussed in chapter 3, the orientations of each ‘ply’ being arranged in
accordance with design requirements, and hot pressed to consolidate the laminate. This process is able to
cope with curved surfaces, provided the degree of curvature is not too great, but there may be a
possibility of local wrinkling of the fibres when prepregs are pressed into doubly curved shapes. One
means of overcoming this problem is to use the reinforcement in the form of a woven cloth since textile
materials can readily be ‘draped’ over quite complex formers. Many of the fine filamentary reinforcing
fibres like glass, carbon and SiC can be readily woven into many kinds of cloths and braids, the fibres
being effectively placed by the weaving process in the directions required by the designer of the final
composite structure. In simple designs, this may call for nothing more elaborate than an ordinary plain
weave or satin weave, with fibres running in a variety of patterns but in only two directions, say 0° and
90°, but weaving processes to produce cloths with fibres in several directions in the plane of the cloth are
all readily available. Fibres of different types may also be intermingled during the weaving processes to
produce mixed-fibre cloths for the manufacture of some of the ‘hybrid’ composites that will be
discussed later.

Most of the continuous fibres that we have considered are expensive raw materials, and it is often
only the fact that the overall cost of a manufactured composite product may nevertheless be lower than a
competing product made from cheaper, conventional materials by more costly processes that makes a
composites design solution an attractive alternative. Thus, although large quantities of glass fibres are
supplied in chopped form for compounding with both thermoplastic and thermosetting matrix polymers,
it may not seem economical to chop the more expensive types of reinforcement. Nevertheless, there are
some advantages in using even these fibres in chopped form, provided they can be arranged in the
composite in such a way as to make good use of their intrinsically high strengths and stiffnesses. Parratt
and Potter (1980) described a process for producing both chopped fibres, like glass and carbon, and
naturally short filaments, like whiskers or asbestos fibres, in the form of prepreg sheets with fibres that
were very well aligned in either unidirectional or poly-directional patterns. These prepregs also have
excellent ‘drapability’ and can be used to form complex shapes, as discussed by Tsuki et al. (1997). As
will be seen in chapter 4, provided the short fibres are well above some critical length, which for carbon,
for example, may be of the order of only a millimetre, they are able to contribute a high fraction of their


intrinsic properties to the composite without the loss that occurs with woven reinforcements as a result of
the out-of-plane curvature of the fibres.

1.4 THE SCOPE FOR REINFORCEMENT OF CONVENTIONAL MATERIALS
The composite matrix is required to fulfil several functions, most of which are vital to the
performance of the material. Bundles of fibres are, in themselves, of little value to an engineer, and it is
only the presence of a matrix or binder that enables us to make use of them. The rôles of the matrix in
fibre-reinforced and particulate composites are quite different. The binder for a particulate aggregate
simply serves to retain the composite mass in a solid form, but the matrix in a fibre composite performs a
variety of other functions which must be appreciated if we are to understand the true composite action
which determines the mechanical behaviour of a reinforced material. We shall therefore consider these
functions in some detail.

1.4.1 Functions of the matrix
• The matrix binds the fibres together, holding them aligned in the important stressed directions.
Loads applied to the composite are then transferred into the fibres, the principal load-bearing
component, through the matrix, enabling the composite to withstand compression, flexural and shear
forces as well as tensile loads. The ability of composites reinforced with short fibres to support loads
of any kind is dependent on the presence of the matrix as the load-transfer medium, and the
efficiency of this load transfer is directly related to the quality of the fibre/matrix bond.
• The matrix must also isolate the fibres from each other so that they can act as separate entities. Many
reinforcing fibres are brittle solids with highly variable strengths. When such materials are used in
the form of fine fibres, not only are the fibres stronger than the monolithic form of the same solid,
but there is the additional benefit that the fibre aggregate does not fail catastrophically. Moreover,
the fibre bundle strength is less variable than that of a monolithic rod of equivalent load-bearing
ability. But these advantages of the fibre aggregate can only be realized if the matrix separates the
fibres from each other so that cracks are unable to pass unimpeded through sequences of fibres in
contact, which would result in completely brittle composites.
• The matrix should protect the reinforcing filaments from mechanical damage (eg. abrasion) and from
environmental attack. Since many of the resins which are used as matrices for glass fibres permit
diffusion of water, this function is often not fulfilled in many GRP materials and the environmental
damage that results is aggravated by stress. In cement the alkaline nature of the matrix itself is
damaging to ordinary glass fibres and alkali-resistant glasses containing zirconium have been
developed (Proctor & Yale, 1980) in an effort to counter this. For composites like MMCs or CMCs
operating at elevated temperature, the matrix would need to protect the fibres from oxidative attack.
• A ductile matrix will provide a means of slowing down or stopping cracks that might have
originated at broken fibres: conversely, a brittle matrix may depend upon the fibres to act as matrix
crack stoppers.
• Through the quality of its ‘grip’ on the fibres (the interfacial bond strength), the matrix can also be
an important means of increasing the toughness of the composite.
• By comparison with the common reinforcing filaments most matrix materials are weak and flexible
and their strengths and moduli are often neglected in calculating composite properties. But metals
are structural materials in their own right and in MMCs their inherent shear stiffness and
compressional rigidity are important in determining the behaviour of the composite in shear and
compression.

The potential for reinforcing any given material will depend to some extent on its ability to carry out
some or all of these matrix functions, but there are often other considerations. We consider now the
likely qualities of various classes of matrix materials.


1.4.2 Metals
Metals owe their versatility as engineering materials to the fact that they can be plastically deformed
and can be strengthened by a variety of methods which, by and large, act by inhibiting the motion of
dislocations. As a consequence of the non-directional nature of the metallic bond, dislocations are highly
mobile in pure metals which are therefore very soft. But by controlling the number and distribution of
dislocations the materials scientist can adjust the properties of a metal or alloy system to suit specific
requirements. There are limitations, however. Increases in strength can usually be achieved only at the
expense of the capacity for plastic deformation, with the consequence that the strongest alloys often lack
tolerance of defects or other stress-concentrators. Since brittleness is a drawback no designer dares
underestimate, this leads to the use of large safety factors which, in turn, means that the full potential of
high-strength alloys is often not utilisable in practice.

Many solid-state hardening methods used in alloys involve producing a material in a metastable
state which may subsequently revert to a more stable but unstrengthened condition if sufficient thermal
energy is provided. Alloys strengthened by precipitation hardening, such as the strong aluminium alloys,
alloys depending on phase transformations of the martensitic type, such as steels, and heavily cold-
worked metals which depend simply on the presence of a high dislocation density, as in piano wires and
lamp filaments, will all soften at elevated temperatures. The strongest aluminium alloys lose their
strength at temperatures little over 150°C, for example.

Many conventional metallic materials are relatively heavy. For land-based engineering projects this
may be of no consequence, but economic arguments relating to pay-loads (in civil aircraft) and tactical
arguments relating to manoeuvrability (in military aircraft) have always been a powerful incentive for
the use of low-density materials in aerospace engineering, and in these energy-conscious times the
economic incentive for lightening automobiles has considerably influenced the motor car designer.

For structural applications involving compression or flexural rather than tensile loads, the relevant
structural stiffness index is not simply Young's modulus, E, or the modulus/density ratio, E/ρ, but may
instead be either of the ratios E/ρ2 or E/ρ3 (Gordon, 1968 and 1978, Ashby and Jones, 1980). For
aerospace applications modifications to materials that result in lowered density may therefore be more
profitable than attempts simply to improve their strength or stiffness.

These features show why it is worthwhile to attempt to use light, strong, stable fibres to reinforce
some of the lighter engineering metals and alloys.

1.4.3 Polymeric materials
Few polymers are thermally stable by comparison with metals or ceramics and even the most stable,
like the polyimides, or poly(ether ether ketone) (known as PEEK) are degraded by exposure to
temperatures above about 300°C, as illustrated by some of the data in Table 1.2. There is nothing that
reinforcement can do to combat chemical degradation, but the associated fall in strength and increase in
time-dependent (creep or visco-elastic) deformation, a feature common to all polymers, though less
serious in cross-linked resin systems than in thermoplastics, can be delayed by fibre reinforcement.


Table 1.2. Thermal Stability of some Matrix Polymers

Type and Polymer Symbol Crystallinity Glass transition Max use
temp, Tg, ° C temp, °C
Thermosets:
Polyester PE no 80 -100 50
Epoxy Ep no 120-180 150
Phenolic Ph no 130-180 200
Bismaleimide BMI no 180-200 220
Polyimide PI no 300-330 280
Thermoplasts:
Polyamide (Nylon) PA yes 80 125
Poly(phenylene sulphide) PPS yes 100 260
Poly(ether ether ketone) PEEK yes 143 250
Polycarbonate PC no 145 125
Polysulphone PS no 190 150
Poly(ether imide) PEI no 210 170
Poly(ether sulphone) PES no 230 180
Thermoplastic polyimide TPI no 270 240

A more serious problem in polymers is their very low mechanical strength and stiffness in bulk form:
and, like metals, the weakest plastics tend to be ductile but the strongest tend to be brittle, although there
are exceptions.

Polymers are traditionally insulators and in their application as such strength is usually a secondary
consideration. The electrical conductivity of plastics reinforced with carbon fibres is of importance in
many aeronautical applications, however, where protection of avionics systems from external electrical
activity (eg. lightning strike) is of importance. Most polymers are already low-density materials, and the
addition of fibres confers no density advantage.

It is in the reinforcing of polymers that most developments have been made so far, and it is likely
that there is still scope for improvement. There is a thriving, international reinforced plastics Industry,
for which both the science and technology are highly advanced. And although the level of awareness of
the merits of reinforced plastics on the part of designers in general engineering seems to be low, the
same is not true of the aerospace industry for which the potential benefits of very high strength and
stiffness, combined with low density, are easily recognized.

1.4.4 Glasses, ceramics and cement
Glasses have high chemical stability, but many lose their mechanical strength at relatively low
temperatures as they pass through the glass transition. Special glasses have been developed with high
transition temperatures (Tg), however, and many are at least as resistant as some of the less stable steels.
The principal problem with glassy materials is that they are always brittle and their measured strengths
are very variable at ordinary temperatures. They are not able to relieve stress concentrations at crack tips
by plastic deformation and they are not, therefore, fail-safe. A constant disadvantage, aggravated by low
thermal conductivity, is that glass usually has poor thermal shock resistance unless, like borosilicate
glass, it also has a low thermal expansion coefficient. Many of these difficulties can be overcome by
reinforcement with fibres like carbon, and with the added bonus of a saving in weight (Phillips et al,
1972).

Most ceramics suffer from the same defects as glasses in that, although they are potentially high-
strength solids, they are also brittle and notch-sensitive. Most ceramics retain their strength to very high
temperatures, however, unlike glasses, and several have good resistance to thermal shock. Improving
toughness and reducing notch sensitivity are the main reasons for attempting to reinforce ceramics since
their stiffnesses are not very different from those of the best reinforcing fibres. For high-temperature
stability we might expect that the best possibility for reinforced ceramics would be an equilibrium
system in which whisker-like crystals of a reinforcing phase were grown within the polycrystalline


matrix of a chemically-related ceramic. Most recent work has followed the more established route,
however, in attempting to reinforce a variety of glasses (such as borosilicate) and glass-ceramics (such as
calcium aluminosilicate (CAS) and lithium aluminosilicate (LAS)) with fibres like carbon, silicon
carbide and alumina (Prewo and Brennan, 1980; Phillips et al, 1972), the fibres being impregnated by a
slurry of fine glass powder and subsequently hot-pressed (Sambell et al, 1970, 1972). Critical control of
manufacturing conditions is needed in order to produce appropriate interfacial conditions for the
optimum combination of strength and toughness. Once control of the glass-making technique is
established, development of the process to include a glass-ceramic transformation is the logical next step
in the production of a continuous-fibre composite. The advantage of the glass-ceramic route is that
relatively modest processing temperatures are involved. For high-temperature exposure of such
composites, however, or for work involving the production of CMCs by conventional pressing methods,
there remains the problem of compatibility. At the time of writing, the concept of all-oxide CMCs is
being strongly argued, although we already have comparative examples of stable pairings in the French
SiC/SiC materials (Frety and Boussuge, 1990). After a period of intense activity in fibre-reinforced glass
and glass-ceramic research with an eye on the gas-turbine industry, it has still not been possible for
engineers to come to terms with the concept, already explored in the 1970s (Aveston et al, 1971), that in
these materials the matrix will crack at worryingly low stress levels (Harris et al, 1992) leaving the
reinforcing filaments exposed to any hostile environment.

Concrete is a particulate ceramic composite in which aggregate particles of graded sizes are
embedded in a glassy or microcrystalline silicate matrix. Like most ceramics and glasses it is brittle and
exhibits a very low tensile failure strain which engineers cope with by using it in compression or with
macroscopic reinforcing bars to carry tensile loads. In design, the tensile load-bearing ability of the
concrete is ignored. Reinforcement of concrete in the composite sense has been widely studied, with
attempts to produce stronger, stiffer and tougher structural materials by adding fibres of asbestos, glass,
steel, polymeric materials, and carbon. Improved properties can be obtained, although usually only at the
expense of a severe economic penalty, given the fact that concrete is the cheapest structural material
available. The addition of even a few percent of the cheapest reinforcing fibres may still raise the price
sufficiently to lead the engineer to use a thicker section of plain concrete in preference to a more
expensive thinner section of reinforced concrete. Such small additions do little to increase the cracking
stress or elastic modulus of cement, but by introducing complex cracking and failure modes they can
substantially improve the work of fracture. The scope for improvement of bulk poured concrete is
probably negligible: it is much more likely that the present trend towards changes in design philosophy
and the use of speciality products in fibre-reinforced concrete will continue. High volume fractions of
well-aligned fibres can be obtained by vacuum dewatering and filament-winding and the products of
these processes can be used in much thinner sections than is normal with concrete. Traditional attitudes
and design procedures are of course no longer applicable for such materials.

1.4.5 Carbon
Carbon is a unique material with many attractive engineering qualities. It can be prepared in a variety
of forms — conventional hot-pressed carbons and graphites, densified impermeable graphite, pyrolitic
graphite, and vitreous carbon — with a wide range of engineering properties. It is valuable for its
lubricating properties, its electrical properties, its nuclear characteristics and, in the pyrolytic and
vitreous forms, for high strength and resistance to oxidative and chemical attack (Cahn and Harris,
1969). The opportunity to improve the mechanical properties of such an important material and reduce
its brittleness somewhat has been the driving force for the development of an invaluable material,
carbon-fibre-reinforced carbon (Savage, 1993) that has been used for rocket nozzles, aerospace
components (including the ablative shields of space vehicles), and surgical implants.

1.5 REFERENCES TO CHAPTER 1
Ashby MF and Jones DRH, 1980, Engineering Materials, (Pergamon, Oxford)
Atzori B, Quaresimin M and Trattenero G, 1994, in Proc 2nd International Seminar Experimental Techniques
and Design in Composite Materials (editor MS Found) Sheffield, Sept 1994, (Sheffield Academic Press,
Sheffield, UK), 193-211.


Aveston J, Cooper GA and Kelly A, 1971, Proceedings of NPL Conference on The Properties of Fibre
Composites, (IPC Science and Technology Press Ltd, Guildford), 15-26
Cahn RW and Harris B, 1969, Nature, 221, 132-141
Dhingra AK, (1980), in New Fibres and their Composites, Watt W, Harris B and Ham AC (editors), (Royal
Society of London), Phil Trans Roy Soc Lond, A294, 411-417.
Fitzer E and Heine M, 1988, in Fibre Reinforcements for Composite Materials, Composite Materials Series,
volume 2, (editor AR Bunsell), (Elsevier, Amsterdam), 73-148.
Frety N and Boussuge M, 1990, in Ceramic-Matrix Composites: Components, Preparation, Microstructure
and Properties, Naslain R and Harris B (editors), Special Issue of Composites Science and Technology,
37, 177-190.
Gordon JE, 1968, Structures, (Penguin, Harmondsworth, Middlesex)
Gordon JE, 1978, The New Science of Strong Materials, (Penguin, Harmondsworth, Middlesex)
Griffith AA, 1920, Phil Trans Roy Soc, A221, 163-198.
Harris B, Habib FA and Cooke RG, 1992, Proc Roy Soc (Lond), A437, 109-131.
Parratt NJ and Potter KD, 1980, Advances in Composite Materials (Proc ICCM3) vol 1 (editors AR
Bunsell, C Bathias, A Martrenchar, D Menkes and G Verchery), (Pergamon, Oxford), 313-326.
Phillips DC, Sambell RAJ, Bowen DH, 1972, J Mater Sci, 7, 1454-1464,
Prewo KM, Brennan JJ, 1980, J Mater Sci, 15, 463-468,
Proctor B and Yale B, 1980, in New Fibres and their Composites, Watt W, Harris B and Ham AC (editors),
(Royal Society of London), Phil Trans Roy Soc Lond, A294, 427-436.
Sambell RAJ, 1970, Composites, 1, 276-285.
Sambell RAJ, Briggs A, Phillips DC and Bowen DH, 1972, J Mater Sci, 7, 676-681.
Savage G, 1993, Carbon-Carbon Composites, (Chapman & Hall, London).
Toreki W, Batich CD, Sacks MD, Saleem M, Choi GJ and Morrone A, Compos Sci & Technol, 51, 145-159.
Tsuji N, Springer GS and Hegedus I, 1997, J Compos Mater, 31, 428-527.
Wallenberger FT and Brown SD, 1994, Compos Sci & Tech, 51, 243-264.
Ward IM, 1980, in New Fibres and their Composites, Watt W, Harris B and Ham AC (editors), (Royal
Watt W, 1970, Discussion Meeting on Strong Fibrous Solids, Proc Roy Soc Lond, A319, 5-15.
Yajima S, 1980, in New Fibres and their Composites, Watt W, Harris B and Ham AC (editors), (Royal
Yang HH, 1993, Kevlar Aramid Fibre, (John Wiley, Chichester)

1.5.1 General bibliography
Agarwal BD and Broutman LJ, 1990, Analysis and Performance of Fibre Composites, second edition, (John
Wiley, New York).
Astrom BT, 1997, Manufacturing of Polymer Composites, (Chapman & Hall, London).
Bunsell AR, (editor), 1988, Fibre Reinforcements for Composite Materials, Composite Materials Series,
volume 2, (Elsevier, Amsterdam).
Chawla KK, 1993, Ceramic Matrix Composites, (Chapman & Hall, London).
Chou TW, 1992, Microstructural Design of Fibre Composites, (Cambridge University Press).
Clyne TW and Withers PJ, 1993, An Introduction to Metal-Matrix Composites, (Cambridge University
Press).
Eckold G, 1994, Design and Manufacture of Composites, (Woodhead, Abington, UK).
Everett RK and Arsenault RJ (editors), 1991, Metal-Matrix Composites; volume 1, Processing and
Interfaces, volume 2, Mechanisms and Properties, (Academic Press, Boston).
Hull D, 1981, An Introduction to Composite Materials, (Cambridge University Press).
Jones RM, 1975, Mechanics of Composite Materials, (Scripta Books, Washington DC)
Kelly A and McMillan NH, 1986, Strong Solids (3rd Edition), (Clarendon Press, Oxford).


Mai YW (editor), 1994, Advances in Inorganic Fibre Technology, Special Issue of Composites Science and
Technology, 51, 123-296.
Matthews FL and Rawlings RD, 1994, Composite Materials: Engineering and Science, (Chapman & Hall,
London).
Naslain R and Harris B (editors), 1990, Ceramic-Matrix Composites, (Elsevier Science Publishers, London
and New York).
Piggott MR, 1980, Load-Bearing Fibre Composites, (Pergamon Press, Oxford)
Powell PC, 1994, Engineering with Fibre-Polymer Laminates, Chapman & Hall, London).
Warren R (editor), 1992, Ceramic Matrix Composites, (Blackie, Glasgow).
Watt W, Harris B and AC Ham (editors), 1980, New Fibres and their Composites, (The Royal Society of
London), Phil Trans Roy Soc Lond, A294.
1.5.2 Sources of data
Hancox NL and Meyer RM, 1994, Design Data for Reinforced Plastics, (Chapman & Hall, London).
Kelly A (Editor), 1989, Concise Encyclopaedia of Composite Materials, (Pergamon, Oxford).
Lubin G (Editor), 1982, Handbook of Composites, (Van Nostrand Reinhold, New York).

2 MAKING COMPOSITE MATERIALS

2.1 THE COMBINING OF MATERIALS

I
n considering the scope for reinforcing conventional materials, we have already examined the
functions of the matrix in a composite and the general nature of available reinforcing filaments. The
problem facing the manufacturer of composites is to develop suitable methods for combining the
matrix and the reinforcement so as to obtain the required shape of component with properties
appropriate to the design requirements. In the early days of the subject, attention was focused on first
producing a ‘piece’ of the composite — a sheet, or rod, or billet, with the fibres arranged in one or more
directions — and then using that intermediate product to construct a finished ‘component’ in much the
same way as steel castings were bolted together to build a car engine or steel plates were welded together
to make a pressure vessel. Early use of composites in practical applications was on the basis of piece-
meal substitution of the composite for a metallic part of identical shape without any consideration of the
special nature of the composite, notably its anisotropic mechanical properties. The idea of bolting GRP
‘top-hat’ stiffeners to the underside of GRP ship decking seems ridiculous now, but the procedure was
certainly investigated in the 1970s.

It is essentially the reinforcement ‘architecture’ that determines the load-bearing characteristics of a
fibre composite, and the beauty of modern composites design and manufacturing procedures is that in
many cases the composite material and the finished component can be created and final in a single
operation. A typical example is in the process of resin-transfer moulding (RTM) which we shall discuss
later. A fibre preform — a skeleton of the approximate shape of the finished component with the fibres
arranged in the directions determined by the design requirements — is placed into a closed mould, and
pre-catalysed resin is injected into the mould, which is often pre-heated. The resin cures in a short time,
and the finished component, requiring only final cleaning up, is removed from the mould, the composite
material and the finished article being formed in a single operation. The rate-controlling step in this near-
net-shape manufacturing process is probably the manufacture of the fibre preform. In the filament-
winding of hollow containers, fibres wetted by catalysed resin are wrapped onto a mandrel and, again, as
the resin cures the component and the material are formed simultaneously. In this case, the locations of
the fibres may have been determined by a computer programme and the same computer code can then be
used to control an automatic filament-winding machine — a good example of CADCAM.

The range of processes used for manufacturing composites is now wide. Almost all of the basic
metal, polymer and ceramic forming processes have been adapted, with greater or less success, and for
each class of composites — fibre-reinforced plastics (FRPs), MMCs and CMCs — there is a range of
preferred techniques, depending on the required product. In many cases, there has been technology
transfer from one field to another, so that similar processes, suitably modified, may be used for different
types of composite. The RTM process for FRPs is clearly related to the liquid-metal infiltration process
which is used for MMCs and to the sol-gel process for CMCs. Almost all successful processes involve
liquid-phase reactions — molten metal, liquid glass, ceramic sol — although many processes for ceramic
composites are gas-phase processes, as in the chemical vapour deposition (CVD) processes for preparing
carbon/carbon composites and some of the monofilament-reinforced ceramics like the SiC/SiC
composites made by SEP in France.

What is feasible for the manufacturer is often limited by the available forms of the reinforcing
filaments. The thick monofilamentary reinforcements like boron and SiC are produced in continuous
single filaments and because of their diameters they cannot easily be handled except as regular arrays of
continuous fibres. But the majority of the fine, continuous reinforcements are manufactured as
continuous tows, often containing thousands of individual fibres, and these can be manipulated by well-
established textile handling techniques. They can be chopped into short lengths for mechanical blending
with matrix resins and fillers, for example, or they can be woven into many kinds of cloths and braids,
the fibres being effectively placed by the weaving process in the directions required by the designer. In
simple designs, this may call for nothing more elaborate than an ordinary plain weave or satin weave,

Chapter two Making composite materials 19

with fibres running in a variety of patterns but in only two directions, say 0° and 90°. But weaving
processes to produce cloths with fibres in several directions in the plane of the cloth — eg. triaxial
weaves with three groups of fibres at 120° to each other, and so-called 4D and 5D fabrics — are all
readily available. The study of textile-based composites is currently a much researched topic (see, for
example, Masters and Ko, 1996). Fibres of different types may also be intermingled during the weaving
processes to produce mixed-fibre cloths for the manufacture of some of the ‘hybrid’ composites that will
be discussed later.

2.2 THE INTERFACE
In materials where the mechanical response depends on loads being shared between two or more
separate constituents or ‘phases’ and where paths for the propagation of cracks will be affected by the
different mechanical properties of the components, the manner in which these adhere to each other
becomes an important consideration. When we come to discuss the strength and toughness of
composites, we shall see that the strength of the interfacial bond between the fibres and the matrix may
make all the difference between a satisfactory material and an inadequate one. The problem is that the
ideal situation for load sharing — a perfect bond between the fibre and the matrix, forcing them to
deform as one, which is the first assumption that we shall make in attempting to calculate the elastic
modulus of a fibre composite — often results in a lower tensile strength than expected because of the
way cracks run in such a composite. In the early days of carbon-fibre technology, it was quickly
established that the greater the degree of order in the graphitized fibre structure the poorer was the
adhesive bond between the new high-modulus fibres and the existing polyester and epoxy resins. As a
consequence, the higher the fibre stiffness the poorer the extent to which the composite was able to resist
shear forces acting in the plane of a laminate ie. the lower the interlaminar shear strength (ILSS). Hasty
attempts were made to improve the ILSS by a variety of surface treatments, including coating with
polymers, high-temperature and electrolytic oxidation, and vapour deposition of other compounds onto
the fibre surface. Electrolytic treatment became the established method, but the need for control over the
extent of the treatment was not appreciated and some early fibres were supplied by manufacturers with
heavily oxidised surfaces that were very well wetted by the resins. This facilitated manufacture, but
resulted in composites that were disastrously brittle, causing a serious set-back for aeronautical use of
carbon fibres. In early work on the use of carbon fibres as reinforcements for aluminium alloys, the
choice of an aluminium/silicon alloy as matrix (on account of the well-known fluidity of these alloys)
led to a rapid reaction between the silicon and the carbon fibre, with the result that a very strong bond
was created but the fibre properties were ruined. This kind of interaction between the fibre surface
characteristics and the fibre/matrix bond, and frequently also the effect of processing, has probably
exercised the minds of composites workers more than most other aspects of the technology.

It would be wrong, then, to consider the fibres and the matrix as the only important constituents of a
composite, even of a simple two-phased composite where no other actual component is intentionally
present. The interface between two solids, especially when thermal or chemical processes have been
involved in putting them together, is rarely a simple boundary between two materials of quite different
character. Over a certain range, which will vary in dimensions depending upon the chemical and
physical natures of the fibres and the matrix, there will be some modification of either chemical or
physical characteristics, or both, resulting in a region, perhaps only of molecular dimensions, perhaps
many microns, which has properties quite different from those of either of the two major components.
Sometimes this occurs fortuitously, as when the two components in a ceramic-matrix composite, which
has to be fired at elevated temperature to achieve compaction, react chemically together, or it may be
deliberately engineered, as when, in the case of GRP materials, we apply a size coating to the fibres after
drawing in order to make them adhere more readily to the resin matrix. Whatever the origin of this
region, which we often refer to as an ‘interphase’, it is important in that its properties determine the
manner in which stresses are transferred from matrix to fibres and, in consequence, many of the
chemical, physical and mechanical properties of the composite itself. Control of this interphase, or
interface region, is a major concern to developers and suppliers of commercial composite materials.


2.3 MANUFACTURING PROCESSES
This book is not about manufacturing, yet it is necessary to know something of the nature of the main
processing methods used in order to appreciate how what happens during processing may affect the
properties of the product. In this chapter, therefore, we shall take only a very general look at the major
processes currently being used to produce composite artefacts. There will be no attempt to discuss the
basic engineering of the individual processes, and further details of manufacturing processes may be
obtained from publications listed in the General Bibliography for Chapter 1.

2.3.1 Polymer-matrix composites
The wide range of processes used to produce reinforced plastics is partly new, and partly derived
from established methods of processing ordinary polymeric materials. The manner of combining fibres
and matrix into a composite material depends very much on the particular combination in question and
on the scale and geometry of the structure to be manufactured.

The commoner varieties of thermoplastic-based materials, like glass-filled Nylon and glass-filled
polyacetal, are made largely by the injection moulding of granules of material in which the chopped
fibres and matrix have been pre-compounded. The principal problem in such cases is that the flow of
material during moulding may be non-uniform, especially in moulds of complex geometry. There may
be regions in which the fibres are highly oriented and others where the degree of orientation is almost
nil, as well as variations in fibre content from place to place. Preferential fibre orientation is often useful,
but only if it can be adequately controlled. Some very sophisticated recent developments in multiple
live-feed injection moulding by Bevis and co-workers have enabled the inventors to produce short-fibre-
filled thermoplastics with very well aligned fibre arrays in different parts of a moulding with the result
that the properties of these materials can compete favourably with those of many continuous-fibre
composites (Allan & Bevis, 1987).

Continuous-fibre thermoset composites are produced by quite different methods. Cylindrically
symmetric structures such as pressure vessels, tanks, rocket-motor casings, centrifuge cylinders, and a
variety of pipes, can be made by winding fibres or tapes soaked with pre-catalysed resin onto expendable
or removable mandrels. Winding patterns may be simple or complex and may be accurately calculated to
resist a prescribed stress system. (eg. a given ratio of hoop stress to longitudinal stress) in service.
Variations in winding pattern or in the combination of stresses subsequently applied to the structure will
clearly change the extent to which the fibres are loaded purely in tension or to which shear stresses are
introduced between separate layers of winding. After the resin has hardened the mandrel is removed and,
if size permits, the product may be post-cured at an elevated temperature. Extremely large vessels can be
made by this method, but these must usually be left to cure at ambient temperature. Since the winding
procedure can be closely controlled, a high degree of uniformity is possible in the fibre distribution of
filament-wound structures, but planes of weakness sometimes occur between winding layers, especially
if resin-rich pockets are allowed to form. An important feature of this process is that after a structure has
been designed according to well-defined principles based on stress analysis of composite materials, the
design software can be extended to provide instructions to a computer or numerically-controlled winding
machine to manufacture the vessel to close tolerances (Eckold, 1994; Jones et al, 1996).

Large panels and relatively complex open structural shapes are easily constructed by hot-pressing
sheets of pre-impregnated fibres or cloth between flat or shaped platens, or by vacuum autoclaving with
the aid of atmospheric pressure to consolidate a stack of ‘prepreg’ sheets against a heated, shaped die.
Woven reinforcements are particularly useful for constructing shapes with double curvature since they
can readily be ‘draped’ over quite complex formers, unlike unidirectional prepregs which may wrinkle
because of their anisotropy. Pressing must be carried out carefully to produce intimate association of the
fibres in different layers, with expulsion of trapped air (and of excess resin unless the prepreg is of the
‘zero-bleed’ variety containing no excess resin), and the time/temperature cycle must be controlled so as
to ensure final curing of the resin only when these conditions have been met. Higher fibre contents are
obtainable with non-woven laminates than with woven-cloth composites, and the characteristics of the
two types of material are quite different. The orientation of the fibres in the separate laminations is


varied to suit the specific load-bearing characteristics required of the laminate or moulding. Although
this is a common process and a great deal of experience has been accumulated in its use, control of the
autoclave cycle is sometimes difficult to establish since there is a complex interaction between the
kinetics of the resin cure, the resin curing exotherm, and the externally applied pressure and temperature.
Research is currently in progress to develop the use of in situ fibre-optic sensors for closed-loop control
of the autoclave (Roberts and Davidson, 1993; Crosby et al, 1995).

For complex aeronautical and similar structures, the relationship between computer-based design
procedures and robotic or computer-controlled manufacturing processes has considerably improved the
production engineering of composites. In particular, the construction of the stacked laminates ready for
processing or autoclaving may be carried out by a computer-controlled tape-laying machine which can
lay up components rapidly, accurately, and with minimum waste of raw materials. Such a system permits
of greater design flexibility than is possible in manual production and can consistently produce parts of
higher strength. Most other operations, including prepreg trimming, transfer between operations,
autoclaving, component trimming, and assembly can also be carried out with a high level of software
control (see, for example, Hadcock and Huber, 1982).

In the pultrusion process, tightly packed bundles of fibres, soaked in catalysed resin, are pulled
through a shaped die to form highly aligned, semi-continuous sections of simple or complex geometry.
Curing of the resin in the die may be achieved either by heating the die itself or by the use of dielectric
heating. Solid and hollow sections may be produced by this process, and because of the high fibre
content (70% by volume is achievable) and the high degree of fibre alignment resulting from the tensile
force used to pull the fibre bundle through the die, extremely high mechanical properties can be
obtained. In more recent developments of the process, off-axis fibres may also be introduced into the
structure. A detailed description of this process has been given by Spencer (1983).

Another familiar process for producing high-quality mouldings of complex shape is resin-transfer
moulding or RTM. Pre-catalysed resin is pumped under pressure into a fibre preform which is contained
in a closed (and often heated) mould. The preform may be made of any kind of reinforcement, but
usually consists of woven cloths or fibre mats, arranged as required by the design. Thick components
containing foamed polymer cores can also be produced in this way. The ease with which this can be
done successfully (Harper, 1985) belies the complexity of the process which involves non-Newtonian
flow of a resin (which may be curing at a significant rate during short-cycle operations) through a
preform in which the fibre distribution results in flow paths of markedly different geometries in the in-
plane and through-thickness directions. In high-speed versions of the process favoured by the
automobile industry the cycle time may be reduced to a minute or less, and there are problems in
obtaining complete wetting and uniform cure throughout the heated mould cavity. The mould cavity is
often evacuated to assist wetting out. This is an important, high-volume, automatable process which,
when used in association with automated fibre-preform manufacture, offers considerable advantages in
the production of low-cost engineering components from GRP (Rudd et al, 1990; Turner et al, 1995). A
wide range of low-viscosity resins — polyesters, vinyl esters, epoxies, urethane-acrylates, etc. — are
now commercially available for use in this process.

A commercially important class of thermoset moulding compounds, the polyester dough-moulding
compounds or DMC, consist of chopped glass fibres blended into a dough with a pre-catalysed resin and
a substantial quantity of inert filler, like chalk. These compounds (also referred to as bulk moulding
compounds or BMC) may be manufactured to final shape by injection moulding or, more commonly, by
the simpler transfer-moulding process — squeezing to shape in hot dies. As in the case of the injection-
moulded reinforced thermoplastics mentioned earlier, problems resulting from non-uniformity and fibre-
orientation effects can also occur during the moulding of these materials.

Lower quality composites reinforced with random-fibre chopped-strand mat (CSM) or continuous-
filament mat (CFM) reinforcements may be press-laminated like the higher-performance CFRP and GRP
laminates but are more frequently laid up by hand methods, especially for irregularly shaped structures.
Large structures, tanks, boats and pipes are often made in this way. The usual procedure is to coat a


shaped former or mandrel with resin, allow the resin to gel and then build up the required shape and
thickness by rolling on layer upon layer of resin-soaked cloth or mat. A final gel-coat is then applied and
the finished structure may be post-cured. The distribution of fibres in such structures will usually be
uneven. Resin pockets and voids are invariably present in these materials.

2.3.2 Metal-matrix composites
The basic attributes of metals reinforced with hard ceramic particles or fibres are improved strength
and stiffness, improved creep and fatigue resistance, and increased hardness, wear and abrasion
resistance, combined with the possibility of higher operating temperatures than for the unreinforced
metal (or competing reinforced plastics). These properties offer potential for exploitation in a range of
pump and engine applications, including compressor bodies, vanes and rotors, piston sleeves and inserts,
connecting rods, and so forth. Components of this type are still under development, but few are in
commercial production (Feest, 1986) apart from one or two limited applications in Japanese
automobiles. Thus, although a wide range of manufacturing methods has been used over the past twenty
years on a laboratory or development scale, at this stage relatively little can be said about large-scale
production processes for MMCs.

Some of the techniques that have been described in detail are

• Unidirectional solidification of eutectics or other constitutionally-appropriate alloys
• Liquid-metal infiltration, often under vacuum, of pre-packed fibre bundles or other preforms
• Liquid-phase infiltration during hot pressing of powder compacts containing whiskers or fibre
bundles
• Hot pressing of compacts consisting of matrix alloy sheets wrapped or interleaved with arrays
of reinforcing wires
• Hot pressing or drawing of wires pre-coated with the matrix alloy (eg. by electroplating,
plasma spraying or chemical vapour deposition (CVI)
• Co-extrusion of prepared composite billets

The first of these processes relates to the manufacture of the so-called in situ composites which began
with model systems, like Cu/Cr and Al/Al2Cu, in most of which thermodynamic constraints impose
serious restrictions on the range of materials that can be treated and in which the fibre contents are small.
Better results were subsequently obtained with more complex systems such as the monovariant ternary
eutectic alloy (Co, Cr)-(Cr, Co)7 C3 which, in the directionally-solidified form, contains about 30 vol%
of the fibrous carbide phase in a Co/Cr solid solution (Thompson et al,1970). Further developments were
based on complex superalloy systems in which the composition of the phases can vary widely without
inhibiting the formation of an aligned two-phased structure during processing. Typical of these materials
are the carbide-reinforced systems exemplified by the NITAC (TaC in nickel) and COTAC (TaC in
cobalt) alloys developed at ONERA in France (Bibring et al, 1971). From the thermodynamic point of
view, composites of this kind possess the clear advantage that they are intrinsically stable under
operating conditions involving elevated temperature. There seems to be little current enthusiasm for this
process.

With both liquid infiltration and solid-state bonding techniques, one of the main problems is that of
ensuring the optimum degree of chemical contact (or wetting) between the fibres and the matrix. In
many systems with practical potential, wetting is inhibited by oxide films or surface chemistry features
of the reinforcing phase, and measures to improve the interfacial characteristics often result in an
undesirable degree of fibre/matrix interaction and damage. Similar consequences of inherent
thermodynamic incompatibility followed when many of the early MMCs were exposed to elevated
temperatures. As a consequence of these wetting and stability problems, a considerable proportion of the
early research and development in this field was concerned with fibre treatments both to improve


adhesion and to provide diffusion barriers — two clearly conflicting requirements — and the
establishment of conditions needed to obtain optimum interfacial bonding.

Short-fibre-reinforced MMCs are commonly made by liquid-metal infiltration (LMI) or squeeze
casting into fibre preforms, a net-shape forming technique that has much to recommend it for localised
or general reinforcement of components such as automotive pistons, connecting rods, etc. Powder
processing is much less satisfactory because of the possibility of fibre damage. A useful practical
development in the liquid-state processing of metal-matrix composites was the introduction in 1984 of a
low-cost short-fibre reinforcement consisting of δ-alumina containing a few percent of silica, called
Saffil by the manufacturers, ICI plc. These fibres, said to be some 45% stronger than other available
continuous α-alumina fibres, are easily wetted by a range of aluminium alloys and can be fabricated into
practical engineering composite components by pressure die-casting or squeeze-casting an alloy matrix
into pre-formed fibre arrays located in the mould. A number of announcements of practical applications
of the material followed, but ICI’s interest in composite materials proved as transitory as those of almost
every other major UK company which had a composites programme in the 1970s and 1980s, and the
material is no longer obtainable.

Products requiring continuous reinforcement, such as boron/aluminium composite panels and
sections that have been under consideration for aerospace applications (Renton, 1977) are preferably
produced by solid-state processes such as diffusion bonding or hot pressing in which critical
pressure/time/temperature cycles are needed to obtain optimum quality. Structures can be fabricated
relatively easily (by comparison with thermoset resin matrix composites) by brazing, welding, diffusion
bonding or mechanical fastening. There is now a wide range of continuous fibres available that are
suitable for reinforcement of MMCs, and a variety of production techniques, including liquid-metal
infiltration, squeeze casting, electroplating, diffusion bonding of plasma-sprayed tapes, and even
explosive welding, has been developed to make use of them, in addition to conventional roll-bonding
and hot-pressing. At the time of writing. however, there are few genuine large-scale applications for
composites of this kind. unlike their resin-matrix equivalents.

Of the recently renewed activity in MMCs, not all is associated with fibre-reinforced materials, much
work having been done on particulate MMCs, a class of materials that was formerly familiar to
metallurgists as ‘cermets’, although the classical cermet — the cemented carbides and other metal-
bonded ceramic materials mostly used for cutting tools — contained a very low proportion of the
metallic phase by comparison with modern MMCs. The most familiar of the modern MMCs are made by
the incorporation of SiC particles (after treatment to improve particle/matrix adhesion) at volume
fractions of 0.1 to 0.3 into aluminium alloys by various routes, including co-spraying, powder
metallurgy, and liquid infiltration. These materials are claimed to show significant increases in specific
stiffness and strength, although at the expense of ductility and toughness, with the advantage (eg. over
reinforced plastics) that they can be processed by conventional metal-forming methods. A recent survey
of the manufacturing processes used for MMCs has been published by Everett and Arsenault (1991).

2.3.3 Ceramic-matrix composites
Fabrication processes are complex and need to be carefully optimised because of the inevitable
sensitivity of materials properties to microstructures controlled by processing conditions and
interactions. Much of the recent work on CMCs in the US, Japan and Europe has largely followed
relatively familiar routes in attempting to reinforce glasses (like borosilicates) and glass-ceramics (like
lithium aluminosilicate or LAS and calcium aluminosilicate or CAS) with fibres such as the commercial
Nicalon and Tyranno varieties of silicon carbide. Substantial improvements in mechanical properties
have been achieved, by comparison with early carbon-fibre/glass composites. The fibres are usually
impregnated with a slurry of fine glass powder and subsequently hot-pressed (Dawson et al, 1987). If the
matrix is a glass-ceramic, the final stage in manufacture is the ‘ceramming’ process, the final heat
treatments to convert the glass to a fully dense ceramic. Companies like Corning in the US offer such
materials for commercial sale. Critical control of manufacturing conditions is needed to produce
appropriate interfacial conditions for the optimum combination of strength and toughness. The
advantage of the glass-ceramic route is that relatively modest processing temperatures are involved,


although even so a typical residual thermal stress of the order of only 200MPa may still be sufficient to
cause cracking of the matrix in as-manufactured or lightly-loaded composites. Much of the research on
monofilament-reinforced ceramics like the French SiC/SiC materials, for example (Naslain and Langlais,
1986), and on the production of carbon/carbon composites (Fitzer, 1987), has been directed towards
chemical vapour infiltration (CVI) and related processes. Commercial quantities of these materials have
been available for a considerable time although CVI is normally a very slow process and components
produced in this way are very costly.

However, for all work involving the production of ceramic-fibre-reinforced ceramic composites by
conventional pressing methods, there remains the problem of compatibility. Nicalon fibre with a starting
strength as high as 3GPa, for example, may lose half its strength when pressed into composites with a
CAS or borosilicate glass matrix, and following high-temperature exposure in service this load-bearing
ability may be further impaired. Current developments in the field involve the search for reliable
diffusion-barrier coatings (plus ça change!).

It is perhaps also worth remarking that although the driving force for developments in the CMC field
has most frequently been the needs of high-temperature applications in the aerospace industry, there are
many other branches of engineering—automotive, chemical, marine, and general engineering, for
example—where there is a need for reliable economic components possessing good mechanical
properties and reasonable wear and corrosion resistance, in combination with adequate impact and
thermal shock resistance at ordinary or only slightly raised temperatures. For many such applications
there is no need for high-temperature stability, and the problems relating to fibre/matrix interactions at
high temperatures are not as serious as in gas-turbine applications, for example, provided processing
methods can be developed which do not require the high firing temperatures normally associated with
ceramics manufacture. One such process is the sol-gel process in which low-viscosity sols are converted
to solid ceramics in a manner analogous to the curing of organic resins. And by extension of the analogy,
composite materials may be made by techniques of casting, filament winding and sol-transfer moulding,
as described by Russell-Floyd et al (1993).

2.3.4 Hybrid composites
Reference to hybrid composites most frequently relates to the kinds of fibre-reinforced materials,
usually resin-based, in which two types of fibres are incorporated into a single matrix. The concept is a
simple extension of the composites principle of combining two or more materials so as to optimise their
value to the engineer, permitting the exploitation of their better qualities while lessening the effects of
their less desirable properties. As such, the definition is much more restrictive than the reality. Any
combination of dissimilar materials could in fact be thought of as a hybrid. A classic example is the type
of structural material in which a metal or paper honeycomb or a rigid plastic foam is bonded to thin skins
of some high-performance FRPs, the skins carrying the high surface tensile and compressive loads and
the core providing lightweight (and cheap) structural stability. The combination of sheets of aluminium
alloy with laminates of fibre-reinforced resin, as in the commercial product ARALL (aramid-reinforced
aluminium, Davis, 1985) is a related variety of layered hybrid, and the mixing of fibrous and particulate
fillers in a single resin or metal matrix produces another species of hybrid composite.

Some hybrids of current interest represent attempts to reduce the cost of expensive composites
containing reinforcements like carbon fibre by incorporating a proportion of cheaper, lower-quality
fibres such as glass without too seriously reducing the mechanical properties of the original composite.
Of equal importance is the reverse principle, that of stiffening a GRP structure with a small quantity of
judiciously placed carbon or aromatic polyamide fibre, without inflicting too great a cost penalty. In
high-technology fields the question of cost may be insignificant by comparison with the advantages of
optimising properties. In aerospace applications, a familiar purpose of using hybrids is to utilise the
natural toughness of GRP or of Kevlar-fibre-reinforced plastics (KFRP) to offset a perceived brittleness
of typical CFRP. From the designer's point of view the important aspect of using hybrids is that provided
there is adequate understanding of the underlying mechanisms of stiffening, strengthening and
toughening, they allow even closer tailoring of composite properties to suit specific requirements than
can be achieved with single-fibre types of composites.


2.4 DEFECTS IN MANUFACTURED POLYMERIC COMPOSITES
All practical reinforced plastics composites are likely to contain defects of various kinds arising from
the processes of manufacture. Indeed, composites are noted for the variability of mechanical properties
they exhibit unless they have been produced under the most rigorously controlled conditions. The
variability of material produced by hand lay-up methods is more marked than that of composites made
by mechanised processes. The specific nature and severity of the defects found in any manufactured
product will also be characteristic of the manufacturing process. In addition, any composite consisting of
materials of widely different thermal expansion coefficients which is heated during manufacture may, on
cooling, develop residual stresses sufficiently high to crack a brittle matrix. The defects that may be
present in manufactured composites include:

• incorrect state of resin cure, especially that resulting from variations in local exotherm
temperatures in thick or complex sections during autoclaving
• incorrect overall fibre volume fraction
• misaligned or broken fibres
• non-uniform fibre distribution, with resultant matrix-rich regions
• gaps, overlaps or other faults in the arrangement of plies
• pores or voids in matrix-rich regions
• disbonded interlaminar regions
• resin cracks or transverse ply cracks resulting from thermal mismatch stresses
• disbonds in thermoplastic composites resulting from failure of separated flows to re-weld
during moulding
• mechanical damage around machined holes
• local bond failures in adhesively bonded composite components

and we shall discuss some of these in more detail in the context of particular manufacturing processes.

In the moulding of shaped artefacts from materials such as dough-moulding and sheet-moulding
compounds (DMC and SMC) there are two main kinds of possible defects. The first results from failure
of the process to control resin cure and flow, perhaps as a consequence of the use of a moulding
compound that is past its shelf-life or because of incorrect control of the temperature/pressure cycle. The
consequences are poor consolidation, incomplete filling of the mould, and perhaps also the failure of
separated flow streams to re-amalgamate in complex mouldings. The second problem may arise either
from the non-uniform filling of the mould cavity (a manual operation) prior to pressing or from poor
attention to the flow patterns that will occur during pressing. In either case there may again be non-
uniformity of fibre content in the finished artefact, and/or local preferred orientation of the short fibres,
with resultant anisotropy in mechanical properties. If this anisotropy is not allowed for in the design, the
component may fail to meet the design specification. Similar considerations apply, of course, to the
injection moulding of short-fibre-reinforced thermoplastics.

Typical defects in RTM mouldings include porosity resulting from bubbles injected with the resin,
‘washing’ or movement of the preform, especially near injection points or corners, failure of the resin to
penetrate the preform uniformly, poor wetting-out of the fibres by the resin, distributed porosity, and
premature gelation. Porosity is a particular problem, since the process relies on the flow of resin to
sweep out the air contained within the mould and the fibre preform, and when filled matrix resins are
being injected into a tightly packed fibre preform, there is sometimes a tendency for the filler particles to
be filtered out of resin. The quality of RTM mouldings is considerably improved, however, by slightly
reducing the air pressure in the mould cavity. In the so-called vacuum-assisted RTM process (VARTM)
it is sufficient to maintain the mould cavity pressure at only 0.7 bar in order to improve considerably the
wetting out of the preform and reduce the level of residual porosity in the moulding to a fraction of a


percent for relatively little extra cost. Figure 2.1 1.6
shows how the mechanical properties of a typical RTM with
glass/polyester composite vary with the levels of vacuum

Property value, relative to the
1.4
porosity characteristic of different processing

value for hand lay-up
hand compression RTM
conditions. The average porosity levels in moulding
lay-up
VARTM mouldings are comparable with those 1.2
obtainable in high-quality autoclaved materials.
1.0
A schematic illustration of some of the typical
faults that occur in the laying up of prepreg Flexural strength
0.8 Flexural modulus
materials is shown in Figure 2.2. The finite
Short-beam shear strength
dimensions of the prepreg will require, from time
to time, that joins (a) will have to be made. 0.6
4 3 2 1 0
Closely butted joins are clearly more satisfactory Porosity, vol%
than overlapped (b) or widely separated (c)
prepreg ends since these will result in resin-rich Figure 2.1. Variation of mechanical properties of GRP
regions, as shown. Rigorous quality-control materials as a consequence of variations in porosity levels
procedures should prevent the incorporation of introduced during manufacturing. In each case the fibre
inclusions such as bits of rubbish (d), but a volume fraction is approximately 0.4. (Hayward and Harris,
1990)
common problem with prepregs is the occurrence
of small bundles of fibre debris, known as ‘whorls’, which are sometimes attached to the surface of the
sheet and subsequently become incorporated in the laminate as small regions of random fibre alignment.
These are considered to be damaging to composite strength, and an aerospace manufacturer may
typically specify that such defects should not occur more than once in, say, ten metres of prepreg roll.
Their effect is modest, however. In samples of [(0/90)2]S glass/epoxy laminate hot-pressed from
commercial prepreg, with deliberate concentrations of whorls to the extent of 3 defects per prepreg ply
in a 150mm x 150mm x 1mm plate, the tensile strength was reduced by only about 6%, and the failure
strain by about 2%, the stiffness being unaffected (Ellis, 1985).

Failure to achieve a consistent level of pressure over the whole surface of a sheet during autoclaving
may result in small regions where the plies are imperfectly bonded (e). These disbonds are particular
sources of weakness in laminates subjected to compression or impact forces. Voids or imperfectly cured
resin regions near the surface of a laminate may absorb moisture during service in wet conditions, with
the result that hydrostatic pressure could develop inside the region and push out the surface layer or gel
coat to create blisters (f). Poorly stored prepregs or prepregs that are draped over a former which is too
sharply curved may become wrinkled (g), and this may again lead to poor surface quality or weaknesses
resulting from resin-rich regions in the laminate. The last feature shown in Figure 2.2 is a deliberately
introduced ‘defect’ rather than an adventitious manufacturing fault. It is often required to change the
dimensions of a structure, and since the individual ply effectively represents a ‘quantum’ of laminate
thickness, such changes can only be made by gradually reducing or increasing the number of plies in a
cross section. This design procedure by the introduction of ‘ply drop-offs’ (h) nevertheless presents a
problem for the stress engineer since the ply ends result in local stress concentrations the effects of
which must be accounted for (see, for example, Curry et al, 1985; Cui et al, 1994). The prepregs
themselves may also be a source of defects if the strictest quality control procedures are not employed.
Broken or damaged fibre-tows, accumulations of debris (eg. the ‘whorls’ mentioned above), and out-of-
date resin are examples of prepreg defects that can lead to poor quality in the final laminate.


Since most composites consist of materials of widely
different thermal expansion coefficients they may, if
heated and cooled during manufacture, develop residual
stresses sufficiently high to crack a brittle matrix. Early
high-performance CFRP laminates were frequently
found to have suffered multiple internal cracking unless
proper care was taken to control the production cycle.
This type of problem has been eliminated to a large
extent for polymer composites by the advent of
‘modified’ or ‘toughened’ resins with high failure
strains, but such problems will inevitably be of more
concern for users of such materials as CMCs.

The advantages of using hybrid laminates may be
both structural and economic, as we have seen, but
there is also a potential disadvantage. Over the
temperature range of about 150°C from conventional
Figure 2.2. Some typical defects in composite curing temperatures down to ambient, the axial thermal
laminates manufactured from prepreg sheets. expansion coefficient of most carbon fibres is close to
zero, while that of an aramid fibre like Kevlar-49 is
negative, about –3 x 10-6 K-1, and that of E-glass is positive, 0.7 x 10-6 K-1. Thus, in hybrid laminates
containing various mixtures of these fibres, there will be a tendency to develop thermal stresses which
may become great enough to cause marked undulations in the plies subject to compression — the Kevlar
plies in a carbon/Kevlar hybrid, for example. A consequence of this is that the wavy plies are unable to
contribute their full share of load-bearing in tension, and the compression strength of a hybrid may also
be reduced because the waviness facilitates a common compression kinking mechanism (see chapter 4).

One of the virtues of polymer composites is that complex shapes can often be manufactured in single-
shot operations, thereby eliminating much of the need for the joining operations that characterise most
conventional manufacturing with metallic materials. Joining and machining operations may not always
be completely avoidable, however, and in such cases there is always likely to be the possibility of
bonding failures in imperfectly prepared adhesive joints (in thermoset composites), disbonds in welded
thermoplastic composite plates, and machining damage (in sawn or drilled components, for example).
But while machining damage will usually be at cut edges, and therefore visible, poor joints will rarely be
detectable from the exterior.

Any of these defects can locally reduce the strength of the material below the required component
design stress. They may then act as sites for the initiation of fatigue damage, or may facilitate the growth
of a fatigue crack during cyclic loading. A comprehensive assessment of the quality of a composite
material prior to putting it into service is therefore as important as the monitoring of the levels of damage
accumulated in a composite structure during service. The use of non-destructive evaluation methods is
thus of great importance in composite materials technology, and modern aerospace design procedures
increasingly relate to the establishment of zero-growth thresholds for manufacturing and service defects.
Many of the defects associated with hand lay-up or manual operation may be avoided, where the extra
cost is justifiable, by the use of computer-controlled tape-laying machines which are able to construct a
prepreg stack for autoclaving to very high standards of quality and repeatability. Similarly, errors of
control in pressing can be avoided to some extent by the use of automated autoclaves with pressure-
temperature cycles carefully programmed from detailed chemical knowledge of the gelation and
viscosity characteristics of the resin in use. It is largely only in the aircraft industry that the high initial
costs of such equipment can be borne, however.

Of particular importance to the manufacturer and user is the associated question of what are likely to
be the consequences of the presence of a particular defect. At present, there is no easy way to quantify
this. A few isolated spherical pores a micron or so in diameter may have no effect on any physical
property, and may not therefore affect the tensile or flexural mechanical performance of a composite.


But a distribution of innocuous looking pores may markedly reduce the interlaminar shear strength of a
material, and by providing sites for accumulation of moisture may also ruin the electrical or dielectric
performance of the material. A minor delamination between plies in a complex laminate may have no
effect on the tensile strength of the material, but such defects are frequently injurious to the compression
performance of laminates and, as indicated above, may grow rapidly to eventual ruin of the composite
under cyclic loading conditions. The question of critical size or quantity of defects must therefore be
addressed with specific reference to the particular composite and service conditions involved in a given
application.

2.5 METHODS OF NON-DESTRUCTIVE EVALUATION FOR POLYMER COMPOSITES
Since the presence of defects in reinforced plastics, either ab initio or as a consequence of damage in
service, is to some extent inevitable, it follows that manufacturers and users need sensitive techniques for
the detection of these defects and damage. Much of the technology previously developed for metallic
engineering materials and structures has been transferred across, with appropriate modifications, for use
with fibre composites. Some of these techniques are more useful than others, and it is often good
practice to use a back-up technique where possible rather than to rely on a single method. Harris and
Phillips (1983, 1990) have given summaries of some of the more common methods in current use, but at
the time of writing there is a high level of activity in R & D work on fibre-optic sensors specifically for
use with reinforced plastics. Brief details of some of these NDT tools are given below.

2.5.1 Optical inspection
In translucent GRP, inspection by transmitted light can give indications of the presence of pores, poor
wetting-out, delaminations, and gross inclusions. Light transmission can also sometimes be correlated
with the fibre content, Vf. Loss of transparency (stress-whitening) is associated with the development of
fibre/resin debonding and resin cracking. In non-translucent composites the only feasible visual-
inspection techniques relate to observation of surface damage. The detection of surface cracking may be
enhanced by dye-penetrant methods. Moiré methods require photographic grids to be printed on the
surface of the composite and irradiated with coherent light. The resulting interference fringes give clear
indications of local stress concentrations and deformation, including those arising from sub-surface
damage. Laser holographic methods and electronic speckle-pattern interferometry (ESPI) are also used
for non-translucent materials.

2.5.2 Radiographic methods
Radiography is not easily applied in the field, but information about composite quality can be
obtained by x-ray inspection. Contact radiography with sources of 50kV or less yields high-contrast
photographs from low-density materials like GRP because of their low inherent filtration. The linear
absorption coefficient of glass is about twenty times that of most resins, and film density measurements
can be correlated with Vf if the material is unpigmented. The fibre distribution, quality of weave, and the
presence of large laminating defects can be easily investigated. Contact radiographs showing fibre
distributions have also been analyzed by optical diffraction methods to give quantitative information
about the fibre distribution. The sensitivity of contact radiography can be improved by impregnating the
composite with a radio-opaque material, usually an alcoholic solution of zinc iodide. By this technique,
resolution of cracks of the order of a few mm long and 0.1 mm deep is feasible in GRP and CFRP, and
delaminations are easily resolved.

2.5.3 Thermal imaging
When a uniform heat flux is supplied to a plate, any anomalous variations in the resulting temperature
distribution in the plate are indications of structural flaws in the material. Similarly, in an otherwise
uniform material working under variable loads, local damage will give rise to changes in the hysteretic
thermal losses in the body of the material. Thermal imaging, with limiting detection of temperature
differences of about 0.2°C, is easily carried out by means of infra-red television photography. Damage
has been detected in glass/epoxy laminates at low stresses and frequencies, but in materials of higher
thermal conductivity detection is more difficult. Testing frequencies greater than 5Hz are needed in


order to detect damage in CFRP because of their high thermal conductivity. Thermal imaging equipment
is routinely available and is widely used in the aerospace industry. The technique has the advantage over
many other methods that it is a remote monitoring technique. The use of thermography has been
extended in recent years by the use of transient heat sources and rapid scanning of the induced thermal
fields by TV/video-compatible infra-red imagers which forms the basis of the pulse video thermography
(PVT) technique. This is a convenient and versatile extension of conventional thermography capable of
rapid detection of near-surface defects. An associated thermal NDE method is that of thermoelastic stress
analysis which makes use of the familiar thermoelastic effect, where a temperature change accompanies
adiabatic elastic deformation of a body. This technique is said to offer spatial resolution down to 1mm
and temperature discrimination of about 0.002°C for stress changes of about 2MPa in steel, although in
composites there are potential problems of interpretation resulting from their anisotropic nature.

2.5.4 Ultrasonic techniques
Ultrasonic inspection techniques are perhaps the most widely-used NDE methods for composites.
The velocity and attenuation of an ultrasonic pulse passing through a material provide information about
the general physical properties (ie. the stiffness) and the structure (ie. the defect/damage state) in a
material, provided certain basic features of the material, such as the anisotropy, are taken into account.
Most composites are highly dispersive, often having regular structural patterns at several levels of scale
(fibres, fibre tows, weave of cloth, laminate stacking sequence), an array of interfaces (fibre/resin
interface, interlaminar interface) and an inherent distribution of defects. Ultrasonic waves in composites
are therefore highly attenuated and their velocity and attenuation are dependent on frequency, this
frequency-dependence being affected by the construction of the laminate. The propagation of ultrasonic
waves may be affected by frequency, composite structure, structural defects, and damage induced by
loading. The problem of high attenuation is overcome by the use of broad-band transducers and low
wave frequencies which give shorter, more highly penetrating pulses with less pronounced near-field
interference effects. As a result of the low pulse intensities used in composites testing it is generally
necessary to use higher amplification than is needed for other materials, but most of the common
ultrasonic techniques, including pulse-velocity measurements, echo methods, goniometry, and ultrasonic
interferometry, are used with composites.

The most highly developed ultrasonic NDE method, based on measurements of through-thickness
attenuation, is the C-scan technique, which is used routinely for inspecting large panels. Synchronized
raster-scanning motions of the transmitter and receiver (either in a water bath or with irrigated water-jet
probes) on opposite sides of the plate enable measurements of the intensity of the transmitted wave to be
made as a function of position. It is common to use a focused transmitter, focusing on the back face of
the plate, with the receiver as close to the front face as possible The transmitted intensity is used to
modulate the brightness of a visual display, or the density of ink on a diagram, so as to build up a picture
of plate quality. The resolution of the technique is limited by dispersion in the composite and by the
beam dimensions. It is necessary to use something like a focused 4MHz probe to detect defects with
dimensions of the order of 1mm, and as a consequence C-scanning is more reliable as an indicator of
general quality than as a detector of specific defects. Large voids, distributions of fine porosity, areas of
variation in fibre content, and delaminated regions will usually be revealed by a C-scan, but for
assessment of the severity of specific defects it is necessary to make standard test plates containing flat-
bottomed drilled holes of various depths and diameters for calibration purposes. Conventional C-
scanning is a slow process, but new techniques for real-time operation are now on the market.

2.5.5 Optical fibre sensors
Optical fibre sensors have a number of advantages over other types of sensor in that they are immune
from electromagnetic interference, they can be embedded in a composite material during manufacture
rather than simply being surface mounted, they can be used to monitor several parameters, including
temperature, pressure, strain, and chemical characteristics, and they offer important possibilities for on-
line application, both for process optimization and in-service health monitoring (Fernando, 1998).


2.5.5.1 Cure monitoring
The mechanical properties of composites are strongly dependent upon the chemical and rheological
events occurring during the cure cycle. There is considerable current interest in the possibility of using
fibre-optic sensors embedded in the composite itself to carry out in situ measurements of cure state, cure
kinetics, resin viscosity and residual stress build-up in real time during manufacture. It is expected that
the signal processing procedures that are involved could, eventually, form the basis for intelligent
processing of composites (Roberts & Davidson, 1993). Techniques that are being explored include
refractive-index-based sensors (Crosby et al, 1995), evanescent-wave spectroscopic sensors (Powell et
al, 1996), and transmission or reflection infra-red spectroscopy sensors (Crosby et al, 1996). These
sensors are able to evaluate changes in the optical properties, and hence the state of cure, of the resin
surrounding a stripped length of embedded fibre, or of the resin in a gap between two lengths of fibre, or
of resin at the end of a single embedded fibre.

2.5.5.2 Sensing of deformation and damage (health monitoring)
The simplest form of assessment of impact damage is based on the fact that optical fibres will fracture
when the composite in which they are embedded suffers an impact event, and this reflects the related
damage that will be sustained by the main load-bearing fibres. More sensitive assessment of the presence
of strain or a rise in temperature of the composite can be obtained by the use of fibres which have Bragg
gratings written into their cores or the somewhat cheaper Fabry-Perot sensors in which changes in the
length of an air gap are measured by interferometry (Greene et al, 1995). These provide a means of
monitoring strain with a high degree of accuracy and resolution. They can also be used for on-line
location of the position of impact events — Greene et al cite an ability to locate an impact with a
resolution of 0.5mm and an accuracy better than 5mm. Provided the separate effects of thermal
expansion and elastic strain can be isolated, sensors of this kind can also provide information about
temperature changes in the material. They also therefore offer the possibility, mentioned in section
2.5.5.1, of closer monitoring and intelligent control of processing procedures, feedback methods being
used to control the severity of the resin curing exotherm and therefore the quality of the finished product.

2.5.6 Microwave methods.
The dielectric constant of glass is much greater than those of resins at microwave frequencies, and
any technique for measuring the average dielectric constant of a GRP composite can therefore be
correlated reasonable satisfactorily with the glass content of the material. A typical instrument consists
of an open-ended coaxial resonator which can be applied to a GRP surface in order to close the resonant
cavity. As the instrument is moved over the surface, changes in the local average glass content are
indicated by changes in the resonant frequency. Since any defect which affects the microwave
penetration will also cause a resonance shift, the method is capable of monitoring the deterioration of a
structure under load, once its dielectric response in the undamaged state is known.

2.5.7 Dynamic mechanical analysis.
When stresses and strains can be continuously monitored in service, we might expect that changes in
elastic modulus would give non-destructive indications of deterioration of properties caused by the
accumulation of damage. However, the change in stiffness of a composite as a result of cycling depends
on the type of composite and the extent to which the matrix, fibres and interface are directly loaded.
Measurement of the modulus itself, either by strain gauges or by ultrasonic pulse-velocity
measurements, may therefore prove to be relatively insensitive to changes in the damage state in certain
cases, although relative changes in resonant frequency and of the damping capacity can give more subtle
indications of damage accumulation. If a structure can be made to resonate in a reproducible manner, its
frequency spectrum can be rapidly analysed to establish its characteristic ‘sound’. Any damage to the
structure will change this characteristic spectrum in a recognisable way and this provides the basis of a
simple ‘coin-tap’ test which can be used to monitor damage in composites.

One of the oldest-established NDE methods is the principle embodied in the Fokker debond tester
manufactured by Wells-Krautkramer. This machine uses swept-frequency probes generating shear waves
of constant amplitude and compares the response of the test material with that of a standard. The


resonant frequency and amplitude of the transducer response are recorded in contact with the standard,
and any shifts that occur when the probe is subsequently coupled to the test sample are noted. These
shifts must be calibrated against some known or measurable property. The technique is said to be
capable of giving more sensitive indications of the presence of flaws, voids, weak bonds, porosity,
delaminations, incorrect cure conditions, poor wetting, and poor gap-filling in joints than any other
ultrasonic method.

2.5.8 Acoustic emission methods.
Any sudden structural change within a composite, such as resin cracking, fibre fracture, rapid
debonding, or interlaminar cracking, causes dissipation of energy as elastic stress waves which spread in
all directions from the source. The technique of detection of these acoustic emissions (AE) by suitable
transducer/amplifier systems is now well-established and triangulation methods may be used to locate
flaws in large structures and to assess their severity. There are several ways of analysis of the
information obtained by AE monitoring of structures under load, some of which offer suitable
quantitative procedures for proof testing or life prediction, and some of which provide deeper insight
into the mechanisms of damage accumulation in composites.

2.6 THE USE OF COMPOSITES
In order to make good use of engineering materials, we have to understand their behaviour
sufficiently well to be able predict performance in both the short term and the long term. In the context
of composite materials, this means developing mathematical models which represent reasonably closely
the known experimental response of real composites to applied stresses and environmental conditions. If
we can predict the behaviour of any given composite accurately as a result of this modelling, then we
can have confidence that the material we have designed will meet the service requirements of the
application concerned. One of the problems with composites is that there is such a diverse range of
materials, and it is necessary to have a thorough understanding of each class: we cannot simply read
across from the characterisation of one group of composites to another without carrying out extensive
checks of the validity of the old models for the newer materials. Theories of the elastic behaviour of
composite materials are very well developed, and predictions of elastic response are often very
satisfactory. Problems arise, however, when attempts are made to predict processes of failure. The
failure of composites is almost always a complex process. Damage accumulates in a widespread fashion
in composites, and many individual processes occur at the microstructural level. The strength of a
composite is determined by the conditions which determine the climax of this damage accumulation
process, and modelling of these conditions is far from perfect at the present time. As a consequence,
there is still a certain lack of confidence on the part of designers, a reluctance to use some of these new
materials for which we have not, as yet, the operating experience that instils confidence, and this leads to
a measure of inefficient (and often uneconomic) design for which the only cure is better understanding.

From the point of view of the designer who is using composites, there is an important constraint:
composites do not behave like metals. Conventional notions of materials behaviour and design methods
are therefore unlikely to be appropriate. The use of composites offers the designer a range of new
solutions to engineering problems of many kinds, but it is vital to use the opportunity to rethink the
approach. It is rarely a good idea simply to carry out a piece-meal substitution of composites parts for
existing metallic parts. This kind of approach will certainly not liberate the designer from the old
constraints, will often lead to more expensive rather than cheaper solutions, and may result in unsafe
designs.

In the application of composites, as in other areas of materials engineering, it is important to consider
the selection of the materials and the choice of manufacturing process as a vital and integrated part of the
design process.

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3 ELASTIC PROPERTIES OF FIBRE COMPOSITES

3.1 SIMPLE MICROMECHANICAL MODELS

T
he simplest method of estimating the longitudinal modulus, E c
stiffness of a composite in which all of
the fibres are aligned in the direction of fibre
matrix
the applied load (a unidirectional load,
composite) is to assume that the structure is a P c
simple beam, as in Figure 3.1, in which the two
components are perfectly bonded together so Figure 3.1. Simplified parallel model of a unidirectional
composite.
that they deform together. We shall ignore the
possibility that the polymer matrix can exhibit time-dependent deformation. The elastic (Young) moduli
of the matrix and reinforcement are Em and Ef, respectively. We let the cross-sectional area of the fibre
‘component’ be Af and that of the matrix component be Am. If the length of the beam is L, then we can
represent the quantities of the two components in terms of their volume fractions, Vf and Vm , which is
more usual, and we know that their sum Vf + Vm = 1. The fibre volume fraction, Vf, is the critical
material parameter for most purposes. The subscript ‘c’ refers to the composite.

The load on the composite, Pc, is shared between the two phases, so that Pc = Pf + Pm, and the strain in
the two phases is the same as that in the composite, εc = εf = εm (ie. this is an ‘iso-strain’ condition).
Since stress = load/area, we can write:

σcAc = σfAf + σmAm

and from the iso-strain condition, dividing through by the relevant strains, we have:

σc A c σ f A f σm A m
= +
εc εf εm

or Ec = Ef Vf + Em(1 - Vf ) ................................................................................ [3.1]

This equation is referred to as the Voigt estimate,
but is more familiarly known as the rule of mixtures.
400
It makes the implicit assumption that the Poisson W/Al-4%Cu
Young modulus, E c , GPa

ratios of the two components are equal (νf = νm), glass/epoxy
thus ignoring elastic constraints caused by 300
differential lateral contractions. More sophisticated
models have been developed which allow for such
effects, the most familiar being that of Hill (1964) 200
which shows that the true stiffness of a
unidirectional composite beam would be greater than
100
the prediction of equation 3.1 by an amount which is
proportional to the square of the difference in
Poisson ratios, (νf – νm)2, but for most practical 0
purposes this difference is so small as to be 0 0.2 0.4 0.6 0.8 1.0
negligible. For example, for high-performance Fibre volume fraction, Vf
reinforced plastics, νm ≈ 0.35 and νf ≈ 0.25, and for a
fibre volume fraction, Vf, of about 0.6, the correction Figure 3.2. Confirmation of the rule-of-mixtures relationship for
needed to account for the Poisson constraints is only the Young moduli, Ec, of undirectional composites consisting
about 2%. A good indication of the validity of the of tungsten wires in Al-4%Cu alloy and glass rods in epoxy
resin.
mixture rule for the longitudinal moduli of two

chapter three Elastic Properties 34

composites of quite different kinds is given in transverse modulus, E t
Figure 3.2.
fibre matrix
To estimate the transverse modulus, Et, we load,
use a similar approach with a block model such Pc
as that shown in Figure 3.3, with the same
constraints as before, ie. well-bonded
Figure 3.3. Simple series model of a composite.
components with similar Poisson ratios, and no
visco-elastic response from the matrix. This is now an ‘iso-stress’ model, so that σc = σf = σm. The total
extension of the model is the sum of the extensions of the two components:

εcLc = εf Lf + εm Lm

If the cross-sections of both phases are the same, L ≡ V, so dividing through by the stress (and
remembering that Vf + Vm = 1) we have:

εc ε f Vf εm Vm
= +
σc σf σm

or 1 Vf Vm
= +
Et Ef Em

This is referred to as the Reuss estimate, sometimes called the inverse rule of mixtures: the transverse
modulus is therefore:

Ef Em
Et = ............................................................................................ [3.2]
Em Vf + Ef (1 − Vf )

and the relationship between the Voigt and Reuss models can be seen in Figure 3.4. The Reuss estimate
is sometimes modified to account for the Poisson effect in the matrix by introducing a ‘constrained’
matrix modulus, effectively by dividing Em by
(1 – ν2m) , so that:
80
1 Vf Vm (1 − νm ) ………..........….[3.3]
2
= + rule of mixtures, or
Et Ef Em Voigt estimate, Ec
Young modulus, E, GPa

60
The relatively small effect of this correction is
also shown in Figure 3.4.
40 modified
The problem with the Reuss model for the series model
transverse case is that the geometry shown in
Figure 3.3 in no way resembles that of a fibre 20 Reuss series
composite perpendicular to the fibres. And in model, Et
assuming that the Poisson ratios of the phases
are the same, it ignores constraints due to 0
strain concentrations in the matrix between 0 0.2 0.4 0.6 0.8 1.0
the fibres. As a consequence, although it Volume fraction of reinforcement, Vf
appears to give the correct form of the
variation of Et with Vf, the values predicted
seldom agree with experimental Figure 3.4. Predicted variations of the longitudinal elastic
modulus, Ec, and the transverse modulus, Et, of composites of
measurements. The model also implicitly glass and epoxy resin connected in parallel and in series. The
assumes that the transverse stiffness of the solid curves give the Voigt (parallel) and Reuss (series)
fibre is the same as its longitudinal stiffness, estimates. The dashed line represents the Reuss estimate
and while this is true of isotropic fibres like modified to include the effect of matrix Poisson constraint.


glass, it is not true of reinforcements with a textile
origin like carbon and Kevlar.

The need for a more refined procedure is apparent
if we look at even an idealised model of the structure
of a composite loaded transverse to the fibres, as
shown in Figure 3.5. A model based on hexagonal
(close-packing) geometry would clearly give different
results from this square-packing model, and both
Figure 3.5. Idealised square packing geometry in a
would be different from the true packing geometry of transverse section of a unidirectional composite.

any real composite (see Figure 3.6, for example) which
will almost always tend to be much less regular than
either of these idealised geometries. In practice the true
transverse modulus lies some way above the lower bound
given by the Reuss estimate.

In discussing the behaviour of anisotropic materials, it
becomes inconvenient to continue using subscripts like c
and t and we resort to using a numerical index system
associated with orthogonal axes, x1, x2 , and x3. We
choose the x1 direction to coincide with the fibre axis and
the x2 direction to coincide with the transverse in-plane
direction (assuming that we are dealing, as is usually the
case, with a thin plate of material). The longitudinal
Figure 3.6. Real packing geometry: the cross-
section of a unidirectional SiC/CAS laminate of Vf stiffness of a unidirectional composite is then referred to
≈ 0. 4. The magnification can be judged from the as E1 and the transverse stiffness is E2. Since in most
fibre diameter, 15μm. practical reinforced plastics composites, Ef » Em the Reuss
and Voigt estimates of stiffness given in equations 3.1 and
3.2 can be usefully approximated (in the new notation) by the relationships:

E1 ≈ Ef Vf........................................................................................................... [3.4]a

E2 ≈ Em (1 – Vf )-1 ............................................................................................. [3.4]b

from which it is clear that the stiffness in the fibre direction is dominated by the fibre modulus, while
that in the transverse direction is dominated by the matrix modulus.

The Poisson ratio, ν, of an isotropic material is defined as the (negative) ratio of the lateral strain, ε2,
when a stress is applied in the longitudinal (x1) direction, divided by the longitudinal strain, ε1,
ie. ν = –ε2/ε1. Consideration of equations 3.1 and 3.2 shows that in a unidirectional composite lamina
there will be two in-plane Poisson ratios, not one as in isotropic materials, and it is convenient to label
these ν12 , called the major Poisson ratio (relating to the lateral strain, ε2, when a stress is applied in the
longitudinal (x1) direction) and ν21, the minor Poisson ratio (relating to the strain in the x1 direction when
a stress is applied in the x2 direction). The two are not the same, since it is obvious that ν12 must be much
larger than ν21. By means of arguments similar to those above for the determination of E1, it can be
shown that for a stress applied in the x1 direction only, the major Poisson ratio is given by:

ν12 = νf Vf + νm(1 - Vf).......................................................................................... [3.5]

ie. it also obeys the rule of mixtures. Further thought will show that the minor Poisson ratio must be
related to ν12 by the equation:

ν21 / E2 = ν12 / E1 .................................................................................................. [3.6]


Values of the fibre and matrix Poisson ratios rarely differ by
a great deal, so that neither matrix nor fibre characteristics
dominate these two elastic constants.

When a unidirectional fibre composite is loaded by an in-
plane (x1x2) shear force, it distorts to a parallelogram, as
shown in Figure 3.7. The shear stress in the fibre direction, τ
12, is matched by its complementary shear stress, τ21. The
simplest model assumes that the fibres and matrix carry the
same stress:

τ12 = G12 γ12 = Gf γf = Gm γm
Figure 3.7. Definition of a shear relative to
the x1 x2 Cartesian axes.
and the (apparent) in-plane shear modulus, G12, is given by:

1 V (1 − Vf ) ..................................................................................................................... [3.7]
= f +
G12 Gf Gm

By analogy with equation 3.2 for the transverse Young's modulus, it can be seen that the matrix shear
stiffness again dominates the composite shear modulus unless the fibre volume fraction is very large.

3.2 THE HALPIN-TSAI EQUATIONS
The models in the last section are simple models, although, if treated with care, they can give useful
approximations to the behaviour of many composites. There have been many more formal treatments,
however, based on more realistic models of the transverse fibre distribution, which yield results of
varying degrees of complexity. These more rigorous approaches may give predictions of elastic
properties that are closer to experimentally observed values than the simple models of section 3.1, but
they are seldom easy to use in practice. For design purposes it is more useful to have simple and rapid
computational procedures for estimating ply properties rather than more exact but intractable solutions.
Convenient interpolation procedures have been developed by Halpin and Tsai (1968; see also Halpin,
1992) who showed that many of the more rigorous mathematical models could be reduced to a group of
approximate relationships of the form:

E1 ≈ EfVf + Em(1 - Vf) ........................................................................................... [3.8]

ν12 ≈ νfVf + νm(1 - Vf)............................................................................................. [3.9]

(1 + ζηVf ) ............................................................................................... [3.10]
E2 = Em
(1 − ηVf )

(1 + ζηVf )
G12 = Gm ........................................................................................... [3.11]
(1 − ηVf )

The first two of these equations are the same rules of mixtures that we have already discussed. In the
second two equations, ζ is a factor, specific to a given material, that is determined by the shape and
distribution of the reinforcement (ie. whether they are fibres, plates, particles, etc. and what kind of
packing geometry), and by the geometry of loading. The parameter η is a function of the ratio of the
relevant fibre and matrix moduli (Ef/Em in equation 3.9 and Gf/Gm in equation 3.10) and of the
reinforcement factor ζ, thus:


⎛ Ef ⎞
⎜ − 1⎟
η= ⎝ m ⎠ ..................................................................................................... [3.12]
E
⎛ Ef ⎞
⎜ + ζ⎟
⎝ Em ⎠

The parameter ζ is the only unknown, and values must be obtained empirically for a given composite
material, although they are also sometimes derived by a circular argument involving comparison of
equations 3.9 and 3.10 with one of the exact numerical solutions mentioned previously. ζ may vary from
zero to infinity, and the Reuss and Voigt models are actually special cases of equation 3.9 for ζ = 0 and ζ
= ∞, respectively. A number of analyses have been carried out to compare the predictions of equations
3.9 and 3.10 with elasticity-theory calculations, often with a great degree of success, and it is frequently
quoted from the early work of Halpin and Tsai that for practical materials reasonable values of ζ are 1
for predictions of G12 and 2 for calculations of E22. It is dangerous, however, to accept these values
uncritically for any given composite. This can be illustrated by taking a specific example.

Sensitive measurements have been made of the elastic properties of a sample of unidirectional
carbon-fibre-reinforced plastic (CFRP) consisting of Toray T300 carbon fibres in Ciba-Geigy 914
epoxide resin with a fibre volume fraction of 0.56 (Pierron & Vautrin, 1994). Measurements were also
made of the properties of the unreinforced matrix resin. Table 3.1 gives the properties of the plain matrix
and fibres.

Table 3.1. Properties of T300 carbon fibres and 914 epoxy resin.

Property Fibres Matrix
Young's modulus, E, GPa 220 3.3
Shear modulus, G, GPa 25 1.2
Poisson ratio, ν 0.15 0.37

The experimental values are first compared in Table 3.2 with the predictions of the simple
micromechanical models of section 3.1. It can be seen that the fibre-dominated modulus, E1, is well
predicted by the simple rule of mixtures, while the other properties, E12, G12 and ν12, are less well
predicted. It is to some extent accidental that the prediction of E2 in Table 3.2 is as good as it is. The
very rigid carbon-carbon bonds in carbon fibres are well-aligned with the fibre length, but in the
transverse direction, the fibre is far less rigid, ie. it is highly anisotropic, with an anisotropy ratio (the
ratio of the axial to transverse moduli) of about 10. Inserting a more appropriate value of Ef (which is
difficult to measure) into equation 3.2 would actually result in a value of E2 for the composite that was
even lower than that predicted in Table 3.2.

Table 3.2. Predictions of composite properties by simple micromechanics models

Equation Relationship Predicted values Experimental values
(moduli in GPa) (moduli in GPa)

3.1 E1 = Ef Vf + Em(1 - Vf ) (RoM) 124.7 125.0

3.2 1 V (1 − Vf ) 7.4 9.1
= f +
E2 Ef Em

3.6 1 V (1 − Vf ) 2.6 5.0
= f +
G12 Gf Gm

3.4 ν12 = νfVf + νm(1 - Vf) (RoM) 0.25 0.34


Calculations of E2 and G12 by the Halpin-Tsai approximate model (equations 3.9 and 3.10) depend on
knowing ζ. By solving these equations for a range of ζ we can then see what values of ζ must be used in
order to obtain predicted moduli that agree with the experimental values. Taking the data from table 2,
but substituting a more appropriate transverse fibre stiffness of about 22GPa for the axial stiffness given
in the table we find that the appropriate values of ζ are about 2 for E2 and 2.5 for G12, for this particular
material. Use of the usually recommended values of 2 and 1, respectively, would thus be accurate for E2
but would result in a 25% underestimate for G12. This emphasises the importance of calibrating the
model against experimental data rather than against predictions of other models if the method is to be
valid for design purposes.

As a second example, we consider results obtained from ultrasonic pulse-velocity measurements on
model composites consisting of 3mm diameter glass rods arranged in regular hexagonal arrays in cast
blocks of epoxy resin. The velocity, v, of a compressional pulse travelling through a solid is given by:

v = √(C/ρ)

where C is an elastic constant and ρ is the density of the material. C is not exactly equal to the
conventional engineering Young modulus, E, although it has been shown that for ultrasonic rod waves C
and E are practically identical for a wide range of GRP materials (Harris and Phillips, 1972). Figure 3.8
shows some experimental values for the longitudinal and transverse moduli of the model composites,
determined by this method, as a function of
80
volume fraction of the glass rods (Shiel, E 1 (glass rods)
1995). The E1 values are well represented by E 2 (glass rods)
the rule-of-mixtures line joining the E (glass beads)
Elastic modulus, GPa

60 Tsai-Halpin model (ζ = 2)
experimental moduli of the plain resin and
the glass rods, but the E2 values lie well T-H model (fitted: ζ = 5.37)
above the line representing the Reuss
40
estimate (equation 3.2) and significantly
above the predictions of the Halpin-Tsai
equation (3.9) with the recommended value
20 Voigt
of ζ of 2. In order to obtain a good fit to the
Reuss
data points, a value of ζ of 5.37 is required.
It is interesting to note that the stiffnesses of 0
model ‘particulate’ composites consisting of 0 0.2 0.4 0.6 0.8 1.0
3mm diameter glass beads randomly Reinforcement volume fraction, V f
dispersed in the same resin (Birch, 1995) are
almost indistinguishable from the transverse Figure 3.8. Comparison of experimental data for the elastic
stiffnesses of the rod-reinforced resin moduli of model glass/resin composites with various theoretical
composites, as might be expected. Nielsen predictions. The upper and lower bounds (rule-of-mixtures and
(1978) has discussed at length the inverse RoM) correspond to equations 3.1 and 3.2. The Halpin-
applicability of the Halpin-Tsai equations to Tsai curve for ζ = 2 represents the commonly used function for
the transverse modulus, E2 (equation 3.9) but a value of ζ = 5.37
particulate composites. is necessary to obtain the best fit to the data.

3.3 THE COMPRESSION MODULUS
We might expect that the compression stiffness of a unidirectional composite would be similar to its
tensile stiffness, until we remember that the fine-scale structures of certain fibres — particularly those
derived from textiles — are known often to result in unpredictable behaviour of these fibres under shear
and bending conditions. We might take as an analogy the behaviour of a piece of rope: tensile forces
draw the fibres into tighter bundles, but any attempt to squeeze a length of rope axially will force the
fibres apart. Piggott and Harris (1980) illustrated the problem by measuring the compression moduli, Ec,
of polyester-based composites reinforced with glass, Kevlar-49 and two varieties of carbon fibres. Their
results, illustrated in Figure 3.9, show that the variation with Vf is linear up to only about 50 vol% of
reinforcement, and that beyond this the rate of increase of stiffness falls or even becomes negative,
depending on the fibre. The change in behaviour is evidently related to the diminishing ability of the


300 1:1
matrix to constrain the fibre bundles to HMS

dEc /dVf , GPa
deform axially as Vm falls. As the inset 200
figure plotting the initial slopes of these glass
HTS
curves against the fibre tensile modulus 100
shows, the rule of mixtures is not properly 200 Kevlar-49
obeyed except by the homogeneous and 0

Compression modulus, GPa
0 100 200 300 400
isotropic glass fibre. The slopes, dEc/dVf, of Fibre tensile modulus, GPa
the curves for the two carbon-fibre 150
composites fall slightly below the 1:1 rule-
of-mixtures values, while that for the aramid HMS carbon
fibre composite falls well below the RoM. 100
These results confirm what we know of the HTS carbon
structures of these reinforcing fibres. Some
recent elegant experiments by Young et al 50 E glass
(1996), who used laser Raman spectroscopy
to study the strains in single filaments Kevlar-49
embedded in transparent resin, clearly 0
0 0.2 0.4 0.6 0.8
showed the markedly non-linear
compression behaviour of carbon and Fibre volume fraction, Vf
aramid fibres, in keeping with the behaviour
Figure 3.9. Variation of the compression moduli of polyester-matrix
of the composites illustrated in Figure 3.9. composites with volume fraction of four species of reinforcing fibre.

3.4 THE ANISOTROPIC PROPERTIES OF A THIN LAMINA
Having considered the properties of a lamina parallel with and perpendicular to the fibres, it is clear
that the elastic properties of a piece of unidirectional composite must vary continuously with orientation
between these two extremes, and that in order to be able to design with such anisotropic materials it is
necessary to have a means of predicting this angular variation.

Most common engineering materials — metals, plastics or ceramics — are usually considered to be
isotropic, which is to say that they have the same properties in whichever direction the property is
measured. This is not strictly true: wood is an excellent example of an anisotropic material in which the
stiffness in the axial direction (parallel to the trunk centre) is greater than the stiffness in any other
direction. Cold-worked metals also often have different properties parallel with and perpendicular to the
direction of working, although designers usually treat them as isotropic. An isotropic material has four
characteristic elastic constants, the Young modulus, E, the shear or rigidity modulus, G, the bulk
modulus, K, and the Poisson ratio, ν. In isotropic bodies, however, only two of these are independent
properties. If, for example, we know E and ν, we can calculate the other two constants from the familiar
relationships (see, for example, Dugdale, 1968):

E E
G= K=
2(1 + ν ) 3 (1 − 2 ν )

In an anisotropic fibre composite material these simple relationships cannot be valid. But in order to
discuss anisotropy, we first need a frame of reference, and we choose for convenience to use an
orthogonal set of axes, (x1,x2,x3), in which the fibres are aligned in the x1 direction and transverse to the
x2 direction, as shown in Figure 3.10.

For this thin plate of composite, we might guess that there must be three values of Young's modulus,
E1, E2, and E3, three shear moduli, G12, G13, and G23, and six values of the Poisson ratio, ν12 and ν21, ν13
and ν31, and ν23 and ν32, (not all of these are shown in Figure 3.10, for the sake of clarity) instead of the
two independent constants that we need for an isotropic material. However, consideration of the
geometry of the laminate suggests that E3 is not likely to be very different from E2 and G13 is similar to
G12, although G23 will probably be different from the other two shear moduli. Similarly, ν12 will be close
in value to ν13 . By contrast with the shear moduli, however, for which symmetry implies that G12 = G21,


it is clear, from the definition of Poisson's ratio given
earlier, that ν12 » ν21, and so forth. The response of a
unidirectional lamina, such as that shown in Figure 3.10, to
applied stresses must therefore be more complex than that
of an isotropic material.

The general state of stress on a body can be written in
the form of a group of six independent components, σ1, σ2,
σ3, τ12, τ23, τ31, where the single suffices represent normal
components (tension or compression) and the double
suffices represent shear components (see Dugdale, 1968).
Likewise, the strains form a similar series of six
components, ε1, ε2, ε3, γ12, γ23, γ31. When a general stress
system consisting of the six stress components acts on an Figure 3.10. Definition of elastic constants for an
isotropic body, the familiar Hooke's-law relationship anisotropic thin composite lamina.
between stress and strain is usually written in the form of a
family of equations for the strain components:

1
ε1 =
E
[σ1 − ν(σ2 + σ3 )]

1
ε2 =
E
[σ2 − ν(σ3 + σ1 )]

1
ε3 =
E
[σ3 − ν(σ1 + σ2 )] ....................................... [3.13]

1 1 1
γ12 = τ12 γ 23 = τ23 γ 31 = τ31
G G G

These equations can also be written in matrix format:

⎡ ε1 ⎤ ⎡ 1 E −ν E −ν E ⎤ ⎡ σ1 ⎤
⎢ε ⎥ ⎢ −ν E 1 E −ν E ⎥ ⎢ ⎥
⎢ 2⎥ ⎢ ⎥ ⎢ σ2 ⎥
⎢ ε3 ⎥ ⎢ −ν E −ν E 1 E ⎥ ⎢ σ3 ⎥ ..................................... [3.14]
⎢ ⎥ = ⎢ ⎥ x ⎢ ⎥
⎢ γ 23 ⎥ ⎢ 1G ⎥ ⎢ τ23 ⎥
⎢ γ 31 ⎥ ⎢ 1G ⎥ ⎢ τ31 ⎥
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ γ12 ⎥
⎣ ⎦ ⎢
⎣ 1 G⎥ ⎢ τ12 ⎥
⎦ ⎣ ⎦

The order of the stress and strain components is arbitrary, and the sequences above are those
conventionally used. In this form, the individual terms in the matrix of constants relating strain and stress
are referred to as compliances and, as we have already said, there are only two independent elastic
constants in the matrix. The blanks in the matrix, which are really zeros, show that the normal stress
components do not cause shear deformations, and the shear stresses do not cause tensile or compressive
deformations. For a thin rod of isotropic material under a simple tension stress, σ1, equations 3.12 and
3.13 reduce to the familiar form of Hooke's law given by:

σ1 = E ε1

and for a thin plate subject to a plane-stress state only, since no stress components containing a suffix 3
are then of any importance (σ3 = τ13 = τ23 = 0), the Hooke's law relationship of equation 3.13 reduces to:


⎡ ε1 ⎤ ⎡ 1 E −ν E 0 ⎤ ⎡ σ1 ⎤
⎢ ε ⎥ = ⎢ −ν E 1 E 0 ⎥ ⎢ σ ⎥ ......................................................................... [3.15]
⎢ 2⎥ ⎢ ⎥ ⎢ 2⎥
⎢ γ12 ⎥ ⎢ 0
⎣ ⎦ ⎣ 0 1 G⎥
⎦ ⎢ τ12 ⎥
⎣ ⎦

By contrast, in an anisotropic solid like a fibre composite, each of the strain components has to be
related to all of the stress components, and it is possible that all of the elements in the matrix of
equations 3.13 could have real values — a total of 36 elastic constants. In reality, however, the symmetry
of a unidirectional material makes it possible to reduce the number of constants and since the most
demanding of current applications for composites call for the use of thin plates, the plane-stress state
allows further simplification.

Since we shall want to make some limited use of the formal tensor theory of anisotropic elastic
behaviour, we must first introduce some new definitions. First, a unidirectional lamina, as discussed with
reference to Figure 3.10, is regarded as transversely isotropic, that is to say that it has one major axis of
anisotropy — the fibre direction — and perpendicular to this the properties are independent of
orientation (ie. E2 = E3, etc.). By convention we take the fibre axis to be parallel to x1 and the lamina is
then referred to as orthotropic. Second, in tensor notation the definitions of shear strains are slightly
different from the engineering definitions that we have been using so far. Although the engineering shear
stress component identified as γ12 is the same as the tensor stress σ12, the tensor strain ε12 is only half the
engineering strain γ12. For the orthotropic lamina loaded in plane stress, as we have seen, no stress
component with a suffix 3 is present, and the Hooke's law tensor relationship, modifying equations 3.13,
is then written:

⎡ ε1 ⎤ ⎡ S11 S12 0 ⎤ ⎡ σ1 ⎤
⎢ ε ⎥ = ⎢S 0 ⎥ ⎢ σ 2 ⎥ ..................................................................... [3.16]
⎢ 2 ⎥ ⎢ 12 S22 ⎥⎢ ⎥
⎢ ε12 ⎥ ⎢ 0
⎣ ⎦ ⎣ 0 S66 ⎥ ⎢σ12 ⎥
⎦⎣ ⎦

or

εij = Srs σkl.......................................................................................................... [3.17]

where the Srs terms are referred to as the elastic compliances. A similar relationship gives stress as a
function of strain:

σij = Crs εkl ........................................................................................................... [3.18]

where the Crs terms are referred to as elastic constants or elastic coefficients, and it should be borne in
mind that in general C is not simply the reciprocal of S.

Note: although the suffices for stress and strain, as in ε1, ε12, etc. are directly related to the orthogonal
axis designations, x1, x2, those used for the Srs terms are not: they simply reflect the rows and columns in
the compliance tensor — hence, S66 simply corresponds to the last of the compliance components in
equations 3.13.

Again we see that the elasticity tensor contains some zero values: shear stresses do not cause normal
strains and normal stresses do not cause shear strains, and we describe this formally by saying that there
is no tension/shear coupling. The significance of this will become apparent later. The compliances, Srs,
in equations 3.15 are related to our normal engineering elastic constants for the composite lamina:

S11 = 1/E1 S22 = 1/E2 S66 = 1/2G12

S12 = –ν12/E1 = –ν21/E2


3.5 ORIENTATION-DEPENDENCE OF THE ELASTIC PROPERTIES OF A UNIDIRECTIONAL LAMINA
Composites are rarely used in the form of unidirectional laminates, since one of their great merits is
that the fibres can be arranged so as to give specific properties in any desired direction. Thus, in any
given structural laminate, predetermined proportions of the unidirectional plies will be arranged at some
specific angle, θ, to the stress direction. In order to calculate the properties of such a multi-ply laminate,
it is first necessary to know how the elastic response of a single unidirectional lamina, such as that which
we have been considering so far, will vary as the angle to the stress direction is changed. This is done by
transforming the x1x2 axes through some arbitrary angle, θ, a procedure that will familiar to anyone who
has studied the derivation of the Mohr’s circle construction. The derivation in given in Appendix 1, and
for our present purpose we need only the result of this analysis to illustrate the full anisotropic behaviour
of a single unidirectional lamina. If both the stresses and the strains are transformed through an angle θ
from the x1,x2 co-ordinates to an arbitrary pair of axes (x,y), we find that the appropriate form of
Hooke’s Law is now:

⎡ ε x ⎤ ⎡ S11 S12 S16 ⎤ ⎡ σ x ⎤
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ ε y ⎥ = ⎢S12 S22 S26 ⎥ x ⎢ σ y ⎥ ...................................................................... [3.19]
⎢ε xy ⎥ ⎢S16 S26 S66 ⎥ ⎢σ xy ⎥
⎣ ⎦ ⎣ ⎦ ⎣ ⎦

Except for the case when the (x,y) axes coincide with the composite orthotropic axes (the fibre
direction, x1, and the transverse direction, x2) the compliance matrix, S rs , now contains six elastic
constants instead of the four in the matrix of equations 3.15. Since there are now no zeros in the
compliance matrix each component of stress
will contribute to all of the components of
strain. A shear stress will therefore induce fibres at 30°
normal strains and normal stresses will to plate edge
induce shear strains. This shear-tension
coupling has important consequences for
designers since a thin plate of a
unidirectional composite will show complex
out-of-plane distortion if loaded off-axis.
This is illustrated in Figure 3.11 for a plate
of a unidirectional lamina with fibres at 30°
to the loading axis which is subjected to a
central point bending load. The out-of-plane
distortion is calculated by the commercial Figure 3.11. Out-of-plane distortion when an off-axis
laminate analysis programme LAP unidirectional laminate is loaded in bending.
(Anaglyph Ltd, UK).

The compliance terms in the matrix in equations 3.18 are barred to remind us that these are no longer
material elastic constants like the Srs in equations 3.15, but are functions of stress, as shown by the
presence of the matrix functions T and T-1 in their definitions (Appendix 1). It can be seen that if only a
tensile stress, σx, is applied, the strain measured in the direction of the stress, εx, is related to σx, by the
familiar Hooke's law, ie.

ε x = S11σ x

and the compliance S11 therefore represents the reciprocal Young modulus for this deformation:

1
S11 =
Ex


But, clearly, S11 , and therefore also Ex, are functions of the orientation, θ. The matrix algebra needed to
evaluate the barred compliances in equations 3.18 can be carried out painlessly with Mathcad, and it can
be shown that the full expression for S11 is:

S11 = S11 cos4 θ + S 22 sin4 θ + 2(S12 + S66 )cos2 θ sin2 θ ........................... [3.20]

where the compliances S11, S22, S12, and S66 are the four elastic constants for the orthotropic lamina
relative to the (1,2) axes, as defined in terms of the normal engineering constants at the end of section
3.3. If those definitions are substituted in equation 3.19, we have the orientation dependence of the
elastic modulus, which we shall call E(θ):

1 cos 4θ sin4θ ⎛ 1 2ν ⎞
= + +⎜ − 12 ⎟ cos θ sin θ
2 2 ................................................ [3.21]

E(θ) E1 E2 ⎝ G12 E11 ⎠

Similarly, the shear compliance component, S 66 , can be evaluated to obtain the orientation dependence
of the shear modulus, G(θ):

1 ⎛ 1 + 2ν12 1 1 ⎞ 1
= 4⎜ + − ⎟ cos θ sin θ +
2 2
(cos4θ + sin4θ ) ..................................... [3.22]
G(θ) ⎝ E1 E2 2G12 ⎠ G12

Equations 3.20 and 3.21 are
plotted to show the angular SiO /epoxy
2
dependence (and the high degree 150 15 T300/914 carbon/epoxy
of anisotropy) of two typical
E(θ), GPa

G(θ), GPa

unidirectional laminates in Figure
3.13. It can be seen that the 100 10
stiffness of the unidirectional
composite falls rapidly as the
loading direction shifts away from 50 5
the fibre axis, the effect being
more marked the higher the
stiffness of the reinforcing fibres. 0
0 30 60 90
0
0 30 60 90
By contrast, the shear stiffness of
the composite is less sensitive to Orientation, θ, degrees
the fibre characteristics, as would
be expected, being largely Figure 3.13. Dependence of Young's modulus, E(θ) and the shear modulus,
controlled by the matrix rigidity. G(θ), on the angle, θ, between the fibres and the stress axis for a carbon-
fibre composite and a silica/epoxy composite.
The shear modulus is a maximum,
60 however, in a direction at 45° to the fibre axis. The
curve predicted by equation 3.20 validity of equations 3.20 and 3.21 for predicting the off-
50 experimental data for silica/epoxy axis elastic properties of laminates is usually very good, as
can be judged from Figure 3.12 which compares the
40 predicted values of E(θ) for the silica/epoxy composite in
E(θ), GPa

Figure 3.13 with experimental data for this material
30 (Pabiot, 1971).
20 For most reinforced plastics composites, where the
modulus ratio Ef/Em is large and the axial lamina
10
properties are dominated by the fibre stiffness, the
influence of the values of in-plane shear modulus and
0
0 30 60 90 Poisson ratio on the form of the orientation dependence
Orientation, θ, degrees defined by equation 3.20 is not significant. But in certain

Figure 3.12. Comparison of predicted and experimental
results for the angular variation of Young's modulus of a
unidirectional silica/epoxy composite (data of Pabiot,
1971)


materials, like ceramic-matrix composites where
the stiffnesses of the fibres and matrix are often
similar, variations in the value of Poisson’s ratio,
caused by changes in processing conditions, for 140
example, may modify the form of E(θ)
substantially, as illustrated by the surface plot of 130
E(θ)
E(θ) for a range of values of ν12 for a SiC-
GPa 120
reinforced calcium aluminosilicate CMC
manufactured by Corning (as shown in Figure
110
3.14).
0.4
100
ν
3.6 MULTI-PLY LAMINATES 0 0.3 tio
,
30 ra
n
Unidirectional composites are rarely used in Orientati
on, θ, de
60
90 0.2 iss
o
grees Po
practice because of the high level of anisotropy
demonstrated by Figure 3.13. In practice it is more
sensible to make use of the versatility of Figure 3.14. Effect of the value of Poisson's ratio on the
composite materials by arranging the fibres in orientation dependence of Young's modulus for SiC/CAS
different directions to suit the design requirements. ceramic-matrix composites.
There are various ways in which this is done, depending on the level of sophistication of the required
product. It is useful at this stage, however, to refer to the most common procedure for producing high-
performance laminates, which is to stack groups of single-ply laminae cut at various angles from the
continuous sheet of 'prepreg' — a continuous layer of fibres pre-impregnated with resin which is then
dried (but not cured) — and then hot-press them between the heated platens, flat or shaped, of a press.
The stacking sequence of such a laminate is described in short-hand form by formulae of the following
kind:

(0)12 ─ a unidirectional composite with 12 plies 200
+θ
(0,90)2s ─ a cross-plied laminate with four pairs of
0/90 laminae arranged symmetrically (indicated
150 -34%
cross-ply
-θ
E, GPa

by the ‘s’) (0,90,0,90,90,0,90,0)
100
ud
[(±45,02,902)2]s ─ a quasi-isotropic laminate with cross-ply θ
24 plies arranged symmetrically 50 ud
(+45,-45,0,0,90,90,+45,-45,0,0,90,90,90,90,0,0,-
45,+45,90,90,0,0,-45,+45), the external ±45 plies 0
0 30 60 90
providing protection against impact damage.
60
The choice of laminating sequence may vary
widely, depending upon the particular design
G, GPa

40 cross-ply
requirements, but an indication of the potential for
modifying the lay-up to achieve a desired result
can be seen by comparing the orientation 20
+225%
dependence of the stiffness of a unidirectional ud
laminate with the stiffness of a series of
0
composites of lay-up ±θ, where in each case θ 0 30 60 90
varies from 0 to π/2, as shown in Figure 3.15. Orientation, θ, degrees
Cross-laminating is a compromise solution which
results in improved torsional and transverse
Figure 3.15. The effect of cross-laminating on Young's
rigidity at the expense of some loss in longitudinal modulus and the shear modulus of a T300/5208 carbon-
stiffness. It can be seen, for example, that by using fibre/epoxy composite.
a ±20° laminate instead of a unidirectional
composite, for only a 34% loss in axial stiffness, the torsional rigidity may be doubled. The extent of the
axial stiffness reduction caused by cross-laminating is greater the stiffer the reinforcing filaments.


Nielsen and Chen (1968) demonstrated the significance of this by showing the effect of randomising the
fibre distribution in the plane of the laminate. The modulus, E(θ), given by equation 3.20, is randomised
over all θ:

π/2

∫ E(θ)dθ 1.0
( E( = 0
π/2
…………...................[3.23]

∫ dθ
0
0.8

<E>/E1
0.6
The solution to this equation, also obtained in
GRP (Ef /Em = 22)
Mathcad, for a pair of composites with 0.4
different modulus ratios, Ef/Em is shown in
Figure 3.16. For most practical purposes (Hull, 0.2
1981) the solution to equation 3.22 can be CFRP (Ef /Em = 70)
approximated reasonably closely by: 0
0 0.2 0.4 0.6 0.8
〈E〉 = E1 + E 2 ...........................[3.24]
3
8
5
8
Fibre volume fraction, Vf

where E1 and E2 are the known or calculated Figure 3.16. Effect of randomising the fibre orientations on the
longitudinal and transverse moduli for a stiffnesses of CFRP and GRP laminates.
unidirectional lamina. An analogous
approximate expression can be obtained for the shear modulus of a random continuous-fibre composite:

〈G〉 = 8 E1 + 4 E2 ...........................[3.25]
1 1

Multi-ply laminates usually, but not always, consist of regular arrays of plies of the same material
and of the same thickness. When a laminate contains equal numbers of plies in each orientation, it is said
to be balanced. A (0,90)3 laminate is thus a balanced laminate, but such a laminate would have 0° and
90° plies on opposite faces and may behave in unexpected ways if loaded in other than the 0° and 90°
directions. Laminates that are arranged to have the plies on either side of a mid-plane matching in both
material and orientation are described as symmetric: such laminates have the advantage that they remain
flat after curing and during deformation. Symmetric laminates are therefore preferred for most purposes,
but careful tailoring of a laminate lay-up can be used deliberately to produce a material that will change
shape in some desired manner under stress.

For design purposes, it is necessary to have accurate methods of calculating the elastic properties of a
composite with a given lay-up as part of the process of tailoring a material to suit a given set of
requirements. For a simple 0/90 lay-up it is a simple matter to treat the laminate as a parallel
combination of materials with two different elastic moduli and apply the mixture rule as given by
equation 3.1, provided accurate values of the two moduli E1 and E2 are known. Thus, from dynamic
measurements on an 11-ply unidirectional GRP composite of Vf = 0.64, values of E1 and E2 of 46.1GPa
and 19.6GPa were measured (Harris et al, 1978: see Table 3.3). For an 11-ply 0/90 laminate from the
same manufacturer with the same raw material specifications, we may rewrite the rule of mixtures:

E0 = 6
11
E1 + 11 E2
5

E90 = 5
11
E + 11 E2
6

to give predicted values of E0 and E90 in the two orthogonal directions. As Table 3.3 shows, the estimates
would be adequate for most practical purposes.


Table 3.3. Comparison of some predicted and experimental elastic modulus values for typical GRP laminates.

Material Vf Density Orientation Dynamic Predicted Model
modulus, modulus,
3 -3
10 kg.m GPa GPa
pultruded rod 0.7 2.09 0 54.4 51.3 RoM
0
90 24.7 9.1 inverse RoM
90 24.7 24.6 Halpin-Tsai (ζ = 4.6)

ud 11-ply 0.6 2.04 0 46.1 47.2 RoM
4
Laminate 90 19.6 7.8 inverse RoM
(Permali, UK) 90 19.6 19.6 Halpin-Tsai (ζ = 4)
45 20.4 12.6 Krenchel

0/90 11-ply 0.6 2.03 0 (6 plies) 35.3 34.0 RoM
3
Laminate 90 (5 plies) 30.5 31.6 RoM
(Permali, UK) 45 21.6 12.6 Krenchel

0/90 woven 0.4 1.85 0 26.5
7
35-ply 90 27.5
laminate 45 16.7
(Permali, UK)

A preliminary estimate of the elastic response of a general multi-ply laminate can often be obtained
by a netting analysis (Cox, 1952; Krenchel, 1964) which sums the contributions from each group of
fibres lying at a specific angle, θ, to the applied stress. If we assume that the fibres are continuous and
that there are no elastic Poisson constraints (νf = νm), each group of fibres (a proportion an of the total
fibre content) can be considered to have a reinforcing efficiency ancos4θ , and an overall composite
efficiency factor, ηθ, can be defined as:

ηθ = Σan cos4 θ.......................................[3.26]

The approximate composite modulus can then be
obtained from a modified rule of mixtures as:

Ec = ηθEf Vf + Em(1 - Vf) ………...........................[3.27]

The fourth-power geometrical term in the efficiency
factor clearly indicates its relationship with equation
3.19. Shear and transverse behaviour is ignored, however,
and the limitation of the model is therefore that it allows
no stiffness contribution from transverse fibres and
ignores interlayer constraints. For composites containing
high volume fractions of high-stiffness fibres, none of
which are at 90° to the stress axis, it can give reasonable
Figure 3.17. Values of the Krenchel efficiency
estimates of the laminate stiffness, but it is not acceptable factor for various fibre groupings.
for detailed design purposes. For the GRP materials
referred to in Table 3.3, for example, the predictions of the stiffnesses E45 are poor. Values of the
efficiency factor, ηθ, for a range of fibre distributions are shown in Figure 3.17. For a 3-D random array
ηθ is about 1/6. It can be seen from Figure 3.16 that although the Krenchel factor is approximately in
agreement with the more general randomisation model for GRP, it gives a poor prediction for CFRP.


In real laminates the individual plies which
are stacked together to produce the multi-ply
structure each have their own characteristic
anisotropy. If a ply is at an angle to the load
axis the simplified form of Hooke's law for the
orthotropic case (equation 3.15) is no longer
applicable and the more complex form of
equation 3.18 must be used. As we have seen,
the zeros in equation 3.15 are replaced by non-
zero values and there is then tension/shear
coupling — ie. the application of a tensile
force will induce shear strains and the
application of shear forces will induce normal Figure 3.18. Out-of-plane distortion in an unsymmetric
laminate consisting of a single pair of off-axis (±θ )plies.
strains. This coupling can lead to unexpected The anisotropic shape changes induced in each separate
distortions of composites under load and ply can only be accommodated in the laminate by the
unless these are carefully balanced in distortion shown, accompanied by the development of
constructing the laminate interlaminar shear interlaminar shear stresses.
forces may be generated that are high enough
to cause delamination (separation of the individual plies). A schematic illustration of the effect of
shear/tension coupling in causing out-of-plane distortion in a simple ±θ laminate is shown in Figure 3.18
(Soden and Eckold, 1983).

A full treatment of the elastic behaviour of thin composite laminates is beyond the scope of this book
(readers are referred to the standard texts listed in the bibliography to Chapter 1) but an indication of the
tk approach used in the classical theory may be given.
Assuming symmetry about the centre line and in-plane
t/2 th loads only, as shown in Figure 3.19, the elastic
k ply response of the kth ply will, in the general case, be
t/2 load described by equations A6 and A6a of Appendix 1, ie.

[ ε ] k = ⎣S ⎦ k [ σ ] k
⎡ ⎤
Figure 3.19. In-plane loading of a laminate of thickness
t, and the location of the kth ply relative to the mid-
plane.
[ σ ] k = ⎣Q ⎦ k [ ε ] k
⎡ ⎤

in simplified form. The second of these relates each of the stress components to all of the strain
components through a matrix of elastic coefficients:

⎡ σx ⎤ ⎡ εx ⎤
⎢ ⎥ ⎢ ⎥ ............................................................................................. . [3.28]
⎢ σ y ⎥ = ⎡ A ij ⎤
⎣ ⎦ ⎢ εy ⎥
⎢σ xy ⎥ ⎢ε xy ⎥
⎣ ⎦ ⎣ ⎦

where [Aij] is a stiffness matrix whose elements (the Q coefficients in equation A6a) are functions of the
unidirectional lamina elastic constants E1, E2, G12, ν12, and the transformation angle θ. If the total
composite is composed of a large number of evenly stacked layers, its properties can be determined by
summing the stiffness matrices, [Aij], of each layer proportional to the number of plies lying in each
direction:

1 k =n
⎡ A ij ⎤ = ∑ ( A ij )k .t k .......................................................................... [3.29]
⎣ ⎦ composite t
k =1

where n is the number of plies, tk is the thickness of the kth ply, and t is the total laminate thickness.


On the other hand, if the laminate contains relatively few plies, the effect of stacking sequence
becomes important. Unsymmetric arrangements of plies, for example, may cause out-of-plane
deformations (twisting or bending) such as that illustrated in Figure 3.18. These effects are mostly
undesirable, but on occasion they may be used to advantage, as in the aeroelastic tailoring of aircraft
structures, designing the lay-up so that it changes shape (and therefore its aerodynamic performance) in a
predictable way in response to aerodynamic forces. In the full classical theory of thin laminates,
therefore, it is necessary to obtain the resultant forces and moments acting on the laminate by integrating
the stresses in each ply through the laminate thickness. In addition to the A matrix of equation 3.27
which relates in-plane forces to in-plane deformations, the full constitutional equation for laminate
deformation then includes relationships between in-plane forces, N, moments, M, in-plane deformations,
ε, and curvatures, x, which are conveniently represented in the form of a partitioned matrix:

⎡N ⎤ ⎡ A B ⎤ ⎡ ε ⎤
⎢M⎥ = ⎢ B D ⎥ ⎢ x ⎥ ............................................................................................ [3.30]
⎣ ⎦ ⎣ ⎦⎣ ⎦

where A is the extensional stiffness matrix that we have already met, D represents the bending
stiffnesses, and B represents the stiffness matrix for coupling between these two.

The calculations involved in analysing a complex laminate fully are tedious but straightforward, and
are ideally carried out on one of the many pieces 60
of computer software currently available. The SiO2/epoxy, ud
LAP programme by Anaglyph, UK, previously 50 glass/epoxy, 0/90, non-woven
mentioned is one such programme which glass/epoxy, 0/90, woven
CSM/polyester, random
operates under Microsoft Windows, but there 40
E(θ), GPa

are many others, of different degrees of
sophistication, offering a variety of calculation 30
facilities and graphics.
20
As an illustration of the general effects of
10
structure and orientation on the elastic stiffness
of glass-fibre laminates, Figure 3.20 shows the
0
variation with orientation of Young's modulus 0 30 60 90
for three types of laminate, the unidirectional Orientation, θ, degrees
silica/epoxy material of Figure 3.12, the 0/90
non-woven and woven glass/epoxy laminates Figure 3.20. Orientation-dependence of Young's modulus for
referred to in Table 3.3, and an isotropic GRP four types of glass-fibre-reinforced plastic. The curves fitted
laminate made with randomly arranged to the upper three data sets correspond to equation 3.20.
chopped-strand mat (CSM) reinforcement.

3.7 SHORT-FIBRE COMPOSITES
High-performance composites are generally made from continuous fibres, but there are many
applications for which the requirements are less demanding, or for which the appropriate manufacturing
route cannot handle long fibres: it is then natural to consider using short fibres. Some reinforcing fibres
are also available only in the form of short filaments.

The ends of a fully-embedded short fibre cannot be fully loaded by shear at the fibre/matrix interface.
This can be seen by embedding different lengths of a fibre in a resin block and attempting to pull the free
end out. The pull-out force is resisted by interfacial friction or bonding, and if the embedded length, x, is
greater than some critical length, lc the tensile stress in the fibre will break the free fibre before the
embedded end can be pulled out. This critical length can be determined by considering the force required
to pull an embedded length, x, of fibre of diameter, d, from a block of matrix (Figure 3.21). A force P
applied to the fibre develops a tensile stress σ = 4P/πd2 in the unembedded length and this is balanced by
a mean shear force at the interface equal to τπdx. Thus, the ratio of embedded length to fibre diameter,


x/d = σ/4τ. If the fibre can just be withdrawn from
the matrix by a tensile load only marginally
smaller than the fibre breaking load, x must be
equal to half the critical length, lc/2, since only
one fibre end is concerned. Thus, the critical
aspect ratio is:
Figure 3.21. Forces on a fibre being pulled from a block of
l c σf …………….....................[3.31] matrix material.
=
d 2τi

where σf is the breaking stress of the fibre and τi is the interfacial shear strength. This is a very simplistic
model which assumes that all deformations are entirely elastic and that the shear stress is independent of
embedded length, but it gives useful insights into composite behaviour.

We imagine an aligned array of short fibres of
identical length embedded in a composite which is
loaded parallel with the fibres, as shown in Figure
3.22. Since the load on the composite must be
transferred into the fibre by shear at the interface, it
can be seen that no transfer can occur at the Figure 3.22. Idealized short-fibre composite containing
extremities of the fibre, and that the tensile stress in discontinuous but aligned fibres.
the fibre will build up from zero at the ends. At the
same time, the interface shear stress will be high at the fibre ends, and will fall to zero when the full
extent of load transfer has been achieved, as
shown in Figure 3.24.

The calculations on which this figure is
based are the result of what is referred to as
a ‘shear-lag’ analysis (Cox, 1952) which
involves determining the conditions for
equilibrium of the shear and tensile forces in
a simple twin-cylinder model such as that
Figure 3.23. Concentric cylinder model used to determine the
shown in Figure 3.23. For the results shown stress distributions, shown in Figure 3.24
in Figure 3.24, Cox’s equations for the fibre
tensile stress and the interfacial shear stress have been used to simulate the situation for a 1mm length of
carbon fibre (Young’s modulus 300GPa) embedded in an epoxy resin matrix. Other solutions for the
stress distributions also exist, but the
general validity of the functions which give
lc/2 lc/2
3 rise to Figure 3.24 have been demonstrated
fibre tensile stress, σ
both by photoelastic experiments and, in
2 recent years, by means of laser Raman
σ, GPa

0.2 spectroscopy which is able directly to
1 determine, at the molecular scale, the strain
interfacial shear stress, τ 0.1
at any point along a single fibre embedded
τ, GPa

0 0
in a resin block under external load (see, for
0 200 400
x, μm
600 800 1000 example, Galiotis, 1991).
-0.1
Distance, x, from fibre end
In the central portion of a long fibre, the
-0.2
Cox stress functions vary only very slowly
fibre length, l towards the fibre centre, so that we can
usually make the assumption that there is a
Figure 3.24. Variation of the tensile stress, σ, in the fibre and the plateau, as illustrated in Figure 3.24. The
shear stress, τ, at the interface along the length of a short fibre ends of a short fibre thus constitute a
embedded in a matrix. The calculations, based on the shear-lag notional 'ineffective' length which reduces
model of Cox (1952), relate to a 1000μm length of carbon fibre in the fibre reinforcement efficiency. The
an epoxy resin matrix.


contribution of these short, aligned fibres to the stiffness of the composite will therefore be less than the
rule-of-mixtures contribution, Ef Vf, by an amount that will depend on the efficiency of the stress
transfer, which is determined by the relative stiffnesses of the fibre and the matrix.

Thus, the mixture-rule equation needs to be modified to allow for this fibre-end effect by introducing
another efficiency factor, usually designated ηl, so that:

Ec = ηlEf Vf + Em(1 - Vf).................................................................................... [3.32]

The Cox shear-lag model leads to a value for the efficiency factor, ηl,

tanh ⎡ β2l ⎤
⎣ ⎦
ηl = 1 − βl
........................................................................................... [3.33]
2

where β, which determines the rate at which the tensile stress builds up at the fibre ends, is a function of
the fibre and matrix stiffnesses and the composite geometry, viz.

2Gm Ef ............................................................................................... [3.34]
β= 2
ro loge (R / ro )

In a regular fibre arrangement, the fibre radius, ro, and the matrix cylinder radius, R, are related to the
fibre volume fraction. For example, if we assume a regular array, as in Figure 3.25, it can be seen that
the triangle defines a unit cell which, if duplicated many times, would result in hexagonal symmetry.
The area of the triangle is ½ (2R)2 sin60. This area contains three fibre sectors, each of area (πro2)/6, and
the total area of fibre is ½ πro2. From the definition of fibre volume fraction (and assuming Af ≈ Vf), it
can be seen that:

½ πro2 πro2 ....................................[3.35]
Vf = 2
=
½( 2 R ) sin 60 2 3R 2

and the value of R in equation 3.33 can therefore be written in terms
of the fibre distribution. If equations 3.30 to 3.34 are now used to
determine values of the stiffnesses of typical GRP and CFRP
composites reinforced with short fibres, the results can be plotted as
functions of the fibre length or the fibre aspect ratio, l/d, as in Figure
3.26, to show these stiffnesses as percentages of the moduli of
unidirectional laminates of the same composites but reinforced with
continuous fibres, as given by the normal rule of mixtures, equation
3.1. We see that ηl → unity as l → ∞, and the efficiency of stress
Figure 3.25. Fibre packing in a simple
transfer is marginally higher in the GRP than in the CFRP. For hexagonal arrangement.
aspect ratios of the order of 100 (ie. for fibre lengths of the order of
only 1mm) some 95% of the full potential stiffening effect of fine reinforcing fibres like glass and
carbon can be achieved in composites reinforced with chopped, rather than continuous, fibres.

In deriving Figure 3.26, we have ignored certain problems in applying the Cox model. It has been
pointed out (Galiotis and Paipetis, 1998) that in reality the matrix shear stiffness in equation 3.33 is
affected by interface chemistry, fibre surface treatments, elastic constraints due to fibre volume fraction,
etc., and R is much more difficult to define for a real composite than is implied by equation 3.34. An
analytic derivation of β is therefore questionable. They propose instead that β should be treated as an
empirical index of the efficiency of stress transfer which can be considered as an ‘inverse length’. For
values of βl ≥ 10 the Cox hyperbolic equation for the fibre tensile stress, σ(x), which is plotted
in Figure 3.24 can be simplified to give:


σ(x) = σ∞ (1 − exp( −β x) ................................................................................ [3.36]

Direct estimates of β can be made by the laser-Raman spectroscopy technique mentioned earlier.

If the short-fibre composite consists of layers of material with the fibres arranged at different angles
to the stress direction, equation 3.31 can be further modified by the inclusion of the appropriate value of
the Krenchel factor, ηθ, thus:

Ec = ηlηθEf Vf + Em(1 - Vf )............................................................................... [3.37]

to give an estimate of the composite stiffness.

The short-fibre composite model described above is clearly based on an idealised structure consisting
of short fibres, all of identical length, arranged
in a unidirectional array. Although there are
manufacturing processes which can achieve such 100
95%
structures (see chapter 1, section 1.3.6), they are
rather specialised and composites reinforced 80
with chopped or short fibres are, for the most Eshort / E long %
part, likely to be of the type which are produced 60
by the injection moulding of chopped-fibre
reinforced thermoplastics or by the hot-press- 40 GRP
moulding of dough moulding compounds (eg.
glass-fibre-filled polyester resins). During such
20
moulding processes, there may well be some CFRP
fibre alignment as a result of the high shear
0
forces acting in the injection die or in the mould, 1 10 100 1000
but the fibres will be imperfectly oriented, so
that the mean angle between the fibres and the Fibre aspect ratio, l/d
material flow direction will be greater than zero Figure 3.26. Variation with fibre aspect ratio, l/d, of the
and the fibre lengths will be distributed about stiffnesses of aligned short-fibre composites of GRP and
some mean value as a result of damage leading CFRP for a fibre volume fraction of 0.5. The results are
to fibre breakage during processing. Thus, in given as percentages of the normal continuous-fibre
composite stiffnesses.
order to estimate the elastic modulus of a short-
fibre composite, the spreads of both the fibre orientations and the fibre lengths have to be taken into
account.

3.8 HYBRID COMPOSITES
Fibres of different types are sometimes mixed in a single matrix to produce hybrid composites. The
mixing may occur at several structural levels, as illustrated in Figure 3.27, from 'skin-core' structures,
through ply-by-ply laminates to mixed-tow laminates.
Random, intimately mixed fibre hybrids are rarely
considered. The purpose of hybridisation is to extend the
concept of 'tailoring' the material’s properties to suit
particular design requirements, and to offset the
disadvantages of one component by the addition of
another. For example, GRP are cheaper and tougher than
CFRP, but they are significantly less rigid, as we have Figure 3.27. Schematic illustration of three
possible types of hybrid laminate structure: a)
seen. In principle, then, the addition of a proportion of skin-core b) ply-by-ply c) tow-by-tow
glass fibres to a CFRP composite offers the possibility of
making the material cheaper while improving its toughness, ideally without paying too great a penalty in
terms of loss of stiffness. Reinforcing cloths and braids are available which are woven as hybrid
reinforcements with two or more species of fibre appropriately placed.


Skin-core structures are common in structural engineering and behave, in a sense, like I-beams, the
core playing the same rôle as the web of the I-beam and therefore being required to carry only shear
forces, while the skin layers carry the major tensile and compression forces. In composite structures of
this kind, the core may be a structural polymer foam or a paper or metal honeycomb. The bending
stiffness, EI, of a skin-core beam can be found in the usual way:

n
(EI)beam = ∑ EIi ............................................................................................... [3.38]
i
i=1

provided the core is an isotropic material and the skin materials are symmetric and balanced laminates
(Stavsky & Hoff, 1969; Marshall, 1982). For example, for a skin-core sandwich beam of width b
consisting of thin skins of a fibre composite of thickness t and a foamed polymer core of thickness c, the
second moment of area of the core about the neutral plane is:

bc 3
Ic =
12

and that of each of the skins about the neutral plane is:

bt 3
Is = + btz2
12

where z is the distance from the neutral plane to the mid-plane of the skin. The bending stiffness of the
sandwich beam is thus:

(EI)beam = 2EsIs + EcIc

When the skin thickness is small relative to that of the core, t2 << z2, and since Ec << Es the expression
can be approximated to:

(EI)beam ≈ 2Esbtz2

which is sufficiently accurate for most purposes. Significant benefits are obtained by separating two
skins of a fibre composite with a low
stiffness core, as illustrated by the graph in 10
5

Figure 3.28 which shows the variation of
beam bending stiffness, given by equation 10
4

3.37, as a function of the ratio of the
EIbeam /EIskin

thicknesses of the core and the skin, c/t. 3
10
The illustration is for a skin thickness, t, of
1mm, a beam width, b of 10cm, and a
2
polyurethane foam core of stiffness 10
20MPa. The beam bending stiffness,
(EI)beam, is normalised with respect to the 10
1

bending stiffness, (EI)s, of the skin
material, and the graph is independent of 10
0

the actual skin stiffness. For a 1mm skin 0 10 20 30 40 50
and a 25mm core, the beam is over 4000
Ratio of thicknesses, core to skin, c/t
times as stiff as the skin materials. Since
the overall density of the beam falls as the
Figure 3.28. The bending stiffness of a sandwich beam of width 10cm,
ratio c/t increases, there is an optimum normalised with respect to the bending stiffness of the skin panels, as
skin thickness for a beam of given weight, a function of the ratio of the thicknesses of the core and skin. The
core is a polyurethane foam and the skin panels are 1mm thick. The
normalised bending stiffness of the beam is independent of the
stiffness of the skin material.


as discussed by Farebrother and Raymond (1977).

For simple hybrids consisting of unidirectional plies or mixed, aligned fibre tows, the tensile moduli
obey the familiar rule of mixtures. With appropriate change of symbols, equation 3.1 becomes:

Ehybrid = EAVA + EBVB + ECVC + ................................................................................................... [3.39]

where EA , EB , EC ...... are the moduli of the individual component unidirectional composites (eg. CFRP
and GRP in a carbon/glass/epoxy hybrid) and VA , VB , VC are the volume fractions of the components
A, B, C, etc. As in the case of a simple composite, VA + VB + VC + = 1. An illustration of the validity of
the mixtures rule for two varieties of unidirectional CFRP/GRP hybrid composites is given in Figure
3.29. Two quite different composite structures are represented, one being an early pultruded polyester
rod material with intimately mixed E-glass and HM carbon fibres (Harris and Bunsell, 1975), and the
other a laminate with alternate plies of E-glass/epoxy and XAS-carbon/epoxy (Dickson et al, 1989).
When a mixed-ply laminate is tested in bending, the resulting flexural stiffness will clearly be affected
by the distributions of the different species of fibre. This was studied by Wagner et al (1982a and b) who
found apparent deviations from the mixtures rule in CFRP/GRP hybrid laminates. These deviations
were, however, predictable by their theoretical analysis based on the Hoff composite-beam model of
equation 3.37. Chamis and Lark (1977) have shown that fair agreement can be obtained between
measured values of the flexural modulus of hybrid beams and those calculated by the formula:

1 k
Ehybrid = ∑ ⎡zi3+1 − zi3 ⎤ Ei ......................................................................................... [3.40]
3t i=1 ⎣ ⎦

where t is the laminate thickness, zi is the distance to the bottom of the ith ply and zi+1 is the distance to
the top of the ith ply.

The tensile and flexural stiffnesses of most laminate hybrids are predicted satisfactorily by the methods
of classical thin-laminate theory, and most computer packages allow the construction of laminates from
plies of any given material for which the properties of the basic unidirectional ply are known. A detailed
discussion of models of hybrid laminates is given by Chou (1992).

3.9 RESIDUAL STRAINS
Residual thermal strains occur in newly
manufactured composites as a result of 5
differential thermal contraction during cooling ud composites (HM carbon)
Normalised Young's modulus,

from the moulding, post-curing, or final heat- 4 [(±45,02)2]S laminates (XAS carbon)
treatment temperature (eg. the ‘ceramming’
temperature for a CMC with a glass-ceramic
Ehybrid / EGRP

3
matrix). The state of thermal stress will also
change as a consequence of any temperature 2
changes during service. Since the axial
thermal expansion coefficient of many modern
1
reinforcing fibres is small (slightly negative in
the case of carbon) and that of a resin matrix
0
relatively large, the fibres in a composite 0 0.2 0.4 0.6 0.8 1.0
laminate will be put into compression and the VCFRP/(VGRP+ VCFRP)
matrix into tension during cooling from a
typical curing temperature of 140°C. A rough
estimate of the level of the axial stress in a Figure 3.29. Variation of the Young's moduli of two families of
unidirectional composite can be obtained by unidirectional carbon/E-glass hybrid composites with
composition. The rule of mixtures fits both sets of data
application of simple compound beam reasonably well.
mechanics. The difference in the thermal
strains in the two components, Δε, is given by:


(εm – εf) = (αm – αf)ΔT

or
σm σf
− = (αm − α f )ΔT
Em Ef

where αm and αf are the coefficients of thermal expansion (CTEs) of the matrix and fibres, respectively,
and ΔT is the temperature change. With the condition for equilibrium that no external load, P, is applied,
the forces in the fibres and matrix will balance:

P = Pf + Pm = σf Af + σm Am = 0

where Af and Am are the cross-sectional areas of the two components, and equal to the respective volume
fractions. Solving these two equations for the fibre and matrix residual stresses, σfr and σmr, we have:

Ef Em Vm (α f − αm )ΔT
σfr =
Ec ...................................................................................... [3.41]
Ef Em Vf (αm − α f )ΔT
σmr =
Ec

Values of these residual stresses for a group of polymer- and ceramic-matrix composites are given in
Table 3.4. It can be seen that in each case the fibre is in compression and the matrix in tension, as
expected. The fibre stresses are low compared with their tensile strengths, although it has been shown
that in very low-Vf resin-based composites fibres may be buckled or broken as a result of the residual
strains (Ferran and Harris, 1971). The matrix strains in the resin-based composites are also low, but it
can be seen that in the CMCs the residual matrix tensile stresses are of the order of 100MPa, which is
close to the expected matrix tensile strengths of these materials. As a consequence, matrix microcracking
occurs very readily in CMCs under very small applied tensile loads (Harris et al, 1993), a matter of some
concern for users of these materials.

Table 3.4. Residual thermal stresses in some typical unidirectional composites.

Matrix Fibre Vf Temperature Fibre residual Matrix residual
range, ΔT, K stress, MPa stress, MPa
epoxy (high T cure) T300 carbon 0.65 120 -19 36
epoxy (low T cure) E glass 0.65 100 -15 28
epoxy (low T cure) Kevlar-49 0.65 100 -16 30
borosilicate glass T300 carbon 0.50 520 -93 93
CAS glass-ceramic Nicalon SiC 0.40 1000 -186 124

In addition to the axial residual stresses, differential thermal contraction also results in residual radial
stresses which may make an important contribution to the mechanical behaviour of the composite since
they provide one of the mechanisms whereby the matrix grips the fibre and therefore allows the transfer
of shear stress from matrix to fibre, as we saw in section 3.6. If there is no mechanical gripping of this
kind, the composite must rely exclusively on a chemical fibre/matrix bond. A rough estimate for the
radial residual stress, Emαm ΔT, when a resin matrix of Em = 3GPa and CTE 100x10-6 K-1 contracts onto a
fibre of infinite rigidity as it cools through a temperature range, ΔT, of 100°C, is about 30MPa. If the
environment surrounding the fibre were to be a homogeneous one having the transverse properties of a
composite instead of pure matrix, the effective stiffness of the matrix would be higher and its CTE
would be lower. The net effect on the radial residual pressure would therefore depend on relative
changes in E2 and α2 with increasing Vf. Harris (1978) used a simple micro-mechanical model to show
that the radial pressure was of the order of 25MPa in glass/resin systems, but a reworking of his
calculation to incorporate more realistic models for the effects of Vf on the transverse stiffness (Halpin-
Tsai) and CTE (Schapery, 1968, see below) suggests values as high as 40MPa for practical levels of Vf.


These values are considerably higher than those
commonly quoted from other models, but are in 1.0
accord with experiments involving the pulling
0.8 ΔT = 100K
out of glass fibres from blocks of resin (Harris et

Residual thermal strain, %
90° ply
al, 1975). 0.6
0° plies
One of the difficulties in carrying out such 0.4
calculations is that of knowing precisely what
0.2
value of ΔT to use. Although it is common
practice to select the glass-transition temperature, tension
0
Tg, or the post-cure temperature for a thermoset- compression
matrix composite, it is known that some stress -0.2
relaxation can occur in resins like epoxies some
0 0.2 0.4 0.6 0.8 1.0
way below these temperatures. The actual
residual stress levels in manufactured composites
are therefore likely to be considerably lower than
those predicted from simple models. Figure 3.31. Residual thermal strains in the 0° and 90°
plies in a 0/90/0 glass/epoxy laminate.
The thermal expansion coefficients in
composite laminae will, like other elastic properties, be anisotropic. Schapery (1968) derived the
following expressions for the CTEs parallel with and transverse to the fibres in a unidirectional lamina:

E f α f Vf + Em αm Vm ...................................................................................... [3.42]
α1 =
Ec

α2 = (1 + νm )αm Vm + (1 + ν f )α f Vf − α1(ν f Vf + νm Vm ) ........................................ [3.43]

α1 and α2 being, respectively, the axial and transverse CTEs. In polymer-matrix composites α1 is
strongly fibre-dominated and falls much more
1.0
rapidly than a mixtures-rule prediction as Vf
increases, whereas α2 initially rises slightly before
Normalised residual stress

ambient conditions
0.8 following an approximate mixtures rule between the
fibre and matrix values. The maximum difference
0.6 between the CTEs occurs at a volume fraction of
about 0.2 for a typical laminate of E-glass and epoxy
0.4 temperature/humidity resin. A consequence of the expansion anisotropy is
cycling that laminates containing plies at different angles
0.2 will distort in unexpected ways unless care is taken
to ensure that the constraints are always matched. A
0 single pair of unidirectional CFRP plies bonded
0 5 10 15 20 25
together at 0° and 90° will behave like a bimetallic
Time, days strip during unconstrained cooling from the curing
temperature — a phenomenon that can be used to
Figure 3.30. Reduction in residual thermal stresses in a determine the residual stress levels in laminates by
unidirectional carbon/glass/epoxy hybrid laminate under
different hygrothermal conditions. direct measurements of the curvature that occurs.

Equations 3.40 may also be used to calculate the residual stresses in 0/90 lay-ups or in mixed-ply
hybrid laminates which can develop significant levels of thermal stress if the expansion coefficients of
the two species are significantly different, as in a CFRP/GRP hybrid, for example. As an illustration,
Figure 3.31 shows the predicted levels of thermal strain, as functions of the ply fibre content, in the plies
of a 0/90/0 GRP laminate. The values of the CTEs were obtained from equations 3.41 and 3.42 and the
value of E2 required to determine the transverse strain in the 90° ply was obtained from the Halpin-Tsai
equation (equation 3.9) with ζ = 4. It can be seen that the 0° plies are in a modest state of compression,
while for a practical Vf of 0.6 the strain level in the transverse ply is of the order of 0.5%. Similar results
were obtained experimentally for model glass/resin laminates by Jones and Mulheron (1983). The


development of residual thermal stresses in hybrid laminates has sometimes been observed to result in a
waviness of the fibre tows in the laminae that are under compression as a consequence of in-plane
buckling of the tows before the resin ceases to be capable of flow. Such a defect would be likely to have
serious consequences for the compression strength of the material (see Chapter 4).

The thermal strains arising during either post-cure cooling or subsequent service at raised
temperatures in both single-fibre and hybrid laminates can be determined satisfactorily by means of most
commercial software applications, and the theoretical basis for the calculations can be found in standard
texts, such as that of Jones (1975). It should be born in mind, however, that most matrix resins are visco-
elastic to some extent, and their ability to undergo time-dependent stress relaxation is accentuated by
even slightly elevated temperatures or exposure to moisture which plasticizes some thermosets. Residual
stresses arising during initial manufacture may therefore fall to insignificant levels during storage under
ambient conditions. An illustration of this is shown in Figure 3.30 where the stresses determined from
the curvature of two-ply strips of a unidirectional glass/carbon/epoxy laminate are observed to be
dissipated as a result of hygrothermal conditioning treatments.

3.10 TEXTILE COMPOSITES
Although many high-performance composites are manufactured from prepreg sheet consisting of thin
layers of aligned fibres preimpregnated with semi-cured resin, there
are many practical advantages in using woven fabrics as the
reinforcing medium. Textiles can be produced in a wide variety of
forms, starting from simple plain-weave cloth, with equal numbers
of tows of fibre in the warp and weft directions, via tubular braided
structures, to cloths with fibres in three or five directions
(sometimes confusingly called 3D and 5D weaves) and genuinely
three-dimensional fabrics with a proportion of fibres running
through the thickness. The characteristics of these materials as
reinforcements may be of great value to composites designers,
particularly since woven fabrics drape much more satisfactorily
over complex curved surfaces than unidirectional prepregs can. A
detailed mathematical treatment of textile composites manufactured
from woven products is given by Chou (1992), and an important
recent collection of papers on the subject has been edited by
Masters and Ko (1996). Figure 3.32. Photograph of a piece of
commercial plain-weave glass cloth.

Since, in woven cloths, varying proportions
of the reinforcing filaments will be lying out of
the plane of the laminate into which they are
incorporated, the elastic reinforcing efficiency
of the fabric will be somewhat lower than that of
a ply derived from a prepreg sheet. This can be
illustrated by way of a simple example for
composites containing a 0/90 textile. The
general appearance of a typical balanced plain-
weave glass fabric is shown in Figure 3.32. The
fibre tows in this fabric are flattened, and the
extent to which they deviate from a single plane
depends both on the degree of flattening and the
tightness of the weave. In satin-weave fabrics,
the ratio of the numbers of tows (or ends) lying
in the 0° and 90° directions may be greater than
unity, and the use of a satin-weave
Figure 3.33. (a) Two-unit representative volume element reinforcement therefore allows the designer
of a composite reinforced with a six-harness satin-weave greater flexibility in meeting service
cloth. (b) Mechanical model of the RVE.


requirements. A simple model to represent the structure of such a weave is shown in Figure 3.33 which
depicts an idealised weave described as a six-harness satin weave. For a harness number of two, the
model is equivalent to a plain weave. Many models of woven-fabric composites treat the undulations of
the warp fibres as they pass over and under the weft tows as a sine wave, but for a rough estimate we can
reasonably adopt the simpler linear approach shown in Figure 3.33.

The ‘representative volume element’ (RVE) shown in this figure can be modelled as two series-
connected elements — the kinked section, component 1, and the unkinked section, component 2. The
first of these is itself a parallel-connected trio of elements, one at the out-of-plane angle θ to the main
composite axis, the second at 90°, and the third a plain resin element. Component 2 is equivalent to a
simple non-woven 0/90 laminate, also parallel connected. Thus, the mechanical analogue of the weave is
illustrated in Figure 3.33.

Component 2 is modelled as in section 3.5 of this chapter, the required values of E1 and E2 required
to calculate the component stiffness, E(2), being determined from the Halpin-Tsai equations with ζ = 5
for a notional Vf of 0.65. Simple geometrical considerations allow us to determine the volumes of the
elements in component 1 and their effective fibre volume fractions. The kinked component is then
analysed by a modified Krenchel approach:

E(1) = cos4(θ) E1Vf(θ) + E2 Vf(90) + Em Vm

where Vf(θ) and Vf(90) are the proportions of fibre at the two specified angles. Unlike the conventional
Krenchel estimate, this therefore includes a contribution from the 90° fibre tow. The overall modulus of
the RVE is now obtained as a Reuss (series) 40
estimate for the stiffness contributions of the
appropriate proportions of components 1 and 2. 0/90 non-woven
The harness number will clearly affect these 35
Elastic modulus, GPa

proportions and the degree of flattening of the
tows determines the angle θ and the overall 30
t
thickness of the RVE. Figure 3.34 shows the
w
results of these calculations, plotting the
25
composite Young modulus as a function of θ° w/t
harness number for various tow-flattening ratios, 14 4
w/t (ratio of major to minor axes of the elliptical 20 18.5 3
tow shape). Crude though this model is, the 26.5 2
agreement with the stiffnesses of the plain- plain weave
15
weave and 0/90 non-woven GRP laminates 1 10 100
referred to in Table 3.3 is adequate for many Harness number
design purposes.
Figure 3.34. Predicted moduli for satin-weave GRP composites
3.11 PLIANT COMPOSITES as a function of harness number. The angle θ is the mean angle
of deviation from the laminate long axis of the fibre tows in the
Pliant composites are, generally speaking, weave cross-over sections. The flatter the fibre tows, the higher
textile fabrics coated with waterproof polymers the aspect ratio, w/t, and the lower the misalignment angle, θ.
which are used in various applications requiring
flexibility, toughness, gas-tightness, etc. Typical applications include air-house membranes, dirigible air-
ship envelopes, and rubber boats. Depending upon requirements, the reinforcement may be fabrics
woven from either glass fibres or from conventional textiles, such as polyesters (eg. Terylene/Dacron) or
polyamides (Nylons). For special requirements, aramids like the lower-performance Kevlar fibres may
be appropriate. The range of possible fibre stiffnesses thus covers an order of magnitude or so, from
about 5GPa for Nylon to 73GPa for glass. The coatings may be low-cost materials like PVC,
polyurethanes and rubbers, or more expensive fluoroelastomers like Hypalon. The more costly and more
rigid fluoropolymers like PTFE have excellent long-term weathering and fire-resistance properties.


The elastic behaviour of pliant composites is more difficult to characterise than that of rigid
composites. When a typical uncoated, woven fabric is stretched the effective stiffness will include
apparent strain contributions from several mechanisms other than the true elastic extension of the fibres
themselves. These mechanisms, which do not occur to the same extent in rigid composites, may be
identified as follows.
i) Yarn shear or rotation may occur if the principle stresses are not aligned with the warp and
weft of the fabric.
ii) Crimp interchange. If a highly crimped weft yarn is directly loaded it will not begin to extend
elastically until it has lost the crimp which will in some measure be transferred to the originally
straight warp yarns. A similar type of deformation occurs, although to a lesser extent, in woven
cloth-reinforced resins.
iii) Yarn flattening, leading to a reduction in effective yarn diameter.
iv) Yarn compaction, resulting in an increase in packing density.
v) Fibre straightening and/or rotation with the yarn structure.

The first two occur at low stresses and depend on the principal stress ratios (not on their magnitudes),
whereas the other mechanisms become active only
at high stresses. In normal fabric behaviour true 500
elastic extension of the individual filaments
probably never occurs, and since this is the only 400 PVC/polyester
mechanism considered in conventional netting PTFE/glass
analyses like that of Cox (1952) such analyses are
Load, kg

300
not useful for predicting the behaviour of fabrics
or pliant composites. Shanahan et al (1978) have
200
shown that in some respects fabrics may be treated
as sheets of an elastic continuum and have applied
the basic framework of laminate theory. For 100
warp
membrane strains the tensile modulus, E(θ), from
weft
equation 3.18 gives a reasonable approximation to 0
0 5 10 15 20
measured properties of some types of fabric, but
coupling between membrane modes of Strain, %
deformation, bending and twisting, and large non-
linear elastic strains cannot be dealt with by the Figure 3.35. Tensile load/strain curves for two types of
continuum model. architectural coated fabrics. Test strips were 5cm wide.

The application of an extensible or rubber coating to a woven fabric is considered, for design
purposes, not to change the cloth characteristics. For small-strain extensional behaviour in the warp or
weft directions this may be an acceptable assumption, but even a non-rigid matrix will significantly raise
the in-plane shear resistance of a coated fabric. The shear stiffness may also be affected by a stress-
system which increases the inter-yarn frictional forces. The wall of an internally pressurised tube, for
example, appears to become stiffer as the internal pressure is increased and rotation of the yarns is
inhibited by the increased inter-yarn friction (Topping, 1961). An illustration of the comparative
stress/strain behaviour of two quite different types of coated fabric is shown in Figure 3.35 (Ansell and
Harris, 1980): the effect of the matrix rigidity is clear.

The generalised two-dimensional form of Hooke's Law for a textile or coated fabric (Alley and
Faison, 1972) contains nine stiffness coefficients. Since the straining of such materials is not necessarily
conservative (ie. the values of the coefficients may be dependent on the state of strain in the material and
its loading history) all nine coefficients are independent, necessary, and multi-valued. Furthermore, in a
coated fabric some of the coefficients are likely to be time dependent. In a non-linear anisotropic
membrane, then, it is not surprising that the setting up of practicable constitutive relationships should be
an unwieldy process and of little practical use for design purposes. Useful surveys of the problems of
designing with coated fabrics are given in two published conference proceedings on air-supported
structures (Evans et al, 1980, and Happold et al, 1984).


Alley VL and Faison RW, 1972, J Aircraft, 9, 55-60 and 211-216
Ansell MP, Harris B, 1980, Air-Supported Structures: the State of the Art, (Institution of Structural
Engineers London), 121-130.
Birch SJ, 1995, Unpublished data, School of Materials Science, University of Bath.
Chamis CC and Lark RF, 1977, Hybrid and Select Metal-Matrix Composites (ed. WJ Renton), (AIAA New
York), 13-51.
Chou TW, 1992, Microstructural Design of Fibre Composites, (The University Press, Cambridge).
Cox HL, 1952, Brit J Appl Physics, 3, 72-79.
de Ferran EM and Harris B, 1971, J Mater Sci, 6, 238-251.
Dickson RF, Fernando G, Adam T, Reiter H and Harris B, 1989, J Mater Sci, 24, 227-233.
Dugdale DS, 1968, Elements of Elasticity, (Pergamon, Oxford).
Evans CJ, Bate SCC, Happold E and Piall GHA (editors), 1980, Proc. Symposium on Air-Supported
Structures: the State of the Art, (Institution of Structural Engineers, London).
Farebrother TH and Raymond JA, 1977, Polymer Engineering Composites (editor MOW Richardson),
(Applied Science Publishers London), 198-235.
Galiotis C, 1991, Compos Sci and Technol, 42, 125-150.
Galiotis C and Paipetis A, 1998, J Mater Sci, 33, 1137-1143.
Halpin JC and Tsai SW, 1969, Effects of Environmental Factors on Composite Materials, Report AFML-TR
67-423, (US Air Force Materials Laboratory, Dayton, Ohio).
Halpin JC, 1992, Primer on Composite Materials Analysis (2nd edition), (Technomic Publishing Lancaster
Pennsylvania USA).
Happold E, Dickson MGT, Halstead DH, Tietz SB and Williams CJK, (editors), 1984, Proc. Symposium on
The Design of Air-Supported Structures, (Institution of Structural Engineers London).
Harris B and Bunsell AR, 1975, Composites, 6, 197-199.
Harris B and Phillips MG, 1983, Developments in GRP Technology (ed. B Harris), (Applied Science
Publishers, London), 191-247.
Harris B, 1978, J Mater Sci, 13, 173-177.
Harris B, Guild FJ and Brown CR, 1979, J Phys D: Applied Phys, 12, 1385-1407.
Harris B, Morley J and Phillips DC, 1975, J Mater Sci, 10, 2050-2061.
Hill R, 1964, J Mech Phys Solids, 12, 199-212 and 213-218.
Hull D, 1981, An Introduction to Composite Materials, (University Press, Cambridge).
Jones FR and Mulheron M, 1983, Composites, 14, 281-287.
Jones RM, 1975, Mechanics of Composite Materials, (Scripta Book Co, Washington, & McGraw-Hill).
Krenchel H, 1964, Fibre Reinforcement, (Akademisk Forlag, Copenhagen, Denmark).
Marshall A, 1982, Handbook of Composites (ed. G Lubin), (Van Nostrand Reinhold New York), 557-560.
Masters JE and Ko F, (editors), 1996, Compos Sci and Technol, 56, 205-386, Textile Composites Special
Issue, (Elsevier Science, Oxford).
Nielsen LE and Chen PE, 1968, J Materials, 3, 352-358.
Nielsen LE, 1978, Predicting the Properties of Mixtures, (Marcel Dekker, New York).
Pabiot J, 1971, Composite Materials Conference proceedings number 63, (AGARD/NATO, Neuilly, Paris),
paper 6.
Pierron F and Vautrin A, 1994, Compos Sci & Technol, 52, 61-72.
Piggott MR, Harris B, 1980, J Mater Sci, 15, 2523-2538.
Schapery J, 1968, J Compos Mater, 2, 380-404.
Shanahan WJ, Lloyd DW and Hearle JWS, 1978, Textile Research J, 48, 495-505.
Shiel D, 1995, Unpublished data, School of Materials Science, University of Bath.


Soden PD and Eckold GC, 1983, Developments in GRP Technology (ed. B Harris), (Applied Science
Publishers, London), 91-159.
Stavsky Y and Hoff NJ, 1969, Engineering Laminates (ed. AGN Dietz), (MIT Press, Cambridge, Mass), 5-
59.
Topping AD, 1961, Aerospace Engng, 20, 53-58.
Wagner HD, Roman I and Marom G, 1982a, Fibre Sci and Technol, 16, 295-308.
Wagner HD, Roman I and Marom G, 1982b, J Mater Sci, 17, 1359-1363.
Young RJ, Yallee RB and Andrews MC, 1996, Proc. 7th European Conference on Composite Materials:
ECCM7, London, May 1996, (Woodhead Publishing, Abington, UK), vol. 2, 383-388.

4 THE STRENGTH OF FIBRE COMPOSITES

S
trength is more difficult to predict than elastic properties because it depends on the mechanisms of
damage accumulation and failure as well as on the properties of the constituents, and the failure
behaviour of fibre composites is often complex. The manner in which damage occurs in the
material, and the way in which it accumulates to reach some critical level which precipitates final
failure depends on many aspects of the composite construction, including the fibre type and distribution,
the fibre aspect ratio, l/d, and the quality of the interfacial adhesive bond between the fibres and the
matrix. Much of the software currently available for predicting
the strengths of fibre composites must be treated with
circumspection on account of this uncertainty relating to the
modelling of failure. Because of the complex nature of failure
in many types of composite, considerations of strength and
toughness are closely inter-related. And although in this book
we have opted, for the sake of clarity, to treat them in separate
chapters, we should bear in mind when designing with
composites that the strength and toughness may not be Figure 4.1. A simple unidirectional composite
independent. lamina under tensile load.

4.1 TENSILE STRENGTH
4.1.1 Unidirectional continuous-fibre composites
Consider a simple unidirectional composite lamina reinforced with continuous fibres which are
initially well-bonded to the matrix so that under load fibres and matrix deform together (Figure 4.1). The
load is shared between them and, following the argument at the beginning of chapter 3, we can write the
stress on the composite, (σ)c, as:

(σ)c = (σ)f Vf + (σ)m (1 – Vf)

where (σ)f and (σ)m are the stress levels in fibres and matrix (as opposed to being properties of fibre and
matrix), and are equal to Efε and Emε, respectively, where ε is the composite strain. What happens as
loading continues depends on the nature of the individual components. There are two common situations
which may be taken as examples, as illustrated in Figure 4.2. In most practical composites, the high-
performance reinforcing fibres can usually be regarded as being brittle, ie. they deform elastically to
failure, showing little or no non-linear deformation. In metal- and polymer-matrix composites, the
unreinforced matrix is usually capable of some irreversible plastic deformation and in such materials the
matrix failure strain is usually
much greater than that of the
fibres. By contrast, in ceramic-
matrix composites the matrix is
also brittle, and although much
weaker than the reinforcing
fibres it is often almost as rigid as
the fibres. The matrix failure
strain is therefore usually less
than that of the fibres.

As the diagram in Figure 4.2
shows, when the stress in the
ductile matrix reaches the matrix
Figure 4.2. Schematic illustrations of the stress/strain curves of two types of yield stress, σmy, the matrix
composite, (a) with a ductile matrix and (b) with a brittle matrix, as derived from continues to bear load, although
the stress/strain behaviour of the constituents. the slope of the stress/strain

chapter four Strength 62

curve falls somewhat. If the fibres are carrying most of the load, when the stress on the fibres reaches σfu
(the fibre tensile strength), failure will occur and the stress on the composite at this point, (σ)c, defines
the composite strength, σc, thus:

σc = σfu Vf + σ'm (1 – Vf) ........................................................................................ [4.1]

where σ'm is the stress in the matrix at the fibre failure strain. With the minor modification, then, that the
matrix stress, σ'm, is not a normal property of the matrix, equation 4.1 is another rule of mixtures,
although it is only valid above a certain critical value of Vf. Failure in this kind of composite may be
gradual if the fibre content is not too high but will be rapid at high Vf levels.

In all-brittle systems, such as most CMCs, the matrix reaches its own failure stress, σmu, at relatively
low composite stress levels, and will begin to crack
throughout the body of the material (Harris et al, 1994).
During this stage, the composite may deform without
increase in stress, as shown in Figure 4.2b, and only
when the matrix is completely permeated by an array of
closely spaced cracks does the composite stress begin to
increase again. Beyond this point, only the fibres are
carrying load, and failure occurs when the fibres reach
their failure stress. In such materials, the problem for
the designer is whether or not the development of an
array of cracks during service can be tolerated.

Considering now the brittle-ductile system, equation
4.1 can be represented on a diagram showing composite
Figure 4.3. Schematic illustration of the variation of strength as a function of fibre content, Vf (Figure 4.3).
the strength of a unidirectional composite with fibre When there are very few fibres present (ie. near the left-
content..
hand axis, for Vf < Vmin), the stress on the composite
may be high enough to break the fibres. But even if the
fibres rupture and cease to carry load, because of its work-hardening ability the matrix is still able to
support the load on the composite until its tensile strength, σmu, is reached. In this simplistic approach,
we consider the effect of the broken fibres to be similar to that of an array of aligned holes, so that, to
begin with, as the number of fibres increases, the strength of the composite actually falls, following the
equation:

σc = σmu (1 – Vf ) .................................................................................................... [4.2]

in Figure 4.3. Up to Vmin it is therefore the matrix that exerts control of failure behaviour. When Vmin is
reached, there are sufficient fibres to bear some of the load when the matrix reaches its failure stress, and
the rule-of-mixtures relationship defined by equation 4.1 then begins to operate. An increase in strength
beyond the matrix tensile strength only occurs above the critical Vf level, Vcrit, which can be found from
the geometry of Figure 4.3 to be:

σmu − σm©
Vcrit = ......................................................................................................... [4.3])
σf − σm©

In practical composites where σfu » σmu, Vcrit is very low — of the order of only 0.01 — and the
relationship between strength and composition therefore approximates to a rule of mixtures.


4.1.2 The problem of the strength of brittle fibres
The simple models that we have been
considering make the assumption that the 25
150 sam ples of Enka HTA carbon fibre
reinforcing filaments have a unique failure
m ean fibre strength:
stress, σfu, and that they all fracture 20
2.7GPa
simultaneously when the load on the

Frequency
composite reaches this critical level. In reality, 15
since almost all of the common reinforcing
filaments exhibit brittle behaviour, there is a 10
statistical distribution of fibre strengths rather
than a unique value of σfu. The mean filament 5
strength, σf u , and the spread of values is
determined by the statistical distribution of 0
defects or flaws in the fibre or on its surface. 0 1 2 3 4 5
In general, the spread of strengths in any Fibre strength, GPa
commercial tow of fine fibres is quite large, as
shown in Figure 4.4 by some typical results Figure 4.4. Distribution of tensile strengths of 150 samples
for commercial HTA carbon fibres of commercial Enka HTA carbon fibre. The sample test
manufactured by Enka. length was 3cm (Harris et al, 1992a

Although we often speak of the ‘mean strength’ and the ‘standard deviation’ of the strengths of the
fibres, the distributions are seldom Gaussian (or Normal) distributions. We usually use an exponential
distribution described by Weibull (1951) which gives the probability of failure at a stress level, σ, of any
sample from a given batch of fibres of identical length as:

⎡ ⎛ σ ⎞m ⎤
P(σ ) = 1 − exp ⎢ − ⎜ ⎟ ⎥ .................................................................................... [4.4]
⎢ ⎝b⎠ ⎥
⎣ ⎦

The parameters m and b are known as the shape and scale parameters, respectively*. The shape factor,
m, sometimes called the Weibull modulus, relates to the uniformity of the distribution of flaws in a
brittle material: a high value of m implies a highly uniform distribution of defect sizes and therefore a
low level of variability of fibre strengths. Conversely, a low value of m implies highly variable flaw
sizes and a large spread of measured strengths, as in Figure 4.4. When the data are ranked and plotted as
described in Appendix 2 (see Figure 4.5), it can be
seen that the two-parameter Weibull model of 2
equation 4.4 provides a reasonable fit to the data,
1
despite the fact that for carbon fibres one would
expect to find a minimum stress level (the location 0
Ln[-ln(1 - P)]

parameter) below which no failures would occur.
gauge length = 2cm
The parameters of the distribution shown are m = 6.3 -1 m = 6.3
and b = 2.96GPa (cf. the arithmetic mean of
2.76GPa). A value of 6.3 is very low when -2
compared with the behaviour of normal engineering
materials (m ≈ 30 to 100), but it is not untypical of -3
ceramics.
-4
For composites users, there are two important 0 0.5 1.0 1.5
consequences of the weakest-link model upon which
Ln(strength, GPa)
the statistical analysis is based. First, the breaking
load of a large bundle of brittle fibres (tows of
Figure 4.5. Two-parameter Weibull plot of strengths of
carbon fibres typically contain between 3,000 and untreated ENKA carbon fibres: gauge length = 2cm.

__________________________________________________________________________________
*A brief introduction to the statistical background to this method of analysis is given in Appendix 2


10,000 individual filaments) is lower than the arithmetic mean strength of all the filaments in the bundle.
This can be shown by the following simplified argument.

We assume that all of the filaments in a large bundle have the same length and the same elastic
modulus and are equally loaded. As a bundle is loaded, the weaker fibres in the bundle will fail at low
loads and the applied load must then be redistributed among the remaining fibres. At some point, the
failure of one more filament will mean that the load carried by those remaining is too great for them to
support, and the bundle fails. Suppose that at a load, PF, on the bundle, n fibres out of a total of N have
already failed. The stress on the remaining filaments is then:

σF = PF /AN-n......................................................................................................... [4.5]

AN-n being the cross-sectional area of the remaining unbroken fibres. The ratio of unbroken fibre area to
total area, AN-n /AN , is 1 - n/N, and since this describes the same failure probability as that given by the
Weibull function, P(σ), in equation 4.4, we can rewrite the expression for PF, the load on the remaining
fibres in the bundle, in equation 4.5, as:

⎛ n⎞ ⎡ ⎛ σ ⎞m ⎤
PF = σF A N ⎜ 1 − ⎟ = σF A N exp ⎢ − ⎜ F ⎟ ⎥ ................................................................... [4.6]
⎝ N⎠ ⎢ ⎝ b ⎠ ⎥
⎣ ⎦

To find the critical load to fracture the bundle, we maximize PF by setting the derivative dPF/dσF = 0,
with the result, after some arithmetic, that:

σF = b(m)-1/m

Inserting this into equation 4.6, we obtain the maximum bundle load:

PF(max) = b AN (m)-1/m e-1/m

and the strength of the bundle, σB, which is given by PF(max)/AN, is therefore:

σB = be-1/m m-1/m .................................................................................................... [4.7]

In Appendix 2 we see that the mean (or first moment) of a Weibull distribution is given by the function
bΓ(1 + 1/m), Γ being the Gamma function. The ratio of the bundle strength to the mean filament
strength, σf u , can therefore be found:

−1 −1 −1
σB be m m m (m.e) m ...................................................................... [4.8]
= =
σ fu ⎛ 1⎞ ⎛ 1⎞
bΓ ⎜ 1 + ⎟ Γ ⎜1+ ⎟
⎝ m⎠ ⎝ m⎠

and this ratio is plotted as a function of m in Figure 4.6. As we see from the graph, the values of m for
brittle fibres like carbon and glass are typically less than 10, so that the strength of a bundle of such
filaments may be only 70% of the mean fibre strength.

Another consequence of the statistical model of failure of brittle solids is that their strengths depend
on the size of the piece being tested and for samples of different sizes equation 4.4 contains a volume
term, as shown in Appendix 2. For a batch of samples of fibres of reasonably constant cross-section, this
size-dependence manifests itself as a variation of strength with sample length. This can be shown by
restating equation 4.4 to include a term for the volume, V:

⎡ ⎛ σ ⎞m ⎤
P(σ) = 1 − exp ⎢− V ⎜ ⎟ ⎥ ……..Equation 4.1)
⎢ ⎝b⎠ ⎥
⎣ ⎦


Assuming some fixed failure probability for two
1.0
groups of fibres of lengths l1 and l2, it is then easy to
see that the ratio of the strengths of these two groups σB
must be: σf
0.8
75%
−m
1

σ(l1 ) ⎡ l1 ⎤
= ⎢ ⎥ …………….Equation 4.2) 65%
σ(l 2 ) ⎣ l 2 ⎦
0.6

The value of m alone thus determines the range for carbon
magnitude of the strength/length effect. This effect is and glass fibres

a feature common to all brittle filaments, and is 0.4
1 10 100
illustrated for some commercial carbon fibres in Weibull shape parameter, m
Figure 4.7. It has always been a problem for
designers of composites structures that when dealing Figure 4.6. Dependence of the strength of a bundle of
fibres on the Weibull shape parameter, m.
4
with data from fibre manufacturers the fibre strength
line predicted from m value should only be quoted in relation to the sample test
Fibre strength, GPa

3 length.

We see now that when trying to calculate the
2 composite strength the choice of the value of σfu that
should be used in equation 4.1 is a matter of some
Courtauld HMS carbon fibres
concern. It is clearly not the mean fibre strength, and
1
m = 7.1 although it may seem that the fibre bundle strength,
σB, is the appropriate value to use, this may also
0 prove to be an incorrect assumption. To try to
5 10 20 50 100 answer this question, we have to consider in more
Fibre gauge length, mm detail the mechanism by which a composite breaks
down under load.
Figure 4.7. Variation of fibre strength with test length for
samples of Courtauld HMS carbon fibres (Dickson, 1980). 4.1.3 Progressive damage and failure in
The straight line is predicted from the value of m of 7.1
obtained from individual fibre tests. unidirectional composites
A fibre bundle embedded in a matrix does not
behave like a free fibre bundle. When the local load level reaches the failure stress at a weak point in a
given fibre it breaks and the load it carried is transferred back into the neighbouring matrix regions. But
away from the broken ends the fibre carries its full share
of the load, by contrast with what happens in an
unbonded bundle. The stress carried by neighbouring
fibres in the vicinity of the break will be perturbed, as
shown schematically in Figure 4.8, but the stress
concentration may not be great enough to break a
neighbouring fibre (or fibres). As the load on the
composite increases other fibre breaks will occur, but
each fibre may break many times without seriously
damaging the overall load-bearing ability of the
composite since the tensile load supported by the broken
fibre within a short distance, δ, known as the ineffective
length, from each broken end, will rapidly build up again
to its original level.
Figure 4.8. Schematic illustration of the
perturbation of the stresses in the neighbourhood
of a broken fibre.


If the scatter in filament strengths is small (large value of m) and the fibres are closely spaced, the
extra load carried across to a neighbouring fibre may be sufficient to break that one too. The local stress
concentration is then even higher. If the process is repeated a number of times, the cross section where
the breaks are occurring will rapidly become too weak to support the applied load and catastrophic
brittle failure will follow. In reinforced plastics containing brittle fibres like glass or carbon, however,
this mode of fracture rarely occurs. The second fibre seldom breaks at the same cross section as the first
because the stress concentration adjacent to the first break is unlikely to coincide with a weak point in
the second fibre. Instead, fibre breaks accumulate randomly throughout the whole sample or structure,
and final failure may
then occur when the
number of breaks in
any one cross section
has effectively
reduced the local Vf
below that required to
support the applied
load at that instant. Figure 4.9. Model composite for studying the random failure of fibres leading to composite
failure. Three adjacent fibre failures is assumed to be the failure condition (x = fibre failure).
This process of
damage accumulation may be illustrated in a simple fashion by considering the model composite shown
in Figure 4.9. It represents a single ‘ply’ consisting of ten parallel fibres. In order to model the statistical
distribution of flaws in the fibres, each fibre ‘chain’ is divided into ‘links’, each of which is of length lc
(= 2δ): as we shall see later, this is the smallest length of fibre that can just be broken by a tensile force.
In a realistic statistical model, such as the Monte Carlo treatment of Fukuda and Kawata (1977), each
fibre section would be allocated a ‘strength’ value from a Weibull distribution, and a notional ‘load’
acting on the composite would be increased in small steps. The links would fail as the local stress
reached the Weibull ‘strength’, and this process would be allowed to continue until some preset failure
criterion was reached. A similar result can be demonstrated by simply using a random-number generator
to cause random failures throughout the model, as displayed by the crosses in Figure 4.9.

A suitable failure condition can be deduced from the work of Jamison (1985) who deconstructed
samples of composite material that had been loaded to failure, or just below, and searched for groups of
adjacent fibre fractures throughout the material. He concluded that since the largest groups that could be
found were triplets, a group of four must constitute a critical-sized nucleus that led immediately to
failure. If the computer running the simple stochastic failure model is programmed to recognise an
appropriate group of adjacent failures, then the time to reach that condition (which may equally well
represent an applied load on a composite being
2
loaded at a constant rate) is the composite life (or
failure stress). The result of repeating such a
simulation 100 times is shown in Figure 4.10: for
0
this simple model, the failure condition was three
Ln[-ln(1 - P)]

adjacent breaks, as illustrated, and, to
-2
accommodate the effective-length concept, in-line
chain links immediately adjacent to an existing
failure were not allowed to break. The total
P(x;b,m) = 1 - exp[-(x/b)m]
-4 number of fibre breaks that may occur prior to
m = 2.85
failure varies widely, from 19 to 147 in the case
b = 34.5
of this series of simulated tests, and the times to
-6 failure (or failure stresses) are also very widely
1.5 2.0 2.5 3.0 3.5 4.0 4.5 distributed as the figure shows. Somewhat
Ln(time to failure), arbitrary units surprisingly, the data fit a two-parameter Weibull
distribution. This is not a very realistic model: we
Figure 4.10. Two-parameter Weibull plot of times to failure might ask, for example, whether three breaks in a
generated by a random-break computer program for a model row would be more damaging than two in a row
fibre composite (100 'tests'). and one slightly oblique, and such matters have to


be addressed in serious calculations. However, we can easily appreciate the progressive nature of
damage from such a model and the reasons for the variability in composite strengths and fatigue lives
(see chapter 6).

4.1.4 Calculation of the tensile strength of a unidirectional composite
As we see, damage builds up progressively in the composite until sufficient fibre breaks occur in one
local region to constitute a crack which can then propagate catastrophically and cause fracture. This
progressive failure has been investigated by a variety of methods, including acoustic emission studies,
and by microscopy or x-ray techniques. If we ignore the (low) matrix contribution in the strength
equation (4.1) we can see that the mean filament strength and the fibre bundle strength might well define
the upper and lower bounds on the strength of a unidirectional fibre composite:

σB v f < σc < σ fu Vf ............................................................................................. [4.9]

But there still remains the problem of knowing what is the relevant ‘test length’ for which we should
choose the value of σf u . The whole length of a fibre embedded in a composite matrix is never stressed
uniformly, and it is usually assumed (Rosen, 1965) that the effective ‘test length’ of fibres breaking
within a matrix is of the order of the critical length, lc, or 2δ (see Figure 4.8). Rosen’s cumulative
weakening-model assumes that fibre flaws are distributed randomly throughout the composite and it
extends the fibre-bundle model to the situation inside the composite. As in the simple composite
simulation described above, fibre breaks occur singly and randomly throughout the material. The load
near the breaks in the fibres must be redistributed because of the ineffective end sections, and this
redistribution will be controlled by the strength of the fibre/matrix interface and the relative stiffnesses
of the fibre and matrix, as we have already seen.

In the expression for the bundle strength (equation 4.7) the characteristic strength b relates to
whatever fibre test gauge length has been used. However, for a given fibre population with shape factor
m, the value of b appropriate to a ‘gauge length’ lc can be obtained by applying equation 4.10.
Combining equations 4.7 and 4.10, therefore, we obtain an expression for the strength, σc*, of the bundle
in the composite:
−m
1

⎛l ⎞
σc = b ⎜ c me ⎟
* ......................................................................................................... [4.10]
⎝L ⎠

where the strength of the reinforcing filaments was determined on samples of gauge length L. The upper
bound on composite strength is then equal to σf u Vf. Dividing this estimate of composite strength with
that predicted from the mean filament strength (ignoring any contribution from the weak matrix) we
obtain the ratio:
−1
⎡ lc ⎤ m
σ* ⎣ ⎢ L .m.e ⎥ ......................................................................................................... [4.11]
⎦
c
=
σ fu ⎛ 1⎞
Γ ⎜1+ ⎟
⎝ m⎠

As we saw in chapter 3, the fibre critical length, lc, is determined by the elastic characteristics of the
fibre and matrix and by the strength of the interfacial bond between the two. The fundamental
reinforcing potential of the fibres is therefore affected by a number of factors other than their nominal
tensile strength. In Figure 4.11 the strength ratio of equation 4.13 is plotted as a function of the critical
aspect ratio, lc/d, for different values of the Weibull shape parameter and assuming a fixed fibre test
length (L) of 5cm. For many practical composites the values of m and lc/d fall within the ranges in this
graph, and it is not surprising, therefore, that the strengths of some unidirectional composites may rise
above a mixture-rule prediction based on the single-fibre strength. An illustration of this coated-bundle
effect is shown by some results of Dickson (1980) in Figure 4.12. In this figure, the failure strains of the


bundles are plotted instead of stresses, and the
4
bundle failure load is used as an indicator of
bundle size in order to avoid the tedious business fibre strength test gauge length = 5cm
of counting large numbers of fine filaments. It
3
can be seen that the failure of dry bundles (filled m=5
symbols) is subject to substantial scatter, but that σc*/ σf m=7
the degree of variability falls as the bundle size m=9
2
increases. This is what would be expected of the
bundle-failure model: the dry-bundle failure
strain of 0.41% is predicted from the single- 1
filament data of Figure 4.7. By contrast, the
variability of the resin-coated bundles, which
resemble real composites, is much greater and not 0
markedly dependent on bundle size. The mean 0 20 40 60 80 100
failure strain of the resin-coated bundles is Critical fibre aspect ratio, l c/d.
0.55%, 20% greater than that of the dry bundles
and over 95% of the dry bundles failed at strains Figure 4.11. Variation of predicted composite strength,
below the mean coated-bundle failure strain. The σc*Vf, normalized with respect to the value predicted from
mean filament failure strain, however, is 0.62% the mean fibre strength, σf Vf, as a function of the critical
and it is therefore clear that although many of the aspect ratio, lc/d.
resin-coated bundles do indeed fail above the line
represented by σ∗ / σfu = 1 in Figure 4.11, on average they do not do so.
c

In practice experimental composite strengths are never as high as the Rosen model would lead us to
expect. And although the model was reasonably successful in a number of cases, it is limited by the fact
that it takes no account of the stress concentrations that result from single and multiple fibre breaks, ie.
of the interactions between individual damage events which depend on the surface characteristics of the
fibre, the interfacial adhesion, and the resin flexibility, as well as on the fibre strength statistics. It has
also been suggested that the notional ‘test length’ of lc is far too small and that the Weibull model cannot
be extrapolated to such small test lengths. Hughes et al (1980) and Harris et al (1992a) have shown,
nevertheless, that for some types of carbon-fibre composite the extrapolated bundle strength gives good
agreement with measured tensile strength values.
1.0
mean failure strain There are perhaps two extreme situations. If there
of resin-coated bundles is a great deal of scatter in the strengths of individual
0.8
fibres, failures occur randomly throughout the
Failure strain, %

composite and are non-interacting. On the other hand,
0.6 if the fibre failure strength is very uniform, there will
be a high degree of interaction between individual
0.4 fibre failures in one cross section of the composite. In
large dry-bundle
general, most composites will behave in a complex
0.2 failure strain fashion between these two extremes and it will
usually be difficult to predict the composite strength
predicted dry-bundle failure strain = 0.41% accurately. Bader (1980) has developed a model for
0
10 100 1000 the tensile failure of unidirectional composites which
Bundle failure load, g (ie. increasing bundle size) suggests that the critical size of the group of fibres
dry bundles which initiates failure is of the order of 9 fibres (ie.
resin-coated bundles about three in a row in any given direction (cf. the
results of Jamison described earlier) but that the final
Figure 4.12. The breaking strains of fibre bundles failure sequence is determined by the relative
containing different numbers of Courtauld HMS carbon
fibres. The filled symbols are for dry bundles and the
resistance of the laminate to splitting parallel to the
open symbols are for resin-coated bundles. The fibres which occurs as a consequence of the shear
breaking load of the bundle is used as a rough indicator stress developed around the groups of failed fibres.
of the number of fibres in the bundle (approx. 10g per Manders and Bader (1981) have also described a
fibre). statistical model to predict the strengths of mixed-


fibre (hybrid) composites in which they suggest that the critical number of fibre fractures is of the order
of three, and Curtis (1986) published details of a computer model for tensile failure of unidirectional
composites which showed the sequential development of fibre failures and suggested that four adjacent
broken fibres is the maximum stable group that could be tolerated in standard high-strength-fibre
composites.

4.1.5 Transverse strength
The strengths of unidirectional composites are highly anisotropic.
Perpendicular to the reinforcing fibres most composites are weak and
failure is controlled by rupture or plastic flow of the matrix, or by
fibre/matrix decohesion: the precise mechanism will depend on the
capacity of the matrix for plastic deformation and the strength of the
fibre/matrix bond. If the matrix is a metal or polymer, its intrinsic
yield strength will govern the transverse behaviour of a composite
containing only a few fibres. If this yielding is restricted to the
ligaments between the fibres, the transverse strength of the
composite might be estimated from a crude rule-of-mixtures model
in which the deforming matrix cross-section is obtained from a
knowledge of Vf and the reinforcement geometry. For the square
array shown in Figure 4.13, for example, the effective volume Figure 4.13. Kies’s model (1962) for
fraction of the deforming matrix normal to the fibres is: strain magnification during transverse
loading of a unidirectional composite.

4Vf
Vm (eff ) = 1 −
π

Assuming no contribution from the fibres, the transverse strength of the composite is then:

⎛ 4Vf ⎞ ......................................................................................................... [4.12]
σt = σmy ⎜ 1−
⎜ ⎟
⎝ π ⎟⎠

As the volume fraction increases, however, the relatively undeformable fibres impose an increasing
degree of constraint on the matrix. Contraction in the fibre direction (the width) is prevented, and
extension under stress can only be accompanied by deformation in the thickness direction, ie. the
deformation is under plane-strain conditions. A consequence of this is that the effective yield strength of
the matrix rises from the ‘free’ value of σmy to the plane-strain yield stress. 1.15σmy (Cottrell, 1964).
Despite this, the transverse strengths of practical composites fall rapidly with increasing fibre content
while their axial strengths rise linearly, and even if corrections are made in equation 4.14 (vide Cooper
and Kelly, 1969) to allow for the effect of a
positive adhesive bond between the fibres and 1.0
matrix, the degree of anisotropy of a 2024 aluminium/Nextel
unidirectional composite rises dramatically with 0.8 T300 carbon/epoxy
increasing Vf, as illustrated in Figure 4.14. Two
materials are represented in this figure, a typical
σc / σt

0.6 σf /σm= 6.6
MMC (mullite-fibre-reinforced aluminium) and
a common epoxy-based CFRP, for which the
ratios of fibre strength to matrix strength are 0.4
different by an order of magnitude (about 7 for σf /σm=88
the MMC and 88 for the CFRP). A crude 0.2
correction has been made to allow for the plastic
constraint just mentioned (the effective matrix 0
yield strength increases in proportion to √(Vf)) 0 0.2 0.4 0.6 0.8
but the effect is insignificant by comparison Fibre volume fraction, Vf
with the geometrical effect of equation 4.14.
Figure 4.14 illustrates an important advantage of Figure 4.14. Variation of predicted ratio of composite
transverse strength to axial strength with fibre content for two
types of composite with different ratios of fibre-to-matrix
strengths. The matrix strength varies from σmy at Vf = 0 to 1.15
σmy at Vf = 0.7.


typical MMCs over reinforced plastics which derives from the much higher inherent matrix strengths of
the former. Typical ratios of transverse to longitudinal strengths of GRP, for example, are only of the
order of 0.05 for glass/epoxy (Harris et al, 1979) and 0.04 for glass/polyester (Guild et al, 1982). Such
low levels of transverse strength mean that unidirectional reinforced plastics are incapable of bearing
transverse loads, and even cross-plied laminates may develop cracks during processing as a result of
thermal stresses normal to the fibres if careful control is not exercised.

Strain concentrations in the matrix may also lead to brittle failure at low strains, even in composites
with otherwise ductile matrix materials. The magnitude of this strain concentration may be obtained
from an approximate elastic model by Kies (1962) which defines a strain magnification factor, SMF, due
to composite strains in a direction transverse to the fibres (Figure 4.13). For this simple rectangular
array, the increase in local matrix strain in the x direction, εm, as a fraction of the mean composite strain,
εc, is given by the SMF:
−1
εm ⎡⎛ a ⎞ ⎛ Em ⎞ ⎛ a ⎞ ⎤ ......................................................................... [4.13]
SMF = =⎢ ⎜ ⎟ + 1− ⎥
ε c ⎣⎜ s ⎟ ⎝ E f ⎠ ⎜ s ⎟ ⎦
⎝ ⎠ ⎝ ⎠

The geometrical ratio, s/a, defined in Figure 4.13, depends on the fibre diameter and volume fraction,
and on the geometrical arrangement of the fibres. In typical GRP materials with 65vol% of 10μm
diameter fibres the strain magnification factor given by Kies's model would be about 5 for a hexagonal
fibre array and 7.5 for a square array. Thus, even in resin matrices with failure strains of the order of 4%,
transverse resin cracking perpendicular to the fibres could
occur at applied composite strains as low as 0.5% at a
very small percentage of the full composite potential
strength. Indeed, in the Permali 0/90 GRP which we have
already used as an example in discussing elastic
properties, transverse cracking was shown to occur in the
90° plies at strains as low as 0.2%, only a tenth of the
composite failure strain (Harris et al, 1979): such
cracking is shown in the photomicrograph in Figure 4.15.
And while this transverse cracking may not impair the
load-bearing ability of the material in a major fibre
direction, it could have serious consequences, if, for
example, the material were required to contain liquid or Figure 4.15. Photomicrograph of transverse-ply
cracking in a 0/90 GRP laminate.
gas under pressure.

There is some doubt, however, whether failure in composites of high Vf ever occurs in the simplistic
manner described by these simple models for, as Bailey et al (1979) have shown, it is the fibre/matrix
bond that is usually the weakest element and transverse cracking occurs by the linking up of local
fibre/resin decohesion failures, which is how we also expect a ceramic- or glass-matrix composite to fail.
The transverse strength of brittle-fibre/brittle-matrix laminates is probably best treated by a modified
fracture mechanics approach where the critical flaw size is related to the transverse dimension of the
fibres.


4.1.6 Orientation-dependence of composite strength
When the fibres in a loaded unidirectional composite lie at an angle to the stress axis the effective
strength of the composite is reduced. In chapter 3 we examined the variations of stress and strain as
functions of orientation, and it is easy to see how some kind of failure criterion could be combined with
the stress or strain functions to provide indications of the variation of strength with orientation. The
strength of a composite is highly dependent on the local damage mechanisms, however, and these must
change as the active stress system affects different components of the composite. In our discussion of the
axial tensile strength, we emphasised the rôle of the
fibres to the exclusion of all other effects. But when
the level of shear stress builds up in a composite this
can lead to failure modes of other kinds, including
fibre/matrix interfacial shear failure and interlaminar
shear. And we have already discussed the possibility
of tensile failure of the interface or of the matrix
between the fibres during transverse loading. It is
vital in using composite strength models to ensure
that the appropriate failure mode is being
considered. This is shown by studying one of the Figure 4.16. Resolution of the force, F, acting on a
composite of arbitrary orientation. The strength of the
simplest of the models, the maximum-stress theory composite in the fibre direction is σc.
of Stowell and Liu (1961).

The central assumption is that failure will occur when the stress in a given direction reaches some
critical value (ie. the critical stress for the operative failure mode). If the measured in-plane strength
properties of a unidirectional composite lamina are the normal axial tensile strength, σc, the transverse
tensile strength, σt, and the in-plane shear strength, τc, the maximum-stress model is developed as
follows.

Consider, first, the case where the angle between the stress and the fibres is relatively small, as shown
in Figure 4.16, so that composite failure is still going to be dominated by fracture of the fibres. The fibre
tensile stresses reach the fibre fracture stress before any other failure event can occur. Resolving the
applied force, F, and the area over which it acts, we have:

σc = Fcosθ/A1 = Fcosθ.cosθ/Ao = σθcos2θ

and hence the failure stress, σθ, of the composite at the angle θ to the fibre direction is given by:

σθ = σcsec2θ........................................................................................................ [4.14]

For larger values of θ, the shear component of stress acting at the fibre/matrix interface will gradually
increase until shear failure occurs. Resolution again yields the angular-dependence of the composite
strength appropriate to an in-plane shear mode of failure:

σθ = τc/sinθcosθ = 2τc.cosec2θ........................................................................... [4.15]

At very large angles, the level of applied tensile stress resolved normal to the fibre/matrix interface
results in a failure passing partly through that interface and partly through the matrix between the fibres,
which occurs at relatively low stresses, like interface shear failure. Further resolution leads to the result:

σθ = σtcosec2θ .................................................................................................... [4.16]

The three equations 4.16, 4.17 and 4.18 define the three régimes of failure of the maximum-stress
criterion and their predictions are in reasonably good agreement with experimental results for many
unidirectional composites. One difficulty with this model is that it is appropriate for unidirectional fibre
composites only and cannot allow for constraints resulting from the presence of fibres at other angles (ie.
in a real composite). It is therefore likely to underestimate real composite properties.


There are many other models of composite failure, and many arguments concerning the validity of
the underlying reasoning. One example is that derived from the von Mises maximum-distortion-energy
criterion for the failure of isotropic metals which was modified by Hill (1964) for anisotropic materials.
The Hill argument was adapted by Azzi and Tsai (1965) for use with transversely isotropic laminates
under plane stress, and in one of its forms it predicts the orientation dependence of composite strength
as:

1 cos 4 θ sin 4 θ ⎛ 1 1 ⎞
= + + cos 2 θ sin 2 θ ⎜ 2 − 2 ⎟ .................................................. [4.17]
σθ
2
σc
2
σt2
⎝ τc σ c ⎠

where the stress terms are as already defined.
10
This model, which is almost universally known
as the Tsai-Hill criterion, has the advantage σθ = σcsec2θ σc = 1.50GPa
over the maximum-stress criterion that it is a σt = 0.04GPa

Composite Strength, GPa
mathematically continuous function, and the τc = 0.068GPa
1 σθ = σtcosec2θ
two models are almost indistinguishable under
the simple tensile conditions which we have
been discussing, as shown in Figure 4.17. In
σθ = 2τccosec2θ
this graph, experimental results for σc , σt , and 0.1
τc for a standard CFRP material, T300/5208,
have been used to generate the maximum-stress Tsai-Hill criterion
and the Tsai-Hill failure criteria, and it can be Stowell and Liu model
seen that they agree well except in the range 0.01
20° < θ < 40° and for this material even this 0 10 20 30 40 50 60 70 80 90
difference is relatively small. It is interesting to Orientation, θ, degrees
note that for slight changes of θ away from 0°
the maximum-stress model predicts a slight Figure 4.17. Comparison of the maximum-stress and Tsai-
increase in strength as a consequence of the Hill failure criteria for a T300/5208 unidirectional
resolution of forces. However, it is very carbon/epoxy composite

difficult to manufacture composites with perfect
1.0 Si02/epoxy; Vf = 0.67 (Pabiot) fibre alignment and equally difficult to test them at a
glass/polyester; Vf = 0.43 (Guild et al)
precise angle to the fibres, so that experimental
errors will make it almost impossible to distinguish
Tensile strength, GPa

between the two models at small θ. The Tsai-Hill
criterion is a common feature of commercial
0.1 computer software programs for strength prediction
(see section 4.4).

The predictions of the Tsai-Hill criterion are
compared in Figure 4.18 with strength
measurements for two types of unidirectional
0.01 composite, SiO2/epoxy (Pabiot, 1972) and E-
0 30 60 90
Orientation, θ, degrees
glass/polyester (Guild et al, 1982). The mean
measured values of σc, σt, and τc for the SiO2/epoxy
Figure 4.18. Comparison between experimental data and are similar to those for the T300/5208 CFRP
the predictions of the Tsai-Hill criterion(full and dashed material just referred to, namely 1.46GPa, 0.32GPa,
lines) for the orientation-dependence of the strength of a and 0.68GPa, respectively, while those for the lower
ud SiO2/epoxy composite (Pabiot, 1971) and a ud E- Vf glass/polyester are proportionately lower. The
glass/epoxy composite (Guild et al, 1982).
comparison between theory and experiment for both
kinds of material is seen to be excellent.

4.1.7 The strengths of multi-ply laminates.
When a 0/90 (non-woven) laminate is stressed parallel to one set of fibres the transverse plies will
begin to crack at low strain levels, as we have seen. This cracking will continue over a range of


composite strains determined by the statistical spread of transverse strengths of the individual laminae,
and each transverse lamina may crack many times, the crack spacing gradually being reduced until each
block of transverse ply is too short to be cracked any further, as illustrated by Figure 4.15. Aveston et al
(1980) have proposed that for 0/90 glass/resin laminates the spacing, c, of transverse ply cracks as a
function of applied stress, σa is given by the relationship:

ELETbd2 ⎡ ε E ⎤
c=− ln ⎢1− Tu C ⎥ ................................................................................ [4.18]
GTEC (b + d) ⎣ σa ⎦

where EC , EL, and ET are the moduli of the composite and of the longitudinal and transverse plies, with
respect to the stress direction, GT is the shear modulus, and εTu is the failure strain of the transverse plies.
The transverse plies are of thickness d while the longitudinal plies are of thickness b. Cracking of the
transverse plies could be supposed to start when
40
the composite strain reaches the normal failure
strain, εTu, of the transverse plies.

Young's modulus, GPa
30
Transverse-ply cracking releases some of the
inter-ply coupling constraint and residual thermal
stresses, and impairs the load-bearing ability of the 20
laminate. It is often assumed that the fully cracked
transverse plies no longer contribute to the
stiffness of the laminate. On this basis, the 0° axial 10
0° orientation
modulus, E0, of the Permaglass XE6 laminate 45° orientation
referred to earlier should fall by about 45% after
0
preloading whereas, as Figure 4.19 shows, the 0 20 40 60 80 100
reduction in 0° stiffness after loading to 95% of the
Stress, σ/σc
failure load in only about 11% and there is no
reduction at all in the 45° stiffness. The fully-
cracked transverse plies therefore continue to Figure 4.19. Elastic modulus of Permali XE6 non-woven
glass/epoxy laminate as a function of pre-stress. The
contribute to the stiffness of the damaged
stress level is normalized with respect to the composite
composite, both through direct stress transfer from failure stress, σc (results of Harris et al, 1979).
the longitudinal plies and because of the lateral
(Poisson) constraint they exert on the extensional strain of the longitudinal plies. The models of Bailey et
al (1979) and Aveston et al (1980) predict that the critical strain for first cracking of the transverse plies
depends on the ply thicknesses, and that the thinner the plies the higher the strains to which cracking is
deferred by the elastic constraint.

Approximate estimates of the strengths of multi-ply laminates may be obtained by the use of the
Krenchel model discussed in chapter 3, with the same values of the efficiency factor, ηθ, defined by
equation 3.25, as were used to calculate elastic response. This factor is then used in a mixture-rule
calculation:

σc = ηθσfu Vf + σm´(1 – Vf) ................................................................................... [4.19]

The approach may again be criticized because it ignores contributions of fibres at 90° to the stress axis
and elastic interactions between the individual anisotropic laminae, but these are less important in
calculating the strength of a laminate than in determining its elastic properties. The choice of an
appropriate value for σm´ is also of no great significance in a fibre-dominated laminate.


Another useful approximate approach is that known as the ‘Ten-Percent Rule’. Hart-Smith (1993)
points out that while, in principle, fibre patterns could vary widely throughout any given structure,
depending on requirements, practical aerospace laminates have for many years been designed within a
relatively limited range of options. They
are usually balanced laminates, with equal
numbers of +45° and –45° plies, and the
ratio of 0° and 90° plies varies to
accommodate local loads. Preferred fibre area of preferred
arrangements usually fall within the fibre patterns
shaded area surrounding the quasi-
isotropic lay-up shown in the ternary quasi-isotropic lay-up
‘phase diagram’ in Figure 4.20. In the ten- (25% 0; 25% 90; 50%

percent rule, each 45° or 90° ply is
considered to contribute one tenth of the
strength or stiffness of an axial (0°) ply to
the overall performance of the laminate.
Justification for this lies in the fact that
there is a rapid fall-off in both the strength
and stiffness of a unidirectional laminate
with orientation, as illustrated by Figure Figure 4.20. Preferred fibre patterns for aerospace composite
3.12 (stiffness) and Figure 4.17 (strength). laminates (adapted from Hart-Smith, 1993).
The effective properties at 45° and 90° are
therefore indeed very similar, although in reality the appropriate value for the anisotropy ratio may be
nearer to 4% than 10% for many unidirectional composites.

Table 4.1. Prediction of laminate efficiencies by laminate theory and approximate methods.

Laminate lay-up Comparison of laminate efficiencies predicted by laminate theory and
approximate methods
Laminate theory Krenchel factor Hart-Smith
software 10% rule
UD 1 1 1
(0,90)S (cross-ply) 0.530 0.500 0.550
(±45)S 0.139 0.250 0.100
[(±45,02)2]S 0.575 0.625 0.550
[(±45)2,03,90)]S 0.489 0.500 0.438
(0,±45,90)S (quasi-isotropic) 0.385 0.375 0.325

The ten-percent rule offers a rapid, and reasonably accurate, means of estimating the strength and
stiffness of any practical laminate consisting only of 0, 45 and 90° plies, without the need for elaborate
calculations or extensive experimental evaluation of properties. For example, for a lay-up consisting of
37.5% fibres in the 0° direction, 50% of ±45 fibres, and 12.5% of 90° fibres, a simple mixture-rule sum
gives the laminate strength and stiffness in the 0° direction as (0.375 x 1 + 0.5 x 0.1 + 0.125 x 0.1) =
0.438 times the value of the relevant property measured in axial tension on the unidirectional laminate.
Table 4.1 gives a comparison of the predictions of the Krenchel and Hart-Smith methods for the
efficiencies of some representative laminates, together with values obtained from analysis by laminate-
theory software for T300/5208 carbon/epoxy composites. Except for the (±45)S laminate, the agreement
is very good, despite the assumptions on which the approximate models are based.

We saw in chapter 3 how the analytical procedures of the classical thin-laminate theory may be used
to determine the levels of stress and strain in a multi-ply laminate. If the level of stress in a ply at some
point in the laminate exceeds the load-bearing ability of that ply, local failure will occur. Because of the
way in which the microstructure controls the accumulation of damage in fibre composites, however, the
composite rarely fails when the first ply breaks. The classical thin-laminate theory provides a means of


estimating the laminate strength. Since all stresses in a loaded laminate can be calculated by the theory, a
single stress analysis serves to determine the stress field that causes failure in any one of the constituent
laminae by comparing the individual lamina stresses, σx, σy, τxy, with some suitable failure criterion,
such as the Tsai-Hill model discussed in section 4.16. The thermal contraction stresses following cure
can also be determined and included in the overall level of stress, mechanical plus thermal. If no laminae
are found to have failed following cure the computer program can adjust the external stress system until
the first lamina fails according to the chosen failure criterion. This lamina is then eliminated, the
stiffness matrices are recalculated to accommodate the redistributed loads and the cycle is repeated. At
the point when the failure of one more lamina results in violation of the failure criterion for the
remaining laminations, the composite is considered to have reached its macroscopic failure stress. The
procedure is thus somewhat similar to that of calculating the strength of a fibre bundle. Hart-Smith
(1996) has criticized this process of ply-by-ply decomposition, followed by assessment of each ply
independently from the others. He argues that such a process cannot be valid when matrix cracking
accompanies the deformation process, and he also considers that values of the transverse strength
obtained by measurements made on a single ply are not relevant for the failure of a 90° ply embedded in
a laminate.

The ply-discounting procedure is probably conservative, however — ie. it predicts lower strengths
than actual values because even after a single lamina has failed one or more times it is still capable of
bearing load, the stresses in the neighbourhood of the break being diffused away into adjacent laminae in
the same manner as those surrounding a break in a single fibre.

A central feature of analytical models and software for the prediction of the strength of laminates is
that they must depend on the use of an appropriate failure criterion. We have already discussed two of
these, the maximum-stress and the Tsai-Hill (maximum-strain-energy) criteria, but there is currently
much disagreement about the validity of many of the commonly used models. It is necessary to make
further reference to this problem, and we shall return to it in section 4.5.

4.1.8 Short-fibre composites
We saw in section 3.7 of chapter 3
that if the fibres are short, stress must fibre
be transferred into them by shear and fibre strength
tensile
their ends are not fully loaded. Only if stress σ
a fibre is longer than the critical
length, lc, can it be broken by loading
the composite and its full reinforcing
potential realized. To illustrate this
further, we consider three situations,
where the fibres in the composite are
assumed to be aligned and of uniform
lengths which are less than, equal to, Figure 4.21. Simplified illustration of the variation of tensile stress in
or greater than the critical length, as short fibres as a function of fibre length. σf is the fibre breaking stress
shown in Figure 4.21. We also make and lc is the fibre critical length.
the simplifying assumption that the
tensile stress in the fibre builds up from the ends in a linear fashion, rather than as given by equation
3.33, but this does not greatly affect the argument.

In the case where l is equal to lc, the tensile breaking stress in the middle of the fibre can just be
reached, and the fibre can therefore be broken, but the load-bearing ability of the whole composite must
be less than that of a continuous-fibre composite containing an identical type of fibre. If the fibre
strength is σfu, ignoring statistical aspects of the problem, the average stress in each fibre of length lc
will be only ½σfu (middle sketch in Figure 4.21). Thus, in a longer fibre, such as that in the right-hand
sketch of this figure, there will be two inefficiently loaded end sections of total length lc in which the
average stress is ½σfu and a central section of length (l – lc) in which the stress is σfu. By a rule-of-


mixtures argument based on these lengths, then, the overall level of stress, σf , in a fibre of length l > lc
will be:

[σfu(l – lc) + ½σfu lc]/l

ie.
( σ)f = σfu ⎛1−
l c ⎞ ................................................................................................. [4.20]
⎜ ⎟
⎝ 2l ⎠

for the linear stress variation model illustrated above, where (1– lc/2l) represents an efficiency factor for
length. The mixture rule may then be modified to give the strength of the composite:

⎛ l ⎞
σc = σfu Vf ⎜1− c ⎟ + σ′ (1− Vf )
m [l ≥ l c ] .................................................. [4.21]
⎝ 2l ⎠

This equation can be used to calculate the strengthening efficiency (the ratio σc/σfu) as a function of fibre
length (l/lc), as shown in Figure 4.22 for a typical composite. It can be seen that provided the fibre
length is about 10 times its critical length, some 95% of the strength obtainable in a continuous-fibre
system can be achieved with short, aligned fibres. The validity of this conclusion is not affected by the
simplistic stress-transfer model assumed above. The 0.8
problem with discontinuous-fibre reinforcement,
0.668
however, as we have suggested, is that it is very
difficult in conventional manufacturing processes to 0.6
keep the fibres well aligned.
σc / σfu

0.4
If the fibres are shorter than lc they cannot be
loaded to their failure stress and the strength of the Vf = 0.65
composite is then determined by the strength of the σfu = 2GPa
0.2
fibre/matrix bond, not by the fibre strength. This is σ'm = 100MPa
sometimes the case in GRPs manufactured by
injection moulding, especially if the lengths of the 0
fibres are reduced by damage during the 1 10 100 1000
manufacturing process. The average stress in the l/ lc
fibres can be estimated from the balance of forces
illustrated schematically in the diagram of Figure Figure 4.22. Efficiency of strengthening in short-fibre
3.21. If the stress in the fibre is identified as (σ)f, the composites, as given by equation 4.23.
2
maximum fibre tensile force, (σ)f.πd /4, is balanced
by an internal frictional force which must exceed the fibre/resin bond strength, τi, or the matrix shear
yield stress, τmy, if failure by a mechanism other than fibre failure is to occur. If we let this limiting shear
stress be equal to τi (the more common failure mode) the force balance at failure will be:

πdlτi πd2 ( σ )f
= [( σ )f < σ fu ]
2 4

and the average tensile stress in the fibre, (σ)f , is therefore:

τil
(σ)f = 2 (σ)f =
1
............................................................................................ [4.22]
d

The composite strength, from the mixtures rule, is therefore:

⎛ τl ⎞
σc = ⎜ i ⎟ Vf + σmu (1 − Vf ) [l < l c ] ........................................................ [4.23]
⎝ d ⎠


where σmu is the matrix tensile strength. This 500
relationship emphasises the importance of good
fibre/matrix bonding if reasonable strength levels 400

Tensile strength, MPa
are to be obtained in short-fibre composites. If the Slope = τiVf
bond is poor, a soft matrix like polyethylene or a 300 τ = 21MPa
pure metal like aluminium will deform readily
around the more rigid fibres and the tensile
200
strength falls as the fibre length is reduced. This is
Short-fibre Cu/W composites
illustrated in Figure 4.23 by some tensile strength Vf = 0.5
measurements on MMCs consisting of aligned 100 σmu
short tungsten fibres in a matrix of pure copper
(Harris and Ramani, 1975). The strength of these 0
0 10 20 30 40 50 60 70
short-fibre composites only approaches that of a
Fibre aspect ratio, l/d
continuous-fibre composite at an aspect ratio of
about 100. Since this is only about twice the
critical aspect ratio, lc/d, however, it appears that Figure 4.23. Relationship between the tensile strength
and fibre aspect ratio in tungsten-wire/copper composites
the stress transfer at the fibre ends is rather more containing short, aligned fibres (Harris & Ramani, 1975).
efficient than is assumed by Figure 4.21.

So far we have assumed the short fibres to have been arranged in a single direction, and comparisons
with the properties of unidirectional, continuous-fibre composites are therefore valid. The shorter the
fibres, however, the more difficult it is to obtain good alignment, and without good alignment close
packing cannot be achieved. The strengths of short-fibre composites are therefore limited both by fibre
misorientation and by the lower fibre contents than those which can be obtained in continuous-fibre
composites in addition to the fibre-length effects already discussed. Methods of producing composites
reinforced with almost perfectly-aligned chopped fibres have been described by Parratt et al (1971) but
short-fibre composites are used most frequently as moulding materials. The polyester dough moulding
compounds (DMCs, sometimes referred to as bulk moulding compounds) and sheet moulding
compounds (SMC) are typical of thermo-set mouldings materials of this kind, while for injection
moulding there is a wide range of glass-fibre-reinforced thermo-plastic composites available.

The strengths of materials of this kind are hard to predict on the basis of equations 4.23 and 4.25
because of the combination of variables, the uncertainty about parameters such as τi and lc (which are
difficult to measure for conditions appropriate to the real composite) and lack of knowledge of the
effects of processing on the strength of the reinforcing filaments. On the basis of assumptions relating to
probable values of lc and τi and with the application of a Krenchel factor of ηθ = 0.375 for randomly-
oriented composites, Harris and Cawthorne (1974) estimated strengths for typical DMC and SMC that
were well within the experimental scatter for
their measured strength values (37MPa and 1.2
100 MPa, respectively, for DMC with fibre size
Frequency (normalized)

15wt% of 6.4mm long fibres and SMC with 1.0
length coating
25wt% of 50mm fibres in chopped-strand
0.8 a) 6mm soluble
mat). The strength of the DMC, with fibres
b) 13mm soluble
shorter than the probable critical length, is in
0.6 c) 13mm insoluble
fact no greater than that of the resin matrix,
but its advantage lies in its much higher
0.4
toughness, as we shall see in chapter 5.
0.2
One of the problems encountered in the
manufacturing of short-fibre composites is 0
that the fibres may sustain damage as a 0 5 10 15 20
result of the mechanical processes involved. Fibre lengths, mm
Very often the lengths of the filaments are
reduced, either as a consequence of the Figure 4.24. Normalized distributions of the lengths of fibres
extracted from moulded DMCs. The mean lengths of the fibres
blending of the constituents in a thermoset
after processing are: a) 1.63mm, b) 1.86mm, c) 3.10mm.


moulding compound, or of the injection moulding of thermoplastic compounds through narrow die
openings. And although the starting lengths of the fibre stock may initially be well above the critical
length, their lengths in the finished mouldings may be well below lc. In DMCs, the final fibre lengths are
also affected by the nature of the sizing compound used on the fibre tows. Figure 4.24 shows examples
of normalized length distributions for fibres extracted from polyester moulding compounds after
processing (Andrews, 1998). Three sets of data are included: two of them for fibres initially 6mm and
13mm long coated with a soluble size, and the third for 13mm fibres with an insoluble size. It can be
seen that when the size dissolves in the resin during processing, the fibre bundles become easily
separated and they are reduced in length so as to have roughly the same modal value of length, only
about 1mm, whatever their initial length. If an insoluble size is used, the bundles do not filamentize to
quite the same extent, but the modal length is only marginally greater. And although 1mm may be equal
to or slightly greater than lc, a large fraction of the fibres are much shorter than this. In typical fibre-
length distributions from injection moulded Nylon 66 with 15vol% of glass fibres, Bowyer and Bader
(1972) showed that some 80% of the fibres were below the critical length, and they emphasised the need
to control moulding procedures to minimize this processing damage. Some materials manufacturers have
developed commercial ‘long-fibre’ glass/thermoplastic moulding compounds to improve on the
properties of conventional mouldings: ICI’s Verton, containing 10mm long fibres, is an example of such
a material.

Bowyer and Bader (1972) attempted to account for the effect of damage to fibres during the injection
moulding of reinforced thermoplastics by studying the fibre length distributions in mouldings. They used
a model in which contributions to composite strength from fibres of sub-critical and super-critical
lengths were summed separately over the effective range of lengths :

⎡ τL ⎛ E ε d⎞ ⎤
σc = ηθ ⎢ ∑ i Vi + ∑ Ef εc ⎜ 1 − f c ⎟ Vj ⎥ + Emεc (1 − Vf ) ........................................... [4.24]
⎢ i d ⎜ 4L j τ ⎟ ⎥
⎣ j ⎝ ⎠ ⎦

The first summation represents the sub-critical fibre distribution, sub-fraction Vi with fibre length
spectrum Li (l < lc), and corresponds to the first term in equation 4.25. The second summation gives the
super-critical contribution, from sub-fraction Vj with fibre-length spectrum Lj (l > lc), and the third term
is the usual matrix contribution. ηθ is the Krenchel factor, 0.375 for a planar-random moulding and 0.167
xy
for 3-dimensional material. In this
po
on
/e equation, τ represents the fibre/matrix
1.4 rb
ca interfacial shear strength that we have
S
HT
1.2
xy previously called τi, the subscript having

po y
1.0 s s/e pox xy been temporarily dropped to avoid
gla on
/e po
S- rb l O
/e
3 confusion with the summation indices, i
0.8 ca A 2

H M
S and j. This model gave reasonable
0.6 ta
l agreement with their experimental results
ce
olya M
C
C
on glass-filled Nylon and polypropylene.
0.4 s/p r S M
las ste rD
-g lye ste
0.2 E po
The strength of short-fibre composites
lye
po
0
is strongly affected by the fibre/matrix
0.5 0.5 0.5 0.14 0.25 0.25 0.12 bond strength, and since the reinforcing
aligned random efficiency of short fibres is low, it is
essential to ensure that they are as
Fibre volume fraction
strongly bonded as possible to the matrix.
Figure 4.25. Tensile strengths of some short-fibre composites, The strengths of some typical short-fibre
aligned and random. composites, commercial and experimental,
are shown in Figure 4.25, together with
those of unreinforced matrix polymers. Improvements in strength by a factor of two or three are
obtainable in practical moulding systems, but the strengths of these materials are very low by
comparison with those of continuous-fibre composites. By contrast, the well-aligned, short-fibre
composites produced by Parratt and Potter (1980) can be seen from Figure 4.25 to have 60 to 70% of the


strength of comparable continuous-fibre composites. The outstanding reinforcing potential of whiskers
indicated in the table is not used in practice because of the cost of this kind of fibre.

The effects of fibre characteristics and
distribution on the expected strengths of
aligned fibre composites are summarized in the
schematic diagram of Figure 4.26. Increasing
the intrinsic fibre strength naturally increases
the expected composite strength in the ways
that we have discussed in this chapter. But if
the fibres are discontinuous, the reinforcing
efficiency is reduced for two reasons: first, as a
result of the aspect-ratio effect described by
equation 4.23 and, second, because the shorter
the fibres the more likely they are to be
misoriented, to a greater or lesser extent. This
Figure 4.26. Schematic illustration summarising the effects
not only results in a further efficiency loss from of choice and arrangement of fibres on the strengths of fibre-
the orientation effect (Krenchel factor), but it reinforced composites.
also limits the maximum fibre content that can
be incorporated into the composite since the closeness of packing is reduced.

4.1.9 Hybrid composites
As we saw in chapter 3 the elastic properties of hybrid composites can be calculated to within limits
acceptable for most design purposes by the application of well-established principles of composite
mechanics and classical thin-laminate theory. But although elasticity calculations are largely
independent of the nature of fibre distributions, predictions of properties such as strength, toughness, and
fatigue behaviour are more difficult because they depend on micromechanisms of damage accumulation
which, in turn, are determined by the construction of the laminate and the scale of dispersion of the
mixed fibres. At the grossest level of hybridization, strips of GRP incorporated into a CFRP laminate can
act as efficient crack arresters (Bunsell and Harris, 1976; Sun and Luo, 1985), provided the width of the
strips is sufficient to dissipate the energy of a crack moving rapidly in the CFRP by localized debonding
and splitting.

The commonest type of hybrid, however, and
that which is most easily achieved in practice, is
made by laminating some balanced sequence of plies
within each of which there is only a single species of
fibres. The composition-dependence of the strength
of a unidirectional composite of this kind is obtained
by analogy with the response of a single-fibre
composite by considering the failure strains of the
two separate constituents (Aveston and Kelly, 1979).
When a small quantity of fibres with low failure
strain, εle, is added to a composite consisting of
fibres of high-failure-strain, εhe, the low-elongation
(le) fibres will experience multiple failure before the
failure strain of the high-elongation (he) fibres is
reached. By analogy with the case of a flexible resin Figure 4.27. Schematic representation of the variation of
containing only one species of brittle fibre, the the strengths of a family of unidirectional two-component
strength of the hybrid composite, σH, will be given hybrid composites with proportions of the two constituents,
at first (Figure 4.27) by the residual load-bearing referred to as the high-extensibility (he) and low-
extensibility (le) materials.
capacity of the he composite, assuming the le
component makes no contribution at all, ie.

σH = σhu Vhe ......................................................................................................... [4.25]


where Vhe is the volume fraction of the he component, Vhe + Vle = 1, and σhu is the strength of the pure he
fibres.

At the other extreme, if a small amount of he fibre is added to an le composite, the strength is again
reduced because the he fibres cannot carry the load when the le component fails. Failure of the le fibres
therefore leads to single, catastrophic fracture of the composite. The hybrid strength in this mode is then:

σH = σ’h Vhe + σlu Vle ............................................................................................ [4.26]

where σlu is the strength of the pure le fibres, and the intercept of this line on the he axis (Vhe = 1) is the
stress σ´h in the he component at the failure strain of the le component, equal to εluEh (εlu and εhu are the
failure strains of pure le fibres and the pure he fibres, 5

Failure strain, %
respectively). The critical composition, Vcrit, for the 4 GRP/CFRP
transition from single to multiple fracture is found by 3 KFRP/CFRP
solving the simultaneous equations 4.27 and 4.28, whence: 2
1
σlu
Vcrit = ..................................[4.27] 0
σ lu + σhu − εluEh 2

and this transition point also defines the hybrid
composition of minimum strength. Figure 4.28 illustrates
1
that this simple model is adequate for two unidirectional
hybrid combinations, CFRP/GRP and CFRP/KFRP. It also
shows that in the case of the KFRP/CFRP combination, for
which the failure strains, εlu and εhu, are markedly 0
0 0.2 0.4 0.6 0.8 1.0
different, the hybrid composite failure strain varies non-
linearly with composition and falls below a straight line Composition, VCFRP /(VCFRP+ VGRP)

joining εlu and εhu. This type of variation is characteristic
and has given rise to the notion of ‘failure-strain Figure 4.28. Variation of tensile strength and
enhancement’, also known as the ‘hybrid effect’. This idea failure strain with composition in ud hybrid
stems from the fact that since there is a continuous laminates of CFRP with GRP and with KFRP. The
full lines in the lower figure represent equations
variation of composite failure strain between the two end 4.27 and 4.28 for the two sets of strength data.
points, some hybrid compositions must inevitably fail at
strains greater than the failure strain of the lower extensibility component, εlu. An analysis by Zweben
(1977) shows that the failure process is affected by the statistical spread of failure strains of the two fibre
species and predicts that the addition of he fibre to an le composite raises the strain level needed to
propagate fibre breaks because the he fibres behave like crack arresters at a microstructural level. The
situation becomes more complicated when the fibres are intimately mixed because micro-mechanisms of
failure change as the pattern of nearest neighbours around a given fibre alters and the strength is
governed by failure either of single fibres or of small bundles of fibres (Parratt and Potter, 1980).

4.2 COMPRESSION STRENGTH
Studies of the compression behaviour of composites
have been complicated by the fact that when testing
small, free blocks of material, failure modes occur which
are not necessarily determined by fibre behaviour, and
the compression strength of a composite often appears to
be lower than its tensile strength. As in the case of wood
compressed along the grain, longitudinal splitting and/or
kinking of the sample often occur, as illustrated
schematically in Figure 4.29, at stresses well below the
equivalent tensile load-bearing ability of the composite.
Figure 4.29. Schematic illustration of
compression failure modes in unidirectional
composites.


Measurements to obtain the true compression strength are difficult to carry out. A range of complex
jigs has been developed for the purpose of preventing macroscopic (Euler) buckling of the sample during
testing, but these often constrain the sample in an unnatural fashion,
and it is uncertain whether the measured value of the compression
strength is valid. Even after many precautions have been taken,
measured compression strengths are often below the tensile strength
of the same material, as shown by some experimental results for a
variety of composites in Table 4.2. By contrast with polymer- and
metal-matrix composites, however, the compression strengths of
reinforced ceramics are often higher than their tensile strengths,
presumably because the rigid ceramic matrix affords a much greater
level of support to the fibres, allowing their reinforcing potential to be
realized, whereas their tensile strengths are often determined by
features other than the fibre strength (Harris et al, 1992b). This is
shown by the data for the unidirectional SiC-reinforced calcium
Figure 4.30. Microstructure of a aluminosilicate composite at the bottom of Table 4.2. In studying
compression failure in a
unidirectional glass/polyester compression behaviour, therefore, there is often uncertainty as to
composite (the compression axis is whether an apparently low value of the compression strength is a
vertical). reflection of material behaviour or a consequence of inappropriate
testing procedures.

Table 4.2. Experimental tension and compression strengths for various composite materials.

Material Lay-up Vf Tensile Compression Ratio
strength, σt strength, σc σc/σt
GPa GPa
GRP ud 0.6 1.3 1.1 0.85
CFRP ud 0.6 2.0 1.1 0.55
KFRP ud 0.6 1.0 0.4 0.40

HTA/913 (CFRP) [(±45,02)2]S 0.65 1.27 0.97 0.77
T800/924 (CFRP) [(±45,02)2]S 0.65 1.42 0.90 0.63
T800/5245 (CFRP) [(±45,02)2]S 0.65 1.67 0.88 0.53

SiC/CAS (CMC) ud 0.37 334 1360 4.07
SiC/CAS (CMC) (0,90)3S 0.37 210 463 2.20

Much effort has been expended on the development of compression test methods to overcome this
problem. Devices commonly used include end-constraint techniques to prevent longitudinal splitting and
lateral supports to prevent premature buckling, in addition to the more obvious precautionary measures
to align the stress axis accurately with the fibres. A recent study by Berg and Adams (1989) suggests that
many of the observed discrepancies between published test results for composites occur as a result of test
methods. They carried out an evaluation of some of the commonly used anti-buckling test fixtures, and
concluded that a method developed at Illinois Institute of Technology (the IITRI fixture), which at the
time was an ASTM Standard test method, D 4310, was the most satisfactory method. By such means it is
possible to measure compression strengths almost equal to the tensile strength in composites other than
those reinforced with Kevlar fibres, as demonstrated by Ewins and Ham (1973) for a series of carbon-
fibre composites and, more recently, by Parry and Wronski (1982). Ewins and Ham suggested that the
mechanism of failure in CFRP is the same in both compression and tension modes and results from a 45°
shear mechanism, but work by Parry and Wronski disagreed with this conclusion on the basis of studies
of the effect of superposed hydrostatic pressure on the compression behaviour. It does not follow,
however, that for any given composite the ‘true’ compression strength must be the same as the tensile
strength.


Since it is not possible to attach any 0.8

significance to the notion of ‘compression E-glass

Compression strength, GPa
HMS carbon
strength’ for a 10μm diameter fibre, when we 0.6 HTS carbon
investigate the effect of reinforcement on Kevlar-49
compression strength we have to take the tensile
strength of the fibre as our point of reference. As a 0.4
unidirectional composite is loaded in compression,
the fibre axis being well-aligned with the stress
axis, the elastic models that we have already 0.2
considered will apply and we would expect the
initial stiffness to be the same as the tensile
0
stiffness, as discussed in chapter 3. But the fibres 0 0.2 0.4 0.6 0.8
can only be maintained in good alignment by the Fibre volume fraction, Vf
matrix, which, in resin-based composites usually
has low shear stiffness, and when local shear or Figure 4.31. Variation of compression strength with fibre
transverse tensile stresses become greater than the volume fraction in unidirectional composites of polyester
matrix can support, a crack may occur or a shear resin reinforced with carbon, glass and Kevlar-49 fibres
(Martinez et al, 1981)
(kink) band, like that shown in Figure 4.30, may
develop and grow as the load increases. Flexible
fibres like glass or Kevlar-49 will usually bend or buckle in this kink band, while brittle fibres like
carbon will often fracture after the initial microbuckling has occurred. These instabilities then lead to
overall failure at stresses below the normal tensile strength of the material. Furthermore, since it is
almost impossible to obtain a unidirectional composite with perfectly aligned fibres — the fibres in most
practical composites are likely to have small kinks or misalignments arising from manufacture — these
defects will act as initiators for the microbuckling mechanisms.

In an early model for compression strength, Rosen (1965) proposed that failure was initiated by
matrix instabilities that lead to co-operative, in-plane buckling of the reinforcement. The model was
developed for lamellar composites with reinforcing plates, rather than fibres, and, except at low Vf, it
predicted failure by a shear mode at a stress:

Gm ...................................................................................................... [4.28]
σcomp =
1− Vf

where Gm is the matrix shear modulus. The deficiency of the model, apart from the fact that it applies to
the buckling of plates, is apparent from the fact that σ
comp → Gm as Vf → 0, and equation 4.30 consistently
3

overestimates the compression strength of real fibre
drawn wire
Compression strength, GPa

composites. Lager and June (1969) suggested that out- annealed wire
of plane buckling occurred in fibre composites, and 2
proposed the use of an empirical ‘influence factor’ to
obtain agreement between their results for wire tensile
boron/epoxy composites and Rosen’s theory. This was strengths
1
a logical idea, since the stored elastic energy in a fibre
which buckles into a helix is bound to be lower than
that in one constrained to preserve a planar buckled
form. It was later demonstrated by de Ferran and 0
0 0.2 0.4 0.6 0.8 1.0
Harris (1970) that helical buckling occurred in fibres
extracted from compressed MMCs. They also showed Fibre volume fraction
that the compression strengths of steel-wire/resin
composites obeyed a mixture rule based on the tensile Figure 4.32. Compression strengths of steel-
strength of the reinforcing wires, as illustrated in wire/polyester-resin composites (de Ferran and
Figure 4.32. These results for two different species of Harris, 1970).
steel wire reinforcements were in good agreement with
a model of Hayashi and Koyama (1971) which allowed for the stress-dependence of the shear modulus.


The implication was that in this case the fibres were controlling compression failure by virtue of their
inherent shear resistance which, in a ductile metal, is the same in compression as in tension.

By contrast, Martinez et al (1981) have shown that in composites reinforced with much finer
filaments of carbon, glass and Kevlar-49 the compression strength appears not to be sensitive in any
straightforward fashion to either the strength or stiffness of the reinforcing filaments, as can be seen
from Figure 4.31. The linear portions of the curves for glass and both varieties of carbon (high-strength
and high-modulus) are largely indistinguishable and extrapolate to stress levels at Vf = 1 of between 1.3
and 1.4GPa, reasonable for the tensile failure of glass or HMS carbon, but only half what might have
been expected for HTS carbon. The curve for the aramid composites lies well below those of the other
composites, and the low strengths are inconsistent with the high tensile strength and stiffness of the
Kevlar-49 fibre itself. The curves for all of the composites are linear up to Vf values of only about 0.4,
the deviations from linearity and subsequent downward trends implying changes in failure mode.

Early models often assumed that fibre buckling was responsible for the compression yielding of
composites and that the Euler buckling theory for a column on an elastic foundation was a suitable basis
for calculating the composite compression strength. Since buckling would be controlled by fibre
modulus and diameter, the results in Figure 4.31, representing fibres of roughly similar diameter but
having a wide range of strengths and moduli, and the earlier results of de Ferran and Harris, representing
experiments with wires of different strengths but having identical moduli and diameters, offer no support
for these ideas. A corollary of the buckling model was that fibres that were already misaligned or kinked
would buckle more readily than misaligned fibres and that the apparently low compression strengths of
many composites could be accounted for in this way. A series of experiments was therefore carried out
by Martinez et al (1981) with composites in which the fibres were deliberately misaligned with respect
to the stress axis by twisting the tows of fibre which were used to reinforce a polyester resin. In this way
a known angle of deviation could be introduced. Their results showed that for misalignments up to 10°
the compression strengths of brittle-fibre composites actually increased and in some cases did not fall
below the aligned-fibre value until nearly twice this degree of misalignment. The increase can be
explained in purely geometric terms in the same way as the maximum-stress model of Stowell and Liu
(qv) explains an initial increase in tensile strength with fibre misorientation.

Although the results described above offer little support for the idea that fibre misalignment controls
compressive failure of unidirectional composites, a recent predictive model suggests otherwise. Barbero
(1998) has derived an explicit mechanics-based model which is able to predict compression strength
with good accuracy on the basis of a non-dimensional function:

G12
χ=
Ωτii

where G12 is the composite shear stiffness and τ11 its shear strength, while Ω is the standard deviation (in
degrees) of the fibre misalignment. The advantage of this model is that values of all three terms in the
definition of χ may be obtained from well-established experimental techniques. Agreement with
experimental data was good for a wide range of unidirectional composites, including glass- and carbon-
fibre-reinforced polyesters, epoxies and thermoplastics.

The factors other than fibre thickness which affect the compression strength are the matrix strength
and rigidity, and the interfacial bond strength. The more rigid the matrix, the greater the compression
strength. Thus, ceramic- and metal-matrix composites are much stronger in compression than resin-based
composites. CMCs, in which the fibres and matrix often have similar levels of stiffness, may have
compression strengths that exceed their tensile strengths, as shown in Table 4.2. Most modern
reinforcing fibres are only about 10-15µm in diameter. Boron, on the other hand, although not now
much used, is much thicker — about 40μm — and thicker fibres resist microbuckling much better than
thin ones, as predicted by the Euler theory. Boron-fibre-reinforced aluminium is currently the composite
material having the highest ratio of compression strength to density. Curtis and Morton (1982) have


shown that the compression strengths of multi-directional coupons of an unspecified CFRP material
were markedly affected by fibre surface treatment, increasing by some 50% as the level of a commercial
treatment was increased from nothing to about 10% of the normal commercial level, and then remaining
constant, at which stage the measured strengths were only some 10% less than the tensile strength of the
same composite. This discrepancy could be removed, however, by taking into account small degrees of
eccentric loading in the test jig.

4.2.1 Compression strength after impact
We have seen that tensile loading introduces impacter
damage into a composite and that the damage +45
accumulation process is partly responsible for -45
0
0
determining the strength of the composite. It is one +45
-45
of the perceived virtues of composite materials that 0
0
they are tolerant of damage and that tensile damage 0
0
does not seriously affect tensile load-bearing -45
+45
ability, but a word of caution is necessary here. 0
0
-45
Composite laminates in many external applications, +45

especially in aircraft, are susceptible to impact
damage from low-velocity projectiles travelling 2cm scale
normal to the surface. The simplest, and most-often
quoted, example is the damage caused by a spanner Figure 4.33. Damage map of the longitudinal section of
dropped by a person standing on an aeroplane a [(±45,02)2]S CFRP composite after it had received a
low-velocity impact of energy 3J (Beheshty, 1997).
wing. A tool weighing 1kg dropped from waist
height delivers a blow with about 10J of energy. In an impact between a round-ended projectile and a
flat plate, tensile stresses may occur in the matrix and across the interlaminar planes of weakness. In a
low-velocity impact the rate of build-up of the Hertzian contact stresses is high. Even a toughened epoxy
resin may therefore appear to be brittle under such conditions and cracks may occur, as shown in Figure
4.33, as planar cracks within the interlaminar regions and as transverse cracks in an apparently conical
distribution spreading outwards through the composite away from the point of contact. The development
of this kind of damage has been modelled by Davies et al (1995). If the impacter has sufficient energy,
there may be serious surface cracking or spalling damage on the back face of the laminate, even when
the damage on the impacted face is scarcely detectable, a condition referred to as BVID — barely-visible
impact damage. The problem for users of composites is that although this kind of damage does not
markedly affect the tensile strengths of most laminates, it has an adverse effect on the compression
strength because the interlaminar cracks are able to extend as delaminations. The CAI, or compression-
1.5 after-impact, test has become a standard way of
assessing the severity of damage induced by low-
velocity impact events. An illustration of the
tension effects of such impacts on the tension and
Strength, GPa

1.0 compression strengths of a CFRP laminate is
shown in Figure 4.34. For this fairly typical
composite consisting of a high-strength fibre in a
modified epoxy resin, an impact of only 5J
0.5
compression reduces the compression strength to some 32%
of its initial value, while reducing the tensile
strength by only 5 or 6%. Such a result has a
0 knock-on effect for other properties of the
0 1 2 3 4 5 6 material, such as the flexural response and
Low-velocity impact energy, J fatigue behaviour. There are two aspects of the
laminate structure that may be considered
Figure 4.34. Effect of low-velocity impact damage on the responsible for this behaviour. First, the fact that
residual tensile and compression strengths of a the matrix resin is an epoxy — even a so-called
[(±45,02)2]S HTA/982 CFRP laminate (Beheshty & Harris, ‘modified’ one — which is naturally rate-
1997).
sensitive means that the matrix is likely to
become brittle at high rates of loading, and if the


stresses are concentrated, either by the presence of notches or by the Hertzian contact effect, the rate is
increased even further. Second, if the laminate is manufactured from prepreg sheet, the interlaminar
planes form natural regions of weakness, as we have seen. Two possible solutions to this problem are a)
to use a thermoplastic matrix instead of a thermoset, and b) to use woven-cloth reinforcement instead of
non-woven fibres. An illustration of the power of this argument is given by some results of
Ghaseminejad and Parvizi-Majidi (1990) who studied the impact behaviour and damage tolerance of two
thermoplastic polyaryl resins, poly(ether ether ketone), known as PEEK, and poly(phenylene sulphide),
or PPS, reinforced with woven carbon-fibre cloth. The compression strength after impact of the PEEK-
based composite remained at 47% of the value for virgin material even after sustaining an impact of 29J.

Guild et al (1993) developed a model for the reduction in compression strength of laminates
following impact. The virtue of their model is that it is based on the premise that failure in post-impact
compression occurs as a consequence of the stress concentration that arises because of the presence of a
damaged zone. It does not require the use of fracture-mechanics concepts which, as the authors point
out, is ruled out by the observed fact that materials whose fracture behaviour is characterized by
different values of the strain-energy release rates GIc and GIIc often show comparable performances in
respect of compression-after-impact. The model predicts, in agreement with experimental results, that
the normalized compression-after-impact strength falls linearly with the width of the damage zone,
whatever the material.

4.3 SHEAR STRENGTH
Most laminated continuous-fibre composites contain planes of weakness between the laminations and
along fibre/matrix interfaces. In shear the composite strength will be dominated by these weaknesses
unless the stress direction intersects the fibres. Shear stresses at the interfaces between the plies can
seldom be avoided by lay-up design because of the anisotropies of neighbouring plies, as discussed in
chapter 3. These interlaminar shear stresses are usually high at edges, as shown by Pipes and Pagano
(1970), and often give rise to delaminations which propagate into the composite from the edges,
significantly reducing the laminate tensile strength (Pagano and Pipes, 1971). Delamination is a major
cause of failure in laminated composites, and one of the concerns of the designer is to ensure that shear
stresses are diffused safely away from stress
concentration points.
transverse
section at AA'
Interlaminar shear failure is most readily seen
in the three-point bending of short beams, a
method commonly used to measure what is interlaminar
usually referred to as the interlaminar shear shear crack
strength, or ILSS, although this is often regarded
as unsatisfactory because the state of stress is
not pure shear. In the diagram in Figure 4.35, if
the level of horizontal shear stress at the mid-
plane point M reaches the interlaminar shear
strength, τIL, of the composite before the tensile Figure 4.35. Schematic illustration of interlaminar shear
stress level at T reaches the composite strength, failure in a unidirectional laminate.
σc, then the beam will fail as shown. If the beam
is longer than a certain critical length, however, it will fail in a normal bending mode by a tensile failure
initiating at the mid-point of the outer face. The variation in failure load as the span-to-depth ratio of the
beam increases is shown schematically in Figure 4.37. As long as the failure mode is interlaminar shear,
the failure load is independent of length, but when the tensile failure mode predominates, the failure load
falls rapidly in inverse ratio to the span of the beam. The critical span-to-depth ratio for the transition can
be found by equating the expressions for the tensile and shear stresses in the beam.

The tensile stress at T is 3PL/2BD2, while the shear stress at the neutral plane is 3P/4BD (the standard
formulae for the bending beam, with B = width, D = thickness, and L = length). Equating these, we
have:


σc 3PL 3P 2L ......................................................................................... [4.29]
= =
τIL 2BD2 4BD D

and the beam will therefore fail in interlaminar
30
shear at the neutral plane if L/D < σc/2τIL. For a
typical unidirectional CFRP laminate with σc ≈ 25
KIIc =A.τIL(v)1/6
1.5GPa and τIL ≈ 50MPa, a shear failure will occur
in bending at span-to-depth ratios less than about 20

ILSS, MPa
15:1, and when the short-beam shear test is used to dry
15
evaluate the ILSS, it is usually recommended that
a span-to-depth ratio of about 5 is used. wet
10

The interlaminar shear strength depends in part 5
on the shear strength of the matrix and in part on
the shear resistance of the interface. The stronger 0
0 5 10 15
the interfacial bonding, then, and the greater the Void content, vol%
amount of interface relative to the amount of
matrix, the greater will be the ILSS: the ILSS
Figure 4.36. The effect of voids and moisture on the
almost always increases with fibre volume fraction interlaminar shear strength of a unidirectional
unless the level of fibre/matrix bonding is very carbon/epoxy laminate (Beaumont & Harris, 1972). The
low. With modern carbon and glass fibres this dashed line represents a suggestion of Corten (1968) that
rarely occurs, however, because the reinforcing the effect of voids on ILSS may be represented by a
constant value of the mode-II critical stress-intensity factor
fibres are usually surface treated after manufacture and the voids as having the effect of cracks (see text).
in order to improve the interfacial bond. In carbon-
fibre composites it was long ago shown that the ILSS was roughly inversely related to the stiffness of the
reinforcing fibres, which reflected the fact that both the fibre mechanical properties and their chemical
activity (wettability) are determined by the level of heat-treatment they receive during processing.

Interlaminar weakness is aggravated by the presence of voids or moisture in the resin, as shown in
Figure 4.36. In some carbon/epoxy composites, for example, 10vol% of voids could reduce the ILSS by
some 25%: moisture absorption will reduce the intrinsic matrix strength and may also cause a
deterioration in the interfacial bond strength, both of which will reduce the ILSS. Corten (1968)
suggested that the effect of voids on the shear strength could be treated in terms of a fracture-mechanics
approach (see chapter 5 for further information on fracture mechanics). If an increase in void content is
notionally equivalent, in terms of stress concentration, to an increase in crack size, and the equivalent
crack dimension, a, is roughly proportional to the
100 cube root of the void content, 3√v, then an
Flexural and shear failure loads, kN

shear failure load expression for the mode II critical stress-intensity
flexural failure load
factor, KIIC, is:
10 tensile
failure K IIc = τIL πa = Aτ6 v ....................[4.30]

As the dashed line in Figure 4.36 shows, an
0 shear
failure
expression of this form fits the data very well down
critical value of L/D to some threshold void content of about 1%. The
constant, A, has a value of about 23 when the ILSS
0.1 is in MPa and v is in %).
0 10 20 30
Span-to-depth ratio, L/D
Although an appreciation of the reasons for the
weakness of composites in shear is of great
Figure 4.37. Transition from shear to flexural (tensile) importance to the designer, it is not necessarily
failure of a thin laminate plate in three-point bending as
the span-to-depth ratio, L/D, increases. The composite
advantageous to use composites with the highest
tensile strength for this sample is 750MPa and its obtainable shear strengths. Cracks running normal
interlaminar shear strength is 50MPa. to fibres in a tension structure may pass at various
times through fibres, resin and interface, but if


nothing changes the crack path, a reasonably flat fracture surface may result and the composite will be
notch-sensitive, even brittle. If the material has an interface of only moderate strength, however, cracks
may deviate through weak regions, such as the interlaminar planes, to become delaminations. As a result,
a crack tip is effectively blunted and the material will appear tougher. Materials suppliers are required to
maintain a fine balance in controlling the fibre, matrix and interface properties in order to optimize
strength (tensile and shear) and toughness. An alternative approach to modifying the interfacial bond
strength is to improve the delamination resistance of a laminate by ‘stitching’ the laminae together with
fibres running in the direction perpendicular to the plane of the laminate. This process has been shown
(Dransfield et al, 1994) to improve both the ILSS (delamination resistance) and also the toughness of the
composite, although Adanur et al (1995) have shown that there is a limiting stitch density beyond which
these improvements are reversed.

Another difficulty often faced by designers is that of assessing the reliability of a reported value of
the shear strength of a composite, which we have previously termed τc. The short-beam shear test
described above is very easy to carry out and is very useful as an experimental tool. But although it
forms the basis of an international standard test, the complex stress state involved makes it unsatisfactory
for obtaining absolute values of the shear strength: τIL is not necessarily a good measure of τc. The ideal
method of measuring τc is perhaps by the torsion testing of thin-walled, hoop-wound tubes, as described
by Puck and Schneider (1969), which gives pure states of shear stress and strain in the tube wall.
Alternative solutions have usually been sought, however, because of the time and expense involved in
carrying out such tests. One such solution is the measurement of the intralaminar shear strength by the
tensile testing of unidirectional test coupons at an angle to the fibres. Chamis & Sinclair (1972) showed,
from a consideration of the relative stress and strain magnitudes in tensile test samples at various angles,
θ, to the fibre direction, that the in-plane shear strain is a maximum when θ = 10° for typical CFRP
laminates. A tensile test on a unidirectional coupon cut at 10° to the fibres should therefore provide an
accurate measure of the intralaminar shear strength. The tensile stress normal to the fibres is not zero at
this angle, however, and locally the composite is effectively in a biaxial stress state. Hart-Smith (1996)
argues that a tensile test on a ±45° laminate provides a more appropriate value of the in-plane shear
strength, and there is much current interest in the Iosipescu test (Iosipescu, 1967), a method involving
transverse shear loading of deeply notched test beams which, when carried out with adequate controls,
generates a state of pure shear and provides good values of shear stiffness, but which still leaves
something to be desired when used to obtain strength data (Pierron et al, 1996).

4.3.1 The shear strength of the interfacial bond
At several points in this book we refer to the importance of the fibre/matrix bond in determining
important physical properties of composites. The economic and safe exploitation of modern high-
performance composites depends directly on our ability to employ the fibres in resin and other matrices
in such a way as to permit the optimization of the major mechanical properties of the composite. Our
ability to tailor composites to a given specification, and to predict accurately the properties of
manufactured materials, calls for an understanding of the nature of the individual components and of the
manner in which they interact when working in combination. Since much of this understanding hinges
on characterization of the strength of the interface, it seems appropriate to digress briefly from dealing
with macroscopic properties in order to consider some of the methods that are used for determining this
all-important characteristic.

One of the difficulties in making valid assessments of the interfacial shear strength, τi, is that the
simplest experimental methods involve interactions between single fibres and blocks or drops of resin
rather than realistic composite structures. Some of the earliest experiments with single fibres embedded
in resin were carried out by Broutman and by McGarry (see Broutman, 1967). These were carried out on
specially shaped specimens which, when loaded in compression, caused shear or tensile debonding of
the fibre from the matrix. These methods have largely been superseded, possibly because of the
difficulty of preparing the samples. A second problem is that the analysis of experimental data often
involves oversimplified assumptions about the stress distribution along a fibre and about the manner of
interfacial failure. Some of these limitations will be apparent in the following discussion of several of the
more familiar test methods.


i) Values from macroscopic shear-strength measurements. It might be anticipated from the schematic
illustration of an interlaminar shear failure shown in Figure 4.35 that, since this kind of failure is a mixed
fracture passing partly through the matrix and partly through the interface, the ILSS would be a rule-of
mixtures sum of the shear strengths of the matrix and the interface. It will not be a linear function of Vf,
however, because the interface failure is associated with a non-planar surface area. As shown by
Hancock and Cuthbertson (1970), the ILSS for a unidirectional composite would be given by:

τIL = aiτi + (1 – ai)τm ....................................................................................... [4.31]

where τm is the shear strength of the resin and ai is the fraction of the fracture surface area which consists
of fibre/resin interface. Since ai → 1 in a composite of high Vf, this relationship may provide a useful
first estimate of τi, but this simplified model is likely to be inadequate because of the shear-stress
concentrations that result from the inhomogeneity of the material. The results of Iosipescu or other
macroscopic shear tests could, of course, be analyzed in a similar manner to that described above.

ii) Values from crack-spacing measurements. When dealing with brittle-matrix composites like CMCs,
estimates of the value of τi may be made from observations of crack spacings in damaged composites. In
such a composite the matrix is usually of lower extensibility than the fibres, so that when the composite
is loaded in tension, multiple cracking of the matrix occurs until a limiting crack density is reached. The
Aveston, Cooper and Kelly theory (Aveston et al,
1971, 1974, 1979), which is discussed in more
detail in section 4.6, shows that the limiting crack
separation in a brittle-matrix composite is between
x' and 2x', where

⎛V ⎞ ⎛σ r⎞
x′ = ⎜ m ⎟ . ⎜ mu ⎟ ..............................[4.32]
⎝ Vf ⎠ ⎝ 2τi ⎠

σmu being the failure stress of the unreinforced
matrix and r the fibre radius. Aveston et al (1974)
show that the mean crack spacing, s = 1.364x'.
Thus, for a typical mean crack spacing of the order c) single-fibre
of 0.2mm in a SiC/CAS composite of Vf = 0.4, fragmentation
(Harris et al, 1992b), x' is about 0.15mm, and the τ
i must therefore be about 3.5MPa if σmu is Figure 4.38. Schematic illustration of four of the most
100MPa. This low value is typical of many CMCs. common experimental methods for determining the
interfacial shear strength.
iii) Fibre pull-out tests. There are two main
variants of this method, as illustrated in Figure 4.38a and b. In the first, single fibres of different lengths
are embedded in resin and loads are applied. If the
embedded length, l, is less than lc/2, the bond will be
Fmax
broken and the fibre extracted. When l > lc/2, the fibre
will break outside the resin (see problem 14, Chapter 8).
When the end of a fibre with l < lc/2 is loaded by pulling,
Load

the load increases to some maximum level at which the
uncoated induced shear forces acting at the interface are able to
silane coated initiate a debond and the load drops to some lower level
as the debond rapidly propagates, as illustrated in Figure
4.39. The equilibrium state identified by the bottom of the
load drop relates to a length of fibre separated from the
Deflexion
surrounding resin and an end ‘plug’ which maintains
Figure 4.39. Load/deflexion curves obtained from contact and exerts an opposing frictional drag as the
single-fibre pull-out experiments on glass/polyester applied force withdraws the fibre from the matrix. The
composites (Harris et al, 1975). The peak loads are peak load must reflect the strength of the combined
not directly comparable because the pulled lengths
of fibre were different.


chemical and physical bond, while the portion of the curve after the drop relates to the work of fibre
pull-out.

The interpretational problem associated with this test is that we have no easy means of knowing
exactly what shear/debond condition the force Fmax or the load at the bottom of the load-drop shown in
Figure 4.39 relate to. We do not know what the relative lengths of the plug and the ‘free’ portion of the
fibre are, and we do not know how the shear stress varies in the plug region, nor how it changes as the
pull-out process progresses. Fmax is often used to obtain an indirect estimate of the maximum interfacial
shear stress required for debonding:

τi = Fmax /πdl.................................................................................................... [4.33]

and the area under the curve after the load drop has been equated with the work of fibre pull-out, which
provides an estimate of the interfacial frictional stress. Outwater and Murphy (1969) and Gao et al
(1988) adopted fracture-mechanics-based models to describe the debonding process, and while these
may well offer more realistic models of the pull-out process, their implementation as tests for
determining τi depends on knowing the critical strain-energy release rate for debonding

Pitkethly and Doble (1989) pointed out that if equation 4.35 is applied to a long fibre then the shear
strength will be underestimated because a part of the fibre is included over which no stress
transfer is taking place. Ideally, the embedded fibre length should be as near to zero as possible,
but for more manageable fibre lengths they took the value of the shear stress obtained from
equation 4.35 as an average value of the shear debond stress and applied a correction derived
from shear-lag theory to obtain a more realistic estimate of the maximum interfacial shear
strength. A recent critique of the test and its analysis has been given by DiFrancia et al (1996). They
cite values of the debonding shear strength for surface-treated carbon/epoxy combinations of about
70MPa, compared to about 25MPa for untreated fibres. Qiu and Schwartz (1993) have tried to
overcome the unrealistic condition of pull-out from a plain resin by designing micro-composite samples
in which the central fibre is withdrawn from a ‘composite’ consisting of seven fibres in resin.

iv) Droplet-debond tests. In the droplet-debond test (Miller et al, 1991) a small droplet of resin is
suspended on the fibre and, after curing, the fibre is withdrawn from the droplet (Figure 4.38b). Because
the interfacial contact is small, there is a high probability that debonding will occur before fibre rupture
or matrix deformation. The interfacial shear strength is again calculated from the maximum debonding
force, Fmax, from equation 4.35. It is usually found that there is a high degree of variability in these tests
(cv ≈ 20%), often as a consequence of the inherent non-uniformity of fibre surfaces. An accurate
assessment of the area of the fibre perimeter within the droplet is required, and this is made either by
wettability measurements or fibre diameter measurements. An accurate assessment of the embedded
length is also sometimes difficult to obtain if there are marked menisci at the ends of the droplet and
these break off at an early stage in the test. Good instrumentation is needed if these dimensional
measurements are not to constitute a major source of error.

v) Micro-indentation (or push-in) tests. This test was first proposed by Mandell et al (1980), and
described for use with CMCs by Marshall and Evans (1985). Analyses have also been given by Chen
and Young (1991) and Ho and Drzal (1996). A cross-section is taken through a real composite and
polished by normal metallographic procedures. A very small diamond indenter, such as that in a micro-
hardness testing machine, is then used to push one of the fibre ends into the body of the composite
(Figure 4.38c). By choosing the load so that the indenter just contacts the matrix, an estimate of τi can
be obtained from the relationship:

τi = F2/(4π2ur3Ef) ............................................................................................... [4.34]

where r is the fibre radius and Ef is the fibre elastic modulus. F is the force applied to the fibre, and this
is calculated from the relationship:


F = 2a2H

H being the fibre hardness which is obtained by indenting fibres under lower loads where no sliding
occurs, and a is the length of the diagonal of the indentation impression in the fibre. The term u is the
distance by which the fibre has been depressed at maximum load and is calculated from the dimensions
of the indentation in the fibre and matrix. Two common difficulties with this test are that it is hard to
assess the effect of the sideways force exerted by the indenter normal to the direction of sliding, and it is
often difficult to obtain a realistic value of fibre ‘hardness’ because the structure (and therefore the
properties) of common fibres are often very variable through the cross section. More sophisticated
purpose-built testing machines provide force/displacement information directly from sensitive transducer
read-outs.

vi) Single-fibre fragmentation tests. This test has
become one of the principal methods for assessing 1.2
fibre/matrix interfacial characteristics. The method, Weibull parameters:
introduced by Kelly and Tyson (1965), consists in 1.0 untreated oxidized
embedding a single fibre in a clear resin sample
(Figure 4.38d) which is loaded in tension so as to 0.8 m 7 4.6

i/(N + 1)
b 75 64
cause the fibre to break down into successively a 0 0
0.6
shorter lengths until all fragments have lengths
shorter than or equal to lc. To ensure that this 0.4
process can be completed it is usually necessary to
ensure that the resin has a high failure strain, thus 0.2
imposing a second element of artificiality on the test.
0
The fragment lengths are measured with an optical 0 20 40 60 80 100
microscope focused on the fibre within the resin (for Fragment aspect ratio, l/d.
detailed descriptions, see, for example, Drzal et al,
1982; Netravali et al, 1989; Feillard et al, 1994).
Figure 4.40. Probability plot of fibre fragment aspect ratios
from a single-fibre fragmentation test. Untreated and
The principle of the test depends on the force- oxidized ENKA HTA carbon fibres.
balance relationship between the critical aspect ratio
of the fibre, lc/d, the fibre strength, σf, and the interfacial shear strength, τi, given by equation 3.30:

lc σ
= f ............................................................................................................. [4.35]
d 2 τi

for rigid/perfectly-plastic behaviour of the composite. In the use of the fragmentation test it is assumed
that the value of critical length obtained from the distribution of broken fibre lengths and the single-
filament strengths obtained from single-filament tests may be used in this equation to obtain a value for
the interfacial shear strength. The situation is less simple than it appears, however. The distribution of
broken fibre lengths is not independent of the fibre strength distribution, and assessment of a true value
of lc is not straightforward.

Assuming a uniform interfacial strength and a fibre strength distribution with little dispersion about
the mean value, the shortest broken length that will be found should be equal to lc/2, and the longest
observed length will be just shorter than lc, the range of lengths being uniformly distributed between
these two extremes, provided the test has been continued to sufficiently large strains. Thus, it is
frequently assumed that the average observed broken length will be 3lc /4, and the value of the critical
length derived from such measurements is 4lmean/3. Two typical distributions of aspect ratios, l/d, of
carbon-fibre fragments in epoxy resin are shown in Figure 4.40 (Harris et al, 1992a). They fit two-
parameter Weibull distributions (see Appendix 2) with wide spreads. The shape factor, m, is about 6,
which is similar to values obtained for the fibre strength distributions, as we saw in section 0. It can be
seen that the effect of carrying out a commercial surface oxidation treatment on the fibre is to shift the
distribution of fragment lengths to the left, so that the characteristic length, obtained from the scale


parameter, b, falls from about 525μm to about to 450μm as a result of this oxidation. Since both the fibre
strengths and the fragment lengths are distributed according to two-parameter Weibull models, it is
therefore necessary to use a statistical approach in order to determine a cumulative distribution function
for the interfacial shear strength, from which an average effective interfacial shear strength can be
obtained.

Table 4.3. Some published values of carbon/epoxy interfacial shear strength
values measured by the single-fibre fragmentation test.

Carbon-fibre/resin Fibre surface Reported value
system treatment of τi , MPa
HMU/828 none 6-14
HMS/828 commercial 20-47

AU/828 none 24
AS/828 commercial 74

AU4/828 none 29-36
AS4/828 commercial 47,48

T300/5208 none 39
T300/5208 commercial 49

In obtaining an estimate of the interfacial shear strength from equation 4.37, the usual assumption is
that the value of strength to be used in the equation is that which is characteristic of the fibre at a gauge
length equivalent to the critical length, lc. If we consider the process of fragmentation, either in a single-
fibre composite or in a real composite, we see from the weakest-link model that each time a fibre
fractures, the remaining pieces of the fibre must become stronger. During the fragmentation process,
therefore, there will be a continuing change in the value of fibre strength that is actually affecting the
fragmentation process and therefore a change in the balance of forces implied by equation 4.37. The
final fracture in a given piece of fibre, will occur when the fibre length, l, is somewhat greater than lc
and somewhat less than 2lc, ie. lc < l < 2lc. Thus, the appropriate value of strength to use in equation
4.37 will be that which is characteristic of a length of about 1.5lc, and this can be obtained from the
strength/length relationship of equation 4.10. Some typical experimental results for values of the
interfacial shear strength in carbon/epoxy systems are given in Table 4.3. These are taken from
published papers, and full references may be found in Harris et al, 1992a.

It is unlikely that the rigid/perfectly-plastic assumption is truly valid for resin-based composites.
Lacroix et al (1992) have made a detailed critique of the method, arguing that lc may not be considered
to be a material constant, since it depends on the shear-stress-transfer model used. The Kelly and Tyson
model(1965) which yields equation 4.37 is valid only for the condition of total debonding and fails to
distinguish between the interfacial bond strength and the frictional shear strength. Predicted values of lc
may be lower than those characteristic of complete debonding if the interfacial bond strength is
sufficiently larger than the frictional shear stress. Despite the obvious experimental difficulties with this
test, the results in Table 4.3 are reasonably self-consistent, and the method can presumably be used for
comparative measurements, even though the absolute values of τi obtained from it may not be accurate.

vii) Direct measurement by laser-Raman spectroscopy. This is essentially a means of measuring strain
by monitoring the frequency of atomic vibrations in a crystalline material which can be probed by
spectroscopic techniques such as Raman or IR spectroscopy. As atomic bond lengths increase under
load, frequency shifts occur which are directly related to the applied load. The use of the laser Raman
method, which is widely applicable to materials such as commercial polymers and ceramics, does not
require physical contact, unlike other strain-measuring techniques, and it is highly selective, being
capable of probing areas as small as 1 or 2μm. Pioneering work in this field has been carried out by
Galiotis and co-workers (1984, 1991).


It is commonly observed that the results obtained from many of the above tests show considerable
variability and there are also usually significant differences between the experimental estimates obtained
from the different tests, as shown by some results published by Herrera-Franco and Drzal (1991) (Table
4.4).

Table 4.4. A comparison of results of interfacial shear-strength measurements obtained by
different test methods for AS4 carbon fibres in 828 epoxy resin (Herrera-Franco & Drzal, 1991)

Experimental method τi, MPa
Short-beam shear test (ILSS) 84.0
Iosipescu shear test 95.6
[±45]3s tension test 72.2
Droplet-debond test 50.3
Micro-indentation test 65.8
Single-fibre fragmentation test 68.3

4.4 FLEXURAL STRENGTH
As discussed in section 4.3, we might expect that when a composite is loaded in bending it would fail
when the stress in the outer fibres on the tensile face reaches the normal composite tensile strength.
However, early flexural tests often gave strength values that were considerably lower than the tensile
strength. One reason for this was that the loading points in flexural test rigs designed for metals often
caused localized damage which initiated premature failures. But even when this defective test procedure
was rectified, low-stress failures still occurred. It was then considered that this was because in materials
with poor in-plane shear resistance and with compression strengths that were lower than the tensile
strengths, it was shear and/or compression damage modes that initiated the premature flexural failures.
Controlled fibre surface treatments have now considerably improved both the interlaminar shear and
compression responses of many commercial materials, and it is usually possible to measure flexural
strengths that are at least equal to the normal tensile strengths.

Direct comparison of tension and flexural strength is somewhat complicated, however, by virtue of
the statistical effect of the different stressed volumes in tensile and flexural test pieces. As discussed in
Appendix 2 and illustrated by equation 4.10, the effective strength of a sample of small volume is greater
than that of a larger sample. Thus, if two identically sized composite test pieces are tested in axial
tension and in bending, the effective strength of the latter will be higher, even though the failure mode in
each case is simple tension, because the volume of the bend-test sample subjected to the highest stress
level is much less than that in a tensile sample. The effect of such geometrical/statistical effects in
ceramics has been demonstrated by Davidge (1979). He shows that, for samples of similar volume, the
ratio of strengths measured in four-point bending and tension is a function only of the Weibull shape
factor, m, thus:
1

σ 4B ⎡ 4(m + 1)2 ⎤ ............................................................................................ [4.36]
m

=⎢ ⎥
σT ⎣ m+2 ⎦

and for a given value of m the ratio of the three-point bend strength to the tensile strength is even greater.

Measurements on a [(±45,02)2]S HTA/913 carbon/epoxy laminate gave values of 1.17±0.05,
0.81±0.02, and 1.00±0.12GPa, respectively, for the axial tensile and compression strengths and the 4-
point flexural strength (Hardiman, 1997). The Weibull shape factor for most good-quality CFRP
materials is usually high, between 20 and 30, by comparison with common values for ceramics which
are less than 10. A mean value obtained at Bath for five other CFRP laminates of the same lay-up as the
above is 22, and the ratio given by equation 4.38 is therefore about 1.23. Thus, whereas the measured
tensile and flexural strengths are statistically almost identical, this apparent equivalence is accidental
since, on this basis, we would have expected the flexural strength to be 23% higher than the tensile
strength, ie. about 1.44GPa, if the flexural failure is determined by the tensile strength. As it happens, the


same factor applied to the experimental compression strength gives a value of 1.0GPa, which is equal to
the measured value of the flexural strength, even though the flexural failure of these samples was not
apparently initiated on the compression face. We note, in passing, that for a group of SiC/CAS CMC
laminates, the ratios of three-point-bend strength to tensile strengths were between 1.5 and 2.5,
consistent with Weibull m values between 5 and 10. Attempts to reconcile the results of measurements of
this kind with the simple statistical model are of course based on the assumption that the characteristics
of the critical flaws on the surface and in the volume of the material are identical, which may not be true.

4.5 FAILURE CRITERIA FOR COMPLEX STRESSES
For most practical purposes, designers require models of behaviour that can predict failure under
realistic combinations of stresses, rather than for the idealised uniaxial stress conditions under which
most laboratory tests are carried out. There has been a great deal of research on complex-stress failure
criteria (twenty or more models have been proposed, although the differences between many of them are
quite small: see the review by Owen and Found, 1972), and designers throughout the world remain in
dispute as to the ‘best’ method. A recent meeting organized jointly by the Institute of Mechanical
Engineers and the Science & Engineering Research Council (now the Engineering and Physical Sciences
Research Council) in the UK (1991) exposed some of the areas of disagreement, and the Elsevier journal
Composites Science and Technology has just published a Special Issue devoted to evaluating a wide
range of popular failure criteria against validated experimental results*. The main difficulty in accepting
one or other of the common methods is that their validity can usually only be tested over limited ranges
of combined stress because of the complexity and cost of the test samples and test procedures for such
experiments — tubes under combined tension, torsion and internal pressure, for example. Most of the
existing failure criteria are in fact restricted to conditions of plane stress (thin plates) and some are only
applicable to orthotropic materials.

We have noted in section 4.1.6 the differences between the maximum-stress criterion and the
maximum-strain-energy criterion in predictions of the orientation-dependence of the strength of a single
lamina. The former takes into account that a unidirectional laminate will fail by different mechanisms at
different angles, but is not able to allow for interactions between failure modes or stresses. The
disadvantage of the strain-energy approach is that it does not take account of different failure modes. In
its formal form it is an interactive model, although the interaction usually has to be allowed for
empirically. The virtue of the strain-energy criterion in the eyes of many designers is that it is
represented by a single equation instead of three (or five if compression failure is included). One of the
earliest applications of the von Mises model was by Norris and McKinnon (1946) for GRP in plane
stress (ie. loading by stresses σ1 , σ2 , τ12, with all other stresses = 0) which predicts that failure will
occur when:
2 2 2
⎡ σ1 ⎤ ⎡ σ2 ⎤ ⎡ τ12 ⎤
⎢ ⎥ + ⎢ ⎥ + ⎢ ⎥ = 1 ....................................................................................... [4.37]
⎣ σc ⎦ ⎣ σt ⎦ ⎣ τc ⎦

the stresses σc, σt, and τc being the composite failure stresses in tension, compression and shear, as
defined in section 4.16. In this form no interaction effects are allowed for. The later model of Azzi and
Tsai (1965), a development of which we have already met, took into account interaction effects between
the in-plane normal stresses:
2 2 2 2
⎡ σ1 ⎤ ⎡ σ1σ2 ⎤ ⎡ σ2 ⎤ ⎡ τ12 ⎤ ............................................................................ [4.38]
⎢ ⎥ +⎢ ⎥ + ⎢ ⎥ + ⎢ ⎥ =1
⎣ σc ⎦ ⎣ σc σt ⎦ ⎣ σt ⎦ ⎣ τc ⎦

although in many situations the second term is relatively small and is often ignored, when the equation
reverts to equation 4.39. Like the maximum-stress theory, these quadratic models are also extendible
__________________________________________________________________________________
* Failure criteria in fibre-reinforced polymer composites, Editors MJ Hinton, P Soden and AS Kaddour, Comp Sci &
Technol, 58, July 1998, 999-1254.


into the second, third and fourth quadrants of the (σ1,σ2) plane-stress framework, provided appropriate
values of the compression strengths are used. Predictions of strength for off-axis loading can be obtained
by substituting the stress definitions used for the maximum-stress model (equations 4.16-4.18) into
equation 4.39, when the form of the maximum-strain-energy criterion given in equation 4.19 is
recovered.

Eckold et al (1978) have developed a laminate theory for the prediction of failure envelopes for
filament-wound structures under biaxial loads. They predict failure, from a maximum-stress criterion
rather than an interaction model, in a ply-by-ply analysis which allows for progressive failure of the
laminate, for shear non-linearity, and for different properties in tension and compression. The composite
is assumed to be composed of a series of homogeneous orthotropic plies, as is usual in laminate theory
calculations. Good agreement between predictions and experimental results was obtained.

Although single-expression models like equation 4.39 are easy to work with mathematically, they are
subject to the very serious criticism that they are truly appropriate only for homogeneous materials and
do not take account of the different physical failure modes that occur in composites as the relative
orientations of the stress and symmetry axes change. Homogenisation methods of this kind result in
single elliptical failure envelopes which can apparently be defined from only three or four data points
obtained under straightforward experimental conditions. But they are incapable of distinguishing
between fibre and matrix failures, or allowing for alternative matrix damage processes, such as cracking
and ductile shear failure. Hart-Smith has been particularly vociferous in condemning the quadratic
models used by Tsai and his collaborators (and embodied in most of the currently available computer
design software). He maintains (1991) that it is scientifically incorrect to use polynomial interaction
failure models whenever the mechanism of failure of the laminate changes with stress and maintains that
separate failure criteria are needed for the fibres, the matrix and the interface. He has argued (1991, for
example) for the adoption of a maximum-shear-stress (Tresca) model because it automatically introduces
truncations related to changes in failure mode which the quadratic models do not, and satisfactorily
explains in-plane laminate test data in all of the stress quadrants.

4.6 THE STRENGTH OF CERAMIC-MATRIX COMPOSITES
Because of the brittleness of the matrices of CMCs, their response to stress differs considerably from
that of reinforced plastics and metals. The matrix cracks at low stresses relative to the final failure stress,
gradually transferring all responsibility for load bearing to the reinforcing fibres. Stress/strain curves for
some unidirectional and cross-plied (0,90)3S laminates of SiC/CAS are shown in Figure 4.41. The right-
hand section of this figure shows the normal axial stress/strain curves, while the left-hand section shows
the variation of the transverse strains during
loading. 500

The axial stress/strain relationships for 400 (0) 12
the unidirectional and (0,90)3s laminates
Stress, MPa

show similar features, with a predominantly 300
linear initial portion, a marked knee, an
almost linear second stage with a much 200
reduced slope, followed by a third non- (0,90) 3S
linear region in which the average slope
100
increases again. The knee indicates the onset
of major matrix cracking.
0
-0.5
10 x ε 22 0 0.5 1.0
ε 11
The transverse strains exhibit unusual
variations: they begin in the normal manner Longitudinal and transverse strains, %
by showing a reduction in lateral dimension
that is proportional to the load — normal Figure 4.41. Longitudinal and transverse strains as a function of
Poisson contraction — but the direction of stress for unidirectional and cross-plied composites of SiC/CAS
this strain is subsequently reversed and (Harris et al, 1992b). The transverse strains are multiplied by ten to
increase visibility on the graph.


eventually becomes positive. This type of behaviour has been observed for a number of CMCs and
although various explanations have been put forward it seems likely that the reversal is due to a gradual
release of residual thermal stresses as matrix cracking develops.

From an energy-based model, Aveston et al (1971) and Budiansky et al (1986) show that the stress at
which matrix cracking should occur, σmc, is given by:
1
⎡ 12τi γ m Vf2E f (1 − ν c )2 ⎤ 3 Ec ............................................................... [4.39]
2
σmc =⎢ ⎥
⎦ (1 − ν c )
2 2
⎣ rVmEcEm

where τi is the interfacial shear strength, r is the fibre radius, and νc is composite major Poisson ratio.
The two unknowns in this equation are γm, the matrix fracture energy, and τi, the interfacial shear
strength. Estimates of the value of τi are often found from observations of crack spacing in damaged
composites which, in a brittle/brittle system, is inversely related to the strength of the interfacial bond.
When the fibres have a greater failure elongation than the matrix, which is almost always the case in
CMCs, multiple cracking of the matrix occurs during loading until a limiting crack density is reached
where the crack spacing is between x´ and 2x´. The condition for multiple fracture is that:

σfu > σmu Vm + σf′ Vf ............................................................................................. [4.40]

where σmu is the matrix tensile strength and σf′ is the fibre stress at the matrix failure strain. The crack
spacing is obtained from a force balance between tensile force in the fibres and shear force at the
interface as:

⎛V ⎞ ⎛σ r⎞
x′ = ⎜ m ⎟ . ⎜ mu ⎟ ............................................................................................ [4.41]
⎝ Vf ⎠ ⎝ 2τi ⎠

σmu being the failure stress of the unreinforced matrix and r the fibre radius. The cracking stress
predicted by equation 4.41 is often very low by comparison with the normal composite tensile strength.
For example, Evans et al (1991) quote a range of 140-160MPa for SiC/CAS composites for which the
tensile strength may well be of the order of 1GPa, and acoustic emission studies have shown that
substantial sub-critical matrix cracking occurs at stresses well below the level predicted by equation 4.41
(Harris et al, 1992b).

The ACK model was used to predict the stress/strain curves for a number of fibre-reinforced cement-
matrix composites with a considerable degree 600
of accuracy (Aveston et al, 1994). The ud SiC/CAS data σfuVf
predicted curve follows three distinct, linear 500 ACK model
stages as shown in Figure 4.42. The first stage slope = EfVf
has a slope which is given by the usual rule of 400
Stress, MPa

mixtures, Ec = Ef Vf + Em(1 - Vf), and rises εmu

to a stress level σmc at the first matrix cracking 300
Eoεmu
strain, εmc. The second stage is horizontal and
200 εmu(1 + α/2)
corresponds to a strain range during which
matrix cracking continues at constant stress
100
until a saturation level is reached at a strain slope = EfVf + EmVm (mixtures rule)
εmc(1 + 0.66α), α being the weighted modulus 0
ratio, EmVm/EfVf. The plateau joins the third 0 0.2 0.4 0.6 0.8 1.0
section which is of slope EfVf, representing a Strain, %
fibre-only contribution, and the failure point is
reached at a stress level σfuVf. Figure 4.42. Illustration of the model of Aveston et al (1974)
to predict the shape of the stress/strain curve for a ceramic-
Not all CMC stress/strain curves are so matrix composite. α is the ratio Em Vm/Ef Vf.


well predicted by the ACK model, however. Figure 4.42 also includes the stress/strain curve for the
unidirectional SiC/CAS laminate referred to in Figure 4.41. It can be seen that matrix cracking, signalled
by the deviation from linearity of stage 1 of the curve, starts well below the predicted level and the stress
continues to rise as the cracking proceeds, rather than remaining constant. The slope of the third stage of
the curve falls well below the predicted fibre-only contribution, and final failure occurs below the
predicted failure stress. Possible explanations of this are that the fibres have been damaged by the
manufacturing process, or sustain damage as matrix cracking proceeds, perhaps as a result of dynamic
crack-growth effects not envisaged in the ACK model.

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5 FRACTURE AND TOUGHNESS OF COMPOSITES

5.1 STRUCTURAL ASPECTS OF COMPOSITES FAILURE

M
any reinforced plastics consist of brittle fibres, such as glass or carbon, in a weak, brittle
polymer matrix, such as epoxy or polyester resin. An important characteristic of these
composites, however, is that they are surprisingly tough, largely as a result of their
heterogeneous nature and construction. During deformation, micro-structural damage is
widespread throughout the composite, as we saw in chapter 4, but much damage can be sustained before
the load-bearing ability is impaired. Beyond some critical level of damage, failure may occur by the
propagation of a crack which usually has a much more complex nature than cracks in homogeneous
materials. Crack growth is inhibited by the presence of interfaces, both at the microstructural level
between fibres and matrix, and at the macroscopic level as planes of weakness between separate laminae
in a multi-ply laminate. The fracturing of a composite therefore involves not only the breaking of the
load-bearing fibres and the weak matrix, but a complex combination of crack deviations along these
weak interfaces. For similar reasons, even all-brittle ceramic/ceramic composites can also exhibit
toughnesses which are orders of magnitude higher than those of unreinforced ceramics.

The microstructural inhomogeneity and anisotropy of fibre composites are together responsible for
the fact that the fracture of these materials is rarely a simple process. Although the complex combination
of micro-failure events that leads to deterioration of load-bearing ability, or to destruction, can often give
rise to surprisingly high levels of toughness, the same complexity makes it difficult or impossible to use
procedures based on fracture mechanics for design purposes. There have been many theoretical and
experimental studies of cracking in composites and of the mechanisms by which toughening is achieved,
but there is still a large measure of disagreement about the contributions to the overall toughness of the
various processes by which cracks are stopped or hindered. This is probably because of the great
diversity of composite types that have been studied. The toughness of a composite is derived from many
sources, and the relative magnitudes of the separate contributions depend not only on the characteristics
of the components, but also on the manner in which they interact. There is thus no simple recipe for
predicting the toughness of all composites. The lay-up geometry of a composite strongly affects crack
propagation, with the result that some laminates appear to be highly notch-sensitive whereas others are
totally insensitive to the presence of stress concentrators. The selection of resins and fibres, the manner
in which they are combined in the lay-up, and the quality of the manufactured composite must all be
carefully controlled if optimum toughness is to be achieved. Furthermore, materials requirements for
highest tensile and shear strengths of laminates are sometimes incompatible with requirements for
highest toughness. Final selection of a composite for a given application is often therefore a matter of
compromise.

In this chapter we examine the cracking of a composite at various structural levels. We begin by
considering how the behaviour of a crack in the matrix phase is modified by the presence of reinforcing
filaments and how the mechanical characteristics of the fibres themselves affect cracking of the
composite. These effects are distinct from those related to true composite action in which it is the
interfacial discontinuity between fibres and matrix that controls crack propagation. At the macroscopic
level there are other discontinuities, the interfaces between laminae for example, or the resin-rich zones
around the boundaries of fibre tows, and these discontinuities also affect crack growth.

5.1.1 Fracture formalisms
The Griffith model of fracture of cracked brittle solids (1920) has had far reaching consequences for
designers, specifically in having laid the foundations of modern fracture mechanics as couched in terms
of the familiar stress-based stress-intensity factor, K, and the associated energy-based parameter, G, the
strain-energy release rate. With appropriate modifications to the basic concepts and definitions of
Griffith (see, for example, Atkins and Mai, 1985; Broek, 1986), we arrive at design methods that can
reasonably account for modest levels of non-linear, inelastic behaviour in a variety of homogeneous

chapter five Toughness 100

brittle and semi-brittle materials — ceramics, metals, and even plastics. It was natural, therefore, that
many attempts should have been made to apply the same concepts to fibre composites in order to obtain
design data that could be directly compared with those already in use for conventional engineering
materials. And, despite the limitations imposed by the microstructural complexity of composites, by the
scale of those structural features relative to the crack geometry, and by the distributed patterns of
damage, there has been a measure of success. However, it is now apparent that the stress-intensity factor
is not an appropriate design parameter for use with fibre composites, except for situations where a
delamination crack spreads between the plies of a conventional laminate (see Hashemi et al, 1990, for
example).

The stress approach is dubious because it is unable to account for mechanistic input in the way that
an energy-based approach can, and perhaps the most successful early studies of composite toughness
have been those based on the Orowan fracture-energy development of the Griffith model (see McGarry
and Mandell, 1972, and Phillips, 1972, for example). Measurements of the fracture energy, or work of
fracture, γF, provided information which related in a fairly obvious way to the observed fracture
processes, permitted comparisons of the performance of different materials, and encouraged deductions
to be made as to the likely effects of materials/processing parameters on the behaviour of a given
composite system. Unfortunately, although some early attempts to correlate γF with the conventional
fracture toughness parameters were moderately successful, γF cannot be thought of as a design parameter
although it fulfils an important rôle in furthering our understanding of composites fracture processes.
The fracture energy idea has also led to the development of the concept of crack-resistance curves (R
curves) which have particular relevance to materials like composites where sub-critical cracking and
progressive failure are common features (Gurney and Hunt, 1967; Mai et al, 1976). In this book, in order
to preserve the analogy with the Griffith surface energy, γS, we define γF as the total energy absorbed
during the complete failure process divided by twice the sample cross-sectional area: it is thus half the
value of R or G, as usually defined.

5.2 FRACTURE PROCESSES IN COMPOSITES
As a starting point, we take the simple, Griffith planar-crack model and the knowledge that during
crack extension an irreversible process takes place in which the net change in the stored elastic energy of
the whole system (testing machine plus sample) caused by an increment of crack growth serves to
provide the energy, γS, needed to produce new surfaces plus the energy, γF, required to activate any other
deformation or damage mechanism that must operate in order for the crack to grow. The second of these
contributions is minute in bulk glass where crack-tip plastic deformation is very limited (though still an
order of magnitude greater than γS,) but is much greater in metallic or polymeric solids where extensive
plastic or viscoelastic deformation may lead to the development of a sizeable plastic zone ahead of the
crack.

This concept of a ‘process’ zone near the crack tip in which we encounter a variety of deformation or
failure events all drawing on the energy of the system is familiar in ceramics where the events in
question may include a distribution of subcritical micro-cracks, crack deviation, and crack blunting.
Ceramic composites are designed to add other, perhaps more effective, damage mechanisms. An
example of a particulate ceramic composite is the family of partially stabilized zirconias, or PSZ
materials which, on account of their improved intrinsic toughness, may provide a better starting point as
a matrix for subsequent fibre reinforcement in CMCs (Evans et al, 1981; Garvie, 1985).

5.2.1 Matrix effects
In practical fibre composites the proportion of fibres present will vary between about 10% (in DMCs,
for example) and 70% (in high-performance CFRP). Crack-resistance effects directly attributable to the
matrix material may not therefore be significant except perhaps in composites reinforced with relatively
low volume fractions of short fibres. Three specific matrix effects might be listed, however:

i) The effective toughness of a non-brittle matrix, such as a metal or thermoplastic, will be
reduced by the presence of a high Vf of rigid, brittle fibres or particles as a result of plastic


constraint which leads to triaxial tensile stress components in the matrix. A cermet, or metal-
bonded ceramic, is a good example of this.
ii) The effective toughness of a soft, flexible matrix may be increased by the presence of a low Vf
of rigid particles or fibres because a high overall stress level on the composite may then be
required to generate a critical degree of crack opening in the stiffened matrix (Harris, 1980).
iii) The effective toughness of a low-toughness matrix may be increased by the presence of fibres
or particles as a result of the slowing up of cracks in the neighbourhood of the filler. The
toughness of certain brittle plastics for example, shows a strong dependence on the crack
speed, often linked to changes in the crack face roughness — the slower the crack, the rougher
the crack face and the higher the associated work of fracture (Harris, 1980).

5.2.2 Fibre effects
Certain types of fibres such as the drawn steel wires used for reinforcing concrete, the polyester
textile fibres used to reinforce rubber tyres, or the aromatic polyamide fibres used in high-performance
laminates, may be classified as ‘tough’ because they are capable of extensive non-elastic deformation
after yielding. Familiar mechanisms of deformation in such fibres are plastic necking in metal wires and
drawing or inter-fibrillar splitting in polymer fibres. Such fibres often show high tolerance of detects and
surface damage, by contrast with brittle fibres like glass or carbon. A bundle of such fibres possesses a
large fracture energy, and a substantial portion of this fracture energy can be transferred into a composite
containing the bundle.

Fibres such as glass, carbon and boron, which
have high breaking strengths and failure strains
(when undamaged) have very low intrinsic
fracture energies, of the order of only 10-100
Jm-2, and their failure is governed by the
statistical nature of flaw distributions. These low
fracture energies make no direct useful
contribution to the composite work of fracture,
even though each fibre may break many times
(multiple cracking). But as a result of this
multiple cracking, damage is widespread, crack
paths in composites become complex, and it is
this that results in very high values of the work Figure 5.1. Schematic illustration of the way in which fibres
of fracture. in a plastically deforming metal-matrix composite constrain
the plastic deformation of the matrix.
5.2.3 Simple fibre/matrix addition
effects
When fibres are incorporated into a matrix of any kind the separate phases may be able to contribute
their individual levels of toughness to the composite in an additive fashion, or they may not, depending
on the kind of interaction that occurs and on the level of constraint that is set up as a result of the
differences in their properties. Early studies (Cooper and Kelly, 1969; McGuire and Harris, 1974) of
metal-matrix composites in which the reinforcements were also metallic, showed that the specific
fracture energy of the composite, (γF)c, could be obtained as a mixture-rule sum of the fracture energies
of the individual components:

(γF)c = (γF)m(1 – Vf) + (γF)fVf ...............................................................................[5.1]

where the subscripts c, m and f refer, as usual, to the composite, matrix and fibres, provided account was
taken of the fact that the presence of the stiffer fibres caused the extent of matrix plastic deformation to
be localized in the neighbourhood of the crack plane. Typically, a metal filament like tungsten or cold-
drawn steel undergoes plastic deformation which is concentrated within a volume of the order of one
fibre diameter, d, in height on either side of the final break, as illustrated schematically in Figure 5.1. If
the fibre failure stress and strain are σfu and εfu, respectively, the fibre fracture energy is therefore


roughly equal to 2d.σfuεfu.πd2/4 per fibre, which amounts to 2dVf σfuεfu for the total fibre contribution.
The matrix contribution is likewise obtained from the stress/strain product for the matrix, σmuεmu, and the
deforming volume which is determined by the geometry of fibre/matrix distribution and the size of the
deformation zone in the fibre. An estimate of this component (Cooper and Kelly, op cit) gives d(1 –
Vf)2Vf–1σmuεmu, and the total work of fracture for the composite is then:

(γF)c = 2dVf σfuεfu + d(1 – Vf)2Vf–1σmuεmu, ............................................................... [5.2]

The terms in d and Vf modify the plastic work terms (stress x strain) to take into account the
localization of the plastic deformation. Both matrix and fibres are subject to large local deformation, and
debonding may also occur. This decohesion relaxes some of the triaxiality that would otherwise prevail
at the interface. Failure to debond would cause conditions
approaching plane strain throughout the composite and failure
surfaces would have the appearance of a flat brittle-looking
failure rather than the tougher, pseudo-plane-stress tensile
failure of the locally separated components shown in Figure
5.2. There is also evidence that the strain to failure of a metal
wire embedded with good wetting in a ductile metal matrix is
rather higher than that of an identical isolated wire, perhaps
because a strong fibre/matrix bond delays the onset of
localized necking in the fibre. Estimates given by equation 5.2
are in reasonable agreement with measured values in
composites where the interfacial bond is not very strong so
that decohesion is able to occur and the two components can
deform in an unconstrained fashion.

Modern composites rarely contain metallic reinforcements,
however, and the more usual situation is that the stiffness
Figure 5.2. Tensile failure surface of a ratio, Ef/Em, is very high. This reduces the deforming volume
tungsten-wire/aluminium composite, showing
plastic deformation of both fibres and matrix of even a highly deformable matrix and limits the extent to
(McGuire, 1972) which it can contribute its plastic work to the toughness of the
composite. When we consider that a composite such as a
GRP, in which both fibres and matrix are brittle (with fracture energies of the order of 0.5 and 100Jm-2,
respectively), may exhibit a work of fracture of the order of 105Jm-2, it is clear that estimates based on
the intrinsic toughnesses of the components have no relevance to real behaviour. It is in the special
nature of composite action that the behaviour of individual components is dramatically modified; new
damage mechanisms are introduced which drain energy from the system and in doing so contribute to
the composite toughness, or crack-growth resistance. This is one reason for the success of GRP and other
high-performance reinforced plastics, and it is the driving force for the intensive search for toughness in
ceramic-matrix composites.

5.3 TOUGHNESS DERIVING FROM COMPOSITE ACTION
5.3.1 Toughening mechanisms in fibre-reinforced plastics
We have seen, for a simple case, how estimates of the energy required for deformation mechanisms
associated with the matrix and fibres provide reasonable models of macroscopic toughness. In real fibre
composites, however, the microstructural inhomogeneity and anisotropy cause the failure process to be
very complex, and the combination of microfailure events — sometimes interacting, sometimes not —
that leads ultimately to ‘destruction’ can often give rise to high levels of fracture energy. We should be
clear that by using the γF definition of toughness we are begging a number of questions. The engineer
used to dealing with K or G will usually be interested only in events up to maximum load. Unlike
metals, most high-performance composites do not exhibit work-hardening phenomena and the load-
bearing ability of a component may fall quite rapidly after peak load, even though the deterioration
process may be gradual, and much energy may be required to complete the destruction process, as
illustrated schematically in Figure 5.3. Such a composite would have a high work of fracture, and be


valuable for many applications where
energy absorption rather than load-
bearing ability (or strength) is called for.
There are also composites in which, as
in plastically deforming metals, a
process zone develops and increases in
size as a single crack grows and the
‘toughness’, now defined as a crack-
growth resistance, R, increases.

Low-energy failures of the kind
implied by a rule-of-mixtures equation
(5.1) for a brittle/brittle composite rarely
occur in modern engineering laminates Figure 5.3. Schematic stress/strain curves illustrating different
except, for example, in certain types of classes of brittle/tough behaviour.
GRP under the combined action of stress
and an aggressive environment (stress corrosion, as in Figure 5.4, for example), although it was a
common feature of early CFRP in which the commercial fibre surfaces were often over-treated. Such
failures are rare, partly because the interface will usually modify the mode of crack propagation and
partly because of the statistical variation of fibre
strengths. At any fibre break the load is shed back, via the
matrix, to the neighbouring fibres, so that the stresses in
these adjacent fibres will be concentrated somewhat
above the average fibre stress level, as illustrated by
Figure 4.8 in chapter 4. It is statistically unlikely,
however, that there will be weak spots at the same points
in these neighbouring unbroken fibres, and in the early
stages of deformation fibre failures are widely distributed
throughout the stressed volume. If the fibre/matrix bond
is not too strong some relaxation of the stresses around
the broken fibre ends can occur and this, together with
local creep relaxation in the matrix, will help to reduce
Figure 5.4. Brittle fibre-to-fibre failure in a high- the level of stress concentration in neighbouring fibres.
performance GRP pultruded rod exposed to stress-
corrosion conditions in dilute hydrochloric acid Thus, in many modern composites a great deal of fibre
solution. damage may occur before the number of fibre breaks in
any cross-section reduces the load-bearing ability of that
cross-section below the current level of applied load. At this point we have, effectively, a crack nucleus
of critical size which could propagate rapidly to cause failure of the composite, and the simple cross-
section argument should be modified to include the effect of the crack-tip stress concentration. If the
fibre/matrix bond is too strong, however, this prevents relaxation, so that failure of whole bundles of
fibres can occur, the crack spreading from one fibre to the next via the matrix with little or no deviation
at the interface, in the brittle mode described earlier. A detailed study of tensile failure due to inter-fibre
stress transfer within the axial plies of notched laminates has been made by Potter (1978).

Practical fibre-reinforced plastics consist of relatively brittle resins (γF values of a few hundred to a
few thousand Jm-2) reinforced with strong, brittle fibres which may be very stiff (eg. carbon) or less stiff
(eg. glass or aramid polymers). The failure strains of these fibres are generally less than that of the
matrix. The fibres may be continuous and straight, continuous and woven, or chopped and randomly
dispersed. When a continuously reinforced composite laminate is loaded, the weaker fibres in the
distribution begin to break at relatively low loads, and this damage is distributed throughout the material,
as illustrated in chapter 4. A brittle fibre fracture is a process which requires an amount of energy
½σfu εfu per unit volume. If we allow all of the fibres in a 0.7Vf carbon-fibre composite to break once,
with a deforming volume of the order of d3, their net contribution to the composite fracture energy can
only be of the order of 1Jm-2. Because of the statistical distribution of fibre strengths, each fibre may
break many times before the composite fails, but even if, in a 10cm long sample, say, each fibre could be


completely broken down to pieces of near
critical length (about 0.5mm in a CFRP), the
total available fracture work from this source
still amounts to only a few hundred Jm-2.

In order to distinguish the separate micro-
mechanisms of toughening it is convenient to
consider a rather artificial model in which a elastic
strain
crack travelling in the matrix approaches an in fibre
isolated fibre (Figure 5.5).The crack is
effectively halted by the fibre (picture b), first,
because the higher stiffness of the fibre inhibits
further opening of the matrix crack, and second,
because the strength of the fibre is too high to
permit of its being broken by the level of stress
currently concentrated at the tip of the matrix
crack. Further crack opening may occur if the
local shearing force acting at the fibre/matrix
Figure 5.5. Schematic illustration of the stages in crack
interface is sufficiently high to allow the fibre to growth in a fibre composite (see text for explanation).
become locally debonded. This debonding is
also encouraged by the fact that there may be a significant level of transverse tensile stress concentration
ahead of the crack tip, parallel with the crack plane, as shown by the work of Cook and Gordon (1964).

The overall consequence is that debonding may occur, the debonded fibre will extend elastically, and
subsequent further crack opening can take place during which the matrix slides relative to the fibre
(stage c): all of these processes require energy. The crack may progress past a number of fibres in this
3 way, leaving them unbroken in its wake and for
fibres: a given fibre/matrix/interface system there may
be some equilibrium state where a stable
spacing = 5mm
number of these bridging fibres continue to
K, MPa.√m

2 support part of the load, modifying the
thermodynamic driving force for crack growth.
This fibre bridging is thus a further toughening
mechanism. Its effect is illustrated in Figure
1 5.6. These are results of a simple double-
plain resin
cantilever beam (DCB) experiment on a
polyester resin sample containing a single array
of glass fibres spaced as indicated in the figure
0
0 50 100 150 200 (Harris and Ankara, 1978). The crack
propagates through the array, and the unbroken
Crack length, mm
fibres behind the crack tip provide a closure
force which opposes the applied stress, so
Figure 5.6. Plot of K against crack length during a test on a
double-cantilever beam sample of polyester resin containing
reducing the driving force for crack growth, an
a linear array of glass fibres 5mm apart (Harris and Ankara, idea proposed very early on by Romualdi and
1978). Batson (1963) for the toughening effect in
fibre-reinforced cement. The apparent
toughness is defined in the graph in terms of a notional ‘stress-intensity factor’, K = σ√a, a being the
length of the crack, and σ the applied stress.


Figure 5.7. Photographs of fibre pull-out in a carbon-fibre-reinforced epoxy composite (right) and a SiC-fibre-
reinforced glass-ceramic. The magnifications may be judged from the fibre diameters, carbon 7μm and SiC 15μm

The extent of this increase in crack resistance as the crack extends, which is commonly termed ‘R-
curve behaviour’, may be limited or extensive, depending on the material. At some point, however, the
rising load on a bridging fibre must reach a level at which the fibre will break (stage d in Figure 5.5).
During loading, the fibre will have been released by Poisson contraction from contact with the matrix
over a debonded length that will depend on the interfacial bond strength, and a weak spot may be
encountered anywhere within this length (or just outside it, where contact and frictional grip is being re-
established). When the fibre failure occurs away from the crack plane, the stored elastic energy is
released and the fibre expands to regain contact with the matrix. Many of the broken fibres will therefore
remain bridging the crack, and the embedded ends must be pulled out of the matrix holes in order for
further crack opening to occur (stage e). This fibre ‘pull-out’ makes further demands on the energy of
the system, and leads to the characteristic brush-like failures of many types of composite, as shown in
Figure 5.7. Since the earliest years of research on fracture-energy mechanisms in composites, there have
been strong disagreements about the major contributing factors. McGarry and Mandell (1972), for
example, were categoric that the primary source of toughening in glass-fibre/resin composites was the
work necessary to break the fibres in a damage zone at the crack tip, and that contributions from the
matrix toughness, fibre debonding or splitting, and fibre pull-out were negligible. Outwater and Murphy
(1969) were equally convinced that it was the debonding process that was the major contributor, while
other researchers believe that fibre pull-out is dominant (Harris et al, 1975). The truth is not easy to
unravel, especially when several mechanisms, which may be inter-dependent, may be making similar
contributions, and it is in any event clear that different situations occur in different kinds of composites.
We now examine the major mechanisms in a little more detail in order to see how simple energy
considerations may offer guidance in designing composite materials.

5.3.1.1 Fibre/resin debonding
Glass fibres are coated prior to incorporation in the resin matrix with a complex mixture that includes
a coupling agent, usually an organo-silane, the purpose of which is to encourage wetting and the
establishment of a direct chemical bond between the glass surface and the resin molecular network. No
such direct bond can easily be produced between a carbon fibre and the resin, so that the coating which
is put on carbon fibres is merely a size which assists handling and encourages wetting-out during
composite manufacture. However, carbon-fibre surfaces are also usually treated, by chemical,
electrochemical, or plasma methods, to modify their surface chemistry and so further enhance wetting.
Depending on the type of fibre, then, and the type and level of coating or surface treatment, the degree of
actual chemical bonding between fibre and resin and the efficacy of any purely physical bond may vary
immensely from system to system. There is uncertainty about the exact nature of the interface in
fibre/resin composites (Drzal et al, 1982, 1983; Kardos, 1985) and in this discussion of debonding we
shall therefore adopt a very simple approach.


As discussed in section 4.3.1, when the end of a
fibre, of length less than half the critical transfer
length, lc, which is embedded in a block of cured
resin, is loaded by pulling (Figure 5.8a), the load
increases to some maximum level at which the
induced shear forces acting at the interface are able
to initiate a debond and the load drops to some lower
level as the debond rapidly propagates. The peak
load must reflect the strength of the combined
chemical and physical bond, and the shaded
triangular area on the diagram in Figure 5.8b
represents the elastic energy that was needed to
cause the debond, ie. the debond energy. The fibre is
not free, however, because the cured resin has
Figure 5.8. A simple pull-out experiment. As the
shrunk onto the fibre during the curing process and
embedded fibre is loaded (a), the resulting
during cooling from the cure temperature. As a load/deflexion curve frequently resembles that shown
consequence, further work is needed to withdraw the in (b).
fibre completely from the resin block, and the full
loading curve is as shown in Figure 5.8b.

The magnitude of the debonding energy can be roughly assessed (Harris et al, 1975) by assuming
that it is equivalent to the energy released when the fibre finally breaks. If a fibre debonds to a total
distance ½ldb on either side of the crack face, the stored elastic energy released at fibre fracture is likely
to be somewhat greater than ½(σ2fu /Ef).(πd2/4).ldb. If N fibres break in the cross section of the
composite, the debonding contribution to the composite fracture work, wdb, normalized with respect to
the sample cross-sectional area, 2A, is:

w db Nπd2σ2 l db .................................................................................................[5.3]
= fu

2A 16AEf

and since the fibre volume fraction Vf = Nπd2/4A, we obtain the debond component of the work of
fracture:

Vf σfu l db ..............................................................................................[5.4]
2
(γ F )debond =
4Ef

This is an under-estimate, since it assumes that the fibre/resin interfacial shear strength falls to zero after
debonding and that the fibre is not strained outside the debonded region, neither of which is likely to be
valid.

In their early attempt to assess the work of debonding, Outwater and Murphy (1969) treated the
problem as a fracture-mechanics case of the propagation of a mode-II shear crack along a cylindrical
interface. For a long fibre of diameter d embedded in a block of matrix, as in Figure 5.8a, and debonded
over a distance x from the surface, the stress needed to continue the debonding process is given by:

4 τx 8E f GIIi
σ fu = + ........................................................................................[5.5]
d d

where τ is the sliding friction stress operating between fibre and matrix in the region already debonded,
and the second term is the energy required to break the bond, GIIi being the fracture surface energy of the
interface in shear. The condition for the start of debonding is given, when τ = 0, as:


8E f GIIi
σ fu = .....................................................................................................[5.6]
d

It is evident that there is a limiting fibre diameter, 8EfGIIi/σ2fu, equivalent to equation 5.3 modified so
as to relate to the debonded surface area of the fibre, above which the fibres will break before they can
debond. In such circumstances a crack would propagate cleanly through fibres and matrix, with no
deviation, resulting in a flat fracture face, such as that shown in Figure 5.9, and a very low fracture
energy. Such a situation could occur if the fibre/matrix
bond were excessively strong, and fibre surface
treatments therefore need to be carefully controlled so
that they are neither too weak (low G → low composite
toughness) nor too strong (no debonding → low
toughness). The interface would be likely to have a
fracture energy intermediate between that for a resin
fracture and that for the failure of a glass/resin adhesive
bond, say a few hundred Jm-2. But Outwater and Murphy
measured values of about 4kJm-2 for GIIi for
glass/polyester models, which suggests that simple
mode-II cracking is not occurring during the debonding
process. Kelly (1970) considered the debonding energy
to be small, although more recent work by Wells and Figure 5.9. Flat surface topography of a typical brittle
Beaumont (1985) suggests that it could be a major composite fracture. The material is a unidirectional
contributor to toughening. carbon/epoxy composite containing plasma-treated
fibres.
The effect of debonding is to permit a large increase
in the volume of fibre that is under stress and, therefore, the amount of stored elastic energy that is
subsequently released when the fibre fails. Harris (1980) suggested that it may well be this consequence
of debonding rather than the debonding itself that adds substantially to the composite fracture energy. It
has also been suggested that in glass/polyester model composites the overall fracture energy is relatively
little affected by the ease or difficulty of debonding (Harris and Ankara, 1978), presumably because the
area under the tail of the load/deflexion curve in Figure 5.8b is usually substantially greater than that in
the debonding triangle indicated.

5.3.1.2 Frictional work following debonding
As implied by equation 5.5, after debonding has taken place the fibres and matrix move relative to
each other as crack opening continues, and work must be done against frictional resistance during the
process. The magnitude of this work is hard to evaluate because we do not know with any certainty what
level of friction force is operating. The work expended will result from the action of the interfacial
frictional force over a distance equal to the differential displacement of fibre and resin which Kelly
(1970) suggested was roughly the product of the debonded length and the differential failure strain,
ldbΔε. In early composites, the matrix failure strains were usually much less than those of the fibres, and
Δε could be taken as being approximately equal to εfu. If the initial friction force, τi πd.½ldb, acts in each
direction from the crack face over a distance ½εfu ldb, the post-debond friction work is (Harris et al,
1975):

⎛ 2τ πdl db ⎞⎛ ε ful db ⎞ .................................................................................[5.7]
w pdb = ⎜ i ⎟⎜ 2 ⎟
⎝ 2 ⎠⎝ ⎠

per fibre. Normalizing again with respect to the sample cross-sectional area, the contribution to the work
of fracture is therefore:

Nτi πdl 2 ε fu
( γ F )post −debond = db
..............................................................................[5.8]
4A


for N fibres in the cross-section, and again converting to Vf, we have:

Vf τil2 εfu
( γF )post−debond = db
..................................................................................... [5.9]
d

Substituting reasonable values for the parameters (Vf = 0.6, τi = 40MPa, ldb ≈ 10d to 50d, d = 7μm, and
εfu ≈ 1.5%), the contribution varies between 250Jm-2 to 6.3kJm-2. For modern fibre/resin combinations
with toughened resins, the resin failure strain may well be much closer to that of the fibre, and the net
contribution to toughening from this source would then become almost insignificant. It is interesting to
note that Marston et al (1974) had some success in attempting to account for the fracture energies of
various types of composite by associating the surface energy of the debonded interface with the actual
debonded surface area. They generalized the Outwater debonding model and showed how interfacial
shear strength might be tailored to obtain reasonable levels of toughness while maintaining rule-of-
mixtures strengths in boron/epoxy and carbon/epoxy composites.

5.3.1.3 Fibre pull-out
The pull-out work has long been considered to be of great significance in providing toughness in
composites, and the earliest solutions were given by Cottrell (1964) and Kelly (1973). When a fibre end
of length x is pulled from the matrix, the effective friction force, τi πd(x/2), acts over the distance x to do
work ½τiπdx2. Since x is normally considered to vary between 0 and ½lc (where lc is the critical fibre
length), the mean work done in pulling N fibres out to a maximum distance ½lc over the composite cross
section is:
lc / 2

∫ x 2dx
w pull−out = 1
2 Nπdτi 0 .............................................................................. [5.10]
lc / 2

∫
0
dx

The pull-out contribution to the work of fracture is then:

Nτi πdl 2 Vf τil 2
( γF )pull−out = c
= c ............................................................................[5.11]

48A 12d

The relationship given in section 4.1.9 of chapter 4:

l c σfu
=
d 2τi

between fibre strength, σfu, effective interfacial bond strength, τi, fibre critical length, lc, and fibre
diameter, d, which was derived by Kelly and Tyson (1965) for metallic composites without interfacial
chemical bonding, may be used to write equation 5.11 in slightly different form:

Vf σ fu l c
( γ F ) pull − out = .........................................................................................[5.12]
24

For typical values (Vf = 0.6, σfu = 3GPa, lc ≈ 10d to 50d), this amounts to between 5 and 25kJm-2,
clearly a potentially powerful source of toughening. We see from equations 5.11 and 5.12 that control of
the interface characteristics in a composite of given fibre/matrix composition is vitally important in
designing toughness into a system. An illustration is shown in the results of some crack growth
experiments of the kind referred to earlier (cf. Figure 5.6) in which the surfaces of glass fibres were
treated in various ways prior to embedding in polyester DCB samples. The curves in Figure 5.11 again
show the increase in apparent toughness as a crack passes through an array of fibres (R curves), and it


can be seen the results are not entirely in accord with conventional ideas. It is true that all of the fibre-
containing samples develop crack resistances greater than that of the plain resin. But the toughening
effect is considerably greater when the fibre surfaces are pristine than when they are coated with a silane
coupling agent, applied in practice to improve the fibre/matrix bond, and the highest toughness is
obtained with a fibres that have been damaged by environmental contamination prior to embedment in
the resin.

3 The parameters σfu, τi and lc are not
damaged fibres independent, and there are limits to the extent to
clean fibres which the interfacial bond strength and the pull-
silane-coated out length (which is related to lc) can be varied. It
Keff, MPa√m

2 fibres has been shown, for example, that as the
fibre/resin bond strength is gradually increased by
fibre surface treatments, the work of fracture first
increases and then decreases as the initial
1
improvement with increasing τi is replaced by a
plain resin deterioration as lc becomes very small. This is
illustrated in Figure 5.10 by some early results for
0 carbon/polyester composites in which the fibres
0 5 10 15 have been exposed to a range of chemical
Crack length, cm
treatments which have modified the strength of the
interfacial bond. The effects of those treatments
Figure 5.11. Notional stress-intensity factor, Keff, as a
are also reflected in changes in the interlaminar
function of crack length for plain polyester resin and three shear strength of the composite (qv, chapter 4), and
composite samples (DCBs) containing glass fibres with this property is used as the abscissa in the graph.
different surface treatments (Harris et al, 1975). Early commercial fibre surface treatments were
sometimes carried to excess in the drive to raise
the composite shear resistance, with the consequence that many brittle structural failures were
encountered (the datum point at the extreme right of the graph was for a commercial treatment). Modern
treatments are designed to optimize strength and toughness. Beaumont and Phillips (1972) showed very
early on that in carbon-fibre-reinforced plastics, untreated fibres produced composites that were notch-
insensitive, whereas treated fibres, especially the somewhat over-treated fibres of that time, led to notch
sensitivity.

There have been many studies of the effects of interface control in modifying the toughness of
composites, and it has often been observed that attempts to improve the work of fracture (as opposed to
the notional fracture-mechanics KC value) often resulted in a reduction in strength. Atkins (1974)
50 showed, however, that if reinforcing filaments were
treatments to γF ∝ (τi)-1
Work of fracture, γF , kJm -2

made to have random alternating sections of high
40 increase bonding and low interfacial bond strength (giving locally
silicone oil to
reduce bonding
low and high interfacial toughness) a desirable
30 combination of strength and toughness could still be
untreated achieved. The strongly bonded regions ensured that
20 fibre the fibre strength could be utilized and the weakly
bonded regions, randomly arranged in the paths of
untreated fibre,
10 composite exposed
running cracks, provided the source of debonding
to steam and pull-out energy. In fact, it was possible to
0 achieve fracture energies of the order of 300kJm-2 in
0 0.02 0.04 0.06 0.08 0.10 0.12 0.14 boron/epoxy composites containing ‘barber’s pole’
Reciprocal ILSS, m2 MN-1 release-agent coatings without loss of strength
relative to composites with uncoated fibres.
Figure 5.10. Effect of fibre surface treatments in
modifying the work of fracture and the interlaminar
shear strength of carbon-fibre/polyester composites The simple models for debonding and pull-out
(Harris et al, 1971) described above both make the incorrect
assumption that the fibre strength has a unique


value, σfu, having been originally derived, like many of the early theoretical studies of composites
behaviour, for metal-fibre/metal-matrix systems. Wells and Beaumont (1985) and Thouless and Evans
(1988) have recognized that the statistical distribution of fibre strengths must be taken into account in
attempting to determine the location of a fibre failure (and hence the pull-out length) and its relationship
with the fibre, matrix and interface properties.

In a composite containing fibres at angles to a
propagating crack, extra work may be needed to withdraw
the fibres (Harris and Ankara, 1978; Harris 1980; Helfet and
Harris, 1972; Piggott, 1970 and 1974). The sources of this
extra work will depend on the constituents of the composite.
Ductile or tough fibres, like metal wires or polymer
filaments, will be capable of deforming inelastically in order Figure 5.12. Model for the plastic shearing
to follow the more complex pull-out path without breaking. during pull-out of a fibre lying at an angle to
the crack plane.
A metal filament that is capable of work-hardening, like a
steel wire in concrete, may contribute a considerable level of
plastic deformation work to the overall composite toughness by being forced to deform first one way and
then another, as illustrated in Figure 5.12. Results of Morton and Groves (1974) support those of Helfet
and Harris in demonstrating that a higher angle of misorientation leads to a higher pull-out stress. They
developed plastic-hinge models to determine extra plastic work and showed that the pull-out force was a
maximum at an angle, θ, of 45°.

Piggott (1970, 1974) suggested that the apparent strength of brittle fibres would be reduced by the
bending process, although the extent of the strength loss was smaller the softer (or more ductile) the
matrix resin since the bending fibres can cause the matrix to yield at the corners formed by the
intersections of the fibre holes and crack faces. However, in recent work Khatibzadeh and Piggott (1997)
has carried out more detailed studies of the effect of loads applied at an angle, and has found that the
apparent strength of brittle fibres is actually initially increased as the angle between fibres and crack
plane increases. At larger angles, though, the combination of geometric effects and mechanical damage
to the fibres will offset the gains described above and the overall work of pull-out will be reduced.

For a brittle-fibre composite, Piggott (1974) 700 fibre spacing = 10mm
proposed that the work of fracture of a composite
containing fibres at an angle θ to the crack plane
γF, Jm-2

was given by: 600
plain resin value = 350Jm-2
(γF)θ = (γF)0(1 – Btan θ)............[5.13]
500
where (γF)0 is the work of fracture of a composite
K, MPa.√m

1.5
with the fibres normal to the crack plane and the fibre spacing = 30mm
parameter B, which is determined by the force 1.0
exerted by the fibres on the matrix, is related to the
ratio of the matrix yield strength to the fibre 0.5
tensile strength, τmy/σfu. For ductile fibres, the plain resin value = 0.5MPa.√m
fracture work increases with increasing angle 0
0 10 20 30 40
provided the breaking stress of the fibre is not
exceeded. Some examples of the effect of fibre Angle between loading direction and fibres, θ o
orientation for metallic and glass fibres in model
composites are shown in Figure 5.13. The results Figure 5.13. Effect of the angle, θ, between the fibre
for the steel-wire/PMMA composites are presented direction and the crack plane on the resistance to crack
as work-of-fracture versus angle, while those for propagation for a) steel wires in PMMA, and b) glass
the glass/polyester, following the style of Figure fibres in polyester resin. The test samples were double
cantilever beams with a single row of spaced fibres
5.6, are given as K (= σ√c). It can be seen that for (Harris, 1980). The dashed curve in the lower plot
both cases the composite toughness first increases represents the Piggott model of equation 5.13, with (γF)0 =
with θ and then falls as the angle increases. 1.16 and B = 0.78.


Although the Piggott model is for work of fracture rather than K, the data agree reasonably well with
equation 5.13, as shown, with values of (γF)0 of 1.16 and B = 0.78. The disagreement at small angles
clearly relates to the issue of whether or not the apparent strength of the glass fibres increases for small
deviations of alignment.
30
If the fibres are discontinuous, as in a DMC, for
example, the situation is changed since the length of

25
the fibre and its orientation may affect the pull-out w = 0.5lc2
behaviour. In a composite containing aligned, short 20

fibres with lengths shorter than lc, the toughness 15
increases with increasing length since the fibres
cannot be broken by the external forces and 10 w = 10 + 90/l
increasing their length increases the extent to which
5 aligned fibres
they can be pulled from the matrix. From equation
random fibres
5.11, it can be seen that the fracture work will be 0
0 5 lc 10 15 20 25 30
proportional to the square of the fibre length for l <
Fibre length, mm
lc. Beyond the critical length, however, a proportion
of those bridging a matrix crack will have an end
Figure 5.14. Variation of work of fracture with fibre
within ½lc of the plane of fracture and these will length in random and aligned composites of chopped
pull out: the rest can fracture and, as a result, the steel wires in polyester resin. The fibre volume fraction
amount of pull-out work required is reduced (Cooper is 0.1. (Helfet and Harris, 1972).
and Kelly, 1969; Harris and Ramani, 1975), the
fracture work being inversely proportional to the fibre length. Thus, for a given system, the toughest
composites will be those with fibres with lengths equal to lc, but, in order to achieve this benefit,
strength and stiffness would have to be sacrificed. If the fibres are randomly oriented instead of being
aligned, the effects described above are less clear cut. The combined effects of fibre length and
orientation on the work of fracture of some model steel-wire/polyester composites are shown in Figure
5.14. The simple model for the effect of fibre length is not entirely satisfactory, since the proportionality
-1
constant for the experimental l relationship shown in this figure is lower than that calculated by Helfet
and Harris (op cit), and the two curves do not intersect at precisely the experimental value of the fibre
critical length, but the argument is at least mechanistically reasonable.

We would normally expect composite toughness to be raised by increasing the fibre volume fraction,
by using fibres of larger diameter, or by using stronger fibres. Contrary to expectation however,
improving the fibre/matrix bond will usually reduce the toughness because it inhibits debonding and
therefore reduces pull-out (Beaumont and Harris, 1972).

Detailed analysis of the individual energy-absorbing mechanisms that are believed to contribute to
the overall toughness of reinforced plastics is useful, as we have said, for guiding us in the design of
composites that may be expected to show improved mechanical properties, particularly in respect of
combined strength and toughness. But it is easy to develop an over-simplified view of composite
behaviour, especially when the analysis is carried out on model systems which are deliberately designed
to prevent interaction of microfailure events.

5.3.2 Toughness of hybrid composites
When more than one species of fibre is embedded in a single resin, the behaviour of the resulting
hybrid composite may not be easy to predict from simple models. Early experiments with
glass/carbon/resin composites led Harris and Bunsell (1975) to conclude that the work of fracture
measured by notched impact followed a simple mixture-rule variation with composition (proportions of
CFRP and GRP), as shown in Figure 5.15, regardless of the distribution of the two constituent fibres.
But more detailed work on carbon/glass and carbon/Kevlar hybrids suggested that this may not always
be the case. Anstice and Beaumont (1983), for example, showed that the presence of the carbon fibres, as
an ‘environment’, as it were, for the glass filaments, effectively modified the debond and pull-out
lengths of the glass so as to reduce the overall hybrid fracture energy below that predicted by the mixture


rule. Stefanidis et al (1985) reported similar 200
results from impact-energy values for

Work of fracture, γF, kJm 2
carbon/Kevlar hybrids. These are examples of a
150
negative ‘hybrid effect’.

An interesting effect that is often overlooked is
100
that since cracks propagating in laminated
composites may be forced to travel
simultaneously through two materials — eg. two 50 GRP/CFRP composites (polyester matrix)
plies of different orientation, or two similarly
oriented plies of different composites (in a hybrid)
— which have different intrinsic fracture 0
0 20 40 60 80 100
toughnesses, this will constrain the crack and
modify the material’s apparent resistance to crack Composition, vol% of CFRP in composite
growth. When a crack is forced to move with a
common front through a sandwich consisting of a Figure 5.15. Work of fracture, measured by notched Charpy
well-bonded high-toughness and low-toughness impact tests, for hybrid CFRP/GRP composites (Harris and
Bunsell, 1975).
pair of layers, since the composite fails at a
common displacement, the failure strain for the
tougher ply will be reduced and that of the less-tough ply will be increased. Guild et al (1978) showed
by a simple mechanics analysis that the value of the fracture energy of the composite, (γF)c, would be the
mixture-rule average, ie.:

(γF)c = ½[(γF)L + (γF)H].......................................................................................... [5.14]

of the toughness of the low- and high-toughness components, (γF)L and (γF)H. Rewriting this in terms of
the strain-energy release rate, G (= 2γF), given by the compliance relationship:

P 2 ⎛ dC ⎞
G= ⎜ ⎟ ................................................................................................... [5.15]
2 ⎝ dA ⎠A

where (dC/dA)A is the rate of change of compliance with crack area, A, it is then easy to show that the
cracking load for the composite beam, PC, is given by:

Pc2 = 1 2 ( PL + PH ) ............................................................................................[5.16]
2 2

from which we conclude that the composite cracking load, and therefore the related critical stress-
intensity factor, KC, must always be greater than a simple mixture-rule sum such as that given for (γF)c.
Atkins and Mai (1976) showed that the cracking load is given by the volume-fraction sum only when
there is no bond between the components. This is an example of an apparent synergistic effect, as
discussed above and in chapter 4, where the composite appears to perform better than would be expected
from simple mixture-rule considerations.

5.3.3 Toughness of laminated structures
5.3.3.1 Delamination
In discussing these microstructural failure events we have taken a simplified view of failure, but
frequently there are further complications which may swamp the effects just described. The debonding
process is effectively a localized crack-blunting and crack-deviation mechanism, and the growing crack
described above may in fact change planes rather than always continuing on the same plane. Multiple
cracks may also be distributed throughout the process zone. But in most practical composites, there is the
added complexity of laminate structure. A laminate is constructed from groups of individual
unidirectional plies which are laid at various angles, depending upon design requirements, or from layers
of woven cloth laid at various angles to the main stress axes. The tension/shear coupling effects


discussed in chapter 3 cause shear stresses to be developed in the plane of the laminate, especially near
free edges, when the material is stressed. And since interlaminar planes in non-woven composites are
always planes of weakness, the interlaminar shear stresses may easily become large enough to disrupt or
delaminate a composite well before its overall tensile strength is reached. A crack travelling through a
given ply may therefore find it energetically favourable to deviate along an interlaminar plane: under
such circumstances, considerations of fibre debonding and pull-out are of little significance.

Although the specific fracture energy for splitting is low (of the same order as that of the resin, ie. a
few hundred Jm-2), the total area of delaminated surfaces created may be very large, so that the net
contribution of a delamination to the overall composite γF may be substantial. Aveston (1971) has
analysed the work of fracture that may be available when the destruction of a bending beam occurs by a
process of repeated fracture arrest and delamination. From strain-energy considerations, he has shown
that the total work for a single delamination may be three times the normal stored elastic energy to peak
load, even in all-brittle composite systems. Most prepreg laminated composites exhibit multiple
delamination, with consequent further increases in fracture work. But delaminations of this kind are not
always able to travel unhindered through fibre-free planes of weakness in modern composites because
the manufacturing process has usually forced the original pre-preg layers into reasonably good contact,
and it is often found that, after some degree of splitting, fibres lying partially across the splitting plane
exert a bridging effect in the form of a tied zone. 1.2
Crack propagation then becomes increasingly
Fractional change in toughness

difficult as the delamination extends. An 1.0
indication of the relative degrees of resistance to
impact damage can be seen in Figure 5.16. Test 0.8
coupons of a quasi-isotropic carbon-fibre
laminate and a woven-roving glass/epoxy 0.6
composite were subjected to repeated impacts by
0.4
a round-nosed tup, the samples being supported
by a solid steel backing during the impacts, and 0.2 CFRP (quasi-isotropic laminate)
the residual levels of toughness were determined GRP (woven-roving/epoxy)
as a function of cumulative incident energy. It 0
can be seen that the prepregged CFRP material 0 100 200 300
begins to lose toughness following the initial Cumulative incident impact energy, J
impact, whereas the woven GRP composite
retains a very high level of toughness after many Figure 5.16. Effect of repeated impact on the notched
impacts. This is of course one of the reasons tensile strength of a quasi-isotropic carbon/epoxy laminate
why crash helmets made of this kind of GRP and a woven-roving glass/epoxy laminate (Harris et al,
composite perform so much better than cheaper, 1991).
unreinforced polycarbonate helmets.

Since delamination appears to be advantageous in improving the overall toughness of composites, it
is natural that much attention should have been paid to methods of increasing the work of delamination,
as characterised by both the mode-I (tensile) strain-energy release rate, GIC, and the mode-II (shear)
characteristic, GIIC (O’Brien, 1982; Hashemi et al, 1990). Results obtained from these two types of
measurement sometimes appear to give contradictory indications. Various methods have been tried to
increase the work of delamination, including:

1. modification of the matrix resin chemistry to increase the basic toughness of the resin.
Alternatively, the use of intrinsically tough thermoplastic matrix polymers like PEEK and
other polyaryls is advantageous, but calls for somewhat more complex manufacturing
processes,
2. modification of the matrix by the incorporation of ‘flexibilizers’ — a process akin to that used
in the production of high-impact polystyrene or ABS. Liquid CTBN rubber added to an
epoxy resin results in a tougher, phase-separated structure containing rubber particles, but the
hot-wet properties of a material are often reduced by ‘flexibilization’,


3. modification of the resin by the incorporation of particles, whiskers or chopped fibres into the
resin, usually in the interlaminar regions of a laminate,
4. interleaving with inert, usually soft, polymer films,
5. through-thickness stitching of the laminate.

The first three of these bring about some improvement, but the degree of improvement is rarely
sufficiently great to make much difference to the overall composite toughness. Srivastava and Harris
(1994), for example, showed that by the incorporation of a number of types of particulate material into
the interlaminar planes of a cross-plied CFRP laminate both the mode-I and mode-II delamination
toughnesses could be increased. GIC was approximately doubled, from about 0.6kJm-2 to roughly
1.3kJm-2, and GIIC was increased from about 120Jm-2 to about 200Jm-2. These improvements, which were
a maximum at particle addition levels of about 3%, falling back thereafter towards the original levels,
were greater the higher the rigidity of the particulate additive. The resultant improvements in laminate
toughness were much smaller, however, amounting to only about a 25% increase in the un-notched
impact strength.

If the composite laminae are deliberately separated by interleaving with inert films which can act as
delamination initiators, the tied zone caused by fibre bridging can no longer form, but the extra energy of
delamination contributes to toughening instead. Favre (1977) showed that the fracture energy of CFRP
laminates may be increased in proportion to the number of deliberately introduced delamination
initiators such as films of aluminium foil or polyimide resin bonded between the carbon/epoxy laminae.
Altus and Ishai (1990) also demonstrated the value of this process of ‘interleaving’ by observing the
effects of soft interleaved layers on delamination. They considered that delamination was triggered by
transverse ply cracking, and that, up to a certain level of thickness, the presence of soft layers reduces
stress associated with transverse cracks and hence delays the start of delamination. Sela and Ishai (1989),
in a useful review of interlaminar toughening methods, showed that interleaving markedly improved the
compression-after-impact strength (CAI) of laminates, but has the disadvantage that it brings a weight
penalty because of the concomitant reduced stiffness. The net Vf is lower than in an equivalent
composite with no interleaving, so that additional plies have to be used to maintain design requirements.
They also recommend selective interleaving as a useful compromise. Mai (1993) shows that the use of
plastic interleaving in which the film is perforated so as to allow intermittent separation of the composite
laminae leads to significant increases in toughness without the concomitant loss in tensile strength which
accompanies the incorporation of non-perforated interleaving films. This effect is seemingly analogous
to that of intermittent fibre coating described in section 5.3.1.3.

We saw in section 4.3 of chapter 4 that third-direction stitching was a useful means of improving the
interlaminar shear strengths of laminates by its effect in improving the delamination resistance.
Dransfield et al (1994) showed that stitching was able to improve some mechanical properties such as
the delamination resistance and the compression-after-impact strength, but that it may cause degradation
of other properties as a result of the creation of pockets of resin or the breakage of load-bearing fibres.
Adanur et al (1995) had recognised the problem of the fibre damage caused by the stitching process, and
they describe an improved stitching method to reduce it. Materials produced by the improved method
then exhibited reduced delamination and increased shear strength and impact energy up to a critical level
of stitch density. It is interesting to note that Mouritz et al (1997) found that stitching increased the
mode-I delamination resistance but not the mode-II resistance. They also noted that the fibre damage
caused by stitching led to reductions in the flexural properties, including the ILSS, especially under
impact conditions.

The interlaminar fracture toughness of composites often shows R-curve behaviour — an increase in
crack resistance as an interlaminar crack grows — as a result of the effect of fibres bridging the growing
crack behind the moving crack tip. Truss et al (1997) demonstrated that the misalignment of short fibres
in a discontinuously reinforced laminate added to this fibre-bridging effect, increasing the fracture
toughnesses for both initiation and growth of interlaminar and intralaminar cracks.


5.3.3.2 Transverse-ply cracking
In the case of an orthogonally laminated composite (eg. of [(0,90)n]S lay-up, or similar), loading along
one of the principal axes will cause cracking of the transverse plies at relatively low strains. In a typical
GRP laminate, for example, this cracking, which is easily detected by techniques such as acoustic
emission monitoring, may begin at strains as low as 0.2%, depending on the thicknesses and distribution
of the plies and the resin toughness (Garrett and Bailey, 1977; Harris et al, 1979) and continue to strains
of about 1% — perhaps half the overall composite failure strain — leaving the transverse plies
completely cracked into blocks of lengths of the order of the ply
thickness, as shown in Figure 4.15 of chapter 4. This cracking
occurs partly through the weak fibre/matrix interface and partly
through the matrix resin, as shown schematically in Figure 5.17
and it reduces the composite stiffness. If a 10cm x 2cm x 3mm
strip of a six-ply (0,90)3S composite were loaded until each of the
90° plies was fully cracked, with a crack spacing of about
0.5mm, and if the γF value for transverse-ply cracking were of
the order of 200Jm-2, there would be about 0.012m2 of crack
surface, and this would require 2.4J of energy. Normalized with Figure 5.17. Schematic illustration of the
respect to twice the sample cross-sectional area, this provides crack path during transverse cracking
about 20kJm-2. This is no doubt a vast over-estimate, but it through the matrix and interface.
certainly suggests that transverse-ply cracking may not be a
negligible contribution to the composite work of fracture.

5.4 TOUGHNESS OF PRACTICAL REINFORCED PLASTIC LAMINATES
The fracture energy of laminated composites depends strongly on both the ply stacking sequence and
on the arrangement of the reinforcing fibres in the individual plies. The degree of anisotropy of the
impact fracture energy of some typical high-quality E-glass/epoxy laminates is shown in Figure 5.18
(Harris, 1980). The high degree of anisotropy of the unidirectional composite is indicated by the ratio of
about 100: 1 between the extreme values for cracking normal to and parallel with the fibres. The fracture
energy for unconstrained cracking in the transverse direction, approximately 2kJm-2, is some four or five
times higher than typical values for cracking of unreinforced resin, an increase partly due to the greater
complexity of the crack path and partly to the higher fracture surface energy of the fibre/resin interface.
For the 0/90 non-woven laminate, which has about the same Vf as the unidirectional composite and
contains approximately equal proportions of
250
fibres in the two directions, the fracture energy

in the two orthogonal directions is roughly half unidirectional, Vf = 0.64
that of the unidirectional composite. This 200 0/90 non-woven, Vf = 0.63
implies that it is the crack/fibre interaction in the 0/90 woven, Vf = 0.47
longitudinal plies that determines the overall 150
toughness of the 0/90 laminate, although this
may be an over-simplification since we observe
relatively little reduction in γF at angles other 100
than 0° and 90° where the fracture within the
plies must be more complex. The woven-cloth 50
laminate, which has a somewhat lower Vf has a
work of fracture only one quarter that of the
0
non-woven 0/90 laminate and it is more nearly
0 30 60 90
isotropic, showing a slight increase in γF at 45°
to the fibre directions. It is interesting to note Orientation, θ°
that the orientation-dependence of the fracture
energy, like those of the elastic modulus and Figure 5.18. Orientation dependence of the work of fracture
tensile strength, can be represented by an of some GRP laminates. The data were obtained from
notched Charpy impact tests (Harris, 1980).
expression of the form:


1 a cos 4 θ b sin 4 θ
= + + csin 2 θ cos 2 θ ........................................................[5.17]
( γ F )θ ( γ F )0 ( γ F )90

and the fitted curves in Figure 5.18 have this form, although we make no attempt to interpret the values
of the constants a, b and c in terms of physical mechanisms.

The differences in γF levels between these three
materials and the degree of anisotropy exhibited by each of
them cannot be explained on the basis of either fibre Vf or
elastic modulus variations. Perhaps the most significant
difference between the macroscopic fracture patterns of
these three composites relates to the volume of material
comprising the fracture zone (Figure 5.19). In the
unidirectional and non-woven 0/90 composites the
continuity of fibres results in an extensive damage zone
around the notch tip, whereas in a woven-cloth laminate
the damage zone is much more restricted. Fracture of a
0/90 non-woven laminate at other than a right angle to the
fibre direction occurs by transverse fracture of the two sets Figure 5.19. Damage accompanying fracture of
of plies, interlaminar shearing failure between the plies, the notched Charpy samples of the GRP
and final separation by a type of pulling-out of the laminates for which the fracture energies are
shown in Figure 5.18. In sequence, left to right,
interlocked plies. The more localized failure of a woven- unidirectional, 0/90 non-woven, and 0/90 woven.
cloth laminate involves fracture and pulling-out of stubby
bundles of fibres, mostly less than 10 fibre diameters in length, the shortness of these pull-outs
presumably being due to the high local curvature of the filaments in the woven cloth. These effects of
style of reinforcement on fracture energy are ascribed to the fact that woven reinforcement constrains the
extension of the damage zone and serves to transfer the stress in a manner analogous to that which
occurs with thin interspersed plies in a non-woven laminate. The tighter the weave, the lower the
toughness.

McGarry et al (1976) have discussed in detail the factors affecting the toughness of practical
laminates. They describe the development of a damage zone ahead of a crack which like the plastically-
yielded zone in a metal, helps to relax the high stresses concentrated at the crack tip. For practical
reinforced plastics the damage zone is a region containing intra-ply cracks propagating parallel to the
fibres and inter-ply delaminations, and its
size and shape exert an important influence
1.0 on the resistance of the laminate to crack
HMS [(±45)2]S
propagation. Sharp cracks in a ply are
Strength ratio, σn / σun

HTS unidirectional
0.8 effectively blunted by the intra-ply cracks
HMS [(0,90)2]S
so that failure requires a higher applied
0.6 stress, but final failure is still determined by
the stress distributions outside the damage
0.4 region.
un-notched strengths:
1.00GPa Growth of a crack through the thickness
0.2
0.60GPa of a laminate is inhibited by the
0.08GPa development of the delaminated region
0
0 0.2 0.4 0.6 0.8 1.0 between cracked and uncracked plies. As a
Notch depth ratio, 2a/W consequence, certain types of laminate
construction may be highly notch-sensitive
Figure 5.20. Notch sensitivity of some CFRP laminates (data of whereas others may be completely
Lee and Phillips, 1981). The stresses are the nominal failure insensitive to the presence of notches. Thus,
stresses, σn (peak load divided by sample cross-section area) notched CFRP laminates with ±45 lay-up
normalized with respect to the un-notched strength, σun . They fail at net section stresses higher than their
are plotted as a function of ratio of the notch length, 2a, to the
specimen width, W. Two types of carbon fibre, high-strength un-notched failure stress, σc, whereas 0/90
(HT) and high-modulus (HM), both surface treated, are
represented.


laminates fail at net section stresses much
1.0
lower than σc. The ±45 laminates are, of Percentage of 0o plies
course, much weaker than the 0/90 laminates 17%
despite their lack of notch sensitivity. 0.8 25%

Strength ratio, σn / σun
Grouping of the plies so as to reduce the 50%
effective number of interfaces between 0.6 100%
differently-oriented plies reduces the notch insensitive:
efficiency of stress transfer from ply to ply σn/σun = [1 - d/2W]
and dramatically increases the fracture 0.4
toughness of the laminate. An illustration of
the differing degrees of notch-sensitivities of 0.2 notch sensitive:
some practical CFRP laminates is shown in σn/σun = Kt-1
Figure 5.20 (Lee and Phillips, 1981). In this
0
diagram, the straight line represents the 0 0.2 0.4 0.6 0.8 1.0
behaviour of a material that is completely Ratio of hole diameter to sample width, d/W
insensitive to the presence of a notch, in this
case a composite of lay-up [(±45)2]S. The Figure 5.21. Results of Soutis (1994) for the compression
extent to which the experimental strengths of T800/924 CFRP laminates of various lay-ups
measurements of notched strength (shown containing holes. Kt = 3 for a circular hole in an infinitely large
here normalized with respect to the un- plate.
notched strength) fall below the straight line is
then an indication of notch sensitivity. A more systematic analysis of the effect of lay-up in one specific
material, a T800/924 carbon/epoxy system (Soutis, 1994), shows quite clearly how increasing the
percentage of 0°plies increases the notch sensitivity (Figure 5.21). The test pieces used by Soutis
contained circular holes rather than notches, but the effect is similar to that shown by Lee and Phillips.
In Figure 5.21 the straight line again represents notch-insensitive behaviour, and the bottom curve,
which is effectively a lower bound to the experimental curves, identifies the fully-notch-sensitive
definition of the stress concentration for a circular hole, Kt (equal to 3 for a circular hole in an infinite
plate).

5.4.1 Short-fibre composites
Thermoset and thermoplastic moulding compounds containing randomly arranged chopped fibres,
usually glass, are cheap materials which are finding wide-spread use in mainly non-load-bearing
applications. Although the stiffnesses of these materials are usefully greater than those of the base resins
from which they are compounded, their strengths are rarely much higher than those of the plain (filled)
matrices. An important virtue of such materials, however, apart from their cheapness and mouldability,
is that they are substantially tougher than unreinforced polymeric materials on account of the toughening
mechanisms described above. They are roughly isotropic, and the lack of obvious planes of weakness
such as occur in prepreg laminates means that
the cracking process resembles that in
5
γF
γF , kJm-2 and KQ , MPa.√m

conventional homogeneous materials more
KQ than that in anisotropic laminates. Because of
4 lp their structure and composition their
behaviour is more matrix- and interface-
3 dominated than that of higher-performance
Pull-out length, lp, mm

composites. Polyester dough-moulding
2 2.0 compounds provide a useful example of the
behaviour of this kind of composite. Some
1 1.0 measurements of the work of fracture, γF, and
the candidate critical stress-intensity factor,
0 0 KQ, for a typical DMC are shown in Figure
-50 0 50 100 150 200 5.22 (Harris and Cawthorne, 1974). The work
Temperature, °C of fracture of the compound is of the order of
2 kJm-2, much greater than that of the
Figure 5.22. Temperature-dependence of the fracture energy, critical polyester/chalk-filler material which
stress intensity, and fibre pull-out length for DMC (Harris and
Cawthorne, 1974)


constitutes the matrix. It is interesting to note that
the two fracture parameters behave quite differently 12
as functions of temperature, and are clearly not

γF , kJm-2 and KQ , MPa.√m
10 γF
measuring the same property even though we use
KQ
both to refer to toughness. The KQ values fall 8
linearly with increasing temperature, presumably
because KQ is influenced by the material strength 6
which falls markedly over the temperature range
shown. On the other hand, γF shows classical tough- 4
brittle transformation behaviour, the transition
2
occurring roughly in the region of the matrix glass-
transition temperature. Thus, the fact that the matrix 0
is becoming less brittle is dominating the variation -6 -4 -2 0
of this property. The figure also shows the observed Log(cross-head speed, m.s-1)
fibre pull-out lengths, and it can be seen that these
vary with temperature in roughly the same way as Figure 5.23. Variation of work of fracture and critical
the γF values. It should be recalled, with reference to stress intensity with testing speed (three-point bend
Figure 5.3, that KQ relates only to behaviour as far as tests) for a polyester dough-moulding compound (DMC)
at room temperature (Harris and Cawthorne, 1974)
the peak load, whereas γF includes the work required
for the whole failure process. In matrix-dominated materials the measured values of toughness
parameters are also more rate-dependent than those in fibre-dominated composites. Figure 5.23 shows,
for example, how the two fracture parameters vary with testing speed in notched three-point bend tests.
In this case, γF is much more sensitive than KQ, but both increase with testing speed.

5.4.2 Design of damage-tolerant composites
Lee and Phillips (1981) have given some rules-of-thumb for the design of damage-tolerant
composites. The requirements are:

i) The fibre/matrix bond strength is most important and must be optimized: too strong a bond
results in brittle composites.
ii) The composite needs to store large amounts of strain energy, ½σf εf, so high-strength,
intermediate-modulus, or high-failure-strain fibres are required.
iii) Thick plies delaminate more easily and reduce interactions between adjacent cracked and
uncracked plies, thereby increasing toughness. But there is a need to compromise because thick
plies increase the susceptibility to transverse cracking, and hence reduce fatigue and
environmental resistance.
iv) The stacking sequence needs to be arranged so as to encourage delaminations between shear
cracks in 45° plies adjacent to load-bearing 0° plies (eg. as in (±45,02)S structures) in order to
increase notched strength by protecting the 0° plies. By contrast, if delamination is suppressed
(as in [0,±45,0]S structures) then the shear cracks which form in 45° plies at low stresses have a
detrimental effect on the 0° fibres, leading to brittle behaviour.
v) The style of reinforcement is important: cloth limits the size of the damage zone, rendering the
composite more brittle.
vi) 45° plies are weak but notch insensitive. They can therefore be used as crack arrestment strips
in a laminate normally reinforced with 0° load-bearing plies.

Mai (1993) has also given a valuable summary of the idea of developing high fracture toughness in
composites by engineering the interfaces in the composite at several levels which encapsulates some of
the main ideas presented in the foregoing sections.

5.5 TOUGHNESS OF CERAMIC-MATRIX COMPOSITES
The current high level of interest in ceramic-matrix composites has largely been generated by users of
materials in high-temperature applications for which existing metallic alloys are increasingly becoming


border-line candidates. In most of the CMCs that have been studied hitherto, both fibre and matrix are
refractory in nature and therefore brittle. Practical applications consequently depend on the ability of
these brittle/brittle combinations to exhibit energy-absorbing mechanisms which are more efficient than
those associated with simple failure of the matrix and the fibres.

In typical CMCs, by contrast with reinforced plastics, the stiffnesses of the reinforcing and matrix
phases are often quite similar, although the strength and failure strain of the fibre will almost always be
higher than those of the matrix. Many of the mechanisms of energy absorption that have already been
described for polymer composites will also occur in CMCs, but the consequence of the extreme
brittleness of the matrix — ie. multiple matrix cracking — is to modify in some respects the extent to
which these other toughening mechanisms can operate. Hillig (1987), following Cottrell (1964), has
summarized the situation for CMCs with reference to the expression for the fracture energy (in terms of
the critical strain-energy release rate, G):
U max

Gc = 2 ∫0
σdU ....................................................................................................5.17)

where U is the non-elastic displacement relative to the bulk material at the crack plane under critical
conditions for crack propagation, and σ is the corresponding stress level in that plane. A large value of
Gc results from a combination of a high crack-tip stress and a broad process zone, which sets the guiding
principles for the development of successful CMCs. The extent of the process zone can be modified, as
with reinforced plastics, by the provision of a group of energy-absorbing mechanisms which, in ceramic
materials, will include incipient microcracking, crack deflexion and branching, fibre/matrix debonding,
and fibre pull-out.

5.5.1 Matrix cracking
One of the seminal treatments of the failure of CMCs was that of Aveston and co-workers (1971,
1973, 1974) which led to what is now almost universally referred to as the Aveston-Cooper-Kelly, or
ACK, model. It is an energy-balance model set up to explain the multiple fracture of CMCs and is
frequently used to predict the occurrence of the first crack as a design parameter. In composites of this
kind first cracking may be more important than the notional strength. The ACK model starts from the
assumption that a crack cannot form in the matrix unless the work done by the applied stress is greater
than the increase in elastic strain energy of the composite plus the fracture surface work of the matrix per
unit area of cross section. The model takes into account the work done by the applied stress on the
composite body, the work expended in debonding the fibres from the matrix, the matrix fracture energy
(usually negligible), the friction work expended as the matrix and fibres slide past each other, the
decrease in the elastic strain energy of the matrix, and the increase in the strain energy of the fibres. The
friction stress controlling the interfacial sliding is assumed to be constant and characteristic of the
system. After cracking of the matrix, the bridging fibres are then assumed to carry the entire load
formerly shared with the matrix. A brief introduction to the derivation of the ACK model is given in
section 4.6 of chapter 4. We refer again, briefly, to matrix cracking in section 5.5.3.

5.5.2 Fibre bridging
In order for crack bridging to occur and contribute to composite toughening, it is essential that the
fibres debond at the interface in preference to fracturing at the crack front. When this condition is
satisfied, the rate of load transfer from fibres to matrix is determined by the frictional resistance to
sliding at the interface. If τi is high, load transfer is efficient, and the load in the fibre decays rapidly with
distance from the crack face. The statistical nature of the strengths of brittle fibres then results in the
occurrence of fibre failures mainly in locations close to the crack face, thereby reducing the potential
pull-out contribution to toughening. A low level of frictional resistance thus promotes toughness and, as
in polymer-matrix composites, control of the interface is therefore of paramount importance in
determining the composite fracture resistance. In a review of the mechanical behaviour of CMCs, Evans
and Marshall (1989) point out that in order for debonding to occur instead of fibre fracture, the ratio of
the interface fracture energy, GIC, to that of the fibre, Gfc, should be less than about 0.25, although, as


they pointed out, there was at the time no experimental validation of this requirement. The effect of
bridging fibres on the composite toughness can be modelled (Romualdi and Batson, 1963; Marshall et
al, 1985) by considering the stresses in the fibres as crack-surface closure tractions which reduce the
stresses at the crack tip.

5.5.3 Fibre pull-out
The fabrication of CMCs almost always
involves the use of high temperatures and/or
pressures in order to achieve good levels of
density and uniform microstructures, and under
conventional processing conditions this may lead
to interfacial reactions which can result in
reductions in the strength of the reinforcing
filaments, increases in the interfacial bond Figure 5.24. Schematic illustration of a unidirectional
strength, and the build-up of differential thermal CMC composite after loading so that the matrix is cracked
stresses. These effects, as we have already seen, into small blocks surrounded by an array of longitudinal
should result in marked reductions in the level of and transverse cracks. See text for explanation.
toughness that can be contributed to the composite
through the pull-out and other mechanisms. Further exposure to elevated temperatures during service
will also lead to deterioration. Results that have been reported range from composites where the bond is
so strong that little or no pull-out can occur to systems where the bonding is so weak that the matrix can
completely disintegrate, leaving dusty bundles of bare fibres. Neither of these extremes offers a useful
engineering composite, and it is clear that control of the interface is therefore just as important an issue
in CMCs as in reinforced plastics and metal-matrix composites.

A good deal of early research on CMCs related to composites with glass or glass-ceramic matrices
reinforced with carbon or SiC fibres which were manufactured by slurry impregnation and hot-pressing.
In much of this work, the composites produced exhibited works of fracture of the order of 3-10kJm-2.
Examples were the work of Phillips (1972, 1974a) on borosilicate glass and a low-expansion lithium
alumino-silicate (LAS) glass-ceramic reinforced with high- and low-modulus carbon fibres. The fracture
energies of the composites were related to differences in the fibre/matrix bond strength and the simple
pull-out model described earlier appeared to account for the greater part of the measured toughness
values. Subsequent CMC development, particularly in the USA, concentrated on glass and glass-ceramic
systems reinforced with Nicalon SiC fibres which have been reported on extensively by Prewo and his
co-workers (1980, 1982a, 1982b) who gave fracture energy values of about 30kJm-2. Values of γF for
SiC/Pyrex composites were also published by Ford et al (1987) who showed that, depending on heat-
treatment conditions, the same composite could have a fracture toughness as low as 2kJm-2, with little
pull-out, or as high as 25kJm-2 when the interfacial bond strength was optimized. When the same
material was heat treated so as to cause extensive delamination, γf values over 40kJm-2 were obtained.

One of the difficulties in using a simple model such as that of equation 5.12 for CMCs is that it is
often impossible to know exactly what is the strength of the fibre after it has undergone the rigours of the
manufacturing process. In resin-based or glass-based composites it is possible to dissolve away the
matrix and carry out strength measurements on the filaments so as to obtain their true strength in situ,
but in composites with more resistant matrices it is impossible to do this without the fibre sustaining
further damage. To remove glass or a silica-based ceramic from a Nicalon-fibre composite, for example,
it is necessary to use hydrofluoric acid or similar aggressive chemicals, and since Nicalon itself contains
some SiO2 it is likely that some attack of the fibres will also occur. For a SiC/CAS composite of Vf about
0.4 manufactured by Corning, experimental measurements gave values of γF of about 50kJm-2 (Harris et
al, 1992). Scanning electron microscope pictures showed pull-out lengths of about 2-3 fibre diameters,
so the contribution of fibre pull-out to the toughness, from equation 5.12, is only about 5kJm-2, if the
fibre strength is about 1.5GPa (a value assumed from the measured strengths of extracted fibres). Harris
et al showed that a substantial part of the remaining fracture energy of the SiC/CAS composite could
have been contributed by the multiple transverse and longitudinal matrix cracking which occurs during
deformation, reducing the matrix to a series of roughly equi-axed blocks as shown schematically in


1.0 Figure 5.24. From an estimation of the total crack
surface area in a typical cracked sample, it was
0.8 (0,±45)2S deduced that the contribution of matrix cracking
(±45,0,90)S to the composite work of fracture must be about
0.6 (90,0,±45)S 250γm, where γm, the specific work of fracture of
σN /σo

the matrix is about 50Jm-2. This works out at
0.4 about 30kJm-2 which, together with the pull-out
work and other small contributions, approximates
0.2
reasonably well to the measured value.

0 5.6 APPLICATION OF FRACTURE MECHANICS
-1 0 1 2 3
TO COMPOSITES
Log(R)
5.6.1 Strength of notched composites and
Figure 5.25. Predicted values of the ratio of notched to un- notch sensitivity
notched strengths of three types of laminate as a function
of hole radius (Pipes et al, 1979). The fitted curve is a Before discussing the possibility of applying a
sigmoidal (Boltzmann) curve. fracture-mechanics formalism to composites
behaviour it is instructive to consider a number of
attempts that have been made to model strength of composites containing holes or notches. In the first
place, experimental work, like that of Waddoups et al (1971) on (0,±45)2S CFRP laminates,
demonstrated that the notched composite strength is often strongly dependent on the size of the hole. In
order to explain observed notch strengths, it is necessary to know the stress distribution in the vicinity of
the notch and to assume that failure occurs when the stress at some distance from the notch (known as
the point-stress criterion) or the mean stress in a given region in the neighbourhood of the notch (the
average-stress criterion) reach the un-notched tensile strength of the laminate. Whitney and Nuismer
(1974) used a two-parameter model, arguing that while the stress-concentration factor is the same for
holes of different sizes in an infinite plate, the stress gradient is different for each. Thus large stresses are
localized more closely to the edge of a small hole than a large hole, and a critical defect is more likely to
occur in a region of high stress for a large hole. The 2-parameter point-stress and average-stress models
are both capable of predicting observed notch size without recourse to fracture-mechanics methods. A 3-
parameter ‘classical elasticity’ model involving a transposition procedure proposed by Pipes et al (1979)
was also successful in being able to predict notched strength versus notch radius relationships for quasi-
isotropic laminates of any stacking sequence, given the relationship for a single sequence, as illustrated
in Figure 5.25. Potter (1978) considers that microcracks that form in the damage zone near a notch as a
result of over-stressing do not inevitably lead to failure, and may even act as stress-relievers. Then,
although the composite may exhibit linear-elastic behaviour, the notched laminate may fail at a stress
greater than that indicated by the linear elastic stress concentration — ie. it will show a notch-size effect.
Potter showed that tensile fracture is governed by inter-fibre stress transfer within the axial plies, and
that limitations on the extent of this stress transfer imposed by matrix strength and fibre/matrix bond
strength result in ‘large’ and ‘small’ notch effects. ‘Large’ notches initiate a sequential fibre failure
process in the axial plies, and their effect is predictable by orthotropic laminate analysis, assuming
inhomogeneity. In the neighbourhood of ‘small’ notches on the other hand, fibre failures in the 0° plies
do not initiate this sequential fibre failure process, but form stable damage zones such as we have
already discussed, which consist of matrix and fibre/matrix interface failures propagating parallel to the
fibres and inhibiting inter-fibre stress transfer. ‘Small’ notches become ‘large’ when the notch geometry
is modified in an appropriate way by the developing damage zone. Early work on unidirectional epoxy-
based composites reinforced with high-modulus carbon and with low-modulus SiO2 suggested that there
is a logarithmic relationship between the fracture energy and the radius of the notch tip in Charpy impact
tests, as shown in Figure 5.26.

5.6.2 The application of fracture
mechanics
Fracture mechanics is universally accepted as a useful discipline for characterising the toughness of
materials that are homogeneous and isotropic on the macroscopic scale — materials like metals and


alloys. In the field of composites, however, there is serious doubt about whether the additional
complexities of heterogeneity and anisotropy do not preclude the practical application of fracture
mechanics. Early questions that were asked in this context (Sih, 1979) are:

• Is it feasible to use the same test procedures (such as critical stress-intensity and compliance
measurements) to characterize composite toughness?
• Can the basic equations of fracture mechanics be modified to include inhomogeneity and
anisotropy?
• To what extent is crack propagation (when it occurs in an appropriate manner) controlled by
initial crack geometry, loading, and material orientation?
• What kinds of theoretical and experimental investigations are needed to develop predictive
techniques to forecast composite behaviour?
Much experimental and theoretical work has indeed been published about a wide variety of
composite types. but it is still far from clear whether fracture mechanics is or is not a useful investigative
tool and predictive model (see, for example,
Sendeckyj, 1975, and Sih and Tamuzs, 1979).
200
It is well known that certain composite or Work of fracture, γF, kJm-2
laminate types give results which can apparently
be treated by linear-elastic fracture mechanics, or 100
extensions of it, whereas others cannot. Several
distinct types of composite stress/strain behaviour SiO2/epoxy; slope 0.15
50
may be observed, for example, as shown in Figure carbon/epoxy; slope 0.09
5.3. The linear, or non-tough, kind of behaviour,
undesirable in engineering structures, is most
likely to allow the application of fracture 20
mechanics whereas the toughest kind of behaviour,
in which the material retains a substantial degree 10
of ‘post-failure’ integrity involving highly 0.05 0.1 0.2 0.5 1 2
complex deformation and cracking mechanisms, Notch root radius, mm
seems least likely to obey fracture-mechanics
principles. And in the failure of such a desirable Figure 5.26. The effect of notch acuity on measured works
engineering material, we have to ask whether the of fracture for unidirectional carbon/epoxy and SiO2/epoxy
more important parameter is that representing the composites (Ellis and Harris, 1973)
initial crack-growth resistance — G or K — or the
total fracture work, γF. The spreading of a delamination crack is perhaps the simplest mode of cracking
in a laminate, and is satisfactorily modelled, both for simple crack growth and for fatigue, by linear-
elastic fracture mechanics, as shown by Vancon et al (1984). On the other hand, the rupturing of a
notched unidirectional GRP rod, which may involve the total disintegration of the resin matrix over a
large part of the sample volume, will be too complex to treat by the usual models. Furthermore, as we
have seen, for any given laminate it is sometimes possible to change from one extreme type of fracture
behaviour to another simply by arranging the plies in different ways.

5.6.3 Application to specific composites
In seeking to apply fracture mechanics to composite materials we are, in essence, attempting to define
a fracture criterion that concerns only the macroscopic behaviour of the material in the presence of
notches, flaws or other design features that are large by comparison with the fibre diameter. At first sight
this does not appear to be feasible because the fracture behaviour of composites can be considerably
modified by the presence of weak interfaces, and the mode of crack propagation does not seem to satisfy
the conditions laid down for the application of fracture mechanics. Experimental results seem to suggest
however that under some circumstances fracture-mechanics ideas can be used. Wu (1968) showed that a
fracture-mechanics approach to the propagation of cracks in homogeneous, isotropic plates could be
applied in a limited way to unidirectionally-reinforced plastics. He determined the critical stress for the


propagation of cracks of various lengths parallel with the fibres under tension and shear loading
conditions, and showed that for each case a relationship of the form k = σ∞√(πa) accurately fitted the
results, k being a constant, a the crack length, and σ∞ the fracture stress. He also showed that the
interaction of stress-intensity factors for combined conditions (tensile and shear mode) was well
described by a relationship of the form
2
K I ⎛ K II ⎞
+⎜ ⎟ = 1 ...............................................................................................[5.18]
K Ic ⎝ K IIc ⎠

This empirical relationship was not in agreement, however, with a general theoretical one given by Paris
and Sih (1965) for anisotropic materials:
2 2
⎛ K I ⎞ ⎛ K II ⎞
⎜ ⎟ +⎜ ⎟ = 1 .........................................................................................[5.19]
⎝ K Ic ⎠ ⎝ K IIc ⎠

although both could be considered to be specific cases of a more general power-law model. Sanford and
Stonesifer (1971) measured the fracture toughness and strain-energy release rate, G, of several types of
glass-fibre/epoxy composites and showed that G is a bulk property of the material, independent of the
mode of loading. For combined tension and shear they give

G = GI + GII ................................................................................................................... [5.20]

so that fracture toughness properties for the in-plane forward-shear mode, which are experimentally
much more difficult to measure than those in tension, could be deduced in a straightforward manner
from opening-mode results. Since G and K are not independent properties of the material, and can be
written

G c = CK c .........................................................................................................[5.21]
2

C being an elastic compliance property. Equation 5.21 can be given in the alternative form

C1K1 + C2K2 = cons tan t ...............................................................................[5.22]
2
2

which is more in keeping with the expression of Paris and Sih. C1 and C2 are functions of the elastic
compliances for the material, viz.
1 1

⎛ C1 ⎞ 2 ⎛ E1 ⎞ 4 ...................................................................................................[5.23]
⎜ ⎟ =⎜ ⎟
⎝ C2 ⎠ ⎝ E 2 ⎠

where E1 and E2 are the elastic moduli in the longitudinal and transverse directions, respectively. The
full form of equation 5.21 for opening-mode cracks in a rectilinearly anisotropic body (as opposed to an
inhomogeneous composite) has been given by Sih, Paris and Irwin (1965):
1

⎛ S S ⎞ ⎡⎛ S ⎞ ⎛ 2S + S66 ⎞ ⎤ .....................................................[5.24]
1 1 2
2 2

G Ic = K 2 ⎜ 11 22 ⎟ ⎢⎜ 22 ⎟ + ⎜ 12
Ic ⎟⎥
⎝ 2 ⎠ ⎢⎝ S11 ⎠ ⎝ 2S11
⎣ ⎠⎥⎦

On making the appropriate substitutions for the Sij in terms of the engineering elastic constants (see
chapter 3), and ignoring a small term involving Poisson’s ratio, we obtain:


1

1 ⎞ ⎡⎛ E1 ⎞ E1 ⎤ .................................................................. [5.25]
1 1
2

2 ⎛
2 2

G Ic = K Ic ⎜ ⎟ ⎢⎜ ⎟ + ⎥
⎝ 2E1E 2 ⎠ ⎢⎝ E 2 ⎠ 4G12 ⎥
⎣ ⎦

This equation can be presented in a form analogous to the conventional fracture-mechanics relationship
for isotropic materials, viz.:

K2
Gc = Ic (isotropic solids) .................................................................... [5.26]
E

K2
Gc = Ic (orthotropic materials ).......................................................... [5.27]
E effective

the effective modulus thus being the reciprocal of the bracketed terms in equation 5.25. It was suggested
in the First Edition of this book (page 75) that for orthotropic composites with a strong anisotropic axis,
such as CFRP, the effective modulus could be approximated by 2√(E1G12). Closer inspection of the
function defined in equation 5.25, however, suggests that this is invalid and that for most practical
purposes the value of the transverse modulus, E2, is a better approximation.

From their work on stress concentrations in flawed (0,±45)2S carbon/epoxy laminates referred to in
the previous section, Waddoups et al (1971) concluded that the experimental behaviour of their
laminates, when tested without deliberately introduced notches, was consistent with predictions of a
fracture-mechanics analysis for a crack dimension of approximately 1mm, even though no
macroscopically detectable crack was present. This and other treatments, including that of Whitney and
Nuismer (op cit), therefore gives rise to the concept of an energy-intense region at each crack tip which
leads to an empirical extension of isotropic linear-elastic fracture mechanics in which the fracture
toughness, KC, for an infinite body under tensile load containing an internal crack of length 2a is given
by

Kc = σ√[π(a + rρ)]......................................................................................................... [5.28]

where rρ is the dimension of the energy-intense region at the crack tip, referred to in other formalizations
as the process zone: rρ is treated as a disposable parameter and is evaluated from experimental data.
Similar approaches have been followed by Owen and Rose (1973) for glass/polyester composites, and by
Beaumont and Phillips (1972) who suggested that the damage zone consists largely of material in which
the fibres were debonded from the resin. Owen and Bishop (1972) have found that as far as the fracture-
mechanics approach is concerned this zone can be treated in the same way as the plastic zone in metallic
materials, and they used measured values of the zone size to correct Kc results obtained from a variety of
GRP laminates. By determining the fracture toughness of plates up to 105mm wide in this way they were
able to show that Kc could be considered to be a genuine material parameter for design purposes. The
work of Owen and his co-workers has largely been carried out by testing wide, notched plates in tension.
One of the difficulties in comparing results obtained by different investigators, however, is that the
various test methods are sometimes found to give quite different data, and it is not clear what type of test
can be said to give absolute values.

Phillips (1974b) measured G and K values for 0/90 carbon/epoxy and carbon/glass composites and
concluded that KC was independent of crack size and could therefore be used to predict composite
strength. A fracture-mechanics analysis gave fracture energies that agreed quite well with those obtained
by other methods, indicating the validity of the Griffith-Irwin criterion for crack-propagation normal to
fibres. He obtained KC values of 6-25 MPa√m for various CFRP materials, GC being between 1 and 20
kJm-2. For the carbon-fibre/glass material he obtained a KC value of about 10 MPa√m and a GC of about
2 kJm-2.


In attempting to use the concept of KC for a unidirectional GRP, Barnby and Spencer (1976) showed
that the compliance calibration is significantly different from that for a continuous elastic medium, and is
non-conservative (ie. the error goes the wrong way). With an experimental calibration involving a
fourth-order polynomial they obtained KC values of about 15MPa√m, similar to those of Owen and Rose
(1973), but observed that the failure mode was not transverse to the fibres.

An example of what can be achieved by a fracture-mechanics type of analysis applied to a series of
carbon-fibre laminates is illustrated by some work reported by Dorey (1977). He shows that when an
attempt is made to compare a range of laminate lay-ups having different notch sensitivities,
measurements of the apparent fracture toughness, KC = σc√(πa), give results that were strongly
dependent on the test-panel width and the notch length and acuity, and which do not therefore permit of
easy evaluation of the relative toughness of different laminates. By contrast, when the results are
adjusted by the application of a sample finite-width correction factor, α, and the damage-zone size
correction factor, rρ, of equation 5.28:

Kc = ασ√[π(a + rρ)] ....................................................................................................... [5.29]

the resulting values become independent of notch and sample geometry and can therefore be used to
make comparisons of materials characteristics such as resin type, ply distribution and layer thickness.
His results illustrate the wide variations in fracture toughness that may be observed in any given class of
composites as a result of the use of different resins and alternative arrangements of the plies in (0,±45)
laminates.

Bishop and McLaughlin (1979), working with laminates similar to those of Dorey, summarized the
materials characteristics that affect the toughness of this type of composite. They showed that the notch-
sensitivity of a (0,±45) CFRP laminate in tension can be reduced by avoiding too high a bond strength,
since in general all lay-ups tend to become brittle if the bond strength is too high. At modest levels of
interfacial bonding, increases in toughness can be obtained by selecting ply angles and stacking
sequences, and by increasing the thickness of the plies. Thicker plies tend to delaminate more readily
than thin plies. Thus, increasing the laminate thickness by repeating the basic stacking sequence has little
effect on failure characteristics, but increasing the thickness of the layers leads to more delamination and
shear cracking, so that toughness is increased. Laminates with four-ply layers, for example, had notched
tensile strengths some 50% greater than similar laminates with single-ply layers. The load-carrying
ability of 0° plies is increased by shear splitting (in the 0° plies), but is reduced by cracks in adjacent 45°
plies: 0/45 ply interactions are of course reduced when delamination occurs. Thus, even if fibres, resins,
and bond strengths are pre-selected on the basis of some optimisation procedure for strength and
stiffness, further optimisation for improvements in toughness can still be done in terms of fibre angles,
stacking sequences and ply thickness.

Some conclusions from work by Wang et al (1975) on through-thickness cracks in cross-plied
laminates may serve as a summary of these ideas. It is clear that the stress field round a crack is severely
modified by the introduction of damage consisting of sub-cracks parallel to the fibres in each ply. A core
zone can be defined around the main crack tip such that outside this zone the stress field is undisturbed
while inside the zone the stress field is relaxed by high levels of inter-laminar stress that are present.
Mixed-mode crack propagation is therefore automatically implied. These sub-cracks serve a function
similar to the plastic zone in metals, and crack resistance results from reduction of the in-plane stress
field at the main crack tip. The relaxed core zone is embedded in a stress field at the main crack tip
which is unaltered by the sub-cracks, and the in-plane stresses follow a classical r-½ stress singularity in
the near-crack-tip region.

5.7 CONCLUSION


Under some circumstances, linear-elastic fracture mechanics can be used directly. Typically this
applies to brittle composites with high fibre/resin bond strengths and brittle matrices. In such materials
crack paths may be planar, or nearly so, and damage zones small (eg. ≤ 1mm), and the effects of
composite heterogeneity are not apparent. Similarly, it is likely that in random composites like chopped-
strand-mat laminates or in short-fibre moulding compounds like DMC and SMC, anisotropy will not be
a problem and fracture mechanics may be used. It has also been observed that for specific cases of cracks
running parallel with the fibres (in a unidirectional laminate) or delaminating cracks in a complex
laminate, crack growth appears to obey fracture-mechanics requirements. In such cases as these the
modified fracture mechanics can often be used for
cracking under both monotonic and cyclic loading
conditions.

By contrast, in tough composites damage is
complex and single-crack failure may not occur. In
cases where a single crack does propagate, a plastic-
zone or process-zone correction factor may work, but
very large samples are often needed to obtain sensible
results. Low Vf composites are also likely to cause
difficulties because the assumption of structural
uniformity is no longer valid.

An important generalisation is that generalisations
should not be made! There is always likely to be
Figure 5.27. Schematic illustration of the general
relationship between fracture toughness and uncertainty over the question of validity of LEFM for
strength for families of metallic materials. any given case.

Designers dealing with metallic materials 100
recognise that, in general, high toughness and
high strength are mutually exclusive
Toughness, KQ , MPa.√m

80
characteristics of materials, and that there is
roughly an inverse relationship between
toughness and strength. This is illustrated 60
schematically in Figure 5.27. Markedly
contrasting behaviour in composites was shown 40
by Harris et al (1988) who examined many
regression line:
published and unpublished results, obtained over 20 KQ = 65σf
a 15-year period, for a wide variety of composite (R = 0.933; n = 67)
types. The range of materials included moulding 0
compounds, cross-plied and quasi-isotropic 0 0.5 1.0 1.5
laminates, unidirectional composites, glass, Tensile strength, GPa
carbon, Kevlar-49 and hybrid laminates, and
glass- and ceramic-matrix composites. They Figure 5.28. Relationship between apparent toughnesses,
showed that it was possible to plot measured KQ, of a wide variety of composite materials and their tensile
values of K (the candidate fracture toughness, strengths. The dashed lines are prediction bands at the 90%
KQ), measured in tension, against the tensile level (Harris et al, 1988).
fracture stress and show, as seen in Figure 5.28,
that there is a reasonably good linear relationship between the two. Clearly a substantial amount of
scatter is to be expected, given the different experimental and analytical methods used by the large
number of authors whose work appears on this diagram, but the correlation is statistically significant. A
crude analysis on the basis of the familiar approximate relationship:

KQ ≈ 1.2.σc √(πa)................................................................................................. [5.30]

indicates that the intrinsic crack dimension for all of this diverse range of materials is about 1 mm, which
is identical with the result of Waddoups et al (1971) for one specific laminate. A similar result was also


shown by Wetherhold et al (1986). We may suppose then that the conclusion of Waddoups et al is
generally valid and a crack-tip process zone of the order of 1mm across forms near cracks in most
composites prior to failure. This is of the order of 100 fibre diameters for composites like GRP and
CFRP, but it is clear from the results of Jamison referred to in Chapter 4 that crack nuclei of this size are
never found in practice. It appears, nonetheless, that regardless of the apparent notch-sensitivity or
otherwise of fibre composites, the micro-structural features which determine their resistance to crack
propagation also control their fracture behaviour in the absence of deliberately introduced notches.

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6. FATIGUE BEHAVIOUR OF FIBRE COMPOSITES

6.1 INTRODUCTION

O
ur familiarity with fatigue failure is so closely linked with the behaviour of isotropic,
homogeneous, metallic materials that we have tended to treat modern fibre composites as though
they were metals. The test methods used to study fatigue in metals have also been applied to
composites, and the interpretation of the results of such tests has often been clouded by our
perception of what constitutes metallic fatigue failure. It is natural that a designer wanting to substitute a
composite for a metal component should want to test the new material by applying a cyclic loading
régime of the same kind as the component would be required to sustain in service in order to prove that
the composite will perform as well as the metal. But it must not be assumed, a priori, that there is some
universal mechanism by which fluctuating loads will inevitably result in failure at stresses below the
normal monotonic failure stress of the material.

Fatigue of metals accounts for a high proportion of engineering failures and has been intensively
studied for more than a century. Design data have been accumulated for every conceivable engineering
metal and alloy, and the engineer has access to a comprehensive set of rules, some empirical and some
based on scientific understanding, with which to cope with any given design requirement, although some
designers often choose to ignore these rules. Fatigue in metals progresses by the initiation of a single
crack and its intermittent propagation until catastrophic failure. This occurs with little warning and little
sign of gross distortion, even in highly ductile metals, except at the final tensile region of fracture. In
ordinary high-cycle (low-stress) fatigue, where stress levels away from the crack tip are low, the
properties of the metal remote from the crack are only slightly changed during fatigue. It is not a general
feature of fatigue in metals and plastics that the strength of the material is reduced by cyclic loading,
although work-hardening or work-softening may occur in metals undergoing low-cycle (high-stress)
fatigue. The usual effect of fatigue at low stresses is simply to harden the metal slightly. Generally
speaking, a stronger material will have a higher fatigue resistance, the fatigue ratio (fatigue limit divided
by tensile strength) being roughly constant.

It is not uncommon for users of composite materials, even in the aerospace industry, to express the
belief that composite materials — specifically, carbon-fibre-reinforced plastics — do not suffer from
fatigue. This is an astonishing assertion, given that from the earliest days of the development of
composites, their fatigue behaviour has been a subject of serious study, and what is usually implied is
that, because most CFRP are extremely stiff in the fibre direction, the working strains in practical
components at conventional design stress levels are usually far too low to initiate any of the local
damage mechanisms that might otherwise have led to deterioration under cyclic loads.

The idea of using composites, especially CFRP, only at very low working strains raises two important
issues. The first is the obvious one that by using expensive, high-performance materials at small
fractions of their real strength, we are over-designing or, in more cost-conscious terms, we are using
them uneconomically. The second is that since anisotropy is a characteristic that we accept and even
design for in composites, a stress system that develops only a small working strain in the main fibre
direction may easily cause strains normal to the fibres or at the fibre/resin interface that may be high
enough to cause the kind of deterioration that we call fatigue damage. In designing with composites,
therefore, we cannot ignore fatigue. And it follows directly that, in addition to needing to understand the
mechanisms by which fatigue damage occurs in composites, we need access to procedures by which the
development and accumulation of this damage, and therefore the likely life of the material (or
component) in question, can be reliably predicted.

In this chapter, we shall concentrate mainly on the fatigue behaviour of polymer-matrix composites.
Since the majority of accessible research studies have concerned these materials, it is inevitable that even
a general picture of composites fatigue will be substantially coloured by our knowledge of fibre-

chapter six Fatigue e
131

reinforced plastics (FRPs). Specific reference to metal- and ceramic-matrix composites will be made
after a discussion of FRPs.

6.2 DAMAGE IN COMPOSITES
Unlike metals, composite materials are inhomogeneous (on a gross scale) and anisotropic. They
accumulate damage in a general rather than a localized
fashion, and failure does not always occur by the
propagation of a single macroscopic crack. The micro-
structural mechanisms of damage accumulation,
including fibre breakage and matrix cracking,
debonding, transverse-ply cracking, and delamination,
occur sometimes independently and sometimes
interactively, and the predominance of one or other
may be strongly affected by both materials variables
and testing conditions.

At low monotonic stress levels, or early in the life Figure 6.1 Degradation of composite strength by
of a composite subjected to cyclic loading, most types wear-out until the residual strength, σR, falls from the
of composite sustain damage, and this damage is normal composite strength, σc, to the level of the
distributed throughout the stressed region, as fatigue stress, at which point failure occurs
described in chapter 4, section 4.1.3. The damage does
not always immediately reduce the strength of the composite although it often reduces the stiffness. Such
strength reductions as might occur (in a process described as wear-out) are sometimes off-set in the early
stages of life by slight increases in strength, known as wear-in. These increases may be caused by such
mechanisms as improved fibre alignment which results from small, stress-induced, viscoelastic or creep
deformations in the matrix. At a later stage in the life the amount of damage accumulated in some region
of the composite may be so great that the residual load-bearing capacity of the composite in that region
falls to the level of the maximum stress in the fatigue cycle and failure ensues, as shown schematically in
Figure 6.1. This process may occur gradually, when it is simply referred to as degradation, or
catastrophically, when it is termed sudden-death. Changes of this kind do not necessarily relate to the
propagation of a single crack, and this must be recognized when attempting to interpret composites
fatigue data obtained by methods developed for metallic materials.

When a pre-existing crack is present in a composite
it may or may not propagate under the action of a
cyclic load, depending upon the nature of the
composite. In a metal-fibre/metal-matrix composite
the fibre itself will be subject to conventional fatigue
processes. Planar (or near-planar) crack propagation
can occur with both fibres and matrix exhibiting
normal signs of fatigue, as shown in Figure 6.2, and
the material behaves simply like a homogeneous metal
of higher strength than the unreinforced matrix. In
highly anisotropic composites of high Vf the crack
will often refuse to propagate normal to the fibres
(mode 1) but will be diverted into a splitting mode. In
composites like unidirectional CFRP this may result in
a brittle, end-to-end splitting failure which simply
eliminates the crack (Figure 6.3) while in a metal-
matrix composite the splitting will be preceded by a
localized but elongated region of intense plastic
Figure 6.2. Fatigue fracture surface of an MMC deformation (McGuire and Harris, 1974). By contrast,
consisting of tungsten filaments in an Al/4%Cu matrix
showing brittle fracture of both fibres and matrix in GRP laminates containing woven-roving or
(McGuire and Harris, 1974). Compare with Figure 5.2 chopped-strand mat reinforcement crack tip damage
which shows a tensile fracture surface of the same
composite.

132

may remain localized by the complex geometry longitudinal splitting
of the fibre array and the crack may proceed
through this damaged zone in a fashion
analogous to the propagation of a crack in a
plastically-deformable metal (Owen and stress
Bishop, 1974).

6.2.1 Definition of failure and Figure 6.3 End-to-end splitting parallel to fibres in a
unidirectional composite, resulting in effective removal of
experimental scatter the stress concentrator.
Smith and Owen (1968) have demonstrated
the extent to which variability affects the results of fatigue tests and have emphasised the value of
replicate testing. This variability stems not only from the statistical nature of the progressive damage
which leads to fracture of a composite, but more particularly from the variable quality of commercial
composite materials. They have also pointed out that in many kinds of composite the number of cycles to
complete separation of the broken halves of a sample is a definition of failure that becomes quite
meaningless if the sample has lost its integrity and its ability to sustain an applied stress as a result of
extensive resin cracking, arguing the need for careful (and relevant) definition of the failure criterion.
They have nevertheless shown that designs based on the use of debonding or the onset of resin cracking
as failure criteria would drastically impair the economic use of GRP.

6.3 REVIEW OF SOME CLASSICAL FATIGUE PHENOMENA
6.3.1 Stress/life curves
A good deal of the work that has been done on the fatigue of fibre composites reflects the much more
extensive body of knowledge relating to the fatigue of metals, and this is not altogether unreasonable
since the established methods of accumulating and analysing
metallic fatigue data provided a reliable means of describing
fatigue phenomena and designing against fatigue. The
formidable treatise of Weibull (1959), for example, is a
valuable source book for modern workers on composites
fatigue. The danger was, and is, in making the assumption that
the underlying mechanisms of material behaviour that give rise
to the stress/life (S/N or σ/logNf) curve are the same in metals
and composites (see Ashby and Jones, 1980, for a description
of the various kinds of metallic fatigue).
Figure 6.4 Schematic illustration of the
features of an S/N curve (sometimes
Before the development of fracture mechanics and its use in referred to as a Wöhler diagram).
treating metallic fatigue as a crack-growth problem, the only
available design information on fatigue behaviour was the σ/logNf curve (Figure 6.4). It represented
directly the perceived nature of fatigue in terms of experimental results, but gave no indication of the
mechanisms of fatigue damage, of the presence or behaviour of cracks, or of changes in the
characteristics of the material as a consequence of the fatigue process. The curve represents the stress to
cause failure in a given number of cycles, usually either the mean or median life of a series of replicate
tests at the same stress. Despite the anomaly (from a mathematical viewpoint) of plotting the dependent
variable on the abscissa rather than vice versa, the σ/logNf curve is nevertheless a useful starting point
for the designer, provided due attention has been paid to the statistical aspects of data generation, so that
the apparently simple failure envelope which it defines is associated with failure probabilities rather than
with some simplistic fail/no-fail criterion. It can then be used, as many designers prefer, without any
consideration of the underlying fatigue damage mechanisms, despite the availability of rather better
fracture-mechanics methods.

There is logic in using the semi-logarithmic form of the σ/logNf plot but there is no reason for
supposing a priori that the failure envelope would be linear on a log-log plot (Basquin’s Law for
metals). Some stress/life plots are linear on both kinds of plot, some on neither. The knowledge that for

133

bcc metals and alloys such as steels, the σ/logNf curve flattened out at long lives (>107 cycles, say) and
gave rise to a fatigue limit, or endurance limit, σe, which could be explained in terms of
dislocation/solute-atom models, and that the fatigue ratio, σt/σe, (σt is the monotonic tensile strength)
was roughly constant for a given class of alloys, gave confidence to users of this kind of design data.
Finally, there was no general idea that the metallic σ/logNf curve should necessarily extrapolate back to
a stress level which was related to any monotonic strength property of the material (at Nf = 1 cycle, for
example).

6.3.2 Constant-life diagrams
In order to use stress/life information for design purposes, a common procedure was to cross-plot the
data to show the expected life (or expected stress for some particular probability of failure) for a given
combination of the alternating component of stress, σalt, defined as half the stress range, ½(σmax – σmin),
and the mean stress, σm, which is ½(σmax + σmin).
envelope representing
The stress ratio, R, is σmin/σmax. In metallic
a given life or life
fatigue it was frequently assumed that
fraction
compression stresses were of no significance
because they acted only to close fatigue cracks,
unlike tensile forces. Master diagrams of this
kind are presented in a variety of forms, all more
or less equivalent, but the most familiar is that
which is usually referred as the Goodman factored line
diagram (Figure 6.5). For design purposes, it
was useful to have an equation to represent the
fail/safe boundary in this diagram, and it is the
linear relationship of Figure 6.5 that is
associated with the name of Goodman (1899)
Figure 6.5 Schematic illustration of a constant-life or
although others have been proposed, including
Goodman diagram. The stress σe is the endurance or fatigue
the earlier parabolic relationship of Gerber limit, and σt is the material tensile strength.
(1874). The linear and parabolic ‘laws’ have
been modified to include safety factors on one or both of the stress components. An important question,
for metals as well as for composites, relates to the minimum amount of test data that is needed to define
the failure envelope with a level of reliability sufficient for engineering design of critical components.

6.3.3 Statistical aspects of the analysis of fatigue data
The mechanical properties of practical composites are almost always variable as a consequence of the
variability of the material quality. It is not surprising, therefore, that the variability of the fatigue
response of a composite is even greater than that associated with metallic materials. Stress/life data may
be obtained by testing single samples at many different stress levels, or by carrying out replicate tests at
rather fewer stresses: the latter is usually considered to be the more satisfactory method because it
provides statistical information at each stress, and provides probability/stress/life curves in addition to
median-life or mean-life curves.

One of the problems is to know how many replicate tests should be done at each stress level since,
given the cost of fatigue-testing programmes, the smaller the number of tests that can be used to
establish a ‘safe’ σ/logNf curve, the better. From a statistical point of view it is usually accepted that at
least 20 replicate tests may be necessary before the user can have any confidence in a statistical analysis
of results (see, for example, Lee et al, 1997). But when stress/life curves are required at, say, five
different R ratios, even five tests at each stress level may be all that can be provided in a reasonable
amount of time, especially at long lives.

A variety of distributions have been used to characterize fatigue lives, but the three-parameter
Weibull function, described in Appendix 2, is usually considered to be the most appropriate model for
this purpose. The form of the cumulative distribution function used for fatigue is:

134

⎡ ⎛ Nf − a ⎞ m ⎤
P(Nf ; a, b, m ) = 1 − exp⎢− ⎜ ⎟ ⎥ ..................................................................... 6.1)
⎢ ⎝ b ⎠ ⎥
⎣ ⎦

where P is the probability of a life Nf, and a, the location parameter, defines a number of cycles for
which there is zero probability of failure. For metallic materials, the value of the shape parameter, m, is
often in the range 2 < m < 6 (Freudenthal and Gumbel, 1953), whereas several recent studies of CFRP
fatigue suggest that for these materials 1 < m < 2 (Whitney, 1981; Harris et al, 1997). It is interesting to
note that for an m value of unity the Weibull model reduces to a simple exponential distribution. The
exponential distribution, which is sometimes used to model distributions of failure times for the
reliability of a product, governs systems where age has no influence on the probability of failure. If a
component survives for a length of service t, the probability of failure within a subsequent time interval
from t to (t + Δt) is λΔt, where λ is a constant for all t. Thus, failure is a random event, and the system
does not deteriorate as a result of service. We would not normally consider this as being applicable to the
fatigue failure of reinforced plastics for which it is known that residual performance is certainly reduced
as a result of the accumulation of damage. One reason why the data may appear to fit an exponential
distribution is that the range of lives at a given stress level is often very wide, spreading sometimes over
two decades or more. Whereas when viewed by eye on a logarithmic life axis this may not appear too
serious, when considered in the Weibull analysis as a linear function, Nf , rather than as a logarithmic
function, logNf, the range could thus easily be judged to be bounded at the lower extreme at Nf = 0, and
hence the distribution may appear exponential. In reality, the shape of the Weibull distribution changes
from being of classical exponential form to being peaked over a very small range of values of m between
1 and 2.

Although we often work in terms of median or mean lives when discussing fatigue data, it is
important to recognize that to the designer of critical structures, such as aircraft components, the
requirements are much more restrictive, and it is likely that the desired failure probability for such a
structure would be at a very low level, say 5%, instead of the 50% level implied by the median life. An
even more appropriate ‘life’ parameter may be the minimum extreme value obtained from an application
of the theory of extreme values (Castillo, 1988). Extreme-value models are appropriate models for
describing many engineering phenomena and systems where the relevant parameters are the
characteristic largest and characteristic smallest values of a distribution. For example, in the case of the
life of a gas-turbine disc fitted with a large number of blades, the turbine life is determined by the life of
the shortest-lived blade — a ‘weakest-link’ model. Such a model, referred to as a Type III asymptote
model, is also applicable to the fatigue behaviour of engineering materials.

One of the characteristics of the Weibull model is what is known as its reproductive property (Bury,
1975). A consequence of this property is that for a population of results that is well modelled by the
Weibull distribution certain other features of the population, such as the minimum extreme values, will
also be described by a Weibull distribution. Put more formally, the exact distribution of the smallest
observations in sets of data that are described by a Weibull model also fits a Weibull model. Thus, if we
have replicate data sets of fatigue lives at a given stress and R ratio which is described by the two-
parameter version of equation 6.1 (see Appendix 2), with shape parameter m and scale parameter b (a
being equal to 0), the smallest observations of these data sets also exhibit a Weibull distribution similar
to that of the parent distribution and with the same value of the shape factor, m, but rescaled by n1/m. The
characteristic minimum value for a test sample of n tests will therefore be (b/n1/m) and the modal value of
the distribution (ie. the most probable value) will be given by:
1

b ⎛ 1⎞m
⎜1 − ⎟
nm ⎝ m⎠
1

If, on the other hand, the desired level of failure probability happens to be the 5% level (q = 0.05), then
the appropriate value of ‘life’ is given by:

135

1

⎡ ⎛ 1 ⎞ ⎤m
b ⎢ln⎜ ⎟⎥
⎣ ⎝ 1 − q⎠ ⎦

Thus, from a series of replicate data sets of fatigue lives at various cyclic stress levels, a
stress/life/probability (or S/N/P) diagram resembling that shown in Figure 6.6 may be drawn, provided
the distribution parameters, b and m, can be obtained for each data set. From a design point of view, the
lower curve in this diagram,
scatter in
representing some given failure strength
probability, is clearly of more scatter in
fatigue lives
interest to the engineer than any
other part of the data, but it can
only be obtained by statistical
means which requires reliable
values of the two distribution
parameters, b and m. This, in turn,
calls for replicate data sets of at
least 20 test results for each stress
level and R ratio, a requirement
which carries with it a serious
time/cost disincentive.
Figure 6.6 Effects of variability in strength and fatigue life on the definition of
However, Whitney (1981) has the σ/logNf curve. The scatterband of lives is jointly defined by the upper
suggested that where only small and lower probability limits of the fatigue data sets and the monotonic
numbers of life values are strength distribution.
available at a number of different
stress levels, the data may be pooled to give an overall value of the Weibull shape parameter, m, this
value then being used to obtain working stress/loglife curves for any given failure probability. This is
done by normalising each test-stress data set with respect to either the characteristic life, b (the scale
factor of the Weibull distribution for the data set) or the median life for the data set, pooling all data sets
for all stress levels and all R ratios, and then re-ranking them in order to allot a new failure probability
function to each point (see Appendix 2 for a discussion of ranking methods). The virtue of this
procedure is that a much larger population is being used to derive the value of the Weibull shape
parameter and calculations of an expected life based on that m value will be much more reliable than
ones obtained from the much smaller data sets for
2 each individual stress level. The Weibull
distribution for a pooled fatigue data set for a
1
carbon-fibre composite laminate is shown in
0 Figure 6.7. The shape parameter for the pooled
Ln[-ln(1 - P)]

-1 data is 1.1, and is similar to values obtained for
other CFRP laminates by, for example, Whitney
-2 (1981) and Harris et al (1997). The value of m
-3 obtained in this way can then be used, as shown
above, to derive stress/life curves for desired
-4
failure probability levels
-5 slope, m = 1.1
-6 6.3.4 Cumulative damage laws
-5 -4 -3 -2 -1 0 1 2 3
In most fatigue environments the mean and
Ln(Nf) alternating stress amplitudes vary and may be
presented as a spectrum of the frequency of
Figure 6.7. Two-parameter Weibull plot for pooled, occurrence of different stress levels. The concept
normalized fatigue data for a [(±45,02)2]S T800/5245 CFRP of deterioration as a result of damage
laminate. The pooled data are for all stress levels at 5
different R ratios (Gathercole et al, 1994). accumulation within the material with stress
cycling is used in combination with the notion of
residual strength, and the damage is quantified such that a parameter Δ represents the fraction of

136

catastrophic damage sustained after n cycles (where n < Nf ) so that at failure Δ = 1 and n = Nf. Thus, Δi
represents the fractional damage after cycling at a stress σi for a fraction of life ni/Ni. Various proposals
for damage laws have been derived, the most common of which is the Palmgren-Miner linear damage
rule (Miner, 1945). The accumulation of damage is supposed to be linear with number of cycles, and is
apparently independent of the value of the stress:

Δ = n/N .................................................................................................................. 6.2)

Thus, if the load spectrum is divided into blocks with ni cycles at stress levels σi, then equation 6.2
applies incrementally to each block and the total damage is the sum of the individual ‘block’ damages,
irrespective of the order of their application, so that:

ni
Δ= ∑N ( = 1 for failure) ................................................................................. 6.3)
i

This well-known, though rarely obeyed, rule was originally proposed for the prediction of the life of
metallic components undergoing fatigue, and although it is a standard feature of text-book presentations
of fatigue it has always been viewed with great suspicion by designers because it is often found to give
non-conservative results, ie. to predict lives greater than those observed experimentally. The simple
concept of damage defined by equation 6.3 is not, of course, a mechanistic concept. It has been applied
to many kinds of engineering materials, regardless of the actual nature of the damage mechanisms that
contribute to the gradual deterioration and ultimate failure of an engineering component subject to cyclic
loading.

The simplest step forward from the linear damage rule is to look for non-linear functions that still
employ the damage parameter Δ, as defined by equation 6.3. In the Marco-Starkey (1954) model, for
example, a simple non-linear presentation suggests an equation of the form:

Δ = (n/N)β ............................................................................................................... 6.4)

where β is a function of stress amplitude. The condition for failure when Δ = 1.0 is satisfied and the
Miner rule is a special case of this law with β = 1.0.

6.5 FATIGUE OF REINFORCED PLASTICS
6.4.1 Constant-stress fatigue curves
The dominant damage mechanisms in FRP vary both with the nature of the composite (the particular
combination of fibres and matrices, reinforcement lay-up, etc.), and with the loading conditions (tension,
bending, compression, etc.), and results for any one material under a given stress condition may appear
to fit no general pattern. This is easily illustrated with reference to the gradual accumulation of results
and materials development over the last two decades or so.

First, the forms of stress/life curves for various kinds of GRP have long been familiar to researchers,
as was the fact that the fall of the curve with reducing stress was considerable. This fall, or average
slope, is an indication of the fatigue resistance of the material, a more rapid fall being indicative of a
greater susceptibility to fatigue damage. Mandell (1982) attempted to show that fatigue damage in GRP
materials can be explained simply in terms of a gradual deterioration of the load-bearing fibres. He
analyses a great range of experimental data to demonstrate that the behaviour of composites with long or
short fibres, in any orientation, and with any matrix, can be explained in this way. He does this by
effectively forcing σ/logNf curves to fit a linear law of the form:

σ = σt – BlogNf ...................................................................................................... 6.5)

137

where σ is the peak cyclic (tensile stress), σt is the monotonic tensile fracture stress, Nf is the fatigue life
of the material at stress σ, and B is a constant, as illustrated in Figure 6.8. For a wide and disparate range
of GRP materials, including moulded reinforced thermoplastics, Mandell obtained a constant value of
about 10 for the ratio σt/B, with very little spread. In principle, the idea that the controlling mechanism
in fatigue of composites is the gradual deterioration of the load-bearing fibres is logical and inevitable.
What controls the actual life of any given sample or composite type is simply the manner in which other
mechanisms, such as transverse-ply cracking in 0/90 laminates, or local resin cracking in woven-cloth
composites, modify the rate of accumulation of damage in the load-bearing fibres. Nevertheless, a single
mechanistic model which includes randomly-reinforced dough-moulding compounds and injection-
moulded thermoplasts as well as woven
and non-woven laminates is unexpected.
1500
The σ/logNf curves for many GRP
Tensile strength, σt , MPa

materials are not linear, however, as
illustrated by the selection of data for a
1000
group of glass-fibre composites shown in
Figure 6.9. With GRP there are two
factors which complicate the appearance
500 of the stress/life curves. As a consequence
σt /B = 10.4
of the high failure strain of the glass fibres
and their sensitivity to moisture,

0 i) the tensile strengths of GRP materials
0 50 100 150 are sensitive to strain-rate and
Slope of stress/loglife curve, B, MPa/decade temperature, and
ii) during cycling at large strains there is
Figure 6.8. Relationship between composite tensile strength and usually a significant rise in temperature
the slope of the fatigue σ/logNf curve (after Mandell, 1982). Data as a result of hysteretic heating which
points taken from the literature represent a wide range of GRP is not easily dissipated by the non-
composites.
conducting constituents of the GRP.

The consequences of this have been demonstrated by Sims and Gladman (1978). First, when fatigue tests
are carried out at constant frequency over the whole stress range of interest, the deformation rate is
usually considerably greater than the rates normally used for measuring the monotonic strength, and as a
result, since it is common for the strength to be included, as shown in Figure 6.9, as the point on the
extreme left of the σ/logNf curve (ie. notionally 2.0
at Nf = 0.5 for a repeated tension test, or Nf = unidirectional
[(0,90)2]S (prepreg)
0.25 for a fully reversed tension/compression
[(±45,02)2]S
test) that measured strength is often actually 1.5
Peak stress, GPa

0/90 (woven)
lower than a value measured at the fatigue-test chopped-strand mat
frequency. Second, if tests are run at constant dough moulding compound
1.0
frequency, the effective strain rate at each stress
level will be different (higher at lower stresses),
and the measured life values for each stress will 0.5
not be associated with a common base-line of
material behaviour: ie. the fatigue curve will
refer to a range of stresses, σ, which are 0
-1 0 1 2 3 4 5 6 7 8
proportions of a variable material property, say
LogNf
σt( ε ), instead of the material property that we
&
normally define as the strength, σt. And third, Figure 6.9. Stress/loglife curves for a range of GRP materials
as the peak stress level falls and the life of the at an R ratio, σ /σ , of 0.1. Data points are omitted for
min max
sample extends, the degree of hysteretic heating clarity, but may be found in the original diagram (Harris,
rises, and the effective baseline strength of the 1994). The stress for a ‘life’ of 0.5 cycles is the monotonic
material therefore also falls, so reversing the tensile strength measured at a loading rate equivalent to that
used for the fatigue tests.
effect of the higher effective deformation rate.

138

As Sims and Gladman pointed out, the only way to avoid the effects of these interacting processes is to
ensure that all σ/logNf curves for GRP materials are determined at a fixed rate of load application
(RLA), that the material strength is measured at the same RLA, and that the hysteretic effects are either
eliminated or accounted for. These corrections can make substantial differences to the shapes of
stress/life curves and to the validity of the design data obtained from them: many otherwise curved
σ/logNf graphs are also rendered more linear by these corrective measures.

By contrast with GRP, carbon-fibre composites are largely rate insensitive and because they deform
less than GRP under working loads and are reasonably highly conducting, hysteretic heating effects are
usually insignificant (Jones et al, 1984). Some of the earliest CFRP, reinforced with the HM (high-
modulus, or Type 1) species of carbon fibre had such low failure strains that fatigue experiments in
tension were extremely difficult to carry out, and some of the first tension σ/logNf curves published
(Beaumont and Harris, 1972; Owen & Morris, 1972) were almost horizontal. The S/N line hardly fell
outside the scatterband of measurements of the monotonic tensile strength, the fatigue response being
effectively dominated by tensile failure of the fibres alone and the slight reduction in resistance after 106
or 107 cycles appeared to be a result of small viscoelastic (creep) strains rather than to cycling effects per
se (Fuwa et al, 1975). Only in torsion or bending tests was the slope of the σ/logNf increased as these
different stress systems allowed other damage mechanisms to occur.

A very small failure strain is often a disadvantage in engineering materials (some of the early CFRPs
were extremely brittle) and developments in carbon-fibre manufacture led to materials that possessed
higher failure strains (lower elastic moduli), although these improvements were often offset by
concomitant reductions in strength. By the end of the 1970s, the most common polyacrylonitrile-based
fibres, designated HMS, HTS and XAS, covered a range of moduli from about 400GPa down to
200GPa, and the tensile stress/loglife data for these materials fell approximately on straight lines with
slopes which increased with decreasing fibre stiffness (Sturgeon, 1973). This is roughly in accord to with
the Mandell model except that Mandell’s correlation was with composite strength rather than stiffness.

Increasing awareness on the part of designers of the qualities of fibre composites, and of carbon-fibre
composites in particular, resulted in demands for fibres with combinations of strength and stiffness
which were different from those which characterized the earlier materials. A particular call was for
higher failure strains in association with high strength, and these demands led to developments in
carbon-fibre manufacture and the availability of a much wider range of fibre characteristics. The fatigue
performance of composites based on these newer fibres is clearly different from that of earlier CFRP, as
illustrated by the range of results shown in Figure 6.10. These data are all for composites with the lay-up
[(±45,02)2]s and the curves are polynomial fits to experimental data. Perhaps the most notable feature of
this collection of data is that the monotonic strength characteristics of the reinforcing fibres are not
directly translated into the fatigue response, ie. the strongest fibres do not always generate the most
1.75 fatigue-resistant composites. And whereas the
earliest of CFRP composites had approximately
5 IM7/977
7
6 T800/924 linear, or even slightly upward turning σ/logNf
1.50 curves, the newer, increasingly ‘high-
Peak stress, GPa

7 T800/5245
6
performance’ composites show downward
5 curvature. We note that none of these curves for
1.25 4
3
the newer materials has reached any kind of
2 stable fatigue limit within the convenient
1 XAS/914
1.00 2 XAS/913 laboratory test window extending to 107 cycles.
3 HTA/913 1
4 T800/6367 This downward curvature also occurs in
0.75 composites reinforced with Kevlar-49 aromatic
0 1 -1 2 3 4 5 6 7 8
LogNf
polyamide (aramid) fibres (Jones et al, 1984;
Fernando et al, 1988). Typical behaviour of a
Figure 6.10. Median stress/loglife curves at R = 0.1 for seven cross-plied KFRP composite is compared with
varieties of carbon-fibre composite, all with the lay-up those of similar GRP and CFRP materials in
[(±45,02)2]S (Harris et al, 1997). Figure 6.11, plotted here in terms of initial peak

139

strain, rather than stress. The composites are comparable in every respect except for the reinforcing
fibres. The performance of the KFRP over the first three decades is similar to that of the CFRP on a
normalized basis in that the slopes of the strain/life curves are similar. At maximum cyclic strain (or
stress) levels only marginally below the tensile-failure scatter band, however, the fatigue resistance
deteriorates rapidly, and a change in controlling failure mode occurs. At the highest stress levels (first
three or four decades in Figure 6.11) failure of the KFRP occurs by the normal fibre-dominated failure
process. At lower stresses, however, (much larger numbers of cycles) the internal structural weaknesses
of the aramid filaments are exposed to the effects
of the cyclic stress and the fibres themselves 5
exhibit much lower fatigue resistance. This GRP
apparent weakness can be overcome to some 4

Initial tensile strain, %
extent by preventing complete unloading of the
composite between cycles. Effectively, the down- 3 KFRP
turn of the σ/logNf curve, which is not seen in the
0/90 CFRP in Figure 6.11, can be delayed by 2
raising the minimum cyclic stress level.
1
These results for carbon, glass and aramid CFRP
laminates illustrate very well a strain-control
0
model of fatigue proposed by Talreja (1981). He -1 0 1 2 3 4 5 6 7 8
suggests that the strain/loglife curves for polymer- LogNf
matrix composites may be thought of in terms of
three régimes within which separate mechanisms Figure 6.11. Strain/life curves for [(0,90)2]S laminates of
control fatigue failure. At high stress levels fibre composites reinforced with HTS carbon, Kevlar and E-
breakage occurs (with or without interfacial glass fibres in a Code-69 epoxy resin (Fernando et al,
1988: repeated tension fatigue, R = 0.1: RH = 65%).
debonding) which leads to failures within the
normal tensile failure scatter band of the 0° plies.
At lower cyclic stress levels, however, although statistical fibre breakage still occurs, it does not lead so
rapidly to composite destruction and other competing mechanisms then have time to occur. These other
mechanisms — matrix cracking, interfacial shear failure, etc. — can then influence the overall damage
state provided the composite working strain level is sufficiently high, and the slope of the strain/loglife
curve therefore begins to fall. There is, however, some notional fatigue-limit strain for the matrix itself
and if working strains do not rise above this level the composite should not, in principle, fail in fatigue.
The strain/loglife curve ought therefore to flatten out again and something in the nature of an endurance
limit should be observed. Whether or not some or all of these stages are observed will clearly depend on
the characteristics of the constituents and the lay-up geometry. In Figure 6.11, where the influence of the
fibre deformation characteristics is more visible than in a stress/life plot, it can be seen that high working
strains in the GRP prevent the establishment of stage 1 of the Talreja model, the curve moving almost
immediately into stage 2, whereas the higher stiffness of the Kevlar fibre delays this transition in the
aramid composites. Working strains in the CFRP are rarely sufficiently high as to exceed the matrix
fatigue limit and so the strain/life curve (like the σ/logNf curve) retains the low slope, characteristic of
stage 1, for the entire stress range shown. It can be seen from the shapes of the σ/logNf curves for
various CFRP materials in Figure 6.10 that many of these high-performance composites exhibit similar
features in tensile fatigue, regardless of the specific nature of the degradation mechanisms, although the
extent to which some or all of the stages of Talreja’s model are visible within a practical experimental
test window depends on the nature of the constituents and of the fibre/matrix bond.

6.4.2 Materials factors affecting fatigue behaviour of reinforced plastics
Brittle fibres like glass, boron and carbon should not show the characteristic weakness of metals
under fatigue loading conditions. Composites containing them should also, therefore, in principle, be
fatigue resistant if the fibres carry the major part of the load and provided the reinforcing phase is not so
extensible as to permit large elastic or viscoelastic deformations of the matrix as a function of time.
These conditions are fulfilled when composites unidirectionally reinforced with high-modulus fibres like
carbon or boron are tested in repeated tension, and their σ/logNf curves are, in general, very flat. As
cycling continues, however, even small viscoelastic movements in the resin lead to local redistributions

140

of stress which permit some random fibre damage to occur, and similar damage will occur in the
neighbourhood of any stress concentrations. As a function of time, then, rather than as a function of the
number of stress reversals, damage levels will build up to some critical level when the composite
strength is no longer above the peak cyclic stress level, and failure will occur.

When the stress is not simply tensile, uniaxial, and aligned with the fibres more severe loads are
placed on the matrix and the fatigue resistance of the composite is reduced. This occurs in simple
flexural fatigue (Beaumont and Harris, 1972) and tension/compression loading of undirectional
composites, and also in fatigue tests on any kind of laminated materials. Ramani and Williams (1976),
for example, tested 18-ply, (0,±30) carbon/epoxy laminates in various combinations of axial
tension/compression cycling and showed an increasingly adverse effect on fatigue resistance as the
compressive stress component was increased. Kunz and Beaumont (1975) and Berg and Salama (1972)
showed that fatigue cracks propagated in CFRP during purely compressive cycling through the
spreading of zones of fibre buckling failure. As we saw in chapter 4, compressive buckling is a
characteristic mode of failure in fibre-reinforced materials of many kinds and it is usually initiated by
local matrix shear failures, sometimes associated with debonding in imperfectly bonded composites.
Ramani and Williams observed extensive shear damage resulting from matrix shear and from
interlaminar and interfacial weakness.

Compression and shear stress systems will
200
also be likely to be even more damaging to
composites containing aramid fibres than to
CFRP
CFRP or GRP, as indicated by the poor
Peak stress, MPa

150
GRP
KFRP compression response of KFRP described in
chapter 4. The shear weakness is also clearly
100
manifested in tensile fatigue of ±45 KFRP
laminates, by comparison with similar samples
50 of CFRP and GRP, as shown in Figure 6.12. In
this mode of stressing the matrix and interface
would normally be expected to dominate
0
-1 0 1 2 3 4 5 6 7 8 composite behaviour, the fibre characteristics
LogN f playing little part. This is demonstrated in
Figure 6.12 by the CFRP and GRP results, but
Figure 6.12. Stress/loglife curves for the same materials as the intrafibrillar weakness of the aramid fibre
those of Figure 6.11 but tested at 45°to the main fibre results in a considerable reduction in composite
directions: R = 0.1; RH = 65% (Jones et al, 1984)
fatigue resistance, and microscopic
observations of KFRP samples that have failed
in this mode clearly show the kinking and splitting of the fibres resulting from this weakness (Figure
6.13). Similarly, when tested in torsion the weak parts of a composite (ie. the resin and the interface) are
loaded directly by the shear forces the strength and rigidity of the fibres contributing relatively little to
fatigue resistance and the torsional rigidity of the material falls substantially during cycling. Some
typical shear strain/loglife curves for two types of
unidirectional carbon/epoxy composites are shown in
Figure 6.14 (Phillips and Scott, 1976). The curves were
obtained under constant angular (torsional) cycling
conditions and the fatigue life is defined as the number of
cycles to the occurrence of cracking.

There is much evidence from fatigue tests on reinforced
plastics that the matrix and interface are the weak links as
far as fatigue resistance is concerned. It appears that lower
reactivity resins give better low-stress fatigue life in GRP,
the optimum resin content being between 25 and 30 vol.%,
although variations in resin content affect the fatigue Figure 6.13. Splitting and kinking damage in
strength at 106 cycles hardly at all by comparison with Kevlar-49 fibres after fatigue cycling (Fernando
et al, 1988)

141

their effect on tensile strength (Owen,
1970). By contrast, variations in resin 0.06
properties appear to have little effect on the epoxy resin
fatigue strength of CFRP (Owen and Morris, 0.05

Shear strain amplitude
1970). Any materials treatment or
0.04
processing that can improve the resistance of HT-carbon/epoxy
the matrix to crack propagation or the
0.03
interfacial adhesion is likely to improve
fatigue properties. Conversely, exposure of 0.02
reinforced plastics to water often results in HM-carbon/epoxy
some degree of plasticization of the matrix 0.01
and weakening of the interfacial bond, and
in GRP and aramid-fibre composites it may 0
-1 0 1 2 3 4 5
also affect the performance of the fibres
LogNf
themselves (Jones et al, 1984). It follows
that the fatigue of almost all FRP will be
environment sensitive, the least sensitive carbon/epoxy 6.14. Strain/loglife curves for unidirectional
Figure
composites together with that of the unreinforced
being those consisting of moisture-resistant matrix resin. The reinforcements are type 1 (high-modulus)
plastics like the polyaryls (PEEK, etc.) and type 2 (high-strength) carbon. The curves were obtained
reinforced with carbon fibres which are under constant angular (torsional) cycling conditions. The
themselves insensitive to degradation in fatigue life is defined as the number of cycles to the
occurrence of cracking. (Phillips and Scott, 1976)
moist atmospheres (Dickson et al, 1985).

For manufacturing reasons (eg. to secure good ‘drape’ characteristics when doubly curved surfaces
are required) there is considerable advantage in using woven fabrics as reinforcements. Some of the
effects of this on strength and toughness have already been seen in previous chapters, and it would be
expected that these effects might follow through into the fatigue response. Curtis and Moore (1985)
showed that both the monotonic tensile strength and the fatigue strengths of CFRP laminates reinforced
with woven fabric were significantly poorer than the properties of similar materials made with non-
woven fibres, largely as a consequence of the distortion in the load-carrying 0° fibres and the stresses
induced at weave cross-over points. However, when the fabric was oriented at 45° to the loading
direction both tensile and fatigue performances were slightly better than for non-woven material.

6.4.2.1 Hybrid Composites
The simplest expectation for the fatigue stress/life curve of a hybrid laminate would be that it would
fall between those of the two single-fibre components, but published results do not offer an obvious
picture on which to base predictive models. It would be expected that the overall fatigue response would
depend upon the extent to which the particular fibre mix controlled strain levels in the composites, and,
if aramid fibres were involved, on the extent to which the intra-fibrillar weakness of that fibre was
exposed to the stress. Some early experiments by Phillips et al (1976) on woven-cloth GRP/CFRP
laminates suggested that adding CFRP to GRP improves its fatigue resistance more or less in proportion
to the amount of CFRP added, ie. in accordance with the rule of mixtures. Hofer et al (1978) also found
that the fatigue stress of unidirectional HTS-carbon/S-glass hybrids obeyed the rule of mixtures when in
the as-manufactured state, but showed a positive deviation from linear when the composites were
hygrothermally aged. Fernando et al (1988) found that the failure stresses for lives of 105 and 106 cycles
for unidirectional carbon/Kevlar-49/epoxy hybrids were linear functions of composition for both
repeated tension and tension/compression loading. But since the tensile strengths of the same series of
hybrids were given by the failure-strain model (Chapter 4, section 4.19), the fatigue ratio (fatigue stress
for a given life divided by the tensile failure stress) showed a marked positive deviation from the linear
weighting rule. The same workers (Dickson et al, 1989) found that unidirectional CFRP/GRP fatigue
strengths fell above a linear relationship, and the fatigue ratio therefore showed an even more marked
positive synergistic effect. This suggests that factors controlling monotonic tensile (and compression)
failure do not necessarily continue to determine failure under cyclic loading conditions, and that for
fatigue applications there appear to be positive benefits in using hybrids in place of single-fibre
composites.

142

6.4.2.2 Short-fibre composites
Although the fatigue resistance of high-Vf
composites reinforced with continuous, 2.5
aligned, rigid fibres is usually high, short-fibre
composites of all kinds are much less resistant 2.0 long fibres
to fatigue damage because the weaker matrix

Peak stress, GPa
is required to sustain a much greater 1.5
proportion of the fluctuating load. Local
failures are easily initiated in the matrix and
1.0 short fibres
these can destroy the integrity of the
composite even though the fibres remain
0.5
intact. The interface region is particularly
susceptible to fatigue damage since the shear
stresses at the interface may be reversing their 0
-1 0 1 2 3 4 5 6 7 8
direction at each cycle and there is always a LogNf
high shear stress concentration at fibre ends. It
is also possible that in both random and Figure 6.15. Stress/loglife data for unidirectional composites of
aligned short-fibre composites the ends of XAS/914 carbon/epoxy laminates reinforced with continuous
fibres and weak interfaces can become sites and discontinuous fibres (R = 0.1) (Harris et al, 1990).
for fatigue-crack initiation.

The beneficial effects of short fibres in materials like injection-moulding compounds have been well-
known for 40 years or more, and although increases in strength in such materials are modest, increases in
toughness are much greater for the reasons discussed in Chapter 5. Since fatigue resistance depends in
part on strength and in part on crack resistance, it would therefore be expected that these benefits would
be advantageously translated into fatigue response. Lomax and O’Rourke (1966) reported that the
endurance limit of polycarbonate at two million cycles is increased by a factor of seven when the
polymer is filled with 40vol.% of 6.4mm long glass fibres. Lavengood and Gulbransen (1969)
determined the number of cycles to failure of short-fibre boron/epoxy composites and found that for
cycling at any given fraction of the failure stress the fatigue life increased rapidly with fibre aspect ratio,
levelling off at l/d ratios of about 200. This therefore represents a critical aspect ratio above which the
fatigue strength is a constant proportion of the flexural strength, as shown in Figure 6.16. These authors
also studied the behaviour of aligned asbestos/epoxy composites and found them to behave in roughly
the same way as boron/epoxy composites containing 25mm-long fibres. Owen et al, (1969) showed that
the fatigue strength at a million cycles of polyester
resin reinforced with random glass-fibre mat varied 106
between 15 and 45% of the ultimate tensile
strength. Huffine et al (1973) studied the tension
Fatigue life, Nf cycles

and flexural fatigue behaviour of glass/epoxy
composites containing 44 vol.% of aligned E-glass 105
fibres 12.5mm long. Their results suggested that
this material was potentially useful for dynamic
60%
structural applications since it appeared to have an
endurance limit of 40% of the ultimate tensile 104 70%
80%
strength. 90%

A comparison of the relative fatigue responses 103 1
of composites containing continuous and 10 102 103
discontinuous fibres is shown in Figure 6.15. The Aspect ratio of fibre
results are for unidirectional composites of the
same carbon/epoxy system, XAS/914, in which the Figure 6.16. Fatigue life versus aspect ratio for an applied
well-aligned prepreg for the short-fibre material, stress of the indicated percentage of the composite tensile
containing chopped fibres 3mm long, was strength. The materials are short-fibre-reinforced
produced by the glycerol-alignment method boron/epoxy composites; repeated tension (Lavengood &
described in Chapter 2: the fibre volume fraction is Gulbransen, 1969).

143

0.35. The continuous-fibre laminate was produced from normal commercial prepreg. It can be seen that
although the strength of the short-fibre composite is much lower than that of the continuous-fibre
laminate, its fatigue response, as characterized by the slope of the σ/logNf curve, is better. In fact, if the
fatigue data are all normalized with respect to the monotonic tensile strengths of the two materials, the
rate of fall-off is much greater for the conventional laminate. Working with similar aligned short-fibre
composites, Moore (1982) also found that the fatigue behaviour of these materials compared favourably
with those of continuous-fibre composites, although progressive cracking occurred from the fibre ends
and fibre surface effects appeared to give more problems. It is assumed, therefore, that less fibre damage
resulting from fatigue can occur if the fibres are already short, and the short-fibre ends apparently do not
act as sources of excessive damage. It is not certain what the effect of fibre length would be in aligned
composites such as these, but Hitchen et al (1995) have shown that in random-short-fibre carbon/epoxy
composites the fatigue life is independent of fibre length at any stress level. They also showed that the
more flexible the matrix, the shorter the fatigue life. Friedrich et al (1986) found that the incorporation
of short glass or carbon fibres into PEEK to produce injection-moulding compounds produced only
slight improvements in the fatigue and fracture toughness of this material.

Harris et al (1977) studied the fatigue behaviour of commercial polyester DMC in both the dry and
wet-aged conditions and for samples with drilled holes. Their results were characterized by a good deal
of scatter, but indicated that in the low-stress (long-life) range the effects of moisture and stress
concentrations were less marked than in the higher stress range. They showed that when their results
were normalized by dividing by tensile strength, the σ/logNf curve for the DMC fell only slightly more
rapidly than the curves for other types of GRP (see Figure 6.9). Atzori et al investigated the effects of a
wide range of material and environmental variables on the fatigue behaviour of moulding compounds
and developed a rule for estimating the fatigue strength from the monotonic tensile strength that was a
modification of that of Mandell described in section 0.

In some cases the fatigue resistance of the matrix plastic has been found to be diminished by
reinforcement with short fibres. For example, Shaver and Abrams (1971) found that randomly reinforced
glass/Nylon composites had only half the endurance of the unfilled polymer. It should also be noted that
the fatigue behaviour of most reinforced plastics will always be worse when the material is exposed to a
wet environment.

6.4.3 Other factors affecting the fatigue of reinforced plastics
6.4.3.1 Machines and specimens
Testing machines of most kinds have been used for reinforced plastics, but testing rates must usually
be carefully chosen to avoid excessive temperature increases caused by hysteretic heating. Heat
dissipation by conduction is poor in many reinforced plastics, and even in CFRP it may be too low to
prevent some temperature rise. In polymeric materials, quite small temperature increases may
significantly change the mechanical properties, and excessive heating will also lead to thermal
degradation. Recommendations of appropriate testing methods for reinforced plastics have been made by
Owen (1970). Choice of appropriate specimen shape is strongly influenced by the characteristic nature
of the composite being tested and has given rise to some difficulties. Early attempts to use test specimens
similar to those used for metals led to unrepresentative modes of failure in composites that (a) were
highly anisotropic and (b) had relatively low in-plane shear resistance, although techniques for testing
most kinds of GRP had been early established by Boller (1959, 1964).

Much early work was carried out on waisted samples of the conventional kind, and designs were
adapted for unidirectional CFRP and similar composites by using long samples with very large radii and
relatively little change in cross-section in the gauge length. An alternative approach that has been used to
prevent the familiar shear splitting in composites of low shear resistance (illustrated in Figure 6.3) was to
reduce the sample thickness rather than the width, but this has now been almost universally superseded
by the technique of using parallel-sided test coupons with end-tabs, either of GRP or soft aluminium,
bonded onto the samples for gripping. Carefully done, this eliminates the risk of grip-damage (and

144

resultant premature failure) without introducing significant stress concentrations at the ends of the test
length and failures can usually be expected to occur in the test section.

Although sample size can have significant effects on the fracture toughness of metals it is not usually
considered that fatigue tests on very small samples are likely to be unrepresentative of the material as a
whole. In composite materials, however, the inhomogeneity resulting from the fibre distribution may be
great enough to influence the fatigue response of a sample the size of which is comparable with the scale
of the inhomogeneity. Thus, in woven-roving laminates, the width of a test piece should be sufficiently
great to include several repeats of the weaving pattern. Similarly, a test piece for a hybrid laminate must
be an order of magnitude or so greater than the scale of the hybridization.

6.4.3.2 Frequency and related effects
Testing has often been carried out at the highest testing rate that could be used without causing
excessive heating of the sample. But even in CFRP, which are fairly good conductors, the occurrence of
progressive damage, leading to lower stiffness and larger strains during a test, will give rise to hysteretic
heating which may affect fatigue behaviour. In the presence of pre-existing design features like holes, or
of growing cracks, high local temperature rises may occur at quite modest applied loads, as shown in
Figure 6.17. The use of available σ/logNf fatigue data for many reinforced plastics for design purposes is
therefore complicated by uncertainty about the separate effects of frequency and autogenous heating of
samples tested at different rates — an uncertainty that is exacerbated by the high degree of scatter
normally found in fatigue-test results for these materials. Normalization of test data in terms of the
monotonic tensile strength is also likely to lead to confusion because of the strong temperature- and rate-
dependence of the monotonic mechanical properties of some reinforced plastics (particularly GRPs).

As mentioned earlier, Sims and Gladman (1978, 1980) have attempted to resolve the problem by
studying the tensile and repeated tension (R = 0.1) behaviour of glass/epoxy laminates at constant rate of
stress application instead of constant test
frequency, a procedure which avoids the confusion 50
of the varying loading rate in conventional
constant-frequency sine-wave fatigue-test routines. 30Hz
40 stress ratio,
By monitoring the specimen temperature rises R = 0.067
ΔT, deg. C

during tests at different rates of loading, they were 30
able to show that the temperature-dependence of
the composite tensile strength transposed directly
20
to the fatigue strength. They showed that fatigue
tests at different rates could be normalized with 10Hz
respect to the tensile strength measured at the 10
3Hz
same rate of stress application to give a single, 1Hz
linear, master fatigue curve. The corrections make 0
2 3 4 5
substantial differences to the interpretation of
fatigue results, especially at longer lives. Their LogN
model is valid for materials of various
constructions and is also able to take into account Figure 6.17 . Rise in sample temperature during cycling at
different frequencies: 8-ply ±45 CFRP laminate (Sun and
the effects of moisture in modifying the tensile Chan, 1979)
properties.

6.4.3.3 Effects of holes and notches
There has been some disagreement about the effects of holes and notches on fatigue resistance of
composites, perhaps because we are often tempted to generalize about the class of materials we call
‘composites’ which, in reality, encompasses a wide range of materials of very different character. A brief
survey of some of the available results will illustrate the danger in such generalisation.

Boller (1957, 1964) found that in GRP sharp notches were more detrimental than drilled holes, but
that notches in general had little effect on fracture strength because of the large number of debonding
sites present in the material to begin with. He also showed that small holes had no effect on long-life

145

fatigue behaviour, presumably because progressive damage quickly removed the stress concentration
due to the hole. Studies of the notch sensitivity of GRP by Prabhakaran and Sridhar (1975) confirm that
values of the fatigue strength reduction factor, Kf, fell during cycling.

Owen and Bishop (1972), on the other hand, showed that holes are effective in initiating fatigue
damage although they do not always affect final failure. They found that a circular hole acts as a stress
concentrator at the onset of damage with reduction of the fatigue strength by a factor between 2 and 4 in
many GRP materials and up to 10 in unidirectional GRP. They were also able to predict the fatigue
stress for the onset of damage fairly accurately from a knowledge of the properties of plain samples and
a stress analysis of the hole. Shütz and Gerharz (1977) showed that the notch sensitivity of
(02,±45,02,±45,90)S carbon/epoxy laminates changed during cycling in such a way as to eliminate the
stress concentration almost completely, whereas the opposite effect occurs during the fatigue cycling of
metals. This progressive neutralisation of notches, which is often accompanied by an increase in residual
strength, occurs by a variety of splitting mechanisms, especially longitudinal splits in 0° plies tangential
to the hole. The accompanying reduction in notch sensitivity can even lead to fractures away from the
cross-section containing the hole. This effect has also been observed in random, short-fibre composites
such as polyester moulding compounds (DMC) (Harris et al, 1977). More recent work on DMCs and
SMCs by Atzori et al (1994) also suggest that the fatigue properties of notched samples are similar to
those of un-notched ones. Curtis and Moore (1985) showed that woven-fabric CFRP composites were
slightly more notch-sensitive in fatigue than non-woven composites. Xiao and Bathias (1994) showed
that the fatigue ratio of woven-fabric-reinforced glass/epoxy composites was unaffected by the presence
of notches. And Kawai et al (1996), comparing the behaviour of Nylon and epoxy composites reinforced
with woven carbon fibres, found that un-notched samples of the carbon/Nylon had a higher fatigue
strength that the carbon/epoxy material, whereas when notches were introduced, the relative
performance of the two materials was reversed.

The crack-stopping ability of composites, which results from their inhomogeneity on a fine scale (the
fibre/matrix interface) and on a gross scale (laminated structure) makes it difficult in many cases to
apply a fracture mechanics approach to fatigue testing and design. The difficulty is most acute with
unidirectional composites or laminates pressed from preimpregnated sheets of continuous and non-
woven fibres. Stress concentrating effects of notches and holes may be almost completely eliminated by
large-scale splitting in the 0° and 45° plies and by delaminations between the plies, the net result often
being disintegration of the composite. In laminates containing woven cloth or CSM, however, or in
mouldings containing random chopped fibres, the scale of this damage is limited by the structure of the
composite and cracks and notches will often propagate in a more normal fashion, especially in wide
plates. Owen and Bishop (1974) have shown that it is possible to use a fracture-mechanics approach,
based on the Paris power law:

da
= A( ΔK )p ......................................................................................................... 6.6)
dN

relating crack growth rate, da/dN, to the range of stress intensity factor, ΔK, for some GRP laminates.
The exponent, p, in this relationship was 12.75 for CSM/polyester composites and 6.4 for plain-weave
fabric GRP, a substantial difference not accounted for by the different tensile strengths of the two
materials. To overcome the problem of accurately defining crack length, a compliance calibration was
used to determine an equivalent crack length, a procedure somewhat similar to that of using a plastic- or
process-zone size correction when applying fracture mechanics to other materials. For constant load-
range cycling Owen and Bishop constructed log(ΔK) versus N curves which were initially flat but turned
up rapidly at certain critical levels. This instability defined a ‘fatigue life’ when the cyclic ΔK level
reached the value of KC for ordinary monotonic crack growth.

Underwood and Kendall (1975) also used a fracture-mechanics approach to study the growth of a
damage zone during the fatigue of notched GRP laminates. They obtained much lower values of the
Paris-law exponent, p, — 2.5 for 0/90 laminates and 4 for (0,±60) laminates — much closer to the values
normally obtained for metallic materials. They found that their results were best described by linear plots

146

of ΔK versus cycles-to-failure which extrapolated back to the monotonic fracture toughness values. They
observed that, depending on the type of material, large diffuse damage zones could increase the residual
strength, while small, sharply defined zones would reduce it.

Vancon et al (1984) have shown that for the very much simpler case of propagation of delamination
cracks in multi-ply laminates, fracture mechanics methods and the Paris power law are straightforwardly
applicable.

6.4.3.4 Effect of stress system and the stress ratio, R
Although interpretation of the results of tests
carried under axial loading conditions (ie. 2.5
repeated tension, repeated compression, and
mixed tension/compression) are perhaps the 2.0

Peak stress, GPa
most straightforward to interpret, a good deal of
flexure
testing has also been done in other modes, 1.5
including shear (torsion) and flexure. What
happens in any kind of test, however, depends 1.0
sensitively on the structure of the composite and tension
manner in which the applied load develops 0.5
stresses within the material. In a unidirectional
rod subjected to torsion, for example, it is the 0
matrix and interface, both sources of weakness -1 0 1 2 3 4 5 6 7 8
in reinforced plastics, that determine the fatigue LogNf
response: the fibres play little part (Beaumont
and Harris, 1972; Phillips and Scott, 1976). The Figure 6.18. Comparison of stress/loglife data for a
situation is similar when a ±45 laminate is tested unidirectional XAS/913 laminate tested in tension and flexure
in repeated tension, as we saw earlier. In flexural (Charrière, 1985).
tests, the stress distribution varies across the
thickness of the sample and it would be expected that the performance of the material in this mode of
testing would be controlled by the maximum tension and compression stresses on the sample faces. In
early tests on high-stiffness materials (see Beaumont and Harris, 1972, for example) it was often
observed that the flexural fatigue behaviour was markedly poorer than that obtained in repeated tension.
This was presumably because in some of these tests, especially those conducted on small test pieces,
significant levels of interlaminar shear loading could be generated, and the final fatigue failure of the
sample was initiated prematurely by interface or matrix damage. In larger samples of more modern
composites, the flexural fatigue response may be very similar to that obtained in tension, as illustrated in
Figure 6.18, a further indication of the fact, mentioned in chapter 4, that compression failures rarely
occur in the flexural loading of thin laminates,
2.0 despite their known weakness under direct
R= compression loading conditions.
1.5
+0.5
Peak stress, GPa

The effects of direct compression loading are
1.0 +0.1
-0.3 best illustrated by the use of stress/loglife curves
0.5
-0.6 at different R ratio, such as the family of curves
-1.0 in Figure 6.19. These median-life curves are
-1.5
0 plotted in terms of peak stress as a function of
logNf and compression stresses are represented
-0.5 +10 as negative. Note that an R ratio of 0 represents
cycling between 0 and some positive stress level
-1.0
-1 0 1 2 3 4 5 6 7 8 (ie. ideal repeated-tension, or T/T, cycling) but it
LogNf is usual to retain a small positive load at the
bottom of the cycle in order to avoid the
Figure 6.19. Median stress/loglife) curves for a [(±45,02)2]S possibility of any overshoot into compression:
T800/5245 laminate at various R ratios (Gathercole et al, hence the use of an R ratio of 0.1 to represent
1994). The plotted curves are best-fit polynomial curves of T/T cycling. Similarly, an ideal repeated-
second or third order.

147

compression cycle would have an R ratio of ∝, which complicates the mathematical handing of R ratios,
so a small negative load is usually retained at the top of a C/C cycle: hence the R ratio of 10 in Figure
6.19. An R ratio of –1 represents a fully reversed T/C cycle.

It is interesting to note that the median σ/logNf curves for repeated tension (T/T) and repeated
compression (C/C) extrapolate back to the related monotonic tension and compression strength values,
and when the two sets of median-life data are normalized with respect to those strength values there is a
high degree of correspondence between them. As the compression component of cycling increases (R
increasingly negative) the stress range, 2σalt , to which the sample is subjected must at first increase, so
that if the data were plotted as stress range versus loglife the data points for R = –0.3 would then
apparently lie above those for R = +0.1. This is a familiar feature of fatigue in fibre composites, and it
leads to the notion that some element of compression load in the cycle can apparently improve the
fatigue response. It also results in a well-known aspect of composites fatigue, viz. that master diagrams
of the constant-life, or Goodman, variety are displaced from the symmetrical location about R = –1 (ie.
the alternating-stress axis) (Owen & Morris, 1972; Schütz and Gerharz, 1977; Kim, 1988). It has
recently been shown (Harris et al, 1997) that the effects of R ratio can be illustrated by presenting the
fatigue data on a normalized constant-life diagram by means of the function:

a = f (1 − m)u (c + m)v ............................................................................................ 6.7)

where a = σalt/σt, m = σm/σt and c = σc/σt
(definitions of σalt and σm were given in section 0,
and σt and σc are the monotonic tensile and
Normalised alternating stress, a

compressive strengths, respectively). The stress 0.8
function, f, depends on the test material, and in
particular on the value of the ratio σc/σt, and the 0.6
exponents u and v, usually quite close to each
other in value, are functions of logNf. Since this is 0.4
a bell-shaped function, it is more akin, as a failure
criterion, to the old parabolic Gerber relationship 0.2
(1874) than to the more frequently used linear
Goodman law (1899). The extreme values of m for 0.03
1.0
a = 0 are 1 on the tension side of the ordinate and 4 0.5 m
5 0
0.0 ess,
6 n str
–c on the compression side: the mean stress range LogN 7 -1.0 -0.5 d mea
alise
f
is thus (1 + c). The setting up of this parametric No rm
form of the constant life curve is in fact a double
normalisation with respect to the monotonic Figure 6.20. Constant-life curves for a T800/924 CFRP
tensile strength of the hybrid material. A family of laminate of [(±45,02)2]S construction (Gathercole et al,
curves representing different constant values of 1994).
life, plotted as a three-dimensional diagram, is
shown in Figure 6.20. There are naturally
HTA/913 differences between the shapes of constant-
E-glass/913 life curves for different kinds of composite.
R = -1.0
0.8
5 -0.3 Figure 6.21, for example, shows curves for
Nf = 10 cycles -1.5
0.6
CFRP (HTA/913) and GRP (E-glass/913)
+0.1
a laminates of identical lay-up and matrix
0.4 +10 resin for a life of 105 cycles. It can be seen
that the GRP curve is much more
0.2 asymmetric than that for the carbon
composite, and that its maximum, which is
0
-1.0 -0.5 0 0.5 1.0 lower than that for the CFRP, lies to the left
m rather than to the right of the alternating-
Figure 6.21. Constant-life plots for HTA/913 CFRP and E- stress axis. These differences may all be
glass/913 GRP laminates of [(±45,02)2]S construction for a life of
105 cycles (Beheshty et al, 1999).

148

ascribed to the difference in the reinforcing fibres.

6.4.4 Damage accumulation and residual strength
The progressive damage which occurs in a composite sample during fatigue will affect the mechanical
properties of the material to an extent which depends on the composition and lay-up of the composite
and on the mode of testing. In a unidirectional, high-modulus CFRP, for example, there may be no
significant changes in
composite stiffness
prior to failure
(Beaumont & Harris,
1972). By contrast, in
cross-plied (0/90) GRP
and CFRP laminates
transverse-ply cracking Figure 6.22. X-ray photograph of a 0/90 carbon-fibre/PEEK composite after 8.5 x105
occurs early in the life fatigue cycles in repeated tension at a peak stress of 200MPa (Dickson et al, 1985).
of the sample, causing a
significant stiffness reduction, perhaps of the order of 10%. Thereafter, for the greater part of the life
following this initial deterioration there may be little further change in stiffness (although longitudinal
cracks may appear as a result of the lateral
constraint imposed by the transverse plies) 800
until close to failure when the elastic
modulus falls rapidly (Reifsnider and AE
Acoustic emission, events

Jamison, 1982: Poursartip and Beaumont, 600
1982). An illustration of the extent of this
progressive damage is shown in the x-ray
photograph of Figure 6.22, showing both

Young's modulus, GPa
400 80
transverse and longitudinal cracking in a
0/90 laminate of XAS-carbon/PEEK after E
cycling in repeated tension. We note that 200 70
even the use of a high-toughness matrix
like PEEK does not eliminate this
0 60
cracking. Boniface and Ogin (1989) have 0 1 2 3 4 5
shown that this kind of transverse-crack Log N
growth can be satisfactorily modelled by
the Paris relationship (equation 6.6) with Figure 6.23. Reduction in stiffness and accompanying acoustic
exponents, p, between 2 and 6. When the emission in the early stages of fatigue cycling of a
reinforcement is in the form of a woven [(±45,02)2]S XAS/914 CFRP laminate at a stress ratio, R = 0.1
(Chen & Harris, 1993)
fabric, the occurrence of transverse-ply
cracking is hindered, and the changes in stiffness accompanying cycling are less marked. And even in
laminates containing only ±45 and 0° plies, there
1.2
may be no fall in stiffness until near to failure, as
strength illustrated by some typical experimental results for a
1.0 scatterband
Residual stress, GPa

[(±45,02)2]S CFRP laminate shown in Figure 6.23.
0.8 The microstructural damage events which lead to
these changes are readily monitored by means of
0.6 Peak stress, such methods as acoustic emission monitoring (Fuwa
MPa et al, 1975; Chen and Harris, 1993), as shown in this
0.4 200 figure.
300
0.2 400
S/logNf curve We referred earlier to the ‘wear-out’ model of
500
0
fatigue damage. It is a matter of common experience
-1 0 1 2 3 4 5 6 7 8 that both the rigidities and strengths of composite
LogNf materials are affected adversely by repeated loading
as damage accumulates in the material. In some
Figure 6.24. Residual strength curves for samples of 0/90
GRP laminate subjected to fatigue cycling at an R ratio of circumstances, both strength and stiffness may
0.1 and various stress levels (Adam et al, 1986).

149

initially increase slightly as the fibres in off-axis plies or slightly misaligned fibres in 0° plies re-orient
themselves in the visco-elastic matrix under the influence of tensile loads, but the dominating effect is
that due to the accumulating damage in the material. The deterioration during cycling usually depends
on the stress as shown in Figure 6.24: the higher the peak stress, the earlier the deterioration starts. In
accordance with the wear-out model, the end-point of each residual-strength curve lies on the normal
σ/logNf curve for the material.

In their work on the comparable group of CFRP, GRP and KFRP cross-plied laminates already
referred to, Adam et al (1986) found that the shapes of the residual strength curves for all three materials
were similar and could be represented by an interaction curve of the form:

ta + sb = 1............................................................................................................... 6.8)

where t is a function of the number of cycles, n, sustained at a given stress, σmax, for which the expected
fatigue life is Nf :

log n − α
t= ...................................................................................................... 6.9)
log Nf − α

The constant α is equal to log(0.5) and simply accounts for the fact the normal strength of the material
corresponds to the lower limit of the cycles scale at ½ cycle. The normalized residual strength ratio, s, is
defined as:

σ R − σ max
s= ....................................................................................................6.10)
σ t − σ max

where σt is again the monotonic tensile strength of the material and σR is the residual strength after
cycling at a peak fatigue stress σmax. The exponents a and b are materials- and environment-dependent
parameters which may have physical significance although they must be obtained by curve fitting. For
rate-sensitive materials like GRP and KFRP, σt and σR must be determined at the same loading rate as
that used for the fatigue cycling. The loss in strength due to cycling, (σt – σR), may then be regarded as a
damage function:
1
Δ = 1 − s = (1 − t a ) b …………............ 6.11) 1.0

The shape of the curve can, in principle, vary from 0.8
a linear form (with a = b = 1), through a circular
Stress ratio, s

quadrant for powers of two, to an extremely angular ta + sb = 1
variation for very high powers. Since both the gradual 0.6
wear-out and ‘sudden-death’ type of behaviour can be a b
accommodated within a single general model, a CFRP (1.8 23.1)
0.4
KFRP (1.8 5.2)
treatment of this kind has advantages over non-
GRP (1.5 4.8)
normalising procedures. Despite apparent mechanistic
0.2
dissimilarities in the fatigue behaviour of the three
types of composite studied in this work, the residual
strength results suggested that a common mechanism 0
0 0.2 0.4 0.6 0.8 1.0
of damage accumulation led to final failure since the
Cycle ratio, t
damage law of equation 6.8, with suitably chosen
values of a and b, fits all of the data for all three Figure 6.25. Normalized residual strength curves for
materials, as shown in Figure 6.25. Having established three comparable 0/90 laminates consisting of HTS
the values of a and b for a given material, the residual carbon, Kevlar-49 and E-glass fibres in a common
strength for any fatigue stress can then be evaluated: epoxy resin (Adam et al, 1986). The values of the
exponents for the fitted curves are shown in brackets.

150

σR = (σ t − σ max )(1 − ta ) + σ max …..……........... 6.12)
1
b
1.8
S/logNf curve
1.6 R = +0.1
or, alternatively, the remaining life, (Nf – n) can be

Peak stress, GPa
1.4
estimated. It seems unlikely that a single
mathematical model could cope with the whole 1.2
range of composite types and lay-ups available, and
1.0
although this model requires validation for each
system of interest — materials, lay-ups, stress 0.8
1 2 3 4 5
systems — this validation would be reasonably 0.6
lines showing onset of specific
damage mechanisms
economical of test time and material, given that the
interaction curve could be deduced from relatively 0.4
-1 0 1 2 3 4 5 6 7 8
few data. LogN

The foregoing discussion suggests that the Figure 6.26. Schematic damage mechanism maps for a
changes in mechanical properties of fatigued T800/5245 [(±45,02)2]S CFRP laminate tested in repeated
composites may be caused by a single damage tension fatigue (R = 0.1) (Chen & Harris, 1993).
mechanism, but this is unlikely to be the case. Key: 1 = fibre fracture in 0° plies; 2 = matrix cracks in
Detailed studies of the damage processes occurring outer 45° plies; 3 = fibre drop-out; 4 = matrix cracks in
inner 45° plies; 5 = delamination at 0/45 interfaces.
during cycling of CFRP materials (Chen & Harris,
1993) show that sequences of damage occur throughout life, and these sequences can be mapped, as
shown in Figure 6.26, on a conventional σ/logNf diagram. The supposition is that each damage-
mechanism curve, including the final failure curve (σ/logNf curve), represents part of an S-shaped decay
curve of the kind postulated by Talreja (1981), although in the experimental window we see only a part
of each curve. It is also supposed that at sufficiently low stresses, corresponding to the notional
endurance limit, all curves will flatten out and converge at large numbers of cycles. One of the problems
of developing damage-growth models for life-
6 prediction purposes (see section 0) is that the
damage development laws are likely to be
5 different for each specific damage mechanism and
will be structure-dependent, as illustrated by the
Crack density, mm -1

4 inner 45° plies
results of crack-density measurements shown in
Figure 6.27. The measurements were made on a
3
T800/5245 CFRP laminate of [(±45,02)2]S lay-up,
outer 45° plies
2
and it can be seen that the development of cracks
in the outer (unconstrained) and inner
1 (constrained) 45° plies proceeds at quite different
rates and that the saturation levels for the two
0 types of crack are also quite different
0 20 40 60 80 100
Percentage of life, % Howe & Owen (1972) studied the
accumulation of damage during cyclic loading
Figure 6.27. Density of cracks in 45° plies in a [(±45,02)2]S
with the object of obtaining useful working
T800/5245 CFRP laminate during cycling at a peak stress of
1GPa and an R ratio of 0.1 (Grimm, 1996). relationships of the Miner-rule type, that might be
used in design. They used optical microscopy to
study the development of debonding sites and resin cracks in CSM/polyester composites. They
suggested that although debonding did not itself cause reductions in strength it served to initiate resin
cracks which did weaken the material. For resin cracking they proposed a non-linear damage law,
independent of stress level, which gives the damage, Δ, as

⎡ ⎛ n⎞ ⎛ n⎞ ⎤
2

Δ= ∑ ⎢ ⎝ N⎠ ⎝ N⎠
A ⎜ ⎟ + B⎜ ⎟ ⎥ ...........................................................................................6.13)
⎢
⎣ ⎥
⎦

where n is the number of cycles sustained by the composite at a stress level which would normally cause
failure after N cycles, and A and B are constants. B is negative and Δ is equal to unity at failure. They

151

used a modification of this law to predict residual strength after cycling of CSM/polyester laminates, this
strength being dependent upon the growth of resin cracks.

6.4.5 Fatigue under conditions of variable stress: block loading
Although studies of damage accumulation processes have been correlated moderately satisfactorily
with fatigue testing régimes involving a limited number of stress levels, it is even more necessary to
make good correlations when variable-amplitude or flight-loading sequences are used for practical
design purposes. Shütz and Gerharz (1977) have reported results of life-prediction experiments with HT-
carbon/epoxy laminates of 0.4Vf and (02,±45,02,±45,90)S construction. They used flight-spectrum tests
typical of the conditions at an aircraft wing root, and they also obtained conventional fatigue data at a
range of different R values from which they made Miner-rule life predictions at two mean stress levels
for comparison with the variable-amplitude test results. The linear-damage rule predicted lives up to
three times those actually measured — far too wide a margin for comfortable design work — and they
concluded that this was probably due to damage contributed by low-load cycles in the compressive
region which is not accounted for in the Miner estimate.

In some more recent experiments, Adam et al (1994) carried out some simple four-unit block loading
sequences on samples of a [(±45,02)2]s T800/5245 CFRP laminate. They first carried out a series of all-
tension (TTTT) experiments for which the numbers of cycles at each of the stress levels were chosen so
that the block should account for 20% of the lifetime of the specimen if Miner's rule were to be obeyed.
Each unit in the sequence should therefore have contributed 5% of the total damage to failure. Since the
contribution to the total life of the specimen by each of the units within the block programme is 5% of
the median life, each block of four units therefore had a notional Miner number, M, of 0.2, ie.

n1 n2 n3 n4
+ + + = M = 0.2 ..................................................6.14)
m(N1 ) m(N2 ) m(N3 ) m(N4 )

where ni is the number of cycles in the unit and m(Ni) is the median life at that particular stress level. If
Miner's rule is followed, the block should repeat itself 5 times before failure occurs. Following the first
all-tension (TTTT) series, the next stage was to introduce a compressive element into the loading
sequence so as to construct TTCT blocks.

It is logical to assume that if the life data for a given constant stress level are randomly distributed
then the Miner sum, M, defined by equation 6.14, should be similarly distributed, regardless of whether
the mean value or median value of M is equal to unity. The Miner sums from all six groups for each
sequence were therefore pooled and analysed in terms of the Weibull model, discussed earlier, which
showed that there was a marked difference between the mean values of the Miner number for the two
groups of tests, 1.1 for the TTTT sequence, and 0.36 for the TTCT sequence. To all intents and
purposes, then, the Miner rule is valid for all-tension block-loading sequences for this material, and the
effect of introducing a single compression block into an otherwise all-tension group was clearly highly
damaging, Miner's law no longer being obeyed and by a substantial margin, the mean life having been
reduced by some 60% as a result of the substitution. Similarly, the substitution of a repeated tension unit
into an all-compression sequence also resulted in a marked reduction in the mean Miner sum. It therefore
appears to be the stress reversal as such which is damaging rather than the actual sequence or the actual
stress levels. This has serious consequences for designers since most practical cycling régimes, are likely
to include stress excursions into both tension and compression. This is certainly true of the familiar
‘FALSTAFF’ and other load-spectrum sequences used in the laboratory to represent actual life
conditions for aircraft components.

6.4.6 Life prediction
There have been many attempts to develop life-prediction methods for reinforced-plastics, but the
complexity of the response of these materials to stress makes it very difficult to develop universally
applicable models. Some of the established methods are based on structural models, while others are less
specific. Most of the structurally based procedures depend on modelling the degradation of material

152

properties as a consequence of microstructural damage (eg. Lee et al, 1989; Hwang and Han, 1989) and
many are based on the concept of a ‘characteristic damage state’ (CDS) developed by Reifsnider and co-
workers (see, for example, Reifsnider et al, 1979). Many of these methods involve both statistical and
laminate-theory analyses and give reasonably good predictions. But many require a good deal of prior
data determination and are structure-specific. It is beyond the scope of this book to treat this subject in
any detail, but we shall refer to two recent approaches which appear to have some generality.

The first is based on the constant-life analysis referred to in section 0. Equation 6.7 is a parametric
description of the stress/R-ratio/life ‘environment’ for composite fatigue which appears to be valid for a
wide variety of material types and structures (Harris et al, 1997). In this equation:

a = f (1 − m)u (c + m)v

of which the terms have already been defined, the shape and location of the fatigue ‘surface’ are
determined by four parameters, the compression and tensile strengths (which define c), the exponents u
and v, and the factor f. Since c is a material property and the alternating and mean components of stress
(a,m) are normalized with respect to the tensile strength, u, v and f must be obtained by a parametric
analysis of fatigue data sets. The exponents u and v (both functions of fatigue life) are usually very
similar in value, unless the constant-life curves are very asymmetric, and for a wide range of CFRP
materials are close to 2. And the parameter f is an inverse-power function of the ratio of the compression
strength to the tensile strength, ie. of c, of the form (Beheshty and Harris, 1999):

f = Ac − p .............................................................................................................6.15)

where A and p are functions of life. The (a,m) data may be obtained from median-life fatigue results, or,
by pooling fatigue data sets as described earlier to obtain valid statistical information, they may be
chosen to represent any required failure probability. If no information other than the tensile and
compression strengths is available for a given composite, a preliminary attempt can still be made to
predict stress/life curves on the basis of accumulated data for other related materials. A limited fatigue-
test programme can then be carried out to provide stress/life data at three or four stresses at, say, three R
ratios and the predictions of the model will become gradually more refined and reliable. The virtue of
this method is that it can provide designers with useful design information at very early stages in the
development of new materials, before detailed descriptions of the fatigue response have become
available, as shown by Harris et al (1997). While this may well be all that is necessary for non-critical
applications, in the case of critical applications such as aircraft structures it would of course always be
necessary to confirm predictions from any model before a design could be validated. The procedure
described above requires data specific to a given type of reinforcing fibre. It appears from these results
that stress/life curves for materials damaged by low-velocity impact can also be predicted from data for
the undamaged laminate provided the changes in the tensile and compression strengths caused by the
impact damage are known (Beheshty & Harris, 1999).

Although a high level of understanding of the mechanisms of fatigue damage accumulation and the
effects of this damage on the ability of a material or structure to sustain the service loads for which it
was originally designed is of importance to materials developers, it must be said that the designer
himself has little interest in the actual physical mechanisms of degradation. What he needs is simply a
way of analysing available data to predict the likelihood of failure under a specified set of conditions. In
recent years artificial neural networks (ANNs) have emerged as a new branch of computing suitable for
applications in a wide range of fields. They offer a means of dealing with many multi-variate properties
for which there is no exact analytical model and fatigue seems to be the kind of materials property that is
suitable for ANN analysis. Neural networks provide a compact way of coping with the large amounts of
characterization data generated in the study of a multi-variate dependent property such as fatigue. They
also provide a very simple means of assessing the likely outcome of the application of a specified set of
conditions, precisely what is required by a designer who needs to make safe use of complex fatigue data
for complex materials like composites.

153

An ANN is a type of computer programme that can be ‘taught’ to emulate the relationships between
sets of input data and corresponding output data and the results of a characterization programme can
therefore be ‘learnt’ by the network and patterns in the data can be established. Once the network has
been ‘trained’ it can be used to predict the outcome of a particular set of input parameters. Unlike other
types of analysis there is no need to know the actual relationship between inputs and outputs and the
question of determining physical laws and relationships need not be addressed. The network simply
compares patterns of inputs with those it has been ‘taught’ and provides an output which takes full
account of its accumulated ‘experience’. It is a highly developed interpolation system and its outputs
provide a good indication of the certainty of an outcome, a property which may be a useful indicator of
where additional experimental information is required. The ability to signal the ‘certainty’ of the
outcome is particularly important for materials like composites for which the fatigue response is often
highly variable even when the monotonic strength properties of the same material may be quite closely
definable.

As we have seen, fatigue in composites involves many variables, including a variety of materials
characteristics (eg. fibre/resin mix, lay-up, moisture content) as well as the specific fatigue variables
(mean and alternating stresses, variable stress characteristics, etc.), all of which may be treated as inputs
to an ANN. One of the first questions that arises relates to the issue of what constitutes ‘failure’. It is
more likely to be the minimum life in a data set than the median
or mean life, and it is important to decide whether the input
should simply be the lowest value of life recorded in a data set,
or some value estimated from the application of extreme-value
theory (Castillo, 1988), as discussed earlier. Some recent results
obtained by Lee et al (1995, 1999) suggest that the ANN
approach may provide a useful method of fatigue data analysis
and life prediction.

6.5 FATIGUE OF METAL-MATRIX COMPOSITES
Much early work on MMCs related to metals reinforced with residualfibre stress
high-strength metallic wires like tungsten and cold-drawn steel.
These were essentially model composites, and were rapidly residual matrix stress

supplanted as experimental materials by modern aluminium and
titanium alloys mostly reinforced with boron, carbon and SiC
fibres of various kinds. Nevertheless, some of the early results Figure 6.28. Schematic illustration of the
provide useful insights into the fatigue response of this kind of first tensile stress/strain cycle of a metal-
material. It will be appreciated that in a metal/metal system, matrix composite loaded beyond the matrix
yield stress, ie. into the elastic-plastic
both components will be susceptible to conventional fatigue range.
failure, whereas in a metal/ceramic system, only the matrix will
be susceptible. However, since metallic fatigue is essentially a brittle-failure process, the difference may
not be great.

When a composite with a low-yield-stress matrix is cycled in repeated tension, the matrix will be
subjected to a more complex stress régime than the level of external applied stress would suggest.
During the first cycle, shown schematically in Figure 6.28, in the régime where both fibres and matrix
deform elastically, the initial elastic modulus of the composite, usually designated EI, is given by the
normal rule of mixtures. At the yield point of the matrix, at the strain εmy, the composite deforms elasto-
plastically with a somewhat lower stiffness (a constant EII if we assume linear work-hardening behaviour
in the matrix) while the fibres continue in their elastic state and the matrix deforms irreversibly. On
reversing the sense of the applied deformation, the composite describes a hysteresis loop, as shown,
while the matrix and fibres unload elastically. The matrix reaches a zero-stress state at a finite level of
strain, and the stresses in the constituents will reach an equilibrium state at the point X, the fibres being
in tension and the matrix in compression, as shown, and the composite is left with a permanent strain.
Further cycling in repeated tension will result in entirely elastic loading and unloading of both fibres and
matrix from this new stress state (Dvorak and Tarn, 1975). However, if the stress cycle is fully reversed

154

(ie. for R = –1), with the matrix stress exceeding the matrix yield stress in each direction, energy will
continue to be dissipated in the material until the plastic strain range has been reduced as a result of
cyclic work-hardening. The effect of the elasto-plastic stress cycle in the matrix may be substantial and
could lead to over-ageing in a peak-aged aluminium matrix, for example, perhaps resulting in more rapid
work-hardening and crack growth than would occur in purely elastic cycling. Thus, although it might at
first sight be expected that in a metal/metal system the fatigue response should be predictable by a
mixtures-rule model for the fatigue stress for a given number of cycles, for example:

σc(Nf) = σf(Nf)Vf + σm(Nf)(1 – Vf)..........................................................................6.16)

the low-cycle fatigue of the matrix may prevent this. McGuire and Harris (1974) studied the fatigue
response of unidirectional composites consisting of a tungsten-wire-reinforced aluminium alloy
(Al/4%Cu) produced by vacuum diffusion bonding, and their results show the effects of heat treating the
matrix alloy to peak hardness or over-ageing it to
10 produce a brittle matrix state. Their σ/logNf
curves showed that increasing Vf increased the
Normalised stress range

fatigue resistance of the composite, and this
increase in fatigue resistance seemed to be a
direct result of the increased strength of the
1 composite since if their stress/life data for each
Al alloy matrix
composite composition are normalized by
Vf = 0.07 dividing the cyclic stress-range values by the
0.12 appropriate sample tensile strength the points for
0.24 all of the composites and the unreinforced matrix
fall on a single line which appears to obey
0.1 Basquin’s Law very well (Figure 6.29). Both
1 2 3 4 5 6 7
matrix and fibres exhibit distinctive signs of
LogNf
fatigue failure, as shown in Figure 6.2, in clear
contrast with the normal tensile fracture mode of
Figure 6.29. Normalized stress/loglife data for an Al/4%Cu
both constituents. The composites with the
alloy and three composites containing tungsten wire
reinforcement (McGuire and Harris (1974). The data are for higher cyclic work-hardening rate (peak aged)
reversed axial tension/compression cycling (R = -1) and the possessed the better fatigue resistance, but the
stress ranges for the data points are normalized with fatigue stresses (for Nf = 107 cycles) for both
respect to the tensile strength of each material. heat-treatment conditions fell well below the
mixture-rule prediction of equation 6.16. In other
metal/metal systems including steel/silver and tungsten/copper (Courtney & Wulff, 1966) and
tungsten/silver (Morris & Steigerwald, 1967) it was shown that the fatigue ratio actually decreased with
increasing fibre content.
1.0
McGuire and Harris also carried out low-cycle
fatigue experiments at constant total strain range, and
Power-law exponent, -q

in these tests the composites showed work-hardening 0.8
and work-softening characteristics that were affected normal range for
by matrix plastic properties. The observed changes did 0.6
metallic materials
not indicate any difference in the nature of
deformation, however, and the exponent, q, in the 0.4
Manson-Coffin law relating plastic strain range to the
number of cycles to failure: 0.2 peak-aged matrix
over-aged matrix
Δε p N q = constant ………………… 6.17)
f 0
0 0.05 0.10 0.15 0.20 0.25
fell smoothly with increasing Vf from values
characteristic of conventional metallic materials, of the
order of 0.5, to something like half that value at Vf ≈ Figure 6.30. Variationfor the Coffin-Manson exponent,
q, with fibre content
of
W/Al-Cu composites in low-
0.2, as shown in Figure 6.30. This indicates that the cycle fatigue (McGuire & Harris, 1974).

155

improvement in fatigue properties results from the effect of 100
the reinforcing fibres in limiting the composite plastic strain matrix
n=5
range of the matrix. Vf = 0.07
Vf = 0.12
The matrix plastic strain range has been identified by Vf = 0.22

da/dN, m/cycle
several workers as being the key parameter determining the
fatigue characteristics not only of metal/metal systems, but
also of metals reinforced with brittle ceramic fibres like 10
carbon and silica. The work of Baker and co-workers, for
example (Baker, 1966; Baker et al, 1966), suggested that in
both types of system, fatigue failure occurred mainly by the
growth of matrix or interface cracks, linking via fractured
fibres in the case of brittle-fibre systems, and that the main
effect of the reinforcing fibres in improving fatigue 1
resistance was to reduce the matrix plastic-strain range. 10 20 50 100
Courtney & Wulff (1966) invoked a fatigue model due to Stress intensity, Δσ.√a, MPa.√m
Orowan (1939) which also ascribed the fatigue failure (of
W/Cu and steel/Ag) to matrix cracking resulting from cyclic Figure 6.31. Paris law plots for fatigue crack
work-hardening as a result of the elastic-plastic deformation growth in W/Al-Cu composites (Harris et al,
1976). The average slope of the curves is
described above. Lee and Harris (1974), working with about 5.
tungsten/copper composites, commented on an associated
effect, recalling the work of Kelly & Lilholt (1969) who showed that during the deformation of an
MMC, the stress levels in the matrix were often considerably greater than would be predicted by the rule
of mixtures, probably because of the highly perturbed and constrained state of stress in the matrix
adjacent to the fibres. The difference between the resulting enhanced matrix flow stress and the
unconstrained matrix yield stress increases as Vf increases and as the fibre diameter decreases. The result
during fatigue is to reduce both the matrix ductility and the matrix plastic strain-range, and thereby
increase the composite life. By contrast with some of the other results referred to above, they showed
positive deviations from the mixtures rule between 4 and 20%, the greater deviations being associated
with finer reinforcing filaments.

Leddet and Bunsell (1979), studying the fatigue of
100
boron/aluminium-alloy composites in rotating
Stress-intensity factor, KQ or ΔK

KQ bending, have shown that fibre/matrix interactions
80 caused by heat treatment change the mechanisms
which lead to progressive deterioration during
60 ΔK cycling. They suggested that with a soft matrix the
predominant damage mechanism is fatigue cracking
40 cyclic: overaged in the matrix parallel to the fibres and originating
cyclic: peak aged
20 tension: overaged
from initial fibre fractures. In a slightly harder matrix
tension: peak aged cracks propagate through the matrix and around the
0 fibres without causing fibre failure, but in a highly
0 0.1 0.2 0.3 0.4
work-hardening matrix successive fibre failure can
occur. The fatigue behaviour of B/Al composites
depends on the intrinsic matrix mechanical properties
Figure 6.32. Variation of effective stress intensity
factor with fibre content for monotonic and cyclic
but is controlled by the fibre strength.
loading conditions (McGuire and Harris, 1974 and
1976). By contrast with reinforced plastics, crack growth
from pre-existing notches often occurs normally in
composites MMCs. Harris et al (1977) showed this to be the case with composites of Vf up to 0.22, but
in samples with 37 vol.% of reinforcement the closely-spaced fibres led to the development of a highly
localized plastic zone parallel with the fibres that inhibited planar crack growth and encouraged the same
kind of longitudinal splitting that occurs in unidirectional reinforced plastics. For their W/Al materials,
the crack growth rate, da/dN, correlated well with a stress-intensity-range parameter, similar to ΔK, of
the form σ√a, following the Paris Law (equation 6.6), as illustrated in Figure 6.31. The slopes of the

156

curves, p, for different composites were identical with
Log(crack growth rate), da/dN in m/cycle 10 that for the unreinforced matrix, about 5. These
p
-6 5 results again indicate that fatigue response was
determined simply by the strength and work-
0
plain copper 0 Vf 0.05 0.10 hardening ability of the matrix, without significant
-7 differences in the controlling damage mechanism. It is
Cu/6%W
clear that the matrix alloy work-hardens sufficiently
to cause fibre failure, but by a conventional fatigue
Cu/9%Cu mechanism rather than brittle (tensile) failure. Similar
-8
effects of work-hardening behaviour would not
necessarily have occurred had the fibres been non-
Paris law slope, p metallic. A comparison between the results shown in
-9 Figure 6.31 and the stress-intensity factor
0.75 1.00 1.25 1.50

LogΔK, ΔK in MPa.√m
measurements for crack propagation in the same
materials under monotonic loading shows the extent
Figure 6.33. Paris-law plots for fatigue crack to which cyclic loading conditions are more damaging
propagation in composites consisting of short, aligned (Figure 6.32).
tungsten fibres in copper (Harris and Ramani, 1975).
Crack propagation in some unidirectional short-
fibre Cu/W composites shows slightly different features (Figure 6.33). In this case, the slope for the
matrix is much higher than that for the Al/4%Cu alloy in Figure 6.31, about 10 instead of 5, and the
intrinsic fatigue resistance of the copper is thus much lower. Addition of a small volume fraction of
tungsten filaments nevertheless reduces the slope to the same level as that for the Al/W composites. The
curious feature of this graph, however, is that for initial cracks below some threshold level, the short
fibres actually cause the growing cracks to slow down as their lengths increase before conventional
Paris-law behaviour is established. Similar results have been reported by Bowen (1993) and Barney et al
(1993) for titanium-alloy MMCs and by Pegoretti & Ricco (1999) for composites of polypropylene
reinforced with short glass fibres. Bowen suggests an approach to the prediction of this crack-arrest
mechanism.

6.6 FATIGUE OF CERAMIC-MATRIX COMPOSITES
Although it has traditionally been considered that ceramics do not suffer from fatigue, this view may
have arisen partly because of the difficulty of carrying out fatigue tests on brittle materials and partly
because fatigue phenomena may easily be obscured by the high level of variability associated with brittle
failure. Certainly, fatigue crack growth
phenomena have been observed in a variety of 1.1
brittle materials and a range of mechanisms,
Normalised modulus, E/Eo

including microcracking, small-scale plasticity 1.0
unidirectional
resulting from dislocation movements and stress- 0.9 peak stress
induced phase transformations, have been (% of
invoked to explain the observations (Suresh et al, 0.8
failure stress)
1988). Nevertheless, although fine-scale effects 0.7 (0,90)3S 57%
such as these will certainly occur in ceramic-
matrix composites, especially those reinforced 0.6
with particulate fillers or fine whiskers, the 0.5 35%
results of fatigue in continuous-fibre CMCs are
on a much larger scale and are due largely to 0.4
1 10 100 1000
cracking. Ritchie et al (1993) have suggested that
Cycle number, N
cyclic fatigue in low-ductility materials such as
these is conceptually distinct from metallic
Figure 6.34. Effect of repeated tension cycling on the
fatigue, and is largely driven by cycle-dependent elastic moduli of unidirectional and cross-plied SiC/CAS
suppression of some of the crack shielding composites. The composite tensile strengths were
mechanisms that normally operate in monolithic 334MPa (ud) and 210MPa (0,90). (Habib et al, 1993)

157

1.5 ceramics.
monotonic flexural failure strain
In the reinforcement of metals and polymers, the
Fatigue strain, %

modulus ratio, Ef/Em, is usually high — 100 or more
1.0
in high-performance reinforced plastics, perhaps less
than 10 in MMCs — whereas in CMCs there may be
little or no difference in the stiffnesses of the
0.5 components. Fibres are added to ceramics to improve
the toughness and reduce the variability, rather than to
improve the rigidity. Thus, although, as we saw in
0 chapter 4, CMCs can be made to support high stresses
3 4 5 6 and many have considerably greater toughness than
LogNf monolithic ceramics, they achieve this at a price,
namely, that extensive cracking occurs in the very
Figure 6.35. Strain/life curve for a (0/90)3S SiC/CAS early stages of loading. In a 0/90 laminate of a GRP or
composite obtained in constant-moment flexural CFRP laminate, the transverse plies are sources of
loading (R = 0), (Habib et al, 1993)
weakness, and these crack easily at low loads when
the stress exceeds their notional transverse strength which may be even lower than the normal tensile
strength of the resin matrix. But since, in CMCs, the modulus ratio may be close to unity, so that there is
no stiffening effect, the matrix even in a unidirectional composite must crack at composite stresses
which correspond to very low loads in the matrix compared to the load-bearing ability of the fibres, as
shown by the Aveston-Cooper-Kelly (ACK) model described in chapter 4. For the SiC/CAS composites
used as an illustration of the ACK model, it was shown that the model predicts that matrix cracking
should start at about 135MPa (about one third of the composite tensile strength). However, the
monitoring of tensile tests by means of acoustic emission (AE) sensors shows that microfailure events
can be detected almost as soon as loading begins, certainly at axial strain levels less than 0.05% (Harris
et al, 1992). Two further comparisons with a 0/90 GRP laminate now invite themselves. The cracking of
the 90° plies in a 0/90 GRP composite reaches its peak rate (as shown by AE activity) at a strain of about
0.7% (Harris et al, 1979), and if the stress is removed and the composite reloaded no new cracking
activity occurs until the previous stress level has been reached (Fuwa et al, 1975). By contrast, if a
unidirectional SiC/CAS composite is reloaded a number of times, the microcracking activity continues
to occur even at the lowest load levels. The potential problems of this early matrix cracking were
apparent from the earliest days of research into fibre-reinforced ceramics and glasses (Phillips, 1988).

Thus, although a cycled 0/90 plastic laminate loses a little of its initial stiffness early in its fatigue
life, this transverse ply cracking — shown in the x-ray photograph of Figure 6.22 — does not seriously
impair the fatigue life of the laminate, the damage sustained in the CMC leads to continuing reductions
in stiffness and residual strength. The changes in the stiffness during a few hundred repeated tension
cycles of a unidirectional and a (0/90)3S SiC/CAS laminate are shown in Figure 6.34. The elastic
modulus of the unidirectional composite falls to about 75% of its initial value by the 100th cycle, while
that of the (0,90)3s material falls by 40% on the initial cycle, and thereafter falls more slowly, and at
about the same rate as that of the unidirectional composite, to about 50% of its initial value. Prewo
(1986) reported similar stiffness losses during repeated loading of SiC/LAS composites, and Sorensen et
al (1993) show how frictional effects related to interfacial sliding result in tangent moduli that are
different on loading and unloading. It appears that the interfacial shear strength decreases during cycling
and that this results in a reduction in the efficiency of stress transfer into the bridging fibres, with
consequent deterioration of fatigue resistance.

Since it is easier to carry out flexural fatigue experiments on ceramics than axial tests, much of the
available data for CMCs are of this kind. For example, strain/loglife results for a (0,90)3s material are
shown in Figure 6.35 and associated residual strength and stiffness curves are shown in Figure 6.36. The
residual stiffness curves of Figure 6.36 show that for strain levels between 0.45% and 0.84%, there is an
initial reduction in stiffness, at a reducing rate over the first 100 or so cycles, which is almost the same
whatever the strain level. Thereafter, the curves become increasingly separated, all taking on a marked
downward curvature the extent of which is now greater the higher the fatigue strain. However, at the

158

lower two strain levels, the observed
Normalised modulus, E/E0 Residual strength, σR /σ0
1.0
reductions are of the order of only 15% by 105
cycles, but at 0.84% strain, the curve is
dropping very rapidly after only about 1000
0.9 reversals. At the higher strain of 0.94%, about
εmax = 0.45% 80% of the monotonic failure strain, the
initially more rapid reduction in stiffness is
0.8 followed by a gradual reduction in the rate of
1.0 loss of stiffness and as a result of the onset of
delamination the curve then crosses over the
0.9
rapidly falling line for cycling at 0.84% strain.
0.8 The effect of this delamination would be to
0.45% reduce the stiffness of the compression side of
0.7
0.64% the sample and shift the neutral axis, thereby
0.6 0.84%
lowering the effective tensile stress on the
0.5 0.94%
tension face and prolonging the life. The loss
0.4 of flexural strength after cycling a sample to a
0 1 2 3 4 5 6
peak strain of 0.45%, equivalent to a peak
Log(number of cycles) tension-face stress of about 200MPa, for
different periods of time was only of the order
Figure 6.36. Residual strength and stiffness of (0/90)3S of 6%, by comparison with a loss of stiffness
SiC/CAS composites during flexural cycling (Habib et al, of 15%, and the residual strength curve
1993).
appears to have flattened out after about 1000
cycles whereas the stiffness loss begins to
accelerate after this point. This pattern of behaviour is somewhat different from that which is usually
found in structural composites in which there is often a proportional change in strength and stiffness
implying a constant failure strain.

The damage mechanisms contributing to these changes appear to be of the same kind as those which
occur during monotonic loading to failure. This has been shown by the extensive AE studies of Harris et
al (1992) and also by measurements of the spacing of matrix cracks in samples deformed in both
monotonic and cyclic loading (Figure 6.37). A simple relationship between the stiffness of a
unidirectional composite and the reciprocal crack spacing may be expected, as can be demonstrated by
an argument similar to that used to obtain the strength of a composite reinforced with aligned, short
fibres. When a resin-based composite containing strong, brittle fibres is loaded, the fibres break into
shorter and shorter lengths until, when they reach the critical length, they are just too short to be broken
down any further and the maximum tensile
load that can be reached in the middle of the 1.1
fibre is just below the fibre strength. The mean montonic
Normalised modulus, Ec /Eo

load supported by the broken fibres is then 1.0
cyclic
only half the fibre breaking load, therefore. In
using the rule of mixtures to determine the 0.9
strength of the composite we apply the
0.8
efficiency factor, (1 - lc/2l), to the fibre stress
component, as shown in section 4.1.8 of 0.7 Ec/Eo = 1 - 0.048s-1
chapter 4. This ensures that when the fibre is
unbroken the efficiency factor is unity, and 0.6
when the fibre is at the critical length, its value
is 0.5. For a brittle-matrix/brittle-fibre 0.5
0 1 2 3 4 5 6 7
composite in which the matrix has a failure
strain lower than that of the fibres, the inverse Reciprocal crack spacing, s-1, mm-1
situation occurs and the matrix breaks into
increasingly short blocks until the critical Figure 6.37. Relationship between crack spacing and elastic
spacing is reached, at which point the blocks modulus for unidirectional SiC/CAS composites loaded
monotonically and cyclically in tension (Habib et al, 1993).
are then too short to allow the transferred

159

stress to break them down any further. We have seen that this critical spacing is close to 0.2mm, for both
monotonic and cyclic loading, in SiC/CAS composites. The short-fibre argument is not, of course,
restricted to fibres: the model makes no assumptions about the composite geometry. We may, therefore,
write the modified rule of mixtures:

⎛ s ⎞
E c = E f Vf + E m ⎜1 − c ⎟ ....................................................................................6.18)
⎝ 2s ⎠

where sc is the limiting value of the matrix crack spacing, s. Inserting known data for this SiC/CAS
material (EfVf = 67GPa; EmVm = 63GPa; and sc = 0.2) and dividing through by the expected value of Ec,
130GPa (given by the unmodified rule of mixtures) in order to normalize the model, as in Figure 6.37,
we have:

Ec/Eo = 1 – 0.048s–1 [for s > sc] ..............................................................6.19)

and when s reaches sc, no further reductions in stiffness occur. This relationship coincides almost exactly
with the best-fit line through the actual measurements. The two sets of data included in Figure 6.37 are
sufficiently similar to suggest that the damage mechanisms occurring during repeated loading and
incremental loading are the same, a suggestion which is corroborated by the similarity of the acoustic
emission patterns obtained from the two kinds of experiment.

The results shown above and those for SiC/LAS composites reported by Prewo (1987) show some
interesting similarities and differences. For the repeated tension cycling of unidirectional composites of
Nicalon-reinforced LAS-I glass ceramic, for example, for peak stresses close to the failure stress the
σ/logNf curve was very flat, and there was no loss in residual strength or stiffness after 105 cycles, even
for cycling at about 80% of the tensile strength. For Nicalon/LAS-II composites fatigued in repeated
tension, below the knee of the stress/strain curve there was no loss in either stiffness or residual strength
up to 104 cycles and the stress/strain loops showed no hysteresis. Above the knee (or, more accurately
reporting Prewo, above the ‘proportional limit’ of his stress/strain curves) no fatigue failures occurred,
but there were marked changes in the shapes of the hysteresis loops. The so-called proportional limit
appeared to be reduced as cycling proceeded, from about 300MPa on the first cycle to about 120MPa by
the third, and the initially wide damping loop rapidly became narrower. However, the initial tangent
modulus on each cycle remained the same as that of the uncycled material. When SiC/LAS-II samples
were cycled in bending, the σ/logNf was again very flat, almost coinciding with the scatter band of
monotonic strength measurements, by contrast with Figure 6.35, and there was again no loss in residual
strength. Furthermore, the fatigue failures were all initiated on the compression side of the test piece. It
is very difficult to reconcile these marked differences in behaviour of two relatively similar types of
composite. The basic properties and damage mechanisms of the two types of material do not differ
appreciably, and even though in some cases we are comparing unidirectional and cross-plied materials
our own results suggest that this difference in lay-up does not greatly influence the overall response of
the composite to mechanical deformation. It does appear from Prewo's published fractographs, however,
that the scale of fibre pull-out is significantly greater than that observed by Harris et al (1992) and it may
well be that it is the nature of the fibre/matrix bond which is responsible for the observed differences in
fatigue response.

Solti et al (1996) have recently developed a simplified model for the fatigue life of unidirectional
CMCs. The model is a modified shear-lag analysis in which the microstructural damage is estimated
from simple damage criteria. It takes into account a number of damage mechanisms, including matrix
cracking, interfacial debonding and slip, fibre fracture and fibre pull-out. The model estimates the
average matrix crack density, discussed above, and predicts the stress/strain hysteresis and racheting
described by several authors, as well as the fatigue response. The predictions of the model are in good
agreement with experimental results.

160

6.7 CONCLUSION
We have attempted to demonstrate the extent to which conventional fatigue testing procedures have
been applied to composite materials and to indicate the limitations, arising from the peculiar nature of
composites and the manner in which damage accumulates in them, that must be imposed on any
interpretation of results. It is important to emphasise that no general conclusions, relating to composites
as a class, can be drawn from any set of fatigue results on one type of composite and that the results
referred to in this chapter are intended to served as illustrations rather than to provide rules.

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7 ENVIRONMENTAL EFFECTS

7.1 INTRODUCTION

T he foregoing chapters deal, for the most part, with the mechanical response of composite
materials under what might be called ‘normal’ conditions, that is, under a variety of régimes of
loading, but with no consideration of possible interactions between the applied stress system and
environments that differ appreciably from ordinary laboratory conditions. This is naturally
limiting, and yet this ‘normal’ performance must always be established before the effects of other more
aggressive environments can be distinguished. By and large, it is the effects of temperature, ultra-violet
radiation, and chemical environments, including oxidising atmospheres, that first come to mind when
considering practical applications of composites in general. But, ironically, it is moist environments that
are likely to be the major source of trouble to designers in most applications.

There are some obvious problems of which the designer must be aware. As far as temperature is
concerned, a number of effects might be
mentioned. The first is that at elevated 1.0
temperatures creep may occur in one or both
Normalised strength

of the components. No measurable creep will CFRP
0.8
occur in glass, carbon, or boron fibres over
the temperature ranges within which most 0.6
resins, even the more thermally-stable ones, KFRP
remain undamaged. In continuously- 0.4
reinforced plastics, therefore, time-dependent GRP
deformation will be limited by the extent to 0.2
which the continuous fibres support the major
part of the load. In short-fibre composite 0.0
0 200 400 600 800 1000
mouldings, however, the upper usable
Temperature/humidity cycles
temperature limit may be much lower than
that at which resin degradation begins to Figure 7.1. Reductions in the strengths of 0/90 laminates of
occur. CFRP, GRP and KFRP as a consequence of
temperature/humidity cycling. The cycle consisted of 12
Leaving aside chemical degradation of hours at 2°C and 0% RH, followed by 12 hours at 85°C and
matrix resins which may begin, in the less 100%RH. (Dickson et al, 1984).
stable polymers, at relatively modest
temperatures, a second effect of temperature is in the material response associated with changes in
temperature, rather than the effects of exposure at constant temperature. We saw in Chapter 2 that
substantial levels of residual stress can be developed as a result of thermal expansion mismatch. Under
such circumstances these can be so great as to cause fibre damage, inter-ply cracks and transverse ply
cracks. Of particular concern in aerospace structures is the deleterious effect of rapid excursions to high
temperature, a phenomenon known as ‘thermal spiking’, which can result in significant reduction of the
residual strength of the spiked laminate. Collings and Stone (1985), for example, observed that the
thermal spiking of dry, cross-plied XAS/914 CFRP resulted in a 7% reduction in the ILSS, although no
visible signs of degradation occurred. On the other hand, they found that similar spiking of laminates
containing moisture produced large-scale damage in the form of microscopic interlaminar cracks which
reduced the ILSS to some 75% of the dry room-temperature value, and almost doubled the equilibrium
moisture content of the laminates. The combined effect of temperature and moisture is frequently
observed to be more damaging than the effect of either separately. The differing responses of cross-plied
epoxy-based composites reinforced with glass, aramid, and carbon fibres to combined
temperature/humidity cycling are illustrated by some results of Dickson et al (1984) shown in Figure
7.1.

Water diffuses readily through many of the thermoset and thermoplastic polymers that are commonly
used as matrix materials. And where the environment has access to the fibre/matrix interface, at the cut

chapter 7 Environmental effects 164

edges of a component, for example, water may be ‘wicked in’ by capillarity along the interface.
Moisture therefore has ready access to those parts of the composite from whence it derives its load-
bearing properties, to a greater or lesser extent, depending on the nature of the matrix polymer
concerned, and in the long term this can result in deterioration of composite properties.

Most forms of high-energy radiation are damaging to polymers because of the relatively low energies
required to cause chemical damage. The most obvious effect in long-chain polymers is the degradation
associated with scission of the main chains, which results in a reduction in the average molecular weight,
and additional cross-linking between chains, resulting in network formation. Scission often causes a
lowering of the polymer viscosity and softening temperature and reduction of mechanical strength, and it
may also lead, in some cases, to an increase in the degree of crystallinity. Cross-linking, on the other
hand, leads to an increase in strength and ductility up to some limiting cross-link density beyond which
the strength is again reduced and the polymer embrittled. These are therefore opposing effects which
nevertheless occur simultaneously. But the long-term effect of irradiation is almost always serious
embrittlement of the polymer. Internal stresses are also developed which, in the presence of an external
stress and an aggressive environment, may result in rapid disintegration. Exposure of thermoset resin
composites to ultra-violet radiation can promote additional cross-linking in the surface resin which may
result in surface microcracks. Most practical reinforcing fibres, with the exception of the aromatic
polyamides and high-stretch polyolefins, are unlikely to be affected by uv radiation, and since the bulk
of the reinforcing fibres in a composite will largely be screened from the uv by the surface layers of resin
and fibre or by pigmented paint coatings, the response of a reinforced plastic laminate to uv will be
largely dominated by what happens to the resin. In artificial exposure experiments lasting some 2500
hours in an ultra-violet cabinet, Dickson et al (1984) found the properties of cross-plied laminates of
glass and Kevlar-49 epoxy-based composites to be substantially unaffected even though long-term
exposure caused considerable darkening of the surface resin. Carbon fibre appears to have the same
effect as lamp-black which is sometimes added to polymers as a means of stabilising the material against
ultra-violet degradation: the fibres, like the particles, efficiently filter the radiation. It is nevertheless
often observed that outdoor exposure results in serious damage to the surface layers of laminates,
sometimes to such an extent that the resin-rich surface layer or gel-coat is completely eroded, leaving the
underlying fibre exposed. This kind of damage, which is perhaps likely to be a result of the combined
effects of ultra-violet, regular wetting and drying, and diurnal temperature cycling, is particularly serious
in the case of glass-fibre- or polymer-fibre-reinforced resins since it leaves the reinforcing elements
totally bare of protection against both mechanical damage and further environmental attack.

A useful illustration of the difficulties facing a designer in relation to environmental effects is that of
the application of composites in space vehicles. Many of the requirements are the same as those for
aeronautical structures, since there is a need to have low weight and high stiffness in order to minimize
loads and avoid the occurrence of buckling frequencies. But the environmental effects are special.
Service temperatures range from cryogenic to many hundreds of degrees, and the temperature changes
are therefore very great: consequently, severe thermal stresses may arise which may cause serious
distortion. Dimensional stability is at a premium, for stable antennae and optical platforms, for example,
and materials need to be transparent to radio-frequency waves and stable towards both uv radiation and
moisture. Many of these effects lead to residual strains, microcracking, and thermal distortion. Porosity
and internal debonds also give problems with moisture, so a high degree of perfection and high-quality
adhesive bonding procedures are needed to ensure dimensional stability. A particular problem for the
space vehicle designer is that reinforced plastics are degraded by atomic oxygen and by radiation.
Paillous and Pailler (1994) showed that the synergistic action of electrons and thermal cycling degrades
the matrices of carbon/epoxy composites by chain scission, cross-linking and microcrack damage, while
the erosion caused by atomic oxygen led to decreases in flexural stiffness. These effects are exacerbated
by impact events in low earth orbit.

The designer will usually be looking to use metal-matrix composites for applications at temperatures
significantly higher than ambient (and higher than temperatures for which there are already plenty of
good reinforced plastics available). In such cases, creep of both components may occur, and a
conventional creep-design approach may be needed. A side effect of high-temperature exposure in


MMCs is that at temperatures sufficiently high to permit creep there will also be a possibility of
diffusion-controlled interfacial reactions which may, at best, modify the load transfer conditions (often
favourably) or, at worst, may cause dramatic reduction in the load-bearing capability of the
reinforcement. McLean (1985) has given a recent review of the high-temperature behaviour of metal-
matrix systems.

In metal-matrix composites an aqueous environment may cause severe corrosion if the
matrix/reinforcement combination forms an appropriate galvanic couple. Carbon-fibre-reinforced
aluminium alloys were identified in the early seventies as being particularly prone to attack (see, for
example, the review by Pfeifer (1977). A wide range of corrosion resistance was observed in various
systems studied at that time, much of the work referred to in this review relating to the effects of marine,
salt-spray and relative-humidity exposures. It was reported that the principal mode of corrosion was
pitting commencing at the interface at points where the fibres were exposed, at cut or machined surfaces
for example, the pitting being often followed by severe exfoliation leading to complete destruction of
test panels. The composite corrosion resistance was strongly affected by the composition and structure of
the matrix alloy, and was highest in composites in which a good fibre/matrix bond was achieved. Similar
corrosion problems may be expected when carbon-fibre-reinforced plastics are used in conjunction with
metallic components in the presence of the almost inevitable aqueous environment.

7.2 HYGROTHERMAL SENSITIVITY OF REINFORCED PLASTICS
7.2.1 Fibre effects
Although reinforcing fibres like carbon, boron, and other ceramic reinforcements are insensitive to
the effects of moisture, others, particularly glass and aramid fibres like Kevlar-49, are affected by
moisture even at low exposure levels. Glass and Kevlar-49 fibres are known to have mechanical
properties that are dependent on time and temperature, giving rise to the familiar ‘static fatigue’ effect
illustrated in Figure 7.2 (Aveston et al, 1980) which is characterized by the fact that the time to failure
under a constant load depends on the load level.

In the case of glass, this is a form of environmental stress cracking, or stress-corrosion, resulting
from the leaching out of the network modifier, NaO2, from the glass structure. The resulting alkaline
environment then attacks the normally-stable SiO2 network and reduces the strength of the glass fibre.
Stress-corrosion essentially results from the conjoint effects of the corrosive environment and the
applied stress. The effect occurs more rapidly in acid or alkaline environments than in water alone, but
even in pure water the rate of loss of strength is high and the extent of the weakening is substantial. The
loss of strength of ordinary E-glass fibres in the
alkaline environment of a damp concrete matrix 8
seriously limited the effective use of glass-fibre-
reinforced cement products (GRC) in the building glass
industry. This problem led to development at the UK Kevlar-49
6
Building Research Establishment, and subsequent carbon
Load, kg

exploitation by Pilkington, of a zirconia-containing
glass known as ‘Cem-FIL’, which is more alkali- 4
resistant than other glass compositions (Majumdar,
1970). Results on the stress-corrosion of this Cem-
FIL glass fibre in aqueous alkaline environments 2
(Proctor and Yale, 1980) suggested that the material
offered a far-from-perfect solution, but it has been
shown that under the more realistic conditions of 0
service exposure of thin-walled Cem-FIL-reinforced -1 0 1 2 3 4 5 6 7
cement pipe the practical life of such materials is Log(t) (failure time, t min.)
greater than the laboratory results of Proctor and
Yale suggested (ARC, 1979). Figure 7.2. Time-dependent failure of carbon, glass and
Kevlar-49 fibres (Aveston et al, 1980).


As the results in Figure 7.2 show, the stress- 6
rupture curve for carbon filaments is horizontal, the
water at 70°C
fibre therefore being completely insensitive to the 5
environment, while Kevlar-49 shows a time-

Weight gain, %
water at 50°C
4
dependent response somewhere between that of
carbon (nil) and that of glass (high). The aramid 3 water at 20°C
fibre, like glass, is therefore also environmentally
sensitive although the stress-corrosion mechanism is 2 65% RH at 23°C
different. The yarn may absorb some 6% by weight
of water in about 30 hours at a relative humidity of 1
96% (Smith, 1979). This moisture absorption is
0
accompanied by dimensional changes which are of 0 1000 2000 3000 4000
importance for the manufacturers of composites. Exposure time, hours
Any absorbed moisture must be removed from the
yarn before prepregging or other laminating Figure 7.3. Moisture absorption in Elf 37SA epoxy resin
procedures since drying out after cure may lead to during exposure to wet or moist environments at various
fibre/matrix decohesion and consequent loss of temperatures (Charrière, 1985).
composite properties. Moisture absorption at
ambient temperatures does not impair the short-term strength of the fibre, but Smith (1979) has shown
that high-temperature wet-ageing seriously reduces its strength.

7.2.2 Resin effects
The open molecular structure of glassy
thermoplastic and thermoset polymers permits 10
relatively rapid diffusion of moisture. Depending
Moisture absorption, %

on the nature of the polymer, considerable 8
Fickian
quantities of moisture may be absorbed into the
material, and both the rate of absorption and the 6 100% RH
saturation absorption level will depend on the
dual-mode absorption
conditions of exposure (ie. temperature and
4
relative humidity). This is illustrated in Figure 7.3 65% RH
by some results for an experimental epoxy resin Mm
manufactured by Elf-Aquitaine. 2
slope = 4Mmh-1√(D/π)
Moisture ingress in polymers usually follows 0
Fickian kinetics, the amount of moisture absorbed 0 20 40 60 80 100
being proportional to √(time) in the early stages of √(t), (min) ½

exposure, as predicted by the one-dimensional
Fickian model of Shen and Springer (1981), for Figure 7.4. Moisture absorption curves for 914 epoxy resin at
23°C and two different levels of relative humidity (RH)
example. A condition for the applicability of (Fernando, 1986).
Fick’s Law is that the diffusion is concentration-
independent, but the diffusion behaviour of many polymers, both glassy and cross-linked, under
particular conditions, may not be adequately described by a law which predicates fixed boundary
conditions, especially when the movement of the diffusing species is linked with molecular relaxations
or morphological effects in the polymer. Deviations from Fickian behaviour are then observed which
may be associated with structural changes, resin degradation, or even simple relaxation in response to
the diffusing species. Frequently, the early response is Fickian and this may be followed by a quasi-
equilibrium stage with a much slower approach to final true equilibrium (dual-mode sorption). True
Fickian and dual-mode sorption effects are illustrated in Figure 7.4 by some results obtained at two
different humidities for Ciba-Geigy 914 epoxide resin (Fernando, 1986).

The equilibrium moisture content depends on the chemical and physical nature of the polymer. The
absorption of water strongly reduces the mechanical properties of hydrogen-bonded polymers like
conventional polyamides, but in polyester and epoxide resins it acts as a diluent, increasing the free
volume and facilitating molecular motion under stress. This plasticizing effect reduces the stiffness of


the resin, lowers the glass-transition temperature, Tg, and 250

Glass transition temperature, Tg , °C
increases the magnitude of time-dependent effects like
stress relaxation. The reduction in Tg by moisture 200
absorption may be by as much as 100°C in some cases,
as Figure 7.5 from the work of de Iasi and Whiteside 150
(1978) shows. Crystalline thermoplasts are much more
resistant to moisture penetration than epoxide resins. The 100
semi-crystalline high-performance polymer poly(ether results for five common epoxy resins
ether ketone) (PEEK), for example, appears to be largely 50

immune from environmental effects. Mensitieri et al
0
(1996) report only a 2°C depression of Tg in samples 0 2 4 6 8 10
equilibrated in water at 60°C. Although PEEK contains Moisture content, %
oxygen atoms that may potentially form hydrogen
Figure 7.5. Effect of moisture content on the glass
bonds, it is not moisture sensitive because these atoms transition temperature of common epoxide resins
are sterically shielded by the aromatic rings. (de Iasi & Whiteside, 1978).

The wetting and drying of a resin are accompanied by dimensional changes which generate residual
stresses in a resin containing rigid reinforcement. Solvent attack on resins can cause both physical and
chemical effects, the latter usually being either hydrolysis (of the ester linkages in a polyester, for
example) or oxidation, although these processes are slow at ambient temperatures. Chemical attack
naturally reduces the mechanical properties of the common resins.

7.2.3 Composite effects
In a normal composite material, moisture will enter by diffusion through the bulk resin, by capillary
flow through pores in the resin, and also along the fibre/resin interface. Fickian kinetics again often
appear to apply, but deviations from Fickian behaviour are frequently observed, such as where
accelerated moisture pick-up occurs as a result of the presence of micro-voids or cracks in the resin. The
moisture diffusion in an anisotropic composite lamina is of course also likely to be anisotropic, the rate
of diffusion being more rapid in the direction of fibre alignment because of the continuity of the resin
diffusion path. The kinetics of moisture uptake are there also markedly influenced by the lay-up of a
laminate (Collings & Stone, 1985).

For the case of Fickian diffusion (ie. assuming that moisture transport is through the resin only) the
diffusion coefficients for a single unidirectional lamina can be approximated by:

D11 = (1 – Vf)Dr ...................................................................................................... 7.1)

D22 = [1 – √(Vf/π)]Dr ............................................................................................... 7.2)

for diffusion parallel and perpendicular to the fibres, respectively, D being the diffusion coefficient for
the matrix resin (Dr » Df). The diffusivity at any angle, θ, to the fibre direction is then

D θ = D11 cos 2 θ + D 22 sin 2 θ ............................................................................... 7.3)

If the response of a single lamina is known (or can be calculated from models such as those of
Springer) the overall moisture absorption behaviour of a multi-ply laminate can also therefore be
calculated. Computer programs for this purpose have been developed (Curtis, 1981).

Bueche and Kelly (1960) developed a theory for the prediction of the Tg of plasticized resin
composites which gives:

α r (1 − Vf )Tgr + α f Vf Tgf
Tg = ............................................................................... 7.4)
α r (1 − Vf ) + α f Vf


where α is a thermal expansion coefficient and Vf 6
is the fibre (or filler) volume fraction, the

Weight gain, % (from dry state)
KFRP; water, 100°C
subscripts r and f referring to the resin and fibre 5
(or filler) components. The theory predicts the
4
trend in Tg reasonably well. The nature of the fibre
can also exert an influence on the process of 3
moisture uptake, as illustrated by the graphs in KFRP; 65% RH, 23°C
CFRP & GRP;
Figure 7.6. The materials are of the same 0/90 lay- 2
H2O, 100°C
up and contain the same epoxide matrix resin, and
1 CFRP; 65%, 23°C
it can be seen that at both ambient temperature in
moist air and in boiling water there are only slight GRP; 65%, 23°C
0
differences between the absorption in composites 0 200 400 600 800 1000
containing carbon and glass fibres, whereas the Exposure time, days
absorption in the aramid-fibre composite is
substantially greater as a consequence of the Figure 7.6. Absorption of water during exposure of
hydrophilic nature of the aromatic polyamide. composite laminates at ambient temperature (65%
relative humidity) and at 100°C (in water). The
laminates are 0/90 laminates of HTS carbon, E-glass
Both reversible and irreversible mechanisms of and Kevlar-49 in Code-69 epoxy resin.
degradation may occur as a result of moisture
ingress. For example, if water travels down the interface or is able to accumulate at the interface after
diffusion through the bulk resin it may cause hydrolytic breakdown of any chemical bonding between
the fibre and the resin and bonds may be disrupted by the
swelling of the resin. Either of these mechanisms of bond
failure will impair the efficiency of stress transfer between
matrix and reinforcement. The strength and stiffness of the
composite will then be recovered on drying out only if new
chemical bonds can be re-established. If the bonding is
simply mechanical, however, and the absorption and
desorption of water are purely physical processes related to
the resin or the interfacial bond (ie. supposing the fibre
strength to be also unaffected), the composite would be
expected to recover its properties on drying out. Thus,
while the tensile strengths of carbon and Kevlar laminates
Figure 7.7. Tensile fracture surface of an E- are little affected by hygrothermal treatments, even as
glass/epoxy GRP laminate after boiling in water severe as boiling in water, a GRP laminate may lose some
(photograph by Dr RF Dickson).
50% of its strength. However, much of this damage is
apparently reversible since even boiled GRP laminates may
regain all or most of their original strengths on re-drying (Dickson et al, 1984). This suggests that in
carbon and Kevlar composites there is only a physical bond between fibres and matrix, and that this is
not affected by hot/wet ageing, whereas in the GRP there is a chemical bond (presumably resulting from
the presence of a silane coupling agent on the fibres) which is destroyed by the hygrothermal treatment
but apparently re-established on removal of the moisture. The marked change in the fracture surface of a
GRP laminate after exposure to boiling water is shown in Figure 7.7: a failure surface of this kind, which
is in marked contrast to the normal bushy fractures shown in Figure 5.7 of chapter 5, is inevitably
associated with brittle behaviour.

The reversibility of this moisture effect, and the fact that the laminate behaviour goes from brittle in
the dry state to ductile when wet and back to brittle on re-drying, supports the general belief in the
plasticizing effect of moisture on the resin. It should be pointed out, however, that other experimenters,
working with other resins and different combinations of environmental and stress conditions, have
observed irreversible damage, such as resin cracking and permanent bond breakdown, with the
inevitable irreversible degradation of composite properties. The chemical stability and resistance to
moisture penetration of the matrix polymer, which depend sensitively on chemical and molecular
structure and state of cure, will usually determine the extent of both reversible and irreversible composite
damage, although resin void-content and composite fibre-content will also exert an effect. It is generally


recognized that laminates based on polyester 4
resins are less moisture resistant than epoxy-
based composites, and that phenolic laminates 913 matrix
are more water-resistant than either of these. But 3

Weight gain, %
within each group of resins there is in fact a
wide range of individual types and
2 914 matrix
compositions, from which the designer may
select according to his requirements, with a
correspondingly wide range of chemical stability 1
and moisture resistance. In the polyester range,
for example, there is increasing chemical
resistance through the sequence orthophthalic, 0
0 200 400 600 800 1000 1200 1400 1600
isophthalic, neopentyl-glycol, and bisphenol
polyester resins. Much of the research on matrix Exposure time, hours
resins has as a general goal the improvement of
Figure 7.8. Moisture uptake in 16-ply undirectional
the polymer tolerance to hot/wet conditions. One
carbon-fibre composites with two different epoxy resin
of the difficulties here is that improvements in matrices exposed to water at 23°C (Charrière, 1985).
hygrothermal tolerance can usually only be
achieved at the cost of a raised processing temperature. In Figure 7.8, for example, we see that the
saturation uptake of water into a carbon-fibre composite based on Ciba-Geigy 913 resin is twice that in a
similar laminate made with the same company’s 914, and as we saw in chapter 1, the former is processed
at 120°C while the latter requires a processing temperature of 170°C.

7.3 TIME-DEPENDENT EFFECTS
The results of Aveston et al in Figure 7.2 showed that the load to cause failure in 105 minutes of E-
glass strands tested in distilled water was less than 50% of the normal breaking load. They also found, to
their surprise, that the situation was not improved when the strands were impregnated with resin. The
apparent strengths of the impregnated strands fell to one sixth of the normal value, regardless of whether
the impregnating resin was epoxy or polyester (although the epoxy resin appeared to delay the onset of
the strength reduction somewhat) presumably because of the lower moisture permeability of the epoxy
resin.

Phillips et al (1983) have reported the results of an extensive series of stress-rupture experiments on a
range of GRP, with a variety of different resins and reinforcement types, in distilled water and in sea-
water at different temperatures. Their results confirm the general idea that the strength of moist GRP is
controlled by the plain moisture sensitivity of
the fibre strands. They were unable to find any 100
effect of variation in immersion conditions, but
Load, % of mean failure load

found that different resin types produced 80 20°C
detectable differences in life at a given stress 40°C
level. Vinyl-ester resin showed considerable 60°C
60
advantage over the less expensive isophthalic
polyester, for example. It has often been
supposed that stress-rupture results for GRP 40
materials fitted a linear stress vs. log time
relationship (Zhurkov, 1965), and attempts at 20
time-temperature superposition based on this
assumption have often been proposed for design 0
purposes. The work of Phillips et al. however, 2 3 4 5 6 7 8 9
suggests that the situation may in reality be Logt (time to failure, t, seconds)
much more complex, and their results were
frequently best fitted by a two-slope curve, as Figure 7.9. Time-dependent failure of GRP materials in
shown in Figure 7.9. Below a certain well- water at various temperatures (Phillips et al, 1983). The
mean ultimate tensile failure load of the material is
defined load level, results were strongly 256N/mm.


temperature-dependent, the effect being greater than could be accounted for solely on the basis of a
thermal-activation model, while above this critical load level, temperature had little effect on sample life.

Jones et al (1983) have studied the more dramatic effects of acid environments on glass filaments and
glass/epoxy laminates. They confirm that the prime cause of stress-corrosion in GRP is the susceptibility
of the glass fibres themselves to stress-corrosion. Thus, while the main function of the resin is to protect
the fibres, they found that the glass/resin interface itself was also subject to attack, with the result that the
environment — sulphuric acid in their experiments — penetrates the laminate. At applied strain levels
greater than 0.15%, failure times for laminates were the same as those for the fibres, while at lower strain
levels the resin protected the fibres and the laminate was more resistant to stress-corrosion. However,
they also found that crystallizable calcium-rich glass corrosion products accumulating in unexposed
areas of the composite could result in more rapid failures than normal at these low strain levels. Stress-
corrosion effects of this kind often markedly change the fracture mode of a GRP composite. Instead of
the kind of brushy fracture that is commonly observed in high-toughness composites, cracks appear to be
able to propagate easily from fibre to fibre without being diverted along the interface, as shown by
Figure 5.4 in chapter 5.

7.3.1 GRP in Corrosive Environments
This brief survey of environmental effects suggests that reinforced plastics components and structures
may not be an obvious first choice for applications where stability against chemical attack is a pre-
requisite. Nevertheless, a great deal of GRP is used for the containment of corrosive and other hostile
liquids and designers specify them with increasing confidence despite the fact that relevant long-term
stress-corrosion data are not always available and extrapolation from laboratory-scale data is always
dangerous. This is usually possible because the manufacturer is able to use expensive, chemically-
resistant and moisture-resistant resins, either for the whole structure if the extra cost can be justified, or
as a thin, renewable surface gel-coat which protects an underlying GRP based on a cheaper (and less-
resistant) matrix resin. Such a gel-coat could be additionally stabilized against ultra-violet if necessary,
by pigmenting with TiO2, for example.

A useful review of corrosion and environmental degradation of GRP composites of various kinds has
been published by Hogg and Hull (1983).

7.4 HIGH-TEMPERATURE STRENGTH AND CREEP OF COMPOSITES
7.4.1 Introduction
The strengthening achieved through the operation of conventional solid-state mechanisms in metals
and plastics is easily destroyed by thermal activation as the temperature is raised and there are few alloy
or polymer systems that can continue to give satisfactory service when atomic or molecular mobility is
enhanced by heating. But since many of the strong solids
used for reinforcement are inherently resistant to this 149
0
form of instability, up to relatively high temperatures, an 204
important advantage of fibre composites is that they may
Weight loss, %

be expected to retain their strengths to much higher 10
260
temperatures than would normally be possible. This 316°C
371
would be true for composites reinforced with inorganic 20
427
synthetic fibres like carbon, boron, SiC or Al2O3,
482
although metallic fibres are naturally subject to the same 30 538
limitations as metallic matrices and glassy fibres lose
their strengths at the glass-transition temperature when 40
molecular networks become freely mobile. -1 0 1 2 3
Log(time, hr)
In reinforced plastics there are few matrix materials
that can withstand temperatures above 300°C (see Figure 7.10. Weight loss during thermal
degradation of glass-fibre/phenolic laminate at
chapter 1), and the limiting factor for the most stable of elevated temperatures (after Boller, 1960). The
matrices is likely to be chemical degradation. On temperatures are all in degrees Celsius,


exposure to air at elevated temperatures, thermosetting materials gradually become degraded through
chemical changes which result in loss of material accompanied by loss of mechanical integrity. Early
work by Boller (1960) illustrated this for glass/phenolic laminates: Figure 7.10, redrawn from his article,
shows that the phenolic matrix is not stable above 200°C for more than two hours or so. Families of
curves of this kind can be shifted along the abscissa to form a continuous master curve having an
exponential form where the weight loss is proportional to exp(–kt), k being a constant. The deterioration
in strength and modulus follows a similar
exponential decay. The same kind of 1.0
time/temperature superposition is used in the ambient
350°C
study of polymers to produce master curves of 0.8

Flexural strength, GPa
creep compliance or relaxation modulus over a
wide range of times on a logarithmic scale (see 0.6
McCrum et al, 1988, for example).
0.4
Most thermoplastic polymers are naturally
unsuitable for high-temperature composites,
0.2
although some of the more sophisticated
polymers like the polyaryls (eg. PEEK) have
0
continuous working temperatures that are quite PBA+ PI+ PBT+ 7075
close to 300°C. Some of the more complex S-glass SiO2 E-glass Al alloy
thermoset polymers, too, possess remarkable
resistance to degradation — polymers like the Figure 7.11. Flexural strengths at ambient temperature and
polyimides. Figure 7.11 compares the room- 350°C of some laminates based on thermally stable
polymers polybenzoxazole (PBA), polyimide (PI) and
temperature and elevated-temperature strengths
polybenzothiazole (PBT), compared with the behaviour of
of composites based on some of these newer an age-hardened 7075 aluminium alloy (Aponyi, 1967)
polymers. It can be seen that, in common with
many other reinforced plastics, the strengths of these materials at ambient temperature are comparable
with those of heat-treated aluminium alloys, but they have a clear advantage over aluminium alloys at
temperatures where the metallic materials over-age as a consequence of solid-state diffusion.

Although aluminium alloys and thermoplastics rapidly lose their load-bearing ability and rigidity as
the temperature rises above ambient, the consequences of this for users of composites based on these
materials will depend to an extent on how the load is distributed relative to the fibres. Consider a
unidirectional composite containing 60vol% of rigid ceramic fibres, for example, in which the matrix
strength is one tenth that of the fibres. A simple application of the mixtures rule shows that if, as a result
of a rise in temperature of, say, 300°C, the matrix loses half its strength, the drop in the uniaxial
composite strength will be only about 3.5%.

100 But while such composites may perform very
Short-beam shear strength, MPa

well by comparison with aluminium alloys when
80 loaded in tension or flexure, even at 300°C, they
913 ud 914 ud will creep readily under the action of interlaminar
60 shear forces when the visco-elastic behaviour of
the matrix and interface govern the composite
40 response. As an illustration, the results of short-
5208 woven beam shear tests on some carbon-fibre composites
914 woven are shown as a function of temperature in Figure
20
7.12. Two types of comparison may be made.
First, the importance of the thermal stability of the
0
0 20 40 60 80 100 resin depends on the composite lay-up: the
Temperature, °C composite based on the high-temperature-cure
914 resin retains a higher proportion of its room-
Figure 7.12. Effect of temperature on the short-beam temperature strength than the low-temperature-
shear strengths of some unidirectional and woven-fabric cure 913 composite as the temperature rises.
carbon-fibre laminates (Charrière, 1985).
Second, when the reinforcement is woven,


1000
resistance to shear loading is still lost as the
Vf: temperature rises, but less rapidly than in the case of
Flexural strength, MPa

800 a unidirectional laminate, and the nature of the
0.4, 0.5
& 0.6
matrix resin is less important.
600

0.2 In metallic composites the high-temperature
400 performance will often be severely limited by
& 0.3
0.1
fibre/matrix interaction. Interfaces in most
200 artificially produced composites with metallic or
0.05 ceramic matrices are unstable at elevated
0
temperatures, and solid-state diffusion often occurs
0 200 400 600 800 1000 1200 1400 unless a thermal/diffusion barrier coating has been
Temperature, °C deposited on the fibre. This often results, at first, in
slightly improved properties as the adhesion
Figure 7.13. The effect of temperature on the flexural
strength of SiC-whisker-reinforced alumina (Tiegs and improves, but this is usually followed by
Becher, 1987). deterioration of composite strength as the fibre
surface is attacked and chemical compounds begin to
form. Work on the development of diffusion barrier coatings to overcome these problems currently has a
high priority in the composites research field,
and some of the ceramic fibres now 120
Creep stress, % of failure stress

commercially available are already protected by
graded-property coatings. Chemical reactions 100
will also occur in CMCs if the exposure
80
temperature is sufficiently high, and functionally
graded diffusion-barrier coatings are also 60
necessary. But although advantageous short-
term performance at elevated temperatures can 40 Tested at 600°C in air
be obtained in CMCs, as illustrated by the
results in Figure 7.13, since brittle-matrix 20
systems inevitably crack under modest loads,
exposing the reinforcing filaments to the 0
0.01 0.1 1 10
working environment, a CMC in service at high
Time to failure, hr
temperature will lose strength over time unless
the fibres are protected by a diffusion barrier. Figure 7.14. Flexural creep-rupture data for a (0,90)3S
This can be seen from the results of some short- SiC/CAS ceramic-matrix composite at 600°C in air.
term flexural creep-rupture tests on (0/90)3S
Nicalon/CAS composites tested in air at 600°C. Figure 7.14 shows that under loads as low as 60% of the
instantaneous breaking load the life of the material is only three or four hours.

7.4.2 Creep
7.4.2.1 Introduction: creep phenomena
Although the mechanisms by which metals, plastics and ceramics deform in creep are different, the
phenomena which are observed in these materials are similar. The basic mathematical laws of creep
often seem to apply across the material classes, even though the physico-chemical phenomena are
different. Studies of creep behaviour of rod-shaped samples in tension show that the creep response,
characterized in terms of the time-dependent strain at constant stress and temperature, may be
represented by a general law of the kind:

ε(t) = ε0 + αt1 3 + β t + κt 3 ………….7.5)

in which the first term represents the initial elastic strain on loading, and the remaining terms are referred
to as primary, secondary and tertiary creep, α, β and κ being constants. An equation of a simplified
form, known as Findley’s equation (Findley and Kholsla, 1955), is also frequently used to describe the
creep behaviour of polymeric materials. Primary creep is sometimes referred to as transient creep, and


secondary creep is usually referred to as steady-state
creep. Which of the time-dependent components are
actually observed in a given creep test depends on the
stress level and the temperature. At temperatures above
half the absolute melting point (½Tm) and fairly high
stress levels, for example, a metal creep curve may
appear to be curving continuously upwards, showing
little, if any, steady-state behaviour and no transient
creep. By contrast, at low temperatures and modest
stresses, many metallic materials show only the
transient component, deformation ceasing after a given
time or appearing to become asymptotic to some
maximum strain level. These features are illustrated in
Figure 7.15. In many creep investigations, it is the Figure 7.15. Schematic illustration of the
steady state that attracts most attention because the appearance of metallic creep curves.
temperature- and stress-dependence of the constant
deformation rate, dε/dt, usually denoted ε II, can be interpreted in terms of the atomic or molecular
&
mechanisms of deformation and the rate theory of chemical reactions (Glasstone et al, 1941). The stress-
dependence of the secondary creep rate of many materials can be written in terms of the power-law
equation:

ε II = Aσn
& T = constant……………………………….7.6)

so that a log/log plot of creep rate versus stress is linear, and this is
often combined with an Arrhenius expression for the temperature
dependence in a general creep law:

ε II = Bσ n e ( ) …………………………………………7.7)
− Q kT
&

In these equations, A, B and n are materials constants, k is Boltzmann’s
constant, T is in degrees Kelvin, and Q is loosely described as being an
‘activation energy’ for whatever deformation process is controlling the
rate of the mechanism causing creep. If a single mechanism is
responsible for the observed strain, then in principle Q may be
calculated from physical knowledge of the mechanism. For example, at Figure 7.16. Schematic illustration
temperatures above ½Tm, solid-state diffusion occurs very rapidly in of the variation of the secondary
creep with stress for metallic
materials.
T°C slope:
10-2
metals, and creep is mostly a result of
Secondary creep rate, hr -1

1371 4.3
dislocation climb as a result of vacancy
1093 11.8
movement through the lattice. The observed
815 15.3
value of Q obtained by plotting log( ε II) &
649 37.3
10-3 versus 1/T is therefore usually equal to the
activation energy for self-diffusion (Dorn,
1957). At high homologous temperatures and
10-4 low stresses the value of n may fall to near
unity and the deformation approximates
Newtonian flow (strain rate proportional to
stress), while at low temperatures and high
10-5 stresses the deformation may be controlled by
0.1 0.2 0.5 1 2
dislocation movements other than climb (ie.
Stress, GPa glide, cross-slip etc.), in which case the value
Figure 7.17. Secondary creep rate of 76μm diameter tungsten of n may be very large. These changes are
wires as functions of stress and temperature (Harris & Ellison, shown schematically in Figure 7.16, and are
1966)


illustrated in Figure 7.17 by some results for the creep of cold-drawn tungsten filaments, 76μm in
diameter, at various temperatures. It can be seen that the creep process at the highest temperature
represented, above 1600K — just about ½Tm, the deformation process appears to be controlled by self
diffusion, whereas at the lowest temperature, about 900K, the value of n is characteristic of non-
diffusion processes.

The modelling of the time-dependent deformation of polymers is done in a slightly different fashion.
Various combinations of elastic springs and viscous dashpots are often used to represent their visco-
elastic response, and there is a highly developed mathematical theory of visco-elasticity (Ferry, 1980).
The three simplest spring/dashpot models are those shown in Figure 7.18. The Hookean spring is
supposed to have an elastic modulus E, and the Newtonian dashpot has a viscosity η. The parallel
element (a) responds like a damped automotive shock absorber, its maximum strain level being limited
by the extension of the spring. The series model, (b), however, has no strain limit and deforms in a
viscous manner after an initial elastic extension. Neither of these represents perfectly the real behaviour
of a polymer, and more complex combinations such as that labelled (c), known as a standard linear solid
(or Zener solid), provides a somewhat better model. The derivation of the creep curve for the Voigt
model (model (a) in Figure 7.18) will illustrate the modelling procedure.

The responses of the separate elements are the normal
definitions of elastic and viscous deformation:

σ = Eε (Hookean), and

dε
σ=η (Newtonian)
dt

The stress acting on the element is shared between the two
components: the total stress on the element is the sum of the
stresses in the two components:

σ = σspring + σdashpot Figure 7.18. Simple spring and dashpot
models that are used to represent time-
and hence we obtain the constitutive equation for the Voigt dependent behaviour of polymers. The
element: parallel model, (a), is known as a Voigt
element, the series model, (b), is known as a
Maxwell element, and the combination of the
dε two represented by (c) is known as a
σ = Eε + η ....................................7.8)
standard linear solid (SLS).
dt

The ratio η/E is called the time constant, τ, and this is referred to as a retardation time or a relaxation
time, depending on the deformation conditions. Rewriting the constitutive equation to incorporate this
definition, we obtain the differential equation:

dε ε σ
+ = ......................................................................................................... 7.9)
dt τ Eτ

which on integration yields the creep equation:

σ
ε (t) =
E
(1 − e − t τ ) .............................................................................................7.10)
This strain, which is fully recoverable, is shown as a time function in Figure 7.19 together with the
responses of the simple elastic and viscous components and the series Maxwell element. The exponential
form is not unlike that of the bottom curve in Figure 7.15, and it is generally considered that the Voigt
model does represent the creep behaviour of polymers to a first approximation (Ward, 1971). It is more
common in discussing these models to present them in terms of a creep compliance, J(t):


J(t) = E-1[1 – exp(-t/τ)] ……………………………..............7.11)

which has the normal units of compliance, mm2.N-1.

The constitutive equation for the standard linear solid (SLS) is more
complicated because there are two time constants instead of one:

dσ ⎛ dε ⎞
σ + τ1 = E ⎜ ε + τ2 ⎟ ………………………………….......... 7.12)
dt ⎝ dt ⎠

where E is now a combination of the stiffnesses of the two springs, (1/E1 +
1/E2 )-1, and the two time constants in the equation are τ1 = η2/(E1 + E2)
and τ2 = η2/E2. The deformation of the SLS, like that of the Voigt element,
is also reversible since the overall control of extension is by the two spring
elements. Figure 7.19. Strain responses
of simple solids to a stress
The foregoing comments about spring/dashpot models relate only to applied at time to.
linear visco-elastic behaviour. The concept of a linear visco-elastic solid
differs from that of a linear elastic material in that it involves the time-dependence characterized by the
derivatives, as in equation 7.12. In a linear visco-elastic material, if a particular stress causes a given
strain in a given time, a stress which is twice as great will cause twice the strain in the same amount of
time. The assumption of visco-elasticity, which in the Voigt model (but not in the Maxwell model)
implies reversibility, is often an invalid assumption for the behaviour of polymers if the strain is not
small. A fuller treatment of the behaviour of non-linear viscoelastic behaviour in polymers and polymer
composites has been given by Schapery (1996) and Emri (1996).

The time-dependent and frequency-dependent behaviour of polymers are closely linked, via the
theory of linear visco-elasticity (Ferry, 1980), and are frequently studied by the experimental method
known as dynamic mechanical and thermal analysis (DMTA). Because of its visco-elastic nature, energy
is dissipated when a polymer undergoes load cycling and this dissipation, characterized by the hysteresis
or energy loss per cycle, reaches a maximum
whenever the frequency of molecular motion is 10
close to the cycling frequency. The energy loss
LogE' and logE"

9
is usually measured as the ratio of the loss
modulus to the storage modulus, E"/E', which is 8 logE'
approximately equal to the phase angle, tanδ.
Thus, measurements of tanδ as a function of 7
temperature will identify the occurrence of
6 plain resin
specific mechanisms of molecular activity. logE"
DMTA is thus a sensitive means of detecting 5 untreated fibres
changes in the mobility of molecules and for 0.8 oxidised and sized fibres
investigating phase structure and morphology.
0.6
Tanδ

For polymeric materials, it is able to identify a
range of relaxations and transitions, and is thus a 0.4
potentially powerful tool for assessing the 0.2
effects of such features of composite materials 0
as matrix polymer modification (eg. 250 300 350 400 450 500
‘flexiblization’), matrix crystallinity,
Temperature, K
transcrystallinity at a fibre/thermoplastic-matrix
interface, and interphase behaviour or other Figure 7.20. Loss and storage moduli and the loss tangent for
effects of fibre surface treatments. epoxy resin and also of samples of unidirectional composites
containing carbon fibres with no surface treatment and in the
An example of the use of DMTA for the oxidized and sized conditions. The resin is Ciba Geigy
LY1927GB epoxy resin and the unidirectional composites
study of fibre composites is shown in Figure contain 10vol% of ENKA HTA carbon fibres. (Harris et al,
7.20. The materials are a plain epoxy resin and 1993)


unidirectional composites of this resin reinforced with 10vol% of carbon fibres, both without surface
treatment and with a normal, commercial, oxidized and sized surface. The upper graph shows the storage
and loss moduli, E' and E". The storage moduli for the three materials all show the same features — a
relatively flat section at low temperatures with a marked drop at about 400K to a second flat region at
high temperature. The fall coincides with a peak in the loss modulus which identifies the glass-transition
temperature, Tg. It can be seen that the stiffness of the composites, as indicated by the real component of
the dynamic flexural modulus, E', is significantly higher than that of the resin, as would be expected,
over the whole temperature range, and the effect of the commercial surface treatment is to increase the
composite stiffness slightly above that for the composite with untreated fibres. The curves for the loss
modulus, E", show that the hysteretic loss in the composites is also higher than that in the plain resin,
and we assume that this is because of something like frictional losses at fibre/resin interfaces. There is no
clear separation between the E" curves for the two surface treatments, however, and this is confirmed by
the tanδ curves in the lower graph. But there is some suggestion that the temperature of the loss peak
(often referred to as the α peak) associated with the glass transition is raised slightly by the presence of
the fibres. The composite thus reflects the visco-elastic features of the matrix resin. Although all of the
graphs exhibit a single, major α peak at about 130°C, the resin curves also show a broad, weakly defined
β transition in the region of 50°C.

This very much simplified sketch can serve as only an introduction to a very extensive subject, and
for further details specialist books on creep should be consulted (eg. for metals: Garofalo, 1965, Evans &
Wilshire, 1993; for plastics: Ward, 1971, McCrum et al, 1988).

7.4.2.2 Creep of composites
Creep, like high-temperature strength, depends upon the stability of the fibre and the extent to which
it can relieve the matrix of direct load. Thus, in composites reinforced with continuous, well-aligned
carbon or boron fibres no creep strains should be observed apart from those which accompany small
amounts of straightening or realignment of the fibres under stress. But even though the fibres themselves
do not exhibit any creep deformation, small redistributions of load due to matrix creep will still lead to
time-dependent fibre breakage because of the statistical distribution of defects in the brittle fibres.
Lifshitz and Rotem (1972) have developed a statistical model for the time-dependent failure of
reinforced plastics which predicts the reduction of strength of composites under constant tensile loads. In
GRP this loss in strength with time can be substantial. Lifshitz and Rotem found that failure occurred at
room temperature in 0.60 Vf unidirectional glass/polyester composites after 105 minutes at 50% of the
ordinary tensile strength, while the value for glass/epoxy composites was about 70%. Fuwa et al (1975)
showed by means of acoustic emission studies that in unidirectional carbon/epoxy composites the
random fibre damage ceased after a few minutes even at stresses greater than 90% of the tensile strength.
On the other hand, creep in laminates under interlaminar shear stresses or in short-fibre composites
under any kind of load will be rapid if the matrix has no inherent creep resistance. Creep will also occur
in laminates with fibres at some angle to the stress axis in a manner characteristic of the constrained
matrix material.

Early creep models were largely developed for all-metal systems in which the creep behaviour
depends on how the phenomena described in the previous section affect the components of the
composite. There are a number of separate possible cases:
i) Continuous fibres; matrix creeping, fibres elastic; stress too low to cause damage to fibres
ii) Continuous fibres; matrix creeping, fibres elastic; stress high enough to cause breakdown of
fibres
iii) Continuous fibres; matrix and fibres creeping; stress too low to cause damage to fibres
iv) Continuous fibres; matrix and fibres creeping; stress high enough to cause breakdown of fibres
v) Short fibres; matrix creeping, fibres elastic
vi) Short fibres; matrix and fibres creeping.

Consider, for example, a metal-matrix composite in which the matrix creeps but the fibre remains
elastic and is undamaged by the stress. The creep strain of the matrix would be constrained by the


capacity of the fibres for elastic deformation, so that the maximum possible creep strain of the system
would be equal to σc/Ef Vf, where σc is the stress acting on the composite. The deformation of the
material would approach this strain level asymptotically, much like the response of the Voigt element
shown in Figure 7.19. Following the spring/dashpot arguments given earlier, the stress on the composite
is divided between the parallel components according to the mixture rule:

σc = σm Vm + σf Vf ................................................................................................7.13)

The strain in each of the two components is the same as that of the composite (the iso-strain model), but
since we are dealing with time-dependent effects we can only write:

εc = εm = εf
& & &

At a given temperature, then, the fibre deformation equation is:

1 ⎛ dσf ⎞
εf =
& ⎜ ⎟ .................................................................................................7.14)
E f ⎝ dt ⎠

and the matrix deformation equation is:

1 ⎛ dσm ⎞
εm =
& ⎜ ⎟ + Bm σ m ....................................................................................7.15)
p

E m ⎝ dt ⎠

the first term being the elastic strain rate and the second the plastic creep rate, assuming power-law creep
(equation 7.6) with an exponent p. Combining equations 7.14 and 7.15 in the differentiated form of
equation 7.13, we obtain an equation for the creep rate of the composite (McLean, 1982):
p
⎛ ε ⎞
ε c = αBm σ p ⎜ 1 − ⎟ .......................................................................................7.16)
&
⎝ εc ⎠
m

where α is given by:

⎡ ⎛ E V ⎞⎤
α −1 = ⎢1 + ⎜ f f ⎟ ⎥ (1 − Vf )
p

⎣ ⎝ E m Vm ⎠ ⎦

McLean showed that this model was able to predict
reasonably accurately the creep behaviour of the in situ matrix
0.3
eutectic composite γ/γ'/Cr3C2. 0.074 (Vf)
0.178
Creep strain, ε(t)

If the temperature is sufficiently high for both fibres 0.264
0.2
and matrix to creep, the deformation equations for the two
components, ignoring the elastic strains now, are: T = 649°C
0.1 σ = 241MPa

εm = B m σ p
& m
T = constant
ε f = Bf σ q
0
& f
0 50 100 150 200
Time, hr

where Bf and q are the power-law constants for the fibres.
Figure 7.21. Curves of creep strain versus time for
The stress on the composite is then given by: unidirectional composites consisting of Inconel
reinforced with tungsten wires (Ellison and Harris,
1966). The stress level was 241MPa and the
temperature 649°C.


1p 1q
⎛ε ⎞
& ⎛ ε ⎞
&
σc = Vf ⎜ c ⎟ + (1 − Vf ) ⎜ c ⎟ .....................................................................7.17)
⎝ Bf ⎠ ⎝ Bm ⎠

This equation cannot be solved explicitly for
300 0.30
the composite creep rate, but it can be used to
derive graphical relationships between creep rate 250 failure strain 0.25
and stress for particular materials if experimental time to rupture

Rupture time, hr
data for Bm, Bf, p and q are available. As an

Failure strain
200 0.20
example, we take some results of Ellison and
150 0.15
Harris (1966) for the creep of an MMC consisting
of an Inconel-600 nickel alloy reinforced with the 100 0.10
cold-drawn tungsten wires for which some creep
data were shown in Figure 7.17. A set of creep 50 0.05
curves for the plain matrix and composites of three
0 0.00
different volume fractions at 649°C and a stress of 0 0.1 0.2 0.3
241MPa are shown in Figure 7.21 and the effect of Fibre volume fraction, Vf
the fibre additions on the rupture time and the
strain to failure can be seen from Figure 7.22. If Figure 7.22. Creep rupture time and failure strain as
the experimental values of Bf, Bm, p and q for the functions of fibre volume fraction for unidirectional
composites of Inconel-600 reinforced with tungsten wires
matrix and for the reinforcing wires are extracted creep-tested at 649°C and 241MPa (Ellison & Harris,
from the publications of Harris and Ellison (1996), 1966).
curves of predicted behaviour of composites based
on the mixture-rule model of equation 7.17 can easily be derived for a given temperature, as in Figure
7.23. As shown by Street (1971) the predicted result for the lowest Vf composite (Vf = 0.074)
corresponds reasonably well with the experimental result, but as the fibre volume fraction increased, the
predictions underestimate the measured creep rates by increasingly large amounts. There are a number of
reasons why the model of equation 7.17 fails to describe the experimental data. The first relates to the
validity of the equation. We have assumed power-law behaviour for both matrix and fibres. In other
words, we are dealing in equation 7.17 with steady-state creep only. Yet it is clear from Figure 7.21 that
the secondary creep regions for fibres and matrix cannot coincide and the basic premise on which the
equation is founded must therefore be invalid. Since we do not have a simple model for the stress-
dependence of transient creep, however, we cannot re-derive the model with more appropriate stress-
dependent functions.

A second possible cause for the discrepancy is that we assume a homogeneous state of stress within
the composite. It has been shown, however, that in
MMCs inhomogeneous deformation resulting from
10-2 matrix the differences in elastic properties of the two
-1

0.10 (Vf) components leads to greater levels of work
Secondary creep rate, hr

0.20 hardening in the matrix close to the fibre than in
10-3 0.40
material more remote from the interface (Kelly and
0.60
fibre
Lilholt, 1969). It may be, then, that the matrix is
10-4 creeping at a higher true stress level than is assumed
in the model of equation 7.17. Kelly and Tyson
T = 649°C
10-5 (1966) had also pointed out that although the
addition of fibres substantially reduced the creep
rate of a short-fibre composite, ε was nevertheless
&
10-6 governed by the creep of the matrix under the action
10 100 1000 of the shear stresses close to the interface.
Creep stress, MPa
In practice the creep resistance of a matrix will
Figure 7.23.Predicted creep behaviour of unidirectional usually be much lower than that of a reinforcing
metal-matrix composites with both fibre and matrix fibre, and we might expect that at high stress and
creeping. The data for fibre and matrix are based on
results of Ellison and Harris (1966) for composites of
high temperature the creep of the composite would
Inconel reinforced with tungsten wires.


be controlled by that of the fibres alone under the fibre stress, σcVf. Street (1971) pointed out that at low
Vf neglect of the matrix contribution could result in significant over-estimates of the composite creep
rate, but Harris (1972) showed that the times-to-failure of these Inconel/tungsten composites at the
equivalent fibre stress σcVf agreed with the creep behaviour of the isolated wires for test times of 100
hours or so at 650°C. When the time to failure was increased well beyond this by increasing Vf,
however, deterioration of the wires by intermetallic diffusion led to a reduction in creep resistance below
that predicted from the isolated fibre properties.

Street also points out that on the basis of equation 7.17, the composite will not exhibit a unique value
of the power-law exponent, nc (ie. n in equations 7.6 and 7.7), although the matrix and fibres do (p and q
in equation 7.17). As implied by Figure 7.23, for most Vf the value of nc is determined by the fibres at
low creep rates, but falls at higher ε as the effect of the matrix increases. The measured values of nc thus
&
depend on both volume fraction and deformation rate.

The creep behaviour of short-fibre composites is governed by the matrix rather than the fibres. A
simplistic approach would be to suggest that the stress component in equation 7.17 is replaced by a term
describing the average stress in the short fibres, similar to that discussed in section 4.18 of chapter 4. For
a non-creeping fibre, the revised equation then becomes:
1q
⎛ l ⎞ ⎛ε ⎞
&
σc = Vf σfu ⎜1 − c ⎟ + (1 − Vf ) ⎜ c ⎟ ................................................................7.18)
⎝ 2l ⎠ ⎝ Bm ⎠

where lc is the fibre critical length, and σfu is the fibre tensile strength. Assuming that the fibres are
longer than or equal to the critical length, the effect is then for the fibres simply to shield the matrix from
part of the load. The creep strain rate can then be obtained explicitly as:
p
⎡ σ − Vf σfu (1 − l c / 2l) ⎤
ε c = Bm ⎢ c
& ⎥ .......................................................................7.19)
⎣ 1 − Vf ⎦

This can only be a lower bound to the composite creep rate, however, because other processes will
occur during the creep of a short-fibre composite. Shear stresses at fibre ends will be relaxed, causing
load redistribution, stress relaxation will occur in the matrix, and load will be transferred by shear into
the fibres. Creep of the fibres themselves may also occur if the temperature and stress are sufficiently
high. Starting from a more rigorous evaluation of the average stress in the fibre than that assumed in
equation 7.18, Kelly and Street (1972b) derived a model which took most of these factors into account.
They found that, although the fibres reduce the effective load on the matrix, the presence of the fibres
actually increases the rate of matrix shear, and this increase is inversely proportional to the fibre spacing.
They showed that for rigid, perfectly bonded fibres of aspect ratio l/d:
−p
ε c = Bm σ c ⎡ Vf Φ (l / d) + Vm ⎤
(1+ 1 p )
& p
...................................................................7.20)
⎣ ⎦

where Φ(l/d) is a load-transfer parameter dependent on Vf and on the power-law exponent, p. For p ≈ 6
and Vf = 0.44, Φ is approximately 0.5: at Vf = 0, equation 7.20 reduces to the creep response of the plain
matrix, as required. This model agrees well with experimental measurements by the same authors
(1972a) on composites of lead reinforced with phosphor-bronze wires.

Mileiko (1970) also developed a model for the creep of short-fibre composites in an attempt to take
these mechanisms into account. His was a simple shear model of steady-state creep which assumed that
the creep ‘character’ of the interface was the same as that of the matrix and that the shear deformation
obeyed the same power-law creep model as the tensile deformation. It treated hexagonal and plate-like
reinforcement arrays, and examined the rôles of Vf, aspect ratio, and fibre distribution. With the


terminology modified to conform with the foregoing discussion, Mileiko’s model gives the composite
creep rate as:

22p+1 ⎛ 1 − V ⎞
ε c = Bm σ c
& p
⎜ ⎟ ................................................................................7.21)
(l / d)p+1 ⎝ Vfp+1 ⎠

Although Mileiko assumed that the matrix load-carrying capacity was negligible, his model nevertheless
gives good agreement with the results of Kelly and Tyson (1966) for silver/tungsten composites
mentioned earlier. The model only agrees with the predictions of the Kelly and Street (1972b) model if a
matrix term is added.

These early models ignored the possibility that fibres may fracture as deformation proceeds. McLean
(1986), however, has modelled the creep of aligned composites containing brittle fibres that fracture as a
result of the interaction between the fibre flaw distributions and the local stress distributions in the
creeping material. A recent analysis by Durodola et al (1994) modifies McLean’s model to take into
account the effects of inherent residual thermal stresses, matrix primary creep, and the effect of matrix
yielding. Good agreement is obtained between the predictions of this model and experimental results for
SiC/Ti-6Al-4V MMCs. Some success has also been obtained in the application of these power-law creep
models to CMCs: a recent summary is given by Chawla (1993). Results are not always what might be
expected, however. For example, Nixon et al (1990), studying the steady-state creep of hot-pressed
composites of Si3N4 reinforced with 20 and 30 vol% of SiC whiskers, found that under some
circumstances the composites crept more rapidly than the unreinforced matrix ceramic. The cause of this
appears to be that grain-boundary sliding by viscous flow was the controlling creep mechanism in these
composites, and this viscous flow (with power-law exponents of the order of unity) was limited by
crystallisation of a glassy phase at the grain boundaries. The SiC contained some silica which was
contributed to the overall grain-boundary glass composition with a resultant decrease in the degree of
crystallisation and hence an increase in the creep rate. It is imperative, when dealing with systems
required for elevated-temperature exposure, therefore, that all compositional and constitutional aspects
of composite behaviour be considered in addition to straightforward micro-structural aspects.

Much of the above discussion is based on the simple mixture-rule assumption that the composite
creep behaviour is a weighted average of the behaviour of the separate constituents and ignores detailed
effects of microstructure. And while this may be satisfactory for many of the synthetic composites
manufactured by combining processes, it may not be valid for materials such as in situ composites which
may contain reinforcements of sub-micron dimensions which provide extra creep resistance, perhaps as a
consequence of dispersion strengthening. In directionally solidified alloys there is evidence that the
creep-resistance of material with fine, closely-spaced fibres is much greater than that of the same alloy
grown with coarse, widely-spaced fibres, even though there is not much difference between their room-
temperature strengths. It is also well known that aspect ratio plays an important rôle: the creep rates of
an aluminium alloy reinforced with SiC
whiskers, for example, are some two orders of 3.0
reinforced
magnitude lower than that of an identical alloy 50MPa
2.5 unreinforced
reinforced with the same volume fraction of
SiC particles (Nieh, 1984).
2.0
Strain, %

20MPa
The creep behaviour of polymer 1.5
composites largely concerns materials in 50MPa
which the matrix is visco-elastic and the fibres 1.0
are inert at most use temperatures. The
‘creeping-matrix/elastic-fibres’ models are 0.5 20MPa
therefore appropriate for unidirectional or 0/90
laminates, but not for the majority of practical 0
10-3 10-2 10-1 1 10 102 103 104
composites where fibres may be random, or at
45° to the stress. In such cases, composite Time, hr
deformation may often be modelled by the
Figure 7.24. Creep curves for unreinforced and glass-filled
Nylon in air at 20°C (after Crawford, 1981)


same kinds of procedures as those used for unreinforced linear visco-elastic polymers (Emri, 1996) as
illustrated by the room-temperature creep curves for some unreinforced and glass-reinforced Nylon
manufactured by injection moulding shown in Figure 7.24. On the strain vs. log(time) basis the curves
have the normal concave-upwards appearance, and the effect of the glass-fibre reinforcement is readily
seen. Many examples have been published of the use of the Findlay equation, referred to earlier, and
time/temperature superposition to obtain master curves of creep strain (or creep compliance) vs.
temperature or the related curve of relaxation modulus vs. temperature (or time). For glass-fabric-
reinforced thermoplastics, Brüller (1996), for example, shows that these materials behave in a linear
visco-elastic fashion when loaded in the warp and weft directions, but are highly non-linear when loaded
at 45° to the fibre directions. Ma et al (1997), working on (±45)4S carbon-fibre/PEEK composites,
obtained results at temperatures up to 200°C for which the Findlay relationship gave a good description,
and they determined a master curve by time/temperature superposition which gave good predictions of
creep behaviour. Miranda Guedes et al (1996) also determined master creep curves in flexure for
carbon/epoxy laminates, despite the fact that bending tests do not produce uniform strains.

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APPENDIX 1

ORIENTATION-DEPENDENCE OF THE ELASTIC PROPERTIES OF A SINGLE UNIDIRECTIONAL LAMINA
Composites are rarely used in the form of unidirectional laminates, since one of their great merits is
that the fibres can be arranged so as to give specific properties in any desired direction. Thus, in any
given structural laminate, predetermined proportions of the unidirectional plies will be arranged at some
specific angle, θ, to the stress direction. In order to calculate the properties of such a multi-ply laminate,
it is first necessary to know how the elastic response of a single unidirectional lamina, such as that we
considered in section 3.5 of chapter 3, will vary with angle to the stress direction. In order to do this we
first transform the x1x2 axes through some arbitrary angle, θ, as shown in Figure 0.1.

Figure 0.1. Transformation of co-ordinates to obtain the orientation-dependence of the elastic properties of a
unidirectional composite lamina.

The initial set of Cartesian axes (1,2 or x1x2) are defined with respect to the orientation of the fibres
in the unidirectional lamina, as shown. For the general case, we need to transform the axes from (1,2)
through an angle θ to the orientation (x,y). Definitions of the stresses with respect to the old and new
axis systems are as given in the diagram. The transformations can be obtained directly by resolution, as
follows, or by the use of Mohr's circle, the two approaches being identical.

The steps in the analysis are as follows:

1. summing in the x direction in triangle 1, and remembering that τ12 = τ21, etc:

σx (A/cosθ) = σ1 (Acosθ) + σ2 (Atanθ.sinθ) + τ21 (Atanθ.cosθ) + τ12 (Asinθ)

Appendix 1 185

or σx = σ1 cos2θ + σ2 sin2θ + 2τ12 sinθ cosθ

2. summing in the y direction in triangle 2:

σy (A/cosθ) = σ1 (Atanθ.sinθ) + σ2 (Acosθ) - τ12 (Atanθ.cosθ) – τ21 (Asinθ)

or σy = σ1 sin2θ + σ2 cos2θ – 2τ12 sinθcosθ

3. summing in the y direction in triangle 1:

τxy (A/cosθ) = –σ1 (Asinθ) + σ2 (Atanθ.cosθ) + τ12 (Acosθ) – τ21 (Atanθ.sinθ)

or τxy = –σ1 sinθcosθ + σ2 sinθ.cosθ + τ12 (cos2θ – sin2θ)

Thus, the transformation matrix, T, for the transformation from the (1,2) axes to the (x,y) axes is:

⎛ cos 2 θ sin 2 θ 2sin θ cos θ ⎞
⎜ ⎟
T = ⎜ sin θ 2
cos θ
2
−2sin θ cos θ ⎟ ............................................. 1)
⎜ ⎟
⎝ − sin θ cos θ sin θ cos θ (cos θ − sin θ) ⎠
2 2

and the transformed stresses are given by the tensor relationship:

⎡ σx ⎤ ⎡ σ1 ⎤
⎢ ⎥ ⎢ ⎥
⎢ σ y ⎥ = T ⎢ σ2 ⎥ ..................................................................................................... 2)
⎢ σ xy ⎥ ⎢ σ12 ⎥
⎣ ⎦
⎣ ⎦

or

σxy = Tσij .............................................................................................................. 2a)

Note: this derivation specifically relates to the transformation (1,2) ⇒ (x,y), ie. it concerns a change
from the material properties framework to the structure framework. Other authors choose to start with
the structural case (x,y) and transform to the material system (1,2), in which case the transformation
matrix will be T-1, the inverse matrix of that in the above equation. This method appears more logical to
engineers dealing directly with structures but it makes no difference which transformation is used
provided that we retain consistency in our approach.

The effect of this transformation can be seen in a 1.0
typical example. If we apply the initial stresses σ1 =
800MPa (tension), σ2 = –200MPa (compression) and σx σy
σx, σy, and σxy, GPa

0.5
σ12 = 150MPa (shear) to a unidirectional lamina and
then rotate the (x1x2) axes through 180°, the resulting
variations in the stress components σx, σy, and σxy, as 0
given by equation 1, are as shown in Figure 0.2.
-0.5
Strain transformations from the (1,2) axes to the σxy
(x,y) axes involve the same transformation matrix, T,
but in this case it is the tensor shear strains, εxy and ε12 -1.0
0 30 60 90 120 150 180
that must be used rather than the engineering shear
Orientation, degrees
strains, γxy and γ12. As before, the transformation
gives: Figure 0.2. Variation of the stress components, σij, as
the x1x2 axes are transformed through 180°.

Appendix 1 186

⎡ εx ⎤ ⎡ ε1 ⎤
⎢ ⎥ ⎢ ⎥
⎢ ε y ⎥ = T ⎢ ε 2 ⎥ ....................................................................................................... 3)
⎢ ε xy ⎥ ⎢ ε12 ⎥
⎣ ⎦
⎣ ⎦

or:

εxy = Tεij ................................................................................................................ 3a)

If we wish to derive the generalised form of the Hooke's law relationship of equation 3.16 of chapter
3, between the strains and stresses in the general axes (x,y), we must substitute equations 2a and 3a into
that equation. As given in chapter 3, the equation is:

εij = Srs σkl ............................................................................................................ 4)

remembering that when writing equations 2a and 3a explicitly for σij and εij we have:

σij = T-1 σxy and εij = T-1 εxy

where T-1 is the inverse of the T matrix (note: T-1 differs from T only in the signs of the terms
containing the function sinθcosθ in equation 1) . Equation 4 then becomes:

T-1 εxy = Srs T-1 σxy or

εxy = T Srs T-1 σxy .......................................................................................... 5)

Note that the two geometric terms T and T-1 do not cancel, and that the product T Srs T-1 thus now
represents the compliance matrix relating the strains εxy to the stresses σxy. The full form of the
generalised Hooke's law for a thin orthotropic lamina is then:

⎡ ε x ⎤ ⎡ S11 S12 S16 ⎤ ⎡ σ x ⎤
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ ε y ⎥ = ⎢ S12 S22 S26 ⎥ x ⎢ σ y ⎥ .......................................................................... 6)
⎢ ε xy ⎥ ⎢ S16 S26 S66 ⎥ ⎢σ xy ⎥
⎣ ⎦ ⎣ ⎦ ⎣ ⎦

and the equivalent set of relationships giving stresses as functions of strains can be obtained in a similar
manner:

⎡ σ x ⎤ ⎡ Q11 Q12 Q16 ⎤ ⎡ ε x ⎤
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ σ y ⎥ = ⎢Q12 Q22 Q 26 ⎥ x ⎢ ε y ⎥ ..................................................................... 6a)
⎢σ xy ⎥ ⎢Q16 Q26 Q66 ⎥ ⎢ε xy ⎥
⎣ ⎦ ⎣ ⎦ ⎣ ⎦

where the Q ij terms are referred to as elastic coefficients or elastic stiffnesses to distinguish them from
the compliances, Sij.

Except for the case when the (x,y) axes coincide with the composite orthotropic axes (the fibre
direction and the transverse direction) the compliance matrix, Srs, now contains six elastic constants
instead of the four in the matrix of equations 3.15 of chapter 3. In this case, since there are no zeros in
the S matrix each component of stress will contribute to all of the components of strain. A shear stress
will therefore induce normal strains and normal stresses will induce shear strains. This is referred to as
shear-tension coupling, and it has important consequences for designers since a thin plate of a
unidirectional composite will distort in complex fashion if loaded off-axis.

Appendix 1 187

The compliance terms in the matrix in equations 6 are barred to remind us that these are not simply
material elastic constants like the Srs in equations 4, but that they are in fact functions of stress, as
shown by the presence of the matrix functions T and T-1 in their definitions. It can be seen that if only a
tensile stress, σx, is applied, the strain measured in the direction of the stress, εx, is related to σx, by the
familiar Hooke's law, ie.

ε x = S11σ x

and the compliance S11 therefore represents the reciprocal Young's modulus for this deformation:

S11 = 1/Ex

but clearly S11 , and therefore also Ex are functions of the orientation, θ. If the matrix algebra needed to
evaluate the barred compliances in equations 6 is carried out, it can be shown that the full expression for
S11 is:

S11 = S11 cos 4 θ + S22 sin 4 θ + 2(S12 + S66 ) cos 2 θ sin 2 θ ....................................... 7)

where the compliances S11, S22, S12, and S66 are the four elastic constants for the orthotropic lamina
relative to the axes (1,2) and defined in terms of the normal engineering constants at the end of section
3.4 of chapter 3. If those definitions are substituted in equation 7, we have the orientation dependence of
the elastic modulus, which we shall call E(θ):

1 cos 4θ sin 4θ ⎛ 1 2ν ⎞
= + +⎜ − 12 ⎟ cos 2 θ sin 2θ ..................................................... 8)
E(θ) E1 E2 ⎝ G12 E11 ⎠

Similarly, the shear compliance component, S66 , can be evaluated to obtain the orientation
dependence of the shear modulus, G(θ):

1 ⎛ 1 + 2ν12 1 1 ⎞ 2 1
= 4⎜ + − ⎟ cos θ sin θ +
2
(cos 4θ + sin 4θ ) ....................... 9)
G(θ) ⎝ E1 E 2 2G12 ⎠ G12

For readers wanting to convince themselves of the validity of these derivations without the tedium
of doing the arithmetic by hand, the commercial software package Mathcad by MathSoft is highly
recommended. All of the manipulations in this Appendix have been carried out with this programme,
which also enables easy graphing of the various functions.

APPENDIX 2

STATISTICAL ASPECTS OF THE STRENGTH OF BRITTLE SOLIDS
The strengths of brittle solids are controlled by small defects or flaws, arising either during
manufacture or through handling, which concentrate stress approximately in proportion to the function
√(a/ρ), where a is the depth of a flaw and ρ the radius of its tip. The distributions of size and acuity of
flaws in any sample will be difficult to characterise by any other than statistical means and the measured
strength values of a batch of supposedly identical samples will thus also be statistically distributed. The
strengths of brittle solids sometimes appear to be distributed in a Normal (ie. Gaussian) fashion, but
usually the distributions are skewed and other
statistical models are therefore more appropriate.
1.0
cdf
The failure of a brittle fibre can be described in pdf

Weibull distribution functions
terms of a ‘weakest-link’ theory by supposing that it 0.8
consists of a series-connected arrangement of
0.6 pdf:
elemental volumes. One of these volumes — one
p(z;m) = mzm-1 exp(-zm)
link in a complete chain — will contain a flaw
0.4
deeper and more damaging than those in any other
elemental volume, and that ‘link’ will fail first at an 0.2
applied load level which, when concentrated by the m=2
flaw, corresponds to the inherent or theoretical 0
0 1 2 3 4
strength of the material. A statistical model which is
Variable, z
frequently used to deal with problems of reliability
and life is that which has come to be known as the
Figure 0.1. Probability distribution function and
Weibull distribution (Weibull, 1951). It is one of a
cumulative probability function for a simple Weibull
family of related models, of which the exponential model with shape parameter, m = 2.
model is another member, which are of interest in
tackling extreme-value problems in engineering. The reader should consult texts by Gumbell (1958),
Bury (1975), Chatfield (1983), or Castillo (1988) to find out more about the statistical background to this
subject.

The Weibull Model
The simplest statement of a mathematical function that may be described as a Weibull function gives
the probability density of the occurrence of an event z as:

p(z;m) = m zm-1 exp(-zm) z, m > 0................................................................... 1)

where m is referred to as a shape parameter. This expression defines a probability distribution function,
or pdf, and it is the derivative of the cumulative distribution function, or cdf, which is the likelihood of
the occurrence of z:

P(z;m) = 1 – exp(-zm) z, m > 0 ................................................................. 2)

Plots of these functions for m = 2 are shown in Figure 0.1 from which it can easily be seen how the
Weibull pdf differs from the familiar Normal distribution, particularly in being bounded at the lower
end. If m is equal to unity, the distribution becomes exponential, in which case there is no lower bound.

In applying such a model to a physical process such as failure, it is clear that the exponential term
must be dimensionless, and this may be achieved by treating z as a reduced or normalized variable,
z = x/b, where b is another characteristic of the distribution and is called the scale parameter. This scale
parameter is the value of the new variable x for which there is a fixed probability given by:

P(x; b,m) = 1 – exp(–1) = 0.632

Appendix 2 189

The parameter b is often treated as a characteristic value analogous to the familiar mean value of a
Normal distribution. Applying the cdf of equation 2 to the specific case of the tensile failure of a set of
samples, all of the same dimensions, of a brittle material, we might write the probability of failure, P(σ),
at a stress level σ, as:

⎡ ⎛ σ ⎞m ⎤
P(σ; b, m ) = 1 − exp ⎢ − ⎜ ⎟ ⎥ σ, b, m > 0 ................................................ 3)
⎢ ⎝b⎠ ⎥
⎣ ⎦

This is known as the two-parameter Weibull model of failure. As shown in Figure 0.1, the pdf
associated with this model is bounded at its lower limit by x = 0. In speaking of the failure of solids, it
would be uncomfortable to have to accept that there was not some finite level of stress below which
there was no possibility of failure, and it is more usual to shift the distributions of equations 1 and 2 to
the right along the z axis so that there is some lower limit, a, to the distribution. The reduced variable is
then defined as:

x −a
z=
b

and the parameter a is called the location parameter. Shifting the distributions in this fashion makes no
difference to their shapes. The Weibull cdf may then be written as:

⎡ ⎛ σ − a ⎞m ⎤
P(σ; a, b, m ) = 1 − exp ⎢ − ⎜ ⎟ ⎥ ................................................................ 4)
⎢ ⎝ b ⎠ ⎥
⎣ ⎦

This is referred to as the three-parameter Weibull model and it is often argued that this is the only
possible valid Weibull function that can be applied to physical processes like tensile failure of brittle
solids, or the fatigue failure of engineering materials. The appearance of the distribution varies widely
with changes in the value of m from the exponential form for m = 1 to shapes resembling Normal

0.08 5
p(x)
⎡ ⎛ x − 1⎞m ⎤
4 p( x; a,b,m) = mb−m ( x − a)m−1 exp⎢− ⎜ ⎟ ⎥
0.06 ⎢ ⎝ b ⎠ ⎥
⎣ ⎦

m:- 3
0.04 1.0
1.1
1.2
0.02
1.5
2.0
0
0 20 40 60 80 100
x
Figure 0.2. Variation of the shape of the Weibull probability distribution function, p(x;a,b,m), with the value of the
shape parameter, m. The pre-selected values of the location parameter, a, and the scale parameter, b, are 10 and
25, respectively.

distributions for higher values of m, as illustrated by the family of pdfs shown in Figure 0.2.

Appendix 2 190

The shape and scale parameters are of great importance in that they provide a means of characterizing
the Weibull distribution in the same way that the average, variance and standard deviation are used to
characterise Gaussian distributions. Some of the parameters of a Weibull pdf are:

i. the expected value of x, or arithmetic mean (called the first moment of the distribution)
⎛ 1⎞
bΓ ⎜1 + ⎟
⎝ m⎠
⎡ ⎛ 2⎞ ⎛ 1 ⎞⎤
ii. the variance (called the second moment of the distribution) b2 ⎢Γ ⎜1 + ⎟ − Γ 2 ⎜1 + ⎟ ⎥
⎣ ⎝ m⎠ ⎝ m ⎠⎦
1
iii. the median (eg. the value for which there is 50% probability of failure) lo(log e 2 ) m
1

⎡ m − 1⎤ m
iv. the mode (most probable value) b⎢
⎣ m ⎥ ⎦
where Γ is the gamma function. From these relationships, it is easy to show that for values of the shape
parameter, m, of the order of 10, there is a useful approximate relationship between m and the familiar
coefficient of variation (cv = standard deviation divided by arithmetic mean) of the form:

m ≈ 1.2/cv

As noted earlier, the characteristic value, b, is often used as a measure of a property like the fracture
strength of a brittle solid in place of the more usual mean value. The difference between the two is often
small, but depends on the value of m through the gamma function Γ(1 + 1/m), as shown above, for which
the argument will usually lie between 1 and 2. The variation of this function, which thus displays the
ratio of the mean to the characteristic value, x /b, is shown in Figure 0.3.

When the Weibull distribution is used to deal
1.00
with weakest-link problems like brittle fracture and
fatigue failure, the parameters b and m have to be
estimated from available experimental data. There 0.95
x
are several methods of doing this, of greater or lesser b
degrees of complexity, and it is by no means easy to
be sure, for given distributions, which method will 0.90
typical CFRP
give the most satisfactory estimates. A discussion of tensile strengths
these methods is beyond the scope of this book, but 0.85
we mention two that are familiar to users of the CFRP fatigue typical carbon
lives fibre strengths
Weibull model. The first, referred to as the method
of moments, uses the conventionally defined values 0.80
1 2 5 10 20
of the arithmetic mean and the variance, deduced Weibull shape parameter, m
from the data, to solve equations 1 and 2 above. This
method is not universally well-regarded. Figure 0.3. The effect of the value of the Weibull shape
parameter, m, on the ratio between the arithmetic mean
The method most familiar to materials scientists of a distribution, x , to the Weibull scale parameter, b.
is that of plotting the probability function against the
measured parameter so as to analyse equations 3 and 4 graphically. The experimental data are first
ranked from lowest value, i = 1, to the highest, i = N, where N is the total number of data values. The
likelihood of failure associated with i = 1 is small while that for i = N is great, and a method is needed
for assigning a probability, P(x), that reflects this range. There are several common methods, but perhaps
the most familiar, the mean-rank assignment, gives the probability as

i
P= ................................................................................................................ 5)
N +1

Appendix 2 191

where i is the rank (ie. the position in a list of ascending order) of a given datum value. For a
distribution of measured strengths, σf, then, to obtain the distribution parameters from equation 3 the
double logarithmic function ln[-ln(1-P)]is plotted against lnσf (Gumbell, 1958). Provided the points fall
on a straight line, the slope and intercept then give the required estimates of m and b. If the resulting line
is curved, the two-parameter Weibull model is not valid and it may be that the three-parameter Weibull
model is more appropriate. A non-linear curve fit of P versus σf will, with care, permit the estimation of
all three distribution parameters. It must be admitted that these simple methods tend to be ill-regarded by
statisticians. More sophisticated treatments, such as the use of maximum likelihood estimators, can be
found in the texts referred to below.

In the foregoing discussion of strength, we have assumed that all samples were of equal dimensions.
A weakest-link model must be able to account for volume effects, however, since it is the dimensions of
flaws that we are dealing with in evaluating the strength of brittle solids. The form of the three-parameter
Weibull model that is most commonly invoked in discussions of strength, therefore, is one which
includes the volume of the sample, V:

⎡ ⎛σ − a ⎞ ⎤
m

P(σ ) = 1 − exp ⎢ −V ⎜ ⎟ ⎥ .......................................................................... 6)
⎢
⎣ ⎝ b ⎠ ⎥ ⎦

REFERENCES TO APPENDIX 2
Bury KV, 1975, Statistical Models in Applied Science, (J Wiley & Sons, London).
Castillo E, 1988, Extreme Value Theory In Engineering, (Academic Press, Boston/London).
Chatfield C, 1983, Statistics for Technology: a Course in Applied Statistics, 3rd edition., (Chapman & Hall,
London).
Gumbel EJ, 1958, Statistics of Extremes, (Columbia University Press, New York).
Weibull W, 1951, J Appl Mech (Trans ASME), 73, 293-297.

APPENDIX 3

SOME APPLICATIONS OF COMPOSITES
This is a brief listing of current and proposed applications of composite materials in various branches
of industry. It is not intended to be comprehensive or all-embracing, but merely to give an indication of
the range of possibilities for designers.

Aerospace
A wide range of load-bearing and non-load-bearing components are already in use in both fixed-wing
and rotary wing aircraft. Many military and civil aircraft now contain substantial quantities of light-
weight, high-strength carbon-, Kevlar- and glass-fibre composites, as laminated panels and mouldings,
and as composite honeycomb structures with metallic or resin-impregnated paper honeycomb core
materials. They are used in air frames, wing spars, spoilers, tail-plane structures, fuel tanks, drop tanks,
bulkheads, flooring, helicopter rotor blades, propellers, and structural components, pressured gas
containers, radomes, nose and landing gear doors, fairings, engine nacelles (particularly where
containment capability is required for jet engines), air distribution ducts, seat components, access panels,
and so forth. Many modern light aircraft are being increasingly designed to contain as much lightweight
composite material as possible. For elevated-temperature applications carbon-fibre-reinforced carbon is
in use. Concord's disk brakes use this material, rocket nozzles and re-entry shields have been fashioned
from it, and there are other possibilities for its use as static components in jet engines. Rocket motor
casings and rocket launchers are also frequently made of reinforced plastics. A particularly interesting
(and important) application of composites is in its development in Australia as a means of repairing
battle damage (patching) in metal aircraft structures.

Space applications offer many opportunities for employing light-weight, high-rigidity structures for
structural purposes. Many of the requirements are the same as those for aeronautical structures, since
there is a need to have low weight and high stiffness in order to minimize loads and avoid the occurrence
of buckling frequencies. Dimensional stability is at a premium, for stable antennae and optical platforms,
for example, and materials need to be transparent to radio-frequency waves and stable towards both uv
radiation and moisture.

Automotive Engineering
There is increasing interest in weight reduction in order to permit both energy conservation and
increased motoring economy. Reduction in the weight of an automobile structure achieves primary
weight-saving and if carried to sufficiently great lengths enables the designer to use smaller power
plants, thus achieving substantial secondary improvements in fuel economy. The majority of automotive
applications involve glass-reinforced plastics because the extra cost of carbon or aramid fibre is rarely
considered to be acceptable in this market. Even so, the cost of using GRP is usually being weighed
against the much lower cost of pressed steel components, and the substitution is often rejected on purely
economic grounds, leaving aside the question of energy saving. A wide range of car and truck body
mouldings, panels and doors is currently in service, including complete front-end mouldings, fascias,
bumper mouldings, and various kinds of trim. There is considerable interest in the use of controlled-
crush components based on the high energy-absorbing qualities of materials like GRP. Leaf and coil
springs and truck drive shafts are also in service, and GRP wheel rims and inlet manifolds have been
described in the literature. Selective reinforcement of aluminium alloy components, such as pistons and
connecting rods, with alumina fibres is much discussed with reference to increased temperature
capability.

BioEngineering
Carbon-fibre-reinforced plastic and carbon components are in use for prosthetic purposes, such as in
orthopaedic fracture fixation plates, femoral stems for hip replacements, mandibular and maxillary
prostheses (jaw remodelling, for example), and for external orthotic supports in cases of limb deformity

appendix 3 Applications 193

etc. Pyrolytic carbon is used to manufacture heart valve components, and the substitution of a
carbon/carbon composite is not unlikely. There have also been developments in the use of particulate
hydroxyapatite as filler in a thermoplastic composite for bone remodelling or replacement.

Chemical Engineering
A substantial amount of GRP is currently in use in chemical plant for containers, pressure vessels,
pipe-work, valves, centrifuges etc. These may be filament-wound or moulded components for
containment of process fluids.

Civil/Structural Engineering
Again the bulk of composites used in this field are glass-reinforced plastics. The low inherent elastic
modulus of GRP is easily overcome in buildings by the use of double curvature and folded-plate
structures: thin GRP panels also offer the advantage of translucency. Glass-reinforced cement (GRC)
products made with Cem-FIL (alkali-resistant glass fibres) are gradually being introduced as structural
cement-based composites, but these GRC are still regarded with some suspicion by architects who prefer
to consider only non-load-bearing applications for glass-reinforced cement. Development of suitable
highly-drawn polymer fibres and net-like polymeric reinforcement has made it possible to produce stable
polymer-reinforced cement for a variety of purposes. But concrete is the cheapest engineering material
available, and it requires very little in the way of expensive reinforcing filaments to be added to it to
make it uncompetitive. The answer is usually to use more concrete! But GRC is perhaps likely to attract
the more adventurous designer with light-weight concrete structures in mind (thin shell structures for
example). A good deal of GRP is used in this industry for folded-plate structures, cladding panels,
decorative ‘sculptured’ panels (like those in the doors of the Roman Catholic cathedral in Liverpool),
services mouldings and ducting, racking, pipework, rainwater mouldings, domestic and industrial water
tanks, form-work for concrete, and complete small structures like foot-bridges. Light-weight composite
panelling for partitioning and similar applications have also been tried. CFRP have been less used until
recently because of the cost, but are increasingly being considered for building light-weight structures,
including a number of bridges.

In recent years there has been a major surge of interest in the use of structural composites in civil
engineering infrastructures. At least two major composites journals have run Special Issues on the
subject in 1996/99, and the 1998 European Conference on Composite Materials, ECCM8, ran a series of
well-attended sessions on civil engineering uses for the first time. Current applications include the use of
pultruded FRP shapes as individual structural elements and shear stiffeners for concrete structures, as
reinforcing bars for concrete, components in composite/concrete structures, as externally applied impact-
containment supports, and as patches for damaged concrete bridgework. A good deal of upgrading of
existing bridge structures by externally applied composite components is being carried out

Domestic
Injection-moulded reinforced thermoplastics and polyester moulding compounds are perhaps the
most common composites used in consumer items for the domestic market, and the range is vast.
Mouldings of all kinds, from kitchen equipment of all kinds to casings for the whole gamut of domestic
and professional electrical equipment, motor-cycle crash helmets, television and computer casings, and
furniture.

Electrical Engineering
Typical applications are radomes, structural components for switch gear, power generator coolant
containment and large-diameter butterfly valves, high-strength insulators (eg. for overhead conductor
systems), printed circuit boards, and casings for electronic equipment. The majority of applications in
this field again use GRP, although the use of composites which are more thermally stable and more
moisture-resistant is increasingly predicated for sensitive, small-scale electronic components. Many
prototype and practical wind-generator designs incorporate GRP or hybrid blading.

appendix 3 Applications 194

Marine Engineering
Marine applications include surface vessels, offshore structures and underwater applications. A vast
range of pleasure craft has long been produced in GRP, but much serious use is also made of the same
materials for hull and superstructure construction of passenger transport vessels, fishing boats and
military (mine-countermeasures) vessels. Sea-water cooling circuits may also be made of GRP as well as
hulls and other structures. Off-shore structures such as oil rigs also make use of reinforced plastics,
especially if they can be shown to improve on the safety of steel structures, for fire protection piping
circuits, walkways, flooring, ladders, tanks and storage vessels, blast panels, and accommodation
modules. High specific compression properties also make composite materials attractive for
submersibles and submarine structures, both for oil exploration and for military purposes, and for towed
transducer arrays for sea-bed sonar mapping.

Sport
Perhaps the most visible development in the use of composites has been in the sports goods industry.
Manufacturers have been quick to seize on the potential advantages of new materials like carbon and
boron fibre composites over conventional wood and metal for sports equipment of all kinds, but whether
the average sportsman (and perhaps even some of the above-average ones) who have been inveigled into
buying this more expensive composite equipment in the hope that it would improve their game have
been able to demonstrate genuine improvement remains uncertain. GRP vaulting poles were perhaps the
earliest of the composite sports gear, but one can now obtain tennis rackets, cricket bats, golf clubs,
fishing rods, boats, oars, archery equipment, canoes and canoeing gear, surf boards, wind-surfers,
skateboards, skis, ski-poles, bicycles, and protective equipment of many sorts in composite materials of
one kind or another. In an industry that is often less directly subject to controls exercised in other areas
of engineering there is often a tendency to dupe customers with the use of names which incorporate the
words ‘carbon’ or ‘graphite’ to describe expensive, black coloured items which may at the present time
legitimately contain little or no carbon fibre.

Engineering composites

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Engineering composites