Chapter 9 Elasticity and its different proprties

ELASTICITY
ELASTICITY
 Elasticity
 Elasticity of Composites
 Viscoelasticity
 Elasticity of Crystals (Elastic Anisotropy)
Elasticity: Theory, Applications and Numerics
Martin H. Sadd
Elsevier Butterworth-Heinemann, Oxford (2005)
MATERIALS SCIENCE
MATERIALS SCIENCE
&
&
ENGINEERING
ENGINEERING
Anandh Subramaniam & Kantesh Balani
Materials Science and Engineering (MSE)
Indian Institute of Technology, Kanpur- 208016
Email: anandh@iitk.ac.in, URL: home.iitk.ac.in/~anandh
AN INTRODUCTORY E-BOOK
AN INTRODUCTORY E-BOOK
Part of
http://home.iitk.ac.in/~anandh/E-book.htm
A Learner’s Guide
A Learner’s Guide

Revise modes of deformation, stress, strain and other basics (click here) before starting this chapter etc.

 When you pull a rubber band and release it, the band regains it original length.
 It is much more difficult (requires more load) to extend a metal wire as compared to a
rubber ‘string’.
 It is difficult to extend a straight metal wire; however, if it is coiled in the form of a
spring, one gets considerable extensions easily.
 A rubber string becomes brittle when dipped in liquid nitrogen and breaks, when one tries
to extend the same.
 A diver gets lift-off using the elastic energy stored in the diving board. If the diving board
is too compliant, the diver cannot get sufficient lift-off.
 A rim of metal is heated to expand the loop and then fitted around a wooden wheel (of say
a bullock cart). On cooling of the rim it fits tightly around the wheel.
Let us start with some observations…
Click here to know about all the mechanisms by which materials fail

Elasticity
Elasticity
Linear
Non-linear
E.g. Al deformed at small
strains
E.g. deformation of
an elastomer like
rubber
 Elastic deformation is reversible deformation- i.e. when load/forces/constraints are released
the body returns to its original configuration (shape and size).
 Elastic deformation can be caused by tension/compression or shear forces.
 Usually in metals and ceramics elastic deformation is seen at low strains (less than ~10–3
).
At higher strains, in addition to elastic deformation we get plastic deformation. However,
other materials can be stretched elastically to large strains (few 100%).
So elastic deformation should not be assumed to be small deformation!
 The elastic behaviour of metals and ceramics is usually linear (i.e. the stress strain curve is
linear).
(or elastic
deformation)

Elastic
Plastic
Deformation
Instantaneous
Time dependent
Instantaneous
Time dependent
Recoverable
Permanent
Anelasticity
Viscoelasticity
 In normal elastic behaviour (e.g. when ‘small’ load is applied to a metal wire, causing a strain within elastic limit) it is
assumed that the strain is appears ‘instantaneously’.
 Similarly when load beyond the elastic limit is applied (in an uniaxial tension test: Slide 7 ), it is
assumed that the plastic strain develops instantaneously. It is further assumed that the load
is applied ‘quasi-statically’ (i.e. very slowly).
 However, in some cases the elastic or plastic strain need not develop instantaneously.
Elongation may take place at constant load with time. These effects are termed Anelasticity
(for time dependent elastic deformation at constant load) and Viscoelasticity. Anelastic
materials are a subset of viscoelastic materials. In anelastic materials the strain is fully
recoverable and in dynamic loading, the stress-strain response depends on the frequency.
Viscoelastic response depends on the rate of loading (strain rate).
 Creep is one such phenomenon, where permanent deformation takes place at constant load
(or stress). E.g. if sufficient weight is hung at the end of a lead wire at room temperature, it will elongate and finally fail.
Time dependent aspects of elastic deformation

m
n
r
B
r
A
U 


A,B,m,n → constants
m > n
Attractive
Repulsive
Atomic model for elasticity
dr
dU
F 

1
1 



 m
n
r
mB
r
nA
F
q
p
r
B
r
A
F
'
'



 At the atomic level elastic deformation takes place by the deformation of bonds (change in
bond length or bond angle).
 Let us consider the stretching of bonds (leading to elastic deformation).
 Atoms in a solid feel an attractive force at larger atomic separations and feel a repulsive
force (when electron clouds ‘overlap too much’) at shorter separations. (Of course, at very
large separations there is no force felt).
 The energy and the force (which is a gradient of the energy field) display functional
behaviour as in the equations below*. This implies under a state of compressive stress the
atoms ‘want to’ go apart and under tensile stress they want to come closer.
The plots of these
functions is shown
in the next slide
* This is one simple form of interatomic potentials (also called Lennard-Jones potential,
wherein m = 12 and n = 6).

r →
Potential
energy
(U)
→
Force
(F)
→
Attractive
Repulsive
r →
Attractive
Repulsive
r0
r0
r0 Equilibrium separation between atoms
m
n
r
B
r
A
U 

 q
p
r
B
r
A
F
'
'



 The plot of the inter-atomic potential and force functions show that at the equilibrium
inter-atomic separation (r0) the potential energy of the system is a minimum and force
experienced is zero.
 In reality the atoms are undergoing thermal vibration about this equilibrium position. The
amplitude of vibration increases with increasing temperature. Due to the slight left-right
asymmetry about the minimum in the U-r plot, increased thermal vibration leads to an
expansion of the crystal.

r →
Force
→
r0
For displacements around r0 → Force-
displacement curve is approximately linear
 THE LINEAR ELASTIC REGION
Near r0 the red line (tangent to the F-r curve at
r = r0)
coincides with the blue line (F-r) curve
 Young’s modulus is the slope of the Force-Interatomic spacing curve (F-r curve), at the
equilibrium interatomic separation (r0). I.e. it is the curvature of the energy-displacement
(interatomic separation) plot.
 In reality the Elastic modulus is 4th
rank tensor (Eijkl) and the curve below captures one
aspect of it.
Young’s modulus (Y / E)
 Young’s modulus is proportional to
the –ve slope of the F-r curve at r = r0
dr
dF
Y 
 2
2
dr
U
d
dr
dF
Y 


Stress
→
strain →
Compression
Tension

 In metals and ceramics the elastic strains (i.e. the strains beyond which plastic deformation
sets in or fracture takes place*) are very small (~103
). As these strains are very small it does
not matter if we use engineering stress-strain or true stress strain values (these concepts will be
discussed later). The stress-strain plot is linear for such materials.
 Polymers (with special reference to elastomers) shown non-linear stress-strain behaviour in
the elastic region. Rotation of the long chain molecules around a C-C bond can cause
tensile elongation. The elastic strains can be large in elastomers (some can even extend a
few hundred percent), but the modulus (slope of the stress-strain plot) is very small.
Additionally, the behaviour of elastomers in compression is different from that in tension.
Stress
→
Strain →
Tension
Stress-strain curve for an elastomer
Due to efficient
filling of space
T due to uncoiling
of polymer chains
C
T
C
T >
Stress-strain curves in the elastic region
* Brittle ceramics may show no plastic deformation and may fracture after elastic deformation.
Compression

Other elastic moduli
 E → Young’s modulus
 G → Shear modulus
 K → Bulk modulus
  → Poisson’s ratio
 We have noted that elastic modulus is a 4th
rank tensor (with 81 components in general in
3D). In normal materials this is a symmetric tensor (i.e. it is sufficient to consider one set of
the off-diagonal terms).
 In practice many of these components may be zero and additionally, many of them may
have the same value (i.e. those surviving terms may not be independent). E.g. for a cubic
crystal there are only 3 independent constants (in two index condensed notation these are
E11, E12, E44).
 For an isotropic material the number of independent constants is only 2. Typically these are
chosen as E and  (though in principle we could chose any other two as these moduli are
interrelated for a isotropic material). More explained sooner.
 In a polycrystal (say made of grains of cubic crystal), due to average over all orientations,
the material behaves like an isotropic material. More about this discussed elsewhere. Mathematically the isotropic material
properties can be obtained from single crystal properties by Voigt or Reuss averaging.
ij ijkl kl
E
 
 In tensor notation

 .
E

.
xy xy
G
 

volumetric
ic
hydrodynam K 
 .


 We have noted that for an isotropic material, the elastic properties are described by two
independent elastic constants (typically, E & ).
 The other elastic constants {like the bulk modulus (K) and the shear modulus (G or )} can be
derived from these constants by standard formulae (e.g. below).
 List of elastic moduli for isotropic materials: Young’s modulus (Y, E), Poisson’s ratio (),
Shear Modulus (G, ), Bulk modulus (K), Lamé’s constant (), P-wave modulus (M).
 The relative magnitudes of these constants can be seen from the table below.
 Similar to the stress-strain relation we saw before, other types of stresses can be related to the
corresponding strain via the appropriate elastic moduli/constants (as below).
Relation between the elastic constants of an isotropic material
Click here to know how isotropy can arise in a material
)
1
(
2 


E
G
)
2
1
(
3 


E
K
Property Magnitude
Young’s modulus
(Y, E)
70 GPa
Poisson’s ratio 0.3
Shear Modulus (G) 26.9 GPa
Bulk Modulus (K) 58.3 GPa
Lamé constant () 40.4 GPa
(1 )
(1 )(1 2 )
E
M

 


 

 .
E

.
xy xy
G
 

volumetric
ic
hydrodynam K 
 .


l
t


 

 When a body is pulled (let us assume an isotropic body for now), it will elongate along the
pulling direction and will contract along the orthogonal direction. The negative ratio of the
transverse strain (t) to the longitudinal strain (t) is called the Poission’s ratio ().
 In the equation below as B1 < B0, the term in square brackets ‘[]’ in the numerator is –ve
and hence, Poission’s ratio is a positive quantity (for usual materials). I.e. usual materials
shrink in the transverse direction, when they are pulled.
 The value of E, G and  for some common materials are in the table (Table-E). Zero and
even negative Possion’s ratio have been reported in literature. The modulus of materials
expected to be positive, i.e. the material resists deformation and stores energy in the
deformed condition. However, structures and ‘material-structures’ can display negative
stiffness (e.g. when a thin rod is pushed it will show negative stiffness during bulking
observed in displacement control mode).
 
 
 
 
0
0
1
0
0
1
/
/
L
L
L
B
B
B
l
t









Initial configuration is
represented by ‘0’ in
subscript

Elastic properties of some common materials
T1
Anisotropic Elastic constants for Cubic System [2]
E11 (GPa) E44 (GPa) E12 (GPa)
Al 108 28.3 62
Au 190 42.3 161
Cu 169 75.3 122
Fe 230 117 135
Ni 247 122 153
Pd 224 71.6 173
Si 165 79.2 64
T2 Specification E (GPa) ν σy
(MPa) y (103
) Ref.
Metals
Aluminum
BS 1490
(99% pure)
71 0.345 240 3.4 [1]
Mild Steel
BS 070M20
>0.2 wt.% C
210 0.293 215 1.0 [2]
Ceramics and
Glass
Aluminum
Oxide (Al2O3)
-
~ 350
344.7-408.8
0.21-0.27
255-260.6
(Fracture)
0.7 [3]
Glass (SiO2) -
~ 73
72.76-74.15
0.166-0.177
69.6-74.53
(Fracture)
1.0 [3]
Polymers Polyester
General purpose
grade
~ 4
1.03-4.48
0.33 51.5-55.1 12.5 [3]
Room temperature data
T1 gives independent elastic constants of a cubic crystal. T2 lists elastic properties of selected
isotropic materials. F1 shows a stress-strain plot in the elastic regime of select materials.
F1

Bonding and Elastic modulus
 Materials with strong bonds have a deep potential energy well with a high curvature
 high elastic modulus.
 Along the period of a periodic table the covalent character of the bond and its strength
increase  systematic increase in elastic modulus.
 Down a period the covalent character of the bonding ↓  ↓ in Y.
 On heating the elastic modulus decreases: on heating from 0 K → M.P there is a decrease
of about 10-20% in modulus (i.e. Young’s modulus is not a sensitive function of T).
Along the period → Li Be B Cdiamond Cgraphite
Atomic number (Z) 3 4 5 6 6
Young’s Modulus (GN / m2
) 11.5 289 440 1140 8
Down the row → Cdiamond Si Ge Sn Pb
Atomic number (Z) 6 14 32 50 82
) 1140 103 99 52 16

Anisotropy in the Elastic modulus
 In a crystal the interatomic and inter-planar distance varies with direction→ this aspect can
be used to intuitively understand the origin of elastic anisotropy. We already know that
linear and planar densities depend on the specific choice (i.e. the [100]CCP direction has different linear
density as compared to the [111]CCP and similarly, the (100)CCP has a different planar density as compared to (111)CCP).
 In a elastically anisotropic material, stretching along certain directions can lead to
shearing.
 Elastic anisotropy is especially pronounced in materials with
► two kinds of bonds, e.g. in graphite E [1010] = 950 GPa, E [0001] = 8 GPa
► Two kinds of ordering along two directions, e.g. in Decagonal QC
E[100000]E[000001]
What is meant by anisotropy (click here: Slide 13)?

 The Young’s modulus (Y) of a isotropic material can be determined from the stress-strain diagram. But, this is
not a very accurate method, as the machine compliance is in series with the specimen compliance. The slope
of the stress-strain curve at any point is termed as the ‘tangent modulus’. In the initial part of the s-e curve this
measures the Young’s modulus. Other kinds of mechanical tests can also be used for the measurement of ‘Y’.
 A better method to measure ‘Y’ is using wave transmission (e.g. ultrasonic pulse echo transmission technique)
in the material. This is best used for homogeneous, isotropic, non-dispersive material (wherein, the velocity of
the wave does not change with frequency). Common polycrystalline metals, ceramics and inorganic glasses
are best suited to this method. Soft plastics and rubber cannot be characterized by this method due to high
dispersion, attenuation of sound wave and non-linear elastic properties. Porosity and other internal defects can
affect the measurement.
 The essence of the ultrasound technique is to determine the longitudinal and shear wave velocity of sound in
the material (symbols: vl vs). ‘Y’ and Poisson’s ratio () can be determined using the formulae as below.
How to determine the elastic modulus?
Specimen ID Aluminium
Cross Sectional Area 30.2 mm2
Gauge Length 28.44 mm
Cross head velocity 2 mm/min
Test mode Stroke
Peak Stress 193.0 MPa
Peak Load 5.83 kN
  → density
Note again that it is not a good idea to calculate Young’s modulus for s-e plot
)
1
(
)
2
1
)(
1
(
2







 l
v
Y
2
2
2
2
2
1



















l
t
l
t
v
v
v
v


What is meant by
anisotropy?
Funda Check
 Anisotropy implies direction dependence of a given property. An isotropic material has the
same value of a property along all directions.
 Neumann’s principle states that symmetry of a property will be equal to or greater than the
point group symmetry of the crystal. Cubic crystals (with 4/m 3 2/m) symmetry are
isotropic with respect to optical properties.
 Amorphous materials are isotropic. Polycrystalline materials with random orientation of
grains are also isotropic. When such a polycrystalline material is plastically deformed, e.g.
by rolling, the material develops crystallographic texture (preferential orientation of the
grains) and this can lead to anisotropy.
 Two important origins of anisotropy are: (i) crystalline, (ii) geometry of sample (shape).
 Couple of effects of crystalline anisotropy: (i) if an electric field is applied along a
direction, the current may flow along a different direction; (ii) if a body is pulled, it may
shear in addition to being elongated.

Property
Material dependent
Geometry dependent
Elastic modulus
 In many applications the elastic properties of the material is the main one to be considered.
However, other factors and properties should also be taken into account.
 More than the modulus, it is the stiffness, which comes into focus as a design parameter.
 Stiffness (as we have noted) of a ‘structure’ is its ability to resist elastic deformation
(deflection) on loading → depends on the geometry of the component. Spring steel (e.g. high carbon
steel with Si, Mn & Cr) will be characterized by its elastic modulus. In the form of the coil the spring (a structure with a certain geometry)
shows more elongation for the same load and stiffness is the relevant property of the spring in design.
 In many applications high modulus in conjunction with good ductility should be chosen
(good ductility avoids catastrophic failure in case of accidental overloading)
 Covalently bonded materials (e.g. diamond has a high E (1140 GPa)) in spite of their high
modulus are rarely used in engineering applications as they are brittle (poor tolerance to
cracks- for more on this refer to chapter on fracture). Ionic solids are also very brittle.
Elastic modulus in design
Ionic solids → NaCl MgO Al2O3 TiC Silica glass
) 37 310 402 308 70
Stiffness

METALS
 Metals are used in multiple engineering applications due to a combination of properties
they possess: (i) reasonable elastic modulus, (ii) good ductility, (iii) ability to be alloyed
to give good combination of properties, (iv) amenable to many types of fabrication
techniques (casting, forging, extrusion, etc.), (v) good electrical and thermal conductivity,
etc. The main issue with metals is often many of them have poor oxidation and corrosion
resistance and not so good tolerance to high temperatures.
 Metals of the First transition series posses a good combination of ductility & modulus
(200 GPa). Second & third transition series have an even higher modulus, but their higher
density is a shortcoming.
POLYMERS
 Polymers are light weight and have become universal. However, they have a low modulus
dependent and cannot withstand high temperatures. They have a poor wear resistance as
well.
 The nature of secondary bonds (Van der Walls / hydrogen) is responsible for their low
modulus. Further aspects which determine the modulus of polymers are:
(i) presence of bulky side groups,
(ii) branching in the chains
 Unbranched polyethylene E = 0.2 GPa,
 Polystyrene with large phenyl side group E = 3 GPa,
 3D network polymer phenol formaldehyde E = 3-5 GPa
(iii) Extent of cross-linking (more cross-linking give rise to higher stiffness).

Increasing the modulus of a material
 The modulus of a metal can be increased by suitable alloying.
 E is a structure (microstructure) insensitive property and this implies that grain size,
dislocation density, etc. do not play an important role in determining the elastic modulus of
a material.
 One of the important strategies to increase the modulus of a material is by forming a
hyrbrid/composite with an elastically ‘harder’ (stiffer) material. E.g. TiB2 is added to Fe to
increase the modulus of Fe.
COMPOSITES
 A hard second phase (termed as reinforcement) can be added to a low E material to
increase the modulus of the base material. The second phase can be in the form of
particles, fibres, laminates, etc.
 Typically the second phase though harder is brittle and the ductility is provided by the
matrix. If the reinforcement cracks the propagation of the crack is arrested by the matrix.
Click here to know more about hybrids Hybrids are designed to improve certain properties of monolithic materials
Laminate
composite
Aligned fiber
composite
Particulate
composite

Young’s modulus of a composite
 The Young’s modulus of the composite is between that of the matrix and the
reinforcement.
 Let us consider two extreme cases.
(A) Isostrain → the matrix and the reinforcement (say long fibres) are under identical strain.
This is known as Voigt averaging.
(B) Isostress → the matrix and fibre are under identical stress. This is known as Reuss averaging.
 Let the composite be loaded in uniaxial tension and the volume fraction of the fibre be Vf
(automatically the volume fraction of the matrix is Vm = (1 Vf)).
Isostrain
Isostress
isostrain
c f f m m
Y Y V Y V
 
1 f m
isostress
c f m
V V
Y Y Y
 
Voigt averaging
Reuss averaging
This is like resistances in series
This is like resistances in parallel
These formulae are derived later

A B
Volume fraction →
Y
c
→
Ym
Yf
Isostress
Isostrain
For a given fiber fraction f, the modulii of
various conceivable composites lie between an
upper bound given by isostrain condition
and a lower bound given by isostress condition
Vf
 The modulus of a real composite will lie between these two extremes (usually closer to
isostrain). The modulus of the composite will depend on the shape of the reinforcement
and the nature of the interface (e.g. in a long aligned fibre composite having a perfectly
bonded interface with no slippage will lead to isostrain conditions.
 Purely from a modulus perspective, a larger volume fraction will give a higher modulus;
however, ductility and other considerations typically limit the volume fraction of
reinforcement in a composite to about 30%.
c f f m m
Y Y V Y V
   Under iso-strain conditions [m = f = c]
 I.e. ~ resistances in series configuration
1 f m
c f m
V V
Y Y Y
 
 Under iso-stress conditions [m = f = c]
 I.e. ~ resistances in parallel configuration
 Usually not found in practice
Voigt averaging
Reuss averaging

c f f m m
Y Y V Y V
 
1 f m
c f m
V V
Y Y Y
 
Derivation of formulae
Isostrain
F
F
m f
F F F
 
m m f f
A A A
  
 
f
m
m f
A
A
A A
  
 
f
m
m m f f
LA
LA
Y Y Y
LA LA
  
 
A
Af Am
L
f
m
m m f f
V
V
Y Y Y
V V
  
 
  
f m
  
 
m m f f
Y V Y V Y
 
Isostress f m
  
 
f m
  
 
f
m
m f
Y Y Y

 
 
f
m
m f
Y Y Y

 
 
1 1 1
m f
Y Y Y
 

 For small stress, strain is proportional (linearly varies with) strain  Hooke’s law.
 The proportionality constants are 4th
order (rank) tensors. The stiffness tensor is Eijkl or Cijkl. The compliance
tensor is denoted as: Sijkl.
Elasticity of Single Crystals and Anisotropy
Usage-1 In short Usage-2 Dimensions
S (or s) Elastic Compliance constant
(Elastic)
Compliance
Elastic Modulus
(N/m2
)1
[stress1
]
C (or c)
Y, E
Elastic Stiffness constant
(Elastic)
Stiffness
Elastic Constant (Young’s
modulus*)
(N/m2
)
[stress]
C
 
 ij ijkl kl
E
 

S
 
 ij ijkl kl
S
 

Hooke’s law
Tensor form Number of components of Eijkl = Cijkl (or Sijkl)
in most general case (in practice the number
of components reduces considerably):
 2D: 2222 = 16, (i = 1,2)
 3D: 3333 = 81, (i = 1,2,3)
* Young’s modulus is an elastic constant and not an elastic modulus!

 If we apply just one component of stress (say 11), this in general will produce not only 11
but also other components of strain (which other components will depend on symmetry of
the crystal- which we shall consider shortly). A glimpse of this we have already seen (in the
topic on understanding stress and strain) if we pull a cylinder along y-direction, it will contract along the
x-direction  11 exists (apart from elongation in the y-direction 22).
 But, this may not be the only other strain produced.
What does the existence of these Cijkl components physically mean?
11 1111 11 1112 12
1121 21 1122 22
C C
C C
  
 
  

 This implies that the body may shear also (12 = 21 exists). This occurs due to
‘anisotropy’ in the crystal.
 So (in general) a bar:  might shear if pulled (in addition to elongating)
 may twist if bent (in addition to bending)
 may bend if twisted (in addition to twisting).
11 1111 11 1112 12 1113 13
1121 21 1122 22 1133 33
1131 31 1132 32 1133 33
C C C
C C C
C C C
   
  
  
   
  
 
2D
3D

 We saw that a single stress component 11 may be related to many strains.
 Similarly, many stress components may give rise to a single strain (11).
11 1111 11 1112 12
1121 21 1122 22
S S
S S
  
 
  

2D 3D
11 1111 11 1112 12 1113 13
1121 21 1122 22 1123 23
1131 31 1132 32 1133 33
S S S
S S S
S S S
   
  
  
   
  
 
There are 4 such equations
(for each component ij) There are 9 such equations

 The total number of expected components are: 2D: 2222 = 16, (i = 1,2);
3D: 3333 = 81 , (i = 1,2,3). But, not all of these are independent.
  Cijkl = Cijlk (also, Sijkl = Sijlk)  Cijkl = Cjikl (also, Sijkl = Sjikl)
 only 36 of the 81 components are independent (for even the most general crystal)*.
 From energy arguments it can further be seen that: Cijkl = Cklij (also, Sijkl = Sklij)*
 of the 36 components only 21 survive (even for the crystal of the lowest symmetry i.e. triclinic crystal).
 Given that crystals can have higher symmetry than the triclinic crystal, the number of
independent components of Cijkl (or Sijkl) may be significantly reduced.
 All cubic crystals (irrespective of their point group symmetry) have only 3 independent
components of Cijkl (or Sijkl) these are C1111, C1122, C2323.
 This is further reduced in the case of isotropic materials to 2 independent constants (C1111,
C1122). Usually, these independent constants are written as E & .
 Hence, for second rank tensor properties (like magnetic susceptibility, ij) cubic
crystals are isotropic; but, not for forth rank tensor properties like Elastic
Stiffness.
No of independent components of Cijkl and Sijkl
* The reasoning is postponed for later.

 
44
11 12
2C
A
C C


Zener Anisotropy factory (A)
A > 1  <111> direction stiffest
A < 1  <100> direction stiffest
 1011
N/m2
Structure C11 C12 C44 A
BCC Li 0.135 0.114 0.088 8.4
Na 0.074 0.062 0.042 7.2
K 0.037 0.031 0.019 607
DC C 10.20 2.50 4.92 1.3
Si 1.66 0.64 0.80 1.6
Ge 1.30 0.49 0.67 1.7
NaCl type NaCl 0.485 0.125 0.127 0.7
KCl 0.405 0.066 0.063 0.37
RbCl 0.363 0.062 0.047 0.31
 The magnitude of anisotropy in cubic crystals is quantified using the Zener anisotropy factor
(A) as defined in the equation below.
 A value of ‘A’ of one (1) implies that the crystal is isotropic.

Chapter 9 Elasticity and its different proprties

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