Rock mechanics for engineering geology part 1

Power point by Jyoti Anischit
• Rock Mechanics.

ROCK CLASSIFICATIONS AND IT’S
USE IN DESIGN
2

INTRODUCTION
• Rock mass classification systems are used for
various engineering design and stability
analysis.
• These are based on empirical relations
between rock mass parameters and
engineering applications, such as tunnels,
slopes, foundations, and excavability.
3

ROCK MASS CLASSIFICATION SYSTEMS
Systems for tunneling: Quantitative
• Rock Mass Rating (RMR)
• Q-system
• Mining rock mass rating (MRMR)
Other systems: Qualitative
• New Austrian Tunnelling Method (NATM)
• Size Strength classification
Systems for slope engineering
• Slope Mass Rating (SMR)
• Rock mass classification system for rock slopes
• Slope Stability Probability Classification (SSPC)
4

PURPOSE
• 1. Identify the most significant parameters influencing the
behaviour of a rock mass.
• 2. Divide a particular rock mass formulation into groups of
similar behaviour – rock mass classes of varying quality.
• 3. Provide a basis of understanding the characteristics of
each rock mass class
• 4. Relate the experience of rock conditions at one site to
the conditions and experience encountered at others
• 5. Derive quantitative data and guidelines for engineering
design
• 6. Provide common basis for communication between
engineers and geologists
5

ADVANTAGES AND DISADVANTAGES OF
DIFFERENT ROCK MASS CLASSIFICATION
SYSTEMS
• RMR classification system
ADVANTAGES:
1. Rock mass strength is evaluated by RMR system.
2. It works well to classify rock mass quality.
3. RMR system is used in many projects as one of the indicators
to define the support or excavation design.
DISADVANTAGES:
1. A great deal of judgment is needed in the application of rock
mass classification to support design.
2. RMR value doesn’t give us rock mass properties.
3. These give only empirical relation & have nothing to do with
rock engineering classification in its true sense.
Jyoti Anischit 6

4. The relatively small database makes the system less applicable
to be used as an empirical design method for rock support.
5. RMR cannot be used as the only indicator, especially when rock
stresses or time dependent rock properties are of
importance for the rock engineering.
• NATM classification system:
ADVANTAGES:
1. NATM can be applied successfully in a large no. of tunnels in
poor and difficult ground conditions.
2. As compared to traditional tunneling, considerable cost
saving is gained, as well as reduced construction time.
7

• Q- system of rock mass classification:
ADVANTAGES:
1. Together with the ratio between the span or height of the
opening and an excavation support ratio (ESR), the Q value
defines the rock support.
8

DISADVANTAGES:
1. The accuracy of estimation of rock support is very difficult to
evaluate.
2. In the poorer rock (Q<1) system may give erroneous design.
3. The true nature of rock mass (e.g. swelling, squeezing or
popping ground ) is not explicitly considered in the Q- system.
4. The value is used as the only indicator to define the classes in
question.
9

• RMi classification system:
ADVANTAGES:
1. Rmi value can be applied as input to other rock engineering
methods to estimate the deformation modulus for rock masses.
2. The system applies best to massive & jointed rock masses where the
joints in the various sets have similar properties.
3. It may also be used as a first check for support in faults & weakness
zones.
DISADVANTAGES:
1. Requires more calculation than RMR & Q- system.
2. For special ground conditions like swelling, squeezing & fault zones,
etc. the rock support should be evaluated seperatlly for each &
every cases.
3. Like other empirical method, it is not possible to evaluate the
accuracy of the system.
10

ROCK MASS CLASSIFICATION USED IN
DESIGN
Q Classification
Q = RQD/Jn x Jr/Ja x Jw/SRF
• I. Relative block size (RQD/Jn)
• II. Inter-block shear strength (Jr/Ja)
• III. Active stresses (Jw/SRF)
11

Temporary mine openings. ESR = 3 - 5
Permanent mine openings,
water tunnels for hydro
power (excluding high
pressure penstocks), pilot
tunnels, drifts and headings
for large excavations.
1.6
Storage rooms, water
treatment plants, minor road
and railway tunnels, surge
chambers, access tunnels.
1.3
Power stations, major road
and railway tunnels, civil
defence chambers, portal
intersections.
1.0
Underground nuclear power
stations, railway stations,
sports and public facilities,
factories.
0.8
Barton et al (1974) defined
an additional parameter
called the Equivalent
Dimension, De, of the
excavation.
De= excavation span
diameter or height (m) /
excavation support ratio,
ESR
ESR is related to the
intended use of the
excavation and to the
degree of security, which is
influence on the support
system to be installed to
maintain the stability of
the excavation
12

RMR Classification
RMR= A.1+A.2+A.3+A.4(E)+A.5+B
Six parameters are used to classify a rock mass
using the RMR system:
1. Uniaxial compressive strength of rock
material.
2. Rock Quality Designation (RQD).
3. Spacing of discontinuities.
4. Condition of discontinuities.
5. Groundwater conditions.
6. Orientation of discontinuities.
14

MINING ROCK MASS RATING (MRMR)
• MRMR = RMR * adjustment factors,
• in which: adjustment factors =factors to compensate
for: the method of excavation, orientation of
discontinuities and excavation, induced stresses, and
future weathering
• The main differentiators of the MRMR 2000 system
compared to previous versions of the Q-system, and
Bieniawski RMR systems are:-
• Scale concept in material strength (intact rock > rock
block > rock mass)
• Inclusion of cemented joints and veinlets
• Abandonment of the Rock Quality Designation (RQD)
as an input parameter
• Mining adjustments (in comparison to Q)
16

• The lack of accountability for the basic rock
mass parameters such as intact rock strength
and strength of defects, the tradeoff against
its simplicity is its poor reliability in highly
fractured, massive, or highly anisotropic
conditions.
• The RMR method simply does not have the
resolution that may be required for a more
accurate assessment of fragmentation,
cavability, and other mine design aspects.
17

Example of the problems with
RQD assessment of highly
fractured or massive rock masses
Example of difference between RQD
and fracture frequency-based IRMR.
The IRMR based on fracture
frequency (solid line) is considered
more representative of actual rock
mass conditions
18

As gravity is the most significant
force to be considered, the
instability of the block depends on
the number of joints that dip away
from the vertical axis.
Adjustments are made where
joints define an unstable wedge
with its base on the sidewall.
The instability is determined by
the plunge of the intersection of
the lower joints,
19

Factors in the Assessment of Mining-induced Stress
The following factors should be considered in the assessment of
mining-induced stresses:
• drift-induced stresses;
• interaction of closely spaced drifts;
• location of drifts or tunnels close to large stopes;
• abutment stresses, particularly with respect to the direction of
advance and orientation of the field stresses (an undercut
advancing towards maximum stress ensures good caving but
creates high abutment stresses, and vice versa)
• uplift;
• point loads from caved ground caused by poor fragmentation
• removal of restraint to sidewalls and apexes.
• increases in size of mining area causing changes in the geometry.
• massive wedge failures;
• influence of major structures not exposed in the excavation but
creating the probability of high toe stresses or failures in the back of
the stope.
• presence of intrusives that may retain high stress or shed stress into
surrounding, more competent rock. 20

Blasting creates new fractures and
loosens the rock mass, causing
movement on joints, so that the
following adjustments should be
applied
Technique Adjustment, %
Boring
100
Smooth-wall blasting
97
Good conventional blasting
94
Poor blasting
80
Adjustments must recognize the
life of the excavation and the
time-dependent behaviour of
the rock mass
Parameters Possible
adjustment, %
Weathering 30-100
Orientation 63-100
Induced stresses 60-120
Blasting 80-100
21

table below shows how the support
techniques in alphabetical symbols,
increases in support pressure with the
decrease in MRMR value
22

• The “Slope Mass Rating” (SMR) is obtained from RMR by adding a
factorial adjustment factor depending on the relative orientation of
joints and slope and another adjustment factor depending on the
method of excavation.
• SMR = RMRB + (F1 x F2 x F3) + F4
(i) F1 depends on parallelism between joints and slope face strike. Its
range is from 1.00 to 0.15. These values match the relationship: F1 =
(1 – sin A)2 where A denotes the angle between the strikes of slope
face and joints.
(ii) F2 refers to joint dip angle in the planar mode of failure. Its value
varies from 1.00 to 0.15, and match the relationship: F2 = tg2Bj
denotes the joint dip angle. For the toppling mode of failure F2
remains 1.00.
(iii) F3 reflects the relationship between slope and joints dips.
(iv) F4 (adjustment factor for the method of excavation has been fixed
empirically.
Slope Mass Rating
24

NEW AUSTRIAN TUNNELING METHOD
• NATM: This method has been developed basically in
Austria
• Its name make use of providing flexible primary lining in
shape of shotcrete , wire mesh, rock bolts ,lattice girder.
• In case of weaker rock mass the use of pipe forepole/pipe
roofing is also resorted for crown support which in turn
lead to less overbreak as well as ensure safety during the
execution.
• The main aspect of the approach is dynamic design based
on rock mass classification as well as the insitu deformation
observed.
• There various approaches classification of the rock mass
most predominantly used here- RQD, RMR and Q factor of
the rock mass.
27

Components of Execution in NATM
i) Sealing Shotcrete – Shotcrete 25-50mm generally( fig 4)
ii) Fixing of Lattice Girder – lattice girder is 3 Bars of steel
reinforcement placed at three corners of triangle with 8mm
steel bar for connection.Easy to handle comparison of steel
ribs. (fig 5)
iii) Fixing of wire mesh –generally used 6mm thick wires (fig 6)
iv) Primary Lining with Shotcrete – In layers each not thicker
than 150mm (fig 7)
v) Rock Bolting (fig 8)
vi) Pipe Forepoling – Used for crown support for next
Excavation cycle ( for Rock Class after III only) ( fig 6)
Note: Wire mesh is not used for Fibre Reinforced Shotcrete
28

Face recently opened sealed
with Shotcrete (Figure 4) Lattice Girder
29

Fixing of Wire Mesh and Pipe
Roofing/Forepoling ( Figure 6)
Shotcreting with CIFA Robotic
Arm (Figure 7)
30

Rock Bolting In Progress with Rocket Boomer
Figure 8
31

Major engineering rock mass
classification systems currently in use
32

APPLICATION OF ROCK MASS
CLASSIFICATION SYSTEMS IN COAL
MINING
33

• Rock Mass Rating (RMR) is the sum of five parameter ratings.
• If there are more than one rock type in the roof, RMR is
evaluated separately for each rock type and the combined RMR
is obtained as:
34

Visualization of rock mass classification
systems
• In most widely used rock mass classification systems, such as
RMR and Q systems, up to six parameters are employed to
classify the rock mass
• Visualization of rock mass classification systems in multi-
dimensional spaces is explored to assist engineers in
identifying major controlling parameters in these rock mass
classification systems
• The study reveals that all major rock mass classification
systems tackle essentially two dominant factors in their
scheme, i.e., block size and joint surface condition.
35

• It is based on the fact that human beings are overpoweringly
visual creatures
• Visualization is the task of generating images that allow
important features in the data to be recognized much more
readily than from processing raw data by other means, for
example like statistics.
36

Visualization in two-dimensional space
37

• The plots from Figures it reveal one common feature of these
widely used rock mass classification systems, that is, the most
important controlling factors are block volume and joint
surface condition
• When parameters are condensed to only these two
parameters, the classification functions are best represented
by planar surfaces in linear (RMR) or log scales (Q), or by
surfaces that are very close to planar surfaces in log scales
(GSI and RMi)
• Thus all the rock mass classification systems are essentially
the same
• It is concluded that any new development of rock mass
classification system should therefore start with careful
consideration of the block size and joint surface condition
characterization.
40

Visualization in three-dimensional space
41

Rock mechanics for engineering geology part 1

FAILURE MECHANICS
Jyoti Anischit Tribhuvan University

• Concepts of Failure
• Tensile Failure
• Shear Failure
– Failure Criteria
– Mohr-Coulomb Failure Criterion

CONCEPTS OF FAILURE
• Failure occurs to any solid material when:
– Sufficiently large stress is applied.
– The material does not return to its original state after
stress relief.
• Mode of failure depends on:
– Stress state
– Type and geometry of material
• Fatigue makes failure to occur below the stress
level.

Uniaxial Test
Stress is applied to the
end faces of the
specimen.
No radial (confining
stress)
Also called Unconfined
Compression Test.

Elastic region
Specimen returns
to its original state
after stress relief.
Yield Point
Permanent
changes beyond
this point.
Specimen does
not return to its
original state after
removal of stress.
Uniaxial
compressive
strength
The peak stress.
Ductile region
Permanent deformation,
but can still support load.
Brittle region
Ability to withstand stress
decreases rapidly as
deformation increases.

Triaxial Test
In addition to axial
stress, confining
pressure of different
magnitude is applied to
the circumference of the
cylinder (by a confining
oil bath).

• Two of the principal stresses are equal.
• Process:
– Axial & confining loads are increased
simultaneously until a prescribed hydrostatic
stress level is reached.
– Confining pressure is kept constant while axial
load increases until failure occurs.

Difference in principal stresses is plotted against axial deformation.
Specimen can still support load after failure due to high confining
pressure. It is called Work Hardening or Strain Hardening.

Uniaxial test
X → abrupt brittle
failure

TENSILE FAILURE
• Tensile failure occurs when
the effective tensile stress
across some plane is the
sample exceeds a critical
limit called Tensile
Strength.

• Tensile failure is caused by the stress
concentrations at the edges of thin cracks
oriented normal to the direction of the least
compressive principal stress.
• For isotropic rocks, conditions for failure will
always be fulfilled first for the lowest principal
stress.
To = tensile strength (in Pa, atm, psi or bar).
3 3 oP T     

• Most sedimentary rocks have a rather low
tensile strength, typically only a few MPa or
less.
• Standard approximation for several
applications is that the tensile strength is zero

SHEAR FAILURE
• It occurs when the shear
stress along some plane in the
sample is too large.

Failure criteria
• Mohr–Coulomb
• Hoek–Brown
• Drucker–Prager
• Griffith (tensile)

Mohr-Coulomb Criterion
 So = cohesion or inherent shear strength of material (in Pa,
atm, psi or bar).
 µ = coefficient of internal friction.
 Shear stress must overcome the cohesion plus the
internal friction in order to produce a macroscopic
shear failure.
 f 
oS  

If the Mohr’s circle lies below the failure line, the rock does not fail and
remains intact.
Failure Line
Mohr
Circle
tan Slope =
cot
tan
o
o
S
A S 

 

 φ = angle of internal friction. It varies from 0 to 90o (approx.
30o)
 A = attraction (in Pa, atm, psi or bar).
 β = angle that fulfils the failure criterion. It gives orientation
of the failure plane. Varies between 45o and 90o.
 At point P:
 Angle 2β gives the position of coincidence of Mohr’s circle
and the failure line.
 Coordinates are given as:
 OR
 1 3
1
sin 2
2
        1 3 1 3
1 1
cos2
2 2
        
2 90o
  
4 2
 
  

– Co = uniaxial compressive strength (in Pa, atm, psi or bar).

 
2 cos
2 tan
1 sin
o
o o
S
a C S



  

 
 
1 sin
tan
1 sin
b




 

 
 
tan 1
sin
tan 1






2
1 3 tanoC   
1 3a b  

Mohr-Coulomb Criterion on Saturated
Rocks
• Principle of effective stress is introduced, i.e.
subtract fluid pressure from the total stress.
– Previously:
– And
– Then:
 
  
 1 3
1 sin2 cos
1 sin 1 sin
o
f f
S
P P

   
 

   
 
1 3a b  
1 1 fP     3 3 fP    
1 3a b   

• Pore fluid can affect the failure of the rock in 2
ways:
– Mechanical effect of pore pressure.
– Chemical interactions between the rock and the fluid.

• Effect of pore pressure on failure:
– Shear stress is unaffected by the pore pressure
– Minimum & maximum principal stresses are
decreased by the same amount.
– Radius of the Mohr circle in unchanged.
– Center of the circle has shifted to the left.
– Circle moves towards the failure line when the
fluid pressure is increased for a material obeying
the criterion.

Axial Stress - Strain Curve
And
Modulus Of Elasticity

In trying to pull the object apart, internal resisting
forces are created and these internal forces are
known as stress.
Stress = Restoring force / area
= (F)/(A) ,
where F is the deforming force acting on an area A of the body.
STRESS

A force acting on a small area such as the tip of a sharp nail,
has a greater intensity than a flat-headed nail!
s= [MLT-2] / [L2]=[ML -1T-2]
s= kg m-1s-2pascal(Pa) = newton/m2
1 bar(non-SI) = 105Pa ~1 atmosphere
1 kb= 1000 bar = 108Pa = 100 Mpa
1Gpa = 109Pa = 1000 Mpa= 10 kb
P at core-mantle boundaryis ~ 136 Gpa(at 2900 km)
P at the center of Earth(6371 km) is 364 Gpa

STRAIN
The ratio of change produced in the dimensions of
a body by a system of forces or couples, in
equilibrium, to its original dimensions is called
strain.

REQUIREMENT OF STRESS –STRAIN
RELATIONSHIP
To analysis and design members.
It is most important while dealing with reinforced
concrete which is a composite material.

STRESS—STRAIN CURVE OF CONCRETE
At first,
As load is applied ,the ratio between
stress-strain is approximate linear.
Concrete behaves almost as an
elastic material.
If load is removed,displacement is
recovered virtually.
Eventually,
The curve is no longer linear.
Behaves more and more as plastic
materia.
The shape of stress-strain curve
is mostly depend on length of time
of loading.

STRESS-STRAIN CURVE OF CONCRETE

It is interesting to note that although cement
paste and aggregates individually have
linear stress-strain relationships, the
behavior for concrete is non-linear. This is
due to the mismatch and micro cracking
created at the interfacial transition zone.

Material behavior is generally represented by a stress-strain diagram,
which is obtained by conducting a tensile test on a specimen of material.

– Stress-strain
responseis linear.
– Slope = Modulus of
Elasticity (Young’s
modulus) = E
Linear region

– Begins at yield stress Σy
– Slope rapidly decreases
until it is horizontal or
near horizontal
– Large strain increase,
small stress increase
– Strain is permanent
Yielding region

– After undergoing large
deformations, the metal
has changed its
crystalline structure.
– The material has
increased resistance
to applied stress
(it appears to be
“harder”).
Strain Hardening

– The maximum supported
stress value is called the
ultimate stress, σu.
– Loading beyond σu
results in decreased
load supported and
eventually rupture.
Necking

It is defined as the slope of its stress-strain curve in the
elastic deformation level.
Modulus of elasticity
E= Stress/Strain

MECHANICAL PROPERTIES
OF MATERIALS
1. Stress-Strain Relationships
2. Hardness
3. Effect of Temperature on Properties
4. Fluid Properties
5. Viscoelastic Behavior of Polymers

Mechanical Properties in
Design and Manufacturing
• Mechanical properties determine a material’s
behavior when subjected to mechanical stresses
– Properties include elastic modulus, ductility,
hardness, and various measures of strength
• Dilemma: mechanical properties that are
desirable to the designer, such as high strength,
usually make manufacturing more difficult

Stress-Strain Relationships
• Three types of static stresses to which
materials can be subjected:
1. Tensile - stretching the material
2. Compressive - squeezing the material
3. Shear - causing adjacent portions of the material
to slide against each other
• Stress-strain curve - basic relationship that
describes mechanical properties for all three
types

Tensile Test
• Most common test for studying
stress-strain relationship,
especially metals
• In the test, a force pulls the
material, elongating it and
reducing its diameter
• (left) Tensile force applied and
(right) resulting elongation of
material

Tensile Test Specimen
• ASTM (American Society for
Testing and Materials) specifies
preparation of test specimen

Tensile Test Setup
• Tensile testing
machine

Tensile Test Sequence
• (1) no load; (2) uniform elongation and area
reduction; (3) maximum load; (4) necking; (5)
fracture; (6) putting pieces back together to
measure final length

Engineering Stress
Defined as force divided by original area:
o
e
A
F

where e = engineering stress, F = applied force, and Ao =
original area of test specimen

Engineering Strain
Defined at any point in the test as
where e = engineering strain; L = length at any point during
elongation; and Lo = original gage length
o
o
L
LL
e



Typical Engineering
Stress-Strain Plot
• Typical
engineering
stress-strain plot
in a tensile test of
a metal
• Two regions:
1. Elastic region
2. Plastic region

Elastic Region in
Stress-Strain Curve
• Relationship between stress and strain is linear
Hooke's Law: e = E e
where E = modulus of elasticity
• Material returns to its original length when
stress is removed
• E is a measure of the inherent stiffness of a
material
– Its value differs for different materials

Yield Point in
Stress-Strain Curve
• As stress increases, a point in the linear
relationship is finally reached when the
material begins to yield
– Yield point Y can be identified by the change in
slope at the upper end of the linear region
• Y = a strength property
– Other names for yield point:
• Yield strength
• Yield stress
• Elastic limit

Plastic Region in
Stress-Strain Curve
• Yield point marks the beginning of plastic
deformation
• The stress-strain relationship is no longer
guided by Hooke's Law
• As load is increased beyond Y, elongation
proceeds at a much faster rate than before,
causing the slope of the curve to change
dramatically

Tensile Strength in
Stress-Strain Curve
• Elongation is accompanied by a uniform
reduction in cross-sectional area, consistent with
maintaining constant volume
• Finally, the applied load F reaches a maximum
value, and engineering stress at this point is
called the tensile strength TS (a.k.a. ultimate
tensile strength)
TS = oA
Fmax

Ductility in Tensile Test
• Ability of a material to plastically strain without
fracture
• Ductility measure = elongation EL
where EL = elongation; Lf = specimen length at fracture; and Lo = original
specimen length
Lf is measured as the distance between gage marks after two pieces of
specimen are put back together
o
of
L
LL
EL



True Stress
Stress value obtained by dividing the
instantaneous area into applied load
where  = true stress; F = force; and A = actual (instantaneous) area
resisting the load
A
F


True Strain
• Provides a more realistic assessment of
"instantaneous" elongation per unit length
o
L
L L
L
L
dL
o
ln 

True Stress-Strain Curve
• True
stress-strain
curve for
previous
engineering
stress-strain
plot

Strain Hardening in
Stress-Strain Curve
• Note that true stress increases continuously
in the plastic region until necking
– In the engineering stress-strain curve, the
significance of this was lost because stress was
based on the original area value
• It means that the metal is becoming stronger
as strain increases
– This is the property called strain hardening

True Stress-Strain
in Log-Log Plot
• True
stress-strain
curve plotted
on log-log
scale.

Flow Curve
• Because it is a straight line in a log-log plot,
the relationship between true stress and true
strain in the plastic region is
where K = strength coefficient; and n = strain hardening exponent
n
K 

Categories of Stress-Strain
Relationship: Perfectly Elastic
• Behavior is defined
completely by modulus of
elasticity E
• Fractures rather than yielding
to plastic flow
• Brittle materials: ceramics,
many cast irons, and
thermosetting polymers

Stress-Strain Relationships: Elastic and
Perfectly Plastic
• Stiffness defined by E
• Once Y reached, deforms
plastically at same stress level
• Flow curve: K = Y, n = 0
• Metals behave like this when
heated to sufficiently high
temperatures (above
recrystallization)

Stress-Strain Relationships: Elastic and
Strain Hardening
• Hooke's Law in elastic region,
yields at Y
• Flow curve: K > Y, n > 0
• Most ductile metals behave
this way when cold worked

Compression Test
• Applies a load that squeezes the
ends of a cylindrical specimen
between two platens
• Compression force applied to
test piece and resulting change
in height and diameter

Engineering Stress in Compression
• As the specimen is compressed, its height is
reduced and cross-sectional area is increased
e = -
where Ao = original area of the specimen
oA
F

Engineering Strain in Compression
Engineering strain is defined
Since height is reduced during compression, value of e is negative
(the negative sign is usually ignored when expressing compression
strain)
o
o
h
hh
e



Stress-Strain Curve in Compression
• Shape of plastic region is
different from tensile test
because cross section
increases
• Calculated value of
engineering stress is higher

Tensile Test vs.
Compression Test
• Although differences exist between engineering
stress-strain curves in tension and compression,
the true stress-strain relationships are nearly
identical
• Since tensile test results are more common, flow
curve values (K and n) from tensile test data can
be applied to compression operations
• When using tensile K and n data for compression,
ignore necking, which is a phenomenon peculiar
to strain induced by tensile stresses

Testing of Brittle Materials
• Hard brittle materials (e.g., ceramics) possess
elasticity but little or no plasticity
– Conventional tensile test cannot be easily applied
• Often tested by a bending test (also called
flexure test)
– Specimen of rectangular cross-section is
positioned between two supports, and a load is
applied at its center

Bending Test
• Bending of a rectangular cross section results
in both tensile and compressive stresses in the
material: (left) initial loading; (right) highly
stressed and strained specimen

Testing of Brittle Materials
• Brittle materials do not flex
• They deform elastically until fracture
– Failure occurs because tensile strength of outer
fibers of specimen are exceeded
– Failure type: cleavage - common with ceramics
and metals at low temperatures, in which
separation rather than slip occurs along certain
crystallographic planes

Transverse Rupture Strength
• The strength value derived from the bending test:
2
51
bt
FL
TRS
.

where TRS = transverse rupture strength; F = applied load at
fracture; L = length of specimen between supports; and b and t are
dimensions of cross section

Shear Properties
• Application of stresses in opposite
directions on either side of a thin element:
(a) shear stress and (b) shear strain

Shear Stress and Strain
Shear stress defined as
where F = applied force; and A = area over
which deflection occurs.
Shear strain defined as
where  = deflection element; and b =
distance over which deflection occurs
A
F

b

 

Torsion Stress-Strain Curve
• Typical shear
stress-strain
curve from a
torsion test

Shear Elastic Stress-Strain Relationship
• In the elastic region, the relationship is defined as
 G
where G = shear modulus, or shear modulus of elasticity
For most materials, G  0.4E, where E = elastic
modulus

Shear Plastic Stress-Strain Relationship
• Relationship similar to flow curve for a tensile
test
• Shear stress at fracture = shear strength S
– Shear strength can be estimated from tensile
strength: S  0.7(TS)
• Since cross-sectional area of test specimen in
torsion test does not change as in tensile and
compression, engineering stress-strain curve
for shear  true stress-strain curve

Rockwell Hardness Test
• Another widely used test
• A cone shaped indenter is pressed into specimen
using a minor load of 10 kg, thus seating indenter
in material
• Then, a major load of 150 kg is applied, causing
indenter to penetrate beyond its initial position
• Additional penetration distance d is converted
into a Rockwell hardness reading by the testing
machine

Rockwell Hardness Test
• (1) initial
minor load and
(2) major load.

Shear Stress
• Shear stress is the frictional force exerted by
the fluid per unit area
• Motion of the upper plate is resisted by this
frictional force resulting from the shear
viscosity of the fluid
• This force F can be reduced to a shear stress 
by dividing by plate area A
A
F


Shear Rate
• Shear stress is related to shear rate, defined
as the change in velocity dv relative to dy
where = shear rate, 1/s; dv = change in velocity, m/s; and dy = change
in distance y, m
Shear rate = velocity gradient perpendicular to flow direction
dy
dv



Elastic Behavior vs.
Viscoelastic Behavior
• (a) Response of
elastic material;
and (b) response
of a viscoelastic
material
• Material in (b)
takes a strain that
depends on time
and temperature

Rock mechanics for engineering geology part 1

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Rock mechanics for engineering geology part 1