Power point by Jyoti Anischit

• Rock Mechanics.

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ROCK CLASSIFICATIONS AND IT’S USE IN DESIGN

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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.

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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 classificationSystems for slope engineering• Slope Mass Rating (SMR)• Rock mass classification system for rock slopes• Slope Stability Probability Classification (SSPC)

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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

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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:4. A great deal of judgment is needed in the application of rock

mass classification to support design.5. RMR value doesn’t give us rock mass properties.6. These give only empirical relation & have nothing to do with

rock engineering classification in its true sense.Jyoti Anischit

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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.

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• 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.

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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.

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• 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:4. Requires more calculation than RMR & Q- system.5. For special ground conditions like swelling, squeezing & fault zones,

etc. the rock support should be evaluated seperatlly for each & every cases.

6. Like other empirical method, it is not possible to evaluate the accuracy of the system.

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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)

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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

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RMR ClassificationRMR= A.1+A.2+A.3+A.4(E)+A.5+BSix 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.

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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)

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• 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.

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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

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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,

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Factors in the Assessment of Mining-induced StressThe 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.

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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

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

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• 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

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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.

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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

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

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Fixing of Wire Mesh and Pipe Roofing/Forepoling ( Figure 6)

Shotcreting with CIFA Robotic Arm (Figure 7)

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Rock Bolting In Progress with Rocket BoomerFigure 8

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Major engineering rock mass classification systems currently in use

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APPLICATION OF ROCK MASS CLASSIFICATION SYSTEMS IN COAL

MINING

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• 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:

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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.

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• 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.

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Visualization in two-dimensional space

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• 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.

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Visualization in three-dimensional space

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Kinematic analysis

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 TestStress is applied to the

end faces of the specimen.No radial (confining

stress)Also called Unconfined

Compression Test.

Elastic regionSpecimen returns to its original state after stress relief.

Yield PointPermanent changes beyond this point. Specimen does not return to its original state after removal of stress.

Uniaxial compressive strengthThe peak stress.

Ductile regionPermanent deformation, but can still support load.

Brittle regionAbility to withstand stress decreases rapidly as deformation increases.

Triaxial TestIn 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 =

cottan

oo

SA 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 31 sin 22

1 3 1 31 1 cos 22 2

2 90o 4 2

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

2 cos 2 tan1 sin

oo o

Sa C S

1 sin

tan1 sin

b

tan 1

sintan 1

21 3 tanoC

1 3a b

© Haimson and Song (1995)

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 cos1 sin 1 sin

of f

SP 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/m21 bar(non-SI) = 105Pa ~1 atmosphere1 kb= 1000 bar = 108Pa = 100 Mpa1Gpa = 109Pa = 1000 Mpa= 10 kbP at core-mantle boundaryis ~ 136 Gpa(at 2900 km)P at the center of Earth(6371 km) is 364 Gpa

Stress tensor

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.

LOAD

STRUCTURE

Stress 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

Stress-strain relationship

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 ofElasticity (Young’smodulus) = E

Linear region

– Begins at yield stress Σy

– Slope rapidly decreasesuntil it is horizontal ornear horizontal

– Large strain increase,small stress increase

– Strain is permanent

Yielding region

– After undergoing largedeformations, the metalhas changed itscrystalline structure.

– The material hasincreased resistanceto applied stress(it appears to be“harder”).

Strain Hardening

– The maximum supportedstress value is called theultimate stress, σu.

– Loading beyond σuresults in decreasedload supported andeventually 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. Hardness3. Effect of Temperature on Properties4. Fluid Properties5. 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 material2. Compressive - squeezing the material3. 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:

oe 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

oL

LLe

Typical Engineering Stress-Strain Plot

• Typical engineering stress strain plot in ‑a tensile test of a metal

• Two regions:1. Elastic region2. Plastic region

Elastic Region in Stress Strain Curve‑

• Relationship between stress and strain is linearHooke'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 lengthLf is measured as the distance between gage marks after two pieces of specimen are put back together

o

ofL

LLEL

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

AF

True Strain

• Provides a more realistic assessment of "instantaneous" elongation per unit length

o

L

L LL

LdL

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

nK

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

Compression Test Setup

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

oAF

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

oh

hhe

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:

251bt

FLTRS .

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

AF

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

AF

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, mShear rate = velocity gradient perpendicular to flow direction

dydv

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