Lecture4(RockMassProperties)

1

Lecture 4

Rock Mass Properties

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Brittle and Ductile Behavior of Rock Mass

Brittle Failure Behavior Ductile Failure Behavior•Resistance to load decreases with increases in deformation •Resistance to load sustain with increases in deformation

•Little or no permanent deformation before failure

•Sudden, catastrophic

•Rock burst in deep hard rock mines

•Most rock in civil and mining behaves as brittle under the usual confining pressure and temperature

•Ductility increases with increase in confining pressure and temperature

•Can occur in weathered rock and heavily jointed rock masses

4

Definition of Failure

4 Stages of Progressive Fracture Development in

Uniaxial

Compression Test on Rock

5

Tangent Young’s Modulus: slope of the axial stress-strain curve at say 50% of the peak strength

Average Young’s Modulus: average slope of the more or less straight line portion of the curve

Secant Young’s Modulus: slope of a straight line joining the origin and the peak

)(

)(

r

a

a

a

εσεσ

ν

ΔΔΔΔ

−=

Corresponding to any definition of the Young’s Modulus, Poisson’s Ratio can be calculated as:

rav εεε 2+=

6

End Effects of Testing Machine and Influence of Height to Diameter Ratio in Uniaxial


Ideal Condition:

Uniform boundary condition

Uniform uniaxial stress

Uniform displacement

Less Ideal Condition:

Prevented from deforming uniformly due to friction between platen and rock specimen

Due to the restraint, shear stresses are developed and axial stress is not principal stress anymore.

To minimize this effect, a H/D ratio of at least 2 is used in practice.

7

Brown and Gonano (1974)

Brush Platens (assembly of 3.2 mm square high-

tensile steel pins)

Solid Steel Platens

Apparent increase in strength and change of behavior when H/D is decreased from normal 2.0 to 0.25

8

Influence of Testing Machine Stiffness in Uniaxial


:To study the post peak behavior

When the peak strength has been reached in a strain-

softening material, the specimen continues to compress, but the load that it can carry progressively reduces.

After that, the machine unloads and its extension reduces.

If the machine stiffness is less than the rock, catastrophic failure occurs because the energy released by the machine (ADEF) is greater than that can be absorbed by the specimen (ADEB)

If the machine stiffness is more than the rock, post-peak behavior can be followed, because the energy released by the machine can be used to deform the specimen along path ABC.

9

For brittle rock, use of servo-controlled testing machine is needed.

Force, pressure, displacement and strain components are pre-set and pre-programmed so that they are varied monotonically increasing with time. The programmed values are compared several thousands of times a second and a servo valves adjust the pressure within the actuator to produce the desired equivalence.

Post peak behavior obtained in limestone by using servo-controlled testing machine

10

Using the servo-controlled testing machine, Wawersik and Fairhurst (1970) halted the tests on specimens of the same rock at different points on the post-peak curve, and then perform thin sectioning and observe the crack development.

Conclude:Class I –

stable fracture propagation, local tensile fracture predominantly parallel to the applied stressClass II –

unstable fracture propagation, local and macroscopic shear fracture

11

Influence of Confining Stresses in Triaxial


With increasing confining stresses:

•The peak strength increases

•Transition from brittle to ductile failure

•Post peak region flatten and widen

•Residual strength reduces and disappear at very high confining stresses

12

•Hoek

and Brown (1980a, 1980b) proposed a method for obtaining estimates of the strength of jointed rock masses, based upon

• an assessment of the interlocking

of rock blocks

• the condition of the surfaces

between these blocks.

•Further development of failure criteria

(Hoek

1983,

Hoek

and Brown 1988)

•For very poor quality rock masses (Hoek, Wood and Shah 1992)

•For a new classification called the Geological Strength Index

(Hoek, Kaiser and

Bawden

1995, Hoek

1995,

Hoek

and Brown 1997).

•Summary of development is given in

Hoek

and Brown (1997).

13

Generalised Hoek-Brown criterion for jointed rock masses

maximum and minimum effective stresses at failure

Hoek-Brown constant m for the rock mass

s and a are constants which depend upon the rock mass characteristics

uniaxial

compressive strength of the intact

rock pieces

Three ‘properties’

of the rock mass have to be estimated. These are:

1. the

uniaxial

compressive strength σci of the intact

rock pieces,

2. the value of the

Hoek-Brown constant mb for the rock mass

3. the value s and a being a function of the Geological Strength

Index GSI for the rock mass.

Empirical Basis

Curve Fitting

14

Original 1988

Generalised Hoek-Brown criterion for jointed rock masses

Empirical Basis

Curve Fitting

2'3

'3

'1 cc sm σσσσσ ++=

( ) 14100−= RMRiemm

( ) 6100−= RMRes

15

σ3

=0

16

17

Brittle-Ductile Transition Mogi (1966)

BehaviorBrittle3.4σσBehaviorDuctile3.4σσ

31

31

<>

Equation Applicable to Brittle Behavior Only

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•Relative size of the opening to the jointing system

•Transition from isotropic intact rock specimen to highly anisotropic rock mass (controlled by joints) to isotropic heavily jointed rock mass

19

•Hoek-Brown failure criterion -

assumes isotropic rock and rock mass behaviour

•

When the structure being

analysed

is large and the block size small in comparison, the rock mass can be treated as a

Hoek-Brown material.

•Where the block size is of the same order as that of the structure being

analysed

or when one of the discontinuity sets is significantly weaker than the others, the

Hoek-Brown criterion should not be used.

•

In these cases, the stability of the structure should be analysed

by considering failure mechanisms involving the sliding or rotation

of blocks and wedges defined by intersecting structural features.

20

Geological strength Index (GSI), Hoek, Kaiser and

Bawden

(1995)

•The strength of a jointed rock mass depends on the properties of

the intact rock pieces and also upon the freedom of these pieces to slide and rotate under different stress conditions.

•This freedom is controlled by the geometrical shape of the intact rock pieces as well as the condition of the surfaces separating the pieces.

•GSI provides a system for estimating the reduction in rock mass strength for different geological conditions.

21

22

•From GSI and mi

, calculate the rock mass strength as follows:

•For GSI > 25, i.e. rock masses of good to reasonable quality, the parameter in the original Hoek-Brown criterion can be estimated from:

•For GSI < 25, i.e. rock masses of very poor quality, the parameter in the original Hoek-Brown criterion can be estimated from:

GENERALIZED HOEK-BROWN CRITERION 2002

GENERALIZED HOEK-BROWN CRITERION 2002, Smooth Continuous Transition

•D: factor depends upon the degree of disturbance subjected by blast damage and stress relaxation

•Varies from 0 for undisturbed in situ rock masses to 1 for very disturbed rock masses

•Based on back analysis of strength and deformation properties of damaged rock mass


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24


25

Deformation modulus

•Based upon practical observations and back analysis of excavation

behaviour

in poor

quality rock masses, the following modification to

Serafim

and Pereira’s equation is proposed for σci <

100

•Serafim

and Pereira (1983) proposed a relationship between the in situ modulus of deformation and

Bieniawski’s RMR classification.

•Based upon back analysis of dam foundation deformations and it has been found to work well for better

quality rocks.

•The deformation of better quality rock masses is controlled by the discontinuities while, for poorer quality rock masses, the deformation of the intact rock pieces contributes to the overall deformation process.


26

Useful guideline for deformation modulus estimation

27

Useful guideline for deformation modulus estimation

Hoek, Kaiser and Bawden

(1995)

Bieniawski

(1978)

Serafim and Pereira (1983)

Grimstad

and Barton (1993)

28

Empirical estimation of rock mass modulus E. Hoek and M.S.

Diederichs

International Journal of Rock Mechanics & Mining Sciences 43 (2006) 203–215

29

Empirical estimation of rock mass modulus E. Hoek and M.S.

Diederichs

International Journal of Rock Mechanics & Mining Sciences 43 (2006) 203–215

Equation (2) can be used where only GSI (or RMR or Q) data are available

Equation (4) can be used where reliable estimates of the intact rock modulus or intact rock strength are available

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34

Mogi’s Line defines the ratio of major and minor effective principal stresses at which there is a transition from brittle to ductile failure. This line is simply defined by sig1/sig3 = 3.4 •

If the principal stress failure envelope lies ABOVE

Mogi’s line, this indicates a brittle failure mode.

•

If the principal stress failure envelope lies BELOW

Mogi’s line, this indicates a ductile failure mode.

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36

37

Estimation of rock mass deformation modulus and strength of jointed hard rock masses using the GSI system

•Linkage between descriptive geological terms and measurable field parameters such as joint spacing and joint roughness

Estimation of mean rock mass deformation modulus

38Based on calibration of published data and back analysis of two caverns

39

Ignoring the effect of intersection angle between joint sets

A Joint Persistent Factor (pi

) is proposed to quantify the degree of interlocking

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

≥

<=Ll

LlLl

pi

ii

i1

si Average joint spacing

li Accumulated joint length of set i

L Characteristic length of the rock mass

40

3

'

i

ii

pss = Equivalent Spacing for discontinuous joint

Equivalent Block Volume considering short joints are insignificant to underground excavation

3213

321

321

sinsinsin γγγpppsssVb =

A Joint Condition Factor is used to quantify the joint surface condition

A

swc J

JJJ =Jw

= large scale waviness

Js

= small scale smoothness

JA

= joint alternation factor

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Scale Effects in Rock Masses

45

Rock masses are basically inhomogeneous and discontinuous media

The variation of the test results with the specimen size is called Scale Effect

Cunha, A. P. 1990. Scale effects in rock mechanics

Increases in sample size affecting results

Results become independent of specimen size (Representative Elementary Volume)

46

Experimental results normalized to 50 mm diameter specimen

Scale Effects on Intact Rock

47

cm

48

Scale Effects on Rock Joints

33.0

500 ⎟⎟⎠

⎞⎜⎜⎝

⎛=

n

nnp L

JRCLδ

Displacement needed to reach the peak shear strength is scaled as:

JRCo and Lo (length) refer to 100 mm laboratory scale samples

JRCn and Ln refer to in situ block sizes.

100 mm1000 mm

49

Bandis (1980)

Brittle Behavior

Ductile Behavior

Decrease in Peak Shear Strength with increase in sample size

50

Application of the compass with base plate method by Richards and Cowland (1982)

i is scale dependent

i decreases with increases in sampling length

51

Scale Effects on Rock Masses

s=1 for intact rock

s=0 as joint intensity increases

m and s decrease as joint intensity increases, and block size decreases, and thus a lower strength; indirect way of predicting strength decrease due to scale effects

52

Scale Effect Investigation with the help of In-situ Stress Measurements

600 mm over-core produced stress results close to the overall mean

大连理工大学岩石破裂与失稳研究中心大连理工大学岩石破裂与失稳研究中心Center for Rock Instability and Seismicity Research, Dalian UnCenter for Rock Instability and Seismicity Research, Dalian University of Technologyiversity of TechnologyCRISRCRISR

Rock FailureRock Failure ProcesProcess Analysiss AnalysisModeling and MonitoringModeling and Monitoring

Presented byChunan TANG

In recent years numerical methods have In recent years numerical methods have been continued to expand and diversify been continued to expand and diversify into the major fields of scientific and into the major fields of scientific and engineering studies. They provide a viable engineering studies. They provide a viable alternative to physical models that can be alternative to physical models that can be expensive, time consuming, and sometimes, expensive, time consuming, and sometimes, extremely difficult to carry out. extremely difficult to carry out.

CRISR, Dalian University of TechnologyCRISR, Dalian University of Technology RFPARFPA

Numerical tools can give an approximate Numerical tools can give an approximate behaviour, in which the boundary and size behaviour, in which the boundary and size effects can be taken into account quite effects can be taken into account quite realistically. realistically.


Progressive failure of rockProgressive failure of rock



Principle of RFPAPrinciple of RFPA

RFPA (Realistic Failure Process Analysis) is a RFPA (Realistic Failure Process Analysis) is a FEM code that can simulate the failure process of FEM code that can simulate the failure process of brittle materials.brittle materials.

•• The inhomogeneity of brittle material is The inhomogeneity of brittle material is considered;considered;

•• The stress analysis is achieved with finite The stress analysis is achieved with finite element program;element program;

•• An An mesomeso scale elastic damagescale elastic damage--based constitutive based constitutive law is proposed for elements.law is proposed for elements.



In RFPA, the macroscopic material behaviour is In RFPA, the macroscopic material behaviour is analyzed at the analyzed at the mesomeso--level, in which the information level, in which the information from the from the mesomeso--scale of the material is incorporated scale of the material is incorporated into the numerical model. Input data for the analysis into the numerical model. Input data for the analysis include the strength and stiffness of the constituents include the strength and stiffness of the constituents of the material. Such properties must be determined of the material. Such properties must be determined from other lower level properties of material. from other lower level properties of material.


Description of the Heterogeneity of Description of the Heterogeneity of Material Properties of RockMaterial Properties of Rock

The rock is composed of many elements with same size, and The rock is composed of many elements with same size, and mechanical parameters (such as strength) of elements is assignedmechanical parameters (such as strength) of elements is assigned according to Weibull distribution:according to Weibull distribution:

( )mm

uu

uu

umuf ⎟⎟

⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛=

−

0

1

00

exp

σσ00 : the mean of strength: the mean of strength

m: a shape parameterm: a shape parameter

With increase of m, the With increase of m, the distribution becomes more distribution becomes more concentrated.concentrated. 0

0.002

0.004

0.006

0.008

0.01

0 100 200 300 400 500

m=1m=1.1m=1.5m=2m=3m=4m=5

( )σP

σ

0σ


————————————————— —————————————————————————————————————————————————————————————————————————————————————


m = 1.5 m = 5.0 m = 10.0

Numerical specimens with elements Numerical specimens with elements distributed according to Weibull distributiondistributed according to Weibull distribution

The grey degree in the specimen indicates the relative The grey degree in the specimen indicates the relative magnitude of strength of elements.magnitude of strength of elements.

The numerical specimens become more homogeneous The numerical specimens become more homogeneous with the increase of Weibull parameter m.with the increase of Weibull parameter m.


Elastic damageElastic damage--based based constitutive law of elementsconstitutive law of elements

εt0

-

ft0

-

ftr

εtu

σ

εc0 ε

fc0

fcr

Constitutive law of element Constitutive law of element (compressive stress is positive)(compressive stress is positive)

MohrMohr--Coulomb Coulomb criterion is met.criterion is met.

Maximum Maximum tensile strain tensile strain

criterion is met.criterion is met.


RFPA modeling procedureRFPA modeling procedure

1.1. First step of the calculation involves the calculation of the stFirst step of the calculation involves the calculation of the stresses resses acting on the elements. Under a quasiacting on the elements. Under a quasi--statically increasing external statically increasing external displacement or force the stress or strain of the elements are gdisplacement or force the stress or strain of the elements are given iven by the solution of the FEM for mechanical equilibrium at each FEby the solution of the FEM for mechanical equilibrium at each FEM M node. node.

2.2. Determining the mechanical property change of the damaged Determining the mechanical property change of the damaged elements according to the constitutive laws and strength criterielements according to the constitutive laws and strength criterion on described above. If the stress of an element attains its prescridescribed above. If the stress of an element attains its prescribed bed breakdown strength, the element fails irreversibly, and its elasbreakdown strength, the element fails irreversibly, and its elastic tic constant is changed according to its postconstant is changed according to its post--failure law. failure law.

3.3. Additional relaxation steps, in which the new equilibrium state Additional relaxation steps, in which the new equilibrium state are are calculated; these steps may lead to the failure of additional calculated; these steps may lead to the failure of additional elements. elements.

4.4. Iterating the procedure leads to fracture propagation. Iterating the procedure leads to fracture propagation.


Experimental and Experimental and numerical modeling numerical modeling of mining induced of mining induced strata failure and strata failure and movement. movement.


Experimental and Experimental and numerical modeling numerical modeling of mining induced of mining induced strata failure and strata failure and movement. movement.



a) 56% peak load b) 65% peak load c) 92% peak load d) 98% peak load e) 100% peak load

Post Peak Behaviour

f) 96% peak load g) 92% peak load h) 78% peak load I) 75% peak load j) 37% peak load

56% peak stress 65% peak stress 92% peak stress 98% peak stress 100% peak stress

96% peak stress 92% peak stress 78% peak stress 75% peak stress 37% peak stress


0

5

10

15

20

25

30

0 20 40 60 80 100 120 140

Loading Rate (0.002 mm / step)

AE R

ate

(cou

nts)

ab

c d e fg h i

j

0

50

100

150

200

250

300

350

400

0 20 40 60 80 100 120 140

Loading Rate (0.002 mm / step)

AE

Rat

e (c

ount

s)

0

500

1000

1500

2000

2500

3000

3500

4000

AE

Acc

umul

atio

n(c

ount

s)

a b c

.

d

ef

g

h

i

j

00.00050.001

0.00150.002

0.00250.003

0.00350.004

0.0045

0 20 40 60 80 100 120 140

Loading Step

AE

Ener

gy (J

)

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

AE

Ener

gyA

ccum

ulat

ion

(J)

a b c

.

d

ef

g

h

i

j


d

σc σt

σt

Clampingaction

Tensile planeOf failure

Idealized deformation, specimen-platen interaction, stress states in the specimen, and failure modes within the specimen: (a) the ratio of platen modulus to specimen modulus Ep/Es>1 (stiff); and (2) the ratio of platen modulus to specimen modulus

Ep/Es<1 (soft)

Effect of constraintEffect of constraint


Ep/Es=0 Ep/Es=0.1 Ep/Es=1 Ep/Es=2 Ep/Es=10

Effect of end constraintEffect of end constraint


Numerically obtained results for specimens with different loadingplatens in terms of Young’s modulus.

0

5

10

15

20

25

30

0 0.5 1 1.5 2Strain (0.0001)

Stre

ss (M

Pa) Ep/Es=10

Ep/Es=2

Ep/Es=1

Ep/Es=0.1

Ep/Es=0

Simulated stressSimulated stress--strain curves for specimens with different strain curves for specimens with different loading platens in terms of Youngloading platens in terms of Young’’s moduluss modulus


Effect of end constraintEffect of end constraint

H/W=3 H/W=2 H/W=1 H/W=0.5

Effect of SlendernessEffect of Slenderness

Numerically obtained results for specimens with different ratio of height to width


05

10152025303540

0 0.5 1 1.5 2Strain (0.0001)

Stre

ss (M

Pa) H/W=3

H/W=1.5

H/W=1

H/W=0.67

H/W=0.5

Simulated stressSimulated stress--strain curves for specimens with different strain curves for specimens with different shape in terms of the ratio of height to widthshape in terms of the ratio of height to width



25

27

29

31

33

35

0 0.5 1 1.5 2 2.5 3 3.5

Ratio of height to width

Stre

ngth

(MPa

)

Simulated strength reduction with specimen size for specimens with different size in terms of the specimen height or width



L×W=30×20 mm2

L×W=100×67 mm2

L×W=120×80 mm2

L×W=150×100 mm2

L×W=190×127 mm2

Numerical simulation on size effect for five specimens with different size but with the same ratio of height to width: failure modes

Effect of specimen sizeEffect of specimen size


23

25

27

29

31

33

35

37

0 10000 20000 30000

Specimen size (mm )

Stre

ngth

(MPa

)

2

Simulated strength reduction with specimen size for specimens with different size

Effect of specimen sizeEffect of specimen size


Brazilian TestsBrazilian Tests


Brazilian TestsBrazilian Tests


σ3 =0 σ3 =12 σ3 =24

Stress-strain curves of specimens Simulated failure envelope of model specimens

0

10

20

30

40

50

60

70

0 0.05 0.1 0.15 0.2 0.25 0.3

ε1/%

σ1/MPa

0

2

4

8

16

Failure envelope

Confinement Confinement and shearand shear


Wing-crack propagation

Influence of heterogeneity on failure modeInfluence of heterogeneity on failure mode


m=1.5

m=3

m=5

Influence of heterogeneity on failure modeInfluence of heterogeneity on failure mode


0

50

100

150

200

250

300

m=1.5m=3m=6

Influence of heterogeneity on failure strengthInfluence of heterogeneity on failure strength

0

10

20

30

40

50

60

70

80

0 0.5 1 1.5 2

Strain (0.0001)

Stre

ss (M

Pa) m=1.1

m=1.5

m=2

m=3

m=5

Influence of material heterogeneity on the stressInfluence of material heterogeneity on the stress--strain curves for strain curves for specimens with different homogeneity indicesspecimens with different homogeneity indices


Influence of heterogeneity on failure event patternInfluence of heterogeneity on failure event pattern

M=5

M=2


m=1.5m=1.5 m=2m=2

m=3m=3 m=5m=5

Precursory microPrecursory micro--fractures prior to main failure fractures prior to main failure



Influence of anisotropic feature on failure modeInfluence of anisotropic feature on failure mode







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Lecture4(RockMassProperties)