4. Rock strength and deformability

4.1 Introduction

• Behavior of excavation in rock mass depends on relative spacing,orientation and strength of discontinuities and stress level.

Fig. 4.1

4.2 Concepts and definitions

– Fracture: formation of planes of separation in rock material– (Peak) Strength: the maximum stress that rock can sustain (point B) – Residual strength: stress that a damaged rock can still carry (point C)– Brittle fracture: process by which sudden loss of strength occurs following

little or no permanent (plastic) deformation. It is associated with strain-softening or strain-weakening ((a)).

– Ductile deformation: deformation that occurs when the rock can sustain further permanent deformation without losing load-carrying capacity ((b)).

4.2 Concepts and definitions

– Yield: the point where a permanent deformation begins (point A).

– Failure: a state/point where a rock no longer adequately supports the load

on it or where an excessive deformation takes place.

– Effective stress: a stress which governs the gross mechanical response of a

porous material. It is a function of applied stress and pore

(water) pressure.

( )' 1, : pore pressureij ij iju uσ σ α δ α= − ≤

4.3 Behavior of isotropic rock material in uniaxial compression

• Influence of rock type and condition– σc is one of the most fundamental property of a rock.– Rock type gives some qualitative indication of its mechanical behavior: a slate

shows an anisotropic behavior by cleavage in it; a quartzite is generally strong and brittle.

– σc depends on nature and composition of rock as well as test condition:σc decreases with increasing porosity, degree of weathering, degree of microfissuring.

• Standard test procedure and interpretation– Refer to the suggestion by ISRM Commission on Standardization of Laboratory

and Field Tests (1979). (p88 of textbook)

4.3 Behavior of isotropic rock material in uniaxial compression

– Four stages of stress-strain response in uniaxial compression: crack closure-elastic deformation-stable crack propagation-unstable/irrecoverable deformation (~peak strength)

– Young’s modulus: tangent -, average -, secant -.– Volumetric strain: εv = εa + 2εr

4.3 Behavior of isotropic rock material in uniaxial compression

• End effects and influence of height to diameter ratio– Different lateral deformation of loading platen and specimen causes lateral

restriction at both ends of the specimen (Fig.4.4).

– This effect is subject to H/D ratio of the specimen (Fig.4.5).

– Brush platen consisting of 3.2 mm2 steel pins is effective to prevent the end effect, but it is too difficult to be prepared and applied in routine testing.

– Inserting a sheet of soft material or applying a lubricant cause tensile stress at the ends.

– ISRM Commission (1979) recommends treatments of the sample ends except by machining to be avoided.

4.3 Behavior of isotropic rock material in uniaxial compression

• Influence of testing machine stiffness– Machine pillars are extended while a specimen is compressed (Fig.4.6).

– Test machine should be stiffer than the specimen to observe the post-peak stress-strain behavior (Fig.4.7).

– For brittle rocks of which stiffness is higher than that of a test machine, servo-controlled equipment is required to record the post-peak behavior of the specimen.

– Strain (displacement) rate is serve-controlled for the post-peak behavior (Fig.4.9).

4.3 Behavior of isotropic rock material in uniaxial compression

• Influence of loading and unloading cycles (Fig.4.13)– Irrecoverable displacement increases as loading-unloading proceeds in the post-

peak region.

– Apparent modulus of rock decreases as loading-unloading proceeds in the post-peak region.

– Few cycles of loading-unloading are recommended in pre-peak range because

they show permanent deformation.

• Point load test (Fig.4.14)– It is carried out when only an approximate measure of peak strength is required.

– Is = P/De2 (=πP/4A) is an Uncorrected Point Load Index which depends on De.

– Is(50) = Is× (De /50)0.45 is the size-corrected Point Load Strength Index.

– σc = (22~24) Is(50)

4.4 Behavior of isotropic rock material in multiaxial compression

• Types of multiaxial compression test– Biaxial: σ1 ≥ σ2 , σ3 = 0.– Triaxial: σ1 > σ2 = σ3.– Polyaxial: σ1 > σ2 > σ3.

• Biaxial compression (σ1 ≥ σ2 , σ3 = 0)– The effect of intermediate principal stress can be neglected so that the uniaxial

compressive strength should be used as the rock strength whenever σ3 = 0(Fig.4.15).

• Triaxial compression (σ1 > σ2 = σ3)– Deformation behavior of rock becomes close to ductile as σ3 increases (Fig.4.18,

19).– Volumetric strain decreases until peak-stress and increases after the peak-stress

under relatively lower confining pressure (Fig.4.18).– Brittle-ductile transition pressure of granite and quartzite is over 1 GPa.

4.4 Behavior of isotropic rock material in multiaxial compression

– Pore pressure makes the rock under confining pressure close to brittle as itincreases (Fig.4.20).

– The classical effective stress law by Terzaghi is not well applied to low permeable rocks nor to loading condition with high strain rate.

• Polyaxial compression (σ1 > σ2 > σ3)– End effect is a main obstacle of the test as in the biaxial test.– σ2 influences the test result, but it is not as great as σ3.

• Influence of stress path– Strength or strain of rock is stress path-independent (Fig.4.21).

4.5 Strength criteria for isotropic rock material

• Types of strength criterion– Pore pressure and σ2 generally affect little on rock failure: σ1 = f(σ3)– Criterion on a particular plane: τ = f(σn).

• Coulomb’s shear strength criterion– s = c + σn tanФ (Ф is an internal friction angle)

( ) ( ) ( )1 3 1 3 1 31 1 1cos 2 , sin 22 2 2nσ σ σ σ σ β τ σ σ β= + + − = −

4.5 Strength criteria for isotropic rock material

( )( )

( )

31

31 3

2 sin 2 tan 1 cos 2 ,

sin 2 tan 1 cos 2 4 2

2 cos 1 sintan

1 sin2 cos 2 cos,1 sin 1 sin

c

c T

c

c

c c

σ β φ β π φσ ββ φ β

φ σ φσ σ σ ψ

φφ φσ σφ φ

+ + − − = = +− +

+ +→ = = +

−

= =− +

4.5 Strength criteria for isotropic rock material

– Discrepancy between reality and Coulomb’s criterion:1) Major fractures in failure are not always based on shear failure.2) Predicted direction of shear failure does not always agree with

experimental observations.3) Experimental failure envelopes are generally non-linear.

• Griffith crack theory– Energy instability concept on crack extension: A crack will extend only when

the total potential energy of the system of applied forces and material decreases or remains constant with an increase in crack length.

2

: tensile stress normal to the crack : surface energy per unit area of the crack having an initial length of 2c

Ecασ

πσα

≥

4.5 Strength criteria for isotropic rock material

- The classical Griffith criterion does not provide a very good model for the peak strength of rock → a number of modification to Griffith’s solution were introduced.

4.5 Strength criteria for isotropic rock material

• Fracture mechanics– Basic modes of distortion: I (extension, opening), II (in-plane shear) and III

(out-of-plane shear)

– Stress intensity factor: a factor to predict the stress state near the tip of a crackcaused by a far field stress.

– Fracture toughness: a property which describes the ability of a material containing a crack to resist fracture = critical stress intensity factors (stress intensity factor at crack extension (KIC, KIIC , KIIIC )

, 2I z IK c K xσ π σ π= =

4.5 Strength criteria for isotropic rock material

• Empirical criteria– Bieniawski (1974)’s criterion

– Hoek & Brown (1980)’s criterion (Fig.4.30, 4.31 & Table 4.2)

31 1 , 0.1k c

m m

c c c c

A Bσ τ σσσ σ σ σ

= + = +

3 31 , varies with rock type and = 1 for intact rock.c c c

m s m sσ σσσ σ σ

= + +

4.6 Strength of anisotropic rock material in triaxial compression

• Peak strength of transversely isotropic rock– Max. differential stress vs. inclination angle (α): Fig. 4.33 – Simplified theoretical approach: Fig. 4.34(b)

( ) ( )( )

( ) ( )

31 3

1 3

2 tan (when a weak plane fails)

1 tan cot sin 2

0 90 , 1 tan cot 0

w ws

w

ow ws

c σ φσ σ

φ β β

σ σ β β φ φ β

+− =

−

− > → ≠ > − >∵

– Plateau in Fig. 4.34(b) exists because the role of weak planes in failure is notconsidered when the failure occurs out of the weak planes.

– Functions of cw and tanφw have been proposed to correct the plateau:

( )

( )0

0

cos2

tan cos2

nw c

m

w

c A B

C D φ

α α

φ α α

= − −

= − −

4.7 Shear behavior of discontinuities

• Shear testing– Direct shear tests with shear force (a) parallel or (b) inclined (10° ~15°) to the

discontinuity: the latter is not proper to the test with very low normal stress.

4.7 Shear behavior of discontinuities

– A core specimen with a discontinuity in a triaxial cell can be used for the test– Stage testing has been devised to overcome the difficulty in preparing several

specimens containing similar discontinuities: disc seats lubricated with a molybdenum disulphide grease are recommended.

4.7 Shear behavior of discontinuities

• Influence of surface roughness on shear strength– Shear strength envelope of a smooth discontinuity surface: Fig.4.38

tanns σ φ=

– Shear strength envelope of a rough joint : Fig.4.39, Fig.4.40.

4.7 Shear behavior of discontinuities

• Influence of surface roughness on shear strength– Shear strength envelope of a sawtooth-shaped joint

( )

tan : a flat joint

cos sin tan : an inclined jointcos sin

cos sin tancos sin

tan

SNS S i N iN N i S iS N i i

i S N iS iN

φ

φ

φ

φ

=

′ −= =′ +

−=

+

= +

4.7 Shear behavior of discontinuities

– Considering the shearing off of asperities

– Shear strength envelope of anisotropic rocks: Fig.4.43

4.7 Shear behavior of discontinuities

• Interrelation between dilatancy and shear strength– Constant normal stress (controlled normal force) vs. constant normal strain

(controlled normal displacement)

4.7 Shear behavior of discontinuities

– Normal stress-displacement & shear displacement-dilatancy-shear stress

4.7 Shear behavior of discontinuities

• Influence of scale

[ ]10

10

tan log tan

log : net roughness component

: geometrical component ( as scale )

: asperity failure component ( as scale )

n r n rn

n

n

JCSJRC i

JCSJRC

JRCJCS

τ σ φ σ φσ

σ

σ

= + → +

↓ ↑

↓ ↑

– Refer to Fig.4.46

• Infilled discontinuities– Filling materials which are soft and weak decrease both stiffness and shearstrength

4.8 Models of discontinuity strength and deformation

tanncτ σ φ= +

• Coulomb friction, linear deformation model– Appropriate for smooth discontinuities such as faults at residual strength

(Fig.4.47)

• Barton-Bandis model– Normal stiffness of a joint is highly dependent on normal stress (Fig.4.48)– Contribution of roughness to shear strength decreases during post-peak shearing

due to mismatch and wear.

10tan log n rn

JCSJRCτ σ φσ

= +

• Continuous-yielding joint model– Plastic shear displacement causes reduction of mobilized friction angle.– Normal and shear stiffnesses of a joint are function of normal stress.

4.9 Behavior of discontinuous rock masses

( )

( )

( )

23 3

/15 20 / 3

100exp28 14

100 exp D = 0 for undisturbed, D = 1 for very disturbed rock9 3

0.5 exp exp 6 ( 0.5 fo

a

n b c c

b i

GSI

m s

GSIwhere m mD

GSIs in situD

a a

σ σ σ σ σ

− −

′ ′ ′= + +

− = − − = −

= + − ≈

1 3

r 50, 0.65 for very low )

, ( from )acm c tm c b tm

GSI a GSI

s s mσ σ σ σ σ σ σ

> →

′ ′= = − = =

• Strength– Overall strength of a multiply jointed rock mass: almost isotropic (Fig.4.49)

• Generalized Hoek-Brown peak strength criterion for rock mass

– Applicable to short-term peak strength criterion of sensibly isotropic rock masses (Fig.4.51)

4.9 Behavior of discontinuous rock masses

( )

( )( )

1040

1/ 3

1040

10

2 100, 10

10100

1 / 2 10100

3.5 log

RMR

M M

cM c c

GSIc

M

p c

E RMR E

E Q Q Q

E D

V Q

σ

σ

−

−

= − =

= =

= −

≈ +

• Deformability– Elastic constants of transversely isotropic rock mass:

– Deformation modulus estimated from rock classification indices (Fig.4.53)

( )12

21 2

2

1 1 1, : spacing

,

1 1 1

n

s

E E SE E K S

EE

G G K S

ν ν ν ν

= = +

= =

= +