4. rock strength and deformabilityocw.snu.ac.kr/sites/default/files/note/4448.pdfcompressive...
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4. Rock strength and deformability
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4.1 Introduction
• Behavior of excavation in rock mass depends on relative spacing,orientation and strength of discontinuities and stress level.
Fig. 4.1
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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)).
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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= − ≤
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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)
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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
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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.
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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).
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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)
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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.
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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).
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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= + + − = −
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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
+ + − − = = +− +
+ +→ = = +
−
= =− +
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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
≥
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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.
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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= =
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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
= + +
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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
= − −
= − −
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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.
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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.
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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.
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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
=
′ −= =′ +
−=
+
= +
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4.7 Shear behavior of discontinuities
– Considering the shearing off of asperities
– Shear strength envelope of anisotropic rocks: Fig.4.43
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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)
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4.7 Shear behavior of discontinuities
– Normal stress-displacement & shear displacement-dilatancy-shear stress
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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
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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.
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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)
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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
= = +
= =
= +