Peng S., Zhang J., Engineering Geology for Underground Rocks

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4.4 Triaxial compressive test 85

increases, as shown in Fig. 4.10. It should be noted that the elastic/Young’s

modulus is directly related to the slope of each stress-strain curve prior to

the peak load. The experimental results indicate that the elastic modulus

and confining pressure have a non-linear relationship for each lithology

(Meng et al. 2006). It can be expressed as follows:

031

bbbE 

(4.14)

where E is the rock elastic modulus in GPa;

is the confining pressure in

MPa; b

, b

, and b

are the parameters depended on lithology. For different

lithologies, the parameters in Eq. 4.14 differ (Table 4.2). Therefore, the

elastic modulus and confining pressure in the rocks in the Huainan coal-

field in eastern China have a pronounced power relation, which best fits

the following equation proposed by Brown et al. (1989), with the exponent

of c = 2:

baE )(



(4.15)

where a, b, and care the constants.

Fig. 4.9. Servo-controlled triaxial compressive test apparatus of MTS Systems in

China University of Mining and Technology.

86 4 Rock strength experiments and failure criteria

Table 4.2. The parameters for different lithologies in Eq. 4.14

Lithology b

(MPa)

Medium-, and fine-

grained sandstone

0.016 -0.121 38.113

d50

Sandy mudstone

0.040 -0.678 26.909

d40

Mudstone

0.006 0.254 28.876

d50

100

120

140

160

0. 0 0. 4 0. 8 1. 2 1. 6

İ (%)

-ı

(MPa)

ı3=0MPa

ı3=5MPa

ı3=10MPa

ı3=15MPa

Fig. 4.10. Complete stress-strain curves for different confining pressures under the

triaxial tests in a medium-grained sandstone.

Applying Mohr-Coulomb strength criterion in the triaxial test results

(refer to Chap. 4, Sec. 4.6), the Mohr circle and rock failure envelope can

be obtained. The Mohr-Coulomb strength criterion indicates that the nor-

mal stress (

), shear stress (

), and principal stresses (

and

) have the

following relationship for dry rocks:







(4.16)

4.4 Triaxial compressive test 87

Obviously, it is a circle in the (

) plane with the center at (



, 0)

and the radius of



. This circle is called Mohr circle as shown in

Fig. 4.11 and can be used to determine rock failure envelope, cohesive

strength (c) and internal friction angle (

) (refer to Sec. 4.6.3 for more de-

tails).

MVW

tan c

, 0)

(





Fig. 4.11. Mohr circle and strength envelope for dry materials.

Figures 4.12 - 4.15 present the Mohr circles and failure envelopes from

series triaxial compressive tests for different rocks sampled from Eastern

China’s Permian coal measures (Peng and Wang 2001). From the tests

rock cohesion and internal angle of friction can be obtained from the Mohr

circles and strength envelopes. Theses experimental results show that the

mudstone and sandy shale have the cohesions ranged from 10.7 to 11.6

MPa and the internal friction angle varied from 30.7 to 40.3q. The cohe-

sions in sandstones ranges from 16.5 to 22.1 MPa; the internal friction an-

gle ranges from 40.6 to 46.23q. It can be seen that mudstone and shale

have much lower cohesion than that in sandstones. Therefore, mudstones

are normally most likely to have compressive/shear failure than sand-

stones.

88 4 Rock strength experiments and failure criteria

Fig. 4.12. Mohr circles and strength envelope from triaxial tests in mudstone.

Fig. 4.13. Mohr circles and strength envelope from triaxial tests in sandy shale.

4.4 Triaxial compressive test 89

Fig. 4.14. Mohr circles and strength envelope from triaxial tests in fine-grained

sandstone.

Fig. 4.15. Mohr circles and strength envelope from triaxial tests in medium-

grained sandstone.

90 4 Rock strength experiments and failure criteria

4.5 Polyaxial compressive test

Mogi (1971) constructed perhaps the first apparatus that enabled the appli-

cation of three independent and mutually perpendicular uniform loads to

the faces of a rectangular prismatic specimen with minimum friction. He

subjected Dunham dolomite and other rocks to different intermediate prin-

cipal compressive stresses for the same least principal stress, and then

raised the maximum principal stress to failure.

The polyaxial cells built by Mogi (1971) and Spetzler et al. (1981) were

mainly suited for testing deformational and strength characteristics of

weak to medium strength rocks. The University of Wisconsin designed and

fabricated a true triaxial testing system suitable for testing both weak and

strong rocks (Haimson and Chang 2000). The true triaxial tests can better

describe the stress-strain relationship for rocks situated in a three dimen-

sional stress domain.

Polyaxial compressive tests demonstrate experimentally that rock

strength is a function of the major principal stress (

) and the minor prin-

cipal stress (

) as well as the intermediate stress (

). Therefore, rock

failure characteristic depends on the effects of all three principal stresses.

4.6 Rock failure criteria

4.6.1 Rock failure types

Rocks fail when the surrounding stress exceeds the tensile, the compres-

sive, or the shear strengths of the rock formation, whichever is reached

first. There are several failure types depended on rock lithology, rock mi-

crostructures, and applied stresses. Jaeger and Cook (1979) described rock

failures at various confining pressures, as shown in Fig. 4.16.

In unconfined compression (Fig. 4.16a), irregular longitudinal splitting

is observed. With a moderate amount of confining pressures, the rock fail-

ure is characterized by a single plane of fracture, inclined at an angle of

less than 45q to the direction of

, as shown in Fig. 4.16b and Fig. 4.17

(Peng and Meng 2002). This is a typical shear failure under compressive

stresses, and it generates a shear displacement along the surface of the

shear fracture. If the confining pressure is increased so that the material

becomes fully ductile (Jaeger and Cook 1979), a network of shear fractures

accompanied by plastic deformation appears, as shown in Fig. 4.16c. The

4.6 Rock failure criteria 91

second basic type of failure is tensile failure, which appears typically in

uniaxial tension. Its characteristic feature is a clean separation with no off-

set between the surfaces (Fig. 4.16d). If a slab is compressed between line

loads as shown in Fig.4.16e, a tensile fracture appears between the loads.

(a) (b) (c)

(d) (e)

(a) (b) (c)

(d) (e)

Fig. 4.16. Rock failure types. a. splitting; b. shear failure; c. multiple shear frac-

tures; d. tensile failure; e. tensile failure induced by point loads.

Fig. 4.17. Shear failure induced by the axial compressive loading in limestone

samples.

There are various failure/strength criteria applied to compare rock

stresses and strength to describe rock failures. For the porous media the ef-

fective stress concentrations near underground engineering due to the far-

field stresses need to be considered for failure initiation. It is commonly

accepted that rock failure is controlled by Terzaghi’s effective stress con-

92 4 Rock strength experiments and failure criteria

cept, because Biot’s effective stress coefficient approaches 1.0 when rock

failure is approached. Terzaghi’s effective stress can be expressed as fol-

lows:

ijijij



(4.17)

where

is the effective stress tensor, p is the pore pressure and

is the

Kronecker delta, and

= 1 when i = j;

= 0 when i  j.

Various failure criteria, such as tensile, Mohr-Coulomb, and Drucker-

Prager, were presented for different types of underground engineering.

4.6.2 Tensile failure and Griffith criterion

Tensile failure will occur when the effective stress becomes tensile and

equals or exceeds the formation tensile strength, that is:

T

(4.18)

where T

is the tensile strength;

is the minimum effective principal

tensile stresses (note that

is negative).

Griffith (1921) proposed that fractures are caused by stress concentra-

tion at the tips of minute Griffith cracks supposed to pervade the material.

A fracture is initiated when the maximum stress near the tip of most fa-

vorably oriented crack reaches a value characteristic of the material. The

Griffith failure takes place if the following equation exists for a dry

material:

!

)(8)(

310

VVVV

  T

(4.19)



T

(4.20)

When

= 0, it is uniaxial compression, so that the uniaxial compres-

sive strength (UCS) predicted by Eq. 4.19 (UCS =

) is:

8TUCS

(4.21)

It should be noted that for some rocks, Eq. 4.21 underestimates the

uniaxial compressive strength. Modified Griffith theory (McClintock and

Walsh 1962) suggested that the uniaxial compressive strength and tensile

strength have the following relationship:

4.6 Rock failure criteria 93

sin1

)sin1(4

TUCS





(4.22)

where

is the internal friction angle of the material.

Another extension of the Griffith criterion gives that the uniaxial com-

pressive strength is 12 times of the tensile strength (Murrell 1963):

12TUCS

(4.23)

4.6.3 Mohr-Coulomb criterion

Shear failure occurs when the shear strength of the formation is exceeded.

The Mohr-Coulomb failure criterion can be used to determine such a fail-

ure mechanism for a porous medium; i.e.:

tan

 c

(4.24)

where

is the normal effective stress;

is the shear stress;

is the angle

of internal friction and c is the cohesion and also called inherent shear

strength of the material.

In the principal space, (

), the Mohr-Coulomb failure criterion

can be expressed as:



qUCS

(4.25)

where

are the maximum and minimum effective principal stresses,

respectively, and

)sin1()sin1(

 q

(4.26)

)sin1(cos2



cUCS

(4.27)

Figure 4.18 shows the Mohr circle and Mohr-Coulomb strength enve-

lope. The failure will not take place if the values of

and

lie below the

strength envelope. In the Mohr-Coulomb criterion, two conclusions should

be noted: (i) the intermediate principal stress

does not affect failure and

this may overestimate failure; (ii) the plane of shear fracture passes

through the direction of the intermediate stress. This conclusion can be ap-

plied to determine the direction of the in-situ intermediate stress, when the

stress regime is known.

94 4 Rock strength experiments and failure criteria

(a)

MVW

tan

 c

UCS

(b)



(a)

MVW

tan

 c

UCS

(b)



Fig.4.18. Mohr-Coulomb strength in effective stress domain. a. shear failure

plane under triaxial compression, and in the shear plane,

tan12tan 

; b.

Mohr circles and strength criterion.

4.6.4 Drüker-Prager criterion

The effect of the intermediate principal stress is not considered in any of

the failure criteria mentioned above. Laboratory data have shown that the

intermediate principal stress plays an important role in the failure of rocks

and can be conveniently represented by the Drüker-Prager yield condition

(Drüker and Prager 1952); that is:



(4.28)

where

and

are material constants; and, I

and J

are the stress invari-

ants, which can be expressed as:

zyx



(4.29)

222222

])()()[(

zxyzxyxzzyyx

WWWVVVVVV



(4.30)

The Drüker-Prager strength criterion is an extended form of the Mohr-

Coulomb failure criterion. In the principal stress space, a section in the

deviatoric plane shows that it is possible in various ways to relate the two

failure criteria (Fig. 4.19). For plane-strain conditions it is recommended

that the inside cone tangent to the pyramid is used (Kovari 1977). In this

case the conversion formulae for the material parameters from the Mohr-

Coulomb criterion to the Drüker-Prager criterion are (Zhang 2002):