Fujii, Y. (2012), Weakness Plane Model to Simulate Effects of Stress States on Rock Strengths, True Triaxial Testing of Rocks(Proc. TTT Workshop, Beijing, Oct, 17, 2011), Kwasniewski, M., Li, X. and Takahashi, M. (Eds.), pp. 243-248, CRC
Press/Balkema, Leiden, The Netherlands.

Weakness plane model to simulate effects of stress states on rockstrengths
Y. Fujii

Hokkaido University, Sapporo, Japan

ABSTRACT: The "weakness plane model" was developed to simulate the effects of various
stress states on the strength of rocks. The model assumes numerous planes of weakness whose
directions are uniformly distributed in a rock; each plane slips or opens based on the Coulomb
criterion with a tension cut-off; and the rock is regarded as failed when the ratio of the failed
plane number to that of all planes reaches a certain value. The same size and strength parameters
were assigned to all planes and no complicated statistical functions are used. The model has been
applied to true triaxial compression, uniaxial tension, Brazilian and extension tests. The model isvery simple but the effects of a stress state are simulated very well. Stress and plane failure
distributions on the Schmidt net has increased the understanding of the mechanism of stress state
effects on rock strength.
1 INTRODUCTION
It is well known that the strength of a rock first increases and then decreases as intermediate
principal stress increases in a true triaxial test (e.g. You, 2009). This behavior would mainly be
due to some effects of intermediate principal stress on microcracks; however, the mechanism of
the strength behavior has not yet been well explained. Koide et al. (1986) developed a statisticalmodel for the failure criterion of true triaxial compression based on the Griffith theory. Thismodel can roughly approximate the effects of intermediate principal stress in true triaxialcompression. However, the model is complicated and applications to other tests such as uniaxial
tension, Brazilian and extension tests have not yet been shown. A simple 2-D model for
simulating the effects of a stress state on rock failure was developed by Fujii & Uehara (2006).
The model successfully simulated the effects of confining pressure on the axial stress value atfailure in extension tests. The model was expanded to 3-D and is called the "weakness planemodel" in this paper. it was applied not only to extension tests but also to true triaxial, uniaxial
compression, uniaxial tension and Brazilian tests. The various effects of stress states on rockstrengths will be explained based on the results of the simulations.
The objective of this research was not to precisely approximate the effects of a stress state on
specific rocks as Koide et al. (1986) or You (2009) did for true triaxial tests but rather to develop
a simple model and to investigate the general mechanisms of the effects of stress states on rockstrengths using the model.
2 WEAKNESS PLANE MODEL
Assumptions are as follows: (1) There are numerous (20000 ) planes of weakness whose
directions are uniformly distributed in rock. (2) Each plane slips or opens based on the Coulomb
criterion with a tension cut-off. Single values of cohesion (C), friction angle ( ) and tensile

strength (T0) are assigned to all planes. (3) A rock is regarded as failed when the ratio of thefailed plane number to that of all planes reaches a certain value.
Directions of 2500 normals of planes of weakness were regularly distributed on a quarter
Schmidt net to assure uniform distribution so that the failure of each plane had the same weighton rock failure. The latitude and altitude of the normal can be calculated asy(1)tan 1xandwhere2 ,(2)cos 1r
2(3)

x2

y2 .

(4)

2andr

Normal and shear stresses for each plane under given principal stresses were calculated from thedirection of the normal for each plane. Maximum shear stress on each plane was calculated fromthe shear stresses on the local coordinate. Failure of each plane was determined based on theCoulomb criterion with a tension cut-off. The number of failed planes was counted regardless ofwhether it had slipped or opened and the number was divided by the total plane number to obtainthe failure ratio (FR). The rock was determined to be failed when the failure ratio reached acritical failure ratio (FRC).
A uniaxial compression test was simulated as an example. The cohesion of the planes ofweakness was set at 10 MPa. The failure ratio increased as the axial stress increased (Fig. 1).
The axial stress value at the beginning of the failure ratio increase increased as the angle ofinternal friction increased. The slope of the failure ratio became less steep as the friction angle
increased. The increase in the slipped plane number with axial stress can be seen in Figure 2.
o0.15= 20B

Failure ratio

30oo

40

0.10.05
A060

80 100 120 140 160
Axial stress (MPa)Figure 1. Simulated failure ratio using the weakness plane model. The ratio increases from certain axialstress levels according to the friction angle. The slopes of the failure ratios also depend on the friction
angle.

S
23(a) Lower axial stress(b) Higher axial stress (c) Directions of principal stresses

Figure 2. Directions of failed planes in a simulated uniaxial compression test using the weakness planemodel on the Schmidt net for = 30 . Axial stress is oriented in the direction normal to the diagram.
The number of slipped (S) planes increased from the lower axial stress (a), A in Figure 1, to the higheraxial stress (b), B in Figure 1.

3 TRUE TRIAXIAL TEST
Typical true triaxial behavior (Fig. 3a), namely, that peak stress first increased and thendecreased as intermediate stress increased was obtained from the simulation assuming basevalues for T0 , C, and FRC to be 1 MPa, 10 MPa, 30 and 0.1, respectively. The effects ofintermediate principal stress significantly increased as friction angle increased (Fig. 3b). Peakstress at ordinary triaxial compression was unchanged but it increased under moderateintermediate principal stress and extensional stress state as FRC increased (Fig. 3c).
Figure 4 shows the shear stress value on the planes of weakness normalized by the required
shear stress for failure on the Schmidt net. The weakness planes slipped when the shear stressreached 1 as is shown in Figure 4. The donut-like distribution of the direction of the slipped plane
can be observed for the case where intermediate principal stress is equal to the confining pressure(Fig. 4a). The plane failures around the direction of the intermediate principal stress werestrongly restricted when a moderate intermediate principal stress was applied (Fig. 4b). The
increase in strength with low-moderate intermediate principal stress was due to the restriction of
the 1- 3 slips. The distribution of the direction of the slipped plane becomes a donut-like shapeagain under extensional stress state (Fig. 4c) but around the direction of the minimum principal
stress this time. The decrease in strength under high intermediate principal stress was caused by
the enhancement of the 2- 3 slips.
800

C
Peak stress (MPa)Peak stress (MPa)

Extension

300
15200
A
100
01053 = 0 (MPa)
0100

200

300

2 (MPa)(a) Effects of400

400

600
404002000o30= 20o0200

Extension

4006002 (MPa)3o(b) Effects of800

Peak stress (MPa)

B

400

300
0.152000.10
FRC = 0.05100

Extension
00100200300

4002 (MPa)(c) Effects of FRC

Figure 3. Simulated peak stress in true triaxial tests using the plane failure model. Peak stress firstincreases and then decreases with intermediate principal stress, which is typical true triaxial strength
behavior. The effects of intermediate principal stress increase as the friction angle increases (b). Peakstress at ordinary triaxial compression is unchanged but it increases under moderate intermediate
principal stress and extensional stress state as FRC increases (c).

S(a) Low

2 (A in Figure 3)

S(b) Moderate

2 (B in Figure 4)

S(c) Extension (C in Figure 4)

Figure 4. Shear stress value on planes of weakness normalized by the required shear stress for failure
plotted on the Schmidt net. Maximum principal stress is oriented in the direction normal to the diagram.
A donut-like distribution can be observed in the direction of the slipped plane for the case whereintermediate principal stress was equal to the confining pressure (a). The failure around the direction of
the intermediate principal stress was strongly restricted when a moderate intermediate principal stress
was applied (b). Distribution of the direction of the slipped plane became a donut-like shape again under
extensional stress state but around the direction of the minimum principal stress this time (c).

4 UNIAXIAL TENSION AND BRAZILIAN TESTS
Base values of T0 , C and were assumed to be 1 MPa, 10 MPa and 30 , respectively. 1 and 2
were set at 0 for uniaxial tension. 2 was set at 0 and 1/ 3 was kept at -3 for Brazilian test. Bothstrengths started from T0 and increased as FRC increased. The rate of increase for the Brazilian
test was larger than that for the uniaxial tension test.
It was realized from the tensile stress distribution (Fig. 6) that the compressive maximum
principal stress in Brazilian tests restricted the opening of the planes whose normals were aroundthe compressive maximum principal stress. This caused an increase in strength for the Brazilian
test. The peak stress in the Brazilian test may depend on cohesion and the friction angle but they
showed no variation for cohesion between 5 MPa and 15 MPa and for a friction angle between
20 and 40 .
Tensile strength (MPa)43Brazilian

210Uniaxial
0T0

0.1

0.2

FRC

Figure 5. Simulated tensile strengths from uniaxial and Brazilian tests. Both strengths start from T0 andincrease as FRC increases. The rate of increase for the Brazilian test is larger than that for uniaxial
tension test.

5 EXTENSION TEST
Extension tests have occasionally been carried out as an extreme stress state of true triaxial test.
Strength under low confining pressure has not, however, been fully clarified although there aremany studies such as Brace (1964), Mogi (1967), Tani (1996), Oikawa & Yamaguchi (2000),Ramsey & Chester (2004), Fujii & Uehara (2006), Takahashi et al. (2009), etc. Strength
behavior in triaxial tension is also unknown.

Large tensile stress(a) Uniaxial tension(b) Brazilian test

Figure 6. Simulated tensile stress value in MPa on the Schmidt net for the uniaxial tension (a) andBrazilian tests (b). Maximum principal stress is oriented in the direction normal to the diagram.
Compressive maximum principal stress in the Brazilian test restricted the opening of planes whose
normals were around the compressive stress (b).

A simulation was carried out using the weakness plane model assuming base values of T0 , C,and FRC to be 1 MPa, 10 MPa, 30 and 0.1, respectively. The peak stresses exhibited bi-linear
functions (Fig. 7). The tensile strength of the rock started at the tensile strength of the weakness
planes at triaxial tension (left end data of Figure 7) regardless of the values of the strengthparameter. Tensile strength increased as confining pressure increased regardless of whether theconfining pressure was compression or not (Region I). Plane failures were all in openings in thisregion (Figure 8a at A in Figure 7a). The increase in tensile strength in this region was due to a
restriction of the opening of the planes caused by an increase in confining pressure. Tensile
strength then began to decrease under moderate confining pressure (Region II). This was due tothe beginning of plane slips (Figure 8b at B in Figure 7a). The minimum principal stress thenbecame positive (compression) and plane failures were only by slip (Region III, Figure 8c at C inFigure 7a).
The slope of the peak stress in Regions II and III decreases as friction angle increases (Fig.
7a). Cross point of Regions I and II moves right-downward as cohesion increases (Fig. 7b). The
failure criterion seems rotated clockwise as critical failure ratio increases (Fig. 7c).

201030

B0CA-10-20oo

400100200

30

20C/T0 = 51010015-10-20

0100

200

3 at failure (MPa)30o= 203 at failure (MPa)3 at failure (MPa)3020
FRC = 0.05

100-10-200.15
00.10

100200Confining pressure (MPa)Confining pressure (MPa)Confining pressure (MPa)(a) Effects of(b) Effect of C/T0(c) Effects of FRC

Figure 7. Simulated minimum principal stress at failure in extension tests. The peak stresses exhibit bilinear functions. The tensile strength of the specimen starts at the tensile strength of the weakness planes
at triaxial tension (left end data) regardless of the values of the strength parameters. The tensile strength
increases as confining pressure increases regardless of whether the confining pressure is compression ornot (Region I). The tensile strength then begins to decreases under moderate confining pressure (RegionII). The minimum principal stress then becomes positive (compression). The slope of the peak stress inRegions II and III decreases as the friction angle increases (a). The cross point of Regions I and II moves
downward to the right as cohesion increases (b). The failure criterion seems to rotate clockwise as
critical failure ratio increases (c).

O(a) A in figure 7aS(b) B in Figure 7aSO(c) C in Figure 7aFigure 8. Failure mode of weakness planes on the Schmidt net. O and S denote the opening and slip,respectively. Plane failures were all in the opening in (a), the opening and slip in (b) and all in the slipin (c).

6 CONCLUDING REMARKS
The "weakness plane model" was developed to simulate the effects of various stress states on thestrength of rocks. The model assumes numerous planes of weakness whose directions areuniformly distributed in a rock; each plane slips or opens based on the Coulomb criterion with atension cut-off; and the rock is regarded as failed when the ratio of the failed plane number to thatof all planes reaches a certain value. The same size and strength parameters were assigned to allplanes and no complicated statistical functions were used. The model was applied to true triaxial
compression, uniaxial tension, Brazilian and extension tests. The model is very simple but theeffects of the stress state were simulated very well. Stress and plane failure distributions on theSchmidt net has increased the understanding of the mechanism of stress state effects on rockstrength.
A precise approximation of the strength behavior of specific test results is possible in thefuture by adapting Hoek and Brown or other nonlinear failure criteria for weakness planes,
weighting the effects of tensile and shear failures of weakness planes on rock failure, assuming a
non-uniform direction distribution for the planes of weakness, etc..
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Fujii, Y. & Uehara, Y. 2006. A study on deformation and behavior of rock subjected to axial extension
under confining pressure. J. MMIJ. 122(6/7): 330-337. (in Japanese)
Koide, H., Takahashi, M., Kinoshita, S., Ishijima, Y. & Nakamura, A. 1986. Effects of Griffith cracks
and inclusions on fracture criteria under a general triaxial state, J. Mat. Sci. Japan 35(392): 14-19.
(in Japanese)
Mogi, K. (1967). Effect of the intermediate principal stress on rock failure. J. Geophys. Res. 72: 51175131.
Oikawa, Y. &. Yamaguchi, T. 2000. Tension test of granite under confining pressure. Proc. MMIJ
Annual Meeting I: 175-176. (in Japanese)
Ramsey, J.M. & Chester, F.M. 2004. Hybrid fracture and the transition from extension fracture to shear
fracture. Nature 428: 63-66.
Takahashi, T., Takahashi, M., Kiyama, T. & Takemura, T. 2009. A triaxial compression and triaxialextension test of Kimachi sandstone. In I. Vrkljan (ed.), Rock Engineering in Difficult Conditions Soft Rocks and Karst: 381-386. Leiden: CRC Press/Balkema.
Tani, K. 1996. Extension failure mechanism of homogeneous soft rock under low confining pressure.
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