Vol. 18 No.3 (322~330)

ACTA SEISMOLOGICA SINICA

May, 2005

Article ID: 1000-9116(2005)03-0322-09

Numerical analysis on the influence of rock specimen size on crack stress field* FU Zhen ( ~ ~) CAI Yong-en (~z'T~,~,) Deparment of Geophysics, School of Earth and Space Sciences, Peking University,Beijing 100871, China Abstract In the simulation o f rupture processes of seismic sources by using either numerical method or rock mechanics ex-

improper setting of the specimen size will influence the stress field near the faults. In this study, 2D finite element method (FEM) was used to calculate the stress field of rock specimens in different sizes with fixed-size elliptic holes. The calculated stress field was compared with analytic solution for elliptic-hole problem in an infinite medium. Numerical results showed that boundary effect of a rock specimen with an elliptic hole on stress field under uniaxial compression cannot be neglected. Critical aspect ratio of the specimen is about 3:2, and critical ratio of distance between the tip of the hole and the border of specimen (d) to the major axis of the elliptic hole (/) is about 7.3. Numerical analysis on rock specimen size can provide theoretical reference for rock specimen experiments, and it is also helpful for setting of model sizes in numerical simulations of fault movement.

periments,

Key words: finite element method; stress field; critical size of rock specimen; boundary effect CLC number: P315.72+7 Document code: A

Introduction Earthquakes are excited by dislocations of rocks, which are generated once the stresses on two sides of rocks accumulate to a certain extent. Numerical simulations and rock mechanics experiments are two important avenues to explore the rupture processes of seismic faults. Several methods are widely used in numerical simulations, such as the boundary element method (BEM), (Das and Aki, 1977; Fukuyama and Madariaga, 1995, 1998), the finite difference method (FDM), (Madariaga, 1976; Olsen et al, 1997) and the finite element method (FEM) (Archuleta and Frazier, 1978; Cai et al, 2000). In rock mechanics, one of the approaches to understand the behavior of seismic faults and seismicity is to investigate the developing process of inner crack of a precast rock specimen with a kerf (Spicak and Lokajicek, 1986; ZHAO et al, 1995; Robina et al, 1998; Robert and Einstein, 1998; TENG et al, 2001). In both numerical and experimental simulations, a proper size of the model (or rock specimen) must be determined. As is known, real earthquakes are caused by the dislocations of rocks under certain tectonic load, therefore the boundaries of mechanical models in numerical simulations must be far enough from the faults so that the boundary effects on stress fields near the faults can be neglected. In principal, the larger the region to be solved is selected, the smaller the boundary effect will be, and thus, however, the larger computation is needed. Special techniques, such as the absorbing boundary technique (Randall, 1989; Peng and Toksrz, 1995), are used in the FDM to reduce the boundary effect on the inner stress field, * Received date: 2003-11-10; revised date: 2004-10-26; accepted date: 2004-12-06. Foundation item: National Natural Science Foundationof China (40234042). E-mail of the first author: [email protected].

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especially near the fault. The same problem exists in rock mechanics experiments. Size of rock specimen is limited, and therefore a proper size must be selected in order to make the boundary effect on the inner stress field as small as possible. Hence for simulation of rupture processes of seismic fault in either numerical or experimental approach, selection of sizes of models or rock specimens must be carefully considered, which will concern the reliability and accuracy of the results. So far, studies on selection of sizes of rock specimens focus mainly on rock mechanics experiments on measurement of rock strength. For uniaxial experiments on specimens without inner holes or cracks, cylinders are usually used (Hansen et al, 1962). Diameter of the specimen must satisfy D/d>lO, in which D and d are the diameters of the specimen and the maximum diameter of grains, respectively (Hawkes and Mellor, 1970; International Society for Rock Mechanics, 1978, 1979). Once D is determined, the length of rock specimen L can be determined by the length/diameter ratio (LDR) L/D. Error due to the imprecision of the loading system is in proportion to LDR, which implies that LDR must be as small as possible. An empirical value L/D=4 can be taken as a safe upper bound (Hawkes and Mellor, 1970). However, there are many debates in the selection of lower bound of LDR. The strength of uniaxial compression decreases with the increase of L/D (Gonnerman, 1925; Thaulow, 1962; Grosvenor, 1963; Hobbs, 1964; Mogi, 1966), till it tends to a stable value, which implies that the lower bound of L/D should be located at the point on which the negative slope of the curve of strength versus L/D increases most rapidly. For sedimentary rock and concrete, this point is located at L/D-1 (Johnson, 1943; Hobbs, 1964). While for some other rocks, this point is at L/D=2.0-2.5 (Mogi, 1966). Balla (1960) permitted the effects of varying friction between platen and specimen to be studied, and calculated the stress inside a cylinder based on the theory of elasticity. Based on these results and the McClintock-Walsh criterion, Hawkes and Mellor (1970) calculated the contour chart of McClintock-Walsh parameter inside the specimen as L/D=2, and indicated the region in which rupture might occur firstly. Their results showed that rough and rigid platens cause significant perturbation of the stress field to a distance of D/2 from each end, and therefore disapproved the choice of L/D=I. Based on the above studies, International Society for Rock Mechanics (1979) suggested that in experiments of measuring rock strength under uniaxial compression, the ratio of length to diameter of specimen should be 2.5-3.0. And American Society for Testing and Materials (ASTM.) suggested a value of L/D>I.O for the same case (Hawkes and Mellor, 1970). However, how to choose the size of precast rock specimen with a kerr or crack is not well studied and reported in literature. What size of model or specimen can deduce the effect of boundary on inner stress field (especially near the crack) to the greatest extent? In this study, we constructed several models for rock specimens in different sizes with elliptic holes, and calculated principal stresses for nodes on the holes by applying a 2D FEM. Errors compared with the corresponding analytic solution to the elliptic hole problem in infinite medium under far field uniaxial compression, in which there is no boundary effect, are also presented. This is a new attempt to investigate how to choose the size of rock specimen, and we believe it is helpful to both rock mechanics experiments with holes or cracks in rock specimens and numerical simulation for fault slip and crack development.

1 Geometrical model Geometry of the numerical model used in this study is illustrated in Figure 1. There is an el-

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liptic hole with a semimajor axis 2.45 mm and a semiminor axis 0.5 mm in the center of the specimen. Distance between the apex of major axis M and the upper boundary is d, and I is the major axis of the elliptic hole. The angle between the major axis and the direction of pressure applied on the boundary, p=110 MPa, is 30 °. In order to study the effect of boundary of specimen on the stress field, we construct several numerical models Table 1 Sizesof eight numerical models as shown in Table 1. The lengths are fixed as 25mm Model Length/mm Width/mm d/l for simplicity, and we will investigate the effect of the 1 25 11 4.275 upper and lower boundaries on the stress field with 2 25 13 5.275 different widths. The widths of models vary from 11 3 25 15 6.275 4 25 17 7.275 nun to 25 mm with a step 2mm. Difference in width 5 25 19 8.275 directly causes difference in d/l, which is an important 6 25 21 9.275 7 25 23 10.275 parameter to measure the size of specimen with hole. 8 25 25 11.275 In all of these models, the size and orientation of the elliptic hole are fixed, and the centers of the holes are coincidence with the centers of specimens.

2 Mathematical physical models for uniaxial rock experiments This is a plane stress problem. The stress components Crx, O'y and and ~'xysatisfy the equilibrium equations

{aer a%

-g-x +-aT-y --o ~%

(1)

ao',

--g-x +-g-y =o The constitutive relation is

O'y

%

=

E 1-v 2 Ell l_v 2

Ev l_v 2 E l_v 2

0

0

0 0

F_.y

(2)

E 2(l+v)

The geometrical relation is ~U

e~ Ox bv

(3)

Ov buwhere, E is the Young's modulus, V is the Poisson ratio, u and v are the displacements in x and y directions, respectively. The boundary conditions satisfy

FU Zhen et al: INFLUENCE OF ROCK SPECIMEN SIZE ON CRACK STRESS FIELD

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325

ulx=0=0,vlx=0,y__o:0,%lx__0,y>0:0 O'x[x=L=-ll0MPa, r~[x=L=0 (4)

' yly=0= y [y o=0

arly=o=ryxl~=o=0 O.IF= .IF=0 in which L and D are the length and width of the specimen, F i s the boundary of elliptic hole, and o'n and ~'n are normal and shear stress on the boundary of the hole, respectively. It is difficult to find analytic solutions to equations (1)-(4), and we solved them numerically by applying FEM. In calculation, E is equal to 46.9 MPa, and V is 0.23, which is corresponding to sandstone.

3 Numerical results For convenience, we selected some fixed nodes around the elliptic holes when dividing meshes in different models, then calculated the stress field on these nodes and compared them. Some of these nodes are distributed on the boundaries of holes, and the others are on line section BC, which is perpendicular to major axis of elliptic hole and the center of the hole is exactly located on the extension line (Figure 1). In Figure 1, O is the center of the hole, A is one of the apexes of the major axis, and otis the angle between OA and the link of node on hole to point O. ct varies from 0 ° to 360 °, with a positive direction counter clockwise. The arrow on line section BC indicates the direction, along which the numbers of nodes on BC increase. According to the analytic solution to elliptic hole problem with the same size and orientation in infinite medium (XU, 1982 •",'~ 2 i a __

m)(2 _ 1

0"* +tYs = q R e [Ze m ~ 2 - 1

.... 2i,t q(mp 4 "[- ¢ 2 ) ( 2 (O'y-a. +zt~Gy)e = p 4 ( m - ~ , ) ( m ( 2 - 1 ) q

I

+ p2 (m - ~,) Le-zia

[2e 2ia - m + m 1 + m ( z - 2e2i"( z-

(5)

m( ~-1

3e2i'~(2 + me2i" - m2 - 1 (2 -I e2ia( 2 + me 2ia - m a - 1 2m(41 (rag"2 -1) 2 m ( 2 -1 3

where ei 2 = ~ 0.t'(~)

P

I ~1 'w(

6°'(()=R(m-1/(2)

(=p(cosO+isinO)

R = -a + b 2

m

=

a- - b a+b

a and b are the semimajor axis and semiminor axis of the ellipse, respectively, t~ is the angle from the direction of major axis to that of load applied on the boundary in a clockwise direction, and q is the value of load (pull is positive, and push is negative). For the problem here, q = - l . l x l 0 8 Pa, a~=30°, a=2.45 mm, b=0.5 mm. Based on the relation of node coordinate in a Cartesian coordinate and polar one

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{~ = R(11 p + m p ) cos t9 = -R(11 p - mp) sin

(6)

we can compare the analytic solution in equation (5) with numerical results, and get insight into the boundary effect on stress field near the hole in a specimen with different sizes. Figure 2 shows the comparisons of principal stress on nodes around the elliptic hole in different models with the corresponding analytic result for infinite medium. Horizontal axis is the ot in Figure 1, and vertical axes are the maximum principal stress oi and the minimum stress tr2 on nodes. Stress curves for different models are illustrated in different markers. In particular, analyric result for elliptic hole problem in infinite Figure 1 Geometry of the rock specimen medium is marked in right triangle. Since numerical results for principal stress on nodes around the hole are close to analytic result, all the principal stress curves for different models agree well with that of analytic one, which implies that the obtained numerical results are stable and reliable. 1.5 1.0 ..~

0.5 0.0 0.5

2 . 0 ~

0

2 -2.0 ~-4 0 -6100

60

.

120

0

180 al(°) ~

240

300

360

~ ~

0

2

4

6

8

10 12 14 Node number

16

18

I - - 70"f ~ 9.0 - _

~-ii.0]

o.o

20

~-13.0 t -15. 0 60

120

180

a/(°)

240

300

360

--w- Model I --+--Model 2 ~ Model 3 ~ - Model 4 ~ - Model 5 -0-Model 6 + Model 7 --o--Model 8 --~ Analytic solution

Figure 2 Comparsions of numerical solutions of the principal stresses in different models with analytical solution in infinite region on the nodes around the perimeter of the elliptic hole

0

2

Model 1 ~ --4+-Model6 +

4

6

Model2 Model7

8

I0

12

14

16

18

20

Node number ~ Model3 ~ - Model4 ~ Model5 -8- Model8 -O- Analyticsolution

Figure 3 Comparsions of numerical solutions of the principal stresses in different models with analytical solution in infinite region on the nodes on the line BC

Principal stress curves for nodes on line section BC (Figure 1) which is perpendicular to the major axis of elliptic hole are shown in Figure 3. Horizontal axis is the number of nodes, and the nodes are off the hole with increase of numbers. Vertical axes are the same as in Figure 2. It can be seen from Figure 3 that the principal stress curves for models are roughly consistent with that of analytic one, however, some of them deviate remarkably from analytic result, especially the nodes with large node numbers for model 1~3. This is because these nodes are close to the boundary, e.g., point C in model 1 and 2, and the boundary will affect the principal stress around these nodes sig-

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nificantly. On the contrary, principal stress of all the nodes for models 4~8 are consistent with that of analytic result very well. If we subtract the corresponding analytic result from the calculated principal for every node, an absolute error for principle stress Ao"i = o'i - try"(i = 1, 2), in which o~ and o"7 are numerical and analytic results respectively, will be obtained. Figure 4 shows the absolute errors of the principal stresses on the nodes around the perimeter of the hole. Since the curvature at the tips of the elliptic hole is much more larger and the mesh is not dense enough in these regions, absolute errors around 0 ° and 180 °, i.e. the tips of the hole, are significantly larger. For other nodes around the perimeter of the hole, absolute errors are very small, and they all fluctuate around do~=0 (i=1, 2). Models 1 and 2 depart from A~=0 (i=1, 2) more remarkably, because the boundaries in these two models are more closer to the hole, and thus affect the principal stress more heavily. For nodes on line section B C (Figure 5), error curves of maximum principal stress for model 1~3 deviate from Atrl=0 significantly. While for minimum principal stress case, results for models 3 is slightly better than those of model 1 and 2, which deviate from A ~ = 0 remarkably, as for case of maximum principal stress. In general, however, all curves for models 4~8 are close to Ao~=0 (i=1, 2), and the absolute errors are less than those of models 1~3. 4.o

2.0
-4,0

6'o

110

60

120

l~o

2~o

3~o

3~o

/~0 a/(')

240

300

380

a/(')

4.0 2.0 2~ 0.0 ,~-2.0

-4.0 -6. 0 -O-

Figure 4

Model 1 Model 5

---4-- Model 2 - O - - Model 6

~ Model 3 --*'-- Model 7

~ Model 4 - - O - Model 8

Absolute errors of the principal stresses on the nodes around the perimeter of the hole

In order to clearly show the effect of boundaries of specimens with different widths on stress field, Figure 6 and 7 shows comparisons of relative errors on different nodes. Since analytic principal stresses on some nodes vanish and hence relative error in ordinary form will be greater than 1, we define the relative error as follows to avoid the occurrence of this special case: re!

e, -

IO"i - - O ' ; I

max(I tri* I)

(i = 1, 2)

(7)

where, o~ and or,.* are numerical and analytic principal stresses on nodes, respectively, and

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max(I o-7 I) is the maximum value of

6.0 4.0

principal stresses for all nodes (e.g., nodes on the perimeter of the hole). In Figure 6a, larger relative errors of maximum principal stress are located on the two tips of hole, while for nodes off the tips, relative errors decrease rapidly. It can be clearly seen that relative error remarkably decreases with increase of specimen width in the intervals of

~,.o 0.0 ~- - 2 . 0 -4.0 -6.0 -8.0

~

~

~

io ;2 l'4 ~'6 18 2'o 2.2 Node number

10.0 5.0

,~ ~,

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o.o 5.0

~' - 1 0 . 0 -15.0

15°-60 ° and 195°-240 ° for o~. Once the width reaches 17 mm or above, relative 2 4 6 8 10 12 14 16 18 20 22 Node number error hardly decreases with increase of - ~ - Model I --4- Model 2 - - ~ Model 3 ~ - Model 4 ~ , - Model 5 + Model 6 --*- Model 7 - O - Model 8 width, and relative errors for most of the nodes except those around the tips of Figure 5 Absolute errors of the principal stresses on the hole are below 5%. Figure 6b shows the nodes on the line BC relative errors for minimum principal stress, which are similar to those in Figure 6a. The difference is that in the intervals of 60°-180 ° and 2400-360 ° for o:, relative error remarkably decreases with increase of specimen width, and if width is greater than 17 mm, relative error is less than 2%. -20.0 -25.0

15" 0%~1~ (a)

-0.0 UV 0 0%w ' ' ~ • 0 v~ 12.0%

..... 6 0 ~ ~

180-~

..... 240 . . . .

180

240

"

120

a/(°)

"

300

36~0

300

360

(b)

9.0%

3.0~ 0.0% "~

~-

60 - -

Model 1 Model 5

~120

~"

---+-- Model 2 - O - Model 6

a/(°)

---x-- Model 3 -"g-- Model 7

"-'W--" Model 4 - O - Model 8

Figure 6 Relative errors of the principal stresses on the nodes around the perimeter of the hole Relative errors for nodes on line section BC (Figure 1) are greater than those around the perimeter of hole. Especially for relative errors for models 1-3 in Figure 7a, which are greater than other models, and the closer the nodes are to the boundary, the greater the relative errors will be. Maximum relative error for model 1 is above 50%, implying a very significant boundary effect.

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While for models 4-8, errors on the be60.0% (a) ginning nodes are relatively large, how50.0% ever, relative error decreases gradually 40.0% 77 30.0% with increase of node number, and rela20.0% tive errors for nodes near the boundary 10.0% are below 10%. Relative errors for O. 0% 2 4 6 8 10 12 14 16 18 20 22 minimum principal stress in Figure 7b Node number 25.0%[ (b) are much less than those in Figure 7a. 20. 0%[ This is because the absolute value for tY2 15, 0%I is greater than o'1, though they are comparable in magnitude. Similar to the case in Figure 7a, boundary effects for 0.0%~ " 2 ~ 6 8 10 12 14 16 18 20 22 models 1-3 are relative larger, e.g., Node number --+-Model 2 ~ Model 3 ~ Model 4 ~ Model 1 maximum relative error for model 1 is up ~ - Model 5 --0-- Model 6 ---*- Model 7 --O-- Model 8 to 20%, while relative errors for models Figure 7 Relative errors of the principal stresses on the 4-8 are below 3%. nodes on the line BC Concerning the above results, if the width of specimen is greater than 17 mm, the boundary effect on the stress field near the elliptic hole will be greatly reduced. For the rock specimen considered in this study, we take the size of 25 mm×17 mm as a critical size, which is corresponding to model 4 in Figure 1, whose aspect ratio is 25/17--3:2, and parameter d/l is 7.275.

4 Conclusions 1) In rock mechanics experiments, when problems such as crack extension under uniaxial compression for a rock specimen with an elliptic hole are studied, errors due to the boundary effect of specimen on inner stress field are usually not negligible. 2) For the specific problem in this study, in order to deduce the boundary effect on the stress field near the elliptic hole to a minimum extent, the width of specimen should be greater than 17mm, that is, the critical size of the specimen is about 25 mm×17 mm, whose aspect ratio is 25/17=3:2, and the parameter d/l is around 7.3. 3) To study the rupture process of a rock with a hole and its displacement or stress field by using experiment method more efficiently, it is helpful to apply numerical method, such as the FEM, to choose a proper size for the specimen, such that the boundary effect is reduced to a minimum extent. 4) When problems such as the dislocation of fault and crack extension are studied by using the FEM are studied, the method for analyzing used in this study can also be applied to determining the size of region for mechanical model. Stress will concentrate near the tips of elliptic hole, and therefore grids should be divided densely enough to ensure a high precision. However, a large computation is required. We compromise these two, and thus the errors for nodes on tips of the hole are relatively large. It can be expected that errors for all nodes will be smaller, if the grids are denser. However, since we focus on the difference among several models, not on the absolute error itself, the results obtained are still reliable. Based on a specific example, we theoretically analyze the boundary effect on the stress field near the hole in a specimen by using the FEM, and propose a criterion for the selection

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of size of specimen. Considering complexities in real experiments, such as the heterogeneities of rock materials, the sizes of grains, and the constraint of apparatus on the size of specimen, it is obvious that there is a limitation in the analysis. It is worth discussing further how to apply the theoretical analysis based on numerical simulation to experiments. References Archuleta R and Frazier G 1978. Three-dimensional numerical simulations of dynamic faulting in a half-space [J]. Bull Seism Soc Amer, 68: 541-572. Balla A. 1960. Stress conditions in triaxial compression [J]. J Soil Mech Found Div, Amer Soc Civ Engrs, 86(SM86): 57-84. Cai Y E, He T, Wang R. 2000. Numerical simulation of dynamic process of the Tangshan earthquake by a new method-LDDA [J]. Pure Appl Geophys, 157(11/12): 2083-2 104. Das S and Aki K. 1977. A numerical study of two-dimensional spontaneous rupture propagation [J]. Geophy J R astr Soc, 50: 643-668. Fukuyama E and Madariaga R. 1995. Integral equation method for plane crack with arbitrary shape in 3D elastic medium [J]. Bull Seism Soc Amer, 85: 614-628. Fukuyama E and Madariaga R. 1998. Rupture dynamics of a plannar fault in a 3D elastic medium: Rate- and slip-weakening friction [J]. Bull Seism Soc Amer, 88: 1-17. Gonnerman H E 1925. Effect of end condition of cylinder in compression tests of concrete [J]. Proc Amer Soc Testing Mater, 24(2): 1036-1 065. Grosvenor N E. 1963. Specimen proportion-key to better compressive strength tests [J]. Mining Eng, 15: 31-33. Hansen H, Kielland A, Nielsen K E C et al. 1962. Compressive strength of concrete-cube or cylinder [J]. Bull Reunion Intern Lab Essais Rech Mater Constr, 17: 22-30. Hawkes I and Mellor M. 1970. Uniaxial testing in rock mechanics laboratories [J]. Eng Geol, 4(3): 177-285. Hobbs D W. 1964. Rock compressive strength [J]. Colliery Eng, 41: 287-292. International Society for Rock Mechanics. 1978. Suggested methods for determining the strength of rocks materials in triaxial compression [J]. lnt J Rock Mech Min Sci, 15(1): 49-51. International Society for Rock Mechanics. 1979. Suggested methods for determining compression strength and deformability [J]. lnt J Rock Mech Min Sci, 16(2): 137-140. Johnson J W. 1943. Effect of height of testing specimen on compressive strength of concrete [J]. Amer Soc Testing Mater Bull, 120: 19-21. Madariaga R. 1976. Dynamic of an expanding circular fault [J]. Bull Seism Soc Amer, 66: 639-667. Mogi K. 1966. Some precise measurements of fracture strength of rocks under uniform compressive stress [J]. Rock Mech Eng Geol, 4(1): 41-55. Olsen K B, Madariaga R, Archuleta R. 1997. Three dimensional dynamic simulations of the 1992 Landers earthquake [J]. Science, 278: 834-838. Peng C B and Toks6z. M N. 1995. An optimal absorbing boundary-condition for elastic-wave modeling [J]. Geophysics, 60(1): 296-301. Randall C J. 1989. Absorbing boundary condition for the elastic wave equation: Velocity-stress formulation [J]. Geophysics, 54(3): 1 141-1 152. Robert A and Einstein H H. 1998. Fracture coalescence in rock-type materials under uniaxial and biaxial compression [J]. Int J Rock Mech Min Sci, 35(7): 863-888. Robina H C, Wong K, Chan T. 1998. Crack coalescence in a rock-like material containing two cracks [J]. lnt J Rock Mech Min Sci, 35(2): 147-164. Spicak A and Lokajicek T. 1986. Fault interaction and seismicity: Laboratory investigation and its seismotectonic interpretation [J]. Pure Appl Geophys, 124: 857-874. TENG Chun-kal, LI Shi-yu, HE Xue-song et al. 2001. An experiment study on dynamic fracture process of the crack system in rocks [J]. Chinese J Geophys, 44(supplement): 136-145. Thaulow S. 1962. Apparent compressive strength of concrete as affected by height of test specimen and friction between the loading surfaces [J]. Bull Reunion Intern Lab Essias Rech Mater Constr, 17: 31-33. XU Zhi-lun. 1982. Elastic Mechanics (Second edition) [M]. Beijing: People's Education Press, 130-146. ZHAO Yong-hong, HUANG Jie-fan, HOU Jian-jun et al. 1995. Experimental study on microcrack of rocks and its implication on seismicity (in Chinese) [J]. Chinese J Geophys, 38(5): 627-635.