Three-dimensional nonlinear strength criterion for rock-like materials based on the micromechanical method

International Journal of Rock Mechanics & Mining Sciences 72 (2014) 54–60

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Three-dimensional nonlinear strength criterion for rock-like materials based on the micromechanical method Xiao-Ping Zhou a,b,n, Yun-Dong Shou a,b, Qi-Hu Qian c, Mao-Hong Yu d a

State Key Laboratory of Coal Mine Disaster Dynamics and Control, Chongqing University, China School of Civil Engineering, Chongqing University, Chongqing 400045, China c Engineering Institute of Engineering Crops, PLA University of Science and Technology, Nanjing, China d Department of Civil Engineering and Mechanics, Xi'an Jiaotong University, Xi'an, China b

art ic l e i nf o

a b s t r a c t

Article history: Received 29 September 2013 Received in revised form 19 July 2014 Accepted 18 August 2014

In this paper, it is assumed the rock-like materials contain penny-shaped microcracks. The modes II and III stress intensity factors at tips of the three-dimensional penny-shaped microcracks are determined. The micro-failure orientation angle α is given out to describe the failure of rock-like materials. The formulation between the micro-failure orientation angle α and stress components is obtained from the mixed-mode fracture criterion. Failure characteristic parameters under true triaxial compression condition are defined, which are an invariant and are relative to the intermediate and minimum principal stresses. A three-dimensional nonlinear strength criterion of rock-like materials, in which the effects of the intermediate principal stress on the failure of rock-like materials is taken into account, is developed based on micromechanical methods. The theoretical nonlinear strength criterion is novel, which is not found in the previous references. By comparison with experimental data, it is shown that the present theoretical strength criterion is in good agreement with the experimental results. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Three-dimensional nonlinear strength criterion Micromechanical methods Three-dimensional penny-shaped microcracks The intermediate principal stress

1. Introduction In the past, a number of strength criteria for rock-like materials have been proposed by various researchers, such as Mohr–Coulomb criterion [1], Griffith criterion [2], Hoek–Brown strength criterion [3–8], unified strength criterion [9], three-dimensional Hoek–Brown strength criterion [10] and so on. Generally, two different approaches, i.e. macroscopic and micromechanical methods, are applied to develop strength criteria of brittle failure for rock-like materials. Most of strength criteria were proposed based on the macroscopic method, in which the initiation and propagation of cracks are not considered. The macroscopic methods follow with interest the macro-failure characteristics of rock-like materials, but most of them rarely consider the physical mechanisms of the failure of rock-like materials, such as Mohr–Coulomb criterion, Hoek–Brown failure criterion and so on. Actually, rock-like materials are discontinuity medium containing many microcracks, initiation and propagation of microcracks significantly affect the failure of rock-like materials. In order to investigate the effects of initiation and propagation of

n Corresponding author at: School of Civil Engineering, Chongqing University, Chongqing 400045, China. Tel.: þ 86 23 6512 0720; fax: þ 86 23 6512 3511. E-mail address: [email protected] (X.-P. Zhou).

http://dx.doi.org/10.1016/j.ijrmms.2014.08.013 1365-1609/& 2014 Elsevier Ltd. All rights reserved.

microcracks on failure of rock-like materials, micromechanical methods must be applied to understand the failure of rock-like materials. Micromechanical methods can improve the understanding of the physical mechanisms of the failure of rock-like materials. On the basis of the two-dimensional penetrated crack model, Zuo et al. [11] proposed a two-dimensional nonlinear strength criterion, in which the effects of the intermediate principal stress on the failure of rock-like materials is not considered. Yang [12] presented anisotropic damage yield criteria based on the effective stress approach. However, rock masses engineering are generally in three-dimensional pressure state and the true triaxial tests on rock by Mogi [13], Lu et al. [14], Xu et al. [15], Michelis [16], and others, showed that the intermediate principal stress significantly affects the rock strength. Higher strength values were observed under the plane strain loading condition and are believed to be due to the strengthening effect of the intermediate principal stress [13–18]. The strength increases considerably in excess of its corresponding value for standard triaxial tests as the intermediate principal stress increases. Wiebols and Cook [19] proposed the energy criterion for the strength of rock in polyaxial compression based on the additional energy stored around Griffith cracks due to the sliding of crack surfaces over each other, in which the effects of the intermediate principal stress were considered. However, the above energy criterion was not derived from the initiation and growth of penny-shaped cracks.

X.-P. Zhou et al. / International Journal of Rock Mechanics & Mining Sciences 72 (2014) 54–60

55

In this paper, on the basis of micromechanical methods, a new three dimensional nonlinear strength criterion of rock-like materials is presented, in which the effects of the intermediate principal stress on the failure of rock-like materials are considered. Moreover, all the parameters have clear physical concept, which can be conveniently determined by experiments. By comparison with experimental data, it is shown that the present strength criterion is in good agreement with the experimental results.

2. The analytical model In fact, brittle failure is the most common failure model for rock-like materials under complex stress condition. The failure pattern is closely related to the intrinsic property and stress condition of rock materials, such as fracture toughness, internal frictional angle, the dip angle of microcracks, the orientation angle of microcracks, Poisson's ratio and so on. In this paper, it is assumed that the failure of rock-like materials is due to the presence of penny-shaped microcracks and there is abundant evidence for the existence of microcracks in many brittle materials [20–22]. Therefore, this model is physically plausible and the following assumptions are made: (1) numerous penny-shaped microcracks are distributed randomly in rock-like materials; (2) interaction with penny-shaped microcracks is neglected; and (3) the matrix material in which penny-shaped microcracks are embedded is isotropic and elastic. 2.1. Propagation of penny-shaped microcracks

where 2

g 0ij

cos θ cos φ 6 ¼ 4  sin θ cos φ sin φ

sin θ

 cos θ sin φ

3

sin θ sin φ 7 5 cos φ

cos θ 0

ð2Þ

Then, σ 022 ; σ 021 ; σ 023 , can respectively be expressed as follows:

Consider a single penny-shaped microcrack in an isotropic body uniformly loaded at far field (Fig. 1). Establish a global coordinate system (O σ 1 σ 2 σ 3 ) and its corresponding local coordinate system (O  σ 01 σ 02 σ 03 ), θ is the dip angle of penny-shaped microcracks, σ 0ij ¼ g 0ik g 0jl σ kl is the orientation angle of penny-shaped microcracks, a is the radius of the penny-shaped microcracks. The stresses in the local coordinate system σ 0ij are given by the following equation [23]:

σ 0ij ¼ g 0ik g 0jl σ kl

Fig. 2. Propagation of wing cracks from the tip of penny-shaped microcrack.

ð1Þ

8 > σ 0 ¼ σ 1 sin 2 θ cos 2 φ þ σ 2 cos 2 θ þ σ 3 sin 2 θ sin 2 φ > < 22 σ 021 ¼ σ 2 sin θ cos θ  σ 1 sin θ cos θ cos 2 φ  σ 3 sin θ cos θ sin 2 φ > > : σ 0 ¼ σ sin θ sin φ cos φ  σ sin θ sin φ cos φ 23

3

1

ð3Þ For three-dimensional penny-shaped microcracks, the effective shear stress is the cause of frictional sliding. When the effective shear is greater than the frictional resistance along the slip surface, frictional slip would lead to the tensile stress at the two tips of the slip surface, which form the “wing cracks”, as shown in Fig. 2. Prior to the onset of crack propagation, the modes II and III stress intensity factors at tips of penny-shaped microcracks can be expressed as follows [24]: 8 0 0 pffiffiffi < K II ¼ 4ðσ 21  μσ 22 Þ a π 2v ð4Þ 0 0 pffiffiffi : K III ¼ 4ð1  vÞðσ 23  μσ 22 Þ a π 2v where μ is the frictional coefficient on the crack surfaces, v is Poisson's ratio, K II is the mode II stress intensity factor, K III is the mode III stress intensity factor. The condition of growth of the mixed mode microcracks can be expressed as follows: K II þ K III Z κ K IC

ð5Þ

where κ can be obtained from experimental results, or approximation pffiffiffi suggested in the literature on the kinked crack, such as κ ¼ 3=2 in maximum-stress criterion [25]. K IC is the mode I critical stress intensity factor, which can be obtained by induced tensile strength and crack length, namely rffiffiffiffi a K IC ¼ 2σ t ð6Þ

π

Fig. 1. Mechanical model for penny-shaped microcrack in rock-like materials.

where σ t is the tensile strength of rock-like materials.

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X.-P. Zhou et al. / International Journal of Rock Mechanics & Mining Sciences 72 (2014) 54–60

2.3. Failure characteristic parameters of rock-like materials

Fig. 3. Wing crack distribution zone.

2.2. The orientation angle of micro-failure in rock-like materials Once the condition of growth of microcracks (5) has been satisfied, wing cracks will nucleate, initiate and propagate at the tip of penny-shaped cracks. A fan-shaped area of wing crack distribution zone in Fig. 3 can be obtained from Eqs. (3) to (5). The included angle of the fan section is defined as the microfailure orientation angle α. Substituting Eq. (4) into Eq. (5), we have h

2μð2  vÞ





i

σ 3 sin 2 θ þ σ 2 cos 2 θ  2ðσ 2  σ 3 Þ sin θ cos θ þ ð2 vÞκσ t tan 2 φ

þ2ð1 vÞðσ 1  σ 3 Þ sin θ tan φ h   2 þ 2μð2  vÞ σ 1 sin θ þ σ 2 cos 2 θ þ2ðσ 1  σ 2 Þ sin

i

θ cos θ þ ð2  vÞκσ t r 0 ð7Þ

where the compressive stresses are positive,σ 1 is the maximum principal stress, σ 2 is the intermediate principal stress, and σ 3 is the minimum principal stress. Eq. (7) can be rewritten as follows: C 1 tan 2 φ þ C 2 tan φ þ C 3 r 0

  8 > C 1 ¼ 2μð2  vÞ σ 3 sin 2 θ þ σ 2 cos 2 θ  2ðσ 2  σ 3 Þ sin θ cos θ þ ð2  vÞκσ t > > < C 2 ¼ 2ð1  vÞðσ 1  σ 3 Þ sin θ   > > > : C 3 ¼ 2μð2  vÞ σ 1 sin 2 θ þ σ 2 cos 2 θ þ 2ðσ 1  σ 2 Þ sin θ cos θ þ ð2  vÞκσ t

From Eq. (8), the tangent of φ can be determined by the following equation: tan φ1 r tan φ r tan φ2

ð9Þ

where tan φ1 ¼

tan φ2 ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C 22  4C 1 C 3 2C 1

 C2 

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C 22  4C 1 C 3 2C 1

ð10Þ

tan φ1  tan φ2 1 þ tan φ1 tan φ2

α¼

ðσ 1  2σ 2 þ σ 3 Þ cos



θ þ ð2 vÞ ð2μσ 2 þ κσ t Þcsc θ þ μðσ 1  2σ 2 þ σ 3 Þ sin θ ωðσ 1  σ 3 Þ

ð12Þ



ð13Þ where rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h iffi

ω¼

ð1  vÞ2 þ cos 2 θ þ μðv  2Þ ðv  2Þμ sin 2 θ  sin 2θ

The above relationship indicates that the cosine of the microfailure orientation angle α increases with the increment of the minimum principal stress σ 3 , and the micro-failure orientation angle α decreases with an increase in the minimum principal stress σ 3 . For an invariable intermediate principal stress σ 2 and an invariable minimum principal stress σ 3 , the relationship between cos α and the maximum principal stress can be defined. Differentiating Eq. (13) with respect to σ 1 , we obtain     ∂ cos α 2ðσ 3  σ 2 Þ cos θ þ ð2  vÞ ð2μσ 2 þ κσ t Þcsc θ þ 2μðσ 3  σ 2 Þ sin θ   rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h iffi  ∂σ  ¼ 1 2 ðσ 1  σ 3 Þ2 ð1  vÞ2 þ cos 2 θ þ μðv  2Þ ðv  2Þμ sin θ  sin 2θ

ð14Þ

ð11Þ

From Eqs. (10) and (11), supposing α ¼ φ1  φ2 , we can obtain tan α ¼ tan ðφ1  φ2 Þ ¼

cos

ð8Þ

where

 C2 þ

The failure characteristic parameter of rock-like materials should be constant when rock-like materials entirely break. Damage mechanics revealed that the initiation of internal microcracks does not indicate failure of rock-like materials [26–28]. Many experiments showed that the maximum principal stress should be further increased to assure that the wing crack continually propagate, while the minimum principal stress can significantly restrain the wing crack to grow [20]. Therefore, the initiation of wing cracks cannot indicate the failure of rock-like materials. As a result, initiation of internal microcracks cannot be selected as the failure characteristic parameters. The larger the minimum principal stress, the smaller the microfailure orientation angle. Microcracks randomly distribute in rocklike materials, and the orientation angle of each microcrack randomly distributes. So the critical micro-failure orientation angle can be used to describe the micro-failure density. An increase in the minimum principal stress leads to a decrease in the micro-failure density. The internal micro-failure density is not constant. Therefore, the micro-failure density cannot also be selected as the failure characteristic parameters. Brady [29] has suggested that the failure of rock-like materials occurs when the volumetric strain due to the internal microfailure density reaches a critical value. Therefore, the failure characteristic parameters of rock-like materials should be relevant to the internal micro-failure density, which is related to the minimum principal stress and the micro-failure orientation angle. Moreover, the failure characteristic parameters should satisfy the following three principles: firstly, the expression of the microfailure characteristic parameter should be a simple mathematic one; secondly, the higher the minimum principal stress, the lower the micro-failure orientation angle; finally, the theoretical result should agree well with the experimental data. Obviously, the expression of the micro-failure orientation angle α is too complicated to be selected as the failure characteristic parameters. According to the second principle and Eq. (12), the cosine of the micro-failure orientation angle α can be written as follows:

From Eq. (13), the maximum principal stress can be expressed as follows: 2 cot θ  þ κ ðv  2Þσ t cscθ þ ðv  2Þμσ 3 σ 1 ¼ cos θ½2σ2  σ 3 þ 2ðv  2Þμσ cos θ þ ð2  vÞμ sin θ  ω cos α

sin θ  ωσ 3 cos α

ð15Þ

X.-P. Zhou et al. / International Journal of Rock Mechanics & Mining Sciences 72 (2014) 54–60

Substituting Eq. (15) into Eq. (14), the following expression can be obtained as:     ∂ cos α ω cos α  cos θ  ð2  vÞμ sin θ 2    ∂σ  ¼ 2ðσ  σ Þ cos θ þ ð2  vÞð2μσ þ κσ Þcscθ  2μðσ  σ Þ sin θ  ω 1 t 3 2 2 2 3

ð16Þ From Eq. (13), the increase of σ 3 results in the increase of cos α. From Eq. (16),  the increase  of σ 3 results in the increase of the denominator of ∂ cos α=∂σ 1 , the increase  of cos α results in   the increase of   the numerator of ∂ cos α=∂σ 1 . The rate of change ∂ cos α=∂σ 1  can be always a constant when the rock fractures no matter what the value of σ 3 is. Therefore, the rate of change  ∂ cos α=∂σ 1  can be regarded as the failure characteristic parameter of rock-like materials. For uniaxial compression condition σ 1 ¼ σ c , σ 2 ¼ 0, and σ 3 ¼ 0, we can obtain   ∂ cos α  ¼  ∂σ  1

ð2  vÞκσ t cscθ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h iffi 2 2 σ ð1  vÞ þ cos θ þ μðv 2Þ ðv  2Þμ sin 2 θ  sin 2θ 2 c

ð17Þ From Eqs. (14) and (17), we have 

2ðσ 3  σ 2 Þcosθ þ ð2  vÞ ð2μσ 2 þ κσ t Þcscθ þ 2μðσ 3  σ 2 Þ sin θ ðσ 1  σ 3 Þ2

 ¼

ð2  vÞκσ t cscθ

σ 2c

ð18Þ From Eq. (18), we find

σ1  σ3 ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ c ðnσ 2 þ mσ 3 Þ þ σ 2c

ð19Þ

where 

  2ð2  vÞμ cscθ þ sin θ  2 cos θ σ c n¼ ; κσ t cscθð2  vÞ   2ð2 vÞμ sin θ þ 2 cos θ σ c m¼ κσ t cscθð2  vÞ It is observed from Eq. (19) that m and n are related to the friction coefficient μ, the coefficient κ of mixed-mode fracture criterion, the uniaxial compressive strength σ c , the uniaxial tensile strength σ t , the dip angle of penny-shaped microcracks and Poisson's ratio.

Fig. 4. Comparison of theoretical and experimental values of the Trachyte.

57

3. Comparison with experimental results The stress angle is defined as follows: " # 2σ 3  ðσ 1 þ σ 2 Þ p ffiffiffi ϕσ ¼ arctan ð  300 r ϕσ r 300 Þ 3ðσ 1 þ σ 2 Þ

ð20Þ

The stress tensor σij expressed by the first invariant of stress tensor I 1 and the second invariant of deviatoric stress tensor J 2 can be written as follows: pffiffiffiffi 8

 σ ¼ I1 þ p2ffiffi3 J 2 sin ϕσ þ 23π > > > 1 3 < pffiffiffiffi σ 2 ¼ I31 þ p2ffiffi3 J 2 sin ϕσ ð21Þ > pffiffiffiffi

 > > : σ ¼ I1 þ p2ffiffi J sin ϕ  2π 3

3

3

2

σ

3

The nonlinear strength criterion Eq. (19) is rewritten in another form: F ¼ 2q cos

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n hpffiffiffi io 3m cos ϕσ þ ðm 2nÞ sin ϕσ ¼ 0

ϕσ  σ c 3σ c þ3pðm þnÞ q

ð22Þ where p ¼ I 1 =3; q ¼

pffiffiffiffiffiffiffi 3J 2 :

The nonlinear strength criterion (19) is compared with experimental results in this paper. It is confirmed by six sets of true triaxial compression test data produced by other researchers [13,16,30–34]. The curves in Figs. 4–9 are computed from Eq. (22), the dots are true triaxial compressive test data. In Fig. 4 and Table 1, the first set of true triaxial compressive test data were obtained from confined compression tests and confined tension tests on the Trachyte by Mogi [13,30]. In Fig. 4 and Table 1, the uniaxial compressive strength is 99.13 MPa, the fitting parameters of the strength are m ¼ 12:08, n ¼ 1:137. In Fig. 5 and Table 2, the second set of true triaxial compressive test data were determined from confined compression tests and confined tension tests on the Dunham Dolomite by Mogi [13]. In Fig. 5 and Table 2, the uniaxial compressive strength is 257 MPa, the fitting parameters of the strength are m ¼ 7:403, n ¼ 3:407. In Fig. 6, the third set of true triaxial compressive test data were obtained from confined compression tests on the marble by Michelis [16,31]. In Fig. 6, the uniaxial compressive strength is 36.45 MPa, the fitting parameters of the strength are m ¼ 24:13, n ¼ 7:12, respectively. In Fig. 7, the fourth set of true triaxial compressive test data were obtained from confined compression

Fig. 5. Comparison of theoretical and experimental values of the Dunham Dolomite.

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X.-P. Zhou et al. / International Journal of Rock Mechanics & Mining Sciences 72 (2014) 54–60

Fig. 6. Comparison of theoretical and experimental values of the coarse grained dense marble.

Fig. 9. Comparison of theoretical and experimental values of the westerly granite.

Table 1 Comparison of theoretical and experimental values of the Trachyte.

Fig. 7. Comparison of theoretical and experimental values of the amphibolite.

Fig. 8. Comparison of theoretical and experimental values of the limestone.

tests and confined tension tests on the amphibolite by Chang and Haimson [32]. In Fig. 7, the uniaxial compressive strength is 164.67 MPa, the fitting parameters of the strength are m ¼ 36:89,

σ3 (MPa)

σ2 (MPa)

σ1 (MPa)

θ (MPa)

p (MPa) qexperimental (MPa)

qtheoretical (MPa)

0.00 15.00 30.00 45.00 45.00 45.00 45.00 45.00 45.00 45.00 45.00 60.00 60.00 60.00 60.00 60.00 60.00 60.00 75.00 75.00 75.00 75.00 75.00 75.00 75.00 100.00 100.00 100.00 100.00 100.00 100.00

0.00 15.00 30.00 45.00 54.55 70.92 95.45 141.80 213.64 289.09 331.81 60.00 90.48 141.90 191.33 228.57 271.43 331.43 75.00 114.29 153.33 228.57 300.00 342.86 390.48 100.00 137.40 185.71 274.29 381.90 411.43

99.13 193.04 253.04 297.39 313.91 326.09 333.39 348.70 360.87 365.22 351.30 339.13 352.17 382.61 395.65 404.35 400.00 382.61 365.22 400.00 417.39 438.26 439.13 424.35 451.30 519.13 460.00 488.70 493.91 521.74 513.04

 30.004  26.294  24.46  23.054  23.573  24.059  24.522  25.240  25.962  26.470  26.624  21.987  22.823  24.002  24.670  25.074  25.369  25.656  20.838  22.243  23.055  24.108  24.711  24.914  25.383  21.348  22.010  22.104  23.124  24.207  24.344

33.043 74.347 104.347 129.130 137.821 147.334 157.947 178.505 206.503 233.102 242.705 153.043 167.550 194.838 215.662 230.973 243.810 258.013 171.739 196.429 215.241 247.277 271.377 280.735 305.593 239.710 232.467 258.138 289.400 334.546 341.490

99.130 171.697 221.659 262.271 259.671 255.809 251.495 248.474 257.420 283.076 303.599 297.387 289.075 280.681 279.805 283.903 293.477 314.738 328.773 317.623 310.277 307.575 319.190 332.301 351.578 375.296 363.616 352.990 347.916 367.376 377.232

99.130 178.040 223.040 252.390 264.265 269.068 266.767 268.711 273.761 289.756 297.035 279.130 278.185 290.453 292.963 298.238 297.349 300.309 290.220 307.245 310.720 315.841 318.255 316.572 349.877 419.130 342.833 353.721 341.888 372.080 372.770

n ¼ 3:77. The fifth set of true triaxial compressive test data in Fig. 8 were determined from confined compression tests and confined tension tests on the limestone by Yin et al. [33]. In Fig. 8, the uniaxial compressive strength is 78.7 MPa, the fitting parameters of the strength are m ¼ 14:07, n ¼ 5:775. The sixth set of true triaxial compressive test data in Fig. 9 were determined from confined compression tests and confined tension tests on the Westerly granite by Haimson and Chang [34]. In Fig. 9, the uniaxial compressive strength is 201 MPa, the fitting parameters of the strength are m ¼ 40:84, n ¼ 3:283. It is shown from Tables 1 and 2, Figs. 4–9 that the theoretical three-dimensional nonlinear strength criterion is in good agreement with experimental results.

X.-P. Zhou et al. / International Journal of Rock Mechanics & Mining Sciences 72 (2014) 54–60

Table 2 Comparison of theoretical and experimental values of the Dunham Dolomite. σ3 (MPa)

σ2 (MPa)

σ1 (MPa)

θ (MPa)

p (MPa) qexperimental (MPa)

qtheoretical (MPa)

0.00 25.00 25.00 25.00 25.00 25.00 25.00 25.00 45.00 45.00 45.00 45.00 45.00 45.00 45.00 45.00 65.00 65.00 65.00 65.00 65.00 65.00 85.00 85.00 85.00 85.00 85.00 105.00 105.00 105.00 105.00 105.00 105.00 125.00 125.00 125.00 125.00 125.00 125.00 145.00 145.00 145.00 145.00

0.00 25.00 67.64 91.18 135.00 176.50 232.40 300.00 45.00 100.00 123.50 155.88 179.40 238.20 267.00 300.00 65.00 117.60 150.00 205.90 302.90 373.50 85.00 132.00 223.00 300.00 364.70 105.00 144.00 205.90 264.70 323.50 347.10 125.00 164.70 182.40 255.90 352.90 420.60 145.00 250.00 300.00 400.00

257.00 400.00 473.50 500.00 552.90 573.50 594.10 626.50 488.20 561.80 582.40 608.80 608.80 670.60 670.50 658.80 567.60 629.40 644.11 690.00 729.40 711.76 623.50 688.20 752.90 733.50 817.60 679.40 723.50 791.20 817.60 832.30 852.90 723.50 785.30 826.50 867.60 905.90 932.40 800.00 900.00 935.30 982.40

 30.000  26.996  27.655  27.857  28.164  28.319  28.476  28.643  25.636  26.512  26.736  26.994  27.086  27.482  27.561  27.617  24.641  25.495  25.773  26.270  26.778  26.939  23.693  24.593  25.491  25.752  26.305  22.918  23.634  24.501  24.954  25.288  25.469  22.158  23.046  23.475  24.174  24.829  25.205  21.810  23.353  23.836  24.524

85.667 150.00 188.713 205.393 237.633 258.333 283.833 317.167 192.733 235.600 250.300 269.893 277.733 317.933 327.500 334.600 232.533 270.667 286.370 320.300 365.767 383.420 264.500 301.733 353.633 372.833 422.433 296.467 324.167 367.367 395.767 420.267 435.000 324.500 358.333 377.967 416.167 461.267 492.667 363.333 431.667 460.100 509.133

257.000 368.108 396.144 410.758 436.986 461.147 493.281 532.189 437.112 464.057 475.344 490.843 502.127 530.661 544.891 561.451 496.619 516.717 529.262 551.497 592.512 624.626 549.721 563.929 593.561 621.372 646.875 598.128 607.463 623.613 640.657 659.477 667.542 642.900 650.448 654.059 670.791 697.420 719.152 684.751 702.292 712.815 738.266

257.000 375.000 428.773 445.611 482.400 490.619 498.857 521.550 443.200 491.613 502.768 517.350 510.058 554.829 549.248 534.094 502.600 540.025 541.636 567.816 583.064 560.308 538.500 581.127 610.708 572.145 640.333 574.400 599.952 641.727 647.689 646.369 660.984 598.500 641.372 674.634 686.574 695.540 707.536 655.000 708.361 725.329 743.456

4. Discussions and conclusions In this paper, it is assumed that rock-like materials contain the three-dimensional penny-shaped microcracks. The modes II and III stress intensity factors at tips of three-dimensional penny-shaped microcracks are obtained from Tada's works. The failure characteristic parameter of rock-like materials and the micro-failure orientation angle are determined. A nonlinear three-dimensional strength criterion for rock-like materials, in which the effects of the intermediate principal stress on the failure of rock-like materials is taken into account, is proposed based on the micromechanical methods. The three-dimensional nonlinear strength criterion derived from the micromechanical methods is novel, which is not found in the previous references. By comparison with experimental results, it is found that the theoretical threedimensional nonlinear strength criterion is in good agreement with experimental results. However, rock-like materials contain different defects. The microcrack is one of these defects and different defects have different shapes, strength and distribution. The effects of different defects on the strength of rock-like materials are different. The effects of different defects on the strength of rock-like materials are not taken into account in this paper. Meanwhile, most of rock failure is caused by the development and influence of microcrack

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groups, not by an individual microcrack. The effects of interaction among microcracks on the strength of rock-like materials are also not considered. In the further studies, the effects of different defects and interaction among microcracks on the strength of rock-like materials will be taken into account.

Acknowledgments This work was supported by the National Natural Science Foundation of China (Nos. 51325903 and 51279218), Project 973 (Grant no. 2014CB046903), Natural Science Foundation Project of CQ CSTC (Nos. cstc2013kjrc-ljrccj0001 and cstc2013jcyjys0005), Research fund by the Doctoral Program of Higher Education of China (No. 20130191110037) and Chongqing University Postgraduates' Science and Innovation Fund, Project number 0218005204101.

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