A three-dimensional long-term strength criterion of rocks based on micromechanical method

Accepted Manuscript A three-dimensional long-term strength criterion of rocks based on micromechanical method Xiao-Ping Zhou, Xiao-Cheng Huang, Filippo Berto PII: DOI: Reference:

S0167-8442(17)30305-1 http://dx.doi.org/10.1016/j.tafmec.2017.07.003 TAFMEC 1908

To appear in:

Theoretical and Applied Fracture Mechanics

Received Date: Revised Date: Accepted Date:

9 June 2017 28 June 2017 4 July 2017

Please cite this article as: X-P. Zhou, X-C. Huang, F. Berto, A three-dimensional long-term strength criterion of rocks based on micromechanical method, Theoretical and Applied Fracture Mechanics (2017), doi: http:// dx.doi.org/10.1016/j.tafmec.2017.07.003

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A three-dimensional long-term strength criterion of rocks based on micromechanical method Xiao-Ping Zhoua,b, Xiao-Cheng Huanga,b, Filippo Bertoc

 (a

b

School of Civil Engineering, Chongqing University, Chongqing 400045, China;

State Key Laboratory of Coal Mine Disaster Dynamics and Control, Chongqing University, Chongqing 400044, China; c

Department of Mechanical Engineering, Norwegian University of Science and Technology, Trondheim 7491, Norway)

Abstract: Rock-like materials are driven not only by applied stresses, but also by time that exhibits creep characteristics. It is significant to establish three-dimensional long-term strength criterion of rocks. In this paper, it is assumed that there exist three-dimensional penny-shaped microcracks in viscoelastic rock matrix. The mode II and mode III stress intensity factors at tips of three-dimensional penny-shaped microcracks in viscoelastic rock matrix are determined. The orientation angle of micro-failure in rock materials is obtained to describe the creep failure of rocks. The relationship between the micro-failure orientation angle and stress components is derived from the creep fracture criterion. Failure characteristic parameters of penny-shaped microcracks under triaxial creep compressive condition are defined, which are an invariant. A three-dimensional long-term strength criterion of rocks is established using micromechanical method, in which the effects of the intermediate principal stress are taken into account. The proposed three-dimensional long-term strength criterion is novel, and never published before. By comparison with the previous experimental data, it is found that the presented three-dimensional long-term strength criterion is in good agreement with the experimental data. Keywords: Three-dimensional long-term strength criterion; Viscoelastic rock matrix; Micromechanical method; three-dimensional penny-shaped microcracks; Stress intensity factor; The intermediate principal stress

1.Introduction Over the years, a number of criteria have been proposed to assess the possible occurrence of rock failure [1-5]. Considering that influence factors of the rock failure include scale, time and characteristics of the defects, it is not surprising that the most popular strength criteria of rocks rely on some significant simplifying assumptions. For example, many well-known criteria, including the often used Coulomb [6] and Hoek–Brown criterion [3,7], usually neglect the effects of time, defects and the intermediate principal stress on the strength of rocks. Such simplifications greatly help researchers face with actual stability analysis of the surrounding rock masses around tunnels, and also reduce the effort needed to obtain the required properties of rock materials. However, for detailed studies required by some of the most challenging rock

*Corresponding

author, Professor, School of civil engineering, Chongqing Univ., Chongqing, China. E-mail address:

[email protected]:+86-23-6512-0720;Fax.+86-23-6512-3511

1

engineering projects, such as underground storage reservoirs and toxic waste disposal facilities, it is usually recognized that the effects of times, defects and the intermediate principal stress on the strength of rock should be taken into account [1, 2]. In the past several decades or more, extensive laboratory creep experiments have been performed to investigate the time-dependent behaviors of many kinds of rocks [8-12]. It is indicated that deformation of rock under a constant stress over extended a period of time generally exhibits a trimodal creep behavior, i.e. primary or transient creep, lately by secondary or steady-rate creep, followed terminating in tertiary or accelerating creep that eventually progresses to dynamic rupture. It is found from laboratory creep experiments that the instability of the surrounding rock mass around tunnels occurs at stresses well below the peak strength of rock mass. Analyses that the short-term strength is applied to estimate the stability of the surrounding rock mass around tunnels have often predicted stable openings even though the failure of rock mass is observed in situ. For example, it is observed that the long-term strength of rock in situ can be as low as 50% of the short-term strength [1]. In order to study the long-term strength of rock, some long-term strength criteria of rocks have been proposed to investigate the time-dependent behaviors of rocks, such as Mises-Schleicher &Drucker-Prager unified(MSDPu) criterion, and so on. However, these strength criteria were proposed based on phenomenological approaches, which can produce the macroscopically observed creep curves of rocks by fitting with experimental data, and the inherent physical mechanisms related to creep deformation are not accommodated in these models, so the key mechanistic parameters remain physically unclear [13]. To authors' knowledge, three-dimensional long-term strength criterion of rock materials, in which the effects of the intermediate principal stress are taken into account, was not derived from micromechanical methods. In fact, rock is a kind of discontinuity medium containing many microcracks, the presence of such microcracks strongly influences the macroscopic mechanical behavior of rocks by serving as stress concentrators and leading to microcracking [14-18]. To overcome the disadvantages encountered in phenomenological models, it is necessary to investigate the effects of initiation and propagation of microcracks on the failure of rocks. In this paper, micromechanical methods are applied to study the effects of time on the strength of rocks. Moreover, a novel three-dimensional nonlinear long-term strength criterion of rocks is proposed, in which the effects of time and the intermediate principal stress on the creep failure of rocks are taken into account. By comparison with experimental data, it is shown that the novel three-dimensional nonlinear long-term strength criterion is in good agreement with the experimental results.

2.The analytical model It is generally accepted that the time-dependent deformation and fracturing process that evolve in rocks are 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 rocks is due to the presence of penny-shaped microcracks and there is abundant evidence for the existence of microcracks in many rock materials [19,20]. Therefore, this model is physically plausible and the following assumptions 2

are made: (i) penny-shaped microcracks are assumed to be randomly distributed in rock matrix; (ii) the interaction between penny-shaped microcracks is neglected before the coalescence of microcracks;(iii) rock matrix in which penny-shaped microcracks are embedded is isotropic and viscoelastic. 2.1 Stress intensity factor of penny-shaped microcracks embedded in isotropic and viscoelastic rock matrix It is assumed that the tensile stress is negative, and the compressive stress is positive. Consider a single penny-shaped microcrack in an isotropic and viscoelastic rock matrix uniformly loaded at far field. Establish a global coordinate system ( O  x1 x2 x3 ) and its corresponding local coordinate system ( O  x1x2 x3 ), as shown in Fig. 1. In a global coordinate system ( O  x1 x2 x3 ), the direction of the maximum principal stress is parallel to the x1 -axis, the direction of the intermediate principal stress is parallel to the x2 -axis, the direction of the minimum principal stress is parallel to the x3 -axis. In the local coordinate system ( O  x1x2 x3 ), the direction of the x2 -axis is parallel to the normal direction of penny–shaped microcrack. The angle between the x2 -axis and the x2 -axis is the dip angle of penny–shaped microcrack  . The angle between the x3 -axis and the x3 -axis is the orientation angle of penny–shaped microcrack  .

x2

x2

x1

x1

x3

x3 Fig. 1. Mechanical model for penny-shaped microcrack in soft rocks. The stresses in the local coordinate system  ij are given by Yu and Feng [21],

 ij  gik g jl kl

3

(1)

s c o s  co  gij    s i n c os  s i n

where

s i n c o s

 c o s  s i n  s i n s i n c o s 

0

(2)

 ,  21  can be respectively expressed as follows:  and  23 Then,  22 2  2 2   1sin 2  cos 2   cos    sin 2 sin 2 2 3     2 sin  cos    1 sin  cos  cos 2    3 sin  cos  sin 2   21     sin  sin  cos    sin  sin  cos  3 1  23

(3)

Tada [22] defined the stress intensity factors at tips of penny-shaped microcracks embedded in isotropic and elastic rock matrix as

 4 ( 2 1   2 2) a  K II  2     ) (2 3   2 2 )a  K  4 ( 1  III  2   where μ is the frictional coefficient on the crack surfaces,

(4)

 is Poisson's ratio, KII is the mode II stress intensity

factor, KIII is the mode III stress intensity factor. 2.2 The creep model In this paper, it is assumed that microcracks are embedded in rock matrix, and rock matrix satisfies Burgers model with the characteristic of instantaneous elastic deformation, primary creep and steady-rate creep. Similarly, the other creep models can also be adopted to study creep behaviors of rock matrix.

Fig. 2. The diagram of Burgers model As shown in Fig. 2, Burgers model can be described as follows

eij 

G2

2

eij 

G2 G 2 1 1 1 Sij  (   )Sij  Sij 2G 1 2 2 2 1 2G  1 2 2  1 2

where G1 is Maxwell shear modulus，G2 is Kelvin shear modulus，1 is Maxwell viscosity ，and

Sij   ij 

 ij 3

( 11   22   33) , eij   ij 

 ij

1 (11   22   33 ) ,  ij   3 0

tensor. The Maxwell shear modulus is equal to elasticity shear modulus, eij  4

(5)

 2 is Kelvin viscosity,

i j ,  ij is stress tensor,  ij is strain i j

d 2 eij dt

2

, eij 

deij dt

, Sij 

d 2 Sij dt

2

, Sij 

dSij dt

.

From Eq.(5), the following expression can be obtained as

 1 t 1 eij  Sij     2G1 2 1 2G

G2 t   2 1  e     2 

(6)

where t is the creep time. From Eq.(6) and works by Yi and Zhu [23], the time factor of the Burgers model under a given load is written as

 fi (t )  H (t )   G2   G1 G1   f ( t )  1  t  1  exp    t   iu 1 G2   2   

(7)

1, t  0 is Heaviside 0, t  0

where fiu (t ) is the time factor for displacement, fi (t ) is the time factor for stress, H (t )   function.

According to works by Zhou [24], energy release rate at tips of the mixed mode I- II-III microcracks in viscoelastic rock matrix can be written as

G(t )  GI (t )  GII (t )  GIII (t )  where fiu (t )  1 

G1

1

t

G1 G2

1  v2 2 1 ( K I  K II 2  K III 2 ) f iu (t ) E 1 v

(8)

  G2   1  exp   t   .  2   

In Eq. (8), G (t ) can be rewritten as

G(t ) fiu t( G)

(9)

where G is energy release rate at tips of the mixed mode I-II-III microcracks in elastic rock matrix. As for the creep fracture, the stress and displacement fields at tips of microcracks can be written as follows:

 (m ) m( K ) m (t )  ij (t )   ij Km   ) m (t ) u ( m )(t )  u m( K i i  Km

(10)

where m =I, II and III, which is respectively denoted by mode I, II and III microcracks,  ij and ui (m)

stress and displacement fields at tips of microcracks in elastic rock matrix,  ij (t ) and ui (m)

( m)

(m)

are respectively the

(t ) are respectively the stress

and displacement fields at tips of microcracks in viscoelastic rock matrix, K m (t ) and K m are respectively stress intensity factor at tips of microcracks in viscoelastic and elastic rock matrix. According to the definition of stress intensity factor, stress intensity factor at tips of microcracks can be expressed as

5

 K  lim  ( m ) 2 x   m x 0  ij y 0    K m (t )  lim  ij( m ) (t ) 2 x y 0 x 0 





(11)

On the basis of the definition of energy release rate, energy release rate at tips of microcracks can be written as

1 a  yy  x,0  u y  x   a,0    yx  x,0  u x  x   a,0    yz  x,0  u z  x   a,0   dx  a 0  a 0 

G  t   lim

(12)

where a is the growth length of microcracks. Substituting Eq. (10) into Eq. (12), the following expression can be obtained as

1 a G (t ) l i m   ij( m ) t u( i m)( t) dx ( )  a 0  a 0 K (t )  K t( 1 a   l i m   ij( m ) m  ui m( ) m  a 0  a 0 Km   Km 

) dx 

(13)

1 a  l i m   ij( m u) i m( )dx  a 0  a 0

 K m (t ) 2  Km 

2



G

From Eq. (13) and Eq. (9), the stress intensity factors of creep cracks can be written as

G(t )  K m fiu (t ) G

K m (t )  K m

(14)

For three-dimensional penny-shaped microcracks, frictional sliding is caused by the effective shear stress. As 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. 3.

2

3 Wing crack

1

1

3

Penny-shaped crack

2 6

Fig. 3. Propagation of wing cracks from the tip of penny-shaped microcrack Substituting Eq. (4) into Eq. (14), the mode II and mode III stress intensity factor at tips of penny-shaped microcracks in viscoelastic rock matrix can be expressed as [25]:

 4 fiu (t )(2 1   2 2 ) a  K II (t )  2     4 fiu (t )(1  )( 2 3   2)2 a  K ( t )  III  2   where  is the frictional coefficient on the crack surfaces,

(15)

 is Poisson’ s ratio, K II is the mode II stress intensity factor,

K III is the mode III stress intensity factor, f (t ) denotes the time factor. According to works by Tada [22], the condition of unstable growth of the mixed mode II-III microcracks can be expressed as

K II (t0 )  K III (t0 )   K IC where as

(16)

 can be obtained from experimental results, or approximation suggested in the literature on the kinked crack, such

  3 / 2 in the maximum-stress criterion [26], t0 is the time of creep failure of microcracks, K IC is the mode I

stress intensity factor, which can be obtained by induced tensile strength and crack length, namely

K IC  2 t

a



(17)

where  t is short-term uniaxial tensile strength of rocks. 2.3 The orientation angle of micro-failure in rocks

1

1

2

 3 Fig. 4. Wing crack distribution zone It is generally accepted that the failure of rocks is lead by the fragment of large amounts of internal microcracks. However, it is very difficult to qualitative the number of microcracks. Therefore, micro-failure orientation angle

 which

represents the number of propagating microcracks is introduced, as shown in Fig. 4. The fan-shaped area of wing crack 7

distribution zone in Fig. 4 can be obtained from Eqs(15)-(16). The included angle of the fan section is defined as the micro-failure orientation angle  . Substituting Eq.(15) into Eq.(16), the following equation could be written as:

  2      tan 2   2 1     sin  tan   2  2    3 sin 2    2 cos 2    2  2   3  sin  cos     1 3  t  fiu  t0   (18)   2       0   2  2    1 sin 2    2 cos 2    2  1   2  sin  cos   t  fiu  t0   where the compressive stresses are positive,  1 is the maximum principal stress,  2 is the intermediate principal stress,

 3 is the minimum principal stress. Eq.(18) can be rewritten as follows:

C1 tan 2   C2 tan   C3  0

where

(19)

  2    2 2 C1  2  2    3 sin    2 cos    2  2   3  sin  cos   t fiu (t0 )   . C2  2 1   1   3  sin   C  2  2    sin 2    cos 2    2     sin  cos    2    1 2 1 2 t  3 fiu (t0 )

From Eq. (19), the tangent of  can be determined by

tan 1  tan   tan 2

 C2   tan 1   where  C2    tan 2  

(20)

C22  4C1C3 2C1 C22  4C1C3 2C1

From Eq.(20), supposing   1   2 , the following equation could be obtained

C2 2  4C 1C tan 1  tan 2 tan   tan(1  2 )   1  tan 1 tan 2 C1  C3

3

(21)

2.4 Failure characteristic parameters of rocks The failure characteristic parameter of rock materials should be constant when rock materials entirely break. Damage mechanics revealed that the initiation of internal microcracks does not indicate failure of rock-like materials [20,27-29]. Many experiments show that the maximum principal stress should be further increased to assure that the wing crack continually propagates, 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 8

cannot be selected as the failure characteristic parameters. The larger the minimum principal stress is, the smaller the micro-failure orientation angle  is. The micro-failure orientation angle  is not constant, tan  , sin  and cos  are not also constant. Therefore, the micro-failure orientation angle  , tan  , sin  and cos  cannot be selected as the failure characteristic parameters. Microcracks randomly distribute in rock-like materials, and the orientation angle of each microcrack randomly distributes. Therefore, the 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. Reference [30] suggested that the failure of rock-like materials occurs when the volumetric strain due to the internal micro-failure 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 micro-failure orientation angle  . Moreover, the failure characteristic parameters should satisfy the following three principles: firstly, the expression of the micro-failure characteristic parameter should be in 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 expressions of the micro-failure orientation angle  , tan  and sin  are so complicated that it cannot be selected as the failure characteristic parameters. Compared with the expressions of

 , tan  and sin  , the

expression of cos  is the simplest. The expression of  cos   1 is also the simplest Therefore,  cos   1 satisfies the first and second principles. According to the second principle and Eq.(21), the cosine of the micro-failure orientation angle  can be written in following form:

    t  ( 1  2 2   )3 cos   (2  ) ( 1 2 2  ) 3 sin    2  2 csc    fiu (t0 )     cos    1   3  (1  )2  cos 2     2  (  2) sin 2  sin 2 

(22)

It is indicated from Eq.(22) that the cosine of the micro-failure orientation angle  increases with increasing the minimum principal stress  3 , while the micro-failure orientation angle  decreases with an increase in the minimum principal stress  3 . For an invariable intermediate principal stress  2 and an invariable minimum principal stress  3 , the relationship between cos  and the maximum principal stress can be defined. Differentiating Eq.(22) with respect to following equation is obtained. 9

 1 , the

    t  2( 3   2 ) cos   (2  )  2( 3   2 )  sin    2  2  csc     fiu (t0 )     cos    2  1  1   3  (1  )2  cos2      2  (  2) sin 2   sin 2  where

(23)

 cos  is defined as the rate of change of cos  to the maximum principal stress.  1

From Eq. (22), the maximum principal stress can be expressed as

cos   2 2   3  2   2   2 cot      2   3 sin    3 cos  

1 

cos    2    sin    cos 

   2   t csc fiu  t0 

(24)

where   (1  )2  cos2      2  (  2)  sin 2   sin 2  . Substituting Eq. (24) into Eq. (23), the following expression can be obtained as

 cos   cos    2    sin    cos    1      t  2    cos   (2   ) 2   csc   2     sin            3 2 2 2 3 fiu (t0 )      2

(25)

From Eq.(22), the increase of  3 results in the increase of cos  . From Eq.(25), the increase of  3 results in the increase of the denominator of  cos   1 , the increase of cos  results in the increase of the numerator of

 cos  1 . The rate of change  cos  1 is always a constant when the rock fractures no matter what the value of  3 is . Therefore, the rate of change  cos   1 can be regarded as the failure characteristic parameter of rocks. If the short-term uniaxial compressive strength of rocks is known, three-dimensional long-term strength criterion of rocks can be expressed by short-term uniaxial compressive strength of rocks. Therefore, for the short-term uniaxial compression condition  1   c ,  2  0 ,  3  0 , we can obtain the rate of change constant  cos  /  1 at t  0 as,

 cos   2  1  c 

  2    t csc fiu (0) (1  ) 2  cos 2      2  (  2)  sin 2   sin 2 

(26)

where  c is the short-term uniaxial compressive strength of rocks, fiu (0) is the time factor when t  0 . Substituting Eq. (23) into Eq. (26) yields

  2( 3   2 ) cos   (2  )  2( 3   2 )  sin    2 2     2  1   3 

   csc   fiu (t0 )    2    t csc   2  c  fiu (0)

 t

From Eq. (27), three-dimensional long-term strength criterion, which is expressed by the short-term uniaxial compressive strength of rocks, can be obtained as 10

(27)

 1   3   c  m 2 n 3 where m 

 2  2    sin   2cos    t  2   csc

fiu (0) c

, n

fiu ( 0 ) fiu (t0 )

 c 

2

(28)

 2  2     csc   sin    2cos    t  2   csc

It is observed from Eq. (28) that m and n are related to the friction coefficient  , the coefficient

fiu (0) c

.

 of mixed-mode

fracture criterion, the short-term uniaxial compressive strength  c , the short-term uniaxial tensile strength  t , the time factor fiu (0) , the dip angle of penny-shaped microcracks θ and Poisson’s ratio

.

If the long-term uniaxial compressive strength of rocks is known, three-dimensional long-term strength criterion of rocks can be expressed by long-term uniaxial compressive strength of rocks. Therefore, for the long-term uniaxial compressive condition  1   cl ,  2  0 ,  3  0 , we can obtain the rate of change constant  cos  /  1 at t  t0 as,

 cos   2  1  cl 

fiu (t0 )

 2    t csc (1  )2  cos 2      2  (  2)  sin 2   sin 2 

(29)

where  cl is the long-term uniaxial compressive strength of rocks, fiu (t0 ) is the time factor when t  t0 , t0 is the time of creep failure of materials under uniaxial compressive loads. Substituting Eq. (29) into Eq. (23), the following expression can be obtained:

  2( 3   2) cos   (2  )  2( 3  )2 sin    2 2    2  1   3 

   csc   fiu (t0 )    2    t csc   2  cl  fiu (t0 )

 t

(30)

From Eq. (30), three-dimensional long-term strength criterion of rocks, which is expressed by long-term uniaxial compressive strength of rocks, can be obtained as

1   3   c l m 2 n 3  where m 



2 cl

(31)

 2  2    sin   2cos   fiu (t0 ) cl  2  2     csc   sin    2cos   , n  t  2   csc  t  2   csc

It is observed from Eq. (31) that m and n are related to the friction coefficient  , the coefficient fracture criterion, the long-term uniaxial compressive strength

fiu (t0 ) cl

.

 of mixed-mode

 cl , the short-term uniaxial tensile strength  t , the time

factor fiu (t0 ) , the dip angle of penny-shaped microcracks θ and Poisson’s ratio

.

3. Comparison with the experimental results The Lode stress angle is defined as follows:

 2 3  ( 1   2 )    3( 1   2 ) 

  arctan 

( 30    30 ) 0

0

(32)

The stress tensor  ij expressed by the first invariant of stress tensor I1 and the second invariant of deviatoric stress tensor 11

J 2 can be written as follows: 2   I1   sin(   )  3   1  3    I1     2 J sin( ) 2    3   2 3    3  2   sin(   )   I1  3   3  

(33)

Three-dimensional long-term strength criterion (28) is rewritten in another form:

   3 fiu (0)  F  2q cos    c   c  3 p  m  n   q  3m cos    m  2n  sin     0    fiu (t0 ) 

(34)

where p  I1 / 3 , q  3J 2 . Similarly, three-dimensional long-term strength criterion (31) is rewritten in another form:





F  2q cos    cl 3 cl  3 p  m  n   q  3m cos    m  2n  sin    0

(35)

3.1 Comparison with the experimental data of coal Three-dimensional long-term strength criterion (28) is compared with the creep experimental results herein. This is confirmed by one set of the confined compression creep test on coal with strength parameters m+ n = 6.94 [31]. The creep parameters of coal are listed in Table 1[31]. According to Eq.(28) and Table 1, the expression of three-dimensional long-term strength criterion of coal is rewritten as follows.

19.519

 1   3  30.661 3 

(36)

1  0.35t0  1.23(1  e1.724t0 )

From Eq. (36), the theoretical unstable time of coal is 145.41h. From Table 1, the experimental unstable time of coal is 131h. Therefore, the theoretical result is in good agreement with the experimental data of coal. Table 1 Creep parameters of coal samples

Specimen No.

RB

Confining pressure /MPa 2

Axial pressure /MPa 10

Short-term uniaxial

Unstable

compressive

time/h

G1

G2

1

2

/GPa

/GPa

/GPa*h

/GPa*h

1.923

1.556

5.463

0.9023

K /GPa

strength/MPa 4.418

131

397.55

3.2 Comparison with the experimental data of different rocks Series sets of triaxial compressive experimental data were obtained from creep tests on the various rocks by Refs[9,10,32-43]. The long-term uniaxial compressive strength of rocks and the fitting strength parameters are listed in Table 2. In Figs 5-21, the curves are obtained from the proposed long-term strength criterion, dots are obtained from the experimental data. It is found from Figs5-21 that the proposed long-term strength criterion agrees well with the 12

experimental data of the different rocks. Table 2 Comparison with the experimental data of different rocks The fitting strength

Long-term uniaxial

Reference

parameters m+n

compressive strength

Barre granite

15.2

158

Kranz [36]

Inada granite

23.1

216

Maranini and Brignoli [38]

Beishan granite

19.2

111

Lin et al. [37]

Takidani granite

23.1

205

Brantut et al. [33]

Busted Butte tuff

18.4

129

Martin et al. [39]

Jinping marble

3.8

80

Yang et al. [43]

Darley Dale sandstone

4.1

67

Heap et al. [9]

Etna basalt

11.1

186

Heap et al. [10]

Tavel limestone

12.9

57

Brantut et al. [33]

Westerly granite

19.7

216

Brantut et al.[32].

Bentheim sandstone

4.6

53

Heap et al.[34]

Beringen siltstone

8.2

18

Hettema et al.[35]

Coal-measure shale rock

9.3

35

Mishra and Verma [40]

Crab Orchard sandstone

16.3

164

Heap et al.[34]

Jintan rock salt

6.2

35

Wang et al. [42]

Gyda sandstone

4. 6

76

Ngwenya et al. [41]

Magus sandstone

5.8

22

Ngwenya et al. [41]

Rocks

13

Fig. 5. Comparison between theoretical results and experimental data of the Barre granite.

14

Fig. 6. Comparison between theoretical results and experimental data of the Inada granite.

15

Fig. 7. Comparison between theoretical results and experimental data of the Beishan granite.

16

Fig. 8. Comparison between theoretical results and experimental data of the Takidani granite.

17

Fig. 9. Comparison between theoretical results and experimental data of the Busted Butte tuff.

18

Fig. 10. Comparison between theoretical results and experimental data of the deep buried marble.

19

Fig. 11. Comparison between theoretical results and the experimental data of the Darley Dale sandstone.

20

Fig. 12. Comparison between theoretical results and experimental data of the Etna basalt.

21

Fig. 13. Comparison between theoretical results and experimental data of the Tavel limestone.

22

Fig. 14. Comparison between theoretical results and experimental data of the Westerly granite.

23

Fig. 15. Comparison between theoretical results and experimental data of the Bentheim sandstone.

24

Fig. 16. Comparison between theoretical results and experimental data of the Beringen siltstone.

25

Fig. 17. Comparison between theoretical results and experimental data of the coal-measure shale rock.

26

Fig. 18. Comparison between theoretical results and experimental data of the Crab Orchard sandstone.

27

Fig. 19. Comparison between theoretical results and experimental data of the Jintan rock salt.

28

Fig. 20. Comparison between theoretical results and experimental data of the Gyda sandstone.

29

Fig. 21. Comparison between theoretical results and experimental data of the Magus sandstone.

4. Discussions and Conclusions In this paper, it is assumed that the three-dimensional penny-shaped microcracks are embedded in viscoelastic rock matrix. The expression of the stress intensity factors at tips of the creep microcracks is determined. A three-dimensional long-term strength criterion for rocks, in which the effects of time and the intermediate principal stress on the creep failure of rocks is taken into account, is proposed based on the micromechanical methods. Three-dimensional long-term strength criterion derived from the micromechanical methods is novel, which is not found in the previous references. By comparison with the previous experimental results, it is found that the novel three-dimensional long-term strength criterion is in good agreement with experimental data. However, rock-like materials contain different defects. The microcrack is one of these defects and different defects have different shapes, distribution and strength. The effects of different defects on the long-term strength of rock-like materials are different. The effects of different defects on the long-term 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 groups, not by an individual microcrack. The effects of interaction among microcracks on the long-term strength of rock-like 30

materials are not considered. In the further studies, the effects of different defects and interaction among microcracks on the long-term strength of rock-like materials should be taken into account.

Acknowledgments This work was supported by the National Natural Science Foundation of China (Nos. 51325903 and 51679017), project 973 (Grant no. 2014CB046903), Graduate Scientific Research and Innovation foundation of Chongqing, China (Grant No. CYB16012) ,Natural Science Foundation Project of CQ CSTC (Nos. cstc2013kjrc-ljrccj0001 and cstc2013jcyjys0005) and Research fund by the Doctoral Program of Higher Education of China(No.20130191110037).

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[33] N.Brantut, M. J. Heap, P. G. Meredith, P. Baud, Time-dependent cracking and brittle creep in crustal rocks: a review. J. Struct. Geol. 52(2013) 17-43. [34] M. J. Heap, P. Baud, P. G. Meredith,. Influence of temperature on brittle creep in sandstones, Geophy, Res, Lett, 36 (2009) L19305. [35] M. H. H. Hettema, C. J. De Pater, K. A. A. Wolf, Effects of temperature and pore water on creep of sandstone rock, U.s.symposium on Rock Mechanics,1991. [36] R. L. Kranz, The effects of confining pressure and stress difference on static fatigue of granite, J. Geophys. Res. D: Atmos. 85(1980) 1854-1866. [37] Q. X. Lin, Y. M. Liu, L. G. Tham, C. A. Tang, P. K. K. Lee, J. Wang, Time-dependent strength degradation of granite, Int. J. Rock Mech. Min. Sci. 46 (2009) 1103-1114. [38] E. Maranini, M. Brignoli, Creep behaviour of a weak rock. experimental characterization, Int. J. Rock Mech. Min. Sci. 36 (1999) 127-138. [39] R. J. Martin, J. S. Noel, P. J. Boyd, R. H. Price, Creep and static fatigue of welded tuff from Yucca Mountain, Nevada, Int. J. Rock Mech. Min. Sci. 34(1997) 190.e1–190.e17. [40] B.Mishra, P. Verma, Uniaxial and triaxial single and multistage creep tests on coal-measure shale rocks, Int. J. Coal Geol. 137(2015) 55-65. [41] B. T. Ngwenya, I.G. Main, S.C. Elphick, B. R. Crawford, B.G.D. Smart, A constitutive law for low-temperature creep of water-saturated sandstones, J. Geophys. Res. B: Solid Earth 106(2001) 21811–21826. [42] G. Wang, L. Zhang, Y. Zhang, G. Ding, Experimental investigations of the creep–damage–rupture behaviour of rock salt, Int. J. Rock Mech. Min. Sci. 66(2014) 181-187. [43] S. Q. Yang, P. Xu, P. G. Ranjith, G. F. Chen, H. W. Jing, Evaluation of creep mechanical behavior of deep-buried marble under triaxial cyclic loading, Arab. J. Geosci. 8(2015) 1-16.

33

Highlights A three-dimensional long-term strength criterion for rocks is proposed. The effects of the intermediate principal stress are taken into account. The proposed three-dimensional long-term strength criterion agrees well with experimental data. The mode II and III stress intensity factors at tips of penny-shaped microcracks are determined.

34