An innovative micromechanics-based three-dimensional long-term strength criterion for fracture assessment of rock materials

Rocks may exhibit time-dependent behaviors. Long-term strength criterion significantly dominates creep failure of rocks. Rocks contain many microcracks, which lead to degrade of long-term strength. In this paper, it is assumed that there exist three-dimensional penny-shaped microcracks in rocks. The mode II stress intensity factors at tips of three-dimensional pennyshaped microcracks in Burgers viscoelastic rock matrix is derived. A novel micromechanics-based three-dimensional long-term strength criterion is established to consider the effects of time and the intermediate principal stress on creep failure of rocks. By comparison with the previous experimental data, it is found that the novel micromechanics-based threedimensional long-term strength criterion is in good agreement with the experimental data.


INTRODUCTION
n the past several decades or more, extensive laboratory creep experiments were conducted to study the creep behaviors of many kinds of rocks [1][2][3][4][5].It is indicated that deformation of rocks under a constant load over extended a period of time generally exhibits primary or transient creep, lately by secondary or steady-rate creep, followed terminating in tertiary or accelerating creep that eventually progresses to dynamic rupture.Moreover, it is observed from laboratory creep experiments that the failure of rocks occurs at stresses well below the peak strength of rocks.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 [6].

I
To investigate the long-term strength of rock, some long-term strength criteria of rocks were established to study the creep behaviors of rocks, such as Mises-Schleicher &Drucker-Prager unified(MSDPu) criterion, and so on.However, these long-term strength criteria were established using phenomenological approaches, which can produce the macroscopically observed creep curves of rocks by fitting with experimental data, and the inherent physical mechanisms related to time-dependent behaviors are not accommodated in these models, so the key mechanistic parameters remain physically unclear [7].To authors' knowledge, three-dimensional long-term strength criterion of rocks, in which the effects of the intermediate principal stress are considered, is not proposed by using micromechanical methods.In fact, rock is a kind of discontinuity medium containing many microcracks and microdefects, the presence of such microcracks strongly influences the macroscopic mechanical behavior of rocks by serving as stress concentrators and leading to microcracking [8][9][10][11][12].To overcome the disadvantages encountered in phenomenological models, it is necessary to study the effects of initiation and propagation of microcracks and microdefects on the creep failure of rocks.In this paper, micromechanical methods are used to investigate the lone-term strength of rocks.Moreover, a novel micromechanics-based three-dimensional nonlinear long-term strength criterion is established to study the effects of time and the intermediate principal stress on the creep failure of rocks.By comparison with experimental data, it is found that the novel micromechanics-based threedimensional long-term strength criterion is in good agreement with the experimental data.

THE ANALYTICAL MODEL
t is generally accepted that the creep deformation and fracturing process that evolve in rocks are closely related to the intrinsic property and stress condition of rocks, such as fracture toughness, internal frictional angle, the dip and orientation angle of microcracks, Poisson's ratio, and so on.In this paper, it is assumed that the creep failure of rocks is due to the presence of penny-shaped microcracks and there is abundant evidence for the existence of microcracks in rocks [13][14].Therefore, this model is physically plausible and the following assumptions are made: (i) penny-shaped microcracks are assumed to be randomly distributed in Burgers viscoelastic rock matrix; (ii) the interaction between penny-shaped microcracks is neglected before the coalescence of microcracks.

Stress intensity factor of penny-shaped microcracks embedded in Burgers viscoelastic rock matrix
It is assumed that the tensile stress is negative, and the compressive stress is positive.Consider a single penny-shaped creep microcrack in Burgers viscoelastic rock matrix uniformly loaded at far field.Establish a global coordinate system (  O x x x 1 2 3 .)and its corresponding local coordinate system ( 3 ), as shown in Fig. 1.In a global coordinate system (  O x x x 1 2 3 ), the direction of the maximum principal stress is parallel to the x 1 -axis, the direction of the intermediate principal stress is parallel to the x 2 -axis, the direction of the minimum principal stress is parallel to the x 3 - axis.In the local coordinate system (     O x x x 1 2 3 ), the direction of the  x 2 -axis is parallel to the normal direction of penny-shaped creep microcrack.The angle between the  x 2 -axis and the x 2 -axis is the dip angle of penny-shaped creep microcrack  .The angle between the  x 3 -axis and the x 3 -axis is the orientation angle of penny-shaped creep microcrack  .

I
The stresses in the local coordinate system are given by Yu and Feng [15], where, Yu and Feng [15] and Tada [16] defined the stress intensity factors at tips of penny-shaped microcracks embedded in isotropic and elastic rock matrix as where μ is the frictional coefficient on the crack surfaces,  is Poisson's ratio, KII is the mode II stress intensity factor.

The Burgers creep model
In this paper, it is assumed that microcracks are embedded in Burgers viscoelastic rock matrix with the characteristic of instantaneous elastic deformation, primary creep and steady-rate creep.As shown in Fig. 2, Burgers model can be expressed as follows where G 1 is Maxwell shear modulus, G 2 is Kelvin shear modulus,  1 is Maxwell viscosity, and  2 is Kelvin viscosity, ,  ij is stress tensor,  ij is strain tensor.The Maxwell shear modulus is equal to elasticity shear modulus,   Eq.( 5) can be rewritten as where t is the creep time.
From Eq.( 6) and works by Yi and Zhu [17], the time factor of the Burgers model under a given load is obtained as where iu f t ( ) is the time factor for displacement, According to works by Zhou [18], energy release rate at tips of the mixed mode I-II-III microcracks in Burgers viscoelastic rock matrix can be written as where ( ) 1 1 exp .
In Eq. ( 8), G t ( ) can be rewritten as 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 obtained as follows: According to the definition of stress intensity factor, stress intensity factor at tips of microcracks can be denoted by Based on the definition of energy release rate, energy release rate at tips of microcracks can be defined by where a is the growth length of microcracks.Substituting Eq. ( 10) into Eq.( 12) yields From Eq. ( 13) and Eq. ( 9), the stress intensity factors of creep cracks can be written as For three-dimensional penny-shaped microcracks, frictional sliding is caused by the effective shear stress.As the effective shear force 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. Substituting Eq. ( 4) into Eq.( 14) yields: where  is the frictional coefficient on the crack surfaces,  is Poisson' s ratio, II K is the mode II stress intensity factor, f t ( ) denotes the time factor.According to works by Tada [16], the condition of unstable growth of the mode II microcracks can be written as where where  t is short-term uniaxial tensile strength of rocks.

THE ORIENTATION ANGLE OF MICRO-FAILURE IN ROCKS
t is generally accepted that the creep failure of rocks is induced by the fragment of large amounts of internal microcracks.However, it is very difficult to quantitatively analyze the number of microcracks.Therefore, microfailure orientation angle  is introduced to define the number of propagating penny-shaped creepmicrocracks, as shown in Fig. 4. The fan-shaped area of wing crack distribution zone shown in Fig. 4 can be obtained from Eqs( 15)-( 16).
The included angle of the fan section is denoted as the micro-failure orientation angle  .Substituting Eq.( 15) into Eq.(16) yields: ,  1 is the maximum principal stress,  2 is the intermediate principal stress,  3 is the minimum principal stress, the compressive stresses are defined to be positive in this paper.Eq.( 18) can be rewritten as : where From Eq. ( 19), the cosine of  can be written as where From Eq.( 20), supposing      2 1 , the following equation can be written


cos cannot be considered as the creep failure characteristic parameters.Microcracks randomly distribute in Burgers viscoelastic rock matrix, and the orientation angle of each microcrack randomly distributes.Therefore, the micro-failure orientation angle  can be adopted to investigate the micro-failure density.An increase in the minimum principal stress leads to a decrease in the micro-failure density.The internal microfailure density does not keep constant.Therefore, the micro-failure density cannot also be chosen as the creep failure characteristic parameters.Reference [24] suggested that the creep failure of rocks occurs when the volumetric strain due to the internal micro-failure density reaches a critical value.Therefore, the creep failure characteristic parameters of rocks should be relevant to the internal micro-failure density, which is related to the micro-failure orientation angle  .Moreover, the creep failure characteristic parameters should satisfy the following three principles: firstly, the expression of the creep 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 chosen as the creep failure characteristic parameters.Compared with the expressions of  ,  tan and  sin , the expression of  cos is the simplest.The expression of    c 1 is also the simplest Therefore,    c 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 expressed in following form: where  22) that the cosine of the micro-failure orientation angle increases with an increase in the minimum principal stress  3 , while the micro-failure orientation angle  decreases with increasing 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  1 yields: From Eq. ( 22), the maximum principal stress can be expressed as Substituting Eq. ( 24) into Eq.( 23) yields 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 where , c is the short-term uniaxial compressive strength of rocks, iu f (0) is the time factor when  t 0 .
Substituting Eq. ( 23) into Eq.( 26) yields From Eq. ( 27), three-dimensional long-term strength criterion expressed by the short-term uniaxial compressive strength of rocks can be denoted by where It is observed from Eq. ( 28) that      is related to the friction coefficient  , the coefficient  of mixed-mode fracture criterion, the short-term uniaxial compressive strength  c , the short-term uniaxial tensile strength  t , the time factor iu f (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 where ( ) , cl is the long-term uniaxial compressive strength of rocks, iu f t 0 ( ) is the time factor when  t t 0 , t 0 is the time of creep failure of rocks under uniaxial compressive loads.Substituting Eq. (29) into Eq.( 23) yields: From Eq. (30), micromechanics-based three-dimensional long-term strength criterion of rocks expressed by long-term uniaxial compressive strength of rocks can written as where n sin sin sin cos 3 is related to the friction coefficient  , the coefficient  of mixed-mode fracture criterion, the long-term uniaxial compressive strength  cl , the short-term uniaxial tensile strength  t , the time factor iu f t 0 ( ) , the dip angle of penny-shaped microcracks θ and Poisson's ratio  .Assumed that Eq.(31) can be simplified to: where s n , and m are the strength parameters which are determined by experiments.
When Eq.( 32) is expressed by the short-term uniaxial compressive strength  c , Eq.( 32) can be rewritten as where  c is the short-term uniaxial compressive strength.

COMPARISON WITH THE EXPERIMENTAL RESULTS
he Lode stress angle is defined as follows: The stress tensor ij  expressed by the first invariant 1 I of stress tensor and the second invariant of deviatoric stress tensor J 2 can be written as follows: Micromechanics-based three-dimensional long-term strength criterion (32) can be expressed by the first invariant I 1 of stress tensor and the second invariant of deviatoric stress tensor J 2 , the following expression can be obtained where Similarly, Micromechanics-based three-dimensional long-term strength criterion (33) can be rewritten in another form: where

COMPARISON WITH THE EXPERIMENTAL DATA OF COAL
eries of triaxial compressive experimental data were obtained from creep tests on various rocks by Refs [25][26][27].The long-term uniaxial compressive strength of rocks and the fitting strength parameters are listed in Tab. 1. Tabs.2-4 show theoretical strength and the experimental data of Barre granite, Inada granite and Jinping marble.Figs 5-7 show that comparison of predicted strength and the experimental data of Barre granite, Inada granite and Jinping marble.It is found from Tabs 2-4 and Figs 5-7 that the proposed long-term strength criterion agrees well with experimental data of different rocks.

Rocks
The strength parameter(s) The fitting strength parameter(m)

Figure 1 :
Figure 1: Mechanical model for penny-shaped microcrack embedded by Burgers viscoelastic rock matrix.

Figure 2 :
Figure 2: The diagram of Burgers model.
10) where m =I, II and III, which are, respectively, denoted by mode I, II and III microcracks,  m ij ( ) and m i u ( ) are, respectively, the stress and displacement fields at tips of microcracks in elastic rock matrix, stress and displacement fields at tips of microcracks in Burgers viscoelastic rock matrix, m K t ( ) and m K are, respectively, stress intensity factor at tips of microcracks in Burgers viscoelastic and elastic rock matrix.


can be obtained from experimental results, or approximation suggested in the literature on the kinked crack, such as   3 / 2 in the maximum-stress criterion[19], t 0 is the time of creep failure of microcracks, IC K is toughness of rocks, which can be obtained by induced tensile strength and crack length, namely

Figure 3 :
Figure 3: Propagation of wing cracks from the tip of penny-shaped microcrack.

 2  1  1  3 
CREEP FAILURE CHARACTERISTIC PARAMETERS OF ROCKShe creep failure characteristic parameter of rocks should be constant when rocks entirely break.Damage mechanics reveals that the nucleation and initiation of microcracks does not imply creep failure of rock-like materials[20][21][22].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 wing crack to grow[23].Therefore, the initiation of wing cracks cannot indicate creep failure of rocks.As a result, nucleation and initiation of internal microcracks cannot be chosen as the creep failure characteristic parameters.The larger the minimum principal stress, the smaller the micro-failure orientation angle .The micro-failure orientation angle  does not keep constant,  tan ,  sin and  cos do not also keep constant.Therefore, the micro-failure orientation angle ,  tan ,  sin and as the rate of change of  cos to the maximum principal stress.

Figure 5 :
Figure 5: Comparison of predicted strength and the experimental data of Inada granite.

Figure 6 :
Figure 6: Comparison of predicted strength and the experimental data of Barre granite.

Figure 7 :
Figure 7: Comparison of predicted strength and the experimental data of Jinping marble

Table 1 :
The fitting strength parameters and uniaxial compressive strength of different rocks.

Table 2 :
Theoretical strength and the experimental data of Inada granite.

Table 3 :
Theoretical strength and the experimental data of Barre granite.

Table 4 :
Theoretical strength and the experimental data of Jinping marble.