Research Article  Open Access
Crack Initiation Criteria and Fracture Simulation for Precracked Sandstones
Abstract
The friction coefficient, tip curvature, and differentwidth crack state influence the stress intensity factor (SIF). The maximum circumferential tensile stress (MTS) and minimum strain energy density criterion (S) face challenges in explaining the modeII fracture propagation of cracks. The maximum radial shear stress (MSS) and modified twin shear stress factor (ITS) criteria are proposed as the brittle modeII fracture criteria. The experiments and numerical analysis are also performed. The results indicate that the fracture angles of the MSS and ITS were similar and different from the results of MTS and S. The equivalent stress intensity factors (ESIFs) from the mixed mode III are proposed to determine the fracture mode. There are different fracture models for different cracks under tensile and compressive stresses. The ratio of the tensile strength to uniaxial compressive strength influenced the fracture angle of ITS. The lateral pressure coefficient (k) had a significant effect on the modeII fracture angle when the angle between the crack and the vertical direction is less than 40° and the lateral pressure coefficient is more than 0. Because the same fracture mode k (k > 0) can inhibit modeI fracturing, conversely, it can also promote modeI fracturing. Experimental results and numerical simulations of fracture propagation under uniaxial compression confirmed that the theoretical results were correct.
1. Introduction
Rock develops in complex geological environments and includes various defects or flaws. These flaws weaken the mechanical properties of the rock mass and modify the stress distributions. Internal cracks in rock mass exert an important effect on determining the fracture mode, initiation, propagation, and rock strength [1–9].
The cracks in a rock mass can be divided into three basic modes: modeI crack (normal load), modeII crack (shear load), and modeIII crack (antiplane shear load). A tensile, shear, and tear crack fracture is defined as a mode I, II, and III fracture, respectively [10]. The mixed mode III is the most common type of a mixed mode fracture. After fracture mechanics was introduced into rock mechanics, the SIF was employed to describe the stress state at the crack tip [11]. When the shear stress acting on the main crack exceeds the friction stress between the cracks, the stress will concentrate at the crack tip. The cracks continue to grow and curve toward the direction of the maximum principal stress, when the stress strength factor meets or exceeds K_{IC}. Based on the maximum shear stress theory, a method has been proposed to compute the crack initiation angle under a mixed mode III fracture [12]. The opentype crack geometry influences crack propagation [12, 13]. The crack initiation and growth on a rock specimen subjected to compressive stress has been investigated experimentally [1, 3, 7, 14–26]. Many numerical methods have been used to analyze the fracture and crack propagation in rocks, such as the finite element method (FEM), boundary element method (BEM), and discrete element method (DEM). By using the finite element fracture software called Franc2D, the energy release rate (G), crack propagation, fracturing time, and static tensile and normaldistributed stresses were calculated to represent the crack initiation and growth in a rock specimen [11, 27]. A realistic failure process analysis has been developed to simulate the cracked rock failure [28]. The PFC2D software has been employed to discuss the effect of the initial flaw orientation in the specimen’s failure mode under compressive loading [29]. A fracture and crack propagation analysis system has been employed in the investigation of crack growth [30]. The maximum tangential stress [31], maximum energy release rate [32], and minimum energy density criterion [33, 34] have typically been considered as the fracture initiation criteria to identify the crack growth mechanism of brittle rocks. The Fcriterion and modified energy release rate criterion have also been used to investigate quasibrittle fracture characteristics [35–37].
These criteria are based on the assumption of a modeI fracture. However, the modeII fracture extension of compressionshear cracks has rarely been investigated. Approaches toward identifying a fracture mode that can determine suitable fracture criteria are lacking. Therefore, it is difficult to determine the fracture mode of a crack and the modes I, II, or III fracture toughness according to experimental results.
The main objective of this paper was to propose a modeII fracture criterion and conduct a numerical analysis of an open and closed brittle rock cracks. Mixed mode III ESIFs are proposed to determine the fracture mode. The relationships between the fracture angle and model of crack propagation with a crack angle and thickness, lateral pressure coefficient, and ratio of tensile strength to compressive strength are discussed. The consistency between the theoretical and numerical results was verified.
2. SIF of Mixed Mode III Crack
According to the linear elastic method, the stress components of the mixed mode III at the crack tip in the polar coordinates can be expressed as follows [38]:where K_{I} is the modeI SIF, K_{II} is the modeII SIF, r is the distance from the crack tip, and is the angle by which the surface deviates from the original crack tip direction.
An infinite plate with a central crack under biaxial loading ( and , ) is shown in Figure 1, where the crack length is 2a. The stress state along the crack plane can be determined as follows:where , , and are the tangential stress, normal stress, and shear stress. The tensile stress is positive, while the compressive stress is negative.
The SIF for cracks with different widths, which causes friction and no friction along the crack plane, is different. and are SIF generated by the transverse compressive stress and normal stress, respectively. is only considered when the transverse compressive stress , and when . In addition, . ModeI SIF (K_{I}) is only considered when the crack is tensile (σ_{N} > 0) for a closed crack (causing friction), and K_{I} = 0 when σ_{N} ≤ 0. ModeI SIF is affected by the crack tip radius of curvature (ρ) and the transverse compressive stress () for a nonclosed crack (no friction along the crack plane) [39] (Figure 1), but for σ_{N} ≤ 0. K_{I} exerts an inhibitory effect on the circumferential stress of modeII SIF (K_{II}). Table 1 lists the results of calculating SIF for closed and nonclosed cracks.

3. Fracture Criterion of Brittle Material
3.1. Maximum Circumferential Tensile Stress Theory (MTS)
The maximum circumferential tensile stress criterion can effectively explain the tensile fracture of brittle rock. The crack propagation direction at the crack tip can be obtained by using the maximum circumferential stress [31]. The equivalent modeI stress intensity factor (K_{Ie}), which is transformed from the mixed mode III fracture, is defined as follows:where the initiation angle .
When K_{Ie} equals the modeI fracture toughness (K_{IC}), the crack initiates, and K_{I} is 0 for a closed crack. Thus, θ_{0} = 70.5° (when , , thus θ_{0} = 70.5°), and K_{Ie} becomes maximum for closed cracks.
3.2. Minimum Strain Energy Density Criterion (S)
The fracture angle of the crack is determined by the direction of the minimum strain energy density, and the strain energy density of a near crack tip element is expressed as follows [40]:where K_{III} is the modeIII SIF (K_{III} = 0) and , , , and can be obtained as follows:where μ is the shear modulus and ν is Poisson’s ratio. The initiation angle (θ_{0}) can be obtained by and .
There exist more fracture criteria, including the maximum potential energy release rate criterion and energymomentum tensor criterion. However, these fracture criteria require a modeI fracture. Therefore, the radial shear stress criterion and modified twin shear stress factor criterion for a modeII fracture are proposed.
3.3. Maximum Radial Shear Stress Criterion (MSS)
The maximum radial shear stress at the crack tip should satisfy the following [41]:
θ_{0} is expressed as follows:where , , , , , and .
3.4. Modified Twin Shear Stress Factor Criterion (ITS)
The twin shear stress factor criterion can be used to analyze the initiation angle of a pure modeI fracture. However, there is significant deviation in the investigation of a mixed mode III fracture. Moreover, an improved twin shear stress factor criterion is proposed to predict the modeII fracture angle.
For 2D plane strain problems, the principal stresses and can be determined as follows:
Based on equation (8), can be obtained by assuming plane strain (). This can satisfy .
The twin shear stress f can be determined by the shear stresses , , and , as follows:where the principle shear stresses can be expressed as , , and and α is the ratio of to of the rocks such that .
By substituting equations (1a)–(1c) and (8) into equations (9a) and (9b), f can be expressed as follows:where is the twin shear stress factor.
The shear stress is constant () on the radial shear stress line, and f on the equal radial shear stress line stress can be expressed as follows:where is given by
The crack fractures along the direction of the twin shear stress minimum value, and the fracture angle are given by
The equivalent modeII SIF that transforms from the mixed mode III fracture can be expressed as follows:
4. Fracture Mode and Fracture Criterion
4.1. Identification of Fracture Mode
The dimensionless stress field at the crack tip can be expressed as follows [42]:
For most rocks, K_{IC} is less than K_{IIC} [43]. The fracture mode can be determined according to the relationship between the stress field and fracture toughness.
The dimensionless stress field of the mixed mode III crack is shown in Figure 2. We can obtain for the modes I () and II crack shown in Figures 2(a) and 2(c), i.e., . Thus, a modeI fracture occurs. However, there exists for the modeI crack without friction along the crack plane (), that is, , which occurs with the modeII fracture.
(a)
(b)
(c)
For a mixed mode III crack, and are superimposed. The circumferential stress field occurs according to Figures 2(a) and 2(c). If , , a modeI fracture occurs. Otherwise, the circumferential stress decreases according to Figures 2(b) and 2(c), when . The circumferential stress field is the same as that shown in Figure 2(c) for a mixed mode III closed crack, when . Thus, and the fracture is modeI. For a mixed mode III crack without friction, when , the fracture mode can be expressed as follows:
4.2. Example: Mixed Mode III Nonclosed Crack under Compression
By assuming that the crack aperture was 2 mm, the crack length was 2a = 10 mm, the curvature radius was ρ = 1 mm at the crack tip, and the crack was nonclosed during loading. The relationship between the crack angles and the fracture angles is shown in Figure 3, where the mixed mode III nonclosed crack was analyzed according to MTS, S, MSS, and ITS. The fracture angles based on MTS and S were similar. The fracture angles based on S were influenced by ; however, the results were quite different when β < 40° and k < 0. Otherwise, the fracture angles between MSS and ITS were similar and the fracture angles from ITS were influenced by . The lateral pressure coefficient (k > 0) exerted a significant effect on the modeII fracture angle when β < 40° (Figures 3(a) and 3(b)).
(a)
(b)
(c)
The mode III fracture regions for the mixed mode III nonclosed cracks (equations (16a)–(16d)) are shown in Figure 4. k exerted a significant effect on the fracture mode. The positive k (k > 0) value inhibited the modeI fracture (Figures 4(a) and 4(b)), while the negative k value promoted the modeI fracture (Figures 4(a) and 4(c)).
(a)
(b)
(c)
The criteria selected to analyze the fracture angles of the crack were based on the fracture mode. In the modeI fracture, the fracture criteria, such as MTS and S, could forecast the fracturing angle. Moreover, the MSS and ITS criteria are suggested for the modeII fracture.
5. Numerical Analysis and Experimental Results of SingleCrack Sample Failure
5.1. Numerical Model
The SIF of a singlecrack sandstone was investigated by using a finite element software ABAQUS. Using the commercial finite element software, the extended FEM and a cohesive model were employed to simulate the crack propagation for a singlecrack rock without considering the progressive process [44, 45]. The Benzeggagh–Kenane (BK) model can be expressed as follows [46]:where is the material parameter and and are the energy release rates of the mode III fracture, , , and , is the stress threshold of the crack fracture, and is the displacement of the cohesive model when fracture occurs.
The computational models of the rock samples are described in Figure 5. The mechanical simulation analysis parameters are listed in Table 2. The SIF of a single crack was obtained numerically based on the elasticity theory, by applying a compressive stress of 10 MPa and k = 0. A collapse element was employed to simulate the singularity of the crack tip.
(a)
(b)

Table 3 depicts the changes in the maximum circumferential stress and the radial shear stress at the crack tip. Figures 6–8 show the computational results of a singlecrack brittle rock fracture with different precracked angles.
 
The tip radius of curvature is 0.25 mm for conditions 1–5 and 1.0 mm for conditions 6–10. 
(a)
(b)
(a)
(b)
(a)
(b)
5.2. Analysis of Closed Crack Fracture
When the crack angle was smaller, the crack extended with a small angle (1–5 in Table 3), rather than with an angle of 70.5°, from the maximum circumferential stress theory. This was caused by the strong influence of material expansion, which was in turn caused by compression. Therefore, equation (1b) cannot be used to calculate the circumferential stress with a small crack angle. However, the material expansion had a smaller effect on the stress field at the crack tip, when the crack angle was greater than 15°. By using the maximum circumferential stress criterion, the fracture initiation angles in the experiments were approximately 70.5°. Therefore, equation (1b) could be used to calculate the circumferential stress in a rock with crack angles greater than 15°.
The results revealed that the material expansion exerted a considerable effect on the stress at the crack tips, when the crack angle was small and the initiation angles gradually approached 70.5°, according to the maximum circumferential stress theory. Therefore, the maximum circumferential stress theory can be used to estimate the fracture criterion and calculate the compressioninduced initiation angles for the closed cracks, if the crack angle is greater than 15°.
5.3. Fracture Analysis of Nonclosed Crack
The modeI SIF of the nonclosed crack was negative when the crack was under compression (). According to equation (1), the modeI circumferential compressive stress with K_{I} < 0 at the crack tip will restrain the circumferential tensile stress caused by the modeII SIF. The maximum circumferential tensile stress at the crack tip was less than the radial shearing stress under certain conditions. Thus, the modeII fracture in a nonclosed crack occurred only if and . Therefore, the mode domain could be divided into two regions, namely, the modeI and modeII fracture regions (Figure 4). Figure 9 represents the crack propagation between numerical and experimental results where crack thickness is 2 mm. The tests were carried out on an MTS815 test system (MTS Systems Corporation, Eden Prairie, MN, USA). The sample was from rocklike material with 2 mm thick crack with a width of 50 mm. The testing and numerical results show that the initiation angle of the crack and fracture propagation are similar. Figure 10 compares the initiation angles between the theoretical analysis and numerical results when and . The testing results indicate that fracture criteria for predicting the crack propagation is closely related to precracked angle.
(a)
(b)
The modeI fracture occurred in a sample with a nonclosed crack, when the crack angle was less than 45° (Figure 4(a)). The crack rupture angles were small, which does not agree with the initiation angles (70.5°) obtained by the MTS and listed in Table 3. For crack angles greater than 30°, there existed considerable differences between the fracture angles of the closed and nonclosed cracks (Figure 8), and the rupture angles of the open cracks were similar to the results obtained by the MSS and ITS (Figure 10). Therefore, the identification of the fracture mode and proposed modeII fracture criteria is correct.
The analysis revealed that the maximum circumferential stress criterion faces challenges in describing the fracture growth of open cracks subject to compression. The initiation angles could be determined by the radial shear stress criterion if and were satisfied.
6. Conclusions
The modeII fracture criteria and a method of fracture mode identification were proposed. We conducted a uniaxial compression numerical analysis on brittle material containing closed and open cracks at various angles and thicknesses. The numerical and theoretical results were analyzed and the following conclusions were drawn:(1)The fracture angles, based on the MSS and ITS, were close. The ITS results were influenced by the ratio of the tensile strength to the uniaxial compressive strength of the rocks. k exerted an important effect on the modeII fracture angle. A positive k value inhibited the modeI fracture, while a negative k value promoted the modeI fracture.(2)Friction caused by crack thickness has an important role in the fracture initiation angle. The initiation direction of the closed crack at the crack tip approximates the direction of the maximum circumferential stress. When the angle between the preloaded crack and the loading direction was large, the crack initiation behavior under loading conformed to the results of the maximum circumferential stress criterion.(3)MTS faces challenges in interpreting the fracture propagation of open cracks subject to compressional stress. The circumferential compressive stress caused by the modeI SIF withheld the circumferential tensile stress from the modeII SIF. For and , the fracture angle of the nonclosed crack was determined by MSS.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors thank the project supported by the Funds for Creative Research Groups of China (no. 41521002) and the National Natural Science Foundation of China (no. 41672282). Huang thanks the Innovative Team of the Chengdu University of Technology.
References
 A. Bobet and H. H. Einstein, “Fracture coalescence in rocktype materials under uniaxial and biaxial compression,” International Journal of Rock Mechanics and Mining Sciences, vol. 35, no. 7, pp. 863–888, 1998. View at: Publisher Site  Google Scholar
 A. Bobet, “The initiation of secondary cracks in compression,” Engineering Fracture Mechanics, vol. 66, no. 2, pp. 187–219, 2000. View at: Publisher Site  Google Scholar
 C. H. Park and A. Bobet, “Crack coalescence in specimens with open and closed flaws: a comparison,” International Journal of Rock Mechanics and Mining Sciences, vol. 46, no. 5, pp. 819–829, 2009. View at: Publisher Site  Google Scholar
 M. Sagong and A. Bobet, “Coalescence of multiple flaws in a rockmodel material in uniaxial compression,” International Journal of Rock Mechanics and Mining Sciences, vol. 39, no. 2, pp. 229–241, 2002. View at: Publisher Site  Google Scholar
 C. A. Tang, P. Lin, R. H. C. Wong, and K. T. Chau, “Analysis of crack coalescence in rocklike materials containing three flawsPart II: numerical approach,” International Journal of Rock Mechanics and Mining Sciences, vol. 38, no. 7, pp. 925–939, 2001. View at: Publisher Site  Google Scholar
 B. Vásárhelyi and A. Bobet, “Modeling of crack initiation, propagation and coalescence in uniaxial compression,” Rock Mechanics and Rock Engineering, vol. 33, no. 2, pp. 119–139, 2000. View at: Publisher Site  Google Scholar
 R. H. C. Wong, K. T. Chau, C. A. Tang, and P. Lin, “Analysis of crack coalescence in rocklike materials containing three flawsPart I: experimental approach,” International Journal of Rock Mechanics and Mining Sciences, vol. 38, no. 7, pp. 909–924, 2001. View at: Publisher Site  Google Scholar
 L. Z. Wu, B. Li, R. Q. Huang, and P. Sun, “Experimental study and modeling of shear rheology in sandstone with nonpersistent joints,” Engineering Geology, vol. 222, pp. 201–211, 2017. View at: Publisher Site  Google Scholar
 L. Z. Wu, H. Deng, R. Q. Huang, L. M. Zhang, X. G. Guo, and Y. Zhou, “Evolution of lakes created by landslide dams and the role of dam erosion: a case study of the Jiajun landslide on the Dadu River, China,” Quaternary International, vol. 503, pp. 41–50, 2019. View at: Publisher Site  Google Scholar
 G. R. Irwin, “Analysis of stresses and strains near the end of a crack traversing a plate,” Journal of Applied MechanicsTransactions, vol. 22, pp. 361–364, 1957. View at: Google Scholar
 A. M. AlMukhtar and B. Merkel, “Simulation of the crack propagation in rocks using fracture mechanics approach,” Journal of Failure Analysis and Prevention, vol. 15, no. 1, pp. 90–100, 2015. View at: Publisher Site  Google Scholar
 N. A. AlShayea, “Crack propagation trajectories for rocks under mixed mode III fracture,” Engineering Geology, vol. 81, no. 1, pp. 84–97, 2005. View at: Publisher Site  Google Scholar
 Y. P. Li and C. H. Yang, “Influence of geometric characteristics of preexisting cracks on mixed mode fractures under compressionshear loading,” Chinese Journal of Rock Mechanics and Engineering, vol. 25, pp. 462–466, 2006. View at: Google Scholar
 M. R. Ayatollahi and M. R. M. Aliha, “On the use of Brazilian disc specimen for calculating mixed mode III fracture toughness of rock materials,” Engineering Fracture Mechanics, vol. 75, no. 16, pp. 4631–4641, 2008. View at: Publisher Site  Google Scholar
 Y. Fujii and Y. Ishijima, “Consideration of fracture growth from an inclined slit and inclined initial fracture at the surface of rock and mortar in compression,” International Journal of Rock Mechanics and Mining Sciences, vol. 41, no. 6, pp. 1035–1041, 2004. View at: Publisher Site  Google Scholar
 H. Horii and S. NematNasser, “Compressioninduced microcrack growth in brittle solids: axial splitting and shear failure,” Journal of Geophysical Research, vol. 90, no. B4, pp. 3105–3125, 1985. View at: Publisher Site  Google Scholar
 R. P. Janeiro and H. H. Einstein, “Experimental study of the cracking behavior of specimens containing inclusions (under uniaxial compression),” International Journal of Fracture, vol. 164, no. 1, pp. 83–102, 2010. View at: Publisher Site  Google Scholar
 Y.P. Li, L.Z. Chen, and Y.H. Wang, “Experimental research on precracked marble under compression,” International Journal of Solids and Structures, vol. 42, no. 910, pp. 2505–2516, 2005. View at: Publisher Site  Google Scholar
 H. Lee and S. Jeon, “An experimental and numerical study of fracture coalescence in precracked specimens under uniaxial compression,” International Journal of Solids and Structures, vol. 48, no. 6, pp. 979–999, 2011. View at: Publisher Site  Google Scholar
 C. H. Park and A. Bobet, “The initiation of slip on frictional fractures,” in Proceedings of the 41st US Rock Mechanics Symposium, pp. 06–923, Golden, CO, USA, June 2006. View at: Google Scholar
 C. H. Park and A. Bobet, “Crack initiation and propagation from frictional fractures,” in Proceedings of the 1st Canada—US Rock Mechanics Symposium, pp. 557–564, Vancouver, Canada, May 2007. View at: Google Scholar
 C. H. Park and A. Bobet, “Crack initiation, propagation and coalescence from frictional flaws in uniaxial compression,” Engineering Fracture Mechanics, vol. 77, no. 14, pp. 2727–2748, 2010. View at: Publisher Site  Google Scholar
 E. Sahouryeh, A. V. Dyskin, and L. N. Germanovich, “Crack growth under biaxial compression,” Engineering Fracture Mechanics, vol. 69, no. 18, pp. 2187–2198, 2002. View at: Publisher Site  Google Scholar
 R. H. C. Wong and K. T. Chau, “Crack coalescence in a rocklike material containing two cracks,” International Journal of Rock Mechanics and Mining Sciences, vol. 35, no. 2, pp. 147–164, 1998. View at: Publisher Site  Google Scholar
 S.Q. Yang, “Crack coalescence behavior of brittle sandstone samples containing two coplanar fissures in the process of deformation failure,” Engineering Fracture Mechanics, vol. 78, no. 17, pp. 3059–3081, 2011. View at: Publisher Site  Google Scholar
 Y. Zhao, “Crack pattern evolution and a fractal damage constitutive model for rock,” International Journal of Rock Mechanics and Mining Sciences, vol. 35, no. 3, pp. 349–366, 1998. View at: Publisher Site  Google Scholar
 L. Z. Wu, G. Q. Shao, R. Q. Huang, and Q. He, “Overhanging rock: theoretical, physical and numerical modeling,” Rock Mechanics and Rock Engineering, vol. 51, no. 11, pp. 3585–3597, 2018. View at: Publisher Site  Google Scholar
 C. Tang, “Numerical simulation of progressive rock failure and associated seismicity,” International Journal of Rock Mechanics and Mining Sciences, vol. 34, no. 2, pp. 249–261, 1997. View at: Publisher Site  Google Scholar
 A. Manouchehrian, M. Sharifzadeh, M. F. Marji, and J. Gholamnejad, “A bonded particle model for analysis of the flaw orientation effect on crack propagation mechanism in brittle materials under compression,” Archives of Civil and Mechanical Engineering, vol. 14, no. 1, pp. 40–52, 2014. View at: Publisher Site  Google Scholar
 H. Dündar and A. O. Ayhan, “Multiple and nonplanar crack propagation analyses in thin structures using FCPAS,” Frattura ed Integrità Strutturale, vol. 10, no. 35, pp. 360–367, 2016. View at: Google Scholar
 F. Erdogan and G. C. Sih, “On the crack extension in plates under plane loading and transverse shear,” Journal of Basic Engineering, vol. 85, no. 4, pp. 519–525, 1963. View at: Publisher Site  Google Scholar
 M. A. Hussian, E. L. Pu, and J. H. Underwood, “Strain energy release rate for a crack under combined mode I and mode II,” in Proceedings of the Fracture Analysis, pp. 2–28, ASTM STP 560, West Conshohocken, PA, USA, 1974. View at: Google Scholar
 R. J. Nuismer, “An energy release rate criterion for mixed mode fracture,” International Journal of Fracture, vol. 11, no. 2, pp. 245–250, 1975. View at: Publisher Site  Google Scholar
 G. C. Sih, “Energydensity concept in fracture mechanics,” Engineering Fracture Mechanics, vol. 5, no. 4, pp. 1037–1040, 1973. View at: Publisher Site  Google Scholar
 M. F. Marji, H. Hosseini_Nasab, and A. H. Kohsary, “On the uses of special crack tip elements in numerical rock fracture mechanics,” International Journal of Solids and Structures, vol. 43, no. 6, pp. 1669–1692, 2006. View at: Publisher Site  Google Scholar
 M. F. Marji and I. Dehghani, “Kinked crack analysis by a hybridized boundary element/boundary collocation method,” International Journal of Solids and Structures, vol. 47, no. 78, pp. 922–933, 2010. View at: Publisher Site  Google Scholar
 M. F. Marji, “On the use of power series solution method in the crack analysis of brittle materials by indirect boundary element method,” Engineering Fracture Mechanics, vol. 98, pp. 365–382, 2013. View at: Publisher Site  Google Scholar
 B. R. Lawn and T. R. Wilshaw, Fracture of Brittle Solids, Cambridge University Press, London, UK, 1975.
 N. I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity, Noordhoff International Publishing, Leyden, The Netherlands, 1953.
 G. C. Sih, Mechanics of Fracture, Noordhoff International Publishing, Leyden, The Netherlands, 1973.
 L. Z. Wu, B. Li, R. Q. Huang, and Q. Z. Wang, “Study on Mode III hybrid fracture criteria for the stability analysis of sliding overhanging rock,” Engineering Geology, vol. 209, pp. 187–195, 2016. View at: Publisher Site  Google Scholar
 M.H. Yu, “Twin shear stress yield criterion,” International Journal of Mechanical Sciences, vol. 25, no. 1, pp. 71–74, 1983. View at: Publisher Site  Google Scholar
 T. Backers and O. Stephansson, “ISRM suggested method for the determination of mode II fracture toughness,” Rock Mechanics and Rock Engineering, vol. 45, no. 6, pp. 1011–1022, 2012. View at: Publisher Site  Google Scholar
 ABAQUS Inc, ABUQUS/Standard User’s Manual Version 6.9, ABAQUS Inc, Waltham, MA, USA, 2009.
 R. Q. Huang, L. Z. Wu, Q. He, and J. H. Li, “Stress intensity factor analysis and the stability of overhanging rock,” Rock Mechanics and Rock Engineering, vol. 50, no. 8, pp. 2135–2142, 2017. View at: Google Scholar
 M. L. Benzeggagh and M. Kenane, “Measurement of mixedmode delamination fracture toughness of unidirectional glass/epoxy composites with mixedmode bending apparatus,” Composites Science and Technology, vol. 56, no. 4, pp. 439–449, 1996. View at: Publisher Site  Google Scholar
Copyright
Copyright © 2019 R. Q. Huang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.