Numerical study on coalescence of two pre-existing coplanar flaws in rock

International Journal of Solids and Structures 50 (2013) 3685–3706

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Numerical study on coalescence of two pre-existing coplanar flaws in rock Louis Ngai Yuen Wong a, Huan-Qiang Li a,b,⇑ a b

School of Civil and Environmental Engineering, Nanyang Technological University, Block N1, Nanyang Avenue, 639798 Singapore, Singapore Zhejiang Provincial Institute of Communications Planning Design and Research, Hangzhou, Zhejiang Province 310006, China

a r t i c l e

i n f o

Article history: Received 4 August 2012 Received in revised form 7 May 2013 Available online 24 July 2013 Keywords: Pre-existing coplanar flaws Inclination angle Ligament length Coalescence AUTODYN

a b s t r a c t Crack propagation and coalescence processes are the fundamental mechanisms leading to progressive failure processes in rock masses, in which parallel non-persistent rock joints are commonly involved. The coalescence behavior of the latter, which are represented as pre-existing coplanar flaws (cracks), is numerically investigated in the present study. By using AUTODYN as the numerical tool, the present study systematically simulates the coalescence of two pre-existing coplanar flaws in rock under compression. The cumulative damage failure criterion is adopted in the numerical models to simulate the cumulative damage process in the crack initiation and propagation. The crack types (shear or tensile) are identified by analyzing the mechanics information associated with the crack initiation and propagation processes. The simulation results, which are generally in a good accordance with physical experimental results, indicate that the ligament length and the flaw inclination angle have a great influence on the coalescence pattern. The coalescence pattern is relatively simple for the flaw arrangements with a short ligament length, which becomes more complicated for those with a long ligament length. The coalescence trajectory is composed of shear cracks only when the flaw inclination angle is small (such as b 6 30°). When the pre-existing flaws are steep (such as b P 75°), the coalescence trajectory is composed of tensile cracks as well as shear cracks. When the inclination angle is close to the failure angle of the corresponding intact rock material, and the ligament length is not long (such as L 6 2a), the direct shear coalescence is the more favorable coalescence pattern. In the special case that the two pre-existing flaws are vertical, the model will have a direct tensile coalescence pattern when the ligament length is short (L 6 a), while the coalescence between the two inner flaw tips is not easy to achieve if the ligament length is long (L P 2a). Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Rock mass is heterogeneous due to the existence of various types of cracks and joints. The initiation, propagation and coalescence of these pre-existing cracks under loading are significant in the study of rock engineering and thermal geology. The coalescences of pre-existing cracks play a decisive role to the mechanical properties of rock mass. In the past several decades, the research work of coalescence of pre-existing flaws, as an extension of the work on single crack initiation and propagation (Bieniawski, 1967; Chen et al., 1998; Chen, 1993; Huang et al., 1990; Lajtai, 1974; Lauterbach and Gross, 1998; Li et al., 2005; Maligno et al., 2010; Ouinas et al., 2009; Pearce et al., 2000; Petit and Barquins, 1988; Potyondy and Cundall, 2004; Rannou et al., 2010; Rispoli, ⇑ Corresponding author at: Zhejiang Provincial Institute of Communications Planning Design and Research, Hangzhou, Zhejiang Province 310006, China. Tel.: +86 15868870631. E-mail addresses: [email protected] (L.N.Y. Wong), [email protected] (H.-Q. Li). 0020-7683/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijsolstr.2013.07.010

1981; Sreeramulu et al., 2010; Tang, 1997; Tang et al., 2000; Wawersik and Fairhurst, 1970; Weber et al., 2010; Willemse et al., 1997; Wong and Einstein, 2009d; Wong et al., 2006; Yang and Jing, 2011; Zhang et al., 2010; Zhao et al., 2010) have been conducted experimentally (Bobet and Einstein, 1998a; Lee and Jeon, 2011; Park and Bobet, 2009, 2010; Sagong and Bobet, 2002; Shen, 1995; Wong and Einstein, 2009b, c; Wong and Chau, 1998; Wong et al., 2001) and numerically (Bobet and Einstein, 1998b; Liu et al., 2004; Ning et al., 2011a; Ning et al., 2011b; Shen and Stephansson, 1993; Tang and Kou, 1998; Tang et al., 2001; Vasarhelyi and Bobet, 2000). Major recent physical experimental work revealed that crack coalescences are affected by the arrangements of the pre-existing flaws (Lee and Jeon, 2011; Wong and Einstein, 2009b; Wong and Chau, 1998), loading conditions (Bobet and Einstein, 1998a), material properties (Wong and Einstein, 2009b,c) and the properties of the pre-existing flaw (Park and Bobet, 2009, 2010). The numerical studies on crack coalescence were also conducted in parallel, by various numerical tools such as DDM (Shen, 1995; Shen and

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Stephansson, 1993), FROCK (Bobet and Einstein, 1998b; Vasarhelyi and Bobet, 2000), RFPA (Liu et al., 2004; Tang and Kou, 1998; Tang et al., 2001), PFC (Lee and Jeon, 2011; Zhang and Wong, in press), NMM (Ning et al., 2011b; Wu and Wong, 2012; Zhang et al., 2010), DDA (Ning et al., 2011a; Pearce et al., 2000), X-FEM (Rannou et al., 2010) and BEM (Chen et al., 1998; Lauterbach and Gross, 1998). These numerical studies were conducted to model the experimental crack coalescence results (Bobet and Einstein, 1998b), unveil the coalescence types (Shen, 1995; Shen and Stephansson, 1993), and establish the coalescence rule in different preexisting flaws arrangement (Liu et al., 2004; Tang and Kou, 1998; Tang et al., 2001). Compared with physical experimental work, numerical studies have the capability to conveniently obtain the mechanics information of crack initiation and propagation. However, due to the complexities of the geomaterial properties, cracking processes, and material plastic deformation, research on the development and refinement of numerical methods is still actively ongoing. In both experimental and numerical research works, the influence of the arrangement of the preexisting cracks is one of the most important aspects to study the crack coalescence phenomena. Among the crack arrangements, the coplanar arrangement is the most common phenomenon in the natural rock masses due to the preferential development of parallel non-persistent joint members belonging to the same joint in response to the crustal stress. When engineering excavation or construction (such as mining, geothermal exploitation and nuclear waste storage) take place in those rock masses, the variance of the stresses will lead to breakage of rock masses due to the initiation and propagation of the rock joints. Therefore, the studies on the coplanar cracks coalescence will be beneficial to the engineering practices in rock masses. Previous experimental studies (Wong and Einstein, 2009b; Wong and Chau, 1998) observed the influences of different factors on the coalescence of the coplanar cracks. To supplement the experimental studies and to obtain a more comprehensive understanding of the mechanics, further numerical studies should be performed. In the present numerical study, coalescence crack types in rock were identified and compared with the experimental results in order to generalize the crack coalescence rule. The study is restricted to models containing only two pre-existing coplanar flaws. Hereinafter, the term ‘‘flaw’’ will be used to describe an artificially created, pre-existing crack or fracture. This paper is structured in the following way. In Section 2, the key findings of our previous study are described. In Section 3, the key aspects of the numerical models and the material models used in the present study are presented. In Section 4, the numerical results are described and categorized according to the coalescence types. In Section 5, the rules of crack coalescence are presented based on the crack coalescence classification. Limitations of the present study and directions of future studies (Section 6), as well as conclusions (Section 7) are provided towards the end of the paper.

2. Previous study Compared with the previous experimental results (Wong and Einstein, 2006; Wong and Einstein, 2009b,c), our earlier numerical simulations (Li and Wong, 2011, 2012; Wong and Li, 2011) indicate that the numerical tool AUTODYN (CenturyDynamics, 2005), which employs a material model composed of the Drucker–Prager strength criterion and Cumulative Damage failure criterion, and uses a triangular unstructured-mesh, is capable of modeling the crack initiation and propagation in rocks. Our previous study (Li and Wong, 2012) unveiled the unique variation characteristics of the pressure, Mises stress and yield

Fig. 1. (a) Physical test specimen and (b) numerical model. Both contain a pair of pre-existing open straight flaws.

Table 1 Material parameters used in the numerical model. Density (g/cm3) Bulk modulus (kPa) Shear modulus (kPa) Drucker–Prager model (kPa)

CD Failure Criterion

Pressure #1 Yield stress #1 Pressure #2 Yield stress#2 Pressure#3 Yield stress#3 pressure#4 Yield stress#4 Pressure#5 Yield stress #5 Pressure#6 Yield stress #6 EPS1 EPS2 Maximum damage factor

2.44 2.7E+07 2.2E+07 1.2e+3 0 1.e+3 2.0e+4 0 2.5e+4 8.0e+4 1.1e+5 1.1e+5 1.6e+5 2.0e+5 1.95e+5 1e 4 1e 3 0.6

stress with time for the tensile failure and shear failure respectively, which can be relied on in the present numerical study to differentiate the initiation of tensile crack and shear crack. For tensile failure, the variations of the pressure*, Mises stress and yield stress with time has the following characteristics: (1) The pressure is negative (tensile stress) immediately before the moment of failure (or yielding). (2) The yield stress decreases rapidly before the moment of failure (or yielding). (3) The pressure, Mises stress and yield stress all become zero after the tensile crack opening event. For shear failure, the variations of the pressure, Mises stress and yield stress with time has the following characteristics: (1) The pressure is positive (compressive stress) immediately before the moment of yielding. ⁄ Pressure (P) is defined as the average of three principal stresses (r1, r 2, r3), i.e. P = (r1 + r 2 + r3)/3.

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(4) The pressure, Mises stress and yield stress do not necessarily return to zero after the cracking event.

3. Numerical analysis 3.1. Numerical models

Fig. 2. Distinction between indirect and direct coalescence (modified from Wong and Einstein (2009a)).

(2) The yield stress has an increasing trend immediately before the moment of yielding. (3) The rate of increase of the Mises stress is higher than that of the yield stress immediately before the moment of yielding.

In the present study, the cumulative damage (CD) failure criterion (Persson, 1991) is chosen to model the gradual crack initiation process in rocklike materials. The hydrocode AUTODYN, which excels in solving a wide variety of problems characterized by both geometric non-linearities and material non-linearities (Tham, 2005), is used. The two-dimensional rectangular numerical models are built in accordance with the previous physical experimental specimens (Wong and Einstein, 2009a,c) of dimensions 150 mm (height)  75 mm (breadth)  32 mm (thickness) (Fig. 1a), containing two coplanar straight flaws through the specimen thickness. The flaw aperture is ‘‘w’’ = 1.27 mm and the flaw length is ‘‘2a’’ = 12.7 mm, where ‘‘a’’ is the half flaw length. The flaws are oriented at different angles (inclination angle, b) with respect to the horizontal direction. The inner flaw tips are separated by a varied ligament length ‘‘L’’. The simulation is carried out for a uniaxial compressive loading condition. A vertical pressure (r0) is applied at the top and bottom boundaries of the specimen (Fig. 1b).

Fig. 3. Direct shear coalescence in model 2a_45 observed in a physical experimental test (Wong, 2008).

Fig. 4. Direct shear coalescence in model 2a_60 observed in a physical experimental test (Wong, 2008).

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Fig. 5. Direct shear coalescence in model a_30 (‘S’ denotes shear crack and ‘T’ denotes tensile cracks. Red trajectory denotes the crack. The same nomenclature is adopted below). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 7. Direct shear coalescence in model a_60.

minimum influence on the crack propagation (Li and Wong, 2012), is adopted in the present study.

Our previous study (Li and Wong, 2012) revealed that the loading rate has a great influence on the crack initiation and propagation, and particularly the crack type. In this study, to simulate the crack coalescence under a quasi-static loading condition, a relatively low loading rate (12.5 MPa/ms) is applied. The triangular unstructured-mesh type, which has been shown to have the

The material model consists of equation of state, material strength and failure criterion, in which, the linear equation of state, the Drucker–Prager (DP) model and the cumulative damage (CD) failure criterion are adopted. Previous studies (Wong and Einstein,

Fig. 6. Direct shear coalescence in model a_45.

Fig. 8. Direct shear coalescence in model 2a_45.

3.2. Material models

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the experimental tests of specimens containing single flaws. By trial and error, a set of parameters associated with the CD failure criterion, which is able to produce the crack types and trajectories closely matching those in the experimental tests, was obtained. EPS in Table 1 stands for the effective plastic strain. EPS1, EPS2 and Maximum Damage Factor are used to constitute the cumulative damage (CD) failure criterion (Persson, 1991). 4. Numerical simulation of crack coalescence The coalescence behaviors of different coplanar flaw pair geometries have been numerically studied in this research. The flaw pair geometry is defined by two geometrical parameters ligament length L and flaw inclination angle b. For example, ‘2a_30’ denotes that the ligament length (L) between the two pre-existing flaw tips is ‘2a’, and inclination angle (b) is 30°, where ‘a’ is the half flaw length. In the present study, the pre-existing flaw inclination angle (b) ranges from 0° to 90° at an interval of 15°, which thus includes seven different inclination angles. At each inclination angle, three models (L = ‘‘a’’, ‘‘2a’’ and ‘‘4a’’) are built. Therefore, twenty-one flaw pair geometries are studied in total. As shown later in this paper, coalescence can be achieved either in a direct or indirect manner. The definitions of direct and indirect coalescence for the cases of tip to tip coalescence have been provided in a previous study (Wong and Einstein, 2009b). As shown in Fig. 2a and b, in indirect coalescence, the coalescence points are relatively far away from the dotted reference line. In contrast, the coalescence point is close to the dotted reference line in direct coalescence (Fig. 2c). Fig. 9. Direct shear coalescence in model 2a_60.

4.1. Direct coalescence 2009a,c) showed that the initiation of macro-crack is not an abrupt event, but associated with a process of cumulative damage. The CD failure criterion, introduced by Persson (1991), has the capability to describe the cumulative damage process. AUTODYN with the material model has the capability to consider the complicated constitutive relation and failure criterion of rock, the cumulative cracking process, mechanics property of the crack and large strain and displacement after the crack initiation. Material parameters used in the simulation are provided in Table 1. The parameters in the equation of state were obtained from the laboratory tests of marble. The parameters in the Drucker– Prager strength Model were obtained by modifying the parameters of the rock-like material embedded in AUTODYN according to the laboratory test. To obtain the parameters in the CD failure criterion, some numerical simulations were conducted according to

According to the present numerical simulation results, direct coalescence for coplanar flaws can be classified into three categories shear coalescence, tensile coalescence and shear-tensile coalescence. Direct shear coalescence refers to the linkage of the two pre-existing flaw tips by a straight shear crack trajectory; direct tensile coalescence refers to the linkage of the two pre-existing flaw tips by a straight tensile crack trajectory; and direct sheartensile coalescence refers to the linkage of the two pre-existing flaw tips by a crack trajectory which is composed of shear crack segment(s) and tensile crack segment(s). The crack trajectories connecting the two pre-existing flaw tips in the above three coalescence categories are respectively referred to as direct shear coalescence crack, direct tensile coalescence crack and shear-tensile coalescence crack. In our simulations, the occurrence of direct shear coalescence is more common than that of direct tensile

Fig. 10. Snapshots of direct shear coalescence process in model a_30.

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Fig. 11. Snapshots of direct shear coalescence process in model a_45.

Fig. 12. Mechanics information of direct shear coalescence process at gauge point #37. PRESSURE is defined in Section 2. MIS = Mises stress. YLD = yield stress.

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Fig. 13. Mechanics information of direct shear coalescence process at gauge point #24.

Fig. 14. Snapshots of direct tensile coalescence process in model a_90.

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Fig. 15. Mechanics information of direct tensile coalescence process at gauge point #44.

coalescence and direct shear-tensile coalescence. The identification of the crack type is based on the mechanics information of these cracks, which will be described in sections 4.1.2, 4.1.4 and 4.1.5. Unless otherwise stated, the letters ‘S’ and ‘T’ denote shear crack and tensile crack, respectively in this paper. 4.1.1. Direct shear coalescence Fig. 3 and Fig. 4 show the crack coalescence processes of two open parallel flaws in models 2a_45 and 2a_60 in Carrara marble. The flaw aperture is ‘‘w’’ = 1.27 mm and the flaw length is ‘‘2a’’ = 12.7 mm. Shear crack development is observed to be associated with a production of shear dust and small scale localized spalling. These phenomena are in contrast to those of tensile crack development, which involves relatively clean crack opening, much less dust and spalling fragments. In physical tests, the direct shear coalescence is found to be one of the common coalescence patterns. Five out of the 21 studied flaw pair geometries display a direct shear coalescence phenomenon (Fig. 5 to Fig. 9). The pre-existing flaw inclination angles of these five models are varied from 30° to 60°. In the simulated results, the shear cracks are found to coa-

lesce with the pre-existing flaws at the same position as where a shear crack would initiate in a specimen containing a pre-existing single flaw (Li and Wong, 2012). Notice that the initiation position of direct shear coalescence crack is slightly offset from the flaw tip, up and down, respectively, at the right and left flaws. The direct shear coalescence crack is hence not parallel to the flaw, but at a slightly steeper inclination relative to the pre-existing flaws. Out of the five direct shear coalescence cases, three are associated with the ligament length equal to ‘a’ (Fig. 5 to Fig. 7), while the other two are associated with the ligament length equal to ‘2a’ (Fig. 8 and Fig. 9). In the two latter cases, a number of tensile branch cracks are found to emanate at an angle from the direct shear coalescence cracks. These branching cracks lead to a wider cracking zone in the bridge area. The crack initiation, propagation and coalescence processes of the two representative cases shown for models a_30 (Fig. 5) and a_45 (Fig. 6) are further examined and described in detail below. As shown in Fig. 10a and Fig. 11a, the direct shear coalescence cracks are found to initiate at the two inner flaw tips. At this time, the tensile wing cracks initiated from the flaw tip regions have already been in the propagation phase. The trajectories of tensile

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Fig. 16. Mechanics information of direct tensile coalescence process at gauge point #55.

Fig. 17. Snapshots of crack coalescence process in model a_75.

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Fig. 18. Mechanics information at gauge point #23 in model a_75.

wing cracks at the outer flaw tips are longer than those at the inner flaw tips. After the initiation of the direct shear coalescence cracks, the shear crack propagates from one initiation point towards the other initiation point (Fig. 10b and Fig. 11b). The crack coalescence is considered to be completed when the shear crack trajectory eventually connects the two initiation points on the flaw tips (Fig. 10c and Fig. 11c). To summarize, the direct shear coalescence process begins with the initiation of direct shear coalescence cracks from the two inner flaw tips, which later propagate from one tip to the other. 4.1.2. Identification of shear crack To identify the crack nature (shear/tensile), some gauge points are set on each coalescence crack to gather the relevant necessary mechanics information. Through analyzing the gathered mechanics information, the coalescence cracks reported in the five cases above (Fig. 5–Fig. 9) are found to be shear cracks. To limit the length of this paper, only the mechanics information gathered at gauge points set on the direct coalescence crack of model a_30 (Fig. 5) is provided and discussed below (Fig. 12 and Fig. 13). In our previous study (mentioned in part 2), the crack type identification is based

on the study of the variation of pressure, Mises stress and yield stress with time. In this paper, another set of mechanics information (Fig. 12b) is also gathered and analyzed to reveal the crack nature more explicitly. Fig. 12a shows the variation of pressure, Mises stress and yield stress with time at gauge point #37 in model a_30 (Fig. 5). Before reaching the failure state, pressure, yield stress and Mises stress at gauge point #37 all increase with time. The rate of the increase of the Mises stress is higher than that of yield stress. At the moment just before failure, the pressure is positive (compressive). Based on the findings of our previous study, the crack thus belongs to a shear crack. Fig. 12b shows the variation of Mises stress and yield stress with pressure at representative time steps at gauge point #37. The two straight lines represent the intact material strength line and the failed material strength line. The yield stress increases in a direct proportion to the pressure. In the early stage, the Mises stress and yield stress increase with the increase of pressure. In a later stage, in spite of the decrease of the pressure, the Mises stress is still increasing until it catches up to the yield stress, leading to the yielding stage and producing plastic strain. The yield stress

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Fig. 19. Mechanics information at gauge point #43 in model a_75.

falling below the intact material strength line represents the occurrence of material failure, which is due to the increase of plastic strain. The failed material still retains a certain shear strength, which is much less than the intact material strength (failed material strength line in Fig. 12b). Based on the above discussion with reference to Fig. 12a and b, the material failure at gauge point #37, which is due to the increase of Mises stress, can be classified as a shear failure. The mechanics information at gauge point #24 in model a_30 (Fig. 5) is shown in Fig. 13. The failure, which is similar to that at gauge point #37 (Fig. 12), is again due to the increase of Mises stress. It reveals that the failure at gauge point #24 also belongs to a shear failure. The mechanics information of the remaining part of the cracks has been checked and found to be similar to that of gauge points #37 and #24 as shown in Fig. 12 and Fig. 13. Therefore, the coalescence crack in model a_30 (Fig. 5) is classified as a shear crack.

4.1.3. Direct tensile coalescence For the pre-existing coplanar flaw models, direct tensile coalescence occurs only for vertical flaws (b = 90°) possessing a small enough ligament length, but not in other cases. To our best

knowledge, relevant physical experimental studies on the coalescence of vertical pre-existing flaws are not available in the literature for reference. Fig. 14 shows a series of snapshots of the direct tensile coalescence process in model a_90. Firstly, tensile cracks initiate at the two inner flaw tips (Fig. 14a). Secondly, shear cracks initiate and propagate from the outer flaw tips (Fig. 14b). The two tensile cracks at the inner flaw tips then propagate very slowly towards each other (Fig. 14c). Finally, the two tensile cracks coalesce to become one continuous crack (Fig. 14d). The tensile crack nature is confirmed by analyzing the mechanics information, which will be discussed later in Section 4.1.4. When the direct tensile coalescence occurs, the shear cracks at the outer tips have already developed substantially.

4.1.4. Identification of tensile crack The mechanics information gathered at the gauge points #44 and #55 set on the crack trajectory in model a_90 (Fig. 14d) are shown in Fig. 15 and Fig. 16. Each figure includes two sets of mechanics information – the variation of pressure, Mises stress and yield stress with time (Fig. 15a and Fig. 16a) and the variation

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Fig. 20. Mechanics information at gauge point #30 in model a_75.

of Mises stress and yield stress with pressure (Fig. 15b and Fig. 16b).

As shown in Fig. 15a and Fig. 16a, the pressure at gauge points # 44 and #55, increases with time. At the moment immediately before material failure, the pressure drops down rapidly to a negative

Fig. 21. Direct shear coalescence in model 2a_75 observed in a physical experimental test (Wong, 2008).

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Fig. 22. Crack coalescence process in (a) model a_75 (aperture w = 0.4 mm) and (b) model 0.8a_75 (aperture w = 0.3 mm).

Fig. 23. Coalescence by an anti-wing crack and a horsetail crack in model a_15.

value (tensile stress), which induces the material failure. Fig. 15b and Fig. 16b show the relationship between yield stress and Mises stress explicitly. In the early stage, the Mises stress and the yield stress increase with the increase of pressure. In the later stage, the Mises stress decreases with the decrease of pressure. In the positive pressure range, the maximum Mises stress is still less than the yield stress, so shear failure does not occur. As the pressure decreases to the negative pressure range, the Mises stress will reach the yield stress, which induces the failure of material. The failure, which occurs under the prevalence of tensile stress (negative pressure), is thus a tensile failure.

Fig. 24. Coalescence by an anti-wing crack and a horsetail crack in model a_0.

Based on the mechanics information shown in Fig. 15 and Fig. 16, the material failure at gauge points #44 and #55 is due to the decrease of pressure (increase of tensile stress). The coalescence crack connecting the two pre-existing flaws is thus tensile in nature.

4.1.5. Direct shear-tensile coalescence Direct shear-tensile coalescence occurs when the coalescence crack involves both shear and tensile crack segments. Fig. 17 shows a series of snapshots of such a direct shear-tensile coalescence

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Fig. 25. Snapshots of crack coalescence processes in model a_15.

process in model a_75. It is interesting to note that although the ligament length is short (L = a), the two flaws are not linked up by a straight crack, but one with a curved trajectory. This coalescence type occurs only in the model containing steeply-inclined

pre-existing flaws (b = 75°) for L = a among our present simulation results. At the early stage of crack development, only two pairs of tensile wing cracks initiate and propagate from the flaw tips (Fig. 17a).

Fig. 26. Mechanics information at gauge point #132 in model a_15.

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Fig. 27. Mechanics information at gauge point #171 in model a_15.

Further loading leads to the shear crack initiation from the two inner flaw tips (Fig. 17b). The shear cracks later propagate along a horsetail crack trajectory, while the tensile wing cracks have no further observable propagation (Fig. 17c). As the crack propagation continues, the horsetail crack initiated from one inner tip propagates towards the other. The two horsetail cracks coalesce at last (Fig. 17d). To reveal the crack type along the coalescence trajectory, a number of gauge points have been set on each coalescence crack segment, three of which are shown in Fig. 17d. The mechanics information gathered at the gauge points is plotted in Fig. 18 to Fig. 20. The approach of crack type identification is similar to the procedures described in earlier sections. At gauge points #23 and #43, which are located on the flaw tips, the pressures are compressive when the material failure occurs. The Mises stresses increases rapidly, which reaches the yield stress to induce the material failure (Fig. 18 and Fig. 19). The failure at gauge points #23 and #43 belong to a shear failure, which confirms that these short crack segments containing gauge points #23 and #43 are the shear crack segments of horsetail cracks. Gauge point #30 is located in the middle of the coalescence crack. From Fig. 20, the pressure is found to be tensile when the Mises stress reaches the yield stress. There-

fore the failure occurred at gauge point #30 belongs to a tensile failure, which confirms that this middle crack segment containing gauge point #30 is a tensile crack segment of the horsetail crack. The coalescence behavior obtained numerically above satisfactorily captures the essence of the physical experimental observations. Fig. 21 shows a series of snapshots of the physical crack coalescence process for model 2a_75. The shear dust produced from the shear crack segments adjacent to the flaw tips (Fig. 21b) can help differentiate the shear cracks from the tensile cracks, since the latter are typically associated with relatively clean open crack trajectories. A short tensile crack segment later appears in the central bridge area (Fig. 21c). The remaining parts of the coalescence crack trajectory associated with production of shear dust can be considered as shear crack segments (Fig. 21d). The present numerical models are built according to the experimental models, whose flaw aperture (w) is 1.27 mm. A very logical question to ask is to what extent the nature and trajectory of the coalescence are dependent on the flaw aperture. In the present numerical study, two finer aperture sizes corresponding to flaw geometries a_75 and 0.8a_75 are examined. The simulated results of the two models of w = 0.4 mm and w = 0.3 mm are shown in Fig. 22a and Fig. 22b, respectively. The mechanics analysis of the

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Fig. 28. Mechanics information at gauge point #137 in model a_15.

two cases reveals that the coalescence crack trajectories also contain a central tensile crack segment. However, when the flaw aperture is very small (Fig. 22b, w = 0.3 mm), the tensile crack segment will become very short and the coalescence crack trajectory appears straight and almost parallel with the flaws. The experimental study by Wong and Chau (1998) indicates a similar situation. Their study unveils that the coalescence pattern in the models containing two coplanar closed pre-existing flaws, whose inclination angle is 75°, belongs to the direct shear coalescence. The central tensile crack segment may be either absent in the coalescence crack or the tensile crack segment is too short to be observed in their closed pre-existing flaw model experiment. In the present numerical study and previous experimental study (Wong, 2008), which examine flaws of a relatively wider aperture, the coalescence crack trajectory is expected to contain a tensile crack segment. According to the direct coalescence analysis above, when the pre-existing flaw inclination angle b is equal to 30° and 60°, the coalescence crack is a shear crack; when the inclination angle b is equal 90°, the coalescence crack is a tensile crack. For b equal to 75°, the coalescence is a mixed shear-tensile type, which appears as a transition between the direct shear and direct tensile coalescence types.

4.2. Indirect coalescence Compared with the direct coalescence type, indirect coalescence is found to be more common in the present numerical study. This section focuses on the indirect coalescence behavior, which involves various common crack types such as tensile wing crack, anti-wing crack and horsetail crack, etc. 4.2.1. Coalescence by anti-wing and horsetail cracks Indirect coalescence by anti-wing and horsetail cracks is observed in models a_15 (Fig. 23) and a_0 (Fig. 24). Three crack segments are involved in each of these two coalescences. The detailed crack coalescence processes of model a_15 are shown in Fig. 25. At the early stage of crack development, cracks leading to the future coalescence initiate from the inner tips of the pre-existing flaws (Fig. 25a). The crack initiation position is found to be at the same position as where a shear crack would initiate in a specimen containing a pre-existing single flaw (Li and Wong, 2012). Subsequently, these two inner tip cracks propagate at an angle towards each other (Fig. 25b) and then coalesce. After the coalescence, the two cracks continue to propagate as one crack trajectory (Fig. 25c).

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Fig. 31. Coalescence by an anti-wing crack and a horsetail crack in model 4a_45. Fig. 29. Coalescence by an anti-wing crack and a horsetail crack in model 2a_30.

Fig. 30. Coalescence by an anti-wing crack and a horsetail crack in model 2a_0.

On each crack involved in the coalescence, some gauge points have been set to gather the mechanics information to identify the crack type. The mechanics information gathered at gauge points #132, #171 and #137 of model a_15 (Fig. 23) are shown in Fig. 26 to Fig. 28. Fig. 26a to Fig. 28a show the variation of pressure, Mises stress and yield stress with time, and Fig. 26b to

Fig. 28b show the variation of Mises stress and yield stress with pressure. Fig. 26a and Fig. 27a reveal that the pressures are compressive when the material failure occurs. Besides, Fig. 26b and Fig. 27b show that it is the increase of Mises stress, which eventually reaches the material strength to cause the material failure. The above mechanics information indicates that the material failure at points #132 and #171 belongs to the shear failure. Hence the cracks containing gauge points #132 and #171 are shear cracks. Fig. 28a reveals that the pressure is negative, i.e. tensile, when the material failure occurs. Besides, Fig. 28b shows the occurrence of material failure, i.e. Mises stress reaching yield stress, is due to a rapid decrease of yield stress associated with a negative pressure regime. Therefore, the mechanics information in Fig. 28 indicates that the material failure at point #137 belongs to the tensile failure. Hence the crack containing gauge point #137 is a tensile crack. The above mechanics analysis reveals that the crack segments propagating from the flaw tips to the coalescence points in models a_15 (Fig. 23) and a_0 (Fig. 24) are shear cracks. The latter can be considered as the shear crack portion of either an anti-wing crack or a horsetail crack. The crack segment propagating beyond the coalescence point is a tensile crack, which can be considered as the tensile crack portion of either an anti-wing crack or a horsetail crack. So this typical coalescence pattern can be regarded to be attributed to the development and interaction of an anti-wing crack and a horsetail crack. Three more similar examples of crack coalescence behavior due to the combinations of different ligament lengths and flaw inclination angles for models 2a_30, 2a_0 and 4a_45 are illustrated in Fig. 29 Fig. 30 and Fig. 31, respectively. Although the crack patterns appear to be more complicated than those in models a_15 and a_0, detailed mechanics analysis at the gauge points on selected crack segments (numbered in Fig. 29 to Fig. 31) reveals that the coalescence is similarly achieved by the development and interaction of an anti-wing crack and a horsetail crack. Similar indirect coalescences by anti-wing cracks and horsetail cracks are also observed experimentally. Fig. 32 shows a series of snapshots of high speed video images illustrating the crack devel-

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Fig. 32. Crack coalescence in model 2a_0 observed in a physical experimental test (Wong, 2008).

Fig. 33. Coalescence by horsetail and shear cracks in model 2a_15. Fig. 34. Coalescence by horsetail and shear cracks in model 4a_0.

opment and coalescence processes in model 2a_0. Shear cracks, associated with a development of shear dust, first initiate from the inner flaw tips. Further propagation of the shear cracks leads to the development of an anti-wing crack and a horsetail crack, which later coalesce. Beyond the point of coalescence, a tensile crack segment trending almost vertically, is observed. 4.2.2. Coalescence by horsetail and shear cracks The previously discussed indirect coalescence type involves no more than two cracks. For certain flaw pair geometries, more complicated coalescence phenomena involving three cracks have been observed experimentally and obtained numerically in the present study (Fig. 33 and Fig. 34). Since the same technique of crack type identification as stated above is used, only the identified crack type is briefly described in this section without detailing the underlying mechanics repeatedly. As shown in Fig. 33 for model 2a_15, five gauge points have been set on the coalescence trajectories. According to the gathered mechanics information, the crack segments containing gauge

points #20, #34 and #43 all belong to shear cracks, and the crack segments containing gauge points #26 and #39 belong to tensile cracks. Therefore, the coalescence in model 2a_15 is due to two horsetail cracks and one medial shear crack. Here, we name the medial shear crack as a ligament shear crack, because it is a shear crack serving as a ligament to connect the two horsetail cracks. Another similar coalescence pattern is shown in Fig. 34 for model 4a_0. The above coalescence pattern has not yet been observed in our experimental studies. However, another similar coalescence pattern, which is composed of two anti-wing cracks and a ligament shear crack, is observed experimentally (Fig. 35). According to our previous numerical study (Li and Wong, 2011; Wong and Li, 2011), the anti-wing crack and horsetail crack, which share similar mechanics properties, are conjugate cracks, i.e. they are symmetrical to each other with regard to crack trajectories. Specifically, they both have a short shear segment adjacent to the flaw tip, while a

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Fig. 35. Coalescence in model 2a_30 observed in a physical experimental test (Wong, 2008).

Fig. 36. Schematic illustration of three basic crack types initiating from a preexisting flaw.

tensile segment farther away. Further physical experimental attempt is warranted to observe any crack coalescence due to the development and propagation of horsetail and shear cracks.

4.2.3. More complicated coalescence The coalescence types illustrated and analyzed above can be described by a limited number of known crack names such as tensile wing crack, horsetail crack and anti-wing crack as shown in Fig. 36 (Cruikshank et al., 1991; Lajtai, 1974; Li and Wong, 2011; McGrath and Davison, 1995; Wong and Einstein, 2009a,d,e; Wong and Li, 2011; Wong et al., 2001, 2006). However, coalescence types are not limited to these simple types. More complicated coalescences composing of more than three cracks are also obtained. In the simulated results, the complicated coalescence, which cannot be easily described by known crack names, can be divided into two categories, namely coalescence by shear cracks, and coalescence by shear and tensile cracks.

Fig. 37. Complicated coalescence composed of shear cracks in mode 4a_15.

Fig. 37 shows a complicated coalescence pattern, which is composed of several cracks linking up the inner flaw tips. The identification of each crack type is based on the mechanics information gathered at the numbered gauge points set on the crack trajectories (Fig. 37). Several coalescence points occur on the coalescence trajectories. The crack segments which connect the coalescence points or crack initiation points are found to be shear in nature. Complicated coalescence patterns composed of shear cracks and tensile cracks are also common. Two of them are shown in Fig. 38a and b. Based on the mechanics information gathered at the numbered gauge points set on the crack trajectory (Fig. 38), the crack type at each gauge point is identified and denoted by ‘S’ or ‘T’ in the figure. The coalescence trajectory linking the two inner flaw tips is composed of tensile cracks as well as shear cracks.

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Ligament length (L) a

2a

4a

0

45

Direct shear

Indirect (AW + HT) Indirect (HTs + S) Indirect (AW + HT) Direct shear

Complicated (S)

30

Indirect (AW + HT) Indirect (AW + HT) Direct shear

60

Direct shear

Direct shear

75

Direct sheartensile Direct tensile

Complicated (S + T) No coalescence

15

90

Complicated (S) Complicated (S) Indirect (AW + HT) Complicated (S + T) Complicated (S + T) No coalescence

Note: ‘AW’ denotes anti-wing crack, ‘HT’ denotes horsetail crack, ‘S’ denotes shear crack, ‘T’ denotes tensile crack and ‘No coalescence’ denotes no coalescence of the inner flaw tips.

Fig. 38. Complicated coalescences composed of shear and tensile cracks in (a) model 2a_75 and (b) model 4a_75.

5. Rule of crack coalescence The crack coalescence patterns obtained from the present numerical study are summarized in Fig. 39. Table 2 classifies the coalescence types according to the coalescence style (direct shear,

direct tensile, direct shear-tensile, no coalescence) and crack names (AW = anti-wing crack, HT = horsetail crack, S = shear, T = tensile crack). As shown in Fig. 39, the crack coalescence trajectories in those models containing a ligament length L = ‘a’ are relatively simple, which are composed of three or fewer cracks. In the models containing a ligament length L = ‘2a’, complicated coalescence trajectories occur for certain flaw inclination angles. Most of the

Fig. 39. Summary of crack coalescence phenomena from simulated results.

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50° and 60° in the intact rock. If the inclination angles of the preexisting flaw pair in a pre-cracked specimen are near the failure angle, direct shear coalescence due to the development of shear cracks generally parallel with the pre-existing flaws is thus reasonably expected.

6. Limitations and future studies

Fig. 40. Crack development in intact model under uniaxial vertical loading.

coalescences in the models containing a ligament length L = ‘4a’ belong to the complicated coalescence style. In other words, the coalescence style becomes more complicated, i.e. more cracks are involved in coalescence, with the increase of ligament length in the model. As summarized in Table 2, when the flaw inclination angle is small (such as b 6 30°), coalescence due to the development and propagation of anti-wing crack(s) and horsetail crack(s) is more favorable (L 6 ‘2a’), and the complicated coalescence is composed of shear cracks only for L = ‘4a’. When the flaw inclination angle is large (such as b P 75°), the coalescence trajectory will be composed of tensile cracks as well as shear cracks. In the special case that the pre-existing flaw is vertical, the inner flaw tips are not easy to coalesce except when the ligament length is small (L = ‘a’). Table 2 also reveals that, for ligament length L = ‘a’, direct shear coalescence occurs when the flaw inclination angles range from 30° to 60°, while for ligament length L = ‘2a’, direct shear coalescence only occurs for flaw inclination angle 45° and 60°. From the above, one can conclude that the direct shear coalescence is favored when the following two conditions are satisfied. First, the ligament length is small (such as L 6 2a). Second, the mid-ranged flaw inclination angle, which is near the failure angle of the intact rock material, is present. The failure angle of an intact rock material is explained below. To complement the present numerical study of crack coalescence of coplanar flaws, an intact model adopting the same material model and failure criterion as those of the pre-cracked models is built and loaded uniaxially (vertically) to failure. The simulated result is shown in Fig. 40, from which the simulated shear crack inclination angles are found to range generally between 50° and 60°. In other words, the most favorable failure angle is between

The present numerical study examines the mechanics of different key crack coalescence patterns observed in the physical experiments. Some rules of crack coalescence of pre-existing coplanar flaws due to the influence of flaw inclination angle and ligament length are generalized. The numerical results presented in this study are based on the Drucker–Prager strength model along with the cumulative damage failure criterion. The simulated cracking phenomena based on this material model are in a good accordance with the physical experimental phenomena, despite that the material model is too simplified in describing the tensile crack initiation. The influence of the mesh on the cracking behavior is recognized. However, adopting a triangular unstructured-mesh in the present numerical study is considered satisfactory for the purpose of crack pattern analysis. Cracking phenomena matching closely those in physical experimental tests can be obtained when the ligament length (L) is small. When L is large, the crack coalescence becomes more complicated as compared with the physical experimental results. Further study is thus required to refine the rule of complicated coalescence when L is large. The numerical results presented in this paper are only based on a set of given material parameters, for a rock-like material containing a pair of pre-existing coplanar open flaws. In the future study, the crack coalescence in the models containing more varied flaw pair geometries, such as stepped flaws, will be examined.

7. Conclusions By engaging a Drucker–Prager strength model and a cumulative damage failure criterion, the present numerical study focuses on the crack coalescence behavior between two pre-existing coplanar open flaws. The simulated crack coalescence patterns are in a general accordance with the physical test results. The mechanics information gathered at the gauge points, which have been set on the coalescence trajectories in advance, provides a quantitative basis for the identification of crack types. The variation of pressure, Mises stress and yield stress with time, and the variation of Mises stress and yield stress with pressure can satisfactorily reveal the mechanics nature of shear crack and tensile crack. Summarizing the numerical simulation results, the ligament length (L) and the flaw inclination angle (b) of the pre-existing coplanar flaw pair are both found to have a great influence on the coalescence pattern. The coalescence pattern is relatively simple when the ligament length is short, while the coalescence pattern becomes more complicated with the increase of the ligament length. The coalescence trajectory is composed of shear cracks only when the flaw inclination angle is small (b 6 30°), and the crack coalescence pattern is most probably composed of anti-wing cracks and horsetail cracks. When the pre-existing flaws are steep (b P 75°), the coalescence trajectory is composed of tensile cracks as well as shear cracks. When the inclination angle is close to the failure angle of the corresponding intact rock material, and the ligament length is not long (such as L 6 2a), the direct shear coalescence is the more favorable coalescence pattern. In the special case that the two pre-existing flaws are vertical, the model will have a direct tensile coalescence pattern when the ligament length is short (L 6 a), while the coalescence between the

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