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Systematic evaluation of cracking behavior in specimens containing single flaws under uniaxial compression
ARTICLE IN PRESS International Journal of Rock Mechanics & Mining Sciences 46 (2009) 239– 249
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Systematic evaluation of cracking behavior in specimens containing single flaws under uniaxial compression L.N.Y. Wong, H.H. Einstein Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
a r t i c l e in fo
abstract
Article history: Received 26 October 2007 Received in revised form 7 March 2008 Accepted 12 March 2008 Available online 12 May 2008
This paper presents the results of an experimental study in which molded gypsum and Carrara Marble specimens containing a pre-existing flaw were tested in uniaxial compression. The main purpose of this study was to observe and characterize the cracks that emanate from a single pre-existing flaw. Seven different crack types were identified based on their geometry and propagation mechanism (tensile/ shear). Specifically, they include three types of tensile cracks, three types of shear cracks, and one type of mixed tensile–shear crack. In addition to the geometry and mechanism, it was also possible to determine the temporal sequence of different crack types. These observations form the basis for a reevaluation of cracking processes reported in the literature. It is possible to apply the classification obtained in the present research to previously conducted experiments by others. This eliminates much of the confusion that has existed when comparing different research results. & 2008 Elsevier Ltd. All rights reserved.
Keywords: Tensile cracks Shear cracks Crack type classification scheme High-speed camera Uniaxial compressive loading test
1. Introduction Since Bombolakis [1] observed the propagation of tensile wing cracks from pre-existing straight flaws under uniaxial compression in Columbia Resin 39, fracturing processes and crack coalescence patterns in pre-cracked samples under compression have been extensively studied experimentally on different materials, including both rock-like brittle/semi-brittle materials (Columbia Resin 39 [1–4], glass [5,6], Plaster of Paris [7,8], polymethylmethacrilate (PMMA) [9,10], molded gypsum [11–17], sandstone-like molded barite [18,19] and sandstone-like concrete mix [20]) and natural rocks (sandstone [9], granodiorite [21], limestone [21], granite [22], marble [22–25] and gabbro [26]). Henceforth, the term ‘‘flaw’’ will be used to describe an artificially created, pre-existing crack or fracture, which can be either open or closed. In these tests, the size of the prismatic specimens ranged from a minimum of 50 mm 32 mm 5 mm [9] to a maximum of 635 mm 279 mm 203 mm [20]. The flaw length and the flaw aperture of open flaws varied between 10 and 50 mm, and between 0.1 and 3 mm, respectively. Note that in some cases closed flaws were also studied, e.g. [3–6,9,12,13,18]. As just mentioned, one of the intentions of all these tests was to observe and characterize the cracks emanating from the flaws. This led to a variety of terminologies, i.e. there are different descriptions for the same crack type and also sometimes not
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E-mail address: [email protected] (H.H. Einstein). 1365-1609/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijrmms.2008.03.006
sufficient differentiation between different crack types. The new experimental techniques, which involved the use of a high-speed camera, detailed in this paper, make it possible to more clearly define the different crack types than was possible in the past. This will make it possible to introduce a crack characterization scheme, which covers previous and present research observations. Specifically, this paper will first briefly review the existing crack characterization terminologies and identify the existing confusing aspects. It will then summarize and discuss the key findings obtained from the present experimental study, which then forms the basis for a re-evaluation of cracking processes reported in the literature.
2. Existing crack terminology and encountered problems In the literature, different terminologies have been used to describe the crack types observed. Since tensile wing cracks were usually found to be the first cracks to appear from the pre-existing flaws (Fig. 1), they were also commonly called primary cracks. Cracks initiating later than the tensile wing cracks were usually referred to as secondary cracks, without specifying the crack mechanism (tensile/shear). Being uncertain whether these additional cracks were tensile or shear in origin, most authors named these cracks secondary cracks. Later studies suggested that secondary cracks are usually shear in origin. As time went by, researchers began using secondary cracks and shear cracks interchangeably, e.g. [13]. However, in certain cases when the authors were unable to determine the
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crack mechanism (tensile/shear), they continued the old practice of using the term secondary with the intention of simply indicating the temporal cracking relationship without implying the crack mechanism. Confusion was thus often created for later readers. It will be later shown with the present experimental study that cracks that initiate after the initiation of conventional tensile wing cracks can be either shear or tensile. Other types of tensile cracks, in addition to wing cracks, have also been observed.
3. Previous studies In this section, the various crack types observed by different authors in specimens containing single flaws under uniaxial loading tests are reviewed. As noted in the previous section regarding the inconsistent use of terminologies, careful interpretation of the crack types identified by different authors is required. The crack types reviewed in this section will be evaluated again in Section 5 based on the classification obtained from the present experimental study presented in Section 4. Lajtai [7] conducted uniaxial compression loading tests on Plaster of Paris specimens of dimensions 300 600 600 (76 mm 152 mm 152 mm), which contain single flaws of various dimensions. All flaws had the same width (aperture) of 0.0200 (0.5 mm), but variable lengths, 2.0, 1.5, 1.0 and 0.5 in, respectively. The flaw inclination angle (angle with the horizontal) varies from 01 to 901
Fig. 1. Initiation of tensile wing cracks from a pre-existing inclined to the uniaxial loading direction.
at 51 intervals. The results, which are schematically summarized in Fig. 2, consist of the following crack initiation sequence (with the original terminologies)—(a) tensile fractures, (b) normal shear fractures, (c) additional normal shear fractures leading to the formation of shear zones, (d) inclined shear fractures. In a literature review, Ingraffea and Heuze [21] identified the influence of material on fracturing behavior under uniaxial compression. They observed that in specimens with a single open flaw inclined to the compression axis, primary cracks, which originated from points of initially highest tension stress, were observed in previous tests on glass, polymethylmethacrilate (PMMA), CR39 and also rocks. Secondary cracks, which originated from points of initially compressive stress concentration, were however absent in glass and plastic, but were only observed in rocks. Notice that Ingraffea and Heuze [21] appeared to have used the term primary cracks interchangeably with tensile wing cracks and the term secondary cracks interchangeably with shear cracks. Ingraffea and Heuze [21] also conducted their own experimental study to observe the crack initiation and propagation from a single pre-existing flaw (0.400 , 10 mm long, 0.00800 , 0.2 mm wide) in limestone (Fig. 3). The authors again used the terms primary cracks to describe the conventional wing cracks, and the term secondary cracks to describe the longer curvilinear cracks. They noticed that, instead of initiating from the flaw tips, the secondary cracks initiated from regions at a distance away from the flaw tips, which are indicated by circles as shown in Fig. 3. After their initiation, the individual secondary cracks propagated towards the flaw tips and towards the top/bottom edges of the specimen (directions indicated by arrows). Petit and Barquins [9] tested low and high-porosity sandstone specimens (50 mm 50 mm 5 mm) both containing 20-mmlong single flaws (o1 mm flaw aperture). The flaws were created by first drilling a 0.15 mm diameter hole in the center and then sawing to either side for an 8-mm long slot with 0.3 mm aperture. In addition to the initiation of branch fractures, shear zones were observed to extensively develop in the specimens (Fig. 4). (Note that these authors used the term branch fractures instead of tensile wing cracks in their discussion.) Some of the shear zones were almost coplanar with the pre-existing flaws, while some of them were almost orthogonal to the pre-existing flaw (indicated by the arrow pair with asterisks *). Huang et al. [23] conducted uniaxial compression loading tests on Fangshan Marble specimens (104 mm 8 mm 6 mm) containing single flaws (20 mm long with a flaw aperture size o1 mm) and observed the following crack development stages (Fig. 5). (1) Initiation and propagation of primary forward tensile cracks (PFTCs). (2) Initiation and propagation of secondary forward tensile cracks (SFTCs). (3) Initiation and intensification of shear belts (backward shear belts (BSBs) and forward shear
Fig. 2. The evolution of fracture from a single open flaw observed by Lajtai [7].
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Fig. 3. Primary and secondary cracks initiated from a pre-existing flaw (0.400 , 10 mm long; 0.00800 , 0.2 mm wide) in limestone under uniaxial compression tested by Ingraffea and Heuze [21].
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Fig. 5. A schematic diagram showing various cracking features in Fangshan marble containing a single open flaw tested by Huang [23]. 1—PFTCs (primary forward tensile cracks); 2—SFTCs (secondary forward tensile cracks); 3—BTCs (backward tensile cracks); 4—FSBs (forward shear belts); 5—BSBs (backward shear belts).
Fig. 4. Crack growth (bf ¼ branch fracture; sz ¼ shear zone) from a single flaw in low- and high-porosity sandstone specimens tested by Petit and Barquins [9]. Sketches labeled ‘‘1’’ on the left show branch fracture (tensile wing crack) formation before maximum stress and sketches labeled ‘‘2’’ on the right show shear zone formation and secondary fractures at or after maximum stress.
belts (FSBs)). (4) Initiation and propagation of backward tensile cracks (BTCs). (5) Specimen failure. Note that Huang et al. [23] were able to determine the crack nature (tensile/shear), the terms primary and secondary were thus solely used for indicating the temporal relationship among different cracks. Chen et al. [24] conducted uniaxial compression loading tests on marble (type not mentioned) specimens (110 mm 80 mm 10 mm) containing 0.1-mm-wide single flaws (Fig. 6a) and identified the following three stages of crack growth: (1) primary cracks propagated perpendicularly to the direction of the flaw (Fig. 6b); (2) secondary cracks propagated in the direction of major principal stress. The secondary cracks developed faster and were longer than the primary cracks. Both primary and secondary cracks were stable and had finite lengths (Fig. 6c); (3) final failure of the
Fig. 6. Schematic representation of the various stages of initiation and propagation of new cracks from a pre-existing flaw under uniaxial compression (vertical) in marble tested by Chen et al. [24]. The flaws were 0.1 mm wide and had various lengths (8, 10, 12, 15, 16 mm).
specimen occurred by the development of an ‘‘X’’-shaped band (Fig. 6d). Although the authors stated that it was a band of microcracks, they did not provide further experimental evidence (e.g. microscopic study) to confirm its identity. Chen et al. [24] described the cracks as either primary or secondary throughout
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their paper. Although they correlated the primary cracks with the tensile cracks, they did not indicate whether the secondary cracks were tensile or shear in nature. They only postulated that the ends of the flaws from which the secondary cracks appeared were ‘‘compressive stress areas’’. The authors also correlated the observed secondary cracks with those observed by Ingraffea and Heuze [21]. However, the secondary cracks reported by Ingraffea and Heuze had a curvilinear shape (Fig. 3), while those observed by Chen et al. [24] were relatively straight (Fig. 6c). In addition, Ingraffea and Heuze [21] observed that the secondary cracks did not initiate from the flaw tips, but in the intact material, which was located at a distance away from the flaw tips, and then propagated both towards the flaw tips and the upper/lower edges
Fig. 7. Schematic representation of cracks developed from single flaws in Huangshi Marble as tested by Li et al. [25]. (a) Wing cracks (tensile cracks) initiated from a flaw with inclination angle 351. (b) Wing cracks (tensile cracks) and secondary cracks (shear cracks) initiated from a flaw with inclination angle 451.
Fig. 8. Layout of a specimen containing a 3-D surface crack as tested by Wong et al. [26].
of the specimen. This type of crack initiation and propagation was not reported by Chen et al. A correlation of the secondary cracks observed by Chen et al. with those observed by Ingraffea and Heuze thus appears unjustified. Li et al. [25] conducted uniaxial compression loading tests on Huangshi Marble specimens (110 mm 62 mm 25 mm), which contained single flaws (0.5–1 mm wide with a 6 mm diameter center hole; the flaw length was not given) oriented at two different inclination angles. The following crack growth phenomena were reported (Fig. 7): (a) Single flaw (351 flaw inclination angle)—wing cracks initiated from the end tips, and (b) single flaw (451 flaw inclination angle)—wing cracks and secondary quasicoplanar cracks initiated from the end tips. The latter led to specimen failure. Once again, the authors did not use the terms tensile and shear to describe the identity of the newly initiated cracks, but wing cracks and secondary quasi-coplanar cracks instead. From the description in their paper and the trajectories of the cracks, it is reasonable to assume that the authors used the term wing cracks to indicate tensile wing cracks, and the term secondary cracks to indicate shear cracks. Wong and her collaborators [26–28] recently conducted uniaxial compression loading tests on prismatic natural rock specimens (gabbro, marble, sandstone and granite) containing single three-dimensional (3D) surface flaws. As illustrated in Fig. 8, the dimensions of a 3D surface flaw are defined by the flaw length 2c and the depth d embedded into the prismatic specimen. The following stages of crack development in gabbro were reported. (1) A tensile crack initiated close to but not at the flaw tip (Fig. 9a). The trajectory of this tensile crack was different from that of a conventional wing crack. More specifically, it was located on the other side of the flaw (compare Fig. 7a with Fig. 9a). (2) The circled region in Fig. 9b indicates a region of local surface spalling. (3) The above-mentioned tensile crack (which was also called anti-wing crack) grew in two different directions—one towards the nearest flaw tip and the other one towards the edge of the specimen. The authors observed that ‘‘compressive cracks initiated at each flaw tips and coalesced with the growing cracks.’’ (Fig. 9c). (4) Additional cracks, such as ‘‘secondary wing cracks’’ as shown in Fig. 9d, initiated from the flaw tips at a later stage. Wong et al. have used the term ‘‘anti-wing cracks’’ to differentiate their observed tensile cracks (Fig. 9) from the conventional tensile wing cracks (Fig. 1). Also, they used the term ‘‘secondary’’ to indicate a temporal relationship. To summarize this review section on the cracking behavior of single flaws, the following key points are noted: In most of the tested specimens, tensile wing cracks as shown in Fig. 1 (which were commonly called primary cracks) were found to be the first cracks to initiate, while shear cracks were never observed to be the first cracks. Additional applied loading led to the initiation of secondary cracks. In some cases, the term secondary was used to indicate such a temporal relationship without implying the mode of crack initiation, e.g. [23], while in some cases, the authors used the term secondary cracks interchangeably with shear cracks e.g.,
Fig. 9. Schematic representation of various stages of crack growth of anti-wing cracks from a 3-D flaw in a gabbro specimen as tested by Wong et al. [26].
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[21,25]. There were, however, cases in which the authors were unable to determine the nature of the cracks, which developed later than the first tensile wing cracks (Fig. 1). These later cracks were simply described as secondary cracks without implying the mode of crack initiation, e.g. [24]. Notice also that the cracks that initiate after the initiation of the first crack are not necessarily shear cracks. According to Huang et al. [23], secondary forward tensile cracks were observed to initiate from the pre-existing flaws as later cracks, and the term secondary here simply implies a temporal relationship [23].
4. Experimental studies conducted by the authors In the present experimental study, unaxial loading compression tests were conducted on prismatic molded gypsum and Carrara Marble specimens containing single flaws to observe the development of new cracks. The specimens had the dimensions 600 (height) 300 (breadth) 1.2500 (thickness) (152 mm 76 mm 32 mm). Straight flaws (0.500 ; 12.5 mm long) of varying inclination angles were created in the specimens parallel to the specimen thickness (Fig. 10).
Fig. 10. (i) Dimensions of a prismatic specimen containing a central straight flaw. Open flaws with rounded tips of aperture size 0.0500 (1.3 mm) in (ii) gypsum and (iii) marble.
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The pre-cracked gypsum specimens were artificially molded according to the same procedures as in previous studies conducted at MIT [11–13,15]. The gypsum specimens were cast from a mixture of Hydrocal B-11 powder, celite powder and water at mass ratios of 700:8:280. The addition of celite powder reduced the amount of bleeding (migration of water to the top of the fluid mix). After being thoroughly blended, the mixture was poured into a steel mold containing a metal shim at the location of the future flaw. Open flaws were created by pulling the metal shims out of the hardened gypsum blocks. By varying the thickness of the metal shims, different flaw aperture sizes were obtained (0.00400 and 0.0500 ). The scope of the present study is limited to open flaws only, and experiments on closed flaws such as those studied by Bobet and Einstein [13] were not carried out. Hardened specimens were then removed from the mold and stored in an oven at 40 1C temperature. Before the compression tests, the surfaces of the specimens were polished with coarse (grit ]60) and then fine (grit ]200) sandpaper. On the other hand, the 0.500 (13 mm) long straight open flaws in Carrara Marble were cut by the OMAX abrasive jet. The OMAX abrasive jet derives its extremely high erosive cutting power from a pressurized stream of garnet abrasive-laden water, which is ejected at a speed of about 100 feet/s. Prepared gypsum and marble specimens were loaded uniaxially in a Baldwin 200 Kips Loading Machine, which was controlled by the computer program MTESTWindowsTM. Load and displacement data were automatically logged at a rate of 2000 samples/min. Please refer to [13,29] for further details on the loading procedures. During the entire loading test, the specimen front face was continuously video-taped by a SONY digital camcorder (DCR-HC65) and was simultaneously monitored by a PHANTOM V5.0 high-speed recording system (high-speed camera). The latter was triggered manually to capture images when cracking occurred. This system was capable of capturing up to 3000 frames/s. The actual span of the recordings depends on the pre-set sample rate (e.g. 500 frames/s vs. 3000 frames/s) and the resolution of the images required. By adopting the appropriate sample rate and image resolution, it was thus possible to view the cracking mechanisms precisely and, with sufficient details, in particular it was possible to determine if tensile cracking (tensile opening observed) or shear cracking (shear displacement observed) was taking place. Also, note that cracking mechanisms were identified based on high-speed video observation, but not by measuring the stress field. It must be emphasized that a proper identification of the crack types relies on the capabilities of the high-speed camera used. The significance of its use is illustrated in the idealized sketches shown in Fig. 11, which correspond to the fracturing and
Fig. 11. Schematic illustration of the occurrence of shearing on newly initiated tensile cracks.
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deformation events observed on a series of high-speed images. In image number 1, no observable cracking has yet occurred. In image number 2, a tensile crack initiates. In image number 3, shearing immediately follows to take place along the crack faces. If the nature of crack is determined solely by the examination of fractographical features without the use of the high-speed camera, it is possible that signs of shearing obfuscate the preexisting tensile cracking [30] and the true identity of tensile cracks could not be identified. Even if a high-speed camera is used, but not of a high enough frame rate, it is possible that only images 1 and 3 are identified. This means that the camera fails to capture the tensile crack opening event associated with the crack initiation mode. The present experimental study showed that seven crack types with different trajectories and initiation mechanism (tensile/ shear) could emanate from the pre-existing flaws in response to the applied loading (Fig. 12). Three of them are tensile and three of them are shear. The remaining one is of mixed tensile–shear nature, with shearing occurring adjacent to the flaw tips and simultaneous tensile opening occurring farther away. In addition to distinguishing the type of the cracks, it was also possible to observe in which sequence they occur, using the previously mentioned observations of tensile opening and shear displacements in the high-speed videos. Table 1 summarizes the types of cracks initiated in gypsum and marble specimens in response to the applied uniaxial compression loading. As indicated in Table 1, type 1 tensile cracks were often the first cracks to emanate from the single flaw in gypsum and marble. In the present paper, tensile wing cracks refer
to those tensile cracks, which display a wing appearance, regardless of their initiation position (Figs. 13a,b). As observed by the authors of this paper in an earlier publication [16], the initiation position of the pair of tensile wing cracks on the preexisting flaw was dependent on the flaw inclination angle. The initiation position moved away from the end flaw tips as the flaw inclination angles decreased from b ¼ 751 (steeper) to b ¼ 01 (shallower). Notice, however, that the horizontal flaws (b ¼ 01) in marble were an exception, in which type 2 tensile cracks initiated as the first cracks instead. (The trajectory of type 2 tensile crack is equivalent to that of type 3 tensile crack for horizontal flaws.) For simplicity, only type 2 tensile crack is mentioned for the relevant discussion. Fig. 13 illustrates the different cracking behavior observed in the two tested materials, both containing horizontal flaws. In response to the uniaxial (vertical) compression (Fig. 13a), the first observable change in gypsum as recorded in the video was the initiation of tensile wing cracks from the middle portion of the pre-existing flaw at which the tangential stresses are maximum along the flaw perimeter. The crack initiation was accompanied by a distinct cracking sound and the overall fracturing process in gypsum was very brittle. Once the cracks initiated, they immediately propagated towards the edges of the specimen. Marble, in contrast, responded differently to the applied loading. As shown in Fig. 13b, wing-shaped white patches and steeply inclined white patches developed from the flaw center and the flaw tips, respectively. When the applied loading further increased (Fig. 13c), the white patches lengthened, widened and intensified in color. Further loading eventually led to an
Fig. 12. Various crack types initiated from the pre-existing flaws identified in the present study. T ¼ tensile cracking opening. S ¼ shearing displacement.
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Table 1 Summary of all crack types observed in gypsum and marble specimens containing single flaws Seriesa
Flaw inclination angle b (deg)
Crack types Type 1 tensile
Type 2 tensile
Type 3 tensile
Mixed tensile–shear
0 15 30 35 40 45 50 55 60 70 75
O1 O1 O1 O1 O1 O1 O1 O1 O1 O1 O1
O O O O O O O O O
O O O O O O O O O
O
Gypsum wide flaws
0 30 45 60 75
O1 O1 O1 O1 O1
O O O O
O O O
O O O O
Marble wide flaws
0 30b 45b 60b 75b
O O1 O1 O1 O1
O1 O1 O1 O1 O1
O O O
O O O
Gypsum narrow flaws
(2/3) (2/3) (3/3) (2/3)
Type 1 shear
Type 2 shear
Type 3 shear
O O O O O O O
O O O
O O O
O1 indicates that the crack is the first crack to initiate. a The aperture of narrow flaws is 0.00400 and that of wide flaws is 0.0500 . b For these flaw geometries, the type 2 tensile cracks were observed to either initiate simultaneously with or later than the type 1 tensile cracks. The first number in parenthesis indicates the number of specimens in which both type 1 tensile cracks and type 2 tensile cracks initiate simultaneously as the first cracks. The second number indicates the number of tested specimens.
Fig. 13. (a) Tensile wing cracks initiate from a horizontal flaw in gypsum in response to uniaxial (vertical) loading. (b) In uniaxial (vertical) compression test on Carrara Marble, multiple white patches first emanate from the horizontal pre-existing flaw, which are then followed by (c) tensile crack opening (T) along the two almost vertical white patches at the two flaw tips (type 2 tensile cracks). The length of the pre-existing flaws is 0.500 (12.7 mm).
observable tensile crack opening along these white patches. Note that the trajectories of the white patches initially emanating from the flaw center are similar to those of the tensile wing cracks in gypsum (Fig. 13a). A similar development of white patches in response to applied loading was also observed by Chen et al. [24], Martinez [22] and Li et al. [25] in their tested marble specimens, and were suspected to be due to the presence of induced microcracks [24] or deviation and failure of crystalline grains [25]. However, no experimental attempts (e.g. microscopic imaging) were made by these authors to confirm their hypotheses. In Wong [29] and in two companion papers [31,32] by the authors of the present paper, the nature of the white patches is studied in detail. The white patch development is found to be due to the development of underlying microcracking zones. Since the objective of the present paper is the observation and generalization of crack patterns, the discus-
sion of white patches and the microstructural changes associated with cracking processes is out of the scope. Interested readers should consult [29,31,32] for further information. Referring again to Table 1 for marble with flaw geometries of b ¼ 301, 451, 601, 751, type 1 tensile cracks were always the first cracks to develop in all of these tested specimens. In addition, type 2 tensile cracks also sometimes initiated simultaneously with the type 1 tensile cracks in some of the tested specimens of each of these flaw geometries. Similar to the horizontal flaws, white patches which delineated the future trajectories of tensile cracks first developed. Later tensile cracking occurred along them leading to the development of type 1 and type 2 tensile cracks as recorded by the high-speed camera. Since all the tested specimens were loaded until specimen failure occurred, after the initiation of the first cracks, additional cracks of different types usually developed in the specimens.
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These crack types are also indicated in Table 1. For the sake of simplification, instead of listing the crack types observed for each individual tested specimen separately, all the crack types observed for the multiple specimens with the same flaw geometry are summarized in each row in Table 1. This means that, for example, even though four different crack types were observed for a flaw geometry, it does not necessarily imply that all of these four crack types were observed in one single specimen The crack types can be categorized as follows:
(1) Type 1 tensile cracks, either initiating as first cracks or later cracks, developed in all single-flaw geometries throughout the whole loading process. (2) Type 2 tensile cracks developed in almost all single-flaw geometries, except b ¼ 701 and 751 for single narrow flaws in gypsum, and b ¼ 751 for single wide flaws in gypsum. For these flaw geometries, the initiation of type 1 tensile cracks, which were the first cracks, was concurrent with the specimen maximum stress (specimen failure). Other crack types thus had no chance to develop afterwards. (3) Type 3 tensile cracks and mixed tensile–shear cracks were very common among the specimens with small and medium flaw inclination angles, but not for large flaw inclination angles (b ¼ 701 and 751 for single narrow flaws in gypsum, b ¼ 751 for single wide flaws in gypsum, and b ¼ 601 and 751 for single wide flaws in marble). For these large flaw inclination angles, the initiation of type 1 and/or type 2 tensile cracks was concurrent with the specimen maximum stress. Other crack types including type 3 tensile cracks thus had no chance to develop afterwards. (4) Shear cracks commonly developed in marble specimens during late stages of loading and their initiation often led to specimen failure (type 1 shear cracks for b ¼ 01, 301, 451; type 2 shear cracks for b ¼ 601, 751). These shear cracks, which initiated from the flaw tips, were generally inclined at an angle of about 55–651 with the horizontal regardless of the orientation of the pre-existing flaw. Therefore, these shear cracks made a varying inclination angle (Fig. 14a) with the shallowly inclined flaws (b ¼ 01, 301 and 451) and trend almost coplanar (Fig. 14b) with the more steeply inclined flaws (b ¼ 601 and 751). The consistent angle of shear cracks (with the horizontal) initiating from flaw tips is compared with the yc (the angle between the failure plane and the
Fig. 14. Shear crack initiation in marble. (a) Type 1 shear crack initiated from a pre-existing flaw with inclination angle 301 (specimen number: CM 30-D). (b) Type 2 shear crack initiated from a pre-existing flaw with inclination angle 601 (specimen number: CM 60-C). In both images, the shear cracks are inclined at 601 with the horizontal.
minimum principal stress) predicted by the Coulomb failure criterion for an originally intact specimen: yc ¼ 451+f/2, where f is the angle of internal friction of the rock. Substituting a value of f ¼ 281 [33] will yield yc ¼ 591, which is close to the observed inclination angle of these shear cracks (Fig. 14). It thus suggests a similarity in the fundamental cracking processes occurring in intact specimens and specimens with a pre-existing artificial flaw. Note also from Fig. 14 that the pre-existing flaws also underwent deformation as cracks propagated (shear in Fig. 14a, opening in Fig. 14b). Since the objective of this paper is on the cracking processes of newly initiated cracks, we did not concentrate on the flaw deformation processes. Please refer to the first author’s thesis [29], which discusses the flaw deformation in detail. In contrast, such an initiation and propagation of shear cracks were less common in gypsum. Although most of the failure cracks in gypsum also originated from the flaw tips, they are mainly of tensile origin (type 2 or type 3 tensile cracks) or mixed shear–tensile cracks. Their initiation readily leads to a tensile splitting of the specimen and an abrupt drop of specimen strength. (5) The initiation of type 3 shear cracks was rare in the tested specimens, and was only observed in marble specimens with flaw inclination angle b ¼ 451, but not in any of the gypsum specimens.
5. Discussion It is noted from the review in Section 2 that the terms primary cracks, tensile cracks and wing cracks were often used interchangeably, while the terms secondary cracks and shear cracks were similarly used interchangeably. The present experimental study, however, finds that such usage of terms is sometimes confusing, and thus very undesirable in describing the cracking behavior. In agreement with previous experiments by others (Section 3), the present experimental studies in gypsum and marble also show that tensile cracks, but not shear cracks, are the first cracks to initiate from the prismatic specimens containing single flaws. However, as discussed above, although type 1 tensile cracks (tensile wing cracks) are always the first cracks to initiate in marble, they are not necessarily the only type. In marble, type 2 tensile cracks, which do not display the conventional wing appearance, can also sometimes initiate as the first cracks simultaneously with the type 1 tensile cracks (Table 1). Therefore, using the terms primary cracks interchangeably with wing cracks is inappropriate. It is hereby suggested that the term primary should only be used to indicate a temporal relationship which refers to all cracks initiating as the first cracks. In addition, tensile cracks should only describe cracks that initiate in a tensile mode, without any implication of the shape of crack trajectory. On the other hand, wing cracks should only imply the shape of crack trajectory and be restricted to describe type 1 tensile cracks only. As shown in Table 1 and discussed above, various types of cracks initiate after the initiation of the first cracks. Most importantly, these later cracks are not only restricted to shear cracks, but also include tensile cracks (e.g. type 2 tensile cracks, type 3 tensile cracks for a majority of gypsum specimens). It implies that using the terms secondary cracks and shear cracks interchangeably is inappropriate. It is hereby suggested that the term secondary should only be used for indicating a temporal relationship to describe those cracks initiated later than the first cracks. Shear cracks should solely be used to refer to the shearing crack initiation mode, without implying any temporal relationship.
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The crack types observed and generalized in earlier studies have been reviewed again, with the objective to unify the terminology. The crack types are reclassified as shown in Fig. 15a–g according to the scheme shown in Fig. 12. In cases where the nature of new cracks was not clearly stated in the original references, several possible crack types are assigned based on the shape of the crack trajectories. For example, in Fig. 15b, there are two types of cracks emanating from the pre-existing flaw. The first type is the conventional tensile wing crack and is assigned as type 1 tensile cracks. The cracking mechanism of the second type, which has a trajectory different from the type 1 tensile crack (tensile wing crack), is not certain. Three possible crack types, namely type 2 tensile crack, type 1 shear crack and
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mixed tensile–shear crack, are thus assigned. Note also that the terms primary and secondary have been omitted in the reevaluation process unless there is a strong indication that such a usage is necessary to indicate a temporal relationship and does not cause confusion (e.g. Fig. 15d).
6. Conclusions The present experimental study, which was based on detailed observations of cracking processes with the high speed camera, shows that seven crack types with different trajectories and crack mechanism (tensile/shear) can initiate from the pre-existing flaws
Fig. 15. Fracturing patterns in specimens containing single pre-existing flaws (a) Lajtai [7], (b) Ingraffea and Heuze [21], (c) Petit and Barquins [9], (d) Huang et al. [23], (e) Chen et al. [24], (f) Li et al. [25], (g) Wong et al. [26].
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Fig. 15. (Continued)
in response to the applied uniaxial loading. Applying this crack type classification scheme to describe the cracks reported previously in the literature makes it possible to avoid the confusion arising from the inconsistent use of terminologies. It is also suggested that the terms primary and secondary not be used to imply the crack nature (tensile/shear), but only to indicate a temporal relationship.
Acknowledgments The project was sponsored by the NSF Geomechanics and Geotechnical Systems Program under Grant CMMI-0555053 and the US Department of Energy Geothermal Program under Grant DE-FG36-06GO16061. The first author is also thankful to the support by the Croucher Foundation Scholarship (Hong Kong) and the Sir Edward Youde Memorial Fellowship (Hong Kong). References [1] Bombolakis EG. Photoelastic stress analysis of crack propagation within a compressive stress field. Ph.D. thesis, MIT, Cambridge, 1963. p. 38. [2] Brace WF, Bombolakis EG. A note on brittle crack growth in compression. J Geophys Res 1963;68(12):3709–13. [3] Nemat-Nasser S, Horii H. Compression-induced nonplanar crack extension with application to splitting, exfoliation, and rockburst. J Geophys Res 1982;87(8):6805–21. [4] Horii H, Nemat-Nasser S. Compression-induced microcrack growth in brittle solids: axial splitting and shear failure. J Geophys Res 1985;90(B4):3105–25. [5] Hoek E, Bieniawski ZT. Brittle fracture propagation in rock under compression. Int J Fract 1965;1:137–55. [6] Bieniawski ZT. Mechanism of brittle fracture of rock, Part II—experimental studies. Int J Rock Mech Min Sci 1967;4:407–23. [7] Lajtai EZ. Brittle fracture in compression. Int J Fract 1974;10:525–36. [8] Nesetova V, Lajtai EZ. Fracture from compressive stress concentrations around elastic flaws. Int J Rock Mech Min Sci 1973;10:265–84.
[9] Petit J, Barquins M. Can natural faults propagate under mode II conditions? Tectonics 1988;7(6):1246–65. [10] Chaker C, Barquins M. Sliding effect on branch crack. Phys Chem Earth 1996;21(4):319–23. [11] Reyes O, Einstein HH. Failure mechanism of fractured rock—a fracture coalescence model In: Proceedings of the seventh international congress on rock mechanics, Aachen,Germany, vol. 1, 1991. p. 333–40. [12] Shen B, Stephansson O, Einstein HH, Ghahreman B. Coalescence of fractures under shear stress experiments. J Geophys Res 1995;100(6):5975–90. [13] Bobet A, Einstein HH. Fracture coalescence in rock-type materials under uniaxial and biaxial compression. Int J Rock Mech Min Sci 1998;35(7): 863–88. [14] Sagong M, Bobet A. Coalescence of multiple flaws in a rock-model material in uniaxial compression. Int J Rock Mech Min Sci 2002;39:229–41. [15] Ko TY, Einstein HH, Kemeny J. Crack coalescence in brittle material under cyclic loading. In: Proceedings of the 41st US symposium on rock mechanics, Golden, CO; 2006 [paper ARMA/USRMS 06-930]. [16] Wong LNY, Einstein HH. Fracturing behavior of prismatic specimens containing single flaws. In: Proceedings of the 41st US symposium rock mechanics, Golden, CO; 2006 [paper ARMA/USRMS 06-899]. [17] Wong LNY, Einstein HH. Coalescence behavior in Carrara marble and molded gypsum containing artificial flaw pairs under uniaxial compression. In: Proceedings of the first Can-US rock mechanics Symposium, Vancouver, 2007. p. 581–9. [18] Wong RHC, Chau KT. Crack coalescence in a rock-like material containing two cracks. Int J Rock Mech Min Sci 1998;35(2):147–64. [19] Wong RHC, Chau KT, Tang CA, Lin P. Analysis of crack coalescence in rock-like materials containing three flaws—part I: experimental approach. Int J Rock Mech Min Sci 2001;38:909–24. [20] Mughieda O, Alzo’ubi AK. Fracture mechanics of offset rock joints—a laboratory investigation. Geotech Geol Eng 2004;22:545–62. [21] Ingraffea AR, Heuze FE. Finite element models for rock fracture mechanics. Int J Num Anal Meth Geomech 1980;4:25. [22] Martinez AR. Fracture coalescence in natural rock. MSc thesis, MIT, Cambridge, 1999. p. 341. [23] Huang JF, Chen GL, Zhao YH, Wang R. An experimental study of the strain field development prior to failure of a marble plate under compression. Tectonophysics 1990;175:283–90. [24] Chen G, Kemeny JM, Harpalani S. Fracture propagation and coalescence in marble plates with pre-cut notches under compression. In: Symp ractured jointed rock masses, Lake Tahoe, CA; 1995. p. 435–9. [25] Li YP, Chen LZ, Wang YH. Experimental research on pre-cracked marble under compression. Int J Solids Struct 2005;42:2505–16.
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[26] Wong RHC, Guo YSH, Li LY, Chau KT, Zhu WS, Li SC. Anti-wing crack growth from surface flaw in real rock under uniaxial compression. In: 16th Eur conf fracture (EFC16), Alexandroupolis, Greece; 2006. p. 825. [27] Wong RHC, Li TC, Chau KT, Li SC, Zhu WS. Crack growth study of a 3-D surface fracture under compression using strain and acoustic emission measurements. In: Proceedings of the first Can-US rock mechanics Symposium, Vancouver, Canada; 2007. p. 565–73. [28] Guo YSH, Wong RHC, Chau KT, Zhu WS, Li SC. Crack growth mechanisms from 3-D surface flaw with varied dipping angle under uniaxial compression. In: Asian Pacific conference fracture & strength, Sanya, China; 2006. p. 335.
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[29] Wong LNY. Crack coalescence in molded gypsum and Carrara marble. Ph.D. thesis, MIT, Cambridge, 2008. p. 876. [30] Einstein HH, Meyer T. Mu¨ller lecture: puzzles in rock. In: Proceedings of the international congress rock mechanics, Paris, vol. 3; 1999. p. 1707–40. [31] Wong LNY, Einstein HH. Crack coalescence in molded gypsum and Carrara Marble: Part 1—macroscopic observations and interpretation. Rock Mech Rock Eng (accepted). [32] Wong LNY, Einstein HH. Crack coalescence in molded gypsum and Carrara Marble: Part 2—microscopic observations and interpretation. Rock Mech Rock Eng (accepted). [33] Pollard DD, Fletcher RC. Fundamentals of structural geology. Cambridge: Cambridge University Press; 2005.
Contents lists available at ScienceDirect
International Journal of Rock Mechanics & Mining Sciences journal homepage: www.elsevier.com/locate/ijrmms
Systematic evaluation of cracking behavior in specimens containing single flaws under uniaxial compression L.N.Y. Wong, H.H. Einstein Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
a r t i c l e in fo
abstract
Article history: Received 26 October 2007 Received in revised form 7 March 2008 Accepted 12 March 2008 Available online 12 May 2008
This paper presents the results of an experimental study in which molded gypsum and Carrara Marble specimens containing a pre-existing flaw were tested in uniaxial compression. The main purpose of this study was to observe and characterize the cracks that emanate from a single pre-existing flaw. Seven different crack types were identified based on their geometry and propagation mechanism (tensile/ shear). Specifically, they include three types of tensile cracks, three types of shear cracks, and one type of mixed tensile–shear crack. In addition to the geometry and mechanism, it was also possible to determine the temporal sequence of different crack types. These observations form the basis for a reevaluation of cracking processes reported in the literature. It is possible to apply the classification obtained in the present research to previously conducted experiments by others. This eliminates much of the confusion that has existed when comparing different research results. & 2008 Elsevier Ltd. All rights reserved.
Keywords: Tensile cracks Shear cracks Crack type classification scheme High-speed camera Uniaxial compressive loading test
1. Introduction Since Bombolakis [1] observed the propagation of tensile wing cracks from pre-existing straight flaws under uniaxial compression in Columbia Resin 39, fracturing processes and crack coalescence patterns in pre-cracked samples under compression have been extensively studied experimentally on different materials, including both rock-like brittle/semi-brittle materials (Columbia Resin 39 [1–4], glass [5,6], Plaster of Paris [7,8], polymethylmethacrilate (PMMA) [9,10], molded gypsum [11–17], sandstone-like molded barite [18,19] and sandstone-like concrete mix [20]) and natural rocks (sandstone [9], granodiorite [21], limestone [21], granite [22], marble [22–25] and gabbro [26]). Henceforth, the term ‘‘flaw’’ will be used to describe an artificially created, pre-existing crack or fracture, which can be either open or closed. In these tests, the size of the prismatic specimens ranged from a minimum of 50 mm 32 mm 5 mm [9] to a maximum of 635 mm 279 mm 203 mm [20]. The flaw length and the flaw aperture of open flaws varied between 10 and 50 mm, and between 0.1 and 3 mm, respectively. Note that in some cases closed flaws were also studied, e.g. [3–6,9,12,13,18]. As just mentioned, one of the intentions of all these tests was to observe and characterize the cracks emanating from the flaws. This led to a variety of terminologies, i.e. there are different descriptions for the same crack type and also sometimes not
Corresponding author. Tel.: +1 617 253 3598; fax: +1 617 253 6044.
E-mail address: [email protected] (H.H. Einstein). 1365-1609/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijrmms.2008.03.006
sufficient differentiation between different crack types. The new experimental techniques, which involved the use of a high-speed camera, detailed in this paper, make it possible to more clearly define the different crack types than was possible in the past. This will make it possible to introduce a crack characterization scheme, which covers previous and present research observations. Specifically, this paper will first briefly review the existing crack characterization terminologies and identify the existing confusing aspects. It will then summarize and discuss the key findings obtained from the present experimental study, which then forms the basis for a re-evaluation of cracking processes reported in the literature.
2. Existing crack terminology and encountered problems In the literature, different terminologies have been used to describe the crack types observed. Since tensile wing cracks were usually found to be the first cracks to appear from the pre-existing flaws (Fig. 1), they were also commonly called primary cracks. Cracks initiating later than the tensile wing cracks were usually referred to as secondary cracks, without specifying the crack mechanism (tensile/shear). Being uncertain whether these additional cracks were tensile or shear in origin, most authors named these cracks secondary cracks. Later studies suggested that secondary cracks are usually shear in origin. As time went by, researchers began using secondary cracks and shear cracks interchangeably, e.g. [13]. However, in certain cases when the authors were unable to determine the
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crack mechanism (tensile/shear), they continued the old practice of using the term secondary with the intention of simply indicating the temporal cracking relationship without implying the crack mechanism. Confusion was thus often created for later readers. It will be later shown with the present experimental study that cracks that initiate after the initiation of conventional tensile wing cracks can be either shear or tensile. Other types of tensile cracks, in addition to wing cracks, have also been observed.
3. Previous studies In this section, the various crack types observed by different authors in specimens containing single flaws under uniaxial loading tests are reviewed. As noted in the previous section regarding the inconsistent use of terminologies, careful interpretation of the crack types identified by different authors is required. The crack types reviewed in this section will be evaluated again in Section 5 based on the classification obtained from the present experimental study presented in Section 4. Lajtai [7] conducted uniaxial compression loading tests on Plaster of Paris specimens of dimensions 300 600 600 (76 mm 152 mm 152 mm), which contain single flaws of various dimensions. All flaws had the same width (aperture) of 0.0200 (0.5 mm), but variable lengths, 2.0, 1.5, 1.0 and 0.5 in, respectively. The flaw inclination angle (angle with the horizontal) varies from 01 to 901
Fig. 1. Initiation of tensile wing cracks from a pre-existing inclined to the uniaxial loading direction.
at 51 intervals. The results, which are schematically summarized in Fig. 2, consist of the following crack initiation sequence (with the original terminologies)—(a) tensile fractures, (b) normal shear fractures, (c) additional normal shear fractures leading to the formation of shear zones, (d) inclined shear fractures. In a literature review, Ingraffea and Heuze [21] identified the influence of material on fracturing behavior under uniaxial compression. They observed that in specimens with a single open flaw inclined to the compression axis, primary cracks, which originated from points of initially highest tension stress, were observed in previous tests on glass, polymethylmethacrilate (PMMA), CR39 and also rocks. Secondary cracks, which originated from points of initially compressive stress concentration, were however absent in glass and plastic, but were only observed in rocks. Notice that Ingraffea and Heuze [21] appeared to have used the term primary cracks interchangeably with tensile wing cracks and the term secondary cracks interchangeably with shear cracks. Ingraffea and Heuze [21] also conducted their own experimental study to observe the crack initiation and propagation from a single pre-existing flaw (0.400 , 10 mm long, 0.00800 , 0.2 mm wide) in limestone (Fig. 3). The authors again used the terms primary cracks to describe the conventional wing cracks, and the term secondary cracks to describe the longer curvilinear cracks. They noticed that, instead of initiating from the flaw tips, the secondary cracks initiated from regions at a distance away from the flaw tips, which are indicated by circles as shown in Fig. 3. After their initiation, the individual secondary cracks propagated towards the flaw tips and towards the top/bottom edges of the specimen (directions indicated by arrows). Petit and Barquins [9] tested low and high-porosity sandstone specimens (50 mm 50 mm 5 mm) both containing 20-mmlong single flaws (o1 mm flaw aperture). The flaws were created by first drilling a 0.15 mm diameter hole in the center and then sawing to either side for an 8-mm long slot with 0.3 mm aperture. In addition to the initiation of branch fractures, shear zones were observed to extensively develop in the specimens (Fig. 4). (Note that these authors used the term branch fractures instead of tensile wing cracks in their discussion.) Some of the shear zones were almost coplanar with the pre-existing flaws, while some of them were almost orthogonal to the pre-existing flaw (indicated by the arrow pair with asterisks *). Huang et al. [23] conducted uniaxial compression loading tests on Fangshan Marble specimens (104 mm 8 mm 6 mm) containing single flaws (20 mm long with a flaw aperture size o1 mm) and observed the following crack development stages (Fig. 5). (1) Initiation and propagation of primary forward tensile cracks (PFTCs). (2) Initiation and propagation of secondary forward tensile cracks (SFTCs). (3) Initiation and intensification of shear belts (backward shear belts (BSBs) and forward shear
Fig. 2. The evolution of fracture from a single open flaw observed by Lajtai [7].
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Fig. 3. Primary and secondary cracks initiated from a pre-existing flaw (0.400 , 10 mm long; 0.00800 , 0.2 mm wide) in limestone under uniaxial compression tested by Ingraffea and Heuze [21].
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Fig. 5. A schematic diagram showing various cracking features in Fangshan marble containing a single open flaw tested by Huang [23]. 1—PFTCs (primary forward tensile cracks); 2—SFTCs (secondary forward tensile cracks); 3—BTCs (backward tensile cracks); 4—FSBs (forward shear belts); 5—BSBs (backward shear belts).
Fig. 4. Crack growth (bf ¼ branch fracture; sz ¼ shear zone) from a single flaw in low- and high-porosity sandstone specimens tested by Petit and Barquins [9]. Sketches labeled ‘‘1’’ on the left show branch fracture (tensile wing crack) formation before maximum stress and sketches labeled ‘‘2’’ on the right show shear zone formation and secondary fractures at or after maximum stress.
belts (FSBs)). (4) Initiation and propagation of backward tensile cracks (BTCs). (5) Specimen failure. Note that Huang et al. [23] were able to determine the crack nature (tensile/shear), the terms primary and secondary were thus solely used for indicating the temporal relationship among different cracks. Chen et al. [24] conducted uniaxial compression loading tests on marble (type not mentioned) specimens (110 mm 80 mm 10 mm) containing 0.1-mm-wide single flaws (Fig. 6a) and identified the following three stages of crack growth: (1) primary cracks propagated perpendicularly to the direction of the flaw (Fig. 6b); (2) secondary cracks propagated in the direction of major principal stress. The secondary cracks developed faster and were longer than the primary cracks. Both primary and secondary cracks were stable and had finite lengths (Fig. 6c); (3) final failure of the
Fig. 6. Schematic representation of the various stages of initiation and propagation of new cracks from a pre-existing flaw under uniaxial compression (vertical) in marble tested by Chen et al. [24]. The flaws were 0.1 mm wide and had various lengths (8, 10, 12, 15, 16 mm).
specimen occurred by the development of an ‘‘X’’-shaped band (Fig. 6d). Although the authors stated that it was a band of microcracks, they did not provide further experimental evidence (e.g. microscopic study) to confirm its identity. Chen et al. [24] described the cracks as either primary or secondary throughout
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their paper. Although they correlated the primary cracks with the tensile cracks, they did not indicate whether the secondary cracks were tensile or shear in nature. They only postulated that the ends of the flaws from which the secondary cracks appeared were ‘‘compressive stress areas’’. The authors also correlated the observed secondary cracks with those observed by Ingraffea and Heuze [21]. However, the secondary cracks reported by Ingraffea and Heuze had a curvilinear shape (Fig. 3), while those observed by Chen et al. [24] were relatively straight (Fig. 6c). In addition, Ingraffea and Heuze [21] observed that the secondary cracks did not initiate from the flaw tips, but in the intact material, which was located at a distance away from the flaw tips, and then propagated both towards the flaw tips and the upper/lower edges
Fig. 7. Schematic representation of cracks developed from single flaws in Huangshi Marble as tested by Li et al. [25]. (a) Wing cracks (tensile cracks) initiated from a flaw with inclination angle 351. (b) Wing cracks (tensile cracks) and secondary cracks (shear cracks) initiated from a flaw with inclination angle 451.
Fig. 8. Layout of a specimen containing a 3-D surface crack as tested by Wong et al. [26].
of the specimen. This type of crack initiation and propagation was not reported by Chen et al. A correlation of the secondary cracks observed by Chen et al. with those observed by Ingraffea and Heuze thus appears unjustified. Li et al. [25] conducted uniaxial compression loading tests on Huangshi Marble specimens (110 mm 62 mm 25 mm), which contained single flaws (0.5–1 mm wide with a 6 mm diameter center hole; the flaw length was not given) oriented at two different inclination angles. The following crack growth phenomena were reported (Fig. 7): (a) Single flaw (351 flaw inclination angle)—wing cracks initiated from the end tips, and (b) single flaw (451 flaw inclination angle)—wing cracks and secondary quasicoplanar cracks initiated from the end tips. The latter led to specimen failure. Once again, the authors did not use the terms tensile and shear to describe the identity of the newly initiated cracks, but wing cracks and secondary quasi-coplanar cracks instead. From the description in their paper and the trajectories of the cracks, it is reasonable to assume that the authors used the term wing cracks to indicate tensile wing cracks, and the term secondary cracks to indicate shear cracks. Wong and her collaborators [26–28] recently conducted uniaxial compression loading tests on prismatic natural rock specimens (gabbro, marble, sandstone and granite) containing single three-dimensional (3D) surface flaws. As illustrated in Fig. 8, the dimensions of a 3D surface flaw are defined by the flaw length 2c and the depth d embedded into the prismatic specimen. The following stages of crack development in gabbro were reported. (1) A tensile crack initiated close to but not at the flaw tip (Fig. 9a). The trajectory of this tensile crack was different from that of a conventional wing crack. More specifically, it was located on the other side of the flaw (compare Fig. 7a with Fig. 9a). (2) The circled region in Fig. 9b indicates a region of local surface spalling. (3) The above-mentioned tensile crack (which was also called anti-wing crack) grew in two different directions—one towards the nearest flaw tip and the other one towards the edge of the specimen. The authors observed that ‘‘compressive cracks initiated at each flaw tips and coalesced with the growing cracks.’’ (Fig. 9c). (4) Additional cracks, such as ‘‘secondary wing cracks’’ as shown in Fig. 9d, initiated from the flaw tips at a later stage. Wong et al. have used the term ‘‘anti-wing cracks’’ to differentiate their observed tensile cracks (Fig. 9) from the conventional tensile wing cracks (Fig. 1). Also, they used the term ‘‘secondary’’ to indicate a temporal relationship. To summarize this review section on the cracking behavior of single flaws, the following key points are noted: In most of the tested specimens, tensile wing cracks as shown in Fig. 1 (which were commonly called primary cracks) were found to be the first cracks to initiate, while shear cracks were never observed to be the first cracks. Additional applied loading led to the initiation of secondary cracks. In some cases, the term secondary was used to indicate such a temporal relationship without implying the mode of crack initiation, e.g. [23], while in some cases, the authors used the term secondary cracks interchangeably with shear cracks e.g.,
Fig. 9. Schematic representation of various stages of crack growth of anti-wing cracks from a 3-D flaw in a gabbro specimen as tested by Wong et al. [26].
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[21,25]. There were, however, cases in which the authors were unable to determine the nature of the cracks, which developed later than the first tensile wing cracks (Fig. 1). These later cracks were simply described as secondary cracks without implying the mode of crack initiation, e.g. [24]. Notice also that the cracks that initiate after the initiation of the first crack are not necessarily shear cracks. According to Huang et al. [23], secondary forward tensile cracks were observed to initiate from the pre-existing flaws as later cracks, and the term secondary here simply implies a temporal relationship [23].
4. Experimental studies conducted by the authors In the present experimental study, unaxial loading compression tests were conducted on prismatic molded gypsum and Carrara Marble specimens containing single flaws to observe the development of new cracks. The specimens had the dimensions 600 (height) 300 (breadth) 1.2500 (thickness) (152 mm 76 mm 32 mm). Straight flaws (0.500 ; 12.5 mm long) of varying inclination angles were created in the specimens parallel to the specimen thickness (Fig. 10).
Fig. 10. (i) Dimensions of a prismatic specimen containing a central straight flaw. Open flaws with rounded tips of aperture size 0.0500 (1.3 mm) in (ii) gypsum and (iii) marble.
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The pre-cracked gypsum specimens were artificially molded according to the same procedures as in previous studies conducted at MIT [11–13,15]. The gypsum specimens were cast from a mixture of Hydrocal B-11 powder, celite powder and water at mass ratios of 700:8:280. The addition of celite powder reduced the amount of bleeding (migration of water to the top of the fluid mix). After being thoroughly blended, the mixture was poured into a steel mold containing a metal shim at the location of the future flaw. Open flaws were created by pulling the metal shims out of the hardened gypsum blocks. By varying the thickness of the metal shims, different flaw aperture sizes were obtained (0.00400 and 0.0500 ). The scope of the present study is limited to open flaws only, and experiments on closed flaws such as those studied by Bobet and Einstein [13] were not carried out. Hardened specimens were then removed from the mold and stored in an oven at 40 1C temperature. Before the compression tests, the surfaces of the specimens were polished with coarse (grit ]60) and then fine (grit ]200) sandpaper. On the other hand, the 0.500 (13 mm) long straight open flaws in Carrara Marble were cut by the OMAX abrasive jet. The OMAX abrasive jet derives its extremely high erosive cutting power from a pressurized stream of garnet abrasive-laden water, which is ejected at a speed of about 100 feet/s. Prepared gypsum and marble specimens were loaded uniaxially in a Baldwin 200 Kips Loading Machine, which was controlled by the computer program MTESTWindowsTM. Load and displacement data were automatically logged at a rate of 2000 samples/min. Please refer to [13,29] for further details on the loading procedures. During the entire loading test, the specimen front face was continuously video-taped by a SONY digital camcorder (DCR-HC65) and was simultaneously monitored by a PHANTOM V5.0 high-speed recording system (high-speed camera). The latter was triggered manually to capture images when cracking occurred. This system was capable of capturing up to 3000 frames/s. The actual span of the recordings depends on the pre-set sample rate (e.g. 500 frames/s vs. 3000 frames/s) and the resolution of the images required. By adopting the appropriate sample rate and image resolution, it was thus possible to view the cracking mechanisms precisely and, with sufficient details, in particular it was possible to determine if tensile cracking (tensile opening observed) or shear cracking (shear displacement observed) was taking place. Also, note that cracking mechanisms were identified based on high-speed video observation, but not by measuring the stress field. It must be emphasized that a proper identification of the crack types relies on the capabilities of the high-speed camera used. The significance of its use is illustrated in the idealized sketches shown in Fig. 11, which correspond to the fracturing and
Fig. 11. Schematic illustration of the occurrence of shearing on newly initiated tensile cracks.
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deformation events observed on a series of high-speed images. In image number 1, no observable cracking has yet occurred. In image number 2, a tensile crack initiates. In image number 3, shearing immediately follows to take place along the crack faces. If the nature of crack is determined solely by the examination of fractographical features without the use of the high-speed camera, it is possible that signs of shearing obfuscate the preexisting tensile cracking [30] and the true identity of tensile cracks could not be identified. Even if a high-speed camera is used, but not of a high enough frame rate, it is possible that only images 1 and 3 are identified. This means that the camera fails to capture the tensile crack opening event associated with the crack initiation mode. The present experimental study showed that seven crack types with different trajectories and initiation mechanism (tensile/ shear) could emanate from the pre-existing flaws in response to the applied loading (Fig. 12). Three of them are tensile and three of them are shear. The remaining one is of mixed tensile–shear nature, with shearing occurring adjacent to the flaw tips and simultaneous tensile opening occurring farther away. In addition to distinguishing the type of the cracks, it was also possible to observe in which sequence they occur, using the previously mentioned observations of tensile opening and shear displacements in the high-speed videos. Table 1 summarizes the types of cracks initiated in gypsum and marble specimens in response to the applied uniaxial compression loading. As indicated in Table 1, type 1 tensile cracks were often the first cracks to emanate from the single flaw in gypsum and marble. In the present paper, tensile wing cracks refer
to those tensile cracks, which display a wing appearance, regardless of their initiation position (Figs. 13a,b). As observed by the authors of this paper in an earlier publication [16], the initiation position of the pair of tensile wing cracks on the preexisting flaw was dependent on the flaw inclination angle. The initiation position moved away from the end flaw tips as the flaw inclination angles decreased from b ¼ 751 (steeper) to b ¼ 01 (shallower). Notice, however, that the horizontal flaws (b ¼ 01) in marble were an exception, in which type 2 tensile cracks initiated as the first cracks instead. (The trajectory of type 2 tensile crack is equivalent to that of type 3 tensile crack for horizontal flaws.) For simplicity, only type 2 tensile crack is mentioned for the relevant discussion. Fig. 13 illustrates the different cracking behavior observed in the two tested materials, both containing horizontal flaws. In response to the uniaxial (vertical) compression (Fig. 13a), the first observable change in gypsum as recorded in the video was the initiation of tensile wing cracks from the middle portion of the pre-existing flaw at which the tangential stresses are maximum along the flaw perimeter. The crack initiation was accompanied by a distinct cracking sound and the overall fracturing process in gypsum was very brittle. Once the cracks initiated, they immediately propagated towards the edges of the specimen. Marble, in contrast, responded differently to the applied loading. As shown in Fig. 13b, wing-shaped white patches and steeply inclined white patches developed from the flaw center and the flaw tips, respectively. When the applied loading further increased (Fig. 13c), the white patches lengthened, widened and intensified in color. Further loading eventually led to an
Fig. 12. Various crack types initiated from the pre-existing flaws identified in the present study. T ¼ tensile cracking opening. S ¼ shearing displacement.
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Table 1 Summary of all crack types observed in gypsum and marble specimens containing single flaws Seriesa
Flaw inclination angle b (deg)
Crack types Type 1 tensile
Type 2 tensile
Type 3 tensile
Mixed tensile–shear
0 15 30 35 40 45 50 55 60 70 75
O1 O1 O1 O1 O1 O1 O1 O1 O1 O1 O1
O O O O O O O O O
O O O O O O O O O
O
Gypsum wide flaws
0 30 45 60 75
O1 O1 O1 O1 O1
O O O O
O O O
O O O O
Marble wide flaws
0 30b 45b 60b 75b
O O1 O1 O1 O1
O1 O1 O1 O1 O1
O O O
O O O
Gypsum narrow flaws
(2/3) (2/3) (3/3) (2/3)
Type 1 shear
Type 2 shear
Type 3 shear
O O O O O O O
O O O
O O O
O1 indicates that the crack is the first crack to initiate. a The aperture of narrow flaws is 0.00400 and that of wide flaws is 0.0500 . b For these flaw geometries, the type 2 tensile cracks were observed to either initiate simultaneously with or later than the type 1 tensile cracks. The first number in parenthesis indicates the number of specimens in which both type 1 tensile cracks and type 2 tensile cracks initiate simultaneously as the first cracks. The second number indicates the number of tested specimens.
Fig. 13. (a) Tensile wing cracks initiate from a horizontal flaw in gypsum in response to uniaxial (vertical) loading. (b) In uniaxial (vertical) compression test on Carrara Marble, multiple white patches first emanate from the horizontal pre-existing flaw, which are then followed by (c) tensile crack opening (T) along the two almost vertical white patches at the two flaw tips (type 2 tensile cracks). The length of the pre-existing flaws is 0.500 (12.7 mm).
observable tensile crack opening along these white patches. Note that the trajectories of the white patches initially emanating from the flaw center are similar to those of the tensile wing cracks in gypsum (Fig. 13a). A similar development of white patches in response to applied loading was also observed by Chen et al. [24], Martinez [22] and Li et al. [25] in their tested marble specimens, and were suspected to be due to the presence of induced microcracks [24] or deviation and failure of crystalline grains [25]. However, no experimental attempts (e.g. microscopic imaging) were made by these authors to confirm their hypotheses. In Wong [29] and in two companion papers [31,32] by the authors of the present paper, the nature of the white patches is studied in detail. The white patch development is found to be due to the development of underlying microcracking zones. Since the objective of the present paper is the observation and generalization of crack patterns, the discus-
sion of white patches and the microstructural changes associated with cracking processes is out of the scope. Interested readers should consult [29,31,32] for further information. Referring again to Table 1 for marble with flaw geometries of b ¼ 301, 451, 601, 751, type 1 tensile cracks were always the first cracks to develop in all of these tested specimens. In addition, type 2 tensile cracks also sometimes initiated simultaneously with the type 1 tensile cracks in some of the tested specimens of each of these flaw geometries. Similar to the horizontal flaws, white patches which delineated the future trajectories of tensile cracks first developed. Later tensile cracking occurred along them leading to the development of type 1 and type 2 tensile cracks as recorded by the high-speed camera. Since all the tested specimens were loaded until specimen failure occurred, after the initiation of the first cracks, additional cracks of different types usually developed in the specimens.
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These crack types are also indicated in Table 1. For the sake of simplification, instead of listing the crack types observed for each individual tested specimen separately, all the crack types observed for the multiple specimens with the same flaw geometry are summarized in each row in Table 1. This means that, for example, even though four different crack types were observed for a flaw geometry, it does not necessarily imply that all of these four crack types were observed in one single specimen The crack types can be categorized as follows:
(1) Type 1 tensile cracks, either initiating as first cracks or later cracks, developed in all single-flaw geometries throughout the whole loading process. (2) Type 2 tensile cracks developed in almost all single-flaw geometries, except b ¼ 701 and 751 for single narrow flaws in gypsum, and b ¼ 751 for single wide flaws in gypsum. For these flaw geometries, the initiation of type 1 tensile cracks, which were the first cracks, was concurrent with the specimen maximum stress (specimen failure). Other crack types thus had no chance to develop afterwards. (3) Type 3 tensile cracks and mixed tensile–shear cracks were very common among the specimens with small and medium flaw inclination angles, but not for large flaw inclination angles (b ¼ 701 and 751 for single narrow flaws in gypsum, b ¼ 751 for single wide flaws in gypsum, and b ¼ 601 and 751 for single wide flaws in marble). For these large flaw inclination angles, the initiation of type 1 and/or type 2 tensile cracks was concurrent with the specimen maximum stress. Other crack types including type 3 tensile cracks thus had no chance to develop afterwards. (4) Shear cracks commonly developed in marble specimens during late stages of loading and their initiation often led to specimen failure (type 1 shear cracks for b ¼ 01, 301, 451; type 2 shear cracks for b ¼ 601, 751). These shear cracks, which initiated from the flaw tips, were generally inclined at an angle of about 55–651 with the horizontal regardless of the orientation of the pre-existing flaw. Therefore, these shear cracks made a varying inclination angle (Fig. 14a) with the shallowly inclined flaws (b ¼ 01, 301 and 451) and trend almost coplanar (Fig. 14b) with the more steeply inclined flaws (b ¼ 601 and 751). The consistent angle of shear cracks (with the horizontal) initiating from flaw tips is compared with the yc (the angle between the failure plane and the
Fig. 14. Shear crack initiation in marble. (a) Type 1 shear crack initiated from a pre-existing flaw with inclination angle 301 (specimen number: CM 30-D). (b) Type 2 shear crack initiated from a pre-existing flaw with inclination angle 601 (specimen number: CM 60-C). In both images, the shear cracks are inclined at 601 with the horizontal.
minimum principal stress) predicted by the Coulomb failure criterion for an originally intact specimen: yc ¼ 451+f/2, where f is the angle of internal friction of the rock. Substituting a value of f ¼ 281 [33] will yield yc ¼ 591, which is close to the observed inclination angle of these shear cracks (Fig. 14). It thus suggests a similarity in the fundamental cracking processes occurring in intact specimens and specimens with a pre-existing artificial flaw. Note also from Fig. 14 that the pre-existing flaws also underwent deformation as cracks propagated (shear in Fig. 14a, opening in Fig. 14b). Since the objective of this paper is on the cracking processes of newly initiated cracks, we did not concentrate on the flaw deformation processes. Please refer to the first author’s thesis [29], which discusses the flaw deformation in detail. In contrast, such an initiation and propagation of shear cracks were less common in gypsum. Although most of the failure cracks in gypsum also originated from the flaw tips, they are mainly of tensile origin (type 2 or type 3 tensile cracks) or mixed shear–tensile cracks. Their initiation readily leads to a tensile splitting of the specimen and an abrupt drop of specimen strength. (5) The initiation of type 3 shear cracks was rare in the tested specimens, and was only observed in marble specimens with flaw inclination angle b ¼ 451, but not in any of the gypsum specimens.
5. Discussion It is noted from the review in Section 2 that the terms primary cracks, tensile cracks and wing cracks were often used interchangeably, while the terms secondary cracks and shear cracks were similarly used interchangeably. The present experimental study, however, finds that such usage of terms is sometimes confusing, and thus very undesirable in describing the cracking behavior. In agreement with previous experiments by others (Section 3), the present experimental studies in gypsum and marble also show that tensile cracks, but not shear cracks, are the first cracks to initiate from the prismatic specimens containing single flaws. However, as discussed above, although type 1 tensile cracks (tensile wing cracks) are always the first cracks to initiate in marble, they are not necessarily the only type. In marble, type 2 tensile cracks, which do not display the conventional wing appearance, can also sometimes initiate as the first cracks simultaneously with the type 1 tensile cracks (Table 1). Therefore, using the terms primary cracks interchangeably with wing cracks is inappropriate. It is hereby suggested that the term primary should only be used to indicate a temporal relationship which refers to all cracks initiating as the first cracks. In addition, tensile cracks should only describe cracks that initiate in a tensile mode, without any implication of the shape of crack trajectory. On the other hand, wing cracks should only imply the shape of crack trajectory and be restricted to describe type 1 tensile cracks only. As shown in Table 1 and discussed above, various types of cracks initiate after the initiation of the first cracks. Most importantly, these later cracks are not only restricted to shear cracks, but also include tensile cracks (e.g. type 2 tensile cracks, type 3 tensile cracks for a majority of gypsum specimens). It implies that using the terms secondary cracks and shear cracks interchangeably is inappropriate. It is hereby suggested that the term secondary should only be used for indicating a temporal relationship to describe those cracks initiated later than the first cracks. Shear cracks should solely be used to refer to the shearing crack initiation mode, without implying any temporal relationship.
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The crack types observed and generalized in earlier studies have been reviewed again, with the objective to unify the terminology. The crack types are reclassified as shown in Fig. 15a–g according to the scheme shown in Fig. 12. In cases where the nature of new cracks was not clearly stated in the original references, several possible crack types are assigned based on the shape of the crack trajectories. For example, in Fig. 15b, there are two types of cracks emanating from the pre-existing flaw. The first type is the conventional tensile wing crack and is assigned as type 1 tensile cracks. The cracking mechanism of the second type, which has a trajectory different from the type 1 tensile crack (tensile wing crack), is not certain. Three possible crack types, namely type 2 tensile crack, type 1 shear crack and
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mixed tensile–shear crack, are thus assigned. Note also that the terms primary and secondary have been omitted in the reevaluation process unless there is a strong indication that such a usage is necessary to indicate a temporal relationship and does not cause confusion (e.g. Fig. 15d).
6. Conclusions The present experimental study, which was based on detailed observations of cracking processes with the high speed camera, shows that seven crack types with different trajectories and crack mechanism (tensile/shear) can initiate from the pre-existing flaws
Fig. 15. Fracturing patterns in specimens containing single pre-existing flaws (a) Lajtai [7], (b) Ingraffea and Heuze [21], (c) Petit and Barquins [9], (d) Huang et al. [23], (e) Chen et al. [24], (f) Li et al. [25], (g) Wong et al. [26].
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Fig. 15. (Continued)
in response to the applied uniaxial loading. Applying this crack type classification scheme to describe the cracks reported previously in the literature makes it possible to avoid the confusion arising from the inconsistent use of terminologies. It is also suggested that the terms primary and secondary not be used to imply the crack nature (tensile/shear), but only to indicate a temporal relationship.
Acknowledgments The project was sponsored by the NSF Geomechanics and Geotechnical Systems Program under Grant CMMI-0555053 and the US Department of Energy Geothermal Program under Grant DE-FG36-06GO16061. The first author is also thankful to the support by the Croucher Foundation Scholarship (Hong Kong) and the Sir Edward Youde Memorial Fellowship (Hong Kong). References [1] Bombolakis EG. Photoelastic stress analysis of crack propagation within a compressive stress field. Ph.D. thesis, MIT, Cambridge, 1963. p. 38. [2] Brace WF, Bombolakis EG. A note on brittle crack growth in compression. J Geophys Res 1963;68(12):3709–13. [3] Nemat-Nasser S, Horii H. Compression-induced nonplanar crack extension with application to splitting, exfoliation, and rockburst. J Geophys Res 1982;87(8):6805–21. [4] Horii H, Nemat-Nasser S. Compression-induced microcrack growth in brittle solids: axial splitting and shear failure. J Geophys Res 1985;90(B4):3105–25. [5] Hoek E, Bieniawski ZT. Brittle fracture propagation in rock under compression. Int J Fract 1965;1:137–55. [6] Bieniawski ZT. Mechanism of brittle fracture of rock, Part II—experimental studies. Int J Rock Mech Min Sci 1967;4:407–23. [7] Lajtai EZ. Brittle fracture in compression. Int J Fract 1974;10:525–36. [8] Nesetova V, Lajtai EZ. Fracture from compressive stress concentrations around elastic flaws. Int J Rock Mech Min Sci 1973;10:265–84.
[9] Petit J, Barquins M. Can natural faults propagate under mode II conditions? Tectonics 1988;7(6):1246–65. [10] Chaker C, Barquins M. Sliding effect on branch crack. Phys Chem Earth 1996;21(4):319–23. [11] Reyes O, Einstein HH. Failure mechanism of fractured rock—a fracture coalescence model In: Proceedings of the seventh international congress on rock mechanics, Aachen,Germany, vol. 1, 1991. p. 333–40. [12] Shen B, Stephansson O, Einstein HH, Ghahreman B. Coalescence of fractures under shear stress experiments. J Geophys Res 1995;100(6):5975–90. [13] Bobet A, Einstein HH. Fracture coalescence in rock-type materials under uniaxial and biaxial compression. Int J Rock Mech Min Sci 1998;35(7): 863–88. [14] Sagong M, Bobet A. Coalescence of multiple flaws in a rock-model material in uniaxial compression. Int J Rock Mech Min Sci 2002;39:229–41. [15] Ko TY, Einstein HH, Kemeny J. Crack coalescence in brittle material under cyclic loading. In: Proceedings of the 41st US symposium on rock mechanics, Golden, CO; 2006 [paper ARMA/USRMS 06-930]. [16] Wong LNY, Einstein HH. Fracturing behavior of prismatic specimens containing single flaws. In: Proceedings of the 41st US symposium rock mechanics, Golden, CO; 2006 [paper ARMA/USRMS 06-899]. [17] Wong LNY, Einstein HH. Coalescence behavior in Carrara marble and molded gypsum containing artificial flaw pairs under uniaxial compression. In: Proceedings of the first Can-US rock mechanics Symposium, Vancouver, 2007. p. 581–9. [18] Wong RHC, Chau KT. Crack coalescence in a rock-like material containing two cracks. Int J Rock Mech Min Sci 1998;35(2):147–64. [19] Wong RHC, Chau KT, Tang CA, Lin P. Analysis of crack coalescence in rock-like materials containing three flaws—part I: experimental approach. Int J Rock Mech Min Sci 2001;38:909–24. [20] Mughieda O, Alzo’ubi AK. Fracture mechanics of offset rock joints—a laboratory investigation. Geotech Geol Eng 2004;22:545–62. [21] Ingraffea AR, Heuze FE. Finite element models for rock fracture mechanics. Int J Num Anal Meth Geomech 1980;4:25. [22] Martinez AR. Fracture coalescence in natural rock. MSc thesis, MIT, Cambridge, 1999. p. 341. [23] Huang JF, Chen GL, Zhao YH, Wang R. An experimental study of the strain field development prior to failure of a marble plate under compression. Tectonophysics 1990;175:283–90. [24] Chen G, Kemeny JM, Harpalani S. Fracture propagation and coalescence in marble plates with pre-cut notches under compression. In: Symp ractured jointed rock masses, Lake Tahoe, CA; 1995. p. 435–9. [25] Li YP, Chen LZ, Wang YH. Experimental research on pre-cracked marble under compression. Int J Solids Struct 2005;42:2505–16.
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[26] Wong RHC, Guo YSH, Li LY, Chau KT, Zhu WS, Li SC. Anti-wing crack growth from surface flaw in real rock under uniaxial compression. In: 16th Eur conf fracture (EFC16), Alexandroupolis, Greece; 2006. p. 825. [27] Wong RHC, Li TC, Chau KT, Li SC, Zhu WS. Crack growth study of a 3-D surface fracture under compression using strain and acoustic emission measurements. In: Proceedings of the first Can-US rock mechanics Symposium, Vancouver, Canada; 2007. p. 565–73. [28] Guo YSH, Wong RHC, Chau KT, Zhu WS, Li SC. Crack growth mechanisms from 3-D surface flaw with varied dipping angle under uniaxial compression. In: Asian Pacific conference fracture & strength, Sanya, China; 2006. p. 335.
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[29] Wong LNY. Crack coalescence in molded gypsum and Carrara marble. Ph.D. thesis, MIT, Cambridge, 2008. p. 876. [30] Einstein HH, Meyer T. Mu¨ller lecture: puzzles in rock. In: Proceedings of the international congress rock mechanics, Paris, vol. 3; 1999. p. 1707–40. [31] Wong LNY, Einstein HH. Crack coalescence in molded gypsum and Carrara Marble: Part 1—macroscopic observations and interpretation. Rock Mech Rock Eng (accepted). [32] Wong LNY, Einstein HH. Crack coalescence in molded gypsum and Carrara Marble: Part 2—microscopic observations and interpretation. Rock Mech Rock Eng (accepted). [33] Pollard DD, Fletcher RC. Fundamentals of structural geology. Cambridge: Cambridge University Press; 2005.