International Journal of Rock Mechanics and Mining Sciences
Crack coalescence in a rock-like material containing two cracks
Introduction
The length scale of fractures found in natural rocks ranges from tens of kilometres down to tens of microns. In addition, fractures or joints in rocks are normally of finite length or are discontinuous in nature. The mechanism of crack coalescence in the rock bridge area (see the definition on Fig. 1) between the preexisting cracks remains one of the most fundamental theoretical problems in rock mechanics to be solved.
From the practical point of view, nearly all rock engineering projects involve, to a certain extent, constructions of structures in or on rock masses, which contain both fractures and rock bridges. The crack coalescence in a rock bridge is usually responsible for the failure of many rock structures. For example, in 1991 a large block of rock of about 2000 m3 slid from a steep rock face at the discussed quarry at Shau Kei Wan in Hong Kong. At the time of the incident, blasting had been taking place above the disused quarry for some time in conjunction with the site formation works for the construction of a new housing estate. Interpreted from the investigation report[1], the preexisting parallel joints, dipping toward the rock slope surface, were not fully persistent before the slide occurred; it is likely that crack initiation and growth at the tip of the joints may have been induced by the vibration of blasting. Therefore, it is believed that crack coalescence was the cause of the rock slope failure. Thus, in rock engineering projects the mechanism of crack coalescence of preexisting fractures plays an important role in controlling the stabilities of slopes, foundations and tunnels. However, the mechanisms of crack coalescence in rock masses, containing joints and rock bridge, have not been fully understood.
Crack initiation and propagation have been subjects of intensive investigation in rock mechanics, both experimentally and theoretically. The first theoretical study on the growth of preexisting two-dimensional cracks was given by Griffith2, 3. Irwin[4] further introduced the concept of critical energy release rate and the crack tip stress intensity factor (K). Relating to the field of rock mechanics, a number of experimental studies have been carried out to investigate the crack initiation, propagation and interaction (e.g.5, 6, 7, 8, 9, 10, 11, 12). For a comprehensive literature review on microcrack studies we refer to Kranz[13]. At the same time, many mathematical models were developed to explain and predict the processes of crack growth, interaction and rock failure (e.g.14, 15, 16, 17, 18, 19, 20). For the discussion on how these shear crack models can be applied to real rocks, we refer to the comparative study by Fredrich et al.[21].
However, most of the previous studies are focused on the mechanisms of crack initiation, propagation and interaction, but relatively few experimental investigations were done to examine the pattern of crack coalescence in the rock bridge area between the preexisting fractures. The pattern of coalescence between two parallel cracks in both modelling materials and natural rocks have been done in direct shear boxes (e.g.22, 23, 24, 25). However, it is not easy to observe the development of the whole process of coalescence inside the shear box. Recently, Reyes[26] and Reyes and Einstein[27] have performed some uniaxial compressive tests on gypsum samples containing two inclined open cracks. The process of crack coalescence was recorded by a microscope connected to a video camera. As expected, the mechanism of coalescence is controlled by the initial geometric setting of the parallel cracks. To incorporate the effect of friction, Shen et al.[28] conducted a series of uniaxial compressive tests on gypsum samples containing both open- and closed-fractures. Related numerical simulation of the failure of rock bridge was also done by Shen and coauthors28, 29, 30, 31. Therefore, Shen et al.'s[28] study seems to be most relevant to the actual process of coalescence in jointed rocks.
However, certain aspects of the study by Shen and coauthors[28] can further be improved. Firstly, the gypsum mixture used by Shen et al.[28] is the one by Nelson and Hirschfeld[32]. However, most of the π-factors for their modelling material do not fall within the range of the π-factors for quartzite and granite, which are the rocks to be modelled (see Ref.[33] or the Appendix Afor the definition of π-factors). Therefore, a more careful dimensional analysis of the modelling material should be done using the π-theorem in order to simulate the patterns of crack coalescence as close as possible to those observed in real rocks (see[33] or the summary in Appendix Afor π-theorem). In addition, we also expect that the pattern of crack coalescence may depend on the types of rocks. Secondly, the effect of the inclination of the preexisting cracks α, the bridge angle β (which is the relative angle between the two inclined cracks), and the frictional coefficient μ of preexisting crack surfaces on the mode of crack coalescence is not fully examined by Shen et al.[28]. In particular, only one frictional coefficient (μ=0.7) has been simulated on the crack surfaces, together with the open crack (μ=0), and only 13 combinations of α/β were examined. In addition, no attempt has been made to compare the observed peak strength (the maximum attainable applied stress under a uniaxial compression) and the prediction by existing crack models.
Therefore, we attempt here to give a more refined study on the pattern of crack coalescence of sandstone-like material and the peak strength. The modelling material to be used here will be analysed by the π-theorem such that it resembles the main characters of sandstone. A total of 87 combinations of α/β and μ will be presented in this study, compared to the 26 combinations by Shen et al.[28]. A regime classification for the patterns of coalescence will also be proposed empirically based upon our experimental observations. The peak strength predicted by using the sliding crack model of Ashby and Hallam[17] is compared to our experimental results and the range of its applicability will be discussed.
Section snippets
Experimental studies
The discussion of our experimental studies is divided into four sections. The first section discusses the physical properties of a sandstone-like modelling material, the second section on the technique in preparing the cracked specimens, the third section on the testing procedure in loading the cracked specimens, and the fourth section on the general experimental observation.
Sliding crack model
Although experimental studies seem to suggest that microcracking is mainly tensile but not shear in nature (e.g.[8]) and some authors also argued that the sliding crack model cannot account for all of the experimental observations (e.g.48, 49), the “sliding crack model” consisting of a sliding shear crack and two tensile wing cracks remains one of the most popular models to describe the inelastic dilatancy of rocks. A myriad number of sliding crack models have been proposed in predicting the
Mode of crack coalescence
When crack coalescence occurs, two main types of cracking can be identified in the rock bridge area: wing cracks, which are tensile in nature; and secondary cracks, which are mainly shear in nature and are normally parallel to the preexisting shear cracks. Based upon these two types of cracks and their combinations, Fig. 6 summarizes nine different patterns of crack coalescence observed in our experiments. In particular, there are three main modes of crack coalescence: (1) shear crack
Regime classification of crack coalescence
As discussed in the previous section, there are main modes of coalescence, namely: the shear mode, the mixed shear/tensile mode and the wing tensile mode. From their appearance, nine different patterns of crack coalescence can be classified as shown in Fig. 6. It is natural to ask: Is there any correlation between α, β and μ and these patterns of coalescence? This section is set forth to answer this question.
Fig. 9, Fig. 10, Fig. 11 plot the regimes of crack coalescence for different modes in
Effect of α, β and μ on peak strength
Another main purpose of this study is to examine how the strength of rock is affected by the geometric setting of the two preexisting cracks and the frictional coefficient on these crack surfaces. The prediction by using the Ashby-Hallam model[17], which is summarized briefly in the previous section, is also presented here for comparison.
Table 2 tabulates 84 observed and predicted peak strengths of our specimens containing different combinations of α, β and μ. Except for two sets of data (α/β
Conclusion
We have presented in this paper an experimental study on the coalescence mechanism in and peak strength of a sandstone-like material containing two inclined frictional parallel cracks subject to uniaxial compression. The observed peak strengths are also compared with the predictions by the sliding crack model of Ashby and Hallam[17].
The modelling material is classified as a sandstone-like material, based upon a dimensional analysis using π-factors. In addition, both brittleness and dilatancy of
Acknowledgements
The study was supported by the Staff Development Program of the Hong Kong Polytechnic University to RHCW. We are grateful to Dr Nick Barton for suggesting this research topic and providing invaluable comments throughout the study. The laboratory assistance by Mr C. W. Leung is also appreciated.
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