International Journal of Rock Mechanics and Mining Sciences
Numerical studies of the influence of microstructure on rock failure in uniaxial compression — Part I: effect of heterogeneity
Introduction
The uniaxial compressive strength of a rock is one of the simplest measures of strength. It may be regarded as the largest stress that a rock specimen can carry when a unidirectional stress is applied to the ends of a specimen. In other words, the unconfined compressive strength represents the maximum load supported by the specimen during the test divided by the cross-sectional area of the specimen. Although the utility of the compressive strength value is limited, the unconfined compressive strength allows comparisons to be made between rocks and provides some indications of rock behavior under more complex stress systems.
Experimentally, researchers have undertaken the task of loading specimens to obtain better knowledge of the compressive failure mechanisms and considerable discussion has been devoted in the literature to this test method [1], [2], [3], [4], [5], [6]. Though this mode of failure has been studied in detail for decades, the details of the failure mechanisms, including the microfracture initiation, propagation, coalescence, axial splitting, shearing, etc., are not fully understood and still remain the subject of considerable scientific interest. One of the key questions is: when can the incipient fracture plane be recognized during the loading stage of an initially intact rock, or at what point does fracture interaction overwhelm the local variations in the stress field or in the local properties and drive the system to fracture, either by coalescence of fractures or by extension of one fracture [4], [7]?
Based on experimental observations, material behavior models can be constructed. The models can be either analytical or numerical. The analytical models lead to correct answers within the framework of axioms underlying the mathematics, but not necessarily for the rock reality. Analytical models have to be simplified and sometimes this simplification ignores important factors influencing the material behavior. Heterogeneity is such an example for rocks. From this point of view, numerical tools incorporating a heterogeneity capability can provide a better simulation of the mechanical behavior. In particular, a numerical approach seems essential for fracture mechanisms because of the important interactions between fracture growth and the structural environment. This implies that the fractures should be modeled preferentially as a real discontinuity. For many cases, analytical solutions which include discontinuities cannot be found and a numerical approximation seems to be the best that can be achieved. As pointed out by Van Mier [8], with the aid of numerical tools, effects from the structural environment (loading system or test conditions) can be studied in great detail. They provide an increased insight into material behavior and structural behavior.
The authors are in agreement with the opinions expressed by Van Mier [8] that the idea of modeling is that we would like to have a numerical tool that can describe in sufficient detail the phenomena observed in experiments. With such a model we would be more confident about making predictions outside the domain that was used for tuning the model, that is, to forecast the more complicated macroscopic response from its relatively simple microscopic behavior.
Statistical modeling has emerged as a promising technique for analysis of fracture in heterogeneous materials such as rock [7]. The combinations of statistical theory with numerical models such as the lattice model [8], the bonded particle model [9], [10], or the RFPA model based on FEM [11] are found to be appropriate for modeling brittle materials such as rocks. At present, lattice models do not seem well suited for modeling compressive fracture. However, by means of the UDEC model or other models such as the rigid particle model, better results are obtained, although with more complex meso-level material laws [8].
In this paper (Part I of a two-part series), uniaxial compressive tests with specimens of brittle disordered rock material were numerically studied by using a Rock Failure Process Analysis code (RFPA2D), developed recently by CRISR at Northeastern University [11]. Following a brief description of the numerical model and the loading procedures adopted in this investigation, some characteristic features of the complete stress–strain curves and the phenomena observed during progressive fracture of several rock-like materials will be summarized in terms of heterogeneity, deformation localization, fracture nucleation or coalescence, and microfracture induced seismic activities. In the companion paper (Part II), other factors affecting the failure behavior of rocks, specimen shape (slenderness), size and boundary restraint, are numerically examined.
Section snippets
Brief outline of the rock failure process analysis numerical code, RFPA2D
Numerical simulation is currently the most popular method used for modeling deformation behavior of rock before failure. Even though progress has been made in numerical simulation of failure occurring in rocks [10], there is a lack of satisfactory models which can simulate the progressive failure in a more visual way, including simulation of the failure process, failure induced seismic events and failure induced stress redistribution.
The demand for new tools, which may contribute to a better
Numerical simulation results
The stress–strain relation for specimen “c-h-2” is given in Fig. 2. The specimen was numerically shortened at a constant displacement rate of 0.002 mm/step, though during the non-linear stages these rates were increased fractionally due to the relaxation of the loading platens. The model predicts the non-linear stress–strain curve similar to the typical curve of brittle rock observed in laboratory tests [2]. It is found that, although the constitutive law for the individual element in the
Complete stress–strain curve
Fig. 2 reveals that the stress–strain characteristics of rock can be modeled well with the RFPA code. As the axial strain is increased, the stress–strain curve is nearly linear at the initial stage. When the load reaches approximately 50% of the peak load, due to the fact that the rock begins to nucleate obvious microfractures, the tangent stiffness of the specimen decreases and it reaches zero at the compressive strength. Above that stress, progressively less and less stress is required to
Conclusions
The following conclusions can be drawn from the numerical simulations described.
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On the basis of the numerical modeling, the stress–strain curve for a heterogeneous specimen of rock generally includes five characteristic regions.
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In region I (up to about 80% strength), the linear elastic portion of the stress–strain curve is associated with microfractures randomly distributed throughout the specimen, and no fabric changes can be observed in comparison with the undeformed specimens.
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At the onset of
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Acknowledgements
The work presented in this paper was mainly undertaken while Professor C. A. Tang paid a visit to the Rock Engineering Research Center, Hong Kong University, Hong Kong, and was partially supported by the Chinese National Key Fundamental Research “973 Programme” (No 95-13-07-01) and the National Natural Science Foundation (No. 49974009).
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