Keywords: boundary elements, discrete elements, crack growth, discontinuity, rock mechanics, numerical simulations, fracturing.

The presence of fractures in rock, or in other materials can have a pronounced influence on both stress and strain distribution in the surrounding material. This disturbance can be accompanied by the opening of a fracture, in shear or in the normal direction. Therefore, a numerical model for fracture growth should be able to account for the physical detachment of both sides of the fracture. Both an existing discontinuity and a new crack can open and undergo a shear displacement. In order to model this properly, discontinuous elements are required. Fracture patterns result from the interaction of crack growth in the intact material and the activation of the discontinuity.

In this paper, two different types of 2D numerical codes are applied for the simulation of a Brazilian tensile test of heterogeneous rock. The first code applied is UDEC, which is a 2D distinct element code [1]. The distinct element model consists of distinct blocks that are mutually connected by contacts. For these contacts tensile and shear failure criteria are defined, allowing them to open and deform upon activation. The UDEC solution scheme is based on a (dynamic) explicit finite-difference method which is used also in continuum analysis.

The second code applied is DIGS, a 2D displacement discontinuity boundary element code [2]. The DIGS models consist of multiple displacement discontinuity elements. These displacement discontinuities function similarly to the contacts in UDEC. DIGS solves iteratively the stress-strain equations while accounting for the displacement discontinuities.

A Brazilian tensile test on a coal sample is modelled. A disk shaped sample is compressed until failure with its axis orthogonal to the direction of loading. The loading of the sample induces tensile stresses in the perpendicular direction. Based on the maximum load, the tensile strength can be calculated using a linear elastic approach. The sample in this study has a predefined cemented fracture (i.e. a discontinuity) that divides the sample in two equal halves, under an angle of 45^o to the direction of loading. Two models are studied both using UDEC and DIGS simulations. In Model 1 only the discontinuity is modelled as such by multiple discontinuity elements (DIGS) or contacts (UDEC), and both halves of the disk are modelled as an elastic continuous material. In Model 2 the central discontinuity is still present, but the background material is further divided by a grid of discontinuity elements. Thus, the occurrence of new cracks in the intact (background) material can be simulated.

For each simulation, the evolution of the stress vectors along the discontinuity, before and during failure are discussed by means of three parameters (i.e. distance to shear failure criterion, minimal principal stress and angle of the stress vector). In addition to this, for Model 2, the simulated fracture patterns are presented.

The simulation of the two models by two different codes resulted in some interesting observations. While the stress levels are of the same order of magnitude, the distribution and orientation of the stresses are dependent on the applied code. On the one hand, this is due to differences in the construction and principles of the codes. On the other hand, this is due to a difference in modelled properties (e.g. contact stiffness, as also demonstrated in reference [3]).

The resulting fracture pattern in Model 2 is a combination of discontinuity activation and tensile fracturing of the intact material. The evolution and the final configuration of the fracture pattern are obviously dependent on the model properties. Decreasing the cohesion of the discontinuity results in a longer zone where the discontinuity is activated. For a very small cohesion value, there is no tensile fracture. Decreasing the tensile strength of the intact material, results in a more intense tensile fracturing and a decreasing zone of discontinuity activation.

The qualitative resemblance of fracture patterns in both numerical models suggests the soundness of the model. One numerical method does not have to prevail over the other, but a combined study can lead to a better insight of the mechanisms involved (e.g. importance of element stiffness).

1

P.A. Cundall, "A computer model for simulating progressive large scale movements in block rock systems", in "Proceedings of the Symposium International. Society of Rock Mechanics (Nancy, F)", Volume 1, paper II-8, 1971.

2

J.A.L. Napier, "Modelling of fracturing near deep level gold mine excavations using a displacement discontinuity approach", in "Proceedings of the International Conference of the Mechanics of Jointed and Faulted Rock", Rossmanith, Balkema Publications, Rotterdam, The Netherlands, 709-715, 1990.

3

B. Debecker, A. Vervoort, "Influence of fracture stiffness on local stress redistribution" in "Proceedings of the International symposium on in-situ rock stress (Trondheim, N)", Balkema Publications, Rotterdam, The Netherlands, 2006. Article in press

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