This paper briefly describes a brittle granular model used in numerical experiments to gain further insight into micromechanical processes which develop into macroscopic brittle failure. The behaviour of the model shows good qualitative agreement with known brittle behaviour, and deviations are shown to be the result of its simplicity. Stress-difference vs strain-difference curves in plane strain show a smooth transition from strain-softening (brittle) to strain hardening (ductile) behaviour with increasing confinement. Anisotropic elasticity theory was employed to show the effect of fracture on the material elastic constants, and a fracture tensor was used to demonstrate that the major fracture direction lies parallel or sub-parallel to the major compressive stress direction. A yield curve for plane strain conditions was found for the model both in tensile and compressive stress space, and features describing the model behaviour are highlighted on the diagram.
A complete theory of brittle behaviour has not yet been found. Existing theories fall into two groups : empirical, such as the Coulomb, Mohr, and Hoek and Brown (1980) theories, and physical, such as the Griffith (1924) theory. Mohr-Coulomb theory now forms the basis of the theory of plasticity, while theorists and experimenters have attempted to take the Griffith (1924) theory further by proposing explicit fracture and damage mechanics theories, for example Brady (1970), Kachanov (1982b), and Stavropoulou (1983). All show discrepancies of one kind or another between predicted and observed behaviour. Some of the fracture-based theories met with success in predicting brittle pre-failure behaviour, but all failed in the post-failure regime for various reasons based mainly on the fact that brittle post-failure behaviour is an exceedingly complex phenomenon, involving fragmentation. Distinct element methods lend themselves to this kind of problem, for example UDEC (Universal Distinct Element Code) was developed to model blocky media and was designed to be equally adept at modelling an intact or fragmented medium, Cundall (1980), Cundall and Hart (1983). Understanding failure and post- failure mechanisms in these rocks would therefore be beneficial to the South African gold mining industry in particular and hard rock mining in general. This paper briefly outlines the numerical model, and then presents some of the results obtained from it. It is concluded that numerical modelling of brittle granular media can give new insights into failure processes which are not obtainable from laboratory experiments, but that further research and development is necessary before a universally applicable brittle fracture criterion can be introduced.
The granular model is two-dimensional, consisting of a square area broken up into straight-sided convex-outward Voronoi Polygons generated numerically. Voronoi polygons were chosen to represent the grains in the quartzites for two reasons, firstly they are generally similar in shape, and secondly they provided a quick and unbiased means of generating granular geometries. The average grain dimension in the model is 0.7 mm, similar to the grain size of many Witwatersrand Quartzites, set in a square area 5mm on a side. The model contains 55 grains, as shown in Figure 1.