Brittle deformation of rocks is a very important deformation mechanism in the upper crust of the Earth and at deeper levels if strain rates are high or fluids are involved [Ranalli 1995; Scholz 2002]. Rocks can only sustain very small amounts of strain in the linear elastic regime and fail afterwards [Means 1976]. Failure takes place by the development and propagation of fractures [Jaeger and Cook 1979]. If failure localizes in planar zones, cataclasites will develop in which cohesion of material along grain boundaries is lost and grain size is reduced by intragranular fractures [Price and Cosgrove 1990].
Fractures are generally classified into extensional fractures (Mode I fractures) and shear fractures (Mode II fractures) [Price and Cosgrove 1990; Pollard and Segall 1987]. Mode I fractures develop parallel to the compressive stress and perpendicular to the tensile stress. Since they are oriented perpendicular to the tensile stress they are opening and can be filled with vein material. Mode II fractures are oriented at an angle of less than 45° to the compressive stress. Sliding takes place along these surfaces but they are not opening so that they seldom contain vein material. A combination of mode I and mode II fractures, so termed hybrid shear fractures [Price and Cosgrove 1990] with a smaller angle to the compressive stress than shear fractures show shear displacement plus opening. Which type of fracture develops depends on the stress field and the microstructure of the specimen [Jaeger and Cook 1979]. During progressive deformation different types of fractures may develop successively [Toussaint and Pride 2002].
In addition to their orientation, shape and spacing of fractures is of general interest. These parameters may give insight into the rheology of fractured material and the degree of deformation it has experienced [Price and Cosgrove 1990]. The distribution of fractures, their shape, spacing and formation of three-dimensional networks are also important in the understanding of fluid flow, waste dispersal and the characterization and modelling of oil reservoirs.
Fracture dynamics is a complex phenomenon since the propagation of fractures can be highly non-linear. Griffith (1920) already ascribed the low strength of material under tension to stress concentrations at tips of micro-cracks that pre-exist in most materials. Once a micro-crack propagates stress concentrations at its tips will increase which eventually leads to large-scale failure of the material. Under compression the material behaviour is more complex [Mogi 1962]. The link between microscopic and macroscopic behaviour may not be simple under these conditions [Lockner 1998; Okui and Horii 1997; Hazzard et al. 2000] and transitions between different failure types probably exist [Toussaint and Pride, 2002]. Crack tips may also become unstable under certain conditions and show branching [Marder 1993].
Several models have been proposed to simulate fracture-development in two-dimensional systems on different scales (e.g. Hazzard et al. 2000; D’Adetta et al. 2001; Mühlhaus et al. 2001 and references therein). Molecular dynamic models are concerned with fractures at atomic scale whereas models in statistical physics and disordered systems investigate macroscopic behaviour. The latter type of model use simple elements that are representing clusters of grains and have average properties. We present another type of model where we use elements that are smaller than the grain scale so that clusters of elements represent single grains. This allows us to model fracture dynamics on a macroscopic scale that includes the effects of the underlying microstructure.