Deformation, fracture, and fragmentation in brittle geologic solids

A model is developed for mechanical behavior and failure of brittle solids of geologic origin. Mechanisms considered include elastic stretch and rotation, thermal expansion, and deformation associated with micro-cracks. Decohesion on preferred cleavage planes in the solid, and subsequent effects of crack opening and sliding, are modeled. Explicit volume averaging over an element of material containing displacement discontinuities, in conjunction with the generalized divergence theorem, leads to an additive decomposition of the deformation gradient into contributions from thermoelasticity in the intact material and displacement jumps across micro-cracks. This additive decomposition is converted into a multiplicative decomposition, and the inelastic velocity gradient is then derived in terms of rates of crack extension, opening, and sliding on discrete planes in the microstructure. Elastic nonlinearity at high pressures, elastic moduli degradation from micro-cracking, dilatancy, pressure- and strain rate-sensitive yield, and energy dissipation from crack growth and sliding are formally addressed. Densities of micro-cracks are treated as internal state variables affecting the free energy of the solid. The mean fragment size of particles of failed material arises from geometric arguments in terms of the evolving average crack radius and crack density, with smaller fragments favored at higher loading rates. The model is applied to study granite, a hard polycrystalline rock, under various loading regimes. Dynamic stress–strain behavior and mean fragment sizes of failed material are realistically modeled. Possible inelastic anisotropy can be described naturally via prescription of cleavage planes of varying strengths.