Visco-Elastic Damage Rheology Model: Theory and Experimental Tests

We present a visco-elastic damage rheology model that provides a generalization of Maxwell visco-elasticity to a non-linear continuum mechanics framework incorporating material degradation and recovery, transition from stable to unstable fracturing, and gradual accumulation of non-reversible deformation. The model is a further development to the damage rheology framework of Lyakhovsky et al. [1997] for evolving effective elasticity. Our approach provides a quantitative treatment for macroscopic effects of evolving distributed cracking with local density represented by an intensive state variable. This assumes a system with a large number of cracks where one can define a smooth distribution over a representative volume that is much larger than the size of a typical crack and much smaller than the size of the entire domain. The present formulation, based on thermodynamic principles, leads to a system of kinetic equations for the evolution of damage. We introduce an effective viscosity inversely proportional to the rate of damage increase to account for gradual accumulation of irreversible deformation due to dissipative processes. A proposed power-law relation between the damage variable and elastic moduli leads to a non-linear coupling between rate of damage evolution and the damage variable itself. This allows the model to reproduce a transition from stable to unstable fracturing of brittle rocks and hysteresis phenomena including the Kaiser effect. Analytical solutions and 3-D numerical simulations based on the model formulation account for the main features of rock behavior under large strain. Model parameters are constrained using triaxial laboratory experiments with low porosity Westerly granite and high porosity Berea sandstone samples. During three of the laboratory experiments, small loading-unloading cycles were carried out. Throughout all of these cycles, acoustic emissions were not recorded and irreversible strain was not accumulated. These and other features of the laboratory data are compatible with the model predictions and provide experimental support for the model.