Mechanics of damage in a random granular microstructure: Percolation of inelastic phases

In this paper a micromechanical approach to the evolution of damage in solids with granular-type microstructures is presented. Damage is defined as an elastic-inelastic transition in the grain boundaries, whereby inelasticity signifies plasticity and breaking. Representation of the microstructure, made up of convex grains of random physical and geometrical properties, in terms of a graph G permits the introduction of grain-grain constitutive interactions. Elastic and inelastic states of the solid are represented in terms of a binary random field Z on the graph G′ dual to G, and the boundary in the stress space between elastic and inelastic response ranges is given by a statistical family of random failure surfaces. The problem of determination of an effective failure surface is reduced to the problem of percolation of inelastic edges on G′. A solution method based on the self-consistent approach to random media, the Markov property of field Z and the percolation theory is outlined. This analysis brings out naturally the size effects-decrease of scatter in strength with specimen size and dependence of average strength on specimen size-as well as the fractal character of percolating sets of inelastic edges. A direct link is found between the entropy of disorder of Z and the thermodynamic entropy; this forms the basis for thermodynamics of damage processes in random media as well as for their experimental investigation.