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

Abstract 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.