Damage in a random microstructure: Size effects, fractals, and entropy maximization

In this paper a micromechanical approach to damage growth in graphrepresentable microstructures is presented. Damage is denned as an elasticinelastic transition in the grain boundaries and is represented in terms of a binary or ternary random field Z on the graph. A method based on the percolation theory brings out the size effects in scatter of strength, and the fractal character of damage geometry, and thus provides a basis for a multifractal model of a range of damage phenomena. The Markov property of field Z leads to a description of Z in terms of Gibbs probability measures and establishes a link between the entropy of disorder of Z and the physical entropy of damage in the ensemble of material specimens. Derivation of stochastic constitutive laws is outlined using the formalism of free energy and the dissipation function extended to random media.