Elastic percolation models for cohesive mechanical failure in heterogeneous systems

We introduce a class of models for microstructural damage and cohesive macromechanical failure in heterogeneous systems. Our models are based on random networks of Hooke-type springs with load limit, such that a spring breaks irreversibly if stretched beyond a critical length u₂. We consider several special cases in which both the spring constant k and u₂ are distributed quantities, and we show that the macroscopic response of the system depends crucially on the form of the probability distribution functions (PDF's) for k and u₂. If the first inverse moment of the PDF is finite, it appears that macromechanical failure occurs by means of a sharp transition, in which a single crack spans the entire system ("brittle failure"). By contrast, if the first inverse moment is infinite, many cracks appear in the system. Then, at a certain microdamage level, as defined by the fraction of broken springs, all moduli of the system vanish ("pseudobrittle failure") and the system undergoes a percolationlike transition.