Effective relations for nonlinear dynamics of cracked solids

A system of effective relations is developed for the mean dynamic response of a solid containing cracks, which introduce nonlinearity arising from unilateral constraints. The system consists of the averaged equation of motion supplemented by a constitutive relation containing an internal variable defining the mean opening of the cracks. This is governed by an evolution law, which connects the macroscopic and microscopic response via a nonlocal relation rendered nonlinear through the requirement that the cracks are traction-free when open but transmit normal stress when closed. An equivalent formulation in the form of a variational inequality with a hereditary term is developed and discussed. This provides an exact formulation of the influence of the cracks on the resulting field. It also assists the development of a simple approximation to the solution of the microscopic problem, which provides a simple “idealised” set of effective relations that retain the main features of the full model and at the same time can be easily treated numerically. In the limit of very slow deformations, the model reduces to one of an elastic body displaying different moduli in tension and compression. Example problems are solved, both for the “bilinear” model and the new model with the hereditary term. This term introduces some new features, in particular the damping of shocks that would be predicted by the bilinear model, transforming them instead into localised bands.