EFFECTIVE RELATIONS FOR NONLINEAR DYNAMICS OF CRACKED SOLIDS

A system of effective relations is developed for the mean dynamic resp onse of a solid containing cracks, which introduce nonlinearity arisin g from unilateral constraints. The system consists of the averaged equ ation of motion supplemented by a constitutive relation containing an internal variable defining the mean opening of the cracks. This is gov erned by an evolution law, which connects the macroscopic and microsco pic response via a nonlocal relation rendered nonlinear through the re quirement that the cracks are traction-free when open but transmit nor mal stress when closed. An equivalent formulation in the form of a var iational 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 appro ximation to the solution of the microscopic problem, which provides a simple ''idealised'' set of effective relations that retain the main f eatures of the full model and at the same time can be easily treated n umerically. In the limit of very slow deformations, the model reduces to one of an elastic body displaying different moduli in tension and c ompression. Example problems are solved, both for the ''bilinear'' mod el and the new model with the hereditary term. This term introduces so me new features, in particular the damping of shocks that would be pre dicted by the bilinear model, transforming them instead into localised bands.