A critical comparison of nonlocal and gradient-enhanced softening continua

Continuous models of material degradation may cease to produced meaningful results in the presence of high strain gradients. These gradients may occur for instance in the propagation of waves with high wave numbers and at str ess concentrators. Adding nonlocal or gradient terms to the constitutive mo delling may enhance the ability of the models to describe such situations. The effect of adding nonlocal or gradient terms and the relation between th ese enhancements are examined in a continuum damage setting. A nonlocal dam age model and two different gradient damage models are considered. In one o f the gradient models higher order deformation gradients enter the equilibr ium equations explicitly, while in the other model the gradient influence f ollows in a more implicit way from an additional partial differential equat ion. The latter, implicit gradient formulation can be rewritten in the inte gral format of the nonlocal model and can therefore be regarded as truly no nlocal. This is not true for the explicit formulation, in which the nonloca lity is limited to an infinitesimal volume. This fundamental difference bet ween the formulations results in quite different behaviour in wave propagat ion, localisation and at crack tips. This is shown for the propagation of w aves in the models, their localisation properties and the behaviour at a cr ack tip. The responses of the nonlocal model and the implicit gradient mode l agree remarkably well in these situations, while the explicit gradient fo rmulation shows an entirely different and sometimes nonphysical response. ( C) 2001 Elsevier Science Ltd. All rights reserved.