A variational formulation for nonlocal damage models

To control localization phenomena exhibited by strain softening constitutive relations, several issues have been proposed by various authors, based on spatial regularization. In this paper, we define a variational framework, thought to encompass some of these issues: the constitutive relations are written at the structural scale and become minimization problems. Such a framework is not only well-suited to the mathematical study of the boundary value problem, but also leads in a natural way to an efficient numerical algorithm. The formulation is first presented, then applied to several classes of models existing in the literature : a homogenization-based constitutive relation, a porosity model and gradient plasticity. Besides the higher degree of generality confered by the formulation, it will be shown that several proper ties can be obtained for these models. (C) 1999 Elsevier Science Ltd. All rights reserved.