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, the authors define a variational framework thought to encompass some of these issues: the constitutive relations are written at the structural scale, and take the form of minimization problems. The attention is restricted to standard generalized materials that depend on internal variables and their gradients. The general framework is defined, appearing as a variational formulation of the constitutive relations, and several related properties are pointed out, especially thermodynamical admissibility and some links with the work \textit{K.~C.~Valanis} [Acta Mech. 116, 1-14 (1996; Zbl 0858.73007)]. A numerical formulation of this variational principle is presented: it is expressed as a minimization problem, giving access to the various tools of mathematical optimization to solve the problem. The proposed framework is applied to several classes of models existing in the literature: a homogenization-based constitutive relation, a porosity model, and gradient plasticity. These models are shown to belong to the general framework, thus gaining a higher degree of generality and a quite simple numerical implementation, while a straightforward application of the variational principle gives rise to some remarkable properties.
scientific article; zbMATH DE number 1440999