LAUSR

Abstract This paper is devoted to the standard theory of gradient plasticity and visco-plasticity, cf. Gudmundson (2004), Fleck and Willis (2009) and the quoted references. Wihin the framework of the generalized standard materials and the generalized principle of virtual work, our attention is focussed on the mathematical basis of the theory in order to underline the basic assumptions, general results and principal difficulties compared to the classical theory of plasticity. Gradient terms can be introduced in the expression of the energy potential or the dissipation potential. Various general results of the literature on the constitutive modeling and on the response of a solid under a given loading are revisited in this unified description. The quasi-static response is governed by a variational inequality which is an alternative form of Biot equation. A full discussion of this evolution equation is given in terms of the energy and dissipation potentials of the solid. The rate problem and the rate variational principle are considered and compared to the incremental problem and the incremental variational principle which result from an implicit discretization of the evolution equation. In particular, rate and incremental minimum principles are related to a stability condition.