Quasi-static response, implicit scheme and incremental problem in gradient plasticity

This paper is devoted to the study of gradient plasticity at small strains. Some time-independent dissipative processes such as brittle damage can also be considered in the same framework. Our attention is focussed on the description of the constitutive equations, on the formulation of the governing equations in terms of the energy potential and the dissipation potential of the solid. A time-discretization by the implicit scheme of the evolution equation leads to the study of the incremental problem which is different from the rate problem. The increment of the response under an increment of the loads must satisfy a variational inequality and, if the energy potential is convex, an incremental minimum principle. In particular, a local minimum of the incremental minimum principle is a stable solution to the variational inequality.