A Dissipative System Arising in Strain-gradient Plasticity

We discuss a nonlocal and fully nonlinear system of partial differential equations which arises in a strain-gradient theory of plasticity proposed by Gurtin (J. Mech. Phys. Solids, 2004). The problem couples an elliptic equation to a parabolic system which exhibits two types of degeneracies: the first one is caused by the nonlinear structure, the second one by the dependence of the principal part on twice the curl of a planar vector field gamma. Furthermore, the elliptic equation depends on the divergence of gamma - which is not controlled by twice its curl - and the boundary conditions suggested by Curtin are of mixed type. We outline two reformulations of the system which are the key to obtain the existence and the uniqueness of solutions (proved elsewhere by M. Bertsch, R. Dal Passo and the authors): the first one enlightens a monotonicity property which is more "robust" than the dissipative structure inherited as an intrinsic feature of the mechanical model; the second one based on a suitable, time-dependent representation of a divergence-free vector field which plays the role of the elastic stress is essential to overcome the lack of control over the divergence of the vector field gamma.