Maximally-dissipative local solutions to rate-independent systems and application to damage and delamination problems

The system of two inclusions ∂uE(t,u(t),z(t))∋0∂uE(t,u(t),z(t))∋0 and ∂R(ż)+∂zE(t,u(t),z(t))∋0 with the dissipation potential RR degree-1 homogeneous and with the stored energy E(t,⋅,⋅)E(t,⋅,⋅) separately convex is considered. The relation between conventional weak solutions and local solutions is shown, and a suitably integrated maximal-dissipation principle is devised to select force-driven local solutions and eliminate solutions with “too-early jumps” as it may occur in energy-driven ones. This is illustrated on scalar examples. An approximation by a simple and efficient semi-implicit time discretization of the fractional-step type is shown to converge to local solutions. On the scalar examples, the approximate solutions are shown to satisfy the integrated maximal-dissipation principle asymptotically, while in general it is devised only to serve as an a-posteriori tool to justify (or possibly adaptively adjust) thus obtained approximate solutions as, in fact, force driven. Applications of such solutions are illustrated on specific examples from continuum mechanics at small strains involving inelastic processes in a bulk or on a surface, namely damage and delamination.