Narrow Results

In this paper, existence and uniqueness results for a class of dynamic and quasi-static problems with elastic-plastic systems are recalled, and a stability result is obtained for the quasi-static paths of those systems. The studied elastic-plastic systems are continuum 1D (bar) systems that have linear hardening, and the concept of stability of quasi-static paths used here takes into account the existence of fast (dynamic) and slow (quasi-static) times scales in the system. That concept is essentially a continuity property relatively to the size of the initial perturbations (as in Lyapunov stability) and relatively to the smallness of the rate of application of the forces (which plays here the role of the small parameter in singular perturbation problems).

We discuss existence, uniqueness, regularity, and homogenization results for some nonlinear time-dependent material models. One of the methods for proving existence and uniqueness is the so-called energetic formulation, based on a global stability condition and on an energy balance. As for the two-scale homogenization we use the recently developed method of periodic unfolding and periodic folding. We also take advantage of the abstract Γ-convergence theory for rate-independent evolutionary problems.