On the stability of elastic–plastic systems with hardening

This paper discusses the stability of quasi-static paths for a continuous elastic–plastic system with hardening in a one-dimensional (bar) domain. Mathematical formulations, as well as existence and uniqueness results for dynamic and quasi-static problems involving elastic–plastic systems with linear kinematic hardening are recalled in the paper. The concept of stability of quasi-static paths used here is essentially a continuity property of the system dynamic solutions relatively to the quasi-static ones, when (as in Lyapunov stability) the size of initial perturbations is decreased and the rate of application of the forces (which plays the role of the small parameter in singular perturbation problems) is also decreased to 0. The stability of the quasi-static paths of these elastic–plastic systems is the main result proved in the paper.