The Riemann problem for a model in elastoplasticity with hysteresis

Abstract

We revisit the 1-D(imensional) case of the 3-D Antman-Szymczak model for the longitudinal motion in an elastic-plastic bar [Ant, Tran]. As far as real-world applications are under consideration the model captures two main behaviour phenomena: limit “total compression”, and tension and compression yield points (elastic-plastic phase transitions, allowing for hysteresis). A major technical difficulty here is on handling the Lipschitz continuous low regularity of the so-called flux function, a consequence of the elastic-plastic phase transition, which imply discontinuous wave speeds. While in the past we solved a general class of Riemann problems for 1st order hyperbolic systems of conservation laws with Lipschitz continuous flux-functions [CLT] capturing the model under consideration we are now concerned with how to handle, directly, the common in applications 2nd order hyperbolic pdes, e.g., what if we rewrite the model as a nonlinear wave equation? Finally, a second reason to revisit this model is because of our (pre)occupation on the proper mathematical modelling of “physical phenomena”: hysteresis has a viscosity like effect, i.e., it is an interesting energy dissipative mechanism.