Perturbed Scale-Invariant initial Value Problems in One-Dimensional Dynamic Elastoplasticity

Abstract

he author considers an initial value problem for equations describing the longitudinal motion of an elastoplastic rod. Conditions on the stress $\sigma $ determine whether the deformation of the rod is plastic or elastic, bdth of which are described by wave equations with different wave speeds. Also, plastic deformation is quasi-linear while elastic deformation is assumed to be linear. The initial conditions are continuous, piecewise $C^1 $, and have a jump in the first derivative only at the origin. This is a generalization of the scale-invariant problem solved by D. Schaeffer and M. Shearer, in which plastic deformation is assumed to be linear and the initial conditions are piecewise linear. The analysis is divided into cases according to the structure of the corresponding scale-invariant problem; the most interesting case reduces to a free boundary problem for the plastic equations on a wedge withh two free boundaries.