# Wave Patterns in A Nonclassic Nonlinearly-Elastic Bar Under Riemann Data

## TL;DR

This paper classifies all wave patterns in a new class of nonlinearly elastic materials with implicit constitutive relations, revealing complex wave interactions and providing solutions to test material properties.

## Contribution

It completely characterizes wave patterns for nonclassic elastic materials under Riemann data using phase plane analysis and dissipation assumptions.

## Key findings

- Multiple wave patterns including composite waves are identified.
- Solutions are provided for physically realizable cases.
- The results can distinguish between classical and nonclassical elastic materials.

## Abstract

Recently there has been interest in studying a new class of elastic materials, which is described by implicit constitutive relations. Under some basic assumption for elasticity constants, the system of governing equations of motion for this elastic material is strictly hyperbolic but without the convexity property. In this paper, all wave patterns for the nonclassic nonlinearly elastic materials under Riemann data are established completely by separating the phase plane into twelve disjoint regions and by using a nonnegative dissipation rate assumption and the maximally dissipative kinetics at any stress discontinuity. Depending on the initial data, a variety of wave patterns can arise, and in particular there exist composite waves composed of a rarefaction wave and a shock wave. The solutions for a physically realizable case are presented in detail, which may be used to test whether the material belongs to the class of classical elastic bodies or the one wherein the stretch is expressed as a function of the stress.