Analytical solution to the Riemann problem of 1D elastodynamics with general constitutive laws

TL;DR

This paper provides analytical solutions to the Riemann problem in 1D elastodynamics with various constitutive laws, including convex and nonconvex cases, and introduces new criteria for initial velocity jumps.

Contribution

It offers a comprehensive analytical framework for solving the Riemann problem in elastodynamics with general constitutive laws, including nonconvex cases.

Findings

Analytical solutions for hyperbola, tanh, and polynomial laws.

New existence criterion for initial velocity jumps.

Determination of admissibility regions for wave solutions.

Abstract

Under the hypothesis of small deformations, the equations of 1D elastodynamics write as a 2 x 2 hyperbolic system of conservation laws. Here, we study the Riemann problem for convex and nonconvex constitutive laws. In the convex case, the solution can include shock waves or rarefaction waves. In the nonconvex case, compound waves must also be considered. In both convex and nonconvex cases, a new existence criterion for the initial velocity jump is obtained. Also, admissibility regions are determined. Lastly, analytical solutions are completely detailed for various constitutive laws (hyperbola, tanh and polynomial), and reference test cases are proposed.