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The equations of 1D elastodynamics write as a $2\times 2$ hyperbolic system of conservation laws. The solution to the Riemann problem (i.e. piecewise constant initial data) is addressed, both in the case of 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 restriction on the initial velocity jump is required to ensure the existence of the solution. Admissibility regions are determined, predicting the occurrence of shocks, rarefactions and compound waves. Lastly, analytical solutions are completely detailed for various constitutive laws (hyperbola, tanh and polynomial), and reference test cases are proposed. A link to a Matlab script is also provided, allowing a direct implementation.
Elastic waves, Nonlinear elasticity, Riemann problem, $p$-system