Hyperbolicity and genuine nonlinearity conditions for certain p-systems of conservation laws, weak solutions and the entropy condition

TL;DR

This paper analyzes specific p-systems of conservation laws from elasticity theory, determining conditions for hyperbolicity and nonlinearity, and examines weak solutions and entropy conditions for various strain-energy functions.

Contribution

It identifies parameter conditions for hyperbolicity and genuine nonlinearity in four known strain-energy functions and studies entropy conditions for weak solutions.

Findings

Conditions for hyperbolicity and nonlinearity are established for each strain-energy form.

Weak solutions are constructed using Rankine-Hugoniot conditions for nonlinear systems.

Solutions satisfy entropy conditions near zero initial velocity for convex entropy functions.

Abstract

We consider a p-system of conservation laws that emerges in one dimensional elasticity theory. Such system is determined by a function $W$, called strain-energy function. We consider four forms of $W$ which are known in the literature. These are St.Venant-Kirchhoff, Ogden, Kirchhoff modified, Blatz-Ko-Ogden forms. In each of those cases we determine the conditions for the parameters $\rho_0$, $\mu $ and $\lambda$, under which the corresponding system is hyperbolic and genuinely nonlinear. We also establish what it means a weak solution of an initial and boundary value problem. Next we concentrate on a particular problem whose weak solution is obtained in a linear theory by means of D'Alembert's formula. In cases under consideration the p-systems are nonlinear, so we solve them employing Rankine-Hugoniot conditions. Finally we ask if such solutions satisfy the entropy condition. For a standard entropy function we provide a complete answer, except of the Blatz-Ko-Ogden case. For a general strictly convex entropy function the result is that for the initial value of velocity function near zero these solutions satisfy the entropy condition, under the assumption of hyperbolicity and genuine nonlinearity.