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

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