  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.

Analysis of PDEs