We consider a p-system of conservation laws that emerges in one dimensional elasticity theory.Such system is determined by a function W .We consider four forms of W .These are St.Venant-Kirchhoff, Ogden, Kirchhoff modified, Blatz-Ko-Ogden forms.In each of those cases we determine the conditions for the parameters µ, λ and f , 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.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.