Equations of nonlinear elasticity theory belong to hyperbolic systems resulting from conservation laws. It is well known that for certain hyperbolic systems of equations the solutions of self-similar boundary problems composed of continuous functions and shock waves are non-unique. In order to choose unique physically feasible solution it is necessary to take into account additional contingencies. In this paper the choice of unique solution from among all the probable solutions is associated with the thermodynamical compatibility condition and the existence condition of evolutionary shock waves. The former condition follows from the second law of thermodynamics for shock waves.
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