Evolutionary behavior of induced discontinuities behind one dimensional shock waves in non-linear elastic materials

The governing differential equation of induced discontinuities behind one dimensinal shock waves in non-linear elastic materials has been derived. This equation depends, in particular, on the shock amplitude itself. Therefore, its solution depends on the solution of the governing equation of the shock amplitudes which, in turn, depend on the induced discontinuities. It is shown in the special case pertaining to a first-order approximation that there exists a critical shock amplitude S_c such that the evolutionary behavior of the induced discontinuities depends on the relative magnitudes of the shock amplitudes and S_c. However, in the special case pertaining to a second-order approximation the evolutionary behavior of the induced discontinuities is monotone.