Further study on one-dimensional shock waves in nonlinear elastic media

One may define the growth of a shock wave by the growth in the amplitude of discontinuity in the velocity (denoted by $\left [ v \right ]$) across the shock wave as the shock wave propagates. One may also define the growth of a shock wave by the growth in the amplitude of discontinuity in the stress $\left [ \sigma \right ]$, strain $\left [ \epsilon \right ]$, or entropy $\left [ \eta \right ]$, It is shown that one definition predicts the growth of the shock wave while others may predict its decay. In this paper we derive the transport equations for one-dimensional shock waves in nonlinear elastic media in which the shock wave can be defined as the amplitude of either ${\left [ \epsilon \right ]^2}$, $\left ( b \right )$, $\left [ v \right ]$ or $\left [ \eta \right ]$. Moreover, the dependent quantity can be any one of, or a linear combination of, the seven quantities behind the shock wave. It is shown that when the region ahead of the shock wave is under a homogeneous deformation, the amplitudes of $\left [ v \right ]$, $\left [ \sigma \right ]$ and $\left [ \epsilon \right ]$ grow or decay simultaneously if $\left ( a \right )$ ${\left [ \epsilon \right ]^2}$ is a strictly increasing function of $\left [ \eta \right ]$, or $\left ( b \right )$ the purely mechanical theory of shock waves is employed in which the effect of the entropy is ignored. Regardless of whether the effect of the entropy is ignored or not, there is no assurance that the amplitudes of $\left [ v \right ]$, $\left [ \sigma \right ]$ and $\left [ \epsilon \right ]$ grow or decay simultaneously if the region ahead of the shock wave is not under a homogeneous deformation.