On wave propagation problems in which $c_f=c_s=c_2$ occurs

The combined longitudinal and torsional plastic waves in a thin-walled tube of rate-independent isotropic work-hardening material are used to illustrate the problems involved when the situation ${c_f} = {c_8} = {c_2}$ occurs. Two examples are presented. In the first example, the stress paths in the $\sigma \sim \tau$ plane for the fast and slow simple waves are examined in the region near the singular point $\left ( {\sigma *,0} \right )$ where ${c_f} = {c_8} = {c_2}$. For $\eta \ge \frac {1}{2}$, where $\eta$ is a nondimensional material constant defined in the paper, there is no stress path passing through the singular point $(\sigma *,0)$ other than the $\sigma$-axis itself. For $0 < \eta < \frac {1}{2}$, there is a family of stress paths emanated from $(\sigma *,0)$ which span an angle of ${\tan ^{ - 1}}{\left ( {1 - 2\eta } \right )^{1/2}}$ with the $\sigma$-axis. In any case, the stress paths for the fast and slow simple waves are not orthogonal to each other at the singular point. In the second example, a study is made of the propagation of the plastic wave front into the tube which is initially prestressed at the stress state $\left ( {\sigma *,0} \right )$. It is shown that the solution in the region next to a region of constant state is not necessarily a simple wave solution. In fact, an unloading can occur at the plastic wave front which changes its speed from ${c_2}$ to ${c_0}$ at the onset of the unloading.