Impact-Induced Tensile Waves in a Rubberlike Material

This paper concerns the propagation of impact-generated tensile waves in a one-dimensional bar made of a rubberlike material. Because the stress-strain curve changes from concave to convex as the strain increases, the governing quasi-linear system of partial differential equations, though hyperbolic, fails to be "genuinely nonlinear" so that the standard form of the boundary-initial value problem corresponding to impact is not well-posed at all levels of loading. When the problem fails to be well-posed, it does so by exhibiting a massive loss of uniqueness, even though an entropy-like dissipation inequality is in force. Because the breakdown in uniqueness is reminiscent of a similar phenomenon that occurs in continuum-mechanical models for impact-induced phase transitions, a mathematically suitable, though physically unmotivated, supplementary selection mechanism fordetermining the solution naturally suggests itself. We describe in detail the solutions determined by two special forms of this selection mec...