Thermodynamic equivalence of steady-state shocks and smooth waves in general media; Applications to elastic-plastic shocks and dynamic fracture

Abstract

By comparing the First Law of thermodynamics in its shock wave form to its smooth wave form, and applying standard continuum mechanical conservation laws and geometrical compatibility, we prove for arbitrary media that a shock wave which propagates without rotating under steady-state conditions is thermodynamically identical to a suitably-chosen steadily propagating smooth wave (and that this is not so in general for nonsteady shocks). This legitimizes the derivation of restrictions on steady-state shock waves by the analysis of suitably-chosen steady smooth waves in purely mechanical material models. Doing so for a broad class of rate-independent elastic-plastic materials rigorously corroborates several recentlypublished shock restrictions whose derivations involved some (now validated) heuristic arguments, and substantially generalizes the material class for which these restrictions apply. Thus, e.g. within smalldisplacement-gradient theory, stress jumps are ruled out across steadily propagating shock waves in quasistatic deformations of any nonsoftening material satisfying plastic normality and positive-definiteness of the elastic modulus tensor (removing the previous limitation of this result to materials that satisfy the global maximum plastic work inequality and whose current yield locus always incorporates all prior yield loci). We also confirm that steady-state shock waves in dynamic anti-plane strain or plane strain deformations cannot exist except at elastic wave speeds for nonhardening materials in the same broad constitutive class unless the yield surface contains a linear segment. Application of these results to steady-state dynamic subsonic plane strain crack growth in elastic-ideally plastic Prandtl-Reuss-Mises material proves that this problem's solution must be shock-free. This implies that certain solutions containing strong discontinuity surfaces, obtained in a recently-published numerical finite element study of this dynamic crack growth problem, are not physically realizable. The conclusion is that either a more robust numerical procedure is necessary which incorporates the thermodynamics-mandated shock restrictions derived here, or that steadystate subsonic dynamic plane-strain elastic-plastic crack growth is not possible in this material model (and potentially not in nature for materials exhibiting plastic normality, purely nonlinear yield surfaces and no hardening).