Thermodynamic Analysis of Shock Waves in General Media

Abstract

A thermodynamic comparison is made between propagating shock waves and smooth waves, in arbitrary media and for arbitrary deformations. for two classes of shock wave: weak shocks; and shocks of arbitrary strength that propagate under steady-state conditions. It is shown that the thermodynamic differences between shocks and smooth waves are slight for weak shocks, and nil for steady-state shocks. This legitimizes both the derivation of restrictions on steadily propagating shock waves by the analysis of suitably-chosen smooth waves, and the use of a purely mechanical constitutive model in the derivation of these restrictions. An important specific application of this approach to deriving shock wave restrictions is provided by analyzing a very broad class of rate-independent elastic-plastic materials. It is shown that positive-definiteness of the elastic modulus tensor and normality of the plastic deformation-rate tensor to the yield surface. together with standard weak conservation laws and compatibility requirements. greatly restrict the permissibility of shock waves. This is especially so in quasi-static deformations, but strong restrictions on the permissible types and speeds of dynamically propagating shock waves are also shown to result. These analyses are first performed within a small-displacement-gradient formulation, for which the results are highly restrictive and easily interpreted. The finite deformation versions of the calculations are then performed. Some examples of the implications of the results to problems in quasi-static and dynamic elastic-plastic crack growth are also provided. for isotropic materials and for ductile single crystals.