Requirements of thermodynamics in the analysis of elastic-plastic shock waves

Abstract

Thermodynamical requirements on elastic-plastic shock waves are investigated to explore the range of validity of, and provide rigorous substantiation for, the previous shock analyses of Drugan and coworkers. These studies assumed (implicitly) that during shock passage, a material particle's stress and deformation history is well-approximated by its history during passage of a smooth wave, and that the material response is purely mechanical. We show precisely the conditions under which these analyses are valid. Courant and Friedrichs' [(1948) Supersonic Flow and Shock Waves (Third Printing: 1985). Springer, Berlin] analysis of the order of entropy effects for weak one-dimensional shocks in mechanically conservative fluids is extended to shocks in general three-dimensional large deformations in a material of arbitrary constitution. Specifically, we prove that the change in thermodynamic state across a suitably-chosen smooth wave coincides with that across a general shock up until third order in material time rates of fundamental field variables, at which point contributions from the shock itself first appear. This result, which is valid even if entropy generation (due to mechanical dissipation) occurs at first or second order, corrects the common mis-conception that a shock may be approximated by a smooth wave only if the entropy generation is small. We further prove that for the special class of shocks that propagate under steady-state conditions with non-rotating reference configuration images, a smooth wave can be constructed whose change in thermodynamic state coincides with that across the shock through all orders of field variable rates. That is, a smooth wave is a potentially exact model of a shock in this class. Having legitimized the representation of a shock by a smooth wave, a large deformation statement of the maximum plastic work inequality is integrated across the shock to give general thermomechanical existence conditions for steady shocks. These conditions reduce to those of Drugan and Rice [(1984) Restrictions on quasi-statically moving surfaces of strong discontinuity in elastic—plastic solids. In Mechanics of Material Behavior (ed. G. J. Dvorak and R. T. Shield), pp. 59–73. Elsevier Science, Amsterdam] and Drugan and Shen [(1987) Restrictions on dynamically propagating surfaces of strong discontinuity in elastic-plastic solids. J. Mech. Phys. Solids 35, 771–787; (1990) Finite deformation analysis of restrictions on moving strong discontinuity surfaces in elastic-plastic materials: quasi-static and dynamic deformations. J. Mech. Phys. Solids 38, 553–574] whenever thermomechanical coupling is neglected (i.e. when the thermal deformation coefficients or the jump in temperature is neglected, or if the jump in strain has a zero inner product with the thermal deformation coefficient tensor); specific situations where such a simplification is sensible are outlined.