REQUIREMENTS OF THERMODYNAMICS IN THE ANALYSIS OF ELASTIC-PLASTIC SHOCK-WAVES

Thermodynamical requirements on elastic-plastic shock waves are invest igated to explore the range of validity of, and provide rigorous subst antiation for, the previous shock analyses of Drugan and coworkers. Th ese 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 respon se is purely mechanical, We show precisely the conditions under which these analyses are valid. Courant and Friedrichs' [(1948) Supersonic F low and Shock Waves (Third Printing: 1985). Springer, Berlin] analysis of the order of entropy effects for weak one-dimensional shocks in me chanically 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 sh ock up until third order in material time rates of fundamental field v ariables, at which point contributions from the shock itself first app ear. This result, which is valid even if entropy generation (due to me chanical dissipation) occurs at first or second order, corrects the co mmon misconception that a shock may be approximated by a smooth wave o nly if the entropy generation is small. We further prove that for the special class of shocks that propagate under steady-slate conditions w ith non-rotating reference configuration images, a smooth wave can be constructed whose change in thermodynamic state coincides with that ac ross 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. Hav ing legitimized the representation of a shock by a smooth wave, a larg e deformation statement of the maximum plastic work inequality is inte grated across the shock to give general thermomechanical existence con ditions for steady shocks. These conditions reduce to those of Drugan and Rice [(1984) Restrictions on quasi-statically moving surfaces of s trong discontinuity in elastic-plastic solids. In Mechanics of Materia l Behaviour (ed. G. J. Dvorak and R. T. Shield), pp. 59-73. Elsevier S cience, Amsterdam] and Drugan and Shen [(1987) Restrictions on dynamic ally propagating surfaces of strong discontinuity in elastic-plastic s olids. J. Mech. Phys. Solids 35, 771-787; (1990) Finite deformation an alysis of restrictions on moving strong discontinuity surfaces in elas tic-plastic materials: quasi-static and dynamic deformations. J. Mech. Phys. Solids 38, 553-574] whenever thermomechanical coupling is negle cted (i.e. when the thermal deformation coefficients or the jump in te mperature is neglected, or if the jump in strain has a zero inner prod uct with the thermal deformation coefficient tenser); specific situati ons where such a simplification is sensible are outlined.