Shock waves in an isotropic elastic space: PMM, vol. 42, no. 4, 1918, pp. 711–717

Abstract

Shock waves being propagated in an isotropic deformed elastic medium are studied within the framework of adiabatic, quadratic elasticity theory. A system of equations in discontinuities is written down which describes the shockwave propagation process from whose solution the velocities of the possible shocks are determined. Conditions for the existence of possible shocks are obtained from the conditions for solvability of the given system, as a function of the properties of the medium and the deformed state in front of the surface of discontinuities, some of the results are extended to the case of an arbitrary dependence of the elastic potential on the strain tensor invariants. Constraints on the existence of shocks imposed by the second law of thermodynamics are studied. An extensive literature (see [1–6], for instance) is devoted to the study of the properties of shocks being propagated in a nonlinear elastic medium. Shocks in an incompressible elastic medium [1] are studied most. Results have been obtained successfuly in the consideration of shocks in a compressible medium for either constraints imposed on the dependence of the elastic potential on the strain tensor invariants [2–4], or on the deformed state ahead of the shock [4–6], or by constraining the analysis just to certain kinds of waves [3, 5, 6]. In this paper no other constraints are introduced, except that taken into account are the highest nonlinear terms (quadratic elasticity theory) and a full study is performed of the properties of shocks in the arbitrary deformed state ahead of the surface of discontinuities.