Lagrangian description of transport equations for shock waves in three-dimensional elastic solids

A set of transport equations for the growth or decay of the amplitudes of shock waves along an arbitrary propagation direction in three-dimensional nonlinear elastic solids is derived using the Lagrangian coordinates. The transport equations obtained show that the time derivative of the amplitude of a shock wave along any propagation ray depends on (i) an unknown quantity immediately behind the shock wave,- (ii) the two principal curvatures of the shock surface, (iii) the gradient taken on the shock surface of the normal shock wave speed and (iv) the inhomogeneous term,which is related to the motion ahead of the shock surface, vanishes when the motion ahead of the shock surface is uniform. Several choices of the propagation vector are given for which the transport equations can be simplified. Some universal relations, which relate the time derivatives of various jump quantities to each other but which do not depend on the constitutive equations of the material, are also presented.