Nonlinear kinematic wave mechanics of elastic solids

The kinematic wave theory due essentially to M.J. Lighthill, G.B. Whitham and W.D. Hayes has rarely been applied to the case of elastic waves, an exception being works by Maugin and Hadouaj on nonlinear surface waves on covered crystal elastic substrates in 1989–1992. Here, basing on the canonical projection of continuum mechanics onto the material manifold (theory of so-called material inhomogeneities and of material or configurational forces) it is first shown that the kinematic wave theory develops in parallel with that “materials mechanics” but in terms of frequency and material wave vector instead of time and material coordinates. A conservation law of material wave momentum is thus deduced involving a material wave Eshelby stress in addition to the conservation of wave action. This formalism and the Whitham–Newell averaging method are then used in an illustration to the case of nonlinear dispersive bulk waves in elastic crystals. A nonlinear (i.e., amplitude dependent), “dispersive” (i.e., dependent on space and time derivatives of the amplitude) dispersion relation is thus constructed by means of asymptotics, which allows one to obtain slowly varying small amplitude, almost monochromatic dynamical solutions such as envelope solitons.