On the energy flux in elastic and inelastic bodies and cross-coupling flux between longitudinal and transversal elastic waves

Abstract

The energy balance equation, including not only the kinetic and deformation energy densities, but also the power of external forces, identifies the energy flux as minus the product of the velocity by the stress tensor: this result does not depend on constitutive relations and applies to elastic or inelastic matter. The simplest case is an isotropic pressure, when the energy flux equals its product by the velocity. In the linear case, the energy flux is obtained in elasticity for crystals and amorphous matter. An independent result is to show that, by inspection of any linear wave equation in a steady homogeneous medium, it is possible to ascertain whether the waves are (a) isotropic or not and (b) dispersive or not, with no need for an explicit solution. An application of this result to linear elastic waves shows that: (i) they are non-dispersive in crystals or amorphous matter; (ii) for the latter material, the longitudinal and transversal waves are isotropic, but their sum is not. A consequence of (ii) is that the superposition of longitudinal and transversal waves: ($\alpha$) adds the two energy densities and powers of external forces; ($\beta$) adds, to the two energy fluxes, a third cross-coupling energy flux that is proportional to the dilatation of the longitudinal wave multiplied by the velocity of the transverse wave.