Relativistic elasticity of dissipative media and its wave propagation modes

Abstract

A phenomenological, general relativistic theory of dissipative elastic solids whose equations form a hyperbolic system is proposed. The non-stationary transport equations for dissipative fluxes containing new cross-effect terms, as required by compatibility with irreversible thermodynamics, have been adopted. As opposed to some conventional theories which are parabolic and predict instantaneous propagation of wavefronts, the theory formulated, consisting of 14 partial differential equations (in the case of special relativity), of total order 17, is hyperbolic and predicts, for all existing propagation modes, finite front speeds. The complete system of special relativistic propagation modes of an elastic solid is determined from the linearised equations. There are four mutually distinct non-trivial propagation modes, two for longitudinal waves and two for transverse waves. If the rigidity modulus decreases to zero one obtains as a special case the normal modes for fluid according to the theory of Muller (1972) and Israel (1976). Weber's equation is also briefly discussed.