Singular surfaces in linear thermoelasticity

Abstract

In this paper we give a detailed account, within the framework of the linear theory of thermoelasticity, of the propagation of surfaces of discontinuity in a homogeneous, isotropic elastic solid which is able to conduct heat. The methods used in the investigation are, in large measure, due to T. Y. Thomas. The early sections of the paper contain a derivation of the principal results of Thomas's theory which enables us to determine, from a consideration of the appropriate Cauchy initial-value problem, the characteristic surfaces of the linear thermoelastic equations. The wavefronts associated with these characteristics are found to propagate with one of the constant speeds υ t= {E t(1−v t) ϱ0(1+V t)(1−2v t)} 1 2 , υ s= {E t 2ϱ 0(1+V t)} 1 2 , ϱ 0, E T , v T being respectively the density, the isothermal Young's modulus and the isothermal Poisson's ratio of the material in its reference state. A discontinuity surface of order r in the displacement and temperature fields is referred to as a weak thermoelastic wave if r⩾2 and a strong thermoelastic wave if r=0 or 1. Concerning the properties of these waves our main conclusions are as follows. Weak thermoelastic waves and strong waves of order 1 are characteristic and may be described as dilatational or rotational according as their speed of propagation is v T or v S . Dilatational strong waves of order 1 are shock waves and rotational waves of this type are propagating vortex sheets. For all thermoelastic waves of order ⩾ 1 the strength (defined in a natural way) is completely determined by its distribution on an initial configuration of the wavefront. Irrespective of the shape of this initial configuration, the strength of a dilatational wave decays rapidly as the wave propagates on account of thermoelastic dissipation. For rotational waves, however, the variation of strength during propagation depends solely upon the geometrical form of the initial wavefront. A strong thermoelastic wave of order 0 is an absolute singular surface in the temperature field, discontinuities of displacement being excluded from consideration. A wave of this type may be characteristic, in which case its speed of propagation is v S ; or it may be non-characteristic, in which case it is a dilatational shock wave. In neither case is the strength of the wave completely determined by its distribution on an initial wavefront, a situation which leads us to argue that thermoelastie waves of order 0 cannot in practice be created. In the final section of the paper the properties of singular surfaces in classical elastokinetics are discussed in the light of the foregoing analysis of discontinuous thermoelastic waves.