© 1968 by Oxford University Press
COMPLETENESS THEOREMS FOR LINEARIZED THEORIES OF INTERACTING CONTINUA
( School of Mathematics and Physics, University of East Anglia Norwich )
A widely used device in classical elastokinetics is the decomposition of the displacement vector into two parts; the first component, representing a dilatational wave, is the gradient of a scalar function satisfying a wave equation; the second component, representing a rotational wave, is a vector function satisfying a wave equation or the curl of such a function. These alternative decompositions, which are associated with the names of Poisson and Lamé, are known to be complete in the sense that every sufficiently smooth solution of the equations of classical elastokinetics admits the indicated representation in terms of a scalar and a vector wave function. In this paper an analogous theory is developed for solutions of linearized equations describing the motion of heterogeneous continua of two types, a mixture of an isotropic elastic solid and a viscous fluid, and an isotropic mixture of two elastic solids. The motion of each type of binary mixture is described by two vectors which can be decomposed in the alternative ways described above, the pairs of scalar and vector functions so introduced satisfying coupled systems of partial differential equations. The completeness of these representations is established and some special cases of the main results are considered. As an application of the theory we examine the behaviour of plane harmonic disturbances in a non-heat-conducting mixture of an isotropic elastic solid and an inviscid fluid.