Vibrations of spherical inclusions in elastic solids

Variational principles are derived for the analysis of dynamical phenomena associated with spherical inclusions embedded in homogeneous isotropic elastic solids. The starting point is Hamilton's principle, with the material properties assumed to vary only with the radial distance r from the origin. Attention is restricted to disturbances that are symmetric about the polar (z) axis, such that the nonzero displacement components in spherical coordinates, u(r) and u(theta), are independent of the polar coordinate phi. The symmetry allows for a decoupling of the polar components, the nth of which is described by U-r,U- n(r, t)P-n(cos theta) and U-theta,U- n(r, t)dP(n)/dtheta. A variational principle is subsequently derived for the field quantities U-r,U-n and U-theta,U-n. Concepts analogous to those of the theory of matched asymptotic expansions are used to embellish the principle in order to allow for the damping associated with the outward radiation of elastic waves. Examples illustrating the use of the variational principle for formulating plausible lumped-parameter models are given for the cases of n = 0 and n = 1. (C) 2005 Pleiades Publishing, Inc.