Integral theorems in linear elastodynamics

We derive ‘antenna theorems’ in elastodynamics for the displacement field generated by a force density located at a spherical surface. On the basis of these theorems we derive an expression for an integral of the Green's function of linear elastodynamics. The integral corresponds to the so-called overlap kernel, which plays a key role in the cluster-expansion approach in the statistical theory of elastic suspensions of spherical particles. In elastostatics, the integral is closely related to mean-field expressions for the effective elastic moduli. A factorized form of the overlap kernel in terms of vector spherical waves is derived.