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The optical theorem is an important tool for scattering analysis inacoustics, electromagnetism, and quantum mechanics. We derive an extendedversion of the optical theorem for the scattering of elastic waves by aspherical inclusion embedded in a linear elastic solid using a vector sphericalharmonics representation of the waves. The sphere can be a rigid, empty cavity,elastic, viscoelastic, or layered material. The theorem expresses theextinction cross-section, i.e. the time-averaged power extracted from theincoming beam per its intensity, regarding the partial-wave expansioncoefficients of the incident and scattered waves. We establish the opticaltheorem for a longitudinal spherically focused beam scattered by a sphere.Moreover, we use the optical theorem formalism to obtain the radiation forceexerted on an inclusion by an incident plane wave and focused beam. Consideringan iron sphere embedded in an aluminum matrix, we compute the scattering andelastic radiation force efficiencies. In addition, the elastic radiation forceis obtained on a stainless steel sphere embedded in a tissue-like medium (softsolid). Remarkably, we find a relative difference of up to $98\%$ between ourfindings and previous lossless liquid models. Regarding some applications, theobtained results have a direct impact on ultrasound-based elastographytechniques, ultrasonic nondestructive testing, as well as implantable devicesactivated by ultrasound.

Extended optical theorem in isotropic solids and its application to the elastic radiation force