  In this work, we introduce an elastic analog of the Purcell effect and show
theoretically that spherical nanoparticles can serve as tunable and robust
antennas for modifying the emission from localized elastic sources. This effect
can be qualitatively described by introducing elastic counterparts of the
familiar electromagnetic parameters: local density of elastic states, elastic
Purcell factor, and effective volume of elastic modes. To illustrate our
framework, we consider the example of a submicron gold sphere as a generic
elastic GHz antenna and find that shear and mixed modes of low orders in such
systems offer considerable elastic Purcell factors. This formalism opens
pathways towards extended control over dissipation of vibrations in various
optomechanical systems and contributes to closing the gap between classical and
quantum-mechanical treatments of phonons localized in elastic nanoresonators.
