  Open phononic systems including resonators radiating inside an unbounded
medium support localized phonons characterized by a complex frequency. In this
context, the concept of elastic quasinormal mode (QNM) arises naturally, as in
the cases of nanophotonic and plasmonic open systems. Based on a complex,
unconjugated form of reciprocity theorem for elastodynamics, the eigenfunction
expansion theorem expressed on the elastic QNM basis yields an accurate
approximation to the response function, for an arbitrary excitation. The
description of the elastic Purcell effect then requires defining a
complex-valued modal volume for each QNM. For validation, we first consider the
case a vibrating nylon rod radiating in water. As a second test example, we
consider a slender nickel ridge on the surface of a fused silica substrate,
before extending our attention to a nanoscale tuning fork composed of two such
ridges. In all cases, the response estimated from only a few elastic QNMs
agrees with the solution to the elastodynamic equation.
