On the eigenvalue problem for free vibrations of a piezoelectric/piezomagnetic body

We present a systematic analysis of the eigenvalue problem associated with free, small-amplitude vibrations in an elastic body possessing piezoelectric, piezomagnetic, and magnetoelectric couplings. An abstract formulation is introduced. The operators in the abstract formulation are shown to be self-adjoint, from which a series of fundamental properties of resonant frequencies and vibration modes are proved concisely. A variational formulation and a perturbation analysis of the eigenvalue problem are also given, based on the abstract formulation.