Elastic-Wave Formulation for Electroelastic Waves in Unbounded Piezoelectric Crystals

The equations describing the elastic behavior of plane-wave disturbances in an infinite piezoelectric crystal are reduced in form to those of a purely elastic medium in the quasistatic approximation. The internal energy of the piezoelectric solid is expressed as one elastic-type term, and an alternative expression for the elastic energy flux yields the combined elastic and electric energy flux on replacing the elastic constant $c^E$ with the stiffened "constants" $c^{\mathrm{kD}}$ without introducing $\overrightarrow{\mathrm{\nabla }}\times \overrightarrow{\mathrm{H}}=\overrightarrow{\mathrm{D}}$ and $\overrightarrow{\mathrm{P}}=\overrightarrow{\mathrm{E}}\times \overrightarrow{\mathrm{H}}$. Similarly, the electric displacement is expressed in terms of the strain variables alone by means of modified piezoelectric "constants" $e^{\mathrm{kD}}$. Properties of this "stiffened-elastic" formalism and of the usual mechanical-electrical formulation are the same, and the two formulations are discussed in relation to each other. The elastic-propagation properties of piezoelectrics are describable by a ray or wave surface, which exists in terms of the $c^{\mathrm{kD}}$, and the techniques of variational elasticity carry over to piezoelectrics. The positive definiteness of the $c^{\mathrm{kD}}$ is asserted for an arbitrary direction to realize physical stability and, whereas no new limiting restrictions among material constants obtain, their use in Rayleigh-Ritz procedures assures its monotonic convergence. The symmetry and transformation properties of the $c^{\mathrm{kD}}$ show that, whereas their symmetry is lower than that of the $c^E$, their centrosymmetric nature requires only the centrosymmetric crystal groups to describe the elastic properties of piezoelectric crystals. Various stiffened-elastic properties are numerically evaluated for an arbitrary nonpure mode and symmetry-related directions for $\alpha$-quartz (class 32), LiNb${\mathrm{O}}_3$ ($3m$), CdS ($6mm$), ${\mathrm{Ba}}_2$Na${\mathrm{Nb}}_5$ ${\mathrm{O}}_{15}$ ($2mm$), ${\mathrm{Bi}}_{12}$Ge${\mathrm{O}}_{12}$ (23), and GaAs ($43m$). The work offers a simplified approach to characterizing bulk and surface elastic-wave properties of electroelastic waves in piezoelectric crystals and of modally analyzing particular classes of piezoelectric structures.