Full solution, for crystal class 3m, of the Holland-EerNisse complex material-constant theory of lossy piezoelectrics for harmonic time dependence

A complex material-constant theory of lossy piezoelectrics is fully solved for crystal class 3 m for harmonic time dependence of the fields and stresses. A new demonstration that the theory's eigen coupling factor equation applies to the lossy alternating current (AC) case also is given. The solution presented for crystal class 3 m provides a complete orthonormal set of eigenvectors and eigenvalues for the eigen coupling factor problem, and it also provides a complete orthonormal set of eigenvectors and eigenvalues for the eigen loss tangent problem, for this crystal class. It is shown that two positive coupling factors are sufficient to express an arbitrary 3 m crystal state. Despite the complex nature of the material constants, the Holland-EerNisse theory produces fully real expressions for the coupling factors. The loss tangent eigenvalues also are fully real and positive. The loss eigenstates are important because driving a crystal in a loss eigenstate tends to minimize the impact of material losses. Given also is a set of loss inequalities for crystal class 3 m. The loss inequalities of crystal class 6 mm are recovered from these when d
₂₂
and s
^E

₄
both vanish.