Electrostatic interactions and the piezoelectric equations

Abstract

In piezoelectricity, the stress equations of motion are coupled to the charge equation of electrostatics by means of the linear piezoelectric constitutive equations which contain the piezoelectric constants that couple the small strain to the electric field. In the case of the more general nonlinear interaction of the electric field with a deformable insulator, the Maxwell electrostatic stress tensor must be included in the description. Furthermore, in the nonlinear case, a rotationally invariant combination of the electric field and the deformation occurs in the constitutive equations in place of the electric field for the same reason that the finite train occurs. The linear electroelastic equations for small dynamic fields, superposed on a static bias obtained from the nonlinear system, are more general than the linear piezoelectric equations in that the linear constitutive equations depend on the small local mechanical rotation, as well as the small strain. Nevertheless, when the biasing stress and electric field vanish, even though a biasing strain and polarization are present, the equations reduce to those of linear piezoelectricity. Consequently, the linear equations for the polarized ferroelectric ceramic, which operates essentially by means of biased electrostriction, are identical with those of linear piezoelectricity for the symmetry of the polarized state. When the biasing electric field is retained in the linear electroelastic equations, they are applicable in the description of the self-biased electret microphone. The influence of any bias on the frequency response of a piezoelectric resonator may readily be obtained from the linear equations for small fields superposed on a bias with the aid of an existing perturbation equation. Some examples are given.