In-plane complex potentials for a special class of materials with degenerate piezoelectric properties

Abstract

In this paper, complex potentials for the solution of two-dimensional, in-plane, linear piezoelectric boundary value problems are presented. These potentials are only valid for a special set of piezoelectric properties that have been identified as being useful in nonlinear ferroelectric constitutive laws. In contrast to more general solution procedures like the Stroh or Lekhnitskii formalisms, the complex potentials derived here are dependent on explicit, closed-form combinations of the piezoelectric material properties. Under either plane strain or plane stress conditions, three complex potentials are required to determine the full set of electrical and mechanical field quantities. The components of stress, strain, displacement, electric field, electric displacement, and electric potential will all be given in terms of these three potentials. To demonstrate the solution to a boundary value problem with these potentials, the asymptotic fields near a crack tip in these materials are presented in closed form.