Alternative expression of piezoelectric representation and its application in inclusion problem

This paper intends to conform the mathematical correspondence between electric and mechanical variables to the physical correspondence in electroelastic problem when a piezoelectric material is subjected to stress field σ_ij and electric field E_i or displacement u and electric displacement D_i. E_i is expressed with an antisymmetric electric field tensor V_ij, and D_i is expressed with an antisymmetric electric displacement tensor P_ij which can be expressed with `induction potential' ψ_i. Then stress equilibrium and electric field equations can be described by a second order canonical differential equation system of u_i and ψ_i. As an example, the coupled electroelastic fields in an inclusion or inhomogeneity embedded in an infinite matrix are obtained in a manner directly analogous to Eshelby's classical elastic solution. The electroelastic fields inside the ellipsoidal inclusion or inhomogeneity and just outside the ellipsoidal inclusion or inhomogeneity are also given. In addition, we analyze the energies of the inclusion. The results in this paper are the backbone in the development of the electric damage theory and ferroelectric constitutive theory.