An arbitrary piezoelectric inclusion with weakly and highly conducting imperfect interface

In the current study, we rigorously analyze an arbitrarily shaped piezoelectric inclusion surrounded by an infinite isotropic piezoelectric matrix subject to antiplane shear and in plane electric field loadings. The inclusion and matrix are separated by a homogeneously imperfect interface that characterizes a spring type interaction between the elastic and electric interfacial boundary conditions. Furthermore, the boundary conditions for a mechanically compliant, weakly conducting and mechanically compliant, highly conducting interface are incorporated into the analysis. Using complex variable techniques the potential function inside the inclusion is formulated as a Faber Series expansion and a system of linear algebraic equations for a closed form solution is developed for the corresponding Faber coefficients under a finite number of terms. Under this approach, expressions for both the elastic and electric fields are developed for the inclusion and matrix. The results are presented in exact form for an elliptic inclusion and numerically simulated for a finite number of terms for purposes of verification. Additionally, the cases of a square and star inclusion geometry are analyzed and results are presented numerically. The results clearly demonstrate that not only is the stress distribution inside the inclusion interface non-uniform, but that the magnitude of the peak stresses are highly dependent on the inclusion shape and imperfect interface condition.