Green function and its application for a piezoelectric plate with various openings

Abstract

For a two-dimensional piezoelectric plate subjected to mechanical and electric load, a Green function satisfying traction free and exact electric boundary conditions along a hole is developed using Lekhnitskii's formalism and the technique of conformal mapping. The critical points for the mapping function used is investigated numerically, and the study indicates that the transformation of a polygonal hole in a piezoelectric plate into a unit circle is nonsinglevalued. A simple approach is presented to treat such a situation. Based on the Green function developed in this paper, a system of singular integral equations for the unknown dislocation de®ned on crack faces is presented to study the interaction between cracks and holes. Numerical results are presented to elucidate the effects of crack orientation on stress and electric displacement (SED) intensity factors and to illustrate the application of the proposed formulation.

Green's function for thermopiezoelectric plates with holes of various shapes

“…For a particular value of z k , it is shown that there exist 2j roots for f k in Eq. (24); half of the roots are located outside the unit circle, the remaining are inside the unit circle [16]. This indicates that the transformation (24) will be single-valued for j 1 (ellipse) since only one root is located outside the unit circle.…”

“…The question is, which transformation should be chosen. This problem has been discussed in [16], and we will omit those details here. As it was done in [16], we choose the root whose magnitude has a minimum value among the j-roots.…”

“…The present paper is a continuation of our previous work [16]. A uni®ed form of thermoelectroelastic Green's functions is presented for an in®nite thermopiezoelectric plate with various openings subjected to thermal loading.…”

“…Later on, thermoelectroelastic Green's functions for bimaterial problems as well as for an elliptic hole embedded in an in®nite plate were presented in [14,15], respectively. Recently, a study in [16] on Green's functions for a polygonal hole in an in®nite piezoelectric plate has shown that the corresponding transformation is not single-valued, and a simple approach was presented to treat this problem.…”

“…Numerical results for SED intensity factors are presented to verify the effectiveness of the proposed formulation. As in [16,17], we effect a plane strain analysis, where the material is transversely isotropic and coupling between in-plane stresses and in-plane electric ®elds takes place. For a cartesian coordinate system x 1 x 2 x 3 , choose the x 3 -axis as the poling direction.…”