Perturbation solution of two arbitrarily-shaped holes in a piezoelectric solid

Abstract

This paper presents a perturbation solution to a two-dimensional problem of two arbitrarily shaped holes within an infinite piezoelectric solid subjected to uniform mechanical and electric loads at infinity. Of particular interest are the stress field and electric field around hole with relation to the distance between the two holes. The shape of each hole is characterized by a truncated mapping which can be regarded as that for an elliptical hole with a perturbation. By using the perturbation method and Faber series, both the complex potentials of the piezoelectric solid and the electric fields inside holes are expressed in a series form with unknown coefficients to be determined by Fourier expansion method. Detailed numerical results are shown for oval, elliptical, triangular and square holes. A basic conclusion is that the hoop stress and electric field strength nearby the point of maximum curvature on the hole׳s boundary increase rapidly with decreasing distance between the point of maximum curvature and the other hole. On the other hand, it is shown that the electric field inside non-elliptical hole is always non-uniform regardless of the distance between the two holes, while that inside elliptical hole tends to be uniform with increasing distance between the two holes. Additionally, if the distance between the two holes is more than four times the hole size, the interaction between the two holes is ignorable and the problem of two holes can be divided into two separate problems of only a single hole.