A New Approach for Piezoelectricity Problems with Circular Inclusions

Abstract

In this paper, we derive the null-field integral equation for piezoelectricity problems with circular inclusions under remote anti-plane shears and in-plane electric fields in two directions. To fully capture circular geometries, separable expressions of fundamental solutions in the polar coordinate for field and source points and Fourier series for boundary densities are adopted to ensure the exponential convergence. Four gains are obtained, (1) well-posed model, (2) singularity free, (3) boundary-layer effect free and (4) exponential convergence. The solution is formulated in a manner of semi-analytical form since error purely attributes to the truncation of Fourier series. Problems with two piezoelectric inclusions for stress and electric displacement distributions are revisited to demonstrate the validity of our method. The main feature of the present paper is that the new formulation can be generalized to multiple circular inclusions in a straightforward way without any difficulty. 本文使用零場積分方程式，求解同時受反平面剪力及平面電場之含圓形夾雜壓電問題。為了充分利用圓形邊界的特性，將基本解及邊界物理量分別展開成退化核及傅立葉級數的形式；因此可以得到四個好處：矩陣良態模式、避免奇異積分、沒有邊界層效應、指數收歛。由於誤差僅來自於擷取有限項的傅立葉級數，故本方法可視為半解析法。文中求解含兩圓形夾雜之應力與電位移分布，以示範驗證本方法的有效性。本方法最大的特色是，可以直接廣泛地求解含多圓形夾雜之壓電問題。