NULL-FIELD INTEGRAL EQUATION FOR STRESS FIELD AROUND CIRCULAR INCLUSIONS UNDER ANTI-PLANE SHEAR

Abstract

In this paper, we derive the null-field integral equation for an infinite medium containing circular inclusions with arbitrary radii and positions under remote anti-plane shear. To fully capture the circular geometries, separable expressions of fundamental solutions in the polar coordinate and Fourier series for boundary densities are adopted. By moving the null-field point to the boundary, singular integrals are transformed to series sums after introducing the concept of degenerate kernels. Not only the singularity but also the sense of principle values are novelly avoided. For the calculation of boundary stress, the Hadamard principal value for hypersingularity is not required and can be easily calculated by using series sums. The solution is formulated in a manner of semi-analytical form since error purely attributes to the truncation of Fourier series. The exact solution for a single inclusion is derived. The problem of two inclusions and the problem of one cavity surrounded by two inclusions are revisited to demonstrate the validity of our method. The proposed formulation has been generalized to multiple inclusions and cavities in a straightforward way without any difficulty. 本文係使用零場積分方程式求解基材含任意大小、位置之圓形置入物或孔洞，受反平面剪力作用下之應力場。將基本解以極座標展開成退化核(分離核)的形式，而以傅立葉級數來完整描述邊界物理量。藉由引入退化核的觀念，將零場點推向邊界時，奇異積分會被轉換成級數和的形式。因此無需面對奇異積分，且在計算邊界應力時，不需處理Hadamard 主值問題，而可輕易地由級數和的形式求得。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"。由於本方法可輕易導得單一置入物的解析解，因此我們分別求解含兩個圓形置入物與兩個圓形置入物圍繞單一圓洞的問題來突顯本方法的一般性。最後，我們提出一套系統性的方法來求解含多圓洞與置入物的反平面問題。