Abstract The crack and the rigid-line inclusion problems under the anti-plane shear are the special case of the elliptic hole and elliptic rigid inclusion, respectively. We revisit this kind problem by using the degenerate kernels in terms elliptic coordinates. The stress intensity factor (SIF) is also addressed. Three ways are employed to determine the SIF. One is the extrapolation approach for the boundary or interior displacement near the tip. Another is the extrapolation approach for the boundary stress or interior stress near the tip. The other is the J-integral enclosing the crack tip. It is interesting to find that a rigid-line inclusion case yields a singular influence matrix due to the degenerate scale of length four in the log kernel. However, double-degeneracy including degenerate scale and degenerate boundary may still result in a singular matrix even though the dual BEM/BIEM is employed. The mechanism is well explained thanks to the introduction of degenerate kernel. Without the introduction of degenerate kernel, the mechanism of the double-degeneracy problem in the BIEM can’t be examined clearly. By using the degenerate kernel, the SIF can be easily determined for the crack or the rigid-line inclusion under the anti-plane shear. The path independence of the J-integral can be derived analytically. The reciprocal relation for the SIF between a crack and a rigid-line inclusion with respect to opposite loading is also addressed. In the numerical implementation, the SIF can be determined using the dual BEM. It is worth noting that BEM shows the advantage that the obtained boundary displacement or boundary stress can be directly used to obtain the more accurate SIF for the crack or the rigid inclusion, respectively.

中文翻译：

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BEM/BIEM中反平面剪切问题的应力强度因子和双简并机制研究

摘要 反平面剪切作用下的裂纹和刚性线夹杂问题分别是椭圆孔和椭圆刚性夹杂的特例。我们通过使用椭圆坐标方面的退化核来重新审视这类问题。压力强度因子 (SIF) 也得到了解决。采用三种方法来确定 SIF。一种是尖端附近边界或内部位移的外推方法。另一种方法是对尖端附近的边界应力或内部应力进行外推法。另一个是包围裂纹尖端的 ​​J 积分。有趣的是，由于对数核中长度为 4 的退化尺度，刚性线包含情况会产生奇异的影响矩阵。然而，包括退化尺度和退化边界的双退化仍然可能导致奇异矩阵，即使采用双 BEM/BIEM。由于引入了退化内核，该机制得到了很好的解释。如果不引入退化核，就无法清楚地研究 BIEM 中双退化问题的机制。通过使用简并核，可以很容易地确定反平面剪切下裂纹或刚性线夹杂物的 SIF。J 积分的路径独立性可以通过解析导出。还讨论了裂纹和刚性线夹杂物之间的 SIF 相对于相反载荷的倒数关系。在数值实现中，可以使用双边界元法确定 SIF。