A singular integral equation method is proposed to analyze the complex plane cracks in one-dimensional (1D) orthorhombic quasicrystals. Using the Somigliana formula, the singular integral equations of the curved crack are derived. Based on the general situation of the curved crack, the singular integral equations of the inclined crack and the arc crack are given. Then the analytical solutions for the singular phonon and phason stresses near the tips of the inclined and the arc crack are obtained. Gauss-Chebyshev quadrature method is introduced to calculate the singular integral equations, and a numerical algorithm for solving the stress intensity factor (SIF) is proposed. Numerical solutions for the phonon and phason SIFs of some examples are solved and discussed.

中文翻译：

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一维斜方准晶复杂平面裂纹的奇异积分方程法分析


提出了一种奇异积分方程方法来分析一维（1D）斜方准晶中的复杂平面裂纹。利用Somigliana公式推导了弯曲裂纹的奇异积分方程。根据弯曲裂纹的一般情况，给出了倾斜裂纹和弧形裂纹的奇异积分方程。然后得到了倾斜和弧形裂纹尖端附近的奇异声子和相子应力的解析解。引入高斯-切比雪夫求积法计算奇异积分方程，提出求解应力强度因子（SIF）的数值算法。求解并讨论了一些例子的声子和相子 SIF 的数值解。