The main purposes of the present paper are as follows: (1) to analytically derive the general solution (stress functions) for an orthotropic elastic half plane with a crack or a notch; (2) to derive the Riemann–Hilbert equation as the analytical method; (3) to solve the present problem using two methods, one is a Cauchy integral method and other is a Riemann–Hilbert method; and (4) to derive the general expressions of the stress intensity factor (SIF) for a crack problem. The stress functions obtained by the Riemann–Hilbert equation are compared with those obtained by Cauchy integral method. Then, it is confirmed that the same stress functions can be obtained. The stress functions are expressed by any elastic constants. Therefore, Mode I and II SIFs can be calculated for any elastic constants. Some examples are shown. It is stated from the results of the examples that the negative Mode I SIFs exist for certain oblique edge crack angles and elastic constants. Because the derived stress functions are expressed using a mapping function, other geometrical shapes can be analyzed by changing the mapping function.

中文翻译：

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具有斜边裂纹和应力强度因子的正交各向异性弹性半平面的应力分析

本文的主要目的如下： (1) 解析推导具有裂纹或缺口的正交各向异性弹性半平面的一般解（应力函数）；(2) 导出黎曼-希尔伯特方程作为解析方法；(3)用两种方法解决目前的问题，一种是柯西积分法，一种是黎曼-希尔伯特法；(4) 推导出裂纹问题的应力强度因子 (SIF) 的一般表达式。通过黎曼-希尔伯特方程获得的应力函数与通过柯西积分法获得的应力函数进行比较。然后，确认可以获得相同的应力函数。应力函数由任何弹性常数表示。因此，可以针对任何弹性常数计算模式 I 和 II SIF。显示了一些示例。实例结果表明，对于某些斜边裂纹角和弹性常数，存在负的 I 型 SIF。由于导出的应力函数是使用映射函数表示的，因此可以通过更改映射函数来分析其他几何形状。