Abstract A solution is derived for a mixed boundary orthotropic elastic plane problem using a mapping function to solve arbitrary configurations. No stress function using a mapping function seems to have been derived. The problem is solved as a Riemann–Hilbert problem. The final exact stress functions are represented by an irrational mapping function as a closed form. Stress components are represented by one complex variable. Arbitrarily shaped hole problems can be solved by changing the mapping function. Once the analytical solution has been derived, calculating the stress components is simpler than for an isotropic problem. It is easier to use an irrational mapping function than to form a rational mapping function for an isotropic problem. As an example, an infinite plane with a square hole subjected to uniform tension is analyzed. The stress distributions are shown for Cases I and III problems corresponding to two characteristic roots of the fundamental equation. In a Case III problem, the symmetry of stress distributions is lost. A solution of an external force boundary value problem can be derived from that of the mixed boundary value problem.

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使用映射函数分析正交各向异性弹性的混合边值问题

摘要 利用映射函数求解任意构型，推导出混合边界正交各向异性弹性平面问题的解。似乎没有推导出使用映射函数的应力函数。该问题被解决为黎曼-希尔伯特问题。最终的精确应力函数由无理映射函数表示为封闭形式。应力分量由一个复变量表示。任意形状的孔问题可以通过改变映射函数来解决。导出解析解后，计算应力分量比计算各向同性问题要简单。对于各向同性问题，使用无理映射函数比形成有理映射函数更容易。作为一个例子，分析了一个无限大的平面，它有一个受均匀拉力的方孔。应力分布显示为对应于基本方程的两个特征根的案例 I 和 III 问题。在案例 III 问题中，应力分布的对称性丢失。外力边值问题的解可以从混合边值问题的解中推导出来。