In this study, complex variable theory was employed to study the stress distribution for rock mass containing a hole with complex shapes. The method of mapping function calculation using optimization technique was investigated and improved. The shape-related mapping function was defined for parameter analysis. A new objective function for mapping function determination was developed to calculate related parameters without the consideration of restrictive conditions between boundary points in z-plane and ζ-plane, which significantly simplified the calculation process. In addition, the mapping function for planes containing a rotated hole was deducted to simplify analytical stress solution process for holes under complex stress conditions. Finally, for six kinds of holes with typical shapes in practical rock engineering, the mapping functions were determined, and the external analytical stress distribution were further presented. The analytical solutions combined with numerical and experimental results show that holes with arched boundaries suffer less stress concentration compared with those with corners. Hole shape mainly affects the stability of rock containing a hole via its influence on stress concentration extent around the hole.

在这项研究中，复杂变量理论被用来研究包含复杂形状孔洞的岩体的应力分布。对使用优化技术的映射函数计算方法进行了研究和改进。定义了形状相关的映射函数以进行参数分析。在不考虑z平面和ζ边界点之间的限制性条件的情况下，开发了用于映射函数确定的新目标函数，以计算相关参数平面，大大简化了计算过程。此外，推导了包含旋转孔的平面的映射函数，以简化复杂应力条件下孔的解析应力求解过程。最后，针对实际岩石工程中典型形状的六种孔，确定了其映射函数，并进一步给出了外部解析应力分布。解析解结合数值和实验结果表明，与具有拐角的孔相比，具有拱形边界的孔的应力集中较小。孔的形状主要通过其对孔周围应力集中程度的影响来影响包含孔的岩石的稳定性。