等向性材料的多邊形孔洞問題雖已有解析解，但要將其擴展至異向性材料卻非常困難。原因在於異向性彈性力學的複變理論涉及三個主要複變數，而等向性彈性力學則只需一個。除了橢圓孔洞外，無法找到能同時適用於這三個複變數的映射函數組。因此，對於非橢圓孔洞，通常會採用一對多映射法或以橢圓孔洞為基礎的微擾法求解。本文採用微擾法推導解析解，並將其對比一對多映射法的解。首先探討在無限域中由橢圓加上單項微擾所形成的簡單多邊形孔洞，而後推廣至任意多邊形孔洞及其退化為裂縫的情況。比較兩種解顯示，一對多映射法在孔洞附近相當準確，但由於映射函數的多值特性，使得位移和應力在分支切割兩側不連續，並使臨界點處的應力趨近無限大。相對而言，微擾法在孔洞附近準確性雖不及一對多映射法，但隨著距離增加，準確性會迅速提升。這兩種解具有互補特性，將其結合使用，可獲得準確的全域結果。此外，利用這兩種方法所得的格林函數可用於建構處理有限域問題的特殊邊界元素法及結合兩者使用的混合邊界元素法。相較於傳統邊界元素法，該方法因為不需在孔洞邊界上劃分網格，故使其計算效率優於傳統邊界元素法。

This dissertation focuses on polygonal holes in two-dimensional anisotropic elastic solids, where physical quantities are independent of the thickness direction but permits coupling between in-plane and anti-plane deformation. The methodology is based on the Stroh formalism, a complex-variable formalism for anisotropic elasticity suitable for both generalized plane stress and generalized plane strain problems. Despite the well-developed solutions for isotropic elasticity, their extension to anisotropic elasticity is nontrivial because mappings for the three variables involved in the Stroh formalism are hard to construct. Proper mappings must (1) be one-to-one (conformal), and (2) produce coinciding images along the hole boundary. Only elliptical holes fulfil both requirements at the same time. Non-elliptical holes are usually addressed using a nonconformal mapping or via perturbation method based on the special case of elliptical holes.
In the present work, solutions are derived using the perturbation method and are compared with available nonconformal solutions. The analysis begins with simple polygonal holes in infinite domains, i.e., elliptical holes with a single-term boundary perturbation, and is then generalized to arbitrary polygonal holes in infinite domains, including their degeneration to cracks. Through numerical examples, nonconformal solutions are found to be accurate near the hole but suffer from discontinuity issues due to the nature of nonconformal mappings, whereas perturbation solutions, though less accurate near the hole, improve as the distance from the hole increases. In regard to their complementary characteristics, the two solutions, when strategically combined, can provide reliable results over the entire domain. For finite domains, two special boundary element methods (using the nonconformal and perturbation solutions, respectively) and a hybrid boundary element method are developed, which are more computationally efficient than conventional boundary elements since meshing on the hole boundary is not required. The proposed boundary elements can be used in conjunction with the boundary-based finite element method to analyse domains containing multiple polygonal holes. The analytical solutions and numerical methods presented in this dissertation are verified through comparison with commercial finite element software Ansys.

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