本研究重點在於將異向性彈性材料中已滿足特殊問題邊界條件之格林函數，透過通用形式之邊界元素分析，將問題由異向性彈性材料延伸至壓電與黏彈性材料，並拓展至動態分析。在廣大的工程研究領域中，奇異性問題的求解與如何提高解的精確度，一直都是研究者深感興趣與極具挑戰的議題。於機械力學，奇異性問題通常發生在幾何不連續或材料不連續處，如同本論文關注的平板含缺陷(孔洞、裂縫與異質)與界面等相關問題，雖然這些問題早已存在著解析解，但若遇到較複雜的幾何外形與外力負載，還是需要藉助數值方法來求解問題。邊界元素法是數值方法中處理此特殊問題(如板含孔洞、裂縫、異質、界面等)較有效且準確的，其最大的特色在於可以減少求解問題的維度，且若使用的基本解滿足問題的邊界條件，在邊界上之分析結果將會完全滿足。針對二維異向性彈性材料，許多基本解如無限域含孔洞、裂縫、異質與界面等問題，已透過史磋複變公式導出，相關邊界元素法之應用也已被證實可改善數值分析的精確度並可節省電腦分析時間。
壓電材料之電彈耦合本構方程式可經由彈性材料方程式之維度擴充而得，由於兩者具有相同的數學模型，故壓電材料的解可直接由相對應之彈性材料的解擴充維度而得到。相同地，藉由彈性體與黏彈性體間之對應法則，彈性材料的解也可應用於黏彈性材料的解，於拉普拉斯域中。因此，透過上述的對應法則，彈性材料的基本解與邊界積分式，皆可應用於壓電與黏彈性材料。此外，藉由邊界元素法中的雙互換定理，若將動態問題的慣性力視為廣義的體力，則靜態問題的基本解也可用於求解動態問題之邊界元素法中。
基於上述論點，為了展現孔洞/裂縫/異質/界面之格林函數成功地應用於異向性彈性體、壓電體與黏彈性體或動態問題中，許多數值分析案例，如異向性彈性/壓電/黏彈性二維平板含孔洞/裂縫/異質/界面之靜態問題與異向性彈性二維平板含孔洞/裂縫/界面之動態問題等，將於本論文中呈現。

The problems concerning with singularity and how to obtain accurate results from them are the topics which greatly interest and challenge to the researchers all the while. In solid mechanics, such singularities are resulted from discontinuities of geometry or material heterogeneity, such as a plate with defects (holes, cracks, inclusions) and interfaces. Although a lot of exact solutions for those critical problems have been derived mathematically, the numerical analyses are still needed for some problems which involve the complex geometry and loading conditions. That is why we endeavor to solve such problems engaged with the boundary element method (BEM). The main advantages of BEM are the reduction of the dimension of the problem by one and the exact satisfaction of certain boundary conditions in some particular problems if their associated fundamental solutions are embedded in boundary element formulation. Because the fundamental solutions satisfy the boundary conditions of the defects and interface, say, no meshes are needed along the boundaries of holes, cracks, inclusions or interfaces in BEM. For two-dimensional anisotropic elasticity, many fundamental solutions, the so called Green’s functions of BEM, have been derived by using the complex variable Stroh formalism, such as an infinite space with holes, cracks, inclusions and interfaces, etc. Several researches in this field have proved that by using such fundamental solutions for BEM it makes the task more efficient and the results more accurate when coping with these problems.
Since the mathematical formulation of piezoelectric elasticity can be organized into the same form as that of anisotropic elasticity by just expanding the dimension of the corresponding matrix to include the piezoelectric effects, the solutions to the problems of piezoelectric materials can be obtained immediately through the extension of the solutions to the associated anisotropic elastic materials. Similarly, with the aid of the correspondence between elastic and viscoelastic materials, solutions for the linear anisotropic elastic solids can be applied directly to the linear anisotropic viscoelastic solids in the Laplace domain. With this understanding, one can acquire the fundamental solutions and boundary integral equations for piezoelectric and viscoelastic problems. Moreover, by using the dual reciprocal theorem for BEM, the static fundamental solutions can be used for dynamic boundary element method if the inertia term of the latter is treated as a general body force, which is the source term remained in the domain integral.
Based on the discussion above, in addition to the dynamic analysis of two-dimensional plates containing holes, cracks, and interfaces, several examples are also demonstrated in this dissertation to show that the static fundamental solutions of holes, cracks, inclusions and interfaces in anisotropic elasticity, derived in the sense of Stroh formalism, and the associated boundary integral equations have been extended to the applications for the piezoelectric and viscoelastic problems successfully with the prominence of accuracy and efficiency of the present BEM.

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