本研究針對智慧型複合材料疊層板，建立偶合分析問題之相關解析解以及邊界元素法。疊層材料可為彈性或是壓電、壓磁、磁電彈等各式智慧型材料及其任意組合，各層順序亦無限制。同時考慮各種材料與疊層順序所引發之偶合現象，如彈性、電與磁效應偶合以及非對稱疊層板拉伸彎矩變形偶合等。目前針對相關磁電彈拉伸彎矩偶合之現有數值方法，大多使用有限元素法來進行，故須對整個結構件進行離散，使用大量網格才可得出收斂數值結果。此外，多數商業有限元素軟體並無提供內建的磁電彈平板或殼元素，使用者須自己額外建立相關的元素來進行偶合分析。藉由本文所提供之邊界元素法，我們只須對平板之中平面邊界進行線段離散，即可得出精準結果。針對特定問題，我們亦可使用對應解析解，無須進行任何離散。
依據古典層板理論以及一階剪切理論，分別建構出薄板與厚板相關偶合分析之基本數學方程式。針對彈性疊層薄板，其通解可表示為類史磋公式之形式。藉此推導出含有橢圓孔洞或裂縫之無限域承受均佈負載及相關格林函數等明示形解析解。利用平面波分解法以及狀態空間法，推導出彈性疊層厚板之格林函數。透過相關物理張量之維度擴充，磁電彈偶合問題之方程式可整理成與對應純彈性方程式相同數學形式。故類史磋公式可進行相關矩陣維度擴充，進一步應用至磁電彈拉伸彎矩偶合疊層薄板。依據此類比關係以及現有之對應純彈性解，得出含孔洞、裂縫、異質磁電彈疊層薄板在不同受力情形下之明示場域解，包括均佈負載以及格林函數。此外，考慮矩形簡支撐磁電彈疊層薄板，相關簡支撐邊界條件藉由傅立葉級數形式之平板位移函數來自動滿足，並導出對應奈維爾型態解以及列維型態解。
得出格林函數後，可進一步求出相關基本解，並用於發展邊界元素法。針對彈性疊層薄板拉伸彎矩偶合分析，我們以上述孔洞裂縫問題格林函數得出對應之基本解，並成功發展特殊邊界元素法。由於不受外力之孔洞或裂縫的邊界條件已由此特殊基本解滿足，因此無須對孔洞或裂縫邊界進行離散。裂縫問題之應力強度因子計算只使用到外圍平板邊界上之節點物理量。使用無限域之彈性疊層厚板基本解，發展出對應的邊界元素法，並進一步與現有的疊層厚樑邊界元素法結合，擴展至疊層加勁厚板分析。此外，考慮淺殼問題所引發的額外曲率效應，我們也將此厚板邊界元素法擴展至疊層淺殼問題的分析。與曲率效應有關之面積分項，則透過雙互換法轉換為邊界積分，以規避面積離散並維持邊界元素法的優勢。另一方面，將純彈性薄板之物理張量以及基本解進行維度擴充，建立磁電彈疊層薄板之邊界元素偶合分析，包含針對孔洞、裂縫、異質等問題之特殊邊界元素，並擴展至多區域問題。相關數值計算上的問題，例如複數型態對數函數之取值、奇異積分、磁電彈材料性質之數值病態係數矩陣，皆在本文裡詳盡地處理。
經由與對應有限元素分析結果進行比較，即便使用大量有限元素網格，本文提供之各式解析解以及邊界元素法(只須使用少量元素)皆較為準確，計算效率亦較佳，並可處理各式智慧型複合材料疊層結構之實際工程問題，包含各種偶合現象、幾何形狀、負載與支撐條件。

For the smart multilayered laminated composite plates, various coupling phenomena may be induced. In this dissertation the associated analytical solutions and boundary element methods (BEMs) are established. The laminates can be stacked using various kinds of materials, including elastic materials and smart materials like piezoelectric, piezomagnetic, and magneto-electro-elastic (MEE), without restrictions on principal directions or stacking sequences. Various coupling effects are incorporated simultaneously, such as the MEE coupling as well as the in-plane stretching and out-of-plane bending coupling. Currently, the coupling analysis for smart laminated plates is mostly implemented using finite element method (FEM), which requires domain discretization to ensure numerical convergency. In addition, most of commercial finite element software does not provide built-in plate or shell element to handle such coupling analysis. Extra efforts are required for the user to construct associated finite element formulations. By using the present BEMs, accurate results can be obtained using only discretization on the mid-plane of the plate. For specific problems the related analytical solutions are applicable without any discretization.
Based upon the classical lamination theory and the first-order shear deformation theory, the mathematical models are constructed firstly for the elastic coupling analyses of laminated thin plates and thick plates, respectively. The associated general solutions for laminated thin plates can be organized into the form of the Stroh-like formalism. The explicit-form field solutions are accordingly obtained, including the one for an unbounded plate with a traction free elliptical hole or straight crack subjected to uniform loads and the associated Green's function for concentrated forces/moments. Through the plane wave decomposition method and the state space approach, the Green's function for laminated thick plates is analytically derived. By suitably expanding the elastic quantity tensors, the Stroh-like formalism can be expanded to adapt for the general smart laminated thin plates with MEE coupling effects. The similar expansion is also made in deriving the analytical solutions for the hole/crack/inclusion problems concerning smart laminated thin plates. In addition, the Navier-type solution and the Levy-type solution are obtained for a rectangular simply supported smart laminated thin plates subjected to surface loads.
With the aforementioned Green's functions, the corresponding fundamental solutions are obtained and employed to develop the related BEMs. For example, the derived Green's function for laminated thin plates with holes or cracks is applied in the special BEM. Since the traction-free conditions along the hole/crack boundary are already satisfied by the special fundamental solutions, no boundary discretization is required there. The calculation of the stress intensity factors for a crack only needs the physical quantities from the remote plate boundary. By using the Green's function for laminated thick plates, the corresponding conventional BEM is developed and further combined with the existing BEM for laminated thick beams to handle stiffened plate problems. In addition, by incorporating the curvature effects raised by shallow shells, the plate BEM is extended to laminated thick shallow shells. The domain integral related to the curvature effects is transformed into boundary integrals via the dual reciprocity method, preserving the advantage of BEM by avoiding domain discretization. For general smart laminated thin plates, the conventional and special BEMs for hole/crack/inclusion problems are developed as the expanded versions of the corresponding elastic BEMs. All issues related to numerical implementation are thoroughly addressed, such as the evaluation of the multi-valued complex logarithmic functions, the singular integrals, and the ill-conditioned coefficient matrices for MEE material constants.
After the comparison to the corresponding finite element analyses (even with very fine meshes used), the present analytical solutions and BEMs (with relatively simple meshes used) demonstrate greater computational accuracy and efficiency. They are applicable to various practical problems concerning smart laminated composite plates, including cases with unrestricted coupling phenomena, diverse geometric shapes, and different loading and support conditions.

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