本研究旨在發展一套適用於三明治複合板結構(Sandwich composite plate structure)於幾何非線性大變形(Geometrically nonlinear large deformation)下的高階轉折理論(Higher-order refined zigzag theory, HRZT)，研究主軸圍繞有限元素方程式的建構(Finite element formulation)，其核心著重於推導板元素之線性與非線性剛性矩陣，以及探討系統於不同邊界條件、幾何尺寸與外部負載組合下，結構之靜態力學響應。
承襲本實驗室過去於三明治複合樑(線性與非線性)以及複合板結構(線性)之研究基礎，本研究進一步將板結構之高階轉折位移理論延伸其應用至幾何大變形領域，以解決高負載情境下所誘發的幾何非線性問題，從而模擬實際工程中更為複雜且具挑戰性的結構行為。
本研究所採用之HRZT 板理論位移場可視為HRZT樑理論之二維推廣，透過於面內位移引入高階轉折函數，使得模型得以滿足各項邊界條件，如層間介面軸向位移連續、層間介面剪應力連續、上下自由表面無剪應力等，對於複合板結構於面內和厚度方向的力學響應提供更精確且靈活的描述。
鑒於板結構問題於高維度空間中求取解析解(Exact Analytical solution)相對複雜，實務上難以取得。本研究基於HRZT理論，針對大變形幾何非線性靜態系統，開發三種不同形式的元素，分別為HRZT4N36、HRZT4N40、HRZT8N72。
元素名稱「4N」與「8N」分別表示為元素內部含有4個以及8個主要節點，而數字「36」、「40」、「72」表示該元素內節點自由度總和。經元素收斂性分析與剪力自鎖分析結果可知，HRZT4N40可在不顯著增加運算量的前提下，也能有效地克服元素剪力自鎖(Shear locking)的效應，具備較佳的實務應用性。
為驗證本研究所建立之HRZT模型的數值準確性，本文將其數值結果與FSDT, HSDT, RZT等各種經典板理論的有限元素模型，以及商用套裝軟體ANSYS三維有限元素模型的結果進行比對，比較項目涵蓋面內位移、撓曲響應、正向應力、面內剪應力與面外剪應力等。結果顯示，本研究發展之HRZT 板元素在幾何非線性靜態響應分析中具備高度數值精確性，能有效捕捉因材料性質與厚度方向不連續所引發之真實力學行為。
此外，針對非線性挫曲分析(Nonlinear buckling analysis)，本研究透過採用預先施加之橫向負載產生初始幾何不完美(Initial imperfection)以模擬真實結構之受壓反應。數值結果證實，HRZT模型不僅能準確預測臨界挫曲負載，更能有效追蹤結構由挫曲前期(Pre-buckling)至挫曲後期(Post-buckling)的完整變形路徑與挫曲階段剛性退化的過程；儘管在薄板大變形的挫曲較為後期區域存在些微數值差異，但整體趨勢仍與ANSYS三維模型展現高度一致性，充分驗證了本模型在評估結構穩定性上的適用性與可靠度。
根據本文呈現的數值結果，證明了HRZT板元素能夠準確描述三明治複合板的靜態非線性系統響應，其有限元素公式及其數值結果可應用於航空工程、海洋工程、土木工程和機械工程等多個領域。

This study aims to develop a Higher-Order Refined Zigzag Theory (HRZT) specifically for sandwich composite plate under geometrical nonlinearity, which is focus on constructing a robust finite element formulation and deriving both linear and nonlinear stiffness matrices for nonlinear system. The static mechanical responses are extensively investigated under various combinations of boundary conditions, geometric dimensions (e.g. aspect ratio), and the level of external loading. Based on the previous laboratory researches of composite beams and plates, this work extends those studies to geometric nonlinear large deformation applications to address complex nonlinear problems in high loading environments and simulate realistic engineering behaviors.
The displacement field for plate structure can be seen the two-dimensional extension of HRZT beam theory. By adopting higher-order zigzag functions, the model can accurately satisfy essential boundary conditions, such as the continuity of interlaminar axial displacements and the continuity of interlaminar shear stresses, as well as no shear stress on the upper and lower surface of face-sheets.
Due to the complexity of obtaining analytical solutions for plate structure, numerical methods are prioritized. Three HRZT-based elements are developed in this study, which are HRZT4N36, 4N40, and 8N72, respectively, where notation 4N/8N denotes primary nodes and number 36/40/72 means total degrees of freedom in a single element).
Convergence and shear locking analysis results demonstrate that HRZT4N40 element effectively overcomes shear locking without significant computational consumption and exhibit superior practical applicability for large-scale structural simulations.
To verify numerical accuracy, results are compared against 3D finite element models with commercial ANSYS software and finite elements model based on various plate theories, such as FSDT, HSDT, and RZT. Comparative analyses of displacement, deflection, and various stress components are validated that HRZT element can effectively capture local effects like zigzag displacement along thickness direction, which is induced by material stiffness variation.
Furthermore, in nonlinear buckling analysis, pre-applied transverse load is used to generate initial geometric imperfections for simulating real-world compressive situation. The results show the model developed by this study accurately track the deformation path from pre-buckling stage to buckling stage, and stiffness degradation through the pre- and post-buckling stages. These results provide essential insights into the structural safety margins. Despite minor discrepancies in advanced post-buckling stage, the results remain highly consistent with ANSYS 3D, validating its reliability in structural stability.
In conclusion, HRZT plate element developed in this study provides an accurate and efficient framework for describing the static nonlinear responses of sandwich composite plates. This formulation and its numerical results offer significant theoretical and practical value for structural application in aerospace, marine, civil, and mechanical engineering.

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