本研究基於三維協合應力偶理論（Consistent Couple Stress Theory, CCST），發展一套微分再生核 (Differential Reproducing Kernal,DRK) 無網格適點法，用以分析單向非均質微板在簡支邊界條件下，承受單軸與雙軸壓縮時之靜態挫屈與自由振動。研究中選取橫向應力與位移分量為主要變數，並結合雙重傅立葉級數展開，建立以厚度方向為主之基本控制方程。進一步將DRK插值函數引入基本方程中，建立一套可處理三維尺寸效應之無網格數值方法。
為驗證所提方法之準確性與收斂性，將其計算結果與文獻中既有之三維解析解進行比較。結果顯示，本研究方法之數值解與三維解具有高度一致性，且具備良好的收斂特性。參數分析結果指出，長寬比 (Aspect Ratio) 、材料尺度參數(Material Length-Scale Parameter) 、長厚比 (Length-to-Thickness Ratio) 以及材料非均質指數 (Inhomogeneity Index) 等因素，皆會顯著影響微板之自然頻率與臨界載重。此外，本研究亦提出CCST理論適用之結構尺寸範圍約為 1 × 10−7 至 1 × 10−3 公尺 。
整體而言，本研究所建立之DRK無網格適點法，不僅可有效分析具尺寸效應之微尺度結構行為，亦在計算效率與精度之間取得良好平衡，對於微機電系統及功能性梯度材料結構之工程應用具有潛在價值。

This study develops a Differential Reproducing Kernel meshless collocation method based on Consistent Couple Stress Theory to analyze the static buckling and free vibration of unidirectional inhomogeneous microplates under simply supported boundary conditions, subjected to uniaxial and biaxial compression. By selecting transverse stresses and displacement components as primary variables and incorporating double Fourier series expansion, the fundamental governing equations focused on the thickness direction are established. The DRK interpolation functions are further integrated into these equations to develop a meshless numerical approach capable of addressing three-dimensional size effects.To verify the accuracy and convergence of the proposed method, numerical results are compared with existing 3D analytical solutions from literature. The results demonstrate that the numerical solutions obtained in this study are highly consistent with 3D analytical solutions and exhibit excellent convergence characteristics. Parametric analysis indicates that factors such as aspect ratio, material length scale parameters, thickness ratio, and the material inhomogeneity index significantly influence the natural frequencies and critical buckling loads of the microplates. Furthermore, this study identifies the applicable structural size range for CCST to be approximately 1 × 10−7 to 1 × 10−3 meters.Overall, the DRK meshless method established in this study effectively analyzes the behavior of micro-scale structures with size effects while maintaining a superior balance between computational efficiency and precision. This work offers significant potential value for engineering applications in Micro-Electro-Mechanical Systems and functionally graded material structures.

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