本研究基於協合應力偶理論(Consistent couple stress theory, CCST)，發展出一套Hermitian C² 微分再生核(Differential reproducing kernel, DRK)無網格方法，此方法選定位移與橫向應力以及其一階與二階導數作為主要變數，利用加權最小二乘法直接計算結果，無需對內插函數直接進行微分。此方法所構建之形狀函數由滿足再生條件的擴增函數(Enrichment functions)，以及符合Kronecker delta性質之原始函數(Primitive functions)所組成，使得形狀函數及其一階與二階導數皆滿足Kronecker delta性質，且在取樣節點上的主要變數、一階與二階導數都具備連續性。本文結合Hermitian C² DRK無網格方法與強形式CCST理論，進一步發展出一套無網格適點法(Point collocation method)，進行簡支承功能梯度(Functionally graded, FG)微板受外力載重作用下之三維尺寸效應之靜態撓曲分析。本文透過與文獻中精確解之比較，驗證其具備良好的準確性與收斂性，並進行參數分析與討論，探討材料尺度參數、非均質性指數與寬厚比等關鍵因子對微板撓度、面內應力、橫向應力及應力偶之影響。研究結果顯示，Hermitian C² DRK無網格方法在強形式CCST尺寸效應微板表現出良好的準確度，在參數分析中，材料尺度參數和非均質性指數增加或是寬厚比降低會使微板整體勁度增加，導致撓度降低。此外，功能性梯度微板的應力與應力偶在厚度方向呈現高次多項式分布，且橫向應力最大值位於微板上半部，其應力偶變化大於均質材料。而均質微板之面內應力呈線性分布，橫向應力分布則為拋物線，最大值位於中平面。此類應力的分布情況研究於現有文獻中較為少見。因此，本研究之分析結果可以做為各類微板理論的驗證依據，亦可提供進一步發展高階剪應變微板理論時之假設參考。

This study presents a Hermitian C² differential reproducing kernel (DRK) meshless method within the framework of strong-form consistent couple stress theory (CCST) for analyzing size-dependent behavior in functionally graded (FG) microplates. Displacement, transverse stress, and their first and second derivatives are used as primary variables, with solutions computed via a weighted least-squares approach that avoids explicit differentiation of interpolation functions. The constructed shape functions satisfy the Kronecker delta property up to second-order derivatives, enabling accurate enforcement of boundary conditions.
A point collocation meshless method is developed and applied to the static bending analysis of simply supported FG microplates under mechanical loading, accounting for three-dimensional size effects. The method's accuracy and convergence are validated against exact solutions, and a parametric study is conducted to examine the effects of material length scale, inhomogeneity index, and aspect ratio.
Results show that FG microplates exhibit high-order stress and couple stress distributions across the thickness, with transverse stress peaking near the upper surface and greater couple stress variation compared to homogeneous plates. In contrast, homogeneous microplates display linear in-plane stress and parabolic transverse stress distributions, with maxima at the mid-plane. The distribution of this type of stress has been rarely investigated in existing literature. Therefore, the analytical results of this study can serve as a benchmark for validating various microplate theories and also provide a reference for assumptions in the further development of higher-order shear deformation microplate theories.

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