本研究基於協合應變梯度理論(Consistent Strain Gradient Theory, CSGT)，發展局部Petrov-Galerkin 無網格法(Local Petrov-Galerkin Meshless Method, LPGM)分析功能性梯度(Functionally Graded, FG)材料微板於簡支承條件下之力學行為。本方法採用雙重傅立葉級數配合應用微分再生核(Differential Reproducing Kernel, DRK)函數，透過選取基底函數與符合 Kronecker delta 性質之原始函數，配合移動最小二乘法建構出具高度連續性且滿足 Kronecker delta 性質之形狀函數，使節點值直接對應位移變數。本文中結合 LPGM 與考慮應力偶(Couple Stress)、體積應變梯度(Dilatational Strain Gradient)以及偏差應變梯度(Deviatoric Strain Gradient)效應之 CSGT 弱形式表述，分析微板之靜態撓曲與自由振動。驗證方面，藉由將 CSGT 退化至僅含應力偶效應之協合應力偶理論(Consistent Couple Stress Theory, CCST)，與修正應力偶理論(Modified Couple Stress Theory, MCST)之結果進行比對，證實 LPGM 之精確性和收斂性，同時也比較了不同權函數(Weight Functions)下所得到的位移值，充分地展現出LPGM的優越計算特性。參數研究顯示，引入上述任一效應皆顯著增強微板勁度，導致最低頻率增加與變形減少。各效應對位移、應力及最低頻率之影響依序由強至弱排列為應力偶、偏差應變梯度、體積應變梯度；對第二低頻率則為偏差應變梯度、應力偶、體積應變梯度。值得注意的是，體積應變梯度對此並無影響。基於上述分析，建議在分析微奈米結構時，應採用較完整之 CSGT 模型以避免低估微結構之勁度。

This study develops a Local Petrov-Galerkin Meshless method (LPGM) based on the Consistent Strain Gradient Theory (CSGT) to analyze the static bending and free vibration of simply supported functionally graded (FG) microplates. The method combines double Fourier series with Differential Reproducing Kernel (DRK) functions. By employing basis and primitive functions satisfying the Kronecker delta property via a least-squares approach, high-continuity shape functions are constructed, equating nodal values directly to displacement variables. The integrated CSGT weak-form formulation accounts for couple stress, dilatational, and deviatoric strain gradients. The LPGM's accuracy and convergence are validated by comparing its degenerated Consistent Couple Stress Theory (CCST) results with the Modified Couple Stress Theory (MCST), alongside evaluations of various weight functions that demonstrate its superior computational characteristics. Parametric studies reveal that incorporating any size effect significantly enhances microplate stiffness, increasing the fundamental frequency and reducing deformation. The influence hierarchy for displacement, stress, and fundamental frequency is: couple stress, deviatoric, and dilatational strain gradients. For the second frequency, the deviatoric gradient dominates, followed by couple stress, while the dilatational gradient has no effect. Consequently, employing the comprehensive CSGT model is recommended for micro-to-nano scale analysis to avoid underestimating structural stiffness.

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