本文基於協合應力偶理論(Consistent couple stress theory , CCST)，發展出一套尺度相關之剪力變形歸一理論，對功能性磁電彈性(Functionally graded magneto-electro-elastic , FG-MEE)微板在完全剪支撐邊界條件下，受電位勢、磁位勢、均勻溫度變化與雙軸壓力變化作用時的自由振動特性進行分析。此理論同時考慮應力偶與厚度伸展效應，並可透過指定厚度方向之剪力變形的形狀函數，重建出各種基於 CCST 的尺度效應理論，包括了尺度相關古典板理論、一階剪力變形板理論、Reddy的優化剪力變形板理論、正弦剪力變形板理論、指數剪力變形板理論與雙曲線剪力變形板理論，並使用此歸一理論來分析簡支撐功能性磁電彈性微板的固有頻率及振動模態。為驗證此歸一理論之準確性，本文將其解與文獻中發表之三維解析解進行比對，並進一步分析不同的幾何參數與材料參數對微板的固有頻率之影響，所考慮到的參數包含了厚度伸展效應、材料尺度參數、長厚比、材料性質梯度因子、電壓值、磁場值、均勻溫度之變化、雙軸壓力。透過對參數進行分析，微板的固有頻率對外加荷載形式極為敏感，當磁、電、熱或機械荷載引致初始拉應力或初始壓應力時，將導致系統頻率上升或下降，顯示磁電焦彈耦合荷載作用對微板固有頻率具顯著影響。

This study develops a unified size-dependent shear deformation theory based on the Consistent Couple Stress Theory (CCST) to investigate the free vibration characteristics of functionally graded magneto-electro-elastic (FG-MEE) microplates subjected to electric potential, magnetic potential, uniform temperature change, and bi-axial mechanical loads under fully simply supported boundary conditions. The proposed model incorporates both couple stress effects and thickness stretching effects, and allows for the reconstruction of various CCST-based size-dependent plate theories by specifying the shear deformation shape functions along the thickness direction, including the size-dependent classical plate theory, first-order shear deformation theory, Reddy’s refined shear deformation theory, as well as sinusoidal, exponential, and hyperbolic shear deformation theories. The unified model is employed to compute the natural frequencies and mode shapes of simply supported FG-MEE microplates, and its accuracy is validated through comparison with three-dimensional analytical solutions reported in the literature. A comprehensive parametric study is conducted to examine the influence of thickness stretching, material length scale parameter, aspect ratio, material gradient index, applied electric and magnetic fields, uniform temperature variation, and bi-axial mechanical loads on the vibrational behavior. The results reveal that the natural frequencies of the microplates are significantly affected by the nature of the applied loads, which may induce initial tensile or compressive stresses, leading to either frequency stiffening or softening, thereby highlighting the critical role of magneto-electro-thermal-mechanical loading conditions in the dynamic response of FG-MEE microstructures.

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