本文基於協合應力偶理論(Consistent couple stress theory , CCST)，發展各種剪切變形尺度相關板之歸一理論，用於對嵌入彈性介質中的簡支撐微、奈米板(Micro-/Nano-scale plates , MNSP)進行靜態撓曲及自由振動分析，其中尺度相關之古典板理論、一階剪力變形板理論、Reddy的優化剪力變形板理論以及正弦、指數和雙曲線剪力變形板理論，均可藉由將一特定之剪切變形厚度分佈函數代入歸一理論中獲得。文中亦將此歸一CCST板理論應用於嵌入彈性介質中的簡支撐功能性微米板(Functionally graded microscale plates , FGMP)和多層石墨烯板(Multilayered graphene sheets , MLGS)之靜態撓曲及自由振動分析，其中對材料尺度參數、長厚比及材料性質梯度因子對FGMP和MLGS之變形、應力及頻率參數的影響均有探討。結果顯示，材料尺度參數對FGMP、MLGS之靜態撓曲和自由振動的影響是顯著的。

Based on the consistent couple stress theory (CCST), the authors develop a unified formulation of various shear deformation plate theories for static bending and free vibration analyses of simply-supported, micro-/nano-scale plates embedded in an elastic medium. The CCST-based classical plate theory (CPT), first-order shear deformation plate theory (SDPT), and Reddy’s refined SDPT, as well as the CCST-based sinusoidal, exponential, and hyperbolic SDPTs can be obtained by assigning specific through-thickness distributions of the shear deformations into the unified formulation, and they are thus included as special cases of the unified consistent couple stress plate theory. The unified CCST-based plate theory is applied to static bending and free vibration analyses of simply-supported, functionally graded microplates (FGMPs) and multilayered graphene sheets (MLGSs) embedded in an elastic medium. The material length scale parameter, aspect ratio, and shear deformation effects on the deformation, stress, and frequency parameters of FGMPs and MLGSs are investigated. It is shown that the solutions for the deformation, the in-plane stress, and the frequency parameters of the FGMPs/MLGSs obtained using the MCST and the CCST are almost identical to each other. The material length scale parameter effects on static bending and free vibration behaviors of the FGMP/MLGS are significant.

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