本研究基於協合應力偶理論（Consistent Couple Stress Theory, CCST）架構之下，發展尺寸效應相關之有限元素方法（Finite Element Method, FEM）。研究重點在於探討簡支功能性（Functionally Graded, FG）石墨烯片（Graphene Platelets, GPLs）加勁複合材料（GPL-Reinforced Composite, GPLRC）圓柱微米殼的三維自由振動特性。在理論推衍中，圓柱微米殼被人為地劃分為許多有限數目的微米尺度層，以進行深入分析。透過傅立葉函數與Hermitian C2多項式，本研究實現了內外表面位移變化的精確插值，確保每一微層在節點表面位移分量的二階導數連續性。研究中考慮了GPLs在厚度方向上的五種分佈形式，包括均勻分佈（Uniform Distribution, UD）及功能性的A型、O型、V型和X型分佈。在材料尺度參數設為零的前提下，透過與現有文獻中相關的FG圓柱形宏觀殼的三維近似解進行比較，驗證了本CCST的FEM在準確性及收斂性方面的效力。數值結果顯示，增加GPLs的重量比例1%，可使FG-GPLRC圓柱微米殼的自然頻率提高至超過均質圓柱微米殼的兩倍。此外，材料尺度參數、GPL分佈形式及GPL的長度與厚度比例對FG-GPLRC圓柱微米殼的自然頻率均產生顯著影響。因此，本研究可以作為評估FG圓柱微米殼各種二維尺寸相關剪切變形理論的準確性參考。

Within a framework of the consistent couple stress theory (CCST), a size-dependent finite element method (FEM) is developed. The three-dimensional (3D) free vibration characteristics of simply-supported, functionally graded (FG) graphene platelets (GPLs)-reinforced composite (GPLRC) cylindrical microshells are analyzed. In the formulation, the microshells are artificially divided into numerous finite microlayers. Fourier functions and Hermitian C2 polynomials are used to interpolate the in-surface and out-of-surface variations in the displacement components induced in each microlayer. As a result, the second-order derivative continuity conditions for the displacement components at each nodal surface are satisfied. Five distribution patterns of GPLs varying in the thickness direction are considered, including uniform distribution (UD) and FG A-type, O-type, V-type, and X-type distributions. The accuracy and convergence of the CCST-based FEM are validated by comparing the solutions it produces with the exact and approximate 3D solutions for FG cylindrical macroshells reported in the literature, for which the material length scale parameter is set at zero. Numerical results show that by increasing the weight fraction of GPLs by 1%, the natural frequency of FG-GPLRC cylindrical microshells can be increased to more than twice that of the homogeneous cylindrical microshells. In addition, the effects of the material length scale parameter, the GPL distribution patterns, and the length-to-thickness ratio of GPLs on natural frequencies of the FG-GPLRC cylindrical microshells are significant.

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