基於協合應力偶理論(Consistent Couple Stress Theory, CCST)，本文發展Hermitian Cn (n=1, 2)有限層狀元素法(Finite Layer Method, FLM)，用以分析具簡支承邊界之功能性(Functionally Graded, FG)壓電材料圓柱三明治微米殼在封閉迴路的條件下受軸壓與電壓之挫屈與自由振動行為。本文基於Hamilton定理進行三維弱形式表述之理論推衍，將微米殼切割成多個層狀元素，並選擇彈性位移場及電位勢作為主變數，各個主變數在面內進行雙重傅立葉級數展開，並在厚度方向上使用Hermitian Cn多項式進行內插擬合。將層狀變位模型代入弱形式表述之理論後，發展出Hermitian Cn FLM，用以分析FG壓電材料之圓柱三明治微米殼。本文藉由與文獻中FG壓電材料圓柱宏觀殼及FG彈性材料圓柱微米殼之參數解進行比較，以此評估Hermitian Cn FLM的準確度與收斂速度。在進行FG壓電材料圓柱宏觀殼的分析時須將材料尺度參數設為零，而進行FG彈性材料圓柱微米殼的分析時則會忽略壓電效應與撓電效應。對於簡支FG壓電材料圓柱三明治圓柱殼，本文亦探討了一些重要因素對其臨界載重、臨界電壓及自由振動頻率的影響，其中包括壓電效應、撓電效應、材料尺度參數、不均勻指數、圓柱殼半徑與厚度之比值、圓柱殼長度與半徑之比值以及外加電壓或軸向載重之大小。研究結果顯示，壓電效應、撓電效應與材料尺度參數上升時，會使得微米殼勁度提升，進而造成臨界載重、臨界電壓與自由振動頻率提升；而徑厚比、長徑比以及外加電壓或軸壓越大，則會使得微米殼整體勁度下降，造成臨界載重、臨界電壓與自由振動頻率的下降。

Within the framework of the consistent couple stress theory (CCST), we develop a Hermitian Cn (n=1, 2) finite layer method (FLM) for carrying out the three-dimensional (3D) analysis of the size-dependent buckling and free vibration behaviors of simply supported, functionally graded (FG) piezoelectric cylindrical sandwich microshells. The microshells of interest are placed under closed-circuit surface conditions and subjected to axial compression and electric voltages. We derive a 3D weak formulation based on Hamilton’s principle for this study. In the resulting formulation, the microshell is artificially divided into nl microlayers, with the elastic displacement components and the electric potential selected as the primary variables. By incorporating a layer-wise kinematic model into our weak formulation, we develop a Hermitian Cn FLM. Each primary variable is expanded as a double Fourier series in the in-surface domain and is interpolated in the thickness direction using Hermitian Cn polynomials. The accuracy and the convergence rate of our FLMs are validated by comparing the solutions they produce for FG piezoelectric cylindrical macroshells and FG elastic cylindrical microshells with the relevant exact and approximate 3D solutions which have been reported in the literature. The material length scale parameter of our FLMs is set at zero in the comparison made with the FG piezoelectric macroshells. In contrast, the piezoelectric and flexoelectric effects are ignored in the comparison made with the FG elastic microshells. The impact of some essential factors on the critical load, critical voltage, and natural frequency of microshells is assessed. The important factors are identified as piezoelectricity, flexoelectricity, the material length scale parameter, the inhomogeneity index, the radius-to-thickness ratio, the length-to-radius ratio, and the magnitude of the applied voltage and the applied load.

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