在固體力學以及平板力學之基礎，本文討論上層為功能梯度材料(FGM)，下層為均質材料的雙層功能梯度矩形板之力學行為。首先，經由公式推導得到FGM平板之統御方程式，並求得四周皆為簡支撐且承受均勻分佈負載的雙層FGM板之位移場和應力場。接著討論在同樣的邊界條件、外型尺寸且承受相同外力的情況下，改變FGM之類型、楊氏係數之比值、函數變化之參數，以及FGM與均質材料在平板中所佔的厚度比例，對於FGM平板之影響。
本文的第二部份是分析FGM平板的振動問題，先推導出FGM平板之振動統御方程式，求解任意邊界條件下之FGM平板的自然頻率，並使用有限元素軟體進行振動模態分析，加以驗證本文求解結果之正確性。另外，由求解自然頻率的過程中，整理並推導得到FGM平板與均質平板自然頻率之對應關係式，亦即能夠經由關係式，從均質板的自然頻率推得相同外型尺寸、邊界條件FGM板的自然頻率。最後舉出幾個典型的個案，證明該關係式之正確性。

Based on the elasticity theory, the mechanics together with the free vibration analysis of a double-layered rectangular plate is studied in this paper. The plate contains two layers with a functionally graded material (FGM) and homogeneous material, respectively. The closed-form solution of the simply supported plate subjected to uniform loading is obtained by Levy’s method. The effects of different FGM property types, ratio of Young’s modulus E2/E1, nonhomogeneous material parameter p, and the layer thickness ratio on the mechanical behavior of the plate will be emphasized.
In the case of a clamped-edge FGM plate, the natural frequencies are derived analytically and validated by the results obtained from finite element analysis. As a conclusion, the correspondence of the natural frequency between FGM plates and homogeneous plates expressed in an equation form is proved to be accurate.

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