中文摘要：

针对功能梯度材料矩形板问题,基于三维弹性理论,将位移和应力分量作为基本变量,通过双三角级数将其控制微分方程转化为常微分方程组的边值问题。采用插值矩阵法直接对常微分方程组边值问题进行求解,得到了功能梯度材料矩形板三维位移、应力场的半解析解。通过算例给出了材料参数按指数形式和幂函数形式变化情况下的功能梯度板的弯曲问题。对比有限元法和状态空间法,结果表明：本文提出的状态空间与插值矩阵法相结合的半解析法能有效地分析材料参数按任意形式连续变化的功能梯度矩形板问题,且具有良好的精度,精度可达10-4量级,能够满足工程需要;与其他方法相比,本文方法具有实施便捷、计算量小等优点,根据其力学场分析结果可设计出满足各种不同需求的功能梯度材料。

英文摘要：

The rectangular plates made of functionally graded materials（FGM） are analyzed by three-dimensional linear elasticity theory.For the functionally graded rectangular plates,some displacement and stress components are acted as the basic variables.Firstly,these basic variables are expressed as the sums of the double trigonometric function expansions in the plane of the plates.Then,they are substituted into the governing differential equations of three-dimensional linear elasticity theory.Consequently,it leads to a series of two point boundary problems of ordinary differential equations with the basic variables.Finally,the interpolating matrix method is applied to directly solve the set of ordinary differential equations.All displacement and stress components of the FGM plate can be obtained.Two examples are given to demonstrate the accuracy and effectiveness of the computed results for the FGM plate by the comparisons with the solutions of the finite element method and state space method.