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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。
According to the three-dimensional elasticity theory, the displacement and stress components are regarded as the basic variables, and the governing differential equations are converted into the boundary value problems of the system of ordinary differential equations by double trigonometric series. The boundary value problem of the system of ordinary differential equations is directly solved by the interpolation matrix method, and the semi-analytical solution of the three-dimensional displacement and stress field of the functionally graded rectangular plate is obtained. The bending problem of functionally graded plate with material parameters changing exponentially and power function is given by an example. Compared with finite element method and state space method, the results show that the semi-analytical method combining state space and interpolation matrix method proposed in this paper can effectively analyze the functionally graded rectangular plate whose material parameters continuously change in any form, and has good accuracy, Compared with other methods, the proposed method has the advantages of convenient implementation and small amount of calculation. According to the result of mechanical field analysis, the functionally graded material can be designed to meet various needs .