功能梯度材料结构分析的半解析数值方法研究

一种典型的半解析数值方法——线法被引入功能梯度材料的结构分析。首先推导了功能梯度材料位移形式的平衡方程和边界条件,然后阐述了线法功能梯度材料结构分析的基本步骤和数值原理。该方法的基本思想是通过有限差分将问题的控制方程半离散为定义在沿梯度方向离散节线上的常微分方程组,然后应用B样条函数Gauss配点法求解该常微分方程组得到问题的解答。为演示线法在功能梯度材料结构分析中的应用,给出了线性梯度和指数梯度功能梯度材料板分别受恒定位移、均匀拉伸载荷和弯曲载荷作用的数值算例。与相应问题解析解和其他数值方法的比较表明,线法的计算结果具有很高的精度,而且不需要任何特殊的考虑就能够有效模拟材料内部物性参数的连续变化,也无需事先选取满足特定条件的待定场函数,是一种非常适合功能梯度材料结构形式和材料特点的半解析数值方法。

Study on semi-analytic numerical method for structure analysis of functionally graded materials

A representative semi-analytical numerical method, method of lines(MOLs), was introduced into structure analysis of functionally graded materials(FGMs). Equilibrium equations and boundary conditions for FGM problems were derived with respect to displacement variables. Then, solution process and numerical fundamental of the approach were expounded. The basic idea of the method is to semi-discretizing the governing equations into a system of ordinary differential equations (ODEs) defined on discrete lines by means of the finite difference method. Solving the ODEs by using B spline collocation method at Gaussian points, solutions of the problem can be obtained. The examples of the FGM plate with exponential or linear gradation under constant displacement, tensile loading or bending loading were presented to demonstrate how to use the method for structure analysis of FGMs. Computational results of MOLs are quite accurate by comparison with ones of analytical solutions of corresponding problems and other numerical approaches. Moreover, it was confirmed that the method can effectively simulate continuous variation of material property parameters without any especial considerations and the approximate field function satisfying special conditions need not be chosen in advance. Therefore, it is an effective semi-analytical numerical method suitable to the structure format and material characteristic of FGMs.