A numerical approach for analyzing the guided wave propagation behavior in a functionally graded material (FGM) hollow cylinder is presented. Based on the three-dimensional linear elasticity theory, the state matrix form of displacement and stress component is introduced into the differential governing equation. Combining with boundary conditions, the dispersion characteristic equations can be constructed. Meanwhile, taking the orthogonal completeness of Legendre series into consideration, the exhausting root findings of transcendental equation can be avoided, and it transforms the redundant integral operators into analytical expressions by means of its recursive properties. Moreover, it is not only unnecessary to stratify FGMs, and introducing the univariate nonlinear regression to modelling the arbitrary gradient distribution, and transforming the redundant integral operator into an analytical expression by means of recursive property. It should be pointed out that the coefficient matrix consists only zero and constant terms, which are related to the elastic constants and normalized wave number. And it provides a fast and effectively approach to extracting the dispersion curves, displacement distribution and stress profile, simultaneously. Meanwhile, the proposed method can get rid of the exhausted root-finding algorithm, and accordingly distinguish the different modes and the dispersion curves simultaneously. Finally, the typical non-stratified computation of dispersion curves of longitudinal guided wave for FGM hollow cylinder is realized. The accuracy of the proposed method is further demonstrated through comparison with the available data from Legendre polynomial method. Several numerical cases about iron based alumina FGM cylinder are studied, and the convergence of the present polynomial approach is discussed. Then the effect of the circumferential wave number, gradient variation, diameter-thickness ratio on the dispersion characteristics are illustrated. Moreover, the distribution characteristics of displacement and stress profiles of F(1,1) mode at a given frequency are discussed in detail. It is shown that the state-vector formalism and Legendre polynomial hybrid approach is a powerful tool in solving wave motions in FGM elastic cylinder, which lays a theoretical foundation for the non-destructive evaluation and quantitative characterization of the gradient structure characteristics of functionally graded hollow cylinder.

提出了一种数值分析方法，用于分析功能梯度材料（FGM）空心圆柱体内的导波传播行为。基于三维线性弹性理论，将位移和应力分量的状态矩阵形式引入微分控制方程。结合边界条件，可以构建色散特性方程。同时，考虑到勒让德级数的正交完整性，可以避免先验方程的穷根发现，并通过其递归性质将冗余积分算子转化为解析表达式。此外，不仅没有必要对FGMs进行分层，而且无需将单变量非线性回归模型引入到任意梯度分布的建模中，并通过递归性质将冗余积分算子转换为解析表达式。应当指出，系数矩阵仅包括零项和常数项，它们与弹性常数和归一化波数有关。它提供了一种快速有效的方法，可以同时提取色散曲线，位移分布和应力分布。同时，该方法可以摆脱穷举的寻根算法，从而同时区分出不同的模式和色散曲线。最后，实现了FGM空心圆柱体纵向导波频散曲线的典型非分层计算。通过与勒让德多项式方法的可用数据进行比较，进一步证明了该方法的准确性。研究了铁基氧化铝FGM圆柱体的几种数值情况，并讨论了该多项式方法的收敛性。然后说明了圆周波数，梯度变化，直径-厚度比对色散特性的影响。此外，详细讨论了在给定频率下F（1,1）模式的位移和应力分布的分布特征。结果表明，状态向量形式和勒让德多项式混合方法是解决FGM弹性圆柱体中波动问题的有力工具，