Based on the Legendre orthogonal polynomial series expansion and the partial wave theory, a Legendre orthogonal polynomial method (LOPM) is proposed to calculate the reflection and transmission coefficients of plane waves at the liquid/solid interface of a liquid-loaded functionally gradient material (FGM) plate. The displacement solutions in FGM plate are fitted approximately by Legendre orthogonal polynomial series. The stresses and the governing differential equations of the FGM plate are derived. Based on the boundary conditions of the liquid/solid interface and the governing differential equations, the linearly independent equations are set up to calculate the reflection and transmission coefficients. Meanwhile, the expansion coefficients of the Legendre orthogonal polynomials are also obtained. The power function establishes the gradient model of the mechanical parameters along the thickness direction. The angular spectrums of reflection and transmission coefficients from LOPM are in well agreement with the calculation results from the transfer matrix method. By analyzing the convergence of the reflection coefficient spectrum, the critical value of the truncated order of Legendre orthogonal polynomials is determined. The mapping relationship between the FGM gradient models and the reflection coefficient angular spectrum, also the frequency spectrums and the displacement/stress distributions can be demonstrated simultaneously, which provides the theoretical fundamentals of the ultrasonic non-destructive testing for the mechanical properties of FGM and extends the application of the LOPM.

基于Legendre正交多项式级数展开和偏波理论，提出了Legendre正交多项式方法（LOPM），以计算平面波在液体加载的功能梯度材料（FGM）的液/固界面处的反射系数和透射系数。 ） 盘子。FGM板中的位移解近似由Legendre正交多项式级数拟合。推导了FGM板的应力和控制微分方程。根据液/固界面的边界条件和控制微分方程，建立线性独立方程，以计算反射系数和透射系数。同时，还获得了勒让德正交多项式的展开系数。幂函数可建立沿厚度方向的机械参数的梯度模型。LOPM反射和透射系数的角谱与传递矩阵法的计算结果非常吻合。通过分析反射系数谱的收敛性，确定勒让德正交多项式的截断阶的临界值。同时可以证明FGM梯度模型与反射系数角谱之间的映射关系，以及频谱和位移/应力分布，为超声对FGM的力学性能进行无损检测提供了理论基础并扩展了应用范围。 LOPM的应用。