A spectral finite element for axial-flexural-shear coupled wave propagation analysis in lengthwise graded beam

A new spectral element ( SE) is formulated to analyse wave propagation in anisotropic in homogeneous beam. The in homogeneity is considered in the longitudinal direction. Due to this particular pattern of inhomogeneity, the governing partial differential equations (PDEs) have variable coefficients and an exact solution for arbitrary variation of material properties, even in frequency domain, is not possible to obtain. However, it is shown in this work that for exponential variation of material properties, the equations can be solved exactly in frequency domain, when the same parameter governs the variation of elastic moduli and density. The SE is formed using this exact solutionas interpolating polynomial. As a result a single element can replace hundreds of finite elements (FEs), which are essential for all wave propagation analysis and also for accurate representation of the inhomogeneity. The developed element is used for eliciting several advantages of the gradation, including mode selection, mode blockage and smoothening of stress waves