An analytical integration Legendre polynomial series approach for Lamb waves in fractional order thermoelastic multilayered plates

Abstract

The Legendre polynomial series approach (LPSA) has been widely used to solve guided wave propagation in various structures since 1999. The LPSA directly introduces the boundary conditions into the control equations through the rectangular window function. However, in the solving process, the Legendre polynomial series, the rectangular window function, and their derivatives are introduced into the integral kernel functions, which results in a lot of CPU time on the abundant numerical integration calculations. To overcome this defect, an analytical integration Legendre polynomial series approach (AILPSA) is presented and is used to solve the Lamb waves in fractional order thermoelastic multilayered plates. Coupled wave equations and heat conduction equation are solved by the AILPSA and the LPSA, respectively. Comparison between two approaches indicates the computational efficiency of the AILPSA is improved by more than 90%. In addition, a backward error estimation is given in the convergency analysis to make up for the deficiency of the numerical experiments. Finally, the influence of fractional order on the thermoelastic Lamb wave is discussed.

Trade-off between ordinary differential equation and Legendre polynomial methods to study guided modes in angle-ply laminate

“…Obviously, the higher the numbers of layers, the larger the errors relevant to the polynomial approach [6]. For more extensive background on the polynomial method we recommend reading references [7][8][9][10][11][12]. The present work is actually motivated by this increased attention to the performance of multi-layered laminates.…”

Double Legendre polynomial quadrature-free method for axisymmetric vibration of functionally graded piezoelectric circular plates

“…Then, Elmaimouni et al (2005) applied the polynomial method for wave propagation in an infinite homogeneous anisotropic cylinder rod. From then on, orthogonal polynomial method has a wide range of development in terms of wave propagation (Othmani et al, 2022), such as lamb waves in piezoelectric-semiconductor sandwich plates (Othmani et al, 2017), the propagation characteristic of guided waves in FGM plates (Gao et al, 2019), lamb wave propagation in anisotropic composite lamina (Liu et al, 2020a), lamb and SH waves in multilayered PZT-4/PZT-5A composites (Othmani et al, 2020), the reflection behaviors of elastic waves in the FGP microstructures (Liu et al, 2020b),circumferential guided waves in a fractional Kelvin–Voigt viscoelastic FGM hollow cylinder (Zhang et al, 2021a), circumferential guided waves in the anisotropic multilayer composite cylinders (Zheng et al, 2021b), and lamb waves for thermoelastic multilayered plates (Yu et al, 2022). In 2011, Elmaimouni et al (2011) extended the mapping of orthogonal function method into a double Legendre orthogonal polynomial method (DLOPM) to solve the piezoelectric disc resonator, and obtained the normalized frequencies and electric input admittance.…”