Thermoelastic guided wave in fractional order functionally graded plates: An analytical integration Legendre polynomial approach

In this paper, an analytical integration Legendre polynomial approach (AILPA) is proposed to investigate the guided thermoelastic wave in functionally graded material (FGM) plates in the context of the fractional order Lord-Shulman (LS) thermoelastic theory. Coupled wave equations and fractional order heat conduction equation are solved by the presented approach, which proposes the analytical integral instead of numerical integration in the available conventional Legendre polynomial approach (CLPA). Comparison of the CPU time between two approaches indicates the higher efficiency of the presented approach. Furthermore, a new treatment of the adiabatic boundary condition for the Legendre polynomial is developed, other than the CLPA can only deal with the isothermal boundary condition. Finally, the phase velocity dispersion curves, attenuation curves, the displacement and temperature distributions for functionally graded plates with different fractional orders are analysed. Both the fractional order and relaxation time have weak influence on the elastic mode velocity, but they have considerable influence on the elastic mode attenuation.