Abstract 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.

中文翻译：

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分数阶功能梯度板中的热弹性导波：分析积分勒让德多项式方法

摘要 在本文中，提出了一种分析积分勒让德多项式方法 (AILPA)，以在分数阶 Lord-Shulman (LS) 热弹性理论的背景下研究功能梯度材料 (FGM) 板中的引导热弹性波。耦合波动方程和分数阶热传导方程由所提出的方法求解，该方法提出了解析积分，而不是可用的传统勒让德多项式方法 (CLPA) 中的数值积分。两种方法之间 CPU 时间的比较表明所提出方法的效率更高。此外，除了 CLPA 只能处理等温边界条件外，还开发了一种新的勒让德多项式绝热边界条件处理方法。最后，相速度色散曲线，衰减曲线，分析了具有不同分数阶的功能梯度板的位移和温度分布。分数阶次和弛豫时间对弹性模态速度影响不大，但对弹性模态衰减影响较大。