An Efficient Legendre Polynomial Series Approach for Thermoelastic Waves in Functionally Graded Piezoelectric Nano Hollow Cylinder

Abstract

The Legendre polynomial series approach (LPSA) has the advantages of simple derivation process, high solving efficiency, and small computational scale, making it an efficient method for solving wave mechanics. However, when solving the nonlocal nano hollow cylindrical structure, the nonlocal integral elastic relationship makes it consume extremely much CPU time for integrals containing the reciprocal of radius, Legendre polynomials, absolute values, and exponential function. To overcome this drawback, an efficiently iterative solution method by using the exponential integral function are proposed to solve these difficult to calculate analytically and time-consuming integrals. Then, a mathematical model is established for axial thermoelastic wave propagation and attenuation in functionally graded piezoelectric nano hollow cylinders based on the presented LPSA and modified nonlocal integral elasticity. The coupling effects of nonlocal parameter, circumferential order, piezoelectric effect, and gradient index on wave characteristics are studied. Numerical results show that both the thermal wave and elastic wave are size-dependent. The nonlocal effect decreases the phase velocity of thermal waves and elastic waves. Meanwhile, it increases the attenuation of thermal waves, but decreases the attenuation of elastic waves. For the phase velocity, the nonlocal effect and circumferential order are mutually reinforcing, although the nonlocal effect and piezoelectric effect are mutually inhibiting.