On vibrations of functionally graded viscoelastic nanobeams with surface effects

This paper addresses the size-dependent free vibration of functionally graded viscoelastic (FGV) nanobeams including the simultaneous effects of the microstructure rotation and surface energy for the first time. Employing the Bernoulli-Euler beam theory, an internal damping mechanism based on Kelvin-Voigt model is adopted to simulate the viscoelastic behavior of the material. The modified couple stress theory and Gurtin-Murdoch surface elasticity theory are reconsidered and harnessed to capture the viscoelastic microstructure rotation and viscoelastic surface energy effects, respectively. The local-Cauchy stress, couple stress and surface stress tensors are obtained incorporating measures for the elastic and the viscous behaviors of the nanobeam. The elastic and viscous material properties of the bulk and surface of the FGV nanobeam are assumed to vary continuously in thickness direction according to a power law. A variational approach on the basis of D'Alembert's principle is employed to derive exactly the size-dependent governing differential equation and the associated nonclassical boundary conditions. An analytical expression is derived for the complex natural frequencies of a simply supported FGV nanobeam. In the context of linear viscoelasticity, the influences of different parameters such as the material damping, gradient index, material length-scale parameter, surface elasticity, surface residual stress, surface mass density, Poisson effect, thickness, and slenderness ratio on the free vibration of simply supported FGV nanobeams are comprehensively discussed. The results highlighted the profound effects of the small size, surface energy and viscoelastic behavior on the free vibration response of FGV nanobeams.