Size-dependent behavior of viscoelastic nanoplates incorporating surface energy and microstructure effects

In this paper, the simultaneous effects of the microstructure rotation and surface energy on the behavior of nanoplates are investigated in the framework of viscoelasticity. Firstly, the modified couple stress elasticity and Gurtin-Murdoch surface elasticity theories are reconsidered and harnessed to incorporate respectively the viscoelastic microstructure local rotation and viscoelastic surface energy effects into the classical viscoelastic plate theory. The couple stress tensor is obtained incorporating measures for the elastic and the viscous behaviors of the plate. Surface stress tensor is derived depending on the surface elastic and viscous parameters. Afterwards, a variational approach on the basis of D'Alembert's principle in conjunction with the Kirchhoff plate theory is utilized to establish the size-dependent integral-differential governing equations and the associated boundary conditions of viscoelastic nanoplate. The developed model accounts for the viscoelastic behavior of the linear non-aging materials using integral-type constitutive relations through the Boltzmann's principle of superposition. Finally, an analytical solution is derived for the simple supported plates according to the aforementioned phenomena using the Navier method and Laplace transformation. In the context of the linear viscoelasticity, a comprehensive parametric study is developed to present the influences of the various material and geometrical parameters on the bending behavior of the viscoelastic nanoplate.