Abstract
In this paper, a three-dimensional (3D) size-dependent formulation is developed for the free vibrations of functionally graded quadrilateral nanoplates subjected to thermal environment. The plate model is constructed within the frameworks of the Gurtin–Murdoch surface and the 3D elasticity theories. In this way, the effect of surface free energy and all the components of stress and strain tensors are considered without any initial assumption on them as there is no need to assume the variation of transverse normal stress inside the bulk material in advance. The variational differential quadrature approach and the mapping technique are applied to derive a discretized weak form of the governing equations. The present solution method bypasses the transformation and discretization of the higher order derivatives appearing in the equations of the strong form. The effects of surface stress, thermal environment, material gradient index and geometrical properties on the size-dependent vibrational behavior of quadrilateral nanoplates are investigated. It is observed that the thermal load intensifies the effect of surface free energy on the natural frequency of the nanoplates. The present model is exact in the extent of the continuum models and can be employed for structures with any thickness–span ratios.
A three-dimensional surface elastic model for vibration analysis of functionally graded arbitrary straight-sided quadrilateral nanoplates under thermal environment