A size-dependent meshfree model based on nonlocal strain gradient theory for trigonometric functionally graded nanoplates on variable elastic foundations

In this paper, a nonlocal strain gradient meshfree model is proposed and developed to explore the bending and vibration behaviours of a novel trigonometric functionally graded nanoplates (TFGNPs). Based on the generalized layerwise higher-shear deformation theory (GL-HSDT) and the nonlocal strain gradient theory (NSGT), a weak form of governing equations for plate motion is derived, with consideration of a two-parameter variable elastic foundation. We employed a cosine function to describe the material gradation of TFGNPs along their thickness while the size-scale effect in nanoplates was effectively captured through the incorporation of NSGT. The radial point interpolation method, which possesses high continuum and Kronecker delta function properties, is employed to develop a meshfree formulation for the discrete solution of governing equations. By comparing the results of the study with those in existing literature, the correctness and high accuracy of present model is verified. It is shown that the material properties of TFGNPs possess high stability and continuous, smooth stress variations. Moreover, a comprehensive parametric study is conducted to determine the sensitivity of the bending and vibration responses of TFGNPs to boundary conditions, geometries, foundation parameters, nonlocal and strain gradient parameters.