Nonlocal Strain Gradient Model for the Nonlinear Static Analysis of a Circular/Annular Nanoplate

Abstract

A nonlinear static analysis of a circular/annular nanoplate on the Winkler–Pasternak elastic foundation based on the nonlocal strain gradient theory is presented in the paper. The governing equations of the graphene plate are derived using first-order shear deformation theory (FSDT) and higher-order shear deformation theory (HSDT) with nonlinear von Karman strains. The article analyses a bilayer circular/annular nanoplate on the Winkler–Pasternak elastic foundation. HSDT while providing a suitable distribution of shear stress along the thickness of the FSDT plate, eliminating the defects of the FSDT and providing good accuracy without using a shear correction factor. To solve the governing equations of the present study, the differential quadratic method (DQM) has been used. Moreover, to validate numerical solutions, the results were compared with the results from other papers. Finally, the effect of the nonlocal coefficient, strain gradient parameter, geometric dimensions, boundary conditions, and foundation elasticity on maximum non-dimensional deflection are investigated. In addition, the deflection results obtained by HSDT have been compared with the results of FSDT, and the importance of using higher-order models has been investigated. From the results, it can be observed that both strain gradient and nonlocal parameters have significant effects on reducing or increasing the dimensionless maximum deflection of the nanoplate. In addition, it is observed that by increasing load values, the importance of considering both strain gradient and nonlocal coefficients in the bending analysis of nanoplates is highlighted. Furthermore, replacing a bilayer nanoplate (considering van der Waals forces between layers) with a single-layer nanoplate (which has the same equivalent thickness as the bilayer nanoplate) is not possible when attempting to obtain exact deflection results, especially when reducing the stiffness of elastic foundations (or in higher bending loads). In addition, the single-layer nanoplate underestimates the deflection results compared to the bilayer nanoplate. Because performing the experiment at the nanoscale is difficult and molecular dynamics simulation is also time-consuming, the potential application of the present study can be expected for the analysis, design, and development of nanoscale devices, such as circular gate transistors, etc.