Nonlinear bending analysis of bilayer orthotropic graphene sheets resting on Winkler–Pasternak elastic foundation based on non-local continuum mechanics

Abstract

In this paper, the nonlinear bending behavior of bilayer orthotropic rectangular graphene sheets resting on a two parameter elastic foundation is studied subjected to uniform transverse loads using the non-local elasticity theory. The non-local theory consider the small scale effects. Considering the non-local differential constitutive relations of Eringen theory based on first order shear deformation theory (FSDT) and using the von-Karman strain field, the nonlinear formulations are derived. Equilibrium partial differential equations are expressed in terms of generalized displacements and rotations. Because of nonlinear partial differential equations, if it is not impossible but it is too complicated to find an analytical solution, so, the differential quadrature method (DQM) that is a high accurate numerical method is investigated to solve the governing equations. The Newton–Raphson iterative scheme is applied to solve the obtained nonlinear algebraic equations system. Different boundary conditions including clamped, simply supports and free edges are considered. Since there is not any researches available for nonlinear bending of bilayer rectangular graphene sheets with FSD theory, so considering the monolayer, the results are compared with available papers. Finally, the small scale effect parameter due to various types of conditions such as thickness ratio, boundary conditions, stiffness of elastic foundation, the van der Waals interactions between the layers, nonlinear to linear FSDT analysis and the differences between non-local and local elasticity theories are investigated.