Exact analytical solutions to the problem of relative post-buckling stiffness of thin nonlocal graphene sheets

Abstract

Abstract The main purpose of this research is to obtain analytical formulations for exact calculation of relative post-buckling stiffness of nonlocal graphene sheets. In addition to calculating the post-buckling stiffness reduction, the buckling and initial post-buckling responses of these structures, when they are subjected to end-shortening strain, have also been studied. To investigate these phenomena, a new technique called semi-Galerkin technique is used in which the out-of-plane deflection function is firstly postulated as the only displacement field and then, exact nonlocal stress function is calculated through a complete solution of the von-Karman compatibility equation. Finally, Galerkin's method is used to solve the unknown parameter considered in the proposed technique. The nano-sheets are modeled as an orthotropic layer with Kirchhoff assumptions and nonlocal differential elasticity theory is employed to achieve the buckling loads and exact relative stiffness values. For in-plane movements of the longitudinal edges of the nano-sheets, two essential and natural boundary conditions are adopted to be “straightly movable” or “freely movable”. The effects of aspect ratio and nonlocal parameter have been studied for each type of boundary conditions and for graphene sheets with different materials. The stress distribution along the length and the width of the nano-sheets is also investigated and discussed by the two local and nonlocal theories and for various values of nonlocal parameter.