(A) plate model for multilayer graphene sheets and their wrinkle structures analysis

Abstract

An equivalent continuum model for multilayer graphene sheets (MLGSs) and its plate model are developed to analyze the deformation behavior of MLGSs. Hyperelastic material models are introduced for the MLGS continuum model by examining the atomistic structures of MLGSs and obtaining their mechanical properties by means of molecular statics simulations. The MLGS plate model, a structural model for MLGSs, is developed by applying kinematics assumptions to the MLGS continuum model subjected to infinitesimal deformation. Finite element methods (FEM) with the corotational formulation are adopted to analyze the mechanical behavior of MLGSs under small-strain deformation and large rotation conditions. The MLGS plate element passes several basic numerical tests, including patch tests, eigenvalue analyses, and geometrically nonlinear benchmark problems. The deflections of a plane-strain cantilever and spherical indentations are analyzed by the proposed MLGS plate element and molecular dynamics (MD) simulations. These results show that the MLGS plate element properly represents the deformation behaviors of MLGSs from the atomic scale to the macroscopic continuum scale.

The buckling of MLGSs subjected to an axial compressive load in plane-strain condition is studied. Closed-form solutions for the buckling load of MLGSs are obtained based on a continuum model for MLGSs. Two different kinematic assumptions, which lead to the MLGS beam and the Euler beam, are used to obtain the buckling loads. The obtained solutions yield significantly different buckling loads when the axial length is small. To validate obtained results, MD simulations are conducted, and they show that the MLGS beam model well captures the buckling load of MLGSs. The buckling solution of MLGS beam model provides two interesting facts. First, the buckling load of MLGSs coincides with the Euler buckling load when the length is large. Second, when the number of layers is large, the buckling strain converges to a finite value and could be expressed as a linear combination of the buckling strain of single-layer graphene and the ratio between the shear rigidity of the interlayer and the tensile rigidity of the layer. The asymptotic behavior of the buckling strain is validated through MD simulations. MD simulations show that the buckling occurs even when the overall thickness is larger than the axial length. We present a diagram that contains the buckling strain of MLGSs according to the boundary conditions, the number of layers, and the axial length.

The buckling and the post-buckling of MLGSs on an elastic foundation subjected to an axial compressive load in plane-strain condition are studied. Using the MLGSs structural model, closed-form solutions for the buckling strain and the characteristic wavelength are obtained and verified through FEM linearized buckling analyses with an interlayer element which could capture arbitrary large relative translational motion between neighboring layer sections. FEM analyses show that the MLGSs beam model well captures the buckling only when the foundation is relatively soft. The buckling of bilayer graphene sheets on an elastic foundation is analyzed by relaxing kinematic assumptions introduced in the MLGSs structural model. The results show that the buckling deformations could be categorized according to the stiffness of the foundation. Delamination could occur when the foundation is stiff. As the stiffness of the foundation becomes soft, various deformed state emerged: closed to in-phase state, transition region, and prevalence of the Euler beam kinematics. The post-buckling analyses are conducted using FE model and obtained various deformation path according to the stiffness of the foundation.