Analytical models for predicting the nonlinear stress-strain relationships and behaviors of two-dimensional carbon materials

Due to having a single atom layer, two-dimensional (2D) materials represented by graphene monolayers exhibit unique and excellent mechanical properties, such as ultrahigh moduli and strengths. A large number of experiments and atomistic simulations have demonstrated nonlinear stress-strain responses. However, there is no theoretical model that analytically describes the relationships between nonlinear mechanical properties and interatomic interaction parameters of 2D materials. Here, we developed a nonlinear stick-spiral model for four typical 2D materials (including graphene, γ-graphyne, β-graphyne, and hexagonal boron nitride) based on a molecular mechanics model. By using the perturbation method, we derived a series of analytical expressions for nonlinear stress-strain relationships and elastic constants of these 2D materials under uniaxial tension along the zigzag and armchair directions. Our analytic models indicated that both Young's moduli and Poisson's ratios of these 2D materials are isotropic and dominate the linear elastic deformation, while their third-order moduli are orientation-dependent and essentially characterize the nonlinear stress-strain responses. The nonlinear stress-strain relationships, elastic constants, and atomic behaviors (such as bond elongation and bond angle variation during deformation) predicted from our analytical models are in good agreement with those from atomistic simulations and previous experiments. Our analytical models further demonstrated that the mechanical properties and behaviors of 2D materials are linked with their bonding and atomic structures (from a quantitative perspective) and are mainly determined by stiffnesses for bond stretching, angle variation, and bond lengths. Our current study provides an effective and accurate analytical approach for investigating the nonlinear behaviors of 2D materials.