Nonlinear elastic response of 2D materials under simultaneous in-plane strains and flexural deformations

In this paper, we study the elastic response of 2D materials that are subjected to simultaneous in-plane strains and flexural deformations. A nonlinear elastic constitutive model is proposed for the in-plane and flexural deformations of 2D materials of arbitrary symmetries. The constitutive model consists of a series expansion of the strain energy density in terms of the strain and curvature components. The influence of in-plane symmetry and out-of-plane reflective symmetry on the number of linearly independent elastic constants is discussed. A methodology is presented for the evaluation of the elastic constants based on the proposed model. The strain energy is evaluated along rays in strain and curvature space to simulate different strain and curvature states of the materials. Subsequently, the elastic constants are systematically evaluated by curve fitting the proposed constitutive model to the sampled strain energy densities. The approach proposed to obtain the elastic constants is independent of the method used for calculating the strain energy density, e.g., classical interatomic potentials or density functional theory, as the curve fitting process is decoupled from the energy calculations. In the present work, the strain energies are evaluated using classical interatomic potentials. The atomic coordinates of the 2D materials are relaxed in a unit cell while preserving the applied strains and curvatures in order to evaluate the strain energy and, subsequently, the elastic constants. We analyze the nonlinear response of graphene to simultaneous strains and curvatures. Furthermore, we analyze black phosphorus which serves as a representative orthorhombic 2D material. The in-plane, flexural and coupling elastic constants are evaluated for both materials, and the effects of bending and twisting on the in-plane strains are studied. We investigate the effect of the in-plane strains on the effective flexural rigidities of the materials. It is found that a tensile strain causes a decrease in the bending rigidity of graphene. The flexural rigidities of black phosphorus are found to be less sensitive to in-plane strains than that of graphene.