Invariant-based hyperelastic constitutive model for hexagonal two-dimensional materials

A general hyperelastic constitutive model for hexagonal two-dimensional (2D) materials is presented, wherein the strain energy density is expressed in terms of the invariants of the Green–St. Venant strain tensor. The functional dependence of the strain energy density on the strain tensor is elucidated through the introduction of new invariants that capture the contributions of dilatational, deviatoric, and strain-induced anisotropy effects. The strain energy density is formulated in terms of these new invariants and the corresponding second Piola-Kirchhoff stress tensor is obtained. A specific form of the constitutive model, characterized by seven elastic constants, is presented and a methodology for systematically evaluating the elastic constants is provided. A general expression for the tangent stiffness tensor is provided based on the new invariants. Additionally, the constitutive model is linearized for infinitesimal deformations and expressions for the Young's modulus, Poisson's ratio, and shear modulus are derived. Expressions for the layer modulus and speed of sound of hexagonal 2D materials are presented in terms of the linearized elastic properties. The nonlinear elastic behavior of graphene and molybdenum disulfide is investigated using the proposed approach. The constitutive model is validated using density functional theory data and the linearized elastic properties of graphene and molybdenum disulfide are compared with values reported in the literature. The proposed constitutive model accurately captures the nonlinear elastic response of graphene and molybdenum disulfide and can be adapted to study other 2D materials of hexagonal symmetry.