Senthil S. Vel

A unified invariant-based hyperelastic framework is presented for hexagonal two-dimensional (2D) materials using the Seth–Hill family of strain measures, encompassing the Green–St. Venant, Biot, logarithmic (Hencky), Almansi, and Swainger strains as special cases. The strain energy density is formulated in terms of physically motivated invariants of the Seth–Hill strains that separately capture dilatational, deviatoric, and deformation-induced anisotropy effects. The conjugate stress tensors associated with the Seth–Hill strains and the corresponding Cauchy stress tensor are subsequently derived. A specific constitutive model characterized by seven elastic constants is proposed for the strain energy density. These constants are determined by fitting the model to strain energy data obtained from density functional theory calculations for combined deformation states involving biaxial stretching with a superimposed shear deformation. Furthermore, the constitutive model is linearized in the infinitesimal strain limit, yielding closed-form expressions for the Young’s modulus, Poisson’s ratio, shear modulus, and in-plane bulk modulus in terms of the elastic constants. The proposed constitutive framework, which is general and applicable to 2D materials with hexagonal symmetry, is illustrated in the present work using graphene and MoS ₂ . When expressed in terms of the physically motivated invariants, different Seth–Hill strain measures yield similar predictions for the strain energy density and elastic response, indicating that the choice of strain measure has minimal impact on the elastic behavior in the present invariant-based formulation.

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.