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 MoS2. 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.
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