Energy landscapes of graphene under general deformations: DFT-to-hyperelasticity upscaling

Profound investigation of the unsurpassed mechanical properties of suspended graphene motivates the link between classical continuum mechanics (where the notions of “mechanical strength”, “mechanical stiffness”, and “elastic energy” have actually been coined) and density functional theory (DFT) rooted in quantum mechanics. Namely, the latter quantifies the energetic ground states of systems consisting of atomic nuclei and electrons; and these ground states, in turn, are directly linked to the elastic energy. While the corresponding state-of-the-art typically concerns graphene mechanics under uniaxial or equally biaxial strain states, we here present a fully anisotropic free (strain) energy function reflecting DFT-simulations associated with tens of thousands of arbitrary biaxial strain states. The latter are realized as sets of primitive unit cell vectors spanning a rhomboid unit cell hosting two carbon atoms. The position of the latter follows from internal energy minimization through the Vienna ab initio simulation package (VASP). As corresponding continuum mechanical representation we employ hyperelastic, structure tensor-based polynomial models up to the fifth order. The corresponding stress-strain relations are of concave nature, and hence, they provide access to the failure limits of the 2D material undergoing arbitrary loading situations. This is expected to introduce a new level of precision in the growing field of the structural mechanics of graphene.