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An Atomistic-based Finite Deformation Continuum Membrane Model for Monolayer Transition Metal Dichalcogenides

A finite-deformation crystal-elasticity membrane model for Transition Metal Dichalcogenide (TMD) monolayers is presented. Monolayer TMDs are multi-atom-thick two-dimensional (2D) crystalline membranes having atoms arranged in three parallel surfaces. In the present formulation, the deformed configuration of a TMD-membrane is represented through the deformation map of its middle surface and two stretches normal to the middle surface. Crystal-elasticity based kinematic rules are employed to express the deformed bond lengths and bond angles of TMDs in terms of the continuum strains. The continuum hyper-elastic strain energy of the TMD membrane is formulated from its inter-atomic potential. The relative shifts between two simple lattices of TMDs are also considered in the constitutive relation. A smooth finite element framework using B-splines is developed to numerically implement the present continuum membrane model. The proposed model generalizes the crystal-elasticity-based membrane theory of purely 2D membranes, such as graphene, to the multi-atom-thick TMD crystalline membranes. The significance of relative shifts and two normal stretches are demonstrated through numerical results. The proposed atomistic-based continuum model accurately matches the material moduli, complex post-buckling deformations, and the equilibrium energies predicted by the purely atomistic simulations. It also accurately reproduces the experimental results for large-area TMD samples containing tens of millions of atoms.