Constitutive modeling of some 2D crystals: graphene, hexagonal BN, MoS${}_2$, WSe${}_2$ and NbSe${}_2$

Abstract

We lay down a nonlinear elastic constitutive framework for the modeling of some 2D crystals of current interest. The 2D crystals we treat are graphene, hexagonal boron nitride and some metal dichalcogenides: molybdenium disulfide (MoS${}_2$), tungsten selenium (WSe${}_2$), and niobium diselenide (NbSe${}_2$). We first find their arithmetic symmetries by using the theory of monoatomic and diatomic 2-nets. Then, by confinement to weak transformation neighbourhoods and by applying the Cauchy-Born rule we are able to use the symmetries continuum mechanics utilizes: geometric symmetries. We give the complete and irreducible representation for energies depending on an in-plane measure, the curvature tensor and the shift vector. This is done for the symmetry hierarchies that describe how symmetry changes at the continuum level: ${\mathcal{C}}_{6\nu }\to {\mathcal{C}}_{2\nu }\to {\mathcal{C}}_1$ for monoatomic 2-nets and ${\mathcal{C}}_{6\nu }\to {\mathcal{C}}_{1\nu }\to {\mathcal{C}}_1$ for diatomic two nets. Having these energies at hand we are able to evaluate stresses and couple stresses for each symmetry regime.

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References in corpus (5)

• Two Dimensional Atomic Crystals
• Structure and electronic transport in graphene wrinkles
• Nonlinear elasticity of monolayer graphene
• Curvature dependent surface energy for a free standing monolayer graphene: some closed form solutions of the nonlinear theory
• Curvature dependent surface energy for free standing monolayer graphene: geometrical and material linearization with closed form solutions

Cited by in corpus (1)

• Graphene as a hexagonal 2-lattice: evaluation of the in-plane material constants for the linear theory. A multiscale approach