Constitutive modeling of some 2D crystals: graphene, hexagonal BN, MoS₂, WSe₂ and NbSe₂

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: $\mathcal C_{6 \nu} \rightarrow \mathcal C_{2 \nu} \rightarrow \mathcal C_1$ for monoatomic 2-nets and $\mathcal C_{6 \nu} \rightarrow \mathcal C_{1 \nu} \rightarrow \mathcal C_1$ for diatomic two nets. Having these energies at hand we are able to evaluate stresses and couple stresses for each symmetry regime.