Method of elliptic functions for determination of deformation and elastic moduli of 2d materials

Charactrization of the mechanical properties of graphene and SWCNTs is a challenging problem. Didderent experimental techniques revealed a deviation in the values of the elastic moduli of van der Waals structures. FEM numerical simulations (Cao, An, Diamond and Related Materials, 146, 2024, 111201) show that the elastic moduli measured via Raman spectroscopy are overestimated. We propose a continuum mechanics model of graphene as an elastic plane weakened by a doubly-periodic system of circular holes. In the model, the area of a hole is significantly smaller than that of a hexagonal cell - an atom of hydrogen for example cannot pass through the dense clouds of electrons forming the cell bonds. The goal of the study is to determine the displacements, stresses. and the elastic bonds of graphene of any chiral vector. The model is governed by plane elasticity, the loading is applied at infinity, while the circular holes are free of tractions. The methodology is based on the Kolosov-Muskhelishvili complex potentials for diobly-periodic structures, the Weierstrass elliptic function, the Natanzon quasi-periodic meromorphic function, the Filshtinsky method and leads to infinite linear lagebraic systems. We discuss the numerical results, the method of reverse homogenization to detrmine the elastic moduli of the bonds, and a generalization of the method to SWCNTs.