Continuum mechanics model of graphene as a doubly-periodic perforated thin elastic plate

TL;DR

This work treats graphene as a doubly-periodic, perforated hexagonal plate under remote tension to connect lattice-scale bond mechanics with continuum properties. It extends the classical Filshtinskii approach by employing doubly-periodic Kolosov-Muskhelishvili potentials and Weierstrass/Natanzon function series to obtain an analytic solution that reduces to an infinite linear system for the series coefficients. The authors derive explicit expressions for stresses and displacements in terms of the bond moduli $E$ and $\nu$, and introduce a reverse homogenization method to express these bond moduli as functions of effective moduli $E^\circ$ and $\nu^\circ$ reported in the literature. Numerical results illustrate the dependence of bond moduli on chirality, hole size, and orientation, showing, for example, that $E$ can be about $74\%$ larger than $E^\circ$ for $\nu^\circ=0.3$ and $\lambda=a/5$, thereby linking microscopic bond properties to macroscopic effective properties and informing defect mechanics and nano-scale graphene models.

Abstract

In this paper, a continuum mechanics model of graphene is proposed, and its analytical solution is derived. Graphene is modeled as a doubly-periodic thin elastic plate with a hexagonal cell having a circular hole at the hexagon center. Graphene is characterized by a general chiral vector and is subject to remote tension. For the solution, the Filshtinskii solution obtained for the symmetric case is generalized for any chirality. The method uses the doubly-periodic Kolosov-Muskhelishvili complex potentials, the theory of the elliptic Weierstrass function and quasi-doubly-periodic meromorphic functions and reduces the model to an infinite system of linear algebraic equations with complex coefficients. Analytical expressions and numerical values for the stresses are displacements are obtained and discussed. The displacements expressions possess the Young modulus and Poisson ratio of the graphene bonds. They are derived as functions of the effective graphene moduli available in the literature.

Figures (6)

• Figure 1: Geometry of the problem
• Figure 2: The total stresses $\sigma^t_r$ and $\tau^t_{r\theta}$ vs $r\in[\lambda, \frac{a}{\sqrt{3}}]$ for $\alpha=0$ (curves 1), $\alpha=\frac{\pi}{8}$ (curves 2), and $\alpha=\frac{\pi}{4}$ (curves 3) when $\lambda=\frac{a}{5}$, $\sigma_1=2$, $\sigma_2=1$, $\theta=0$.
• Figure 3: The total stresses $\sigma^t_r$ and $\tau^t_{r\theta}$ vs $\alpha\in[0,\pi]$ for $\theta=\frac{\pi}{8}$ and $r=\lambda+\frac{1}{16}(\frac{a}{2}-\lambda)$ (curves 1), $r=\lambda+\frac{1}{4}(\frac{a}{5}-\lambda)$ (curves 2), and $r=\lambda+\frac{15}{16}(\frac{a}{2}-\lambda)$ (curves 3) when $\lambda=\frac{a}{5}$, $\sigma_1=2$, and $\sigma_2=1$.
• Figure 4: Poisson ratio $\nu$ and the dimensionless parameter $E/E^\circ$ of the bonds vs radius $\lambda$ when $\nu^\circ=0.3$ and $E^\circ$ is fixed.
• Figure 5: Effective Poisson ratio $\nu^\circ$ and the dimensionless parameter $E^\circ/ E$ of the perforated plane vs radius $\lambda$ when $\nu=0.2668$ and $E$ is fixed.