Buckling of a non‐Euclidean annular plate

Abstract

The effect of free edges of a monoatomic graphene sheet leads to excess edge energy due to the reconstruction of dangling bonds. Molecular static calculations show, that individual carbon atoms near the edge are displaced out of plane for relaxed nanoribbons [1]. In this work we are considering the effect of excess edge energy for almost circular graphene patches. To tackle this problem in the framework of continuum mechanics we are modelling the edge effect with a non-Euclidean plate model. A linear stability analysis of the flat configuration leads to the stability boundary in the parameter plane. Atomistic modelsAb initio calculations [2] give accurate values for the energy cost of creating free edges in a monoatomic graphene sheet. The same procedure was used for calculating the edge energies with two classical potentials AIREBO [3] and REBO [4]. The general form of the used AIREBO potential is Edge energy as a function of the applied strain is depicted in Fig. 1. The slope of the individual lines at = 0 represents the edge stresses. There is a significant difference between the results from ab initio calculations and the classical potentials. Ab initio calculations predict larger edge energies for zigzag edges than for armchair ones, whereas the two classical potentials (REBO & AIREBO) show the opposite order. The order of the edge stresses are interchanged as well. A comprehensive comparision of results from different atomistic models can be found in [6]. Empirical force fields are efficient for simulating larger systems, which makes them appropriate for studying the effect of the excess edge energy on the global configuration. Although the discrepance in the quantitative individual values, the results are qualitatively compareable. In particular the negative edge stress is remarkable, because this is the reason for destabilisation of the flat edge. Figure 2 shows the relaxed configuration of an almost circular graphene patch. The wavy disturbance of the edge decays towards the center.

Asymptotic solutions for the Föppl – von Kármán equations governing deflections of thin axisymmetric annular plates

“…Temporal terms also allow one to couple the plate dynamics to thermal field, in order to study the influence of heat transfer on the bending process; see [52]. The Föpplvon Kármán equations have been considered in order to describe annular sheets at a molecular level, with monoatomic graphene sheets having been studied in [53]. While this study will be concerned with annular plates having radial symmetry, we remark that annular plates with deflections varying both with the radial variable and with the angular variable have been considered, and are studied in order to better understand the process of wrinkling [54][55][56].…”