# Mechanics of thermally fluctuating membranes

## TL;DR

This paper investigates how thermal fluctuations affect the mechanical properties of graphene membranes, showing that standard thin plate theories need modification for accurate modeling at room temperature and micron scales.

## Contribution

The study reveals the limitations of Föppl-von Kármán theory for graphene, providing modified equations and scaling analysis to better understand its mechanics under load.

## Key findings

- Thermal fluctuations significantly reduce Young's modulus in micron-sized graphene.
- Standard FvK theory does not accurately describe graphene mechanics at room temperature.
- Modified models improve the interpretation of indentation experiments.

## Abstract

Besides having unique electronic properties, graphene is claimed to be the strongest material in nature. In the press release of the Nobel committee it is claimed that a hammock made of a squared meter of one-atom thick graphene could sustain the wight of a 4 kg cat. More practically important are applications of graphene like scaffolds and sensors which are crucially dependent on the mechanical strength. Meter-sized graphene is even being considered for the lightsails in the starshot project to reach the star alpha centaury. The predicted strength of graphene is based on its very large Young modulus which is, per atomic layer, much larger than that of steel. This reasoning however would apply to conventional thin plates but does not take into account the peculiar properties of graphene as a thermally fluctuating crystalline membrane. It was shown recently both experimentally and theoretically that thermal fluctuations lead to a dramatic reduction of the Young modulus and increase of the bending rigidity for micron-sized graphene samples in comparison with atomic scale values. This makes the use of the standard F\"oppl-von Karman elasticity (FvK) theory for thin plates not directly applicable to graphene and other single atomic layer membranes. This fact is important because the current interpretation of experimental results is based on the FvK theory. In particular, we show that the FvK-derived Schwerin equation, routinely used to derive the Young modulus from indentation experiments has to be essentially modified for graphene at room temperature and for micron sized samples. Based on scaling analysis and atomistic simulation we investigate the mechanics of graphene under transverse load up to breaking. We determine the limits of applicability of the FvK theory and provide quantitative estimates for the different regimes.