# Indentation metrology of clamped, ultra-thin elastic sheets

## TL;DR

This paper analyzes the indentation of ultrathin elastic sheets using membrane theory to improve measurement accuracy of surface properties, highlighting potential errors from common approximation methods.

## Contribution

It provides a systematic membrane-based approach to analyze indentation data, revealing limitations of approximate formulas in thin film metrology.

## Key findings

- Highlighting geometric nonlinearities in thin sheet indentation
- Demonstrating potential errors in existing approximation methods
- Suggesting improved analysis techniques for experimental data

## Abstract

We study the indentation of ultrathin elastic sheets clamped to the edge of a circular hole. This classical setup has received considerable attention lately, being used by various experimental groups as a probe to measure the surface properties and stretching modulus of thin solid films. Despite the apparent simplicity of this method, the geometric nonlinearity inherent in the mechanical response of thin solid objects renders the analysis of the resulting data a nontrivial task. Importantly, the essence of this difficulty is in the geometric coupling between in-plane stress and out-of-plane deformations, and hence is present in the behaviour of Hookean solids even when the slope of the deformed membrane remains small. Here we take a systematic approach to address this problem, using the membrane limit of the F\"{o}ppl-von-K\'{a}rm\'{a}n equations. This approach highlights some of the dangers in the use of approximate formulae in the metrology of solid films, which can introduce large errors; we suggest how such errors may be avoided in performing experiments and analyzing the resulting data.