Indentation of solid membranes on rigid substrates with Van-der-Waals attraction

TL;DR

This paper investigates how boundary conditions and Van-der-Waals attraction influence the indentation response of thin solid membranes on rigid substrates, revealing a potential shift from cubic to pseudo-linear force-deformation behavior.

Contribution

It introduces a theoretical model showing that sliding boundary conditions and VdW forces can alter the indentation force response from cubic to linear, highlighting the importance of boundary effects.

Findings

Indentation force may be linear rather than cubic in depth.

Sliding boundary conditions significantly affect membrane response.

Van-der-Waals attraction influences the force-deformation relationship.

Abstract

We revisit the indentation of a thin solid sheet of size $R_{sheet}$ suspended on a circular hole of radius $R \ll R_{sheet}$ in a smooth rigid substrate, addressing the effects of boundary conditions at the hole's edge. Introducing a basic theoretical model for the Van-der-Waals (VdW) sheet-substrate attraction, we demonstrate the dramatic effect of replacing the clamping condition (Schwerin model) with a sliding condition, whereby the supported part of the sheet is allowed to slide towards the indenter and relax the induced hoop compression through angstrom-scale deflections from the thermodynamic equilibrium (determined by the VdW potential). We highlight the possibility that the indentation force $F$ may not exhibit the commonly anticipated cubic dependence on the indentation depth ($F\propto \delta^3$), in which the proportionality constant is governed by the sheet's stretching modulus and the hole's radius $R$, but rather a {\emph{pseduo-linear}} response, $F \propto \delta$, whereby the proportionality constant is governed by the bending modulus, the VdW attraction, and the sheet's size $R_{sheet} \gg R$.