We revisit the indentation of a thin solid sheet of size $R_{\mathrm{sheet}}$ suspended on a circular hole of radius $R\ll R_{\mathrm{sheet}}$ 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 pseduolinear response $F\propto \delta$, whereby the proportionality constant is governed by the bending modulus, the vdW attraction, and the sheet's size $R_{\mathrm{sheet}}\gg R$.

• Received 6 August 2020
• Revised 4 March 2021
• Accepted 11 March 2021

DOI:https://doi.org/10.1103/PhysRevE.103.043002