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

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 $R_{sheet} \gg R$.