Indentation of solid membranes on rigid substrates with van der Waals attraction

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