Indentation of a stiff membrane on an incompressible elastic halfspace

Abstract

Indentation of a very stiff membrane (like graphene) on an incompressible elastic material has been suggested as a method to measure the elastic modulus of the membrane, but so far the method is less explored than that based on indentation of a free-standing membrane clamped on the outer boundary, which relies on analytical solutions. However, we analyse the problem rigorously with an energy minimization in the Rayleigh sense with a one term approximation of the vertical displacement, and show that in the fully non-linear regime, the load $F$ has a single term solution increasing as the power 5/3 of the indentation $\Delta$. The solution is corrected only in the prefactor by extensive FEM investigation using a concentrated load resulting finally in $F=\frac{1.45\times 4\pi }{{\left(384\pi \right)}^{1/3}}{\mu }_s^{2/3}{E}_m^{1/3}{\Delta }^{5/3}{h}^{1/3}$, where ${\mu }_s$ is the substrate shear modulus, $h$ the membrane thickness, and $E_m$ its elastic modulus. We also find the effect of a finite membrane outer radius analytically, so that this method is also based entirely on analytical solutions. Comparison with experimental results seems very promising.