Indentation of a stiff membrane on an incompressible elastic halfspace

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 analyze 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 Δ. The solution is corrected only in the prefactor by extensive FEM investigation using a concentrated load resulting finally in F=1.45×4π384π1/3μs2/3Em1/3Δ5/3h1/3, where μs is the substrate shear modulus, h the membrane thickness, and Em 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.