Effects of adhesive and frictional contacts on the nanoindentation of two-dimensional material drumheads

Abstract

Nanoindentation of suspended circular thin films, dubbed drumhead nanoindentation, is a widely adopted technique for characterizing the mechanical properties of micro- or nano-membranes, including atomically thin two-dimensional (2D) materials. This method involves suspending an ultrathin specimen over a circular microhole and applying a precise indenting force at the center using an atomic force microscope (AFM) probe. Classical solutions assuming a point load and a fixed edge, which are referred to as Schwerin-type solutions, are commonly used to estimate Young’s modulus of the membrane material out of load–deflection measurements. However, given the widespread experimental evidence for adhesive and frictional contacts between the probe tip and the membrane, as well as sliding between the membrane and its supporting substrate, quantitative investigations of the effects of these interactions are required. In this paper, we formulate a boundary value problem to rigorously model such effects, ensuring relevance to experimental operations. Our numerical analyses reveal that the adhesive effect at the tip-membrane interface diminishes as the indentation depth increases or the tip size decreases. Furthermore, frictional interactions at this interface shift the maximum membrane stress from the center to the tip-membrane contact line with increasing indentation depth and interfacial shear stress. At large indentation depths, the size of the indenter tip and the sliding of the membrane-substrate are found to have a large effect on the indentation load–deflection relationship. Thus, we propose a new approximate formula for this relationship assuming a non-adhesive and frictionless spherical tip of a finite radius and a slippery contact with the supporting substrate. This formula is more accurate than the widely used Schwerin-type solution. It can be used to simultaneously extract the in-plane stiffness of the membrane and the shear strength at the membrane-substrate interface.