Indentation of a layer on an elastic substrate: influence of thickness and elastic modulus of the coating

The indentation experiment consists in pressing a probe into the surface of a material. The study of the force response of the material depending on the displacement of the indenter (depth of indentation) allows the evaluation of surface mechanical properties. In the case of indentation of inhomogeneous (coated) hyperelastic materials, questions arise about the contact area required for calculating mechanical properties, as well as about the applicability of contact models developed for homogeneous linear-elastic materials. The shape of the indenter in the vicinity of the tip begins to play a role when indenting to a shallow depth and/or studying thin films (the real geometry of the tip). In this work, the finite-element method was used to analyze the indentation of a layer on a hyperelastic substrate by an indenter in the form of a truncated cone or a paraboloid of revolution. In this work, the finite element method was used to analyze the pressing of an indenter in the form of a truncated cone or a revolution paraboloid into the system of layer on a hyperelastic substrate. The size of the tip radius of the indenter, the elastic modulus of the layer and its thickness were varied. The dependences of the force response on the depth of indentation, the size of the contact area and the depth of penetration were obtained. It is shown that, when indenting the layer on a substrate, the contact depth differs significantly from that of a homogeneous material and is a function of the depth of indentation and the parameters of both the indenter and the surface of the system. Analytical approximations of the contact depth for the truncated conical indenter were obtained. The applicability of Sneddon’s equation (indentation of a homogeneous linear-elastic material) for the layer on a hyperelastic substrate was considered. The limitations on its use imposed indenter and material parameters were determined. It is shown that, if the indentation depth is of the order of the tip radius, simple expressions describing the indentation by a cylinder or a paraboloid without taking into account the real contact area can be used to estimate elastic modulus. Recommendations are given on the practical use of the obtained results, in particular, in the treatment of atomic force microscopy data.