LAUSR

Abstract The classic version of the depth-sensing indentation techniques assumes the estimation of the elastic contact modulus of a material sample by measuring the slope (the contact stiffness) of the initial part of the unloading branch of the force-displacement curve. This approach assumes that the curve at loading reflects both elastic and plastic deformations of the material, while the unloading is taking place elastically. Therefore, neglecting the plastic deformations, one can assume that the structure of the material is the same at both branches and the assumptions of the Hertz-type contact theory are valid for the unloading branch. However, the contact problem for an elastic film attached to a substrate depends on the properties of the substrate. Hence, the film contact modulus is usually estimated by measuring the slopes of the initial unloading force-displacement curves obtained for different maximal values of indentation depth, and fitting the experimental points by various empirical analytical dimensionless functions of the ratio between the contact radius, a , and the layer thickness, t . Here, analytical analysis of contact problems for coated materials is performed. Both re-scaling and asymptotic techniques are employed. Asymptotic analysis of the contact at the small-scale indentation range (the ratio a / t is small) shows that the formula of the contact stiffness derived for an elastic half-space, has to be multiplied by the so-called indentation scaling factor that is a function of a / t . Thus, the asymptotic approach allows us to take into account analytically the substrate effect. The analytical fitting function obtained agrees with both some known semi-empirical functional forms and the published experimental results on depth-sensing nanoindentation of thin metallic layers, while the function is in a disagreement with results obtained for inhomogeneous films of brittle materials such as coals. It is argued that the disagreement is caused by structural transformations (crushing) of the coals during loading.