Stress concentration in periodically rough Hertzian contact: Hertz to soft-flat-punch transition.

We report on the elastic contact between a spherical lens and a patterned substrate, composed of a hexagonal lattice of cylindrical pillars. The stress field and the size of the contact area are obtained by means of numerical methods: a superposition method of discrete pressure elements and an iterative bisection-like method. For small indentations, a transition from a Hertzian to a soft-flat-punch behaviour is observed when the surface fraction of the substrate that is covered by the pillars is increased. In particular, we present a master curve defined by two dimensionless parameters, which allows one to predict the stress at the centre of the contact region in terms of the surface fraction occupied by pillars. The transition between the limiting contact regimes, Hertzian and soft-flat-punch, is well described by a rational function. Additionally, a simple model to describe the Boussinesq-Cerruti-like contact between the lens and a single elastic pillar, which takes into account the pillar geometry and the elastic properties of the two bodies, is presented.