We investigate the normal contact stiffness in a contact of a rough sphere with an elastic half-space using 3D boundary element calculations. For small normal forces, it is found that the stiffness behaves according to the law of Pohrt/Popov for nominally flat self-affine surfaces, while for higher normal forces, there is a transition to Hertzian behavior. A new analytical model is derived describing the contact behavior at any force. 1. Introduction Since Bowden and Tabor , it has been known that surface roughness plays a decisive role in contact, adhesion, friction, and wear. The main understanding of the contact mechanics of nominally flat rough surfaces was achieved in the middle of the 20th century due to works by Archard  and Greenwood and Williamson . In the last years, contact mechanics of rough surfaces has once again become a hot topic [4–6]. Most of the previous work was devoted to investigation of nominally flat surfaces. For many tribological applications, however, the contact properties of rough bodies with macroscopically curved surfaces are of great interest. A first analysis of the contact problem including a curved but rough surface was given by Greenwood and Tripp . They applied the Greenwood/Williamson (GW) model  of independent asperities with a Gaussian distribution to a parabolic shape. In this model, the roughness can be seen as an additional compressible layer. They calculated the mean pressures as a function of the radius and for low loads found a reduction in the maximum pressure and an enlargement of the apparent area of contact. For high loads, the indentation behavior found was Hertzian. In the present paper, we will investigate the indentation of a rough sphere into an elastic half-space without the restrictions stemming from the GW model. We calculate the incremental normal stiffness of the contact, which not only determines the dynamic properties of a tribological system, but also its electrical and thermal conductivity [8, 9]. In the interest of purity of results, we assume the bodies to be elastic at all scales and confine ourselves to self-affine roughness without cutoff. Contact stiffness of such surfaces has been recently studied in detail numerically and analytically [10, 11]. We show that there is a pronounced crossover from the behavior which is typical for fractal surfaces [12, 13] to Hertz-like behavior , similar to GT . Furthermore we derive an analytical approximation for the entire range of forces. 2. Methods We consider a rigid rough spherical indenter with the radius , which is
R. Pohrt, V. L. Popov, and A. E. Filippov, “Normal contact stiffness of elastic solids with fractal rough surfaces for one- and three-dimensional systems,” Physical Review E, vol. 86, no. 2, Article ID 026710, 7 pages, 2012.
R. Pohrt and V. L. Popov, “Investigation of the dry normal contact between fractal rough surfaces using the reduction method, comparison to 3D simulations,” Physical Mesomechanics, vol. 15, no. 5-6, pp. 275–279, 2012.
I. A. Polonsky and L. M. Keer, “A numerical method for solving rough contact problems based on the multi-level multi-summation and conjugate gradient techniques,” Wear, vol. 231, no. 2, pp. 206–219, 1999.
N. Sneddon, “The relation between load and penetration in the axisymmetric boussinesq problem for a punch of arbitrary profile,” International Journal of Engineering Science, vol. 3, no. 1, pp. 47–57, 1965.