1 Copyright © 2006 by ASME Proceedings of ASME-IJTC06 International Joint Tribology Conference October 23-25, 2006, San Antonio, Texas, USA IJTC2006-12281 ELASTIC-PLASTIC SPHERICAL CONTACT UNDER COMBINED NORMAL AND TANGENTIAL LOADING Victor Brizmer, Yuri Kligerman, Izhak Etsion Dept. of Mechanical Engineering Technion - Israel Institute of Technology Haifa, 32000, Israel brizmer@technion.ac.il ABSTRACT The behavior of an elastic-plastic contact of a deformable sphere and a rigid flat under combined normal and tangential loading with full stick contact condition is investigated theoretically. This allows static friction modeling under highly adhesive conditions. Combined loading begins with a normal preload that produces an initial contact area and causes elastic or elastic-plastic deformations in the contact zone. The full stick contact condition leads to a junction between the contacting bodies which can bear tangential loading. On the second stage of the loading process, the tangential load is being applied gradually, while the normal load remains constant. The maximum tangential load that can be supported by the junction prior to sliding inception is determined as the static friction. The effect of normal load on this static friction is investigated. The evolution of the contact area during the tangential loading revealed an essential "junction growth" mainly just before sliding inception. INTRODUCTION The elastic-plastic spherical contact under combined normal and tangential loading is a classical problem in contact mechanics (see Johnson [1]) with a wide range of scientific and technological aspects. It is applicable, for instance, in modeling of friction and wear between rough surfaces and in particle flow simulations. The treatment of combined normal and tangential loading of elastic spherical contact stems from the classical work of Mindlin [2]. It was shown that the contact area between two spheres under combined loading consists of a central stick region surrounded by an annular slip zone. As the tangential load increases, the central stick region gradually diminishes and finally disappears. A local Coulomb law with pre-defined friction coefficient was used to determine an upper limit for the surface shear stresses. Hamilton [3] derived explicit equations for the stress field beneath a sliding, normally loaded Hertzian contact. Tabor [4] introduced an improved model of static friction related to material yielding in the contact zone. It was assumed that a local slip starts in the contact area provided the material reaches plastic yield at this point, according to the von Mises criterion. It was found that the contact area substantially increases during the tangential loading (junction growth). Chang et al. [5] calculated the maximum tangential load that a sphere, normally loaded by a rigid flat, can bear before sliding inception. The sliding onset between the sphere and flat was interpreted as the first yield at a single material point of the sphere which greatly underestimates the maximum tangential load. Kogut and Etsion [6] presented a semi analytical approximate solution for sliding inception between a sphere in contact with a rigid flat under combined loading. The present work suggests a completely different approach for solving the problem of sliding inception of a spherical contact applying the full stick contact condition. THE SPHERICAL CONTACT MODEL Figure 1 presents a deformable sphere of a radius R in contact with a rigid flat under combined normal and tangential loading. The bold and thin dashed lines show the contours of the sphere before and after applying the normal load, P, respectively, while the solid lines show the final contours of the contacting bodies after applying the tangential load, Q. The normal load produces an initial normal interference, w 0 , and an initial circular contact area of a diameter d 0 . The succeeding application of a tangential load causes an increase of these initial interference and initial contact area. The final interference and diameter of the contact area are denoted as w and d, respectively. The contact between the sphere and flat is assumed to be under full stick condition (where further relative displacement of points engaged in contact is prevented) throughout the whole process of combined loading. The full stick assumption is realistic when considering the junction formed in the interface of contacting bodies (see Tabor [4]). The boundary conditions consist of zero displacement in the x, y, and z directions at the bottom of the sphere and in the y-direction at the plane of symmetry ( xz-plane). The surface of the sphere is free