D. A. Hills 1 Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, U.K. e-mail: david.hills@eng.ox.ac.uk A. Sackfield School of Science and Technology, Nottingham Trent University Clifton campus, Nottingham NG11 8NS, U.K. R. J. H. Paynter Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, U.K. Simulation of Fretting Wear in Halfplane Geometries: Part 1—The Solution for Long Term Wear The final configuration of a cylindrical Hertzian contact, subject to oscillatory shear and undergoing wear, is studied. It is assumed that wear has proceeded for a long time, so that the final, modified contact is wholly adhered. It is shown that the extent of the final contact corresponds to that of the initial adhered region and the pressure distribution, and state of stress at the new contact edge are all derived, so that the environment in which cracks nucleate is well described. DOI: 10.1115/1.3118785 Keywords: wear, incomplete, Hertz, contact, evolution 1 Introduction Fretting fatigue analysis is concerned with the effect of differ- ential surface tangential displacement, when two bodies are pressed into contact, on accelerating the nucleation of cracks. These conditions, viz. the presence of interfacial contact pressure and slip, are precisely those needed also to cause wear. A question which is often posed, therefore, is “How does wear modify the surface profile of the bodies, and therefore the contact traction distribution?” Clearly the problem is coupled, because changes in the contact pressure distribution and slip zone size will, in turn, modify wear conditions and the conditions responsible for exac- erbating the fatigue problem. In order to solve this problem fully, it is clearly necessary to track out the amount of wear occurring per cycle of loading, and hence to determine the rate of evolution of the profile. This has been done 1,2, but assumptions have to be made about the applicability of, for example, the Archard wear law under conditions of partial slip: this is an environment to which the originally conceived wear law was never envisaged of having relevance, and questions about the role of third body par- ticles in “lubricating” the contact inevitably arise. A simpler prob- lem, and the one addressed here, is to look solely at the final state to which the configuration might be expected to evolve. No at- tempt is made to analyze the transient problem during which wear will occur this is considered in Part 2, but to examine, instead, the final regime when wear has removed material in regions of slip, so reducing the local contact pressure to zero there, and what remains is an adhered contact with adjacent slits where wear has occurred but now ceased. In this sense the calculation is a sequel to work conducted earlier, independently in Oxford 3 and by Goryacheva et al. 4, in which it was assumed that wear would reduce the contact pressure in slip regions to a spatially uniform level. If wear proceeds for long enough the inevitable final solu- tion is one in which the contact pressure in regions of relative slip becomes vanishingly small. The solution as a whole is most appropriate when the rate of wear is relatively high because it follows that, to be relevant in determining the overall life of the contact, it must have occurred early on, so that it is solely responsible for generating the envi- ronment in which the crack nucleates. As will be shown later in the paper, the solution then also provides a very compelling de- scription of the stress environment in which the crack nucleates, using, in a rigorous form, the asymptotic description initially for- mulated by Giannakopoulos et al. 5. The specific case of a Hertzian contact is analyzed in detail, but general arguments are presented, which allow the principles used to be applied to a wide range of incomplete contacts. 2 Formulation The problem to be solved is shown in Fig. 1a. A cylinder, of radius R, indents a halfplane under the influence of a constant normal load, P. If the two bodies are elastically similar and plane strain conditions exist, the initial halfwidth of contact, a, is given by a 2 = 2PAR  1 where A =41-  2  / E and where E and  are the Young’s modulus and Poisson’s ratio, respectively. The contact is then subject to a fully reversing shearing force, of amplitude Q, so that a classical Cattaneo–Mindlin partial slip problem arises, where the halfwidth of the central stick region is b, given by  b a  2 =1- Q fP 2 Wear occurs and, after some time, a new equilibrium problem emerges illustrated in Fig. 1b. Contact is now sustained over a central contact region -c , c, where c  b, and within this region, as there has never been slip, the original profile of the contact is maintained. External to this adhered contact region, slits are present where wear has reduced the profiles to give a glancing contact. The first step in the solution is to determine the interfacial contact pressure within the new contact region. To do this, imag- ine that the normal load is applied in two steps. It is increased gradually from zero, until the contact just spans the interval -c , c, and the corresponding load, P b , is given by Eq. 1, i.e., P b = c 2 2AR 3 with a corresponding normal traction distribution given by the usual Hertz recipe, i.e., 1 Corresponding author. Contributed by the Tribology Division of ASME for publication in the JOURNAL OF TRIBOLOGY. Manuscript received July 18, 2008; final manuscript received February 24, 2009; published online May 22, 2009. Assoc. Editor: Ilya I. Kudish. Journal of Tribology JULY 2009, Vol. 131 / 031401-1 Copyright © 2009 by ASME Downloaded 15 Jun 2009 to 163.1.14.184. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm