Wear 268 (2010) 900–904 Contents lists available at ScienceDirect Wear journal homepage: www.elsevier.com/locate/wear The effect of wear on nucleation of cracks at the edge of an almost complete contact David A. Hills, Robert J.H. Paynter ∗ , David Nowell Department of Engineering Science, University of Oxford, Parks Road, Oxford, OX1 3PJ, United Kingdom article info Article history: Received 17 September 2008 Received in revised form 25 November 2009 Accepted 7 December 2009 Available online 16 December 2009 Keywords: Fretting wear Asymptotic analysis abstract A contact with a flat region and a rounded corner will have adhered and slipping regions and a locally non-singular contact traction distribution. When wear occurs in the slipping region, the geometry and thus the pressure distributions change. An analytical asymptotic analysis of the edge of this “almost complete” contact shows that wear will leave a complete contact with the same extent as the original adhered region and a singular traction distribution. © 2009 Elsevier B.V. All rights reserved. 1. Introduction Most studies of wear consider situations where the whole of the contact is sliding and a change of contact geometry ensues. Equally, most studies of fretting consider mixed adhered and slip- ping regions but do not address changes in geometry. However, there have been some studies that simulate evolution of the contact geometry during fretting where the mechanical analysis is car- ried out by the Finite Element Method [1,2] and other numerical schemes [3]. This paper follows an analytical asymptotic analysis where the focus of the analysis is on the edge of the contact, since for many components contacts are ‘almost complete” and the contact trac- tions are dominated by a stress concentration, but with attendant slip located at the edge of the contact. If truly complete, the nature of the singularity will have similarities with those of cracks and notches, and analysis can then follow into prediction of crack nucle- ation and growth [4]. Real components often have a small rounded region introduced either intentionally, to reduce the singularity in the pressure distribution, or it may occur due to plastic defor- mation of the corner early in the life of the component; in most cases these provide only a local modification of the overall traction distributions which are still dominated by the singularity. The problem we wish to address, here, is the contact-edge condi- tion for an almost complete contact; the analysis is thus asymptotic in the space dimension. The contact is subject to a constant normal ∗ Corresponding author. Tel.: +44 1865 283489; fax: +44 1865 273906. E-mail address: robert.paynter@eng.ox.ac.uk (R.J.H. Paynter). force and an oscillatory shear force, where the material is prone to wear. The solution is to be developed using uncoupled half-plane theory, so that it applies, strictly, only to the case when the indenter itself, defining the size of the contact, is rigid, and the contacting body is incompressible. However, although this would appear to be a major restriction being imposed at the outset, it does mean that the solution can be obtained in closed form, and it provides, also, a valuable calibration for any subsequent numerical treatment of the corresponding problem when both bodies have the same elastic constants. In what follows it is assumed that the normal force is applied first and maintained at a constant level. A cyclic shear force is then imposed and is assumed to be fully-reversing, so that no frictional shakedown occurs. Attention is then concen- trated on the state of stress and partial slip existing at the moments when the shearing force reaches its maximum value, and the shear- ing tractions, extent of slip and slip displacement are all at their maxima. The asymptotic form to be developed can be collocated into the edge of any notionally complete contact, enabling both the effects of slight rounding, and subsequent wear, to be taken into account. An example problem is shown schematically in Fig. 1, which shows a square-edged, flat-faced punch, of half-width a, and where we wish to understand the influence of (a) the presence of slight edge radii on the contact pressure and (b) how this might, in turn, be modified because of the effects of wear. For this example problem, the contact pressure, p p (x), and the shearing traction, q p (x), are related to the normal force, P, and the shear force, Q, by the relations [5] p p (x) P = q p (x) Q = 1   a 2 - x 2 . (1) 0043-1648/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.wear.2009.12.015