ORIGINAL PAPER A Macro-scale Approximation for the Running-in Period I. I. Argatov • Yu. A. Fadin Received: 29 November 2010 / Accepted: 8 March 2011 / Published online: 22 March 2011 Ó Springer Science+Business Media, LLC 2011 Abstract The article presents asymptotic modeling of the running-in wear process with fixed contact zone under a prescribed constant normal load or an imposed contact displacement. The wear contact problem is formulated within the framework of the two-dimensional theory of elasticity in conjunction with Archard’s law of wear. The running-in process is considered at the macro-scale level, while the micro-processes associated with roughness changes, tribomaterial evolution, and microstructural alteration in the subsurface layers as a first approximation are neglected. The setting of the steady-state regime for the macro-contact pressure evolution is chosen as the criterion to characterize the completion of running-in. Simple closed-form approximations are derived for the running-in period and running-in sliding distance. The obtained results can be used for estimating the running-in period in wear processes where the evolution of the macro-shape devia- tions at the contact interface plays a dominant role. Keywords Dynamic modeling  Contact mechanics  Wear mechanisms Nomenclature a Half-width of the contact zone C Asymptotic constant depending on the ratio H/a c 2r , c 2r?1 Integration constants c m , C m Integration constants d 0 Asymptotic constant E Young’s elastic modulus H Thickness of the elastic layer k Dimensional wear coefficient in Archard’s wear law L in Running-in sliding distance P Line normal load in 2D contact problem p(x, t) Contact pressure q(x, t) Residual contact pressure t Time variable T c Characteristic time of the tribological system T in Running-in time period v Sliding speed of the punch x Transverse coordinate in 2D contact problem x 0 Dimensionless transverse coordinate w Linear wear a 2r Eigenvalues of integral equation (4) b Auxiliary parameter, b = kv/# d 0 (t) Variable vertical contact displacement of the punch d 0 Constant vertical contact displacement of the punch DðxÞ Macro-shape function of the punch # Elastic constant, # = 2(1 - m 2 )/(pE) j = 3 - 4m Kolosov’s constant for plain strain m Poisson’s ratio k m Eigenvalues of integral equation (12) n Coordinate integration variable n 0 Dimensionless coordinate integration variable s Time integration variable I. I. Argatov (&) Institute of Mathematics and Physics, Aberystwyth University, Ceredigion SY23 3BZ, Wales, UK e-mail: ivan.argatov@gmail.com Yu. A. Fadin Laboratory of Friction and Wear, Institute for Problems in Mechanical Engineering, V.O., Bolshoy pr., 61, 199178 St. Petersburg, Russia 123 Tribol Lett (2011) 42:311–317 DOI 10.1007/s11249-011-9775-9