F.-K.Chang Maria Comninou Mem. ASME Sheri Sheppard Student Mem. ASME J. R. Barber Department of Mechanical Engineering and Applied Mechanics, University of Michigan, Ann Arbor, Mich. 48109 The Subsurface Crack Under Conditions of Slip and Stick Caused by a Surface Normal Force A solution is given for the elastic stress field in a halfplane containing a plane crack parallel to the surface and subjected to a uniform normal pressure and a con- centrated normal load. Frictional slip according to Coulomb's law is permitted between the crack faces. As the load is increased, a slip zone originates and grows either from a crack tip or from an intermediate point. Various arrangements of slip and stick zones can occur depending on the magnitude of the load and its location relative to the crack. At very high loads, the crack faces start to separate, but this case is not treated in the present paper. Introduction The phenomenon of surface slip due to static and moving surface loads has been studied in a series of publications [1-5], which considered the problem of an elastic layer pressed on an elastic substrate. This geometry approximates the contact problem for cylinders with shrink fitted tires. In particular, the results of [5] can be compared with ex- perimental data obtained by Anscombe and Johnson [6] on the rolling of two steel cylinders, one of which is fitted with a steel tire. A related problem of interest concerns the propagation of a crack parallel to the surface of a solid due to a series of moving loads. Railway wheels, rails, and other surfaces loaded in rolling contact are prone to "spalling failure" in which a subsurface crack propagates parallel to the surface until eventually a thin plate of material is detached, Hundy [7]. The propagation process depends on the stress intensity factors at the crack tips and these will be in turn influenced by frictional contact between the crack faces during the loading cycle as a moving load passes overhead. In this paper we consider the static situation as a preliminary study to the inherently transient problem of the moving load. When the load is sufficiently high, we find that slip occurs between the crack faces in either one or two zones whose location depends on the position of the load and the extent of the crack. If the slip zones extend to one or both of Contributed by the Applied Mechanics Division for presentation at the 1984, PVP Conference and Exhibition, Joint with Applied Mechanics Division and Materials Division, San Antonio, Texas, June 17-21, 1984 of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS. Discussion on this paper should be addressed to the Editorial Department, ASME, United Engineering Center, 345 East 47th, Street New York, N.Y. 10017, and will be accepted until two months after final publication of the paper itself in the JOURNAL OF APPLIED MECHANICS. Manuscript received by ASME Applied Mechanics Division, April, 1983; final revision August, 1983. Paper No. 84-APM-17. Copies will be available until February, 1985. Fig. 1 Geometry of the problem the crack tips, singular shear tractions are developed ahead of the tip giving a mode II stress intensity factor. Otherwise, the stress field near the tips remains bounded and the frictional contact between the faces inhibits crack propagation. Un- fortunately, there are as yet no experimental data with which these results can be compared. Formulation Consider the geometry of Fig. 1. A crack of length L 2 - L x is located at depth a parallel to the surface of an elastic half plane. A compressive force P and a uniform pressure p 0 are applied on the surface of the half plane. In this analysis, the coordinate system is defined so that the force P acts at the point (0, a), while the location and extent of the crack are considered variable, i.e., the parameters L { , L 2 may take any (including negative) values. The force is allowed to increase monotonically in magnitude. The faces of the crack can transmit frictional forces and Coulomb's law of friction is assumed. As long as the crack faces remain in conditions of stick, the Journal of Applied Mechanics JUNE 1984, Vol. 51/311 Copyright © 1984 by ASME Downloaded 04 Sep 2009 to 141.212.141.77. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm