ht. J. Solrds Srrurrures Vol. 35, Nos 31 32, pp. 4077 40X9, 1998 tr) 1998 Elsemer Science Ltd All nghts reserved. Printed in Great Britain Pergamon PI1 : SOO20-76fl3(97)00302-8 0020-7683/98 519 00 + .OO CLOSURE OF A THROUGH CRACK IN A PLATE UNDER BENDING J. P. DEMPSEY,* I. I. SHEKHTMAN Department of Civil and Environmental Engineering, Clarkson University, Potsdam, NY 13699-5710, U.S.A. and L. I. SLEPYAN Department of Solid Mechanics, Materials and Structures, Tel Aviv University, Ramat Aviv 69978, Tel Aviv, Israel (Received 18 April 1997 ; accepted 15 June 1997) Abstract-A plate with a pre-existent through crack is considered under the action of a remote bending moment and a remote in-plane force. The problem statement is reduced to the solution of two coupled integral equations with strongly singular kernels. The independent variables in the latter equations are the closure displacement and rotation angle. The corresponding closure force and moment distributions, and the contact-crack opening boundary (the closure perimeter), are found as functions of the remote bending-compression ratio. The validity of previously stated analytical asymptotics for the contact boundary is examined. The dependence of the extent of closure on the crack length-to-thickness ratio is studied. Comparisons are made with experimental results. 0 1998 Elsevier Science Ltd. All rights reserved. INTRODUCTION Difficulties caused by the closure of a through crack in a thin plate under bending have long been recognized (Smith, 1969 ; Wynn and Smith, 1969 ; Smith and Smith, 1970 ; Jones and Swedlow, 1975 ; Heming, 1980 ; Alwar and Ramachandran Nambissan, 1983). The mechanics of crack closure has received an increased scrutiny in recent years (Joseph and Erdogan, 1989; Young and Sun, 1992, 1993; Cordes and Joseph, 1995 ; Kuo et al., 1995; Dempsey et al., 1995a ; Slepyan et al., 1995). In particular, Joseph and Erdogan (1989) provided a plot of the closure width distribution for the case of a through-cracked plate subjected to pure bending. Slepyan et al. (1995) provided an analytical asymptotic solution to the latter problem valid for long cracks. The material presented therein can be considered to pose an inverse surface crack treatment to that given much earlier by Rice and Levy (1972). The present paper examines the dependencies of the shape and extent of closure on the remote loading and crack length to plate thickness ratio. The formulation provided by Slepyan et al. (1995) is adapted and extended. The title problem reduces to the solution of two hypersingular integral equations that yield the averaged crack opening displacement and crack face rotation. The latter quantities are coupled through the smooth closure condition along the crack front. Given that the shape of the closure region is unknown at the outset, the solution procedure is necessarily iterative. At each iteration, the solution of the integral equations follows the procedure standardized by Kaya and Erdogan (1987). Consider a cracked infinite elastic plate - a3 < X < + co, -cc < Y < + co (Fig. 1) of uniform thickness 2h, 121 < h. The through-the-thickness crack of length 21lies at Y = 0, (XI < 1, (21 < h. The plate is subjected to the action of a self-equilibrated system of the external forces. All the external loads are sufficiently remote with respect to the crack area for their action can be explicitly described by the distribution of the initial extensional inplane force So(X) and bending moment M’(X) acting in the intact plate at Y = 0, 1x1 < I. *Author to whom correspondence should be addressed. Tel. : 00 315 268 6517. Fax : 00 315 268 7985. E-mail : john@sun.soe.clarkson.edu. 4077