ZAMM  Z. Angew. Math. Mech. 81 (2001) 9, 642 –– 647 Vrbik,J.; Singh,B.M.; Rokne,J.; Dhaliwal,R.S. PlasticDeformationattheTipofanEdgeCrack A closed form solution is given for the problem of plastic deformation at the tip of an edge crack forming at the free boundary of a half-space. Using a hypothesis by Dugdale the length of the plastic zone is obtained. The results are further extended for a layer with an edge crack. MSC (2000): 74C20, 74R05 1. Introduction The Dugdale model of a crack in a ductile material was introduced to investigate the inelastic zone at the ends of a stationary slit in steel sheets under static tension. The Dugdale model predictions agree closely with experimental results [1]. Uniform internal stresses equal to the material yield stress are prescribed in the inelastic region so that the stress singularities at the crack tip do not appear. Depending on the size of the plastic zone brittle, quasi-brittle and ductile fractures are differentiated. Studies of plastic deformations around cracks appear to be of fundamental impor- tance for describing the failure from the macroscopic point of view and also for constituting criteria of fracture. References to work on including plastic deformation around the cracks and the governing fracture criteria are found in the books by Parton and Morozov [2], Lardner [3] and in the article by Vitvitskii, Panasyuk, and S. Ya Yarema [4]. Recently some of the authors of the papers [5 – 12] have contributed significantly to penny and line crack prob- lems in elastic solids by using the Dugdale hypothesis. It is also important to mention the work by Olesiak and Shadley [13], Olesiak and Wnuk [14], and Tsai [15] for a penny-shaped crack, by Wang and Shen [16], by Herr- mann and Wang [17] for thermal loading, and by Smith [18] for the stability problem. A number of solutions for notches and cracks under anti-plane mode or by longitudinal shear in the plastic mode at the tip of the crack are obtained as for example in Smith [19], Hult and McClintock [20], Koskinen [21], Field [22], Rice [23], Bilby, Cottrell,and Swinden [24], and Atkinson and Howard [25]. In reviewing the above references it is clear that the analytical solutions developed in this paper are new and that the integral transform method used is more powerful than the methods used in the cited literature. In this paper the crack formed in mode III of fracture mechanics in the two dimensional theory of elasticity is considered. In Section 2 we consider a half-space under anti-plane strain conditions with ductile crack at its boundary as showninFig.1. In Section 3, we consider the layer under anti-plane strain with a ductile crack as shown in Fig. 2. In each case the solution of the problem is reduced to dual integral equations and the length of the plastic zone is thus obtained. 2. Formulation and solution of problem I We consider the distribution of stress in the interior of a two-dimensional semi-infinite elastic medium, when a shallow groove or cut is spread on the boundary of the half-space x> 0 and a crack occupying the segment 0 < x < l; y ¼ 0 along the x axis and is opened by internal shear stress acting along the length of the crack. The plastic zone is de- scribed by the segment of the line l<x<a þ l; y ¼ 0: Because of the assumed nature of the plastic zone, the shear 642 ZAMM  Z. Angew. Math. Mech. 81 (2001) 9 Fig. 1. Plastic deformation at the tip of an edge crack in half-space