Pergamon zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDC J. Mech. Phys. Solids, Vol. 43, No. 8, 1221-1241, 1995 pp. Copyright 0 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0022-5096(!35)00020-8 0022-5096/95 $9.50+0.00 FUNDAMENTAL EIGENSTRAIN SOLUTIONS FOR AXISYMMETRIC CRACK PROBLEMS ALEXANDER M. KORSUNSKY Department of Materials Science and Metallurgy, University of Cambridge, Cambridge CB2 3QZ, U.K. (Received 30 December 1994; in revisedform 20 February 1995) ABSTRACT In this paper the fundamental eigenstrain solutions are derived for axisymmetric crack problems. The solutions are found in terms of Papkovich-Neuber potentials, whichin turn are expressed usingone function from the family of Lipschitz-Hankel integrals. In order to achieve the most concise form,two methods are used in the analysis: integration method for the axial opening eigenstrain ring and direct solution method for the radial opening eigenstrain ring and the ring of shear. The behaviour of the elastic stress fields in the vicinity of each type of eigenstrain ring is analysed. It is shown that the relevant component of stress exhibits a second order of singularity as the point of observatio approaches the eigenstrain ring. It is also demonstrated that the ring curvature a-’ serves as the measure of the deviation of the stress field from the appropriate plane strain solution. Implications of the results for the solution of crack problems are discussed. 1. INTRODUCTION Problems possessing torsionless axial symmetry often arise in the mechanics of fracture. Examples include penny-shaped and annulus-shaped cracks loaded in ten- sion in the axial direction (Clemens and Ang, 1988) and cylindrical cracks developing due to debonding at the fibre-matix interface (Fart-is et al., 1989). Another interesting range of problems relates to brittle fracture of glasses and ceramics from impact or contact pressure of spherical or almost spherical particles on their surface (Frank and Lawn, 1967;Korsunsky et al., 1995). This type of fracture, known as Hertzian cracking, originally develops from the patch of contact in the shapeof a shallow cylindrical crack. Upon further loading the crack tends to change the direction of its propagation away from the symmetry axis and assumes a stable shape of a frustum of a cone of vertex half angle of approximately 70” (Li Yingzhi and Hills, 1990). The problems included in the list above have been mostly addressed individually, i.e. a specific solution was sought in each case. It is obvious, however, that the strong property of axial symmetry puts all such problems into a clearly defined class of their own. In this paper the family offundamental solutions has been derived and analysed as it was needed to provide a unified formulation for axisymmetric crack problems in isotropic homogeneous media. Consider an axisymmetric crack in an infinitely extended elastic solid, subjected to 1221