International Journal of Fracture 115: 193–204, 2002. © 2002 Kluwer Academic Publishers. Printed in the Netherlands. Dominance of asymptotic crack tip fields in elastic functionally graded materials G. ANLAS 1 , J. LAMBROS 2,∗ and M.H. SANTARE 3 1 Department of Mechanical Engineering, Bogazici University, 80815 Bebek, Istanbul, Turkey 2 Department of Aeronautical and Astronautical Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, U.S.A. 3 Department of Mechanical Engineering, University of Delaware, Newark, DE 19716, U.S.A. Received 12 March 2001; accepted in revised form 8 April 2002 Abstract. The stress field surrounding an edge crack in an elastic functionally graded plate is calculated using two dimensional finite element analysis. The property gradient direction is parallel to the crack line and loading is constrained to be symmetric such that a pure mode I situation is achieved. The extent of dominance of asymptotic fields is evaluated by comparing the stress field calculated from the finite element analysis to that calculated by asymptotic equations. Two separate forms of the asymptotic stress fields, one for homogeneous materials and another for continuously nonhomogeneous materials are used. The shape and extent of the dominance regions of each asymptotic field and their dependence on crack length and material nonhomogeneity is also presented. Under the pure mode I conditions considered here, it is seen that both asymptotic fields exist around the crack tip with the one for homogeneous materials in general being embedded in the one for continuously nonhomogeneous materials. The ligament length is seen to primarily control the extent of development of the asymptotic stress field for nonhomogeneous materials. The steepness of the material gradient affects the relationship between the two asymptotic stress fields and therefore the extent of their dominance. 1. Introduction The asymptotic stress field ahead of a crack tip in an isotropic homogeneous linearly elastic material is well known and extensively used by researchers in the area of fracture mechan- ics. In a cracked body, the actual stress will in general deviate from the asymptotic values. However, the asymptotic solution may still be of use if a region can be identified where the difference between actual and asymptotic stresses can be neglected. Such a region is called a K -dominant region. Clearly, for purposes of experimental measurement, it is of interest to identify the extent of such regions in the body. K -dominance has been studied in isotropic and anisotropic homogeneous materials (Krishnaswamy et al., 1991; Prabhu and Lambros, 2000) and at discrete material interfaces (Lee and Rosakis, 1993). Recently, the study of functionally graded materials has brought new challenges and issues to the area of fracture mechanics. One such issue is the determination of the size and the shape of the K -dominant zone when material nonhomogeneity is present. Several researchers have investigated the nature of the asymptotic stress field surrounding the crack tip in functionally graded materials. Delale and Erdogan (1983) studied the crack problem for a nonhomogeneous plane where the nonhomogeneity follows an exponential relationship in the direction of the crack. They showed that the stresses around the crack tip have the conventional square root singularity. Eischen (1987) determined the leading terms ∗ Author for correspondence. E-mail: Lambros@uiuc.edu