13th International Conference on Fracture June 16–21, 2013, Beijing, China -1- Three-parameter approaches for three-dimensional crack-tip stress fields Chongmin She1,*, Junhua Zhao2, Wanlin Guo1,* 1 State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, P. R. China 2 Institute of Structural Mechanics, Bauhaus-University Weimar, 99423 Weimar, Germany * Corresponding authors: cmshe@nuaa.edu.cn, wlguo@nuaa.edu.cn Abstract Two three-parameter descriptions for the three-dimensional (3D) crack-tip stress fields have been introduced. The three-parameter solution K-T-Tz is developed to describe the linear elastic crack-tip stress state, and the J-QT-Tz is to elastic-plastic crack-tip field. The conventional two-dimensional solutions such as K, K-T, HRR and the extended J-Q description which considers the in-plane constraint modification can hardly provide satisfied description for the three-dimensional crack front fields, especially for the out-of-plane stress near the crack front. It is shown that a consideration of the out-of-plane constraint and use of the three-parameter description is necessary and efficient to predict the 3D stress fields near the crack front. Keywords Fracture mechanics, Three-dimensional stress field, Out-of-plane stress constraint 1. Introduction The complicated three-dimensional (3D) stress fields near the crack front play a vital role in the strength of materials [1], and control the initiation and propagation of cracks [2]. The character of the stress fields near the crack front has long been extensively studied. The classical linear elastic and elastic–plastic fracture mechanics are based on the theory stemming from the one singular term of asymptotic expression and its amplitude the stress intensity factor (SIF, K) [3] and HRR solution [4, 5], respectively. Then more accurate two-parameter approaches, such as K–T [6], J–T [7], J–Q [8, 9] and J–A2 [10, 11], have been developed to describe the crack-tip field. These approaches have been applied successfully in engineering designs though they are limited to describe the effect of the in-plane constraint on the crack-tip field and fracture toughness. In fact, fracture toughness depends on the 3D out-of-plane stress level near the crack front also [12]. It is well known that fracture toughness depends highly on the thickness of the test specimen until a threshold thickness, beyond which the toughness does not decrease further. The toughness at this thickness is called plane strain fracture toughness. It is less than the fracture toughness of thinner plates and is a material property. So the variable fracture toughness is inconvenient in the engineering applications if the 3D out-of-plane stress level is not considered accurately. In order to describe the out-of-plane stress level, the out-of-plane stress constraint factor Tz was introduced by Guo [13-15], the factor is defined as 33 11 22 zT , (1, 2, 3)=(x, y, z) or (r, , z) (1) where r, , x and y are coordinates in the conventional polar and Cartesian systems with origin at the crack tip and z is the third coordinate (parallel to the crack front) in both systems. The corresponding coordinate system and a normal sheet element of a through-straight crack are shown in Fig.1. In the state of plane stress, Tz=0. In the state of plane strain, Tz changes from the Poisson’s ratio v of the linear elastic material to 0.5 for elastic-perfectly plastic material.
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