13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- Fig. 1 The coordinate system and a normal sheet element of a through-straight crack. The out-of-plane constraint factor Tz plays an important role in the determination of fracture toughness of a structural element. The effect of Tz on 3D crack-front fields and fracture toughness were systematically studied by Guo, then the 3D two-parameter principles of K–Tz, J–Tz have been proposed [12-16]. Combining with the in-plane constraint T or Q, the 3D three-parameter principles of K-T-Tz and J-QT-Tz have also been proposed [13-15, 17-22]. In this paper, the recent researches on the 3D three-parameter principles of K-T-Tz and J-QT-Tz are emphasisly summarized. The comparisons of the three-parameter principles with two-parameter and one-parameter principles are presented. 2. Three-parameter principle K–T–Tz for the linear elastic material Based on the SIF K, the more accurate two-parameter approach K–T was proposed by Williams [6] for linear elastic material. For the 3D crack case, the three-parameter principle K-T-Tz was developed by Guo [13-15, 17-19] based on the K-T approach, which can be expressed as 1 1 2 ( ) ij ij i j zz z ii jj K f T r T (i, j)= (x, y), (r, θ) or (1, 2) (2) where T is the T-stress, 11 3 cos 1 sin sin 2 2 2 f (3) 22 3 cos 1 sin sin 2 2 2 f (4) 12 3 cos sin cos 2 2 2 (5) The three-parameter principle K-T-Tz can describe not only the in-plane constraint by T but also the out-of-plane stress constraint by Tz. Then the crack-tip stress fields described by the principle K-T-Tz for the typical cracks such as through-the-thickness crack, quarter elliptical corner crack, semi-elliptical surface crack and embedded elliptical crack are discussed. The geometry of a plate with a quarter elliptical crack under uniform tension is presented in Fig.2, the corresponding FE mesh is shown in Fig.3. The 3D singular elements with four mid-side nodes at the quarter points are used around the crack front to simulate the inverse square root singularity at the crack tip. The FE model in Fig.3 can be used to simulate the semi-elliptical surface crack and embedded elliptical
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