13th International Conference on Fracture June 16–21, 2013, Beijing, China -9- 2 sin cos 2 3 ΔK 2 cos Δ 2 K Δσ π II 3 I ϕ ϕ ϕ ϕ ⋅ − = r . (11) The structure in Fig. 10a and Fig. 10b possess different gradation angles ϕM. This difference in the gradation angle ϕM leads to a different kinking angle ϕTSSRB due to the loading condition and the material gradation. In Fig. 10a the gradation angle ϕM is 30°. The point of contact is at this coordinate leading to the kinking angle ϕTSSRB = ϕM = 30°. In Fig. 10b the structure possesses the gradation angle ϕM = 60°. Here the point of contact can be found at the coordinate with the largest stress. Hence the kinking angle ϕ0,MTS = -30° due to the loading condition, according to the MTS-concept of Erdogan and Sih, defines the kinking angle ϕTSSRB in this case. a) b) Figure 10. Occurrence and direction of fatigue crack growth for Mixed Mode loading in a fracture mechanical graded material: a) gradation angle ϕM = 30°, b) gradation angle ϕM = 60°. In graded materials the crack propagation direction depends on the Mixed Mode loading situation and the material gradation. Which crack growth occurs can be determined with the TSSRB-concept. Furthermore the concept enables the determination of the starting of fatigue crack growth and of unstable crack growth. 4. Conclusion Using the presented TSSRB-concept crack propagation as well as the direction of crack growth can be determined in fracture mechanical graded materials. By comparing the stress function Δσϕ√2πr with the material functions ΔKth(ϕ), ΔKC(ϕ) and KC(ϕ) the occurrence, the direction as well as the start of unstable fatigue crack growth and the start of unstable failure due to static loading can be determined. Furthermore this concept can also be modified for the application for an elastic material gradation as well as for the combination of a fracture mechanical and elastic material gradation. Acknowledgements This contribution is based on investigations of the collaborative research centre SFB/TR TRR30, which is kindly supported by the German research funding organisation (DFG).
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