13th International Conference on Fracture June 16–21, 2013, Beijing, China -7- 3.2. TSSRB-concept for the prediction of crack propagation in fracture mechanical graded materials The new developed TSSRB-concept enables the prediction of the start and the direction of the crack propagation as well as the determination of unstable crack growth in fracture mechanical graded materials [10, 11, 12]. Fig. 8 shows a crack-afflicted structure which is loaded periodically resulting in a pure Mode I loading solution. Furthermore the structure consists of two materials whose fracture mechanical properties differ from each other. The presented gradation angle ϕM = 30° defines the position of the material transition in relation to the crack tip. The crack could kink by the kinking angle ϕ0,MTS which depends on the stress situation and can be determined by a crack propagation concept for homogeneous and isotropic materials (for example the MTS-concept of Erdogan and Sih). Another imaginable kinking angle is the gradation angle ϕM itself, due to the fact that the crack strives to take the way of least resistance. Figure 8. Fracture mechanical graded structure with gradation angle ϕM = 30° The occurrence of fatigue crack growth as well as the entrant kinking angle ϕTSSRB can be determined by the TSSRB-concept. This concept is based on the assumption that stable crack growth starts, when the cyclic tangential stress Δσϕ (Eq. 5) reaches a material limit value Δσϕ,th or rather when a cyclic comparative stress intensity factor ΔKV, determined by the means of the cyclic tangential stress Δσϕ, reaches the Threshold value curve ΔKth(ϕ). This material function depends on the coordinate ϕ and consists of the values ΔKth,material1 and ΔKth,material2 for the different regions, see Eq. (8). Similarly the material functions ΔKC(ϕ) and KC(ϕ) can be defined in dependency of the gradation angle ϕM. M M th,material 2 th M M th,material1 180 for ( ) 180 ϕ ϕ ϕ ϕ ϕ ϕ ϕ − °< < Δ Δ ≤ ≤ + ° Δ K K K (8) The determination of the start and the direction of fatigue crack growth is shown in Fig. 9a. The first intersection of the stress function Δσϕ√2πr with the material function ΔKth(ϕ) (Eq. 8) identifies the occurrence and the corresponding direction of crack growth, whereas the intersection of the stress function Δσϕ√2πr with the material function ΔKC(ϕ) defines the occurrence of final failure of
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