13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- 2.1.1. Concept according to Erdogan and Sih The concept of the maximum tangential stress (MTS) by Erdogan and Sih [4, 5] enables the determination of the crack growth direction as well as the start of unstable crack propagation on the basis of the tangential stress σϕ, presented in Eq. (1). 2 sin cos 2 3 2π K 2 cos 2π K σ II 3 I ϕ ϕ ϕ ϕ ⋅ − = r r (1) The concept assumes that the crack propagates in the direction ϕ0,MTS which is perpendicular to the maximum tangential stress σϕmax. Eq. (2) defines the kinking angle ϕ0,MTS according to the MTS-concept which depends on the loading condition. ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟⎟ + + ⎠ ⎞ ⎜⎜ ⎝ ⎛ =− ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + + =− 2 I II 2 I II 2 I II 2 II 2 I 2 II 2 I I 2 II 0,MTS K K 1 9 K K 1 8 K K 3 arccos K 9K 3K K K 8K arccos ϕ (2) The crack grows unstable if σϕmax reaches a material limit value σϕc or if a maximum comparative stress intensity factor KVmax (Eq. 3), determined by the tangential stress σϕ (Eq. 1), reaches the fracture toughness KIC. ( ) ( ) ( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = = = → → 0 II 2 0 I 0 r 0 max r 0 Vmax K sin 2 3 2 Kcos 2 cos 2π σ 2 lim π σ K lim ϕ ϕ ϕ ϕ ϕ ϕ r r (3) Eq. (4) shows the criterion for unstable crack growth. IC Vmax K K= (4) Due to cyclic loading the tangential stress σϕ (Eq. 1) is transformed to the cyclic tangential stress Δ σϕ (Eq. 5) with the cyclic stress intensity factors ΔKI and ΔKII. 2 sin cos 2 3 2π ΔK 2 cos 2π ΔK Δσ II 3 I ϕ ϕ ϕ ϕ ⋅ − = r r (5) Fatigue crack growth starts if the cyclic comparative stress intensity factor ΔKVmax=Δσϕ√2πr reaches the Threshold value ΔKth. In case of cyclic loading unstable crack growth occurs if KVmax = KIC (Eq. 4) or if ΔKVmax = ΔKIC = KIC⋅(1-R), with the stress ratio R = σmin/σmax = Kmin/Kmax. 2.1.2. Concept according to Richard The general fracture concept of Richard [1, 5, 6] is very practical and adaptive and can be used for different materials. The concept is based on a comparative stress intensity factor KV. This value depends on the stress intensity factors KI and KII (Eq. 6). ( )2 1 II 2 I I V Kα K 4 2 1 K 2 1 K + = + (6) The material parameter α1 depends on the ratio of the fracture toughness of Mode I KIC and the fracture toughness of Mode II KIIC. If α1 is set to 1.155 an excellent approximation of the fracture limit curve of the maximum tangential stress criterion is obtained. Unstable crack growth occurs as soon as KV exceeds the fracture toughness KIC for Mode I. Furthermore the concept also enables the determination of the kinking angle ϕ0 (Eq. 7), with
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