13th International Conference on Fracture June 16-21, 2013, Beijing, China [11]. A mixed-mode loading is a superposition of a single mode-I and single mode-II loading. Thus, the values of stresses and strains are split into parts related to mode-I and mode-II loading as follows. The stresses and displacement gradients related to the symmetric crack tip opening are calculated according to: σI 11(xc)= 1 2 (σs 11(xc,+π)+σs 11(xc,−π)) (17a) uI 1,1(xc)= 1 2 us 1,1(xc,+π)+u s 1,1(xc,−π) (17b) The stress and displacement gradient related to the antimetric crack tip opening are determined by subtracting the symmetric parts from the total values, i.e. σII 11(xc,±π)= σs 11(xc,±π)−σI 11(xc) (18a) uII 1,1(xc,±π)=u s 1,1(xc,±π)−u I 1,1(xc) (18b) As the mode-I stresses and strains according to Eqs. (17) exhibit a non-singular behavior on the crack faces, extrapolating σI 11 andu I 1,1 towards the crack tip is feasible. From Fig. 4(a) is becomes obvious, that particularly σI 11 exhibits large numerical errors and thus an extrapolation is reasonable [11]. The mode-I values within [0, δ] are replaced by those, calculated from a linear regression. Rearranging Eqs. (18), the mode-I and mode-II stresses and strains are recombined and the crack face integral is calculated as usual following Eq. (9a), considering the new values. Substituting the resulting values Jk into Eq. (11), this provides accurate SIF for curved cracks under mixed-mode conditions. −20 0 20 40 60 80 100 120 140 160 180 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 σs 11 /MPa ˆxc σI 11 original σI 11 extrapolated σII 11 original on Γ+ σII 11 original on Γ− (a) 0.25 0.255 0.26 0.265 0.27 0.275 0.28 0.285 0.29 0.295 0.3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.025 −0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.02 0.025 J1 / (N/mm) J2 / (N/mm) ˆxc J1 original J2 original J1 extrapolated J2 extrapolated J1 CTEmethod J2 CTEmethod (b) Figure 4. (a) Decomposed mode-I and mode-II stress distributions σI 11 and σII 11, original and extrapolated. (b) Numerically calculatedJk-integral, crack face integral calculated within the range [0, ˆxc], comparison of different methods. In Fig. 4(a) the tangential stress σs 11 on the crack faces is plotted vs. a normalized crack face coordinate ˆxc =(R−xc)/R,with ˆxc =1 at the crack tip and ˆxc =0for xc =R. The considered boundary value problem is that one of a Double Cantilever Bream with dissimilar forces F1 =100N and F2 =99N -6-
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