13th International Conference on Fracture June 16–21, 2013, Beijing, China -8- 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 0 100 200 300 400 500 600 700 800 900 1000 r [micron] GB normal stress [MPa] Analytical model (A nn =0.72 and α = 0.27) FE calculations 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 0 2000 4000 6000 8000 10000 12000 r [micron] GB normal stress [MPa] Analytical model (A nn = 0.72 and α = 0.27) FE calculations 4. Model versus FE calculations 4.1 Parameters adjustment It is worth to note that following the theory of matching expansions [Leguillon, 2002], a same singularity exponent, α, is valid for both GB normal and shear stresses. Therefore, the current section deals with the adjustment of analytical parameters nn A , nm A and αin order to fit FE curves. Figure 10. a) GB normal stress (Ann = 0.72) b) GB shear stress (Anm = 0.64), with respect to the distance to the GB-SB intersection in close field configuration for: analytical model, Eq. 3 and 4, (red) with α = 0.27 and FE calculations (black) with L=10.71 μm, t=0.09 μm, Σ0 = 393 MPa, αGB = 33°, αSB = 45°, τ0 = 60 MPa Fig. 10 a) and b) show a fair agreement between our optimized analytical model and the FE calculations: one suitable value of α (=0.27) is found for GB normal stress and GB shear one. As expected two different values of the factors are found depending on the chosen stress component. It takes 0.71 for the GB normal stress and 0.64 for the GB shear stress. 4.1 Validation Let one now show the validity of our model with respect to the SB characteristic sizes (L and t). Indeed, the parameters adjustments are carried out for a new value of SB length (Fig. 11 a)) and a new one of SB thicknesses (Fig. 11 b)). For L=6.7 μm, for instance, the same parameters, Ann=0.72 and α=0.27, are found and in the same manner, for t=0.04 mm, Ann=0.72 and a=0.27. Therefore, we can conclude that our model is insensitive to the SB lengths for the GB normal stress. Figure 11. a) GB normal stress ( L = 6.7 μm, t = 0.09 μm, Σ0 = 878 MPa, Ann=0.72 and α = 0.27 ) b) GB normal stress ( L = 10.665 μm, t = 0.04 μm, Σ0 = 176 MPa, Ann=0.72 and α = 0.27 ), with respect to the distance to the GB-SB intersection in close field configuration for: analytical model (red) and FE calculations (black) αGB = 33°,αSB = 45° and τ0 = 60 MPa 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 0 1000 2000 3000 4000 5000 6000 r [micron] GB normal stress [Mpa] Analytical model (A nn = 0.71898 and α nn = 0.27) FE calculations 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 0 500 1000 1500 2000 2500 3000 3500 4000 4500 r [micron] GB shear stress [MPa] Analytical model (A nm =0.64 and α=0.27) FE calculations a) b) a) b)
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