13th International Conference on Fracture June 16–21, 2013, Beijing, China -9- -c -a c a Z X P Q (t) sticked zone λ weight function analysis at the « hot spot »(Z=0) ( )) .w ( h,max C eq. C* eq. σ =σ ∇ σλ ( ) h,max ∇ σλ 0 0.2 0.4 0.6 0.8 1 1.2 0 20 40 60 80 100 120 k= 0.0045 R²=0.8574 祄1 λ= 祄 10 λ= k= 0.0142 R²=0.8556 C eq. d σ τ ( ) (MPa/祄) h,max ∇ σ λ 0 100 200 300 400 500 600 0 200 400 600 800 ) (MPa) ( .w( hmax h,max σ ∇ σ λ ) (MPa) J .w ( ( hmax 2,a ∇ σ λ Plain Fretting Fretting Fatigue no crack nucleation cracking 祄 10 λ= ( ) h,max w 1 k. = − ∇ σλ (a) (b) (c) Figure 7: Weight function approach: a) illustration of the methodology- b) identification of the weight function – c) Application of the methodology (ℓ=Ø/2=10µm). Using this weight function approach the predictions are highly improved. The statistical analysis gives C* eq. %Eσ = 1% and C* eq. %Vσ = 6%. However, as shown in Fig. 7b, the “k” coefficient highly depends on the volume dimension over which the hydrostatic stress gradient is computed. The smaller the ℓ value, the higher the stress gradients, and smaller the k factor is. This suggests that the methodology must be calibrated keeping constant the ℓ dimension from the identification of the weight function to the structural fatigue analysis. By contrast, the R² coefficient is very stable which indirectly support the stability of this approach. Theses tendencies are confirmed by Figure 8, where using extensive computations, both “k and R²” variables are plotted versus ℓ variable. The k variable can be expressed using a very simple power function at least for the studied cylinder/plane contact: ⋅ λ = −3 k 5.10 (7) Besides, the related R² coefficient remains unchanged around 0.85 whatever the ℓ value. Hence, it can be intuited that such approach could be transposed for any low spatial stress field resolution like observed in industrial FEM contact analysis. Fixing the ℓ variable equal for instance to the FEM contact mesh size and calibrating the related k weight function factor, very good prediction of the crack risk are expected. k = 0.005 ℓ 0.488 correlation = 0.999 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0 50 100 150 200 stress gradient averaging domain, ℓ (µm) R²=0.85 k (Mpa/µm)‐1 R² /10 h,max σ Figure 8: evolution of the k coefficient and related R² correlation factor as a function of the length scale ℓ.
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