13th International Conference on Fracture June 16–21, 2013, Beijing, China -8- 祄5 K 2 1 2 b 2 d th 0 T ⎟⎟ ≈ ⎠ ⎞ ⎜⎜ ⎝ ⎛ σ Δ π λ = ≈ (3) This strategy previously applied par Araujo et al. in [4] is now considered for the given Crossland’s fatigue analysis (Fig. 7). 0 100 200 300 400 500 600 0 200 400 600 800 J ( ) (MPa) 2,a Tλ ( ) (MPa) H,max T σ λ no crack nucleation cracking Plain Fretting Fretting Fatigue ‐ c ‐ a c a Z X P Q (t) stick zone ( ) ( ) J ,a( ) h,max T 2 T T C eq. λ λ λ +α⋅σ = σ Tλ Fatigue analysis at the critical distance (z = ) 祄5 Tλ ≈ “hot spot” Figure 7: Critical distance approach assuming a constant length scale value ( 祄 b 2 5 T 0 λ = = ). The experimental results are closer to the material boundary and the dispersion is reduced. The statistical analysis gives %E ( )T C eq. σ λ = 11% and ( )T C eq. %Vσ λ = 12%. The global predictions are less conservative but still dispersed. This suggests that the “critical distance” approach which consider a single “material” length scale parameter is not sufficient to fully capture the stress gradient for such very large stress gradient range. 5.3 Weigth function approach An alternative non local approach which consists to consider the “hot spot” fatigue stress value weighted by a linear decreasing function (w) of the hydrostatic stress gradient surrounding the hot spot location is now considered [6, 7]: ( )) .w ( hmax C eq. C* eq. σ =σ ∇ σ λ where ( ) hmax w 1 k. = − ∇ σ λ (4) With ( ) hmax ∇ σ λ the mean stress gradient of the hydrostatic stress over a cubic volume defined by the length scaleλ. Hence for plain strain conditions it leads to ( ) 2 h,max 2 h,max h,max z x (x,z) ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂σ ⎟⎟ + ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂σ = ∇σ (5) which for the studied fretting conditions infers : ( ) 2 h,max h,max 2 h,max h,max hmax ( a ,0) ( a,0) ( a, ) ( a,0) ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛σ − −σ − + ⎟⎟ + ⎠ ⎞ ⎜⎜ ⎝ ⎛σ − −σ − ∇ σ = λ λ λ λ λ (6) The length scaleλis usually related to the grain size so that λ= Ø/2 =10 µm which presently gives k=0.0142 (MPa/µm)-1
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