13th International Conference on Fracture June 16–21, 2013, Beijing, China -9- 1.0) ( 0.44 ( ) 0.545 ( ) 3.28 0.45 0.774 2.364 0.38 ( ) 0.645 ( ) 2.15 0.12 1 3.52 2 2 2 2 , ≤ ⋅ ⋅ ⋅ + ⋅ + + ⋅ + ⋅ + ⋅ ⋅ ⋅ + ⋅ + + ⋅ + ⋅ + = a/r r a a x a x r a a x r a r a a x a x r a a x r a MB S (21b) Figure 2 shows that this analytical expression fits the weight function results very well. The maximum error is 0.36% (for a/r = 0.5). 4.2. Analytic crack surface displacement equation for Dugdale loading For a radial crack emanating from a semi-circular notch in a semi-infinite plate with crack surfaces subject to Dugdale loading, the analytical crack-surface displacement equation is σ σ σ σ σ , , , , , B A B A B M f f u u ⋅ ⋅ = (22a) where σ,Au ， σ,Af and σ,Bf are given by Eq. (19), Eq. (20) and Eq. (14), respectively. The factor σ,BM is determined by fitting the crack opening displacement results obtained from the weight function method, Eq.(15): 2 2 2 2 2 2 , 0.37 ( ) 0.392 ( ) 3.2 ( ) 2.33 1.544 2.91 0.723 0.594 0.718 0.287 0.14 ( ) 0.454 ( ) 3.1 ( ) 2.26 1.65 2.98 0.63 0.728 0.921 0.713 a x a d r a a x a d a x r a a d r a a x a d r a a x a d r a a x a d a x r a a d r a a x a d r a MB ⋅ ⋅ + ⋅ + − ⋅− ⋅+ ⋅+ ⋅⋅+ ⋅⋅− ⋅⋅− ⋅ ⋅ + ⋅ + − ⋅− ⋅+ ⋅+ ⋅⋅+ ⋅⋅− ⋅⋅− =σ (22b) Figure 3 shows some typical comparisons between the fitted expression, Eq. (22), and the weight function results. For a radial crack emanating from a semi-circular notch in a semi-infinite plate, Eq. (22) represented by the circular symbol, is in very good agreement with the results from weight function method (the solid curves) for all points along crack surface. The maximum difference occurs in the loading segment (a/r=0.05, b1/a=0.9), and is about 2.4%. With this analytical equation of crack surface displacements for the Dugdale loading, crack surface displacements for a segment (width Δb) pressure acting at an arbitrary location along the crack faces can readily be obtained by using the same procedure as Ref [12,13], i.e. taking the difference between two Dugdale loadings with b2=a, b1 and b1+Δb in Fig.1. d b b ,σ ,σ ,σ ,σ d b ,σ ,σ ,σ ,σ , b σ M f f u M f f u u Δ B A B A B A B B , A 1 1 ) ( ) ( = + = Δ ⋅ ⋅ − ⋅ ⋅ = (24) Equation (23) will not only significantly improve the accuracy of fatigue crack closure analysis for the specific crack geometry in consideration, but also will improve the computational efficiency for the modified strip-yield-model-based fatigue crack growth predictions by eliminating the time -consuming numerical integrations for crack surface displacements under partial loading.
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