applied fracture mechanics

168 Applied Fracture Mechanics24 Will-be-set-by-IN-TECH2.5k 1− , k2−211.5120.521 x o0−2 −1 0 1 2 3Figure 9. Distributions of the stress intensity factors k − 1 (curve 1) and k− 2 (curve 2) in case of a surfacecrack: α = 0.339837, δ 0 = 0.3, and other parameters as in the previous case of a surface crack (afterKudish [15]). Reprinted with permission of the STLE.In particular, according to (45) and (46) at any point (x, y) there are two angles α 1 and α 2 alongwhich a crack may propagate which are determined by the equation in dimensional variablestan 2α = −a ∫H2y (t−x)T(t,x,y)dt−a Haπ ∫H, T(t, x, y) = yp(t)+(t−x)τ(t) . (47)[(t−x) 2 +y 2 ] 22 q0 + [(t−x) 2 −y 2 ]T(t,x,y)dt−a HAlong these directions k 1 reaches its extremum values. The actual direction of crackpropagation α is determined by one of these two angles α 1 and α 2 for which the value ofthe normal stress intensity factor k 1 (N, x, y, l, α) is greater.A more detailed analysis of the directions of fatigue crack propagation can be found in [1].5. Two-dimensional contact fatigue modelIn a two-dimensional case compared to a three-dimensional case a more accurate descriptionof the contact fatigue process can be obtained due to the fact that in two dimensions it isrelatively easy to get very accurate formulas for the stress intensity factors at crack tips [1, 5].The rest of the fatigue modeling can be done the same way as in the three-dimensional casewith few simple changes. In particular, in a two-dimensional case of contact fatigue onlysubsurface originated fatigue is considered and cracks are modeled by straight cuts withhalf-length l. That gives the opportunity to use equations (45) and (47) for stress intensity