13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- and from the conservation law, , proposed by Eshelby [6] ( ) ∫ ∫ − = = in s i i j j s j j wn Tu ds wn ds J , , (3) where sin is any integration path within the area closed by boundary S+s, and sin+s form a closed integral loop. From the geometrical point of view, boundary crack initiation, regardless whether it occurs at a crack tip, a notch corner or on a general crack-free boundary, can always be defined as a boundary movement in some direction, with the limit taken and the notch-like boundary becomes a crack, as shown in Fig. 2 and Fig. 3. Then, the energy release rate of boundary cracking can be defined as , (4) where denotes the driving force of boundary cracking in direction x1 when the limit taken exists, or the energy release rate with unit boundary movement s in direction x1; denotes the driving force in direction x2, or the energy release rate with unit boundary movement s in direction x2. For a homogenous and isotropic substrate Griffith’s criterion [9] states that the crack will extend when the critical value is reached [10,11] . (5) For a standard cracked specimen subjected to Mode I loading, and can be calibrated as ! = !" = !"! 1− ! / , (6) where KIC is the Mode-I fracture toughness. 3. Asymptotic stress field in sliding contact 3.1. Boundary Condition A typical fretting contact problem of a rigid flat-ended indenter with half width a, sliding on a homogeneous, isotropic, elastic body in half plane is shown in Fig. 3. The Cartesian coordinates (x1, x2), and the polar coordinates (r, θ), both with the origin at the left edge of the indenter, are selected. Normal force P and tangential force Q act on the indenter and the following normal and shear tractions along interface have been solved in closed form [4], ( ) λ λ π λπ − − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − −= a x a x P p x 1 1 1 1 2 sin (7) and ( ) ( )1 1 q x fp x = , (8) where f is the coefficient of friction; λ is determined by ( ) ( ) 0 1 1 2 21 < < − − = λ µ µ λπ , tan f (9) and µis Poisson’s ratio of the substrate. iJ 0→s ( ) ( ) α α | sin | cos 0 2 0 1 → → + = s s J G J ( ) 0 1 | →s J ( ) 0 2 | →s J max G c G G= max 1 max G J = cG
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