13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- y a a1 d r 2a2 W=1 x H a a1 d 2a2 W=1 x H r Plastic zones coalescence s s s s Fig.1 A coalesced center crack in a finite width panel containing three un-equal cracks, the total length of the fictitious crack includes all the strip yield zones 2.3. Weight function method for general collinear cracks In this section, the weight functions for general collinear cracks are presented. The derivation of the weight function method for general collinear cracks was based on the reciprocity theorem and the superposition principle [8]. It was found that the weight functions for general collinear cracks are quite different from that for a single crack configuration. 0 A B C b c a y x Fig.2. Three symmetric collinear cracks in an infinite sheet subjected to remote uniform stress Take a typical MSD configuration, three collinear cracks in an infinite sheet, Fig.2, as an example. The weight functions for the crack tips A, B and C are given in Eqs.(3-5), respectively [8]. ( ) ( ) ( ) [ ] ( ) [ ] 1 2 , , , ; 0, ' , , , , , , ; , , , r a r rA u a b c x a x a E m a b c x u a b c x a x b c f a b c a σ π ⎧∂ ∂ ∈ ⎪ = ⋅ ⎨ ∂ ∂ ∈ ⋅ ⎪⎩ (3) ( ) ( ) ( ) ( ) [ ] ( ) [ ] 1 2 , , , ; 0, ' , , , , , , ; , , , 2 r b r rB u a b c x b x a E m a b c x u a b c x b x b c f a b c c b σ π ⎧∂ ∂ ∈ − ⎪ = ⋅ ⎨ ∂ ∂ ∈ ⋅ − ⎪⎩ (4) ( ) ( ) ( ) ( ) [ ] ( ) [ ] 1 2 , , , ; 0, ' , , , , , , ; , , , 2 r c r rC u a b c x c x a E m a b c x u a b c x c x b c f a b c c b σ π ⎧∂ ∂ ∈ ⎪ = ⋅ ⎨ ∂ ∂ ∈ ⋅ − ⎪⎩ (5) where the non-dimensional stress intensity factors frA (a,b,c), frB (a,b,c) and frC (a,b,c) for remote uniform tension stress were given in Ref.[14]. The corresponding crack opening displacements for center and side crack 1 ( , , , ) ru a b c x and 2( , , , ) ru a b c x are: ( ) ( ) { } ( )( )( ) 2 2 2 2 2 2 2 2 2 2 1 ( ) ( ) 2 , , , d ; ' x a r x x c a E k K k a u a b c x x a x a E a x b x c x σ − ⎡ ⎤ − − + ⎣ ⎦ = ⋅ − ≤ ≤ − − − ∫ (6a)
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