13th International Conference on Fracture June 16–21, 2013, Beijing, China -7- 3. Partition of mixed-mode fracture in layered isotropic DCBs with non-rigid interfaces 3.1. Classical beam partition theory The mode I ERR IE G is considered first. The interface normal stress nσ is found to be [ ] ( ) ( ) [ ] (3) 1 (4) 3 3 3 1 31 2 3(1 ) h u w Eh n γ γ γ γ σ + − + =− (28) where 2 1 w w w = − and 1 2 u u u = − are the relative opening and shearing displacements at interface. The mode I ERR is then found by using J-integral. ∫ ∫ ∫ = ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ = → Bw nB B da w n da IE dw dwdx da G 0 0 0 0 1 lim σ σ (29) Substituting Eq. (28) into Eq. (29) gives (1) 1 1 2 3 ) (1 3 )/[(1 )]( B B B L IE IE P P w b G G β γ γ − + = + + (30) The first term IT L IE G G= in Eq. (20) and the (1) Bw in the second term is the relative crack tip rotation. It is seen that Eq. (30) is not completely analytical due to the second term. It is more important to note that the second set of orthogonal pure modes is not present. The mode II ERR can be considered similarly. The interface shear stress sτ is found to be su s s sP τ τ τ τ σ = + + (31) with ( ) ( ) [ ] γ γ γ τ + + = 2 1 3 1 2 1 2P P bh B B sP , ( ) ( )∫ = − x n s dx h1 0 31 2 σ γ γ τ σ , ( ) [ ]γ γ τ + = 41 (2) 1 Eh u su (32) The mode II ERR IIE G is then calculated by using J-integral. B u sP L IIE IIE du G G B ∫ = + 0 τ (33) The first term IIT L IIE G G= in Eq. (20) and the second term can be calculated for a given cohesive law. 3.2. Shear deformable beam partition theory It is simple to verify that the mode II ERR IIT G remains the same as the IIE G in Eq. (33). However, the mode I ERR IT G needs reconsideration. The governing equation for the interface normal stress nσ is ( ) ( ) ( ) (3) 1 (4) 2 (2) 31 2h u w n n γ γ α σλ σ + − − = (34) where ( ) ( )( )1/ 2 2 1 3 1 h k G E xz γ γ λ= + and ( )γ γ α + = 1 1 2k G h xz . By using the method of parameter variation, the solution to Eq. (34) is found. ( )( ) ( ) [ ] ( ) ( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + − ⎟+ ⎠ ⎞ ⎜ ⎝ ⎛ − + − + + = + + ∫ ∫ ∫ ∫ − − − − − x x x x x x x x x x x x x x n h e uedxe uedx e we dx e wedx h w w u ce c e 0 0 1 2 0 0 3 1 1 (2) 2 2 1 3 1 4 2 3 1 2 λ λ λ λ λ λ λ λ λ λ γ γ αλ αλ γ γ λ α σ (35) The two integration constants 1c and 2c are determined using the conditions ( ) ( ) 0. (1) Δ = Δ = a a n n σ σ ( ) ( )∫ ∫ Δ − Δ − − − =− a x a x h ue dx we dx c 1 0 2 0 3 1 3 1 4 2 λ λ γ γ αλ αλ (36)
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