13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ′ ⎟⎟ − ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − = 1 2 1 2 1 1 2 1 β β β B B Be B B IE IE M M M N G c M (12) ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ′ − ′ ⎟⎟ − ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − = 2 1 1 2 1 2 1 1 2 1 θ θ θ θ Be B B Be B B IIE IIE M N M M N G c M (13) where IE c and IIE c are still given by Eq. (10) and γ B B Be N N N2 1 1 = − . The pure mode relationships are now as follows: 2 1 γ θ =− , 1 2 6 h =− θ , ( ) γ γ γ β 1 3 3 2 1 + + = , ( ) ( 1) 2 3 1 2 − + = γ γ β h for 1≠γ , 1 2 = β for 1=γ (14) 1 1′ =− θ , ( ) ( )3 1 2 1 61 γ γ θ + + ′ =− h , 3 1 γ β′ = (15) The isotropic 1θG and 1βG for use in Eq. (10) are ( )γ γ θ + = 1 24 3 1 2 1 Eb h G , ( ) ( )2 3 1 2 1 3 72 1 1 γ γ γ β + + = b Eh G (16) 2.1.2. Shear deformable beam partition theory In the absence of crack tip shear forces, the total ERR G in a mixed-mode fracture is still given by Eq. (1) within the context of the first order shear deformable laminated composite beam theory. However, the two sets of fundamental orthogonal pure modes now coincide at the first set, i.e. the { }β θ, set and the partitions of the total G are given by 2 3 2 2 1 1 2 1 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − − = β β β B B B B IT IT M N N G c M , 2 3 2 2 1 1 2 1 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − − = θ θ θ B B B B IIT IIT M N N G c M (17) where 2 1 1 1 1 − ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = − β θ θ c G IT , 2 1 1 1 1 − ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = − θ β β c G IIT (18) When crack tip shear forces B B P P2 1 , are present, the following two terms need to be added to the mode I ERR in Eq. (17): ( ) ( )2 1 2 1 2 2 2 1 1 2 2b HH H H H P H P G B B P + − = , ( ) 2 1 2 1 2 2 1 1 2 1 1 2 2 2 1 1 2 1 1 1 1 1 ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + − ⎟ + ⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − Δ = ∗ ∗ ∗ H H D D D HH H P H P b G B B P θ θ α α θ θ θ (19) where 1H and 2H are the through-thickness shear stiffnesses and ( ) ( )1 3 1 2 1 1 2 1 1 1 2 1 1 2 2 1 β θ β β β θ β β β β β αθ − + − − + = B B B B N M N M (20) In the case of layered isotropic DCBs, these partitions reduce to 2 2 1 1 2 1 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − = β β Be B B IT IT M N G c M , 2 2 1 1 2 1 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − = θ θ Be B B IIT IIT M N G c M (21) The mode I contribution from crack tip shear forces reduces to ( ) ( )γ γ γ + − = 1 2 2 1 2 2 2 1 xz B B P b hk G P P G , ( ) ( )( )1 2 2 2 1 2 2 1 1 4 3 1 1 1 / xz B B P k G E b h P P G γ γ α α θ θ θ + − Δ = (22)
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