13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- ( )( ) ( )( )2 2 2 2 1 1 2 2 1 2 2 1 1 1 2 2 1 B AD B h A B AD B h A − − + − =θ , 2 2 1 1 1 1 1 1 2 D hB B h A + + =− θ , ( )( ) ( )( ) 1 1 2 1 2 2 2 2 2 2 2 1 1 1 3 2 2 D h B B AD B h A B AD − − − + = θ (4) ( ) ( )1 2 1 1 2 1 1 2 1 1 θ θ θ β ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ + − + − =− D D D D D D D D , ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ⎟⎟− + − ⎠ ⎞ ⎜⎜ ⎝ ⎛ − + + ⎟⎟+ − ⎠ ⎞ ⎜⎜ ⎝ ⎛ − + = B B B B D h D h B A A Bh B D D B B B D h 1 1 2 2 2 1 2 2 1 1 1 2 2 2 2 4 1 1 1 1 2 θ θ β , ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ⎟⎟− + ⎠ ⎞ ⎜⎜ ⎝ ⎛ + − − ⎟− + ⎠ ⎞ ⎜ ⎝ ⎛ − = B B D h D h B A A Bh B D D B D h 2 4 1 1 1 1 2 1 2 1 2 1 3 1 1 3 3 θ θ β (5) The second set of fundamental orthogonal pure modes, referred to as the { }β θ′ ′ , set, has the same format as that of the first set in Eq. (3), but with 1 1′ =− θ , ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ⎟⎟− + − ⎠ ⎞ ⎜⎜ ⎝ ⎛ ′ − + + ⎟⎟+ − ⎠ ⎞ ⎜⎜ ⎝ ⎛ ′ − + ′ = B B B B D h D h B A A Bh B D D B B B D h 1 1 2 2 2 1 2 2 1 1 1 2 2 2 2 4 1 1 1 1 2 β β θ , ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ⎟⎟− + ⎠ ⎞ ⎜⎜ ⎝ ⎛ ′ + − − ⎟− + ⎠ ⎞ ⎜ ⎝ ⎛ ′ − ′ = B B D h D h B A A Bh B D D B D h 2 4 1 1 1 1 2 1 2 1 2 1 3 1 1 3 3 β β θ (6) * 1 2 1 D D∗ ′ =β , 1 1 2 B A ′ =− β , 1 2 2 3 D B B ∗ ∗ ′ =− β (7) Any four fundamental pure modes from either the first set or the second set can be used to partition a mixed mode. The partitions are given below. ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ′ − ′ − ′ ⎟⎟ − ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − − = 3 2 2 1 1 2 1 3 2 2 1 1 2 1 β β β β β β B B B B B B B B IE IE M N N M M N N G c M (8) ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ′ − ′ − ′ ⎟⎟ − ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − − = 3 2 2 1 1 2 1 3 2 2 1 1 2 1 θ θ θ θ θ θ B B B B B B B B IIE IIE M N N M M N N G c M (9) where 1 1 1 1 1 1 1 1 − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ′ ⎟⎟ − ⎠ ⎞ ⎜⎜ ⎝ ⎛ − = β θ β θ θ c G IE , 1 1 1 1 1 1 1 1 − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ′ ⎟⎟ − ⎠ ⎞ ⎜⎜ ⎝ ⎛ − = θ β θ β β c G IIE (10) and ( ) ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + + − = ∗ ∗ ∗ b D D D G 2 1 2 2 1 1 2 1 1 2 1 1 θ θ θ , ( ) ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + + − = ∗ ∗ ∗ b D D D G 2 1 2 2 1 1 2 1 1 2 1 1 β β β (11) The partitions in Eqs. (8) and (9) use both sets of orthogonal pure modes. The partition theory in Ref. [1] only gives the { } 1 1 , β θ′ ′ pure modes correctly. The partition theories derived in Refs. [6,7] is equivalent to using only the first set of pure modes to partition a mixed-mode. The methodologies used in Refs. [6,7] are not able to find the second set of pure modes. The partitions are easily reduced for isotropic materials. With a thickness ratio 2 1 h h =γ now introduced, they are
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