13th International Conference on Fracture June 16–21, 2013, Beijing, China -6- 2.1.3. 2D elasticity partition theory One averaged partition theory is obtained by averaging the classical and shear deformable partitions. This partition has been found to give an excellent approximation to the partition from 2D elasticity. The mode I and II components of the ERR from the averaged partition theory denoted by IG and II G respectively. They are ( ) P P IT IE IG G G G G 1 1 2 θ θ α + + Δ = + , ( ) 2 IIT IIE II G G G = + (23) 2.1.4. Local and global partition theories When ERR is calculated right at the crack tip, i.e. using an infinitesimally small region around the crack tip, it is called a local calculation. When it is calculated using a finite small region, it is called a global calculation. In terms of the finite element method (FEM), an infinitesimally small region means one element length in a very fine mesh, whilst a finite small region means multiple element lengths. When global ERR calculation is used, the above three local partition theories, i.e. the classical, shear-deformable and 2D partition theories give the same partitions as that of the local classical partition theory. That is, the classical partition theory unifies the three theories in a global partition. The differences between the three local theories arise from the differences of the crack tip stresses in the three theories. However, the global distribution of interfacial stresses is governed by the classical beam and plate theory. 2.2. Clamped-clamped laminated composite beams A clamped-clamped composite laminated beam with a symmetric delamination is considered. The loads 1P and 2P are applied at the mid-span. The pure mode I mode in the first set of orthogonal pure modes in classical beam theory, i.e. the { }β θ, set, is given by ( ) ( ) [ ] 2 2 2 1 1 1 1 2 2 1 2 2 B B hA B B h A P P P − + = =− ∗ ∗ θ (24) Its orthogonal pure mode II mode P P P β = 2 1 is too complex to be presented here algebraically. The second set of orthogonal pure modes in classical beam theory, i.e. the { }β θ′ ′ , set is given by 1 2 1 ′ = =− P P Pθ , * 1 * 2 2 1 P P D D P = ′ = β (25) Within the context of shear deformable beam theory, the expressions for P P P θ = 2 1 pure mode I and P P P β = 2 1 pure mode II are too complex to be presented here algebraically. However, when the through-thickness shear effect is not excessively large, they are very close to those in classical beam theory. 2.3. Clamped circular layered isotropic plates A clamped circular layered isotropic plate with a central delamination and central loads 1P and 2P are considered. The first set of orthogonal pure modes in classical plate theory are found to be 1 2 1 θ θ = = P P P , 1 2 1 β β = = P P P (26) where 1θ and 1β are given in Eq. (14). The corresponding ERRs are given by ( ) ( ) [ ]γ π ν γ θ + − = 1 2 3 1 3 2 1 2 2 1 Eh G P P , ( )( ) ( ) [ ]2 3 2 1 2 2 1 1 3 9 1 1 2 γ π γ ν γ β + − + = Eh G P P (27) The second set of pure mode I and II modes are the same as those in Eq. (15). In the first order shear deformable plate theory, the first set of pure modes is approximately pure and the second set disappears.
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