13th International Conference on Fracture June 16–21, 2013, Beijing, China -8- ( ) ( )∫ ∫ Δ Δ − − = a x a x h ue dx we dx c 1 0 2 0 3 2 3 1 4 2 λ λ γ γ αλ αλ (37) Then, mode I ERR is found using J-integral. ( ) ( ) ( ) ( ) ( ) ( ) 2 2 3 1 2 (1) 2 0 0 1 1 (1) 2 0 2 B I B B B w I w B B B w B nm IT dw w w w h u dw M bdw w G I B I B B α σ σ γ γ α α λ σ σ − + + = + − + =− ∫ ∫ ∫ (38) Note that the first term in Eq. (38) is calculated from a given interface cohesive law with ( ) ( ) (1) 1 2 3 1 2 B nm I h u M b γ γ α λ σ + − =− in which ( ) Be B B nm N M M h M 1 3 2 1 2 1 1 3 (1 ) /[2(1 ) ] (1 3 ) /(1 ) γ γ γ β γ γ + − − + + = + (39) and ( ) ( ) 2 2 1 1 2 1 2 2 1 2 (1) 6 6 γ γ γ h N Ebh M M u Be B B B + + =− (40) In the case of a rigid interface the first two terms in Eq. (40) disappear and the third term reduces to IT L IT G G= in Eq. (20). For a non-rigid interface the first term in Eq. (38) is calculated based on the given cohesive law and the second and third terms are not able to be determined analytically. However, for most of practical engineering problems with hard interfaces the third term in Eq. (38) can be replaced by IT G in Eq. (20). Therefore, in Eq. (38) can be calculated by using a given interface cohesive law and the following: ( ) ∫ − = B B I w w I B nB L IT dw G σ σ σ (41) Therefore, the second term in Eq. (38) is found and the mode I ERR IT G for a hard interface is obtained analytically. 3.3. 2D elasticity partition theory A DCB under crack tip bending moments B M1 and B M2 is considered here. Refer to Ref. [12] for general loading conditions. By using the two sets of fundamental orthogonal pure modes, i.e. { }β θ, in Eq. (14) and { }β θ′ ′ , in Eq. (15), approximate orthogonal pure mode I and mode II modes are ( ) ( ) ( )( )2 3 2 1 3 1 2 log 2 1/2 log 1/2 er N N N er N N N N er k k k θ θ θ θ θ θ θ − + + − = + (42) ( ) ( ) ( )( )2 3 2 1 3 1 2 log 2 1/2 log 1/2 er N N N er N N N N er k k k β β β β β β β − + + − = + (43) where k k E er / = is the ratio of interface stiffness to Young’s modulus. 1Nθ , 2Nθ , 3Nθ , 1Nβ , 2Nβ , 3Nβ are functions of the two sets of fundamental orthogonal pure modes. Detailed expressions for them are given in Ref. [12]. A mixed mode can be partitioned using this pair of pure modes. 4. Numerical and experimental assessments The partition theories presented above have been extensively validated by using FEM simulations and in general excellent agreement has been observed [8–13]. Here, one example is presented for an isotropic DCB with non-rigid interface. The geometric dimensions of the DCB are length 110mm =L , width 1mm =b , total thickness 2mm 2 1 + = h h and crack length 10mm =a . The material Young’s modulus is 1GPa = E . The loading conditions are 1Nm 1 = B M and 0 2 = B M . Mixed-mode partitions from the present 2D elasticity theory and Abaqus FEM are recorded in Table 1 for various thickness ratios γ and interface stiffness to Young’s modulus ratios er k . An
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