13th International Conference on Fracture June 16–21, 2013, Beijing, China -7- where 1 2 4 5 [ , , , ] T b b b b =b , 1 2 4 5 [ , , , ] T c c c c =c and n is the node number in an element. 3.2. Numerical discretization of the interaction integral In numerical computations, the interaction integral in Eq. (17) needs to be discretized in the crack-tip local coordinate system as I II I II 1 2 1 1 2 1 1 1 1 1 1 1 ( ) ( ) ( ) ( ) ( ) ( ) e A aux T T aux T aux T e p p p aux T e p aux T p q q q q x x x x x x I w q x x = = − − ⎧ ⎫ ⎡ ⎤ ⎡ ⎤ ∂ ∂ ∂ ∂ ∂ ∂ + + + ⎪ ⎪ ⎢ ⎥ ⎢ ⎥ ∂ ∂ ∂ ∂ ∂ ∂ ⎪ ⎪ ⎣ ⎦ ⎣ ⎦ = ⎨ ⎬ ∂ ∂ ⎪ ⎪ ⎡ ⎤ − + − ⎣ ⎦ ⎪ ⎪ ∂ ∂ ⎩ ⎭ ∑∑ u u σ σ σ σ J σ σ ε C 0 C x σ . (33) where I 11 12 1 1 [ ] T D B σ σ = σ and II 12 22 2 2 [ ] T D B σ σ = σ ; Ae is the number of elements in the integral domain A; ep is the number of integration points in one element; p J and pw represent respectively the determinant of Jacobian matrix and the corresponding weight factor at the integration point p. Here, the derivative of actual displacement vector is ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 b n i i i i i b i c i i i i i N N N N x x x x x ψ ψ ψ = ⎧ ⎫ ⎛ ⎞ ∂ ∂ ∂ ∂ ∂ ⎪ ⎪ = + + + ⎜ ⎟ ⎨ ⎬ ⎜ ⎟ ∂ ∂ ∂ ∂ ∂ ⎪ ⎪ ⎝ ⎠ ⎩ ⎭ ∑ u u b c . (34) 4. Numerical examples Figure 3. A 2D particulate MEE plate with a crack: (a) geometry and boundary conditions; (b) finite element mesh As shown in Fig. 3(a), a 2D CoFe2O4 particle-reinforced BaTiO3 matrix composite plate is considered. The plate of unit length ( 0.5 W= ) is composed of 16 square cells of length 2 W each of which contains a circular particle of radius 0r at its center. Therefore, the volume fraction of the particles is 2 2 0 4 fV r W π = and in this paper, 0.5 fV = . In the center of the plate, there exists an inclined crack of length 2a and angle θ measured counterclockwise. The poling directions of both
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