13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- complex interfaces. Then, this paper will derive a domain-independent interaction integral for MEE composites. 2. Basic relations and interaction integral 2.1. Basic equations for PE media The field equations for a linear MEE medium in the absence of body forces, volume charge and concentrated magnetic source are given as follows. 2.1.1. Governing equations y Constitutive equations ij ijkl kl lij l lij l i ikl kl il l il l i ikl kl il l il l C e E h H D e E H B h E H σ ε ε κ β ε β γ = − − = + + = + + , (1) y Kinematic equations , , , , 1 ( ), , 2 ij i j j i i i i i u u E H ε φ ϕ = + =− =− . (2) y Equilibrium equations , , , 0, 0, 0 ij i i i i i D B σ = = = . (3) where iu , ijσ and ijε are the elastic displacement, stress, strain tensors, respectively; φ, iD and iE are the electric potential, electric displacement, electric field tensors, respectively; ϕ, iB and iH are magnetic potential, magnetic induction, magnetic field tensors, respectively; ijkl C , ilκ, and ilγ are the elastic stiffness, dielectric permittivity and magnetic permeability tensors, respectively; ikl e , ikl h and ilβ are the PE, PM and electro-magnetic tensors, respectively. The repetition of an index implies summation with respect to the index over its range. 2.1.2. Expanded tensor notation By letting 4 5 4 5 4 5 , , , , 0, 2 , 2 , 0 j j j j i j j j j j i j u u D B E H φ ϕ σ σ σ ε ε ε ′ ′ ′ ′ = = = = = =− =− = , (4) 4 5 4 4 5 5 4 5 , , , , , 0, arbitrary value jkl jkl jkl jkl j l jl j l jl j l jl i j kl i j k l i j k l C e C h C C C C C C κ γ β ′ ′ ′ ′ ′ ′ ′ ′ ′ = = =− =− =− = = . (5) we can extend the tensors iu , ijσ , ijε and ijkl C ( , , , 1,2,3 i j k l = ) respectively into Iu , IJσ , IJε and IJKL C ( , , , 1,2,...,5 I J K L= ) which meet the symmetry relations IJ JI σ σ = , IJ JI ε ε = and IJKL JIKL IJLK KLIJ C C C C = = = . Here, the subscripts , , , 4,5 i j k l ′ ′ ′ ′ = . A Cartesian coordinate system only contains 1x -axis and 2x -axis and thus, 4 5 ,4 ,5 0, (*) (*) 0 n n= = = = . (6) where (*) denotes an arbitrary variable or expression. According to the above relations, Eqs. (1)
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