13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- For nonhomogeneous MEE materials, the auxiliary fields have different choices [6] and an incompatibility formulation is selected in this paper. Namely, in the polar coordinate system ( , ) r θ with the origin at the crack tip, the expanded auxiliary fields are defined as 1 2 ( ), ( ), ( ) 2 aux aux aux N aux N aux aux N J N J IJ IJ IJ IJKL KL r K u K f g C r θ σ θ ε σ π π − = = = x (14) where the index { , , , , } N IIIIIIIVV = denotes different crack opening modes with the value corresponding to a general subscript {1, 2, 3, 4, 5} I = ; aux IK , aux II K , aux III K , aux IV K and aux VK denote the auxiliary mode-I, mode-II, mode-III mechanical SIFs, EDIF and MIIF, respectively. The angular functions ( ) N Jf θ and ( ) NIJ g θ are the standard angular functions for a crack in a homogeneous MEE medium, which depend only on the material properties at the crack-tip location, and their detailed definitions can be found in Ref. [6]. 2.3. Calculations of the interaction integral The infinitesimal contour integral in Eq. (13) can not be obtained directly in numerical calculations and thus, it is usually converted into an equivalent domain integral. To begin, as shown in Fig. 1, consider a closed contour 0 B C C ε − + − Γ =Γ +Γ +Γ +Γ where ε −Γ is the opposite path of the contour ε Γ . According to the assumption that the crack faces are assumed to be mechanical traction-free, electrically impermeable and magnetically impermeable, it can be easily proved that 0 1 ,1 ,1 0 1 lim [ ( ) ( )] 2 aux aux aux aux JK JK JK JK I IJ J IJ J I I u u n qd ε σ ε σ ε δ σ σ Γ Γ → =− + − + Γ ∫Ñ . (15) where q is an arbitrary weight function with value varying smoothly from 1 on ε Γ to 0 on BΓ . When the material properties are continuously differentiable, applying divergence theorem to Eq. (15), one obtains 1 ,1 ,1 , 1 {[ ( ) ( )] } 2 aux aux aux aux JK JK JK JK I IJ J IJ J I A I u u q dA σ ε σ ε δ σ σ =− + − + ∫ . (16) where A is the domain enclosed by the contour 0Γ for 0 ε Γ → . According to Eqs. (7)-(9) and Eq. (14), the interaction integral in Eq. (16) can be simplified as [7] 1 1 ,1 ,1 1 , ,1 {( ) [ ( ) ( )] } aux aux aux aux IJ J IJ J JK JK I I IJ IJKL IJKL KL A I u u q C C q dA σ σ σ ε δ σ σ − − = + − + − ∫ 0 x . (17) Compared with the expression in Ref. [6] for nonhomogeneous MEE media, the expression in Eq. (17) does not contain any term related to the derivatives of material properties with respect to the coordinates. Therefore, the present interaction integral can facilitate the facture analysis of practical MEE materials whose derivatives of material properties are difficult to obtain. Moreover, when the integral domain A contains an interface on which the properties are discontinuous (see the dash line IΓ in Fig. 1), A needs to be divided into two parts for using divergence theorem. Similarly to Ref. [7], the interface has no contribution to the interaction integral and thus, the same expression in Eq. (17) can be obtained. Namely, the present interaction integral does not require the material properties to be continuous, which brings a great convenience to the fracture studies of MEE composites for the integral domain can be chosen to a region containing arbitrary material interfaces. 2.4. Extraction of the fracture parameters This section will introduce how to solve the IFs by using the interaction integral. For linear MEE
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