13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- normal vector m to interface Γ at point q. According to the equilibrium condition on the bi-material interface, the tractions on both sides of the interface should be equal. We have (1) (2) ⋅ = ⋅ m m σ σ (12) Since the interface is perfectly bonded, the derivatives of actual displacements with respect to the curvilinear coordinate 2ξ are equal on both sides of the interface, as a result (1) (2) 2 2 ( ) ( ) ∂ ∂ = ∂ξ ∂ξ u u (13) According to the above assumptions, it can be found that the first term of the interface integral is equal to the third one, and the second term is zero. If the mass density of the materials on both sides of the interface is equal to each other, we can obtain 0 * interface I = (14) In addition, we assign the values auxiliary velocity fields to zero. The interaction integral (I-integral) indicated in Eq. (7) can be simplified as ,1 ,1 , 1 ,1 ,1 , ( ( )) ( ) tip aux aux ij ijkl ijkl kl j j A aux aux aux jk j k i ij j ij j i A I S S x u u qdA u u u q dA σ σ ρ σ δ σ σ ⎡ ⎤ = − + ⎣ ⎦ ⎡ ⎤ − − + ⎣ ⎦ ∫ ∫ & (15) In order to show the advantages of Eq. (15), we will compare it with the traditional J-integral and the M-integral given by Song et al. [4]. The J-integral in the form of the stiffness can be expressed as ,1 1 , ,1 ,1 1 ( ) ( ) 2 ij i j j i i ijkl ij kl A A J u W q dA uu C qdA σ δ ρ ε ε = − + − ∫ ∫ & (16) The resulting M-integral is { } { } ,1 ,1 1 , , ,1 ,1 ,1 ( ) aux aux aux ij i ij i ik ik j j A aux aux aux ij j i i i ijkl kl ij A M u u q dA u uu C qdA σ σ σ ε δ σ ρ ε ε = + − + + − ∫ ∫ & (17) Through rigorous proof, we can conclude that the M-integral is totally equivalent to the I-integral. However, the expressions are quite different. It can be found that the derivatives of material properties exist unavoidably in both the above equations. Differently, the I-integral in Eq. (15) does not involve any derivatives of material properties. Moreover, in certain conditions, the I-integral is still valid even when the integral domain contains material interfaces. Therefore, the applicable range of the present interaction integral is wider than that of the two methods mentioned above for non-homogeneous materials. 2.2. Extended Finite Element Method without Tip Enriched Functions By enriching the standard approximation with additional functions, the extended finite element method (XFEM) allows for the modeling of arbitrary geometric features independently of the finite element mesh. This advance has provided a convenient computational tool for modeling discontinuities and their evolvements. However, if both the strong discontinuities i.e., cracks and the weak discontinuities i.e., inclusions exist in the domain, especially when the crack tip approaches near the inclusions, it is difficult to obtain the accurate solutions i.e., stress intensity factors (SIFs) using the XFEM technique. In addition, in the XFEM modeling of cracked problems, the corresponding analytical results are pre-requisite. If the analytical solutions are difficult to obtain or are very complex themselves, the application is not convenient. Based on the above reasons, Wang et al. proposed a numerical method, named as extended finite element method without tip enriched functions, for modeling crack growth in particle reinforced composite materials [10]. We employ this numerical method to determine the basic solution of the boundary value
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