13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- 2. Numerical Strategy The interaction integral utilizes two admissible fields: auxiliary and actual fields. Auxiliary fields are based on known fields such as Williams’ solution, while actual fields utilize quantities such as displacements, strains and stresses obtained by means of XFEM without tip enriched functions. The choice of the auxiliary fields is discussed firstly. Then, the derivation of the interaction integral and the introduction of the numerical technique will be provided. 2.1. Interaction Integral In this work, the asymptotic fields of Williams’ solution are employed as the auxiliary fields for dynamic non-homogeneous materials, because the dynamic asymptotic fields of non-homogeneous materials show similar behavior to those of quasi-static homogeneous materials around the crack tip [8]. In addition, the incompatibility formulation, proposed by Dolbow et al. [6], is selected. In this formulation, the auxiliary displacements and stresses are obtained directly from Williams’ solution and the auxiliary strains are evaluated from the non-homogeneous constitutive model. The auxiliary displacement is given by Eq. (1) 0 0 ( ) ( ) 2 2 2 2 aux aux aux I II I II i i i K K r r u u u θ θ μ π μ π = + (1) The auxiliary stress is given by Eq. (2) ( ) ( ) 2 2 aux aux aux I II I II ij ij ij K K r r σ σ θ σ θ π π = + (2) Finally, the auxiliary strain is obtained from ( ) aux aux ij ijkl kl S x ε σ = (3) where ( ) ijkl S x is the compliance tensor of the non-homogeneous material. Since the material property involved in the auxiliary displacement is the local value at the crack tip, the auxiliary strain fields are not compatible with the auxiliary displacement fields. Next, we will focus on the derivation of the interaction integral. The dynamic J-integral for cracked homogeneous linear elastic materials is [9] 1 ,1 0 =lim ( ) i ij j i J W L u nd δ σ Γ→ Γ ⎡ ⎤ + − Γ ⎣ ⎦ ∫ (4) Superimposing the actual and auxiliary fields on Eq. (4) and one can obtain the interactional part 1 ,1 ,1 0 1 lim ( ) ( ) 2 aux aux aux aux aux jk jk jk jk i j j ij j ij j i I u u u u nd σ ε σ ε δ ρ σ σ Γ→ Γ ⎡ ⎤ = + + − + Γ ⎢ ⎥ ⎣ ⎦ ∫ && (5) The related definitions of the interaction integral are illustrated in Fig. 1. Here, we call attention to an important assumption, namely, the auxiliary stress and strain fields are assumed to be related through the same elasticity tensor as the actual stress and strain fields ( ) aux aux aux ij ij ijkl kl ij kl kl C x σ ε ε ε ε σ = = (6) Due to the way in which we have defined the auxiliary fields and the material property in-homogeneity, the associated terms do not vanish when we employ the divergence theorem. The contour integral is then converted into an equivalent domain integral which involves the term induced by the interface together ,1 ,1 ,1 ,1 * , 1 ,1 ,1 , int ( ( )) ( ) ( ) tip aux aux aux aux ij ijkl ijkl kl j j j j j j A aux aux aux aux jk j k j j i ij j ij j i erface A I S S x u u u u u u qdA u u u u u q dA I σ σ ρ ρ ρ σ ρ δ σ σ ⎡ ⎤ = − + − − ⎣ ⎦ ⎡ ⎤ − + − + + ⎣ ⎦ ∫ ∫ & & & && && (7) where q is the weight function varying from unity at the crack tip to zero on BΓ , and
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