13th International Conference on Fracture June 16–21, 2013, Beijing, China -1- An interaction integral method for 2D elastodynamic crack problems Zhiyong Wang1, Li Ma1,*, Linzhi Wu1, Hongjun Yu2 1 Center for Composite Materials and Structures, Harbin Institute of Technology, Harbin 150001, PR China 2 Institute of Applied Mathematics, Harbin Institute of Technology, Harbin 150001, PR China * Corresponding author: mali@hit.edu.cn Abstract In this paper, a domain formed interaction integral is derived for the evaluation of dynamic stress intensity factors (DSIFs) for arbitrary 2D cracks in non-homogeneous materials. The interaction integral is formulated by superimposing the actual and auxiliary fields on the path independent J-integral. By selecting the appropriate auxiliary fields, the derived interaction integral does not involve any derivatives of material properties compared to the available expressions in the literature. Moreover, it can be proved that the integrand is valid even when the integral domain contains material interfaces. Therefore, the integrand is simpler in form and it can be applied in more general situations. The numerical implementation of the new expression of interaction integral is then combined with the extended finite element method (XFEM) without tip enriched functions and a benchmark and test problem is presented. Finally, a non-homogeneous cracked body under dynamic loading is employed to investigate dynamic fracture behavior such as the variation of DSIFs for different material properties. Keywords Interaction integral, dynamic stress intensity factors, XFEM without tip enriched functions 1. Introduction Dynamic stress intensity factors (DSIFs) are crucial fracture parameters in understanding and predicting dynamic fracture behavior of a cracked body. To evaluate DSIFs for both homogeneous and non-homogeneous materials, numerous methods have been developed. Among these methods, the numerical techniques may be the most convenient and reliable ones to determine the fracture parameters for more complicated cases, as discussed below. For homogeneous materials, Kishimoto et al. proposed a modified path-independent J-integral, which involves the inertial effects to determine DSIFs combined with the finite element method (FEM), and employed a decomposition procedure for mixed-mode problems [1]. Soon after, Nishioka et al. derived another dynamic J-integral to determine DSIFs for non-homogeneous materials [2]. However, the derived integrand is not well-suited for the finite element method. Kim et al. derived an equivalent domain form of the J-integral by using the divergence theorem and some additional assumptions [3]. As we known, it is difficult to extract mixed-mode DSIFs using J-integral. Instead, the interaction integral, which is known to be superior to both the displacement correlation technique (DCT) and J-integral, may be a suitable choice. Song et al. presented a domain formed interaction integral, namely M-integral, to investigate the DSIFs for homogeneous and smoothly non-homogenous materials [4]. In the formulation, the non-equilibrium formed auxiliary fields are employed, which have been discussed by Kim et al [5] and Dolbow et al. [6]. More recently, Réthoré et al. presented an interaction integral based on Lagrangian conservation for the estimation of DSIFs for arbitrary 2D moving cracks [7]. Most of the previous works are concerned with the materials with continuous and differentiable properties. If the above conditions are not met, the applications of the interaction integral method are impeded. Moreover, very few published papers have considered the cases that there are several material interfaces in the interaction integral domain. Actually, such phenomenon generally exists. In this paper, the derivation of an interaction integral and its associated domain form without any derivatives of material properties is presented. We also present the mathematically rigorous proof that the proposed interaction integral method is still valid even when there are material interfaces in the integral domain. Several test problems and the comments are provided in the last section.
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