13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- Over the last decade, the authors have extended the classical boundary element methods (BEM) for analysis of the fracture mechanics in FGMs. The new BEM method incorporates Yue’s solution into the classical BEM methods. The BEM developed by the authors can be classified into two types: multi-region BEM and single region BEM (i.e., dual BEM). In this paper, we introduce mainly the some developments of the Yue’s solution based dual BEM and their applications in the fracture mechanics of the layered and graded materials. 2 A brief introduction of the multi-region BEM based on Yue’s solution Mathematical formulation and computational procedures of the multi-domain BEM have been published by Yue et al. [4,5]. The BEM discretizes a FGM layer as a system of n number of fully bonded dissimilar sub-layers. The Yue’s solution is used as the fundamental solution to replace the classical Kelvin point load solution in conventional three-dimensional boundary element methods. As a result, any FGMs with arbitrary property gradient in depth can be examined using this BEM. Since Yue’s solution satisfies the continuous conditions at any interface, there is no need to consider any sub-layer interfaces as boundary surfaces or sub-domain interfaces in the numerical formulation of BEM. In other words, the crack problem can be straightforwardly carried out using the similar BEM procedure for the same crack in a homogeneous elastic solid of infinite extent. It is only need to generate the BEM meshes for the crack interfaces and their associated auxiliary surfaces. The auxiliary surfaces are needed since the conventional multi-region method is used in the BEM. In the computational formulation, the eight-node isoparametric elements are usually employed to discretize the boundary surfaces. The so-called traction-singular elements are used to model the singular fields around the crack tip. Authors [5,6] used the multi-region BEM to analyze the stress intensity factors of a penny crack parallel or perpendicular to the interfacial layer of FGMs and the growth of the penny crack under remote inclined loads. Besides, authors [7,8] further analyzed the stress intensity factors of an elliptical crack parallel or perpendicular to the interfacial layer of FGMs and the growth of the penny crack under remote inclined loads. 3. Yue’s solution based dual boundary element method 3.1 General As mentioned in Section 2, the multi-region BEM based on Yue’s solution has been applied for the analysis of penny and elliptical cracks in a FGM system. It can be found that the proposed method has the following drawbacks: (1) The introduction of artificial boundaries is not unique and thus cannot be implemented into an automatic procedure. (2) The method generates a larger system of algebraic equations than strictly required. (3) If artificial boundaries are located in FGMs, the unknown quantities of linear equations increase greatly because of the variations of material properties. Dual BEM can overcome the above drawbacks. Dual boundary element formulation is based on a pair of boundary integral equations (BIE), namely, the displacement and traction BIEs. The method is a single-region based, thus it can model the solids with multiple interacting cracks or damage. Although dual BEMs have been widely applied, few of the crack problems in
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