13th International Conference on Fracture June 16–21, 2013, Beijing, China -8- Fig. 8 shows the variation of the SIF values at the crack fronts x m2 ′ =± andy m1 ′ =− . Herein, the KI and KII values are positive. Along the crack front y m1 ′ =− , the KI and KII values become larger near the sidex m2 ′ = . As the crack surface approaches to the interface, the absolute values of KI and KII decrease. 0 2 4 6 8 0.0 1.0 2.0 3.0 4.0 5.0 6.0 L (a) KI vs. L x'=2m y'=-1m x'=-2m D C B A KI/π1/2(MPa.m-1/2) case 3 case 4 h1=1.0m, h1=0.7m h1=0.5m, h1=0.3m 0 2 4 6 8 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 x'=2.0m y'=-1.0m x'=-2.0m D C B A (b) KII vs. L KII/π1/2 (MPa.m-1/2) case 3 case 4 h1=1.0m, h1=0.7m h1=0.5m, h1=0.3m L Fig. 8 SIF values of rectangular crack subjected to linear loads (Cases 3 and 4) 5 Concluding remarks This paper introduces a novel dual boundary element method associated with Yue’s solution. In analyzing the crack problems in FGMs and layered materials, the advantages of this approach are: (1) it is not necessary to introduce elements at the interface, (2) the method is applicable for multilayered solids with any layer number and (3) the high accuracy can be obtained for the crack in multilayered solids. Only the results of crack problems in infinite domains are presented in this paper and the numerical examples of crack problems in finite domains can be found in [9]. In 1995, Yue [12] developed the fundamental solution of a transversely isotropic bi-material. The authors also used this fundamental solution to develop the BEMs of the bi-material similar to the ones of Yue’s solution. The proposed BEM includes the multi-domain BEM and the single domain BEM (i.e., dual BEM). The application of the BEMs to analyze a penny crack, an elliptical crack and a square crack in bi-materials has been presented [13-15]. In the book [16], the authors introduced the research results by using the fundamental solutions [3, 12] systematacially. The proposed BEMs and the results can be a powerful numerical tool, which can apply to various complex three-dimensional geometries of layered or gradient materials with cracks under mixed-mode loading. Acknowledgements The authors would like to thank the financial supports from the Committee on Research and Conference Grants of The University of Hong Kong and the National Natural Science Foundation of China (Grant No. 51079081). References [1] V. Birman, L.W. Byrd, Modeling and analysis of functionally graded materials and structures. ASME Applied Mechanics Reviews, 60(2007):195-216. [2] M.H. Aliabadi, Boundary element formulations in fracture mechanics. ASME Applied
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