13th International Conference on Fracture June 16–21, 2013, Beijing, China -7- decreases more and more rapidly with increasing Poisson ratio 2ν. However, it is shown in Fig.8 that the influence of modulus ratio out of range of 1 2 10 / 10 E E − < < on the dimensionless stress factor 1F is so small that it can be neglected. In Figs.6-8, the dimensionless expression of 1F is defined as 1 1 1 2 2 K F E T l λ α π− = Δ (9) 5. Concluding remarks A new finite element method was developed to analyze irregular-shaped inclusion problems under thermo-mechanical loads. The method consists of a super polygonal inclusion corner element in conjunction with standard four-node hybrid-stress elements. A benchmark example of a square inclusion problem was discussed. The present results is validated by comparison with the numerical results obtained using the conventional finite element commercial software ANSYS package. The present numerical solutions show that our method provides satisfactory results with coarse meshes and is effective and applicable to thermo-mechanical problems with multiple singular points. In addition, for square inclusion problems, the following useful conclusions are drawn: (1) The dimensionless stress intensity factor 1F decreases more and more rapidly with increasing Poisson ratio 2ν; (2) The influence of modulus ratio out of range of 1 2 10 / 10 E E − < < on the dimensionless stress factor 1F is so small that it can be neglected. Acknowledgements This project is supported by the National Natural Science Foundation of China under Grants 10662004 and 51065008 and also in part sponsored by the Major State Basic Research Program of China (973 Program) (Grants No. 2009CB623203-3 and 2011CB612210) References [1] Chen DH. Stress intensity factors for V-notched strip under tension or in-plane bending. Int J Fract, 70(1995) 81-97. [2] Munz D, Yang YY. Stress singularities at the interface in bonded dissimilar materials under mechanical and thermal loads. J Appl Mech, 59(1992) 857-881. [3] Chen CD, Chue CH. Singular stresses near apex of wedge by finite element analysis. J Chinese Inst Eng, 26(2003) 423-434. [4] Lu NS, Zhang Z, Yoon J, Suo ZG. Singular stress fields at corners in flip-chip packages. Eng Fract Mech, 86(2012) 38-47. [5] Banks-Sills L, Ishbir C. A conservative integral for bimaterial notches subjected to thermal stresses. Int J Num Meth Eng, 60(2004) 1075-1102. [6] Shin KC, Kim WS, Lee JJ. Application of stress intensity to design of anisotropic/isotropic bi-materials with a wedge. Int J Solids Struct, 44(2007) 7748-7766. [7] Nomura Y, Ikeda T, Miyazaki N. Stress intensity factor analysis at an interfacial corner between anisotropic bimaterials under thermal stresses. Eng Fract Mech, 76(2009) 221-235.
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