13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- constant. The temperature boundary of the example is assumed to be 1 2 0 0.5 θ θ θ = = and 0 10 C θ = ° . The following data are used in the FEM analyses: / 8 L W= , / 0.5 a W= , 1 ( ) x W E x E e δ = × , 1 ( ) x W x e γ α α = × , 1 ( ) x W x e β λ λ = × , 2 1 ln( ) E E δ= , 2 1 ln( ) α γ α = , 2 1 ln( ) λ β λ = ， 1 1.0 E = , 2 5 E = , 1 0.01 α= , 2 0.02 α = , 1 1 λ= , 2 10 λ= , 0.3 ν= Different element numbers W L N N× ( 81 648 × , 101 808 × and 121 968 × ) for two different temperature conditions are chosen. The results are shown in Table 2. It can be seen that the present results agree well with the solutions provided by KC and Kim [11]. From these two examples, the convergence and the accuracy of the present method are verified. Table 2 Comparison of the mode-I TSIF and T-stress in FGMs for different thermal loading (Example 2) Present results KC and Kim’s results Loading condition Element numbers 1 0 / K K T-stress 1 0 / K K T-stress 3D T-stress 81 648 × 0.0023 0.00589 101 808 × 0.0023 0.00592 1 0 0.5 T T = 121 968 × 0.0023 0.00594 0.0229 0.0067 0.006 81 648 × 0.00438 0.01118 101 808 × 0.00438 0.01124 1 0 0.05 T T = 121 968 × 0.00438 0.01129 0.00437 0.0126 0.0115 5. Conclusions In this paper, a modified interaction energy integral for thermal loading condition is derived for the T-stress computations. The interaction energy integral is proved to be domain-independent for thermal conditions. It can be found that the numerical results are in good agreement with those in published papers. The present method is effective to analyze the thermal fracture problems of nonhomogeneous materials. Acknowledgements This work is sponsored by NSFC(11072067), NSFC (10802056), NSFC (11202058), NCET(08-0151), Ph. D Programs Foundation of Ministry of Education of China, Science Funds for Distinguished Young Scholar of Heilongjiang Province, China, SRF for ROCS (SEM), Heilongjiang Science Fund for ROCS and China Postdoctoral Science Foundation (20110491071) and the Fundamental Research Funds for the Central Universities (Grant No. HIT. NSRIF. 2013082). References [1] J.R. Rice. Limitations to the small scale yielding approximation for crack tip plasticity. J Mech Phys Solids. 22 (1974) 17-26. [2] Z.Z. Du, J.W. Hancock. The Effect of Nonsingular Stresses on Crack-Tip Constraint. J Mech
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