13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- Then we can get * * * tip tip tip tip tip I T E C E f α θ = ⋅ − Δ (19) Where * 2 (plane stress) (plane strain) 1 tip tip tip tip E E E ν ⎧ ⎪ = ⎨ ⎪ − ⎩ * 1 (plane stress) 1 (plane strain) tip tip C ν ⎧ = ⎨ ⎩ + (20) So, the T-stress can be evaluated easily if we can obtain the interaction energy integral I . As Yu et al. [15] proved, the equivalent domain integral is not referred to in the above analysis and as a result, both the domain size and the material properties in the integral domain are not limited for the equivalent domain integral. 4. Numerical examples and discussions To test the validity of the method developed in the above section, two crack problems in nonhomogeneous materials are considered. Example 1: An edge crack in a homogeneous plate. An edge crack of length ‘‘a’’ is located in a homogeneous plate which subjected to steady-state thermal loading. The temperature boundary of the example is assumed to be 1 0 0 C θ θ = = ° and 2 1 C θ = ° . This problem have been studied by Sladek and Sladek [9] and KC and Kim [11]. The following data are used for the numerical analysis: / 4 L W= , / 0.1~0.8 a W= , 0.3 ν= , 5 ( ) 1.0 10 E x = × , 5 ( ) 1.67 10 xα − = × , 1 λ= Tables 1 present the mode-I TSIFs and the T-stress for various crack lengths. The T-stress obtained is in good agreement with those reported in Sladek and Sladek [9]. Table 1 Comparison of the mode-I TSIF and T-stress in homogeneous materials under thermal loading (Example 1) Sladek and Sladek’s results Present results / a W 1K T-stress 1K T-stress 0.1 0.6454 -0.4317 0.6432 -0.4198 0.2 0.776 -0.2179 0.7756 -0.2196 0.3 0.7951 -0.0314 0.7953 -0.0322 0.4 0.7527 0.1463 0.753 0.1489 0.5 0.6705 0.3258 0.6708 0.3295 0.6 0.5601 0.5075 0.5605 0.5142 0.7 0.4288 0.698 0.4291 0.7064 0.8 0.2825 0.896 0.2828 0.9095 Example 2: An edge crack in a FGMs plate. In this example, an edge crack problem in a FGMs plate is studied. The material properties, Young’s modulus and thermal expansion coefficient are exponential functions of x, while Poisson’s ratio is
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