-5- the crack tip is necessary for the regular finite element method in the typical TBCs model. Eight-node bilinear plane strain quadrilateral reduced integration elements are selected for all three layers except the crack tip region, where very fine mesh of singular elements are constructed. In addition, The J contour integral values are independent to the mesh configuration if the mesh configuration is fine enough around the crack tip. Herein, each layer in TBC is taken to be homogeneous, isotropic and linear elastic materials whose geometry and material properties are 0.2 mm TC h h = = , 0.1 mm BC h = , 30 mm Sh = , 60 GPa TC E = , 200 GPa BC E = , 211GPa SE = , 0.3 S BC TC ν ν ν = = = and / 20 W h = , where the subscripts TC, BC and S represent top coat, bond coat and substrate, respectively. 3. Result and Discussion In fracture mechanics, SERR at the crack tip can regarded as the driving force of the crack. Therefore, obtaining the SERR value of the surface crack for different lengths can detailedly describe the initiation and propagation of it. Fig.2 shows reduced revised function ( , , ) L a c κ as a function of interfacial defect location for different surface crack length. Noted that the length of presetting interface crack is fixed as a uniform value 0.15 L = in order to investigate the separate effect of interface crack location on the driving force of surface crack. As discussed earlier, the SERR for the surface crack in the TBCs with interface defect can be written as Eq. (10) which consist two functions, the revised function and the original SERR function without interface defect, respectively. In Fig.2, the value of ( , , , , )/ ( , , ) Z L Z a c a α β α β in Eq. (10) for a constant 0.15 L = equals to the normalized SERR / G G. Therefore, the reduced revised function ( , , ) L a c κ ( ,α β is constant since the material parameter is uniform) for 0.15 L = can be obtained from the normalized SERR. The ( , , ) L a c κ for 0.15 L = is always larger than 1 for all surface crack length, / a h, which means the interface defect always promotes the initiation and propagation. This promotion is the result of the constrain reducing for the top coating caused by the interface defect. The close analyses have also been made by Tsui et al.[27] and Thouless et al.[28]. Noted that the ( , , ) L a c κ curve for small / a h slightly deviates from the dash line standing for the SERR function without interface defect ( , , ) Z a α β while the ( , , ) L a c κ for relatively large / a h is much bigger ,which means that the interface defect effects the propagation of the surface crack more dramatically than the initiation. That's to say, the initiation of the surface crack appears to be impendent on the interface defect offset while the propagation intensively depend on the interface defect location. In addition, when the surface crack approach to the interface, the ( , , ) L a c κ seems to decrease slightly (i.e. the curves for / a h=0.8 or 0.9 is in a low degree compared to that for / a h=0.6 or 0.7). As shown in Fig.2, the interface defect offset c is a significant factor to affect the propagation of the surface crack. For the large offset c , the ( , , ) L a c κ is approach to 1, indicating the variation of SERR is exactly similar to the solution for perfect interface. Therefore, in this case, the remote interface defect appears to have relatively little influence on the surface cracking behavior. Conversely, when the offset is equal to zero, the ( , , ) L a c κ seems to largely deviate from the dash line(i.e. the solution for perfect interface). In other words, the interface defect under the surface crack may promote the surface crack propagation more vigorously than the remote one. As a result, the surface cracks centralizing in some areas under of which there exists potential interface cracks may propagate much faster. Then they easily penetrate into the interface and coalesce with the interface cracks, which lead the eventual spalling of the ceramic coating. Therefore, we consider the interface defect as a significant reason for the premature failure of TBCs during service. As the absolute value of c enlarges to a critical value, the ( , , ) L a c κ for 0.15 L = becomes
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