-4- expressed as 1 2 1 2 E E E E α − = + (6) 1 2 2 1 1 2 2 1 1 (1 2 ) (1 2 ) 2 (1 ) (1 ) μ υ μ υ β μ υ μ υ − − − = − − − (7) where ( )2 / 1 i i i E E v = − , iE , 1υ and 1μ are the plane strain modulus, Young’s modulus, Poisson’s ratio, and shear modulus of film/substrate systems' components, respectively. For homogeneous materials, the Dundurs’ parameters are written as α = β = 0 . Strain energy release rate (SERR) is another basic concept to study the driving force of an crack. A larger SERR value of the surface crack tip presents the more probability of surface rupturing. The SERR can be calculated by SIF and the Young’s modulus of the top coating 2K G E = (8) In present work, the multiple layers TBCs model with pre-existing interface crack has tiny differences with the typical film/substrate system. In detail, the interfacial defect existence may make it difficult to investigate the fracture mechanism in TBCs by using classical fracture mechanism. Fortunately, the finite element method can calculate the value of SIF and SERR accurately. Therefore, the commercial soft ABAQUS is adopted in present work to obtain the effect of the interfacial defect on the surface crack driving force (e.g. SERR). In the preliminary calculation, we found that a relatively small interfacial defect has little influence on the crack mode. In general, the mode II SIF is less than one-tenth of the mode I SIF, which means that the mode I crack dominates the fracture behavior and mode II crack can be ignored in the following analyses. Therefore, the SERR for the surface crack can be written in a similar form with Eq. (4) / (, ,,,) (, ,,,)(, ,) L L K h w a c a c w a σ α β η α β α β = = (9) where the function w is the value of normalized SIF of surface crack obtained by FEM, and w represents the right expression of Eq. (4), which is determined by the Dundurs’ parameters α , β and surface crack length / a a h = . The revised function η is regarded as the function of the interface crack geometric characteristic, the crack length / L L W = and the crack location / W c c = . Similarly, the SERR can also be written as follow by using the Eq. (8) and Eq. (9) 2 (, ,,,) (, ,,,)(, ,) L L GE Z a c a c Z a h α β κ α β α β σ = = (10) where the function Z, κ and Z represent the SERR value function, the revised function of the interfacial defect and the original SERR function without interfacial defect, respectively. Since it is intractable to calculate SIF and SERR for an crack in TBCs, the more convenient calculation of SERR is applied in this paper. In the linear elastic fracture mechanics, the SIF and the SERR can be represented by the value of J-integral. Therefore, SERR associated with crack length can be calculated by J-integral method. For a virtual crack advance ( )s λ , the value of J-integral can be calculated by [26] ( ) A J s dA λ = ⋅ ⋅ ∫ n H q (11) where dA is the total areas of a layer of elements enclosing the crack tip, n is the outward normal vector to the corresponding integral contour, and q is the direction of virtual crack extension, H is given by W ∂ ⎛ ⎞ = ⎜ − ⋅ ⎟ ∂ ⎝ ⎠ u H I σ x (12) where W is strain energy. The finite element code ABAQUS is employed for numerical calculations. A fine mesh near
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