-3- The coordinate system (x, y) is set at the top surface of the film with surface crack, whose length a is along to y-axis. An interface defect with length l is presetting at the top/bond coating interface, the center of which offsets from the y-direction by c. Figure 1. The TBCs model with surface crack in the TC and interfacial crack at TC/BC interface. Assume a uniform strain is remotely applied to the TBCs system in the x-direction, the remote stress in the top coating is given by 0 1 0 2 1 1 Eε σ υ = − (1) where 1E , 1υ are the Young’s modulus and Poisson’s ratio of top coating. Therefore, the mixed boundary conditions on the surface crack faces can be expressed as (0, ) 0, 0 (0 0, ) (0 0, ), ij i i y y a u y u y a y h σ = < < ⎧ ⎨ + = − < < ⎩ (2) where ijσ (i,j=x,y) is the stress component and iu (i,j=x,y) is the displacement in x-direction or y-direction, respectively. Moreover, the stress on the interface crack face at interfacial delamination area should be zero and in other area at the interface the displacement should be continuous. The mixed boundary conditions at the interface are shown as ( , ) 0, 2 2 (, 0) ( , 0), and 2 2 ij i i l l x h c x c l l u x h u x h x c x c σ ⎧ = − − < <− + ⎪⎪ ⎨ ⎪ − = + <− − >− + ⎪⎩ (3) The problem can be solved by the superposition technique. The film/substrate system without any cracks is solved first and then added the stresses to the crack faces to represent the applied external loads 0σ. Then the numerical solution for mode I stress intensity factor (SIF) of the surface crack in a film/substrate system without any interfacial defect has been given by Beuth[24] ( ) ( ) ( ) 1/2 1/2 0 / 1.1215 / 1 / 1 / s K h a h a h a h σ π λ − = − + (4) where the λ is a curve fitting parameter which is dependent of / a h, The value of λ also been given by Beuth[24]. The s is the singularity exponent which depends on elastic mismatch, and is the root to ( ) ( ) 2 2 2 cos 2 1 0 1 1 s s α β α β π β β − − − − + = − − (5) where α, β are two non-dimensional Dundurs’ parameters representing the material mismatch between adjacent layers[25]. For the plane strain problem, the Dundurs’ parameters α , β can be
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