13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- 2 2 * * 0 2 2 0 12 1 cr h E E E b π σ ν = = − (2) In [9-10] and here estimations for coefficient σγ are obtained with the help of the theory of plates. 3. Model of simple elastic clamping In the framework of von Karman theory the plate deflection is ( ) ( ) ( ) 3 * '' 0 12 IV h E v x x hv x σ + = (3) Here ( )x σ is the compressive stress within the cross-section of the coating. It should be noted that this stress is generally differ from 0σ due to relaxation caused by curving of the delaminated part of the coating. For b y b − < < , ( ) cr x const σ σ = = the general solution of Eq. (3) is ( ) 1 2 cos , v x A kx A x b = + < (4) * 2 12 cr k E h σ = (5) Here constants iA need to be determined from the boundary conditions. As a variant of the solution it may be suggested that the delaminated section of coating could be treated as a plate with elastically clamped ends. Condition of elastic clamping at points x b =± is ( ) ( ) ' '' x b x b v x hdv x = = = (6) Here hd is the coefficient of proportionality between the angle of the plate at clamping and the second derivative of the displacement (proportional, in turn, to the bending moment, acting at this point); d is a dimensionless coefficient that can be not determined in the frame of elementary theory. The presence of plate thickness h is due to necessity of adjusting dimensions: this parameter is the only one of the dimension of length in the model, because the clamping stiffness may not depend on the plate size b (for more information see [9-10]). Substitution of Eq. (4) into Eq. (6) yields tan 0 kb khd + = (7) The expression for coefficient d were obtained in [11] on the base of the model where the coating was considered as a plate and the substrate as half-plane. The solution was found with the help of Fourier transform and Wiener-Hopf technique. By neglecting the action of shear stress it was found 3 * * 0 s d d E E = , 2/3 5/6 0 2 3 0,636 d − = ≈ (8) Here ( ) * 2 1 E E = −ν (9) For rigid clamping
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