13th International Conference on Fracture June 16–21, 2013, Beijing, China -6- displacement u , simultaneously with Eq. (3). That was done in [5] with the final result for the critical stress after writing it in the form convenient for our purposes being: 2 12 22 11 12 tan 0 1 a b a a h σ σ π γ π γ + − = + (18) The coefficient of symmetric matrix ij a of elastic clamping were calculated in [5] by numerical solving of a system of integral equations as functions of ratio of moduli of coating and substrate and ratio of coating thickness to the delamination length. Taking into account that according to Eq. (12) 3 * * 22 0 12 / S a d E E = (19) expression Eq. (18) differ from Eq. (11) only by its last term, which is usually not large; nevertheless its accounting leads to (slight, but systematic) deviation of the results from the master curve, obtained by using more simple Eq. (11). Dependence of σγ on β according to this model is also presented on Fig. 2, coefficient 11 a was calculated with the help of the first formula of Eq. (20) for * * / 1 S E E = . If the initial surface is curved, then in addition to longitudinal force and bending moment, transverse force N appears at the clamping point. Hence, condition Eq. (17) may be generalized: 1 0 11 12 13 1 2 1 0 21 22 23 1 0 13 23 33 EU a F a h M a N a h F a h M a h M EV a F dV E dx a h M a N − − − − − = + + + + = + + = (20) Here 3x3 matrix ij a may be called the extended matrix of coefficient of elastic clamping. Calculating coefficients of the extended matrix ij a may become useful for studying delamination and buckling of initially curved coatings. 6. Calculating coefficients of matrix of elastic clamping Coefficients of matrix (20) were calculated in [4] for a particular geometric parameters numerically, and in [5] by numerical solving a system of integral equations for various ratios geometric parameters and moduli. However it is desirable to have an asymptotical, or at least, approximating formulae. Formulae for 22 a were obtained in [11] on the base of the model where the coating was considered as a plate and the substrate as half-plane by using Fourier transformation and Wiener-Hopf technique. The value is given by Eq. (8), Eq. (19). The value of 23 a were found (ibid.) to be ( ) ( ) 2/3 2/3 7/3 2/3 23 2 3 / 2.52 / S S a E E E E − = = (21) Coefficients 11 12 , a a (as well as some coefficients of the extended matrix Eq. (16)) for the case of
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