13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- directed along layering, axis 2x is directed along the normal. For anisotropic (orthotropic) coating modulus *E should simply be replaced with the longitudinal modulus: * 22 E E= . For multilayered structures, if the thickness of individual layers are much less than the characteristic size of delaminated part of coating, the layered structure may be described as an effective homogeneous anisotropic media [15]. The value of effective longitudinal modulus of coating may be calculated as an averaged over the individual layers forming its delaminated part * * i i i E h E h = ∑ ∑ (14) Here *, i i E h are elastic moduli and thicknesses of the individual layers. Similarly, for the multilayered substrate the effective elastic constants may be expressed as follows 11 12 22 66 * * * * 11 12 22 66 / / i i i i i i i i i i i i c h c h h c h c c c c c h h h h = = = = ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ (15) Coefficients of matrix of elastic compliance ijβ to be substituted to (13) are obtained from here by inversing the matrix of elastic constants. * * * 1 22 11 11 22 66 66 * * * 2 * * * 2 11 22 12 11 22 12 c c c c c c c c c − β = β = β = − − (16) It is worth to note, that the above formulae describe the crack propagation in multilayered coating both between the individual layers and between the whole package forming the coating and substrate. The difference consists in accounting the particular number of layers while calculating the effective properties Eq. (14), Eq. (15). 5. Model of generalized elastic clamping The model of elastic clamping may be generalized in order to account the influence of longitudinal force on the boundary conditions. Such a model was used by [5] for finding the critical stress of buckling of the delaminated coating. In the frame of that model it is supposed that longitudinal displacement u and angle of rotation are linear function of longitudinal force F and bending moment M acting at the point of clamping 1 0 11 12 1 2 0 21 22 EU a F a h M a h F dV E dx a h M − − − = + + = (17) Although condition of elastic clamping Eq. (6) was proven to be asymptotically correct [16], and other terms might be beyond the accuracy of beam (plate) theory, accounting for additional terms in Eq. (17) may be useful in numerical calculations. The direct application of boundary condition Eq. (17) to Eq. (4) is impossible due to presence of unknown parameter F, which, however, may be found by solving the equation for longitudinal
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