PSI - Issue 2_B

K.B. Ustinov / Structural Integrity Procedia 00 (2016) 000 – 000

8

K.B. Ustinov / Procedia Structural Integrity 2 (2016) 3439–3446

3446

Here b is the half length of the delamination length. It is seen that the ratio of the critical stress, corresponding to buckling of the delaminated parts of coatings and the critical stress of a rigidly clamped plate, 2 2 2 2 (2) 0 12 h b E    , is determined by a single nondimensional parameter  , which is a combination of the elastic constants of the coating and substrate, and the ratio of the coating thickness and delamination length. The remaining formulae for components of matrix A appear to be useful for solving problems related to coating delamination from curved substrates (Ustinov and Kasparova, 2015).

Acknowledgements

The work was done under financial support of Program I.33 of the Presidium of RAS

References

Abrahams, I.D., 1998. On the non-commutative of factorization of Wienner-Hopf kernels of Khrapkov type. Proceedings of the Royal Society A 454, 1719-1743. Antipov, D.A., Silvestrov, V.V., 2002. Factorization on a Riemann surface in scuttering theory. The Quarterly Journal of Mechanics and Applied Mathematics 55, 607-654. Chebotarev, G.N., 1956. To Solution of Riemann Boundary-Value Problem in Closed Form for Systems of n Pairs of Functions], Uch Zap Kazan Universitet 1166 (4), 31-58 [In Russian]. Cotterell, B., Chen Z., 2000. Buckling and cracking of thin film on compliant substrates under compression. International Journal of Fracture 104 (2), 169 – 179. Daniele, V.G., 1978. On the factorization of Wiener- Hopf matrices in problem solvable with Hurd’s method. Trans ANTENNAS propagate. 26 614-616. Dundurs, J., 1968. Discussion: “Edge - bonded dissimilar orthogonal elastic wedges under normal and shear loading” (Bogy, D. B., 1968, ASME Journal of Applied Mechanics, 35, 460 – 466). Transactions ASME Journal of Applied Mechanics 36 (3), 650-652. Goldstein, R.V., Ustinov, K.B., Chentsov, A.V., 2011. Influence of substrate compliance on stresses generating stability loss of a delaminated coating. Computational Continuum Mechanics 4 (3), 48-57. Jones, D.S., 1984. Commutative Wiener-Hopf factorization of a matrix. Proceedings of the Royal Society A. 393, 185-192. Heins, A.E., 1950. System of Wiener-Hopf equations. Proceeding of Symposia in Applied mathematics II. McGraw-Hill, 76-81. Hutchinson, J.W., Suo, Z., 1992. Mixed Mode Cracking in Layered Materials. In Hutchinson J.W., Wu T.Y. (Ed.) Advance in Applied Mechanics 29, 63-191. Khrapkov, A.A., 2001. Wiener-Hopf method in mixed elasticity theory problems. St. Petersburg, VNIIG im B.E. Vedeneeva 144p. Moiseev, N.G., 1989. Factorization of Matrix Functions of Special Form. Soviet mathematics - Doklady 39 264-267 [In Russian]. Parry, G., Colin, J., Coupeau, C., Foucher, F., Cimetière , A., Grilhé , J., 2005. Effect of substrate compliance on the global unilateral post buckling of coatings: AFM observation and finite element calculations. Acta Materialia 53, 441-447. Salganik, R. L., 1963. The Brittle Fracture of Cemented Bodies. Journal of Applied Mathematics and Mechanics 27 (5), 1468-1478. Salganik, R.L., Ustinov, K.B., 2012. Deformation problem for an elastically fixed plate modeling a coating partially delaminated from the substrate (plane strain). Mechanics of Solids 47 (4), 415-425. Ustinov, K.B., 2014. On Shear Separation of a Thin Strip from the Half-Plane. Mechanics of Solids 49 (6), 713-724. Ustinov, K.B., 2015. On influence of substrate compliance on delamination and buckling of coatings. Engineering Failure Analysis 48B, 338-344. Ustinov, K.B., 2015a. On separation of a layer from the half-plane: elastic fixation conditions for a plate equivalent to the layer. Mechanics of Solids 50 (1), 62-80. Ustinov, K.B., 2015b. On delamination of a stipe along the boundary between two elastic layers. Part 1, problem formulation, the case of normal crack . № 4. 226 -245 [in Russian]. Ustinov, K.B., Kasparova, E.A. 2015. Influence assessment of curvature and yielding of substrate on parameters of coating delamination. Deformation and fracture of materials. No. 3, 28-35 [In Russian]. Vorovich, I.I., Aleksandrov, V.M., Babeshko, V.A., 1974. Nonclassical mixed problems in theory of elasticity. Moscow, Nauka. 456 p. [In Russian]. Yu, H.-H., Hutchinson, J.W. 2002., Influence of substrate compliance on buckling delamination of thin films. International Journal of Fracture 113, 39-55. Zlatin, A.N., Khrapkov, A.A., 1985. Vector Riemann Problem with Zero Index of Matrix-Coefficient Exponent. Izv VNIIG im B.E. Vedeneeva. 181, 12-16 [In Russian]. Zlatin, A.N., Khrapkov, A.A., 1986. A Semi-Infinite Crack Parallel to the Boundary of the Elastic Half-Plane, Dokl. Akad. Nauk SSSR. 31, 1009-1010. Zlatin, A.N., Khrapkov, A.A., 1990. Elastic Half-Plane Weakened by a Crack Parallel to Its Boundary, Studies in Elasticity and Plasticity, 16: Problems of Contemporary Fracture Mechanics, 68-75 [in Russian].