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World Journal Of Engineering A BOUNDARY PERTURBATION ANALYSIS FOR A MULTILAYER FILM COATING Kostyrko S.A., Grekov M.A. St. Petersburg State University, Universitetski pr., 35, St. Petersburg, 198504, Russia, with H N hN , H N 1 0, H i H i 1 hi , i 1, N j {z z z j x1 iH i }, planar interfaces In recent years, much attention has been focused on development and investigation of the technologies that rely on the use of multilayer thin film structures. In some engineering applications involving multilayers often leads to improvement of mechanical, optical, electrical and magnetic properties. However, high device quality can only be maintained if the defects in films are kept to minimum. It’s well known that during film deposition and subsequent thermal processing the film surface frequently becomes unstable and evolves into an undulating profile with cusp-like valleys [1]. The stress concentration caused by such film roughening is found to create dislocations, microcracks and lead to delamination. Thus, the main interest of this study lies in the stress concentration along a rough surface of a multilayer film structure. The elastic solution for undulating free surface in stressed multilayer film can be found in [2]. In this paper authors used composite beam theory and linear boundary perturbation techniques to analyze the stress distribution along a sinusoidal rough surface. Since a surface evolution forms a wide range of configurations [1], it is of fundamental as well as practical interest to derive a solution for stress state in multilayered structure with an arbitrary undulated free surface. We consider plane strain problem for a thin j 2, N 1 . We assume that the film surface has an arbitrary small perturbation 1 : 1 x1 i H1 f ( x1 ) and free from loads. Here f ( x1 ) f ( x1 ), max f ( x1 ) , f ( x1 ) 1/ , 0 1 . Thereby, the boundary conditions we can write as n ( 1 ) 0, 1 1 u ( zi ) u u 0, n ( zi ) n n u lim u ( z ), n lim n ( z ), zi i z zi i 0 ij lim ij , lim i i n , z k , i 1, N ( z) n nN 1 , z N 1 N i 1 n separate layers deposited on a substrate of thickness hs h f , Young’s modulus E s and u i u i 1 , z j , i 1, N u( z) u N 1 , z N 1 Poisson’s ratio s . Suppose that i -th layer has thickness hi , Young’s modulus Ei and Poisson’s ratio i . The substrate is modeled as an elastic N 1 z : Im z 0,Re z R1 (4) Here n1 , u1 are the vectors of stresses and displacements arisen in a homogeneous half-plane with wavy boundary 1 and elastic properties of the layer 1 under action of unknown surface load under remote loads 11 N 1 . And the film is modeled as coherently bonded strips N (3) x2 where is a rotation angle of a material particle. Following the superposition principle, the displacement vector u u1 iu2 at point z x1 ix2 and stress vector n nn i nt at this point associated with a plane with a normal vector n is represented as film of total thickness h f hi which consists of half-plane (2) z zi i 0 22 i 12 0, 11 n1 , 0 x2 (1) q ; nj , u j are the vectors of stresses and displacements arisen in a two-component plane with elastic properties of the solids j 1 , j under i , i {z Hi 1 Im z H i , Re z R1}, i 1 599 World Journal Of Engineering remote loading 11 j with the unknown jumps of tractions ni and displacements u i at the planar interface i . nim ( x1 ) Boundary conditions at k , k 1, N 1 (1)–(3) lead to following 2 N 1 boundary equations Ki 2 ( x1 , t ) nim (t )dt u im ( x1 ) u 2 ( z2 ) u1 (q, z2 ) u 3 ( n3 , u 3 , z2 ) 0 ( zl ) ( l 1 n u ( zl ) u ( l 1 l 1 n , u , zl ) (5) where l 3, N . The stresses and the displacements are related to Goursat-Kolosov’s complex potentials. We expand these complex potentials as well as unknown functions q , ni , ui as a power series in small parameter m m 0 m! ni ( z ) qm ( 1 ), m m 0 m! m m0 m! u i ( z ) (t )dt nim ( z ), (7) im n (t )dt Ki8 ( x1, t )u im (t )dt A solution of this problem is obtained by expansion in a Fourier series. Thereby, developed a new method which allows to get an approximate solution of a corresponding boundary valueproblem for a thin multilayer film/substrate system with arbitrary perturbed free surface. Note, that the accuracy of analysis can be improved by considering higher orders of perturbation. Stress concentration analysis of two-layer thin film is then performed by means of the method described above. The influence of different shapes of undulated film surface on the stress distribution in the film is examined. N u N 1 ( z N 1 ) u N ( nN , u N , z N 1 ) 0 im ( z N 1 ) ( , u , z N 1 ) 0 N n q 1 Ki3 ( x1, t )u Ki 9 ( x1 , t )u im (t )dt F2 m ( x1 ), m 0,1, , u , zl ) l 1 N n Ki 6 ( x1, t ) u l 1 ( nl 1 , u l 1 , zl ) 0 N 1 n Ki 7 ( x1 , t ) nim (t )dt l 1 nl 1 ( nl 1 , u l 1 , zl ) 0 l (t )dt n2 ( z2 ) n1 (q, z2 ) n3 ( n3 , u 3 , z2 ) 0 l 1 n im n Ki 4 ( x1 , t )u im (t )dt F1m ( x1 ), m 0,1, q ( 1 ) n1 (q, 1 ) 0 l n Ki1 ( x1, t ) Acknowledgements The work was supported by Saint-Petersburg State University under grants 9.0.165.2009 and 9.37.129.2011 and Russian Foundation for Basic Research under grant 10-01-00093. (6) u im ( z ) References After that taking into account boundary equations (5) we derive a system of 2n 2 Fredholm integral equations of the second kind in the unknown expansion coefficients nim , uim (7). Equations (7) is the same for each-order approximation but right hand sides F1m ( x1 ), F2 m ( x1 ) of it depends on all solutions of previous levels of approximation. 1. Gao H., Nix W. D. Surface roughening of heteroepitaxial thin films // Ann. Rev. of Materials Science. 1999. V. 29. P. 173– 209. 2. Kim J.-H., Vlassak J.J. Perturbation analysis of an undulating free surface in a multi-layered structure // Int. Journal of Solids and Structures. 2007. V. 44. P.7924-7937 600