World Journal Of Engineering Equation Chapter 0 Section 1a

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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   n1 ,    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
m0
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
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