acta mechanica et automatica, vol.8 no.2 (2014), DOI 10.2478/ama-2014-0015
STRESS DISTRIBUTION IN A COMPOSITE COATING BY LOCAL LOADING OF THE FREE SURFACE
Roman KULCHYTSKY-ZHYHAILO*, Waldemar KOŁODZIEJCZYK*
*Faculty
of Mechanical Engineering, Bialystok University of Technology
ul. Wiejska 45C, 15-351 Białystok, Poland
[email protected], [email protected]
Abstract: A three-dimensional problem of the theory of elasticity for halfspace with multilayered coating with periodical structure is considered. The fundamental layer consists of two layers with different thicknesses and different mechanical properties. The coating is described
by the homogenized model with microlocal parameters. The solution is derived by using integral Fourier transform. Calculations were conducted with the assumption of elliptical distribution of normal and tangential tractions applied to the surface of the layered system in a circular area. Analysis of the stresses was restricted to the first principal stress distribution.
Key words: Gradient Coating, Three-Dimensional Problem of Elasticity, Normal Loading, Shear Loading
1. INTRODUCTION
In the mechanics of contact interaction, much attention is now
given to coatings used for the improvement of the tribological
characteristics of friction couples. Thus, the coatings formed by
periodically deposited elastic layers are now extensively investigated (Farhat et al., 1997; Voevodin et al., 2001) parallel with the
uniform coatings (Schwarzer 2000; Bargallini et al., 2003;
Kulchytsky-Zhyhailo and Rogowski, 2010) and nonuniform coatings whose mechanical properties are described by continuous
functions of the distance from the surface (Guler and Erdogan,
2007; Liu et al., 2008; Kulchytsky-Zhyhailo and Bajkowski, 2010).
In the analysis of the stressed state, the researchers, as a rule,
focus their attention on the evaluation of the tensile and Huber–
Mises stresses described by the second invariant of the deviator
of the stress tensor.
Considering contact problems for multilayered coatings in
classical approach we need to solve partial differential equations
for every sublayer and satisfy continuity conditions on the interfaces.
z
h=H/a
-1
1
2
1
2
p
1
z
x
E1 ,  1
E2,  2
...
...
E1 ,  1
E2,  2
E0,  0
x
h1=H1 /a
h2=H2 /a
-1
p
1
model
homogenized
homogenizowany
h=H/a
x
A coating with periodical structure can be replaced by a homogeneous one by using e.g. homogenized model with microlocal
parameters (Matysiak and Woźniak, 1987; Woźniak, 1987).
As it was show in earlier papers (Kołodziejczyk and Kulchytsky-Zhyhailo, 2013; Kulchytsky-Zhyhailo and Kołodziejczyk, 2005;
Kulchytsky-Zhyhailo, 2011) the stress distribution in the substitute
homogeneous medium is a good approximation of stress distribution in the multilayered medium when the ratio of thickness
of a fundamental layer to a specific size of contact area is less
than 0.1.
In the present work, we consider a three-dimensional problem
of elastic half space with laminated coating of periodic structure
loaded by normal and tangential tractions.
The multilayered coating is replaced by a homogeneous one
which mechanical properties are described by the homogenized
model with microlocal parameters. The objective of this work is to
determine relations between applied loading and stresses in the
coating and in the substrate as well as analysis of tensile stresses
and Huber-Mises stresses in the coating due to loading applied in
a circular area on the free surface.
medium
1
x
0
E0,  0
x
x


Fig. 1. The scheme of the problem
83
Roman Kulchytsky-Zhyhailo, Waldemar Kołodziejczyk
Stress Distribution in a Composite Coating by Local Loading of the Free Surface
2. FORMULATION OF THE POBLEM
where elastic potentials satisfy equations:
Let us consider a non-uniform elastic half-space loaded by
normal and tangential
traction in the region with the specific size
on the free surface (Fig. 1). The non-uniform half
space is formed by the homogeneous isotropic half space with
Young’s modulus
and Poisson’s ratio
and a system of two
periodically deposited elastic layers with thicknesses
and
(
is the thickness of the fundamental layer),
Young’s moduli
and
, and Poisson’s ratios
and ,
respectively. Assume that the conditions of perfect mechanical
contact are realized between the layers of the coating and between the coating and the substrate.
Since the nonhomogeneous coating is described by the homogenized model with microlocal parameters (Matysiak and
Woźniak, 1987; Woźniak, 1987) the governing equations take the
form:
A1u x, xx  0.5 A1  A2 u x, yy  A5 u x , zz 
1
1
1
0.5 A1  A2 u y1,xy   A3  A5 u z1,xz  0,
0.5 A1  A2 u y1,xx  A1u y1,yy  A5u y1,zz 
(1b)
0.5 A1  A2 u x1,xy   A3  A5 u z1,yz  0,



(1a)

(1c)
where
– macro-displacements vector (displacements averaged in a fundamental layer),
– dimensionless coordinates
(Cartesian coordinates related to a specific size of contact area
a),
,
, ,
– coefficients
calculated from known relations (Matysiak and Woźniak, 1987;
Kaczyński, 1994; Kulchytsky-Zhyhailo, 2011; Kołodziejczyk
and Kulchytsky-Zhyhailo, 2013) on the base of mechanical
and geometrical properties of alternating layers:
~ [ ]
[ ]
A1    2~ 
, A2   
,
ˆ
  2ˆ
ˆ  2ˆ
~
,
(2b)
are roots of the characteristic equation:
A1 A5 4  ( A32  2 A3 A5  A1 A4 ) 2  A4 A5  0 ,
i 
 i2 A1  A5
A3  A5
,  32 
2 A5
.
A1  A2
Considered boundary problem, formulated in dimensionless
coordinates related to the specific size a of the loading area, leads
to equations of theory of elasticity (2b) defined in the appropriate
homogenized medium and equations
(1  2 0 )u 0   grad div u 0   0
(3)
defined in the substrate, with boundary conditions:
 normal and tangential tractions on the surface,
 zz(1) ( x, y, z)   p( x, y) ,  xz(1) ( x, y, h)   x ( x, y) ,
(4a)
 yz(1) ( x, y, h)  0 , (x,y) Ω,
(4b)
 ideal contact between coating and substrate,
A5 u z1,xx  u z1,yy  A4 u z1,zz   A3  A5  u x1,xz  u y1,yz  0,
2
i , xx  i , yy   i2 i , zz  0,
2
u x(0) ( x, y,0)  u x(1) ( x, y,0) , u (y0) ( x, y,0)  u (y1) ( x, y,0) ,
(4c)
u z(0) ( x, y,0)  u z(1) ( x, y,0) ,
(4d)
 zz(0) ( x, y,0)   zz(1) ( x, y,0) ,  xz(0) ( x, y,0)   xz(1) ( x, y,0) ,
(4e)
( 0)
(1)
 yz
( x, y,0)   yz
( x, y,0) ,
(4f)
 vanishing displacement components:
u x(i ) , u (yi ) , u z(i )  0, x 2  y 2  z 2   , i=0,1,
(4g)
where u(0) is the displacement vector in the substrate.
The characteristic feature of the homogenized model is that
it gives different expressions for calculation of the stress tensor
components, which experience a jump on the interfaces between
layers.
~ [ ]([  ]  2[ ])
A3   
,
ˆ  2ˆ
~
([ ]  2[  ]) 2
[  ]2
A4    2~ 
, A5  ~ 
,
ˆ
ˆ  2ˆ
1,k 
 xx
 K k u x1,x  Lk u y1,y  M k u z1,z , k  1,2 ,
(5a)
~
  1  (1   )2 , ~  1  (1  )2 ,
1,k 
 yy
 Lk u x1,x  K k u y1,y  M k u z1,z , k  1,2 ,
(5b)
[ ]   (1  2 ) , [ ]   (1   2 ) ,
1,k 
 xy
  k u x1,y  u y1,x , k  1,2 ,
(5c)

2

2
H
ˆ  1 
2 , ˆ  1 
2 ,   1 ,
H
1 
1 
Ei i
Ei
i 
,  
.
1   i 1  2 i  i 21   i 
where 
is the stress tensor in
layer of the fundamental layer (
),
, ,
– coefficients calculated from
known expressions (Kaczyński, 1994):
Equations (1) have the same form as equations of theory
of elasticity for a transversely isotropic solid. Therefore their solution can be expressed in terms of elastic potentials proper to a
transversely isotropic medium (Elliot, 1949):
u x(1)  1, x  2, x  3, y , u (y1)  1, y  2, y  3, y ,
u z(1)  11, z   2 2, z ,
84
(2a)
K k  k  2 k  hk k
M k  k  hk k
 
ˆ  2ˆ
   2  ,
ˆ  2ˆ
, Lk  k  hk k
h1  1, h2  
 
ˆ  2ˆ
,

.
1 
It is essential to note that discussed stresses differ significantly in the individual sublayers composing a fundamental layer
(Kulchytsky-Zhyhailo, 2011; Kulchytsky-Zhyhailo and Kołodziej-
acta mechanica et automatica, vol.8 no.2 (2014), DOI 10.2478/ama-2014-0015
czyk, 2005; Kołodziejczyk and Kulchytsky-Zhyhailo, 2013).
The remaining stress tensor components can be obtained using the following relations:
 zz(1)   zz1,k   A3 u x1,x  u y1,y   A4 u z1,z , k  1,2 ,

(1)
xz
(5d)



 A u    u   , k  1,2 .
1,k 
1
1
1
y, z
1
z, y
  xz  A5 u x, z  u z , x , k  1,2 ,
1,k
(1)
 yz
  yz
5
(5e)
(5f)
3. SOLUTION OF THE POBLEM

  f ( x, y, z) exp( ix  iy )dxdy .



The solution in the substrate can be expressed as:
~
s 2u~x0   , , z   i10   , , z   i~ 0   , , z  ,
(7a)
s 2u~y0   , , z   i1
(7b)
~ 0 
 ,, z   i~ 0  ,, z  ,
2u~z0   , , z   d 0 za1  ,   2a0  , expsz  ,
where
~
210   , , z   2  d 0 a1  ,   d 0 sza1  ,  
2a0  , s exp sz 
(7c)
,
~ 0   , , z   a7  , expsz  .
(7d)
(7e)
Formulas (6) and (7) contain 9 unknown functions of the transform parameter. Satisfying boundary conditions we obtain two
systems of algebraic equations to determine unknown functions.
They contain 6 and 3 equations respectively:
A  A   a  ,   A  A   a  ,   ~p ,  ,(8a)
2
4 1 1
2
3
4
2
2 2
1  1 11a1  ,    2  1 21a3  ,  
i ~
 x  ,  ,
A5 s
i i1 a2i 1  , ci  a2i  , si   a0 s  0 ,
(9b)
A5 31 a5  , c3  a6  , s3   0a7  ,   0 .
(9c)
a j  ,   
 2  d 0 a1  ,   2a0  , s  0,
 A  
i 1
2
4 i i

 A3 a2i1  , si  a2i  , ci 
 0 a1  ,   2a0  , s   0,


(8b)
(8c)
(8d)
i~x  ,   
a j s , i  5,6,7 .
sA5
(10b)
Solving obtained linear equations and taking into account relationship between displacements and stresses and elastic potentials we obtain solution of the problem in Fourier transform domain.


2
~xx p    S xy p1 s, z   2 S xy p2 s, z  ~p  ,  ,
s


(11a)
~
 2  p
 p    p 

~ yy
  S xy1 s, z   2 S xy
2 s, z  p  ,  ,
s


(11b)
~zz p   S z1p  s, z ~
p ,  ,
(11c)
~xy p  

~xz p  
i  p 
S z 2 s, z ~
p  ,  ,
s
(11e)
~ yz p  
i  p 
S z 2 s, z ~
p  ,  ,
s
(11f)
s
2
 p
~
S xy
2 s, z  p  ,  ,
(11d)
~xx  

i
s
(12a)
  
~
 2  
 
 S xy1 s, z   2 S xy

2 s, z   2 S xy3 s, z   x  , ,
s



 
~yy


~zz  

i  
 
S xy1 s, z   S xy
2 s, z  
s
 
 
~
 2 s  2 S xy
2 s, z   2 S xy3 s, z   x  , ,



2
i 1

where: ci  cosh  i1sh , si  sinh  i1sh , i  1, 2, 3.
Solution of these two systems of equations can be normalized
as follows:
~
p  ,   p 
i~x  ,   
a j  ,  
a j s  
a j s ,
A5
sA5
(10a)
i 1
2 a2i 1  , si  a2i  , ci 
(9a)
a5  , s3  a6  , c3  b0  ,   0 ,
4
2
2
i ~
 x  ,  ,
A5 s
i  1,0,...,4,
The Fourier transforms of the elastic potentials have the following form:
~
i  s 2 a2i 1  , sinh  i1sh  z  
(6)
 s  2 a2i  , cosh  i1sh  z  , i  1,2,3.
3
 31a5  ,  


(8f)
i 1
 0 1  d 0 a1  ,   2a0  , s   0,

General solution of differential equations defined in the coating and in the substrate we obtain using two-dimensional Fourier
transform:
~
1
f ( , , z ) 
2
2
A5   i  1 i1 a2i1  , ci  a2i  , si 
i  
S z1 s, z ~x  ,  ,
s
(12b)
(12c)
~xy  
(8e)

(12d)
~
i   
 2  
 
 S xy3 s, z   2 S xy

2 s, z   2 S xy3 s, z   x  , ,
s 
s



85
Roman Kulchytsky-Zhyhailo, Waldemar Kołodziejczyk
Stress Distribution in a Composite Coating by Local Loading of the Free Surface
  2  

 2  


S
s
,
z

S z 3 s, z ~x  ,  ,
z
2
2
2
s
s

(12e)
S   s, z   S   s, z ~  ,  ,
(12f)
~xz   
~yz  

s
2


z2
z3
x
where the form of introduced functions S depends on the location
of the investigated point. Moreover functions with subscripts
defined in the coating depend on the number of a sublayer
in a fundamental layer and are given by:

2


 j,k 
1
2
S xy
 K k a2 ij 1Si  a2 ij Ci , j  p, , k  1,2
1   A5 M k  i i
i 1
2


 j,k 
1
 j
 j
,
S xy
2  2 A5 k  a2 i 1Si  a2i Ci , j  p, , k  1,2
i 1


 , k 
 
 
1
S xy
3   A5 k a5 S3  a6 C3 , k  1,2 ,

2


S z1j    A51 A3  A4 i i 2 a2 ij 1Si  a2 ij Ci , j  p, ,
i 1

2

S z 2j     i  1 i1 a2 ij 1Ci  a2 ij Si , j  p, ,
i 1


S z3   31 a5 C3  a6 S3 ,
where:
Si  sinh  i1sh  z  , Ci  cosh  i1sh  z  , i  1, 2, 3.




The corresponding functions in the substrate are as follows:
 j
 j
 j
 j
01 A5 S xy
1  2d 0  1a1  d 0 sza1  2a0 s exp sz ,
 j
 j
 j
 j
01 A5 S xy
2  d0  2a1  d0 sza1  2a0 s exp sz ,
1
 j
0
5 z1
 AS


  a1  d0 sza1  2a0 s expsz ,
 j
 j
 j
Tab. 1. Mechanical and geometrical properties elements
of the discussed non-uniform half space
Material
substrate
Steel
TiN
coating
Ti
 AS
 
0.25
0.18
0.32
4.7/1.2
0.6
2z
0.4
0.2
a)
0
-2
-1.5
-1
-0.5
 
Taking the inverse Fourier integral transforms, we obtain relations (in terms of double integrals) between displacement vector
components and stress tensor components in the coating and
in the substrate and functions describing loading distributions.
H1/H2
0.8
  A S z 3  a7 expsz  .
1
0
5

The integrals at internal points of the nonuniform half space
(
) are taken with the help of the Gaussian quadrature.
On the surface
, we take into account the asymptotic
behavior of the solution of the system of equations obtained
as the parameter of the integral transformation tends to infinity.
The integrals in which the integrands are replaced by their asymptotics are taken analytically. To find the remaining integrals, we
apply the Gaussian quadrature.
Fig. 2 illustrates distributions of the first principal stress 
in sublayers of the coating with greater Young modulus in the
plane
. Figs. (2a) and (2b) show an interaction in case of
normal traction. Contours of  distribution caused by tangential
tractions are shown in Figs. (2c) and (2d). It can be seen that
tensile stresses arise on the unloaded part of the surface of the
half-space. The maximum value of 1 appears close to the
point
,
,
and increases with increasing
value of the friction coefficient. For specific thickness of the coating tensile stresses additionally appear in the vicinity of the coating-substrate interface.
Fig. 3 show distributions of the second invariant of deviatoric
stress tensor (
√ ) in sublayers with greater Young
modulus. It can be distinguished two local maxima of : (1) at the
point
,
,
(2) on the coating-substrate interface or in its vicinity.
01 A5 S z 2j   d0  1a 1j   d0 sza 1j   2a0 j s expsz , j  p, ,
1
 
0
5 xy3
E (GPa)
220
440
120
0
0.5
1
1.5
0.5
1
1.5
0.5
1
1.5
0.5
1
1.5
2
x
0.8
0.6
z
0.4
0.2
b)
0
-2
-1.5
-1
-0.5
0
2
x
0.8
4. RESULTS AND DISCUSSIONS
0.6
2z
0.2
Calculations were made under assumptions:
 the tangential tractions are related to the normal tractions
by the Amontons-Coulomb law of friction:
where
is the friction coefficient
 the axisymmetrical pressure in the form ( )
(
) is applied over a circular area of radius .
 mechanical properties of the substrate and the coating as well
as the thickness ratio of sublayers forming fundamental layer
were taken from the literature (Voevodin et al., 2001). Details
are given in Tab. 1.
86
0.4
c)
0
-2
-1.5
-1
-0.5
0
2
x
0.8
0.6
z
0.4
0.2
d)
0
-2
-1.5
-1
-0.5
0
x
2
Fig. 2. Distribution of the first principal stress  in sublayers with greater
Young modulus in regions in which 
:
,
;
,
; c)
,
; d)
,
.
acta mechanica et automatica, vol.8 no.2 (2014), DOI 10.2478/ama-2014-0015
REFERENCES
0.8
0.6
2z
0.4
0.2
a)
0
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
x
0.8
0.6
z
0.4
b)
0.2
0
-2
-1.5
-1
-0.5
0
0.5
1
1.5
0.5
1
1.5
2
x
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0.6
2z
0.4
0.2
c)
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-2
-1.5
-1
-0.5
0
2
x
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0.6
z
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0.2
d)
0
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
x
Fig. 3. Distributions of the second invariant of deviatoric stress tensor
in sublayers with greater Young modulus: a)
,
;
b)
,
; c)
,
;
d)
,
5. CONCLUSIONS
Calculations show that:
 Distribution of the first principal stress 1 in layers with greater
Young modulus is similar to that in a homogeneous coating
when
(Kulchytsky-Zhyhailo and Rogowski, 2007, 2010; Schwarzer, 2000). Values of tensile
stresses (if exist) on the coating-substrate interface are lower
than in a homogeneous coating with the same Young modulus.
 Distributions of the stress tensor components which experience a jump on the interfaces are different in layers with
greater Young modulus from distributions in layers with smaller Young modulus.
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The investigations described in this paper are a part of the research
projects S/WM/1/2013 and S/WM/1/2012 realized at Bialystok University
of Technology.
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