Elastic stress analysis of bi-layered isotropic coatings and substrate subjected to line scratch indentation

j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 1 9 6 ( 2 0 0 8 ) 213–221

journal homepage: www.elsevier.com/locate/jmatprotec

Elastic stress analysis of bi-layered isotropic coatings and substrate subjected to line scratch indentation M. Shakeri, A. Sadough, S. Rash Ahmadi ∗ Mechanical Engineering Department, Amirkabir University of Technology, Tehran, Iran

a r t i c l e

i n f o

a b s t r a c t

Article history:

In this study, a model is developed to predict the stresses in thin coatings induced by a line

Received 4 September 2006

scratch indenter. In this case, the plane problem of surface loading of double elastic layers

Received in revised form 4 May 2007

perfectly bonded to an elastic dissimilar half-plane is considered. Fundamental solutions

Accepted 18 May 2007

are obtained for forces acting perpendicular and parallel to the layer surface. The stress and displacement fields are calculated for the coatings and the substrate due to these forces. These results are compared with the results of finite element method (FEM) by using ANSYS

Keywords:

software, as well as with the results of single isotropic-coated system and an uncoated

Coating

elastic half-plane. These equations are useful for ranking the coating–substrate adhesion of

Scratch

different coated systems, or for estimating the critical mean stress for interfacial failure.

Stress analysis

© 2007 Elsevier B.V. All rights reserved.

FEM Multilayer Adhesion

1.

Introduction

Coating is widely used in optical, microelectrical, biomedical and decorative applications. The coating is designed to prepare favorable mechanical (i.e. low friction, abrasion resistance), chemical (i.e. barrier for gasses), optical, magnetic and electrical properties to various substrates. In general, the functional behavior of coated systems depends on the substrate and coatings properties (Blees and Wenkelman, 2000). Evidently, the durability and function of coating critically depends on adhesion between the coating and substrate (Blees and Wenkelman, 2000). Many situations in engineering require the transmission of loads through contact between different components and parts of assemblies. Often the contacting surface must be allowed to undergo relative sliding with respect to each other, for example, in ball bearing and journal bearing assemblies or between the fan and blades and root disks in aero-engines. Coatings are used in order to

∗

Corresponding author. E-mail address: [email protected] (S.R. Ahmadi). 0924-0136/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2007.05.038

achieve superior mechanical properties that are distinctly different from those of the substrate, e.g. increased hardness and stiffness, or lower stiffness and lower coefficient of friction (Ma and Korsunsky, 2004). Contact between coated bodies increases the complexity of stress states in the coating layers and substrate, affected by the elastic properties of the coatings, substrate, the friction coefficient, etc. (Ma and Korsunsky, 2004). Two main approaches for this type of problem have been used. The first is the analytical method, widely investigated and used for these types of problems (Wu and Chiu, 1967; Bentall and Johnson, 1986; Elsharkawy, 1999; Porter and Hills, 2002). The other method is the finite element method (FEM) employed by many researchers (Ihara et al., 1996; Anderson and Colllins, 1995; Aslantas and Tasgetrian, 2002). FEM can be effectively used for arbitrary complex geometry and complex material constitutive laws, but it requires some significant effort in pre- and post-processing of data. The ana-

214

j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 1 9 6 ( 2 0 0 8 ) 213–221

lytical method is more convenient and straightforward than FEM when is used for simplified contact configurations and linear material response. For material selection and preliminary design, analytical method may be much more efficient than FEM. However, a complete framework is still lacking for calculating contact tractions and stress fields for contact between an indenter and surface in coated systems. The aim of the present study is the development of a general framework for calculating stress and displacement fields between a solid indenter and a system with bi-layered isotropic coatings. The solution procedure is constructed in the following steps. Fundamental solutions are determined for concentrated normal and tangential forces acting at the surface of a coated half-plane. Solutions for the Airy stress functions are obtained for both cases and are used to derive the expressions for the elastic stress and displacement fields in coatings and substrate. These problems are also modeled by ANSYS software and the results of finite element method are obtained. The results of analytical solution are compared with the FEM results. The analytical results are also compared with the result of uncoated half-plane and single-coated system.

Since q1 and q2 are conjugate functions, they satisfy ∂q2 ∂q1 = , ∂x ∂y

∂q1 ∂q2 =− ∂y ∂x

∂q2 = ∂x

and so

1 4 Q1 .

(5)

The stress–strain relations for the plane strain condition may be written as (Timoshenko and Goodier, 1982): 1+ [xx − (xx + yy )] E 1+ = [yy − (xx + yy )] E 1+ xy = E

εxx = εyy εxy

(6)

which can be expressed in term of , q1 and q2 . So:









∂q1 1+ ∂2  ∂u 4(1 − ) = − 2 ∂x E ∂x ∂x ∂q2 ∂2  ∂v 1+ 4(1 − ) − 2 = E ∂y ∂y ∂y

(7)

Therefore, we have:

2.

Fundamental solution

The problem of elastic coatings of uniform thickness perfectly bonded to a dissimilar elastic underlying half-plane is investigated on the basis of two-dimensional theory of elasticity. Under the conditions of plane strain, the stress components  xx ,  yy ,  xy can be expressed as (Timoshenko and Goodier, 1982): ∂2 

xx =

∂y2

,

yy =

∂2  ∂x2

,

xy = −

∂2  ∂x∂y

,

(1)

where  is the Airy stress function. It must be biharmonic 4  = 0. Solution of the equation 4  = 0 can be chosen in the form (Timoshenko and Goodier, 1982):

 =

∞



[(A1 + A2 y)e−wy + (A3 + A4 y) ewy ] (2)

where A1 , A2 , . . ., c1 , c2 are parameters which are only related to parameter w. Writing Q1 for 2  which represent  x +  y we observe that Q1 is the harmonic function, and will have a conjugate harmonic function Q2 . Consequently, Q1 + iQ2 is an analytic function of z = x + iy, and we may write:

v=





1+ ∂ 4(1 − )q2 − , E ∂y (8)

Finally, the plane strain general solution is written as u=





∂ 1+ 4(1 − )q1 − , E ∂x

xx =

∂2  , ∂y2

yy

∂2  , ∂x2

v=

xy = −





1+ ∂ 4(1 − )q2 − , E ∂y

∂2  . ∂x∂y

(9)

In view of the symmetry of the normal load and the requirement that the substrate must be stress-free at large distance from loading point, Airy stress functions for coatings and substrate can be written as (Xie and Hawthorne, 2002; Sull, 2002):



∞

I =

0 II

(3)

[(A1 + A2 y) e−wy + (A3 + A4 y) ewy ] cos wx dw

∞

=



0

Q1 = ∇ 2  = x + y .



∂ 1+ 4(1 − )q1 − , E ∂x

3. Fundamental solution for a concentrated load normal to the surface

0

× (c1 cos wx + c2 sin wx)dw,

u=

III =

[(A5 + A6 y) e−wy + (A7 + A8 y) ewy ] cos wx dw ,

∞

(10)

[(A9 + A10 y) e−wy ] cos wx dw

0

The integral of this function with respect to z is another analytic function, denoted as 4 (z). Then writing q1 and q2 for the real and imaginary pats of (z), we have: ∂q1 = ∂x

1 4 Q1 .

(4)

where Ai are generally functions of Fourier transform variable w, subscripts I, II, III refer to coating No. 1, coating No. 2 and substrate, respectively (Fig. 1). Using the procedure described in the previous section, the expressions for auxiliary functions q1 , q2 are obtained as follows:

215

j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 1 9 6 ( 2 0 0 8 ) 213–221

Fig. 2 – Coating–substrate system.

Fig. 1 – Coating–substrate system. For coating No. 1:

 =

1 2

qI2 =

1 2

qI1

∞

0∞

(A4 ewy − A2 e−wy ) sin wx dw (11)

(A4 ewy + A2 e−wy ) cos wx dw

0

For coating No. 2:

 qII 1 =

1 2

qII 2 =

1 2

∞

0∞

Fig. 3 – Uncoated half-space under concentrated load.

(A8 ewy − A4 e−wy ) sin wx dw (12)

(A8 ewy + A4 e−wy ) cos wx dw

0

And for substrate:



∞

1 qIII 1 = −2



qIII 1 =

0∞

1 2

A10 e−wy sin wx dw (13)

A10 e−wy cos wx dw

0

The stress and displacement fields for the coatings and substrate can now be given in the following form:

Fig. 4 – The mesh used to model a system with two coatings under concentrated normal load.

For coating No. 1, we have:

 xx =

∞



0

[(w2 A1 + A2 (w2 y − 2w)] e−wy + (w2 A3 + A4 (wy + 2w) ewy ] cos wx dw ∞

w2 [(A1 + A2 y) e−wy + (A3 + A4 y) ewy ] cos wx dw

yy = −



∞0

xy = 0

1 + 1 E1 1 + 1 v= E1

u=

w[(−wA1 + A2 − wA2 y) e−wy + (A3 w + A4 + A4 wy) ewy ] sin wx dw



∞

0 ∞

(14)

[2(1 − 1 )(A4 ewy − A2 e−wy ) + w(A1 + A2 y) e−wy + (A3 + A4 y) ewy ] sin wx dw [2(1 − 1 )(A4 ewy + A2 e−wy ) − (−wA1 − A2 wy + A2 ) e−wy + (A3 w + A4 + wA4 y) ewy ] cos wx dw

0

Table 1 – Material properties used in analytic solution (Harry et al., 1999) Section/material

E (GPa) 

Coating No. 1/hard tungsten–carbon (WC)

Coating No. 2/ductile tungsten (W)

600 0.25

400 0.28

h1 = 0.02 mm = 20 ␮m; h2 = 0.02 mm = 20 ␮m; L = 200 N/mm.

Substrate/ stainless steel 210 0.3

216

j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 1 9 6 ( 2 0 0 8 ) 213–221

And for substrate, we obtain:

 xx =

∞



0

[(w2 A9 + 2wA10 − w2 A10 )] e−wy cos wx dw ∞

w2 [(A9 + A10 y) e−wy ] cos wx dw

yy = −



0 ∞

w[(−wA9 + A10 − wA10 y) e−wy ] sin wx dw

xy =



0

Fig. 5 – The mesh used to model a system with two coatings under concentrated tangential load.

u=

1 + 3 E1

v=

1 + 3 E3

(16)

∞

0 ∞

[2(1 − 3 )(A10 e−wy ) + w[(A9 + A10 y) e−wy ] sin wx dw [2(1 − 3 )A10 e−wy − A10 e−wy + A10 ) e−wy cos wx dw

0

The unknown functions Ai (w) are now determined from the boundary conditions at the surface (Fig. 2):



For coating No. 2, we have: y=0→

 xx =

∞



0

∞0

xy = 0

1 + 2 E1 1 + 2 v= E2

u=

(17)

[(w2 A5 + A6 (w2 y − 2w)] e−wy + (w2 A7 + A8 (wy + 2w) ewy ] cos wx dw ∞

w2 [(A5 + A6 y) e−wy + (A7 + A8 y) ewy ] cos wx dw

yy = −



yy = −ı(x) xy = 0

w[(−wA5 + A6 − wA6 y) e−wy + (A7 w + A8 + A8 wy) ewy ] sin wx dw





∞

0∞

(15)

[2(1 − 2 )(A8 ewy − A6 e−wy ) + w(A5 + A6 y) e−wy + (A7 + A8 y) ewy ] sin wx dw [2(1 − 2 )(A8 ewy + A6 e−wy ) − (−wA5 − A6 wy + A6 ) e−wy + (A7 w + A8 + wA8 y)] cos wx dw

0

Fig. 6 – Normal load. (a) Distribution of stress in y direction under L = 200 N/mm. (b) Displacement in y direction (v) under L = 200 N/mm.

Fig. 7 – Tangential load. (a) Distribution of stress in y direction under L = 200 N/mm. (b) Displacement in x direction (u) under L = 200 N/mm.

217

j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 1 9 6 ( 2 0 0 8 ) 213–221

These conditions are applied in Eq. (17), where ı(x) is Dirac delta function. The continuity conditions of stress and displacements along the interfaces are (Fig. 2):

at

y = h1 →

⎧  = yy(coat. No. 2) ⎪ ⎨ yy(coat. No. 1) xy(coat. No. 1) = xy(coat. No. 2)

⎪ ⎩ u(coat. No. 1) = u(coat. No. 2)



and y=0→

v(coat. No. 1) = v(coat. No. 2)

at

y = h2 →

⎧  = yy(substrate) ⎪ ⎨ yy(coat. No. 2) xy(coat. No. 2) = xy(substrate)

(18)

⎪ ⎩ u(coat. No. 2) = u(substrate) v(coat. No. 2) = v(substrate)

In order to show the format of the unknown functions, they are determined from the boundary and continuity conditions for material properties represented in Table 1. Hard sputterdeposited WC layer has been combined with pure tungsten layers in bi-layer and multilayer structures. Bi-layered coatings of this type up to 60 ␮m thick, produced by magnetron sputtering and composed of hard WC and ductile W layers, were found to be very promising erosion-resistant coatings for compressor blades in gas turbine engines (Harry et al., 1999; Quesnel et al., 1993).

4. Fundamental solution for a concentrated load tangential to the surface Similar to the argument of the previous section, according of the anti-symmetry of the tangential load and far-field stressfree condition of substrate, Airy stress functions for coatings and substrate can be written in the form (Xie and Hawthorne, 2002; Sull, 2002):

 FI = FII =

∞

0

[(A1 + A2 y) e−wy +(A3 + A4 y) ewy ] sin wx dw

∞

0

FIII =

[(A5 + A6 y) e−wy +(A7 + A8 y) ewy ] sin wx dw ,

∞

Using the procedure described in the previous section, the stress and displacement field for the coatings and substrate can be found. In this case, the boundary conditions at the surface are as follows yy = 0 xy = −ı(x)

(20)

These conditions are applied in Eq. (14), where ı(x) is Dirac delta function. The continuity conditions of stress and displacements along the interfaces are;

at

y = h1 →

⎧  = yy(coat. No. 2) ⎪ ⎨ yy(coat. No. 1) xy(coat. No. 1) = xy(coat. No. 2)

⎪ ⎩ u(coat. No. 1) = u(coat. No. 2) v(coat. No. 1) = v(coat. No. 2)

at

y = h2 →

⎧  = yy(substrate) ⎪ ⎨ yy(coat. No. 2) xy(coat. No. 2) = xy(substrate)

(21)

⎪ ⎩ u(coat. No. 2) = u(substrate) v(coat. No. 2) = v(substrate)

The unknown functions Ai (w) are now determined from the boundary and continuity conditions.

5.

Total stress and displacement fields

With these solutions for the cases of normal and tangential loads one can obtain displacements and stresses for any inclined force at an arbitrary angle to the boundary. In this way, the total stress and displacement fields are obtained from superposing the results of normal and tangential forces, so we have (Quesnel et al., 1993):

(19) n t xx = xx P + Txx ,

[(A9 + A10 y) e−wy ] sin wx dw

u = un P + Tut ,

0

n t yy = yy P + Tyy ,

n t xy = xy P + Txy ,

v = vn P + Tvt

(22)

Table 2 – Comparison of results for normal and tangential load Stress (MPa)/displacement (mm)

Analytical solution

FEM solution

(a) Normal load  xx  yy  xy u v

−31.2 −31.19 −31.83 0.000748 0.00223

−32.1 −30.5 −32.3 0.000755 0.00226

−31.83 −31.83 −31.83 0.000748 0.00224

(b) Tangential load  xx  yy  xy u v

−31.2 −31.83 −31.2 0.000216 0.00325

−32.1 −30.5 −32.3 0.000213 0.00327

−31.83 −31.83 −31.83 0.000216 0.00325

x = 1 mm; y = 1 mm; load = 200 N/mm.

Timoshenko solution

218

j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 1 9 6 ( 2 0 0 8 ) 213–221

Table 3 – Comparison of results for normal and tangential load Stress (MPa)/displacement (mm)

Analytical solution

FEM solution

Timoshenko solution

(a) Normal load  xx  yy  xy u v

−15.29 −15.28 −15.91 0.0000762 0.001849

−16.1 −15.7 −16 0.0000760 0.001855

−15.91 −15.91 −15.91 0.0000762 0.001849

(b) Tangential load  xx  yy  xy u v

−15.90 −15.91 −15.29 0.000225 0.00265

−16.08 −15.50 −15.80 0.000222 0.00269

−15.91 −15.91 −15.91 0.000225 0.00265

x = 2 mm; y = 2 mm; load = 200 N/mm.

where P is the normal load, T the tangential load and n and t refer to the solution of normal and tangential load, respectively.

Tangential load: xx = −

6.

Stress fields in uncoated half-plane

xy = −

Stress and displacement fields in uncoated half-plane are as follows (Fig. 3) (Sokolnikoff, 1976; Muskhelishvili and Radok, 1975):

xx = − xy = −

2Px3 (x2

2 + y2 )

,

yy = −

2Pxy2 (x2

2 + y2 )

,

2Px2 y (x2 + y2 )

(23)

2

(x2

2 + y2 )

,

yy = −

2Ty3 (x2 + y2 )

2

,

2Txy2 (x2 + y2 )

where r =

7.

Normal load:

2Tx2 y

(24)

2



x2 + y2 and  = arctan(y/x).

Finite element method

In this section, a system of half-space with bi-layered coating is modeled (Figs. 4 and 5) and the stress field was calculated by finite element method using ANSYS software. This problem is solved for both normal and tangential loads. A plane strain finite element method has been developed for calculat-

Table 4 – Comparison of results for normal load Position Stress (MPa) (a)  xx  yy  xy u v

Position Stress (MPa) (b)  xx  yy  xy u v

x = 1 mm, y = 1 mm; load = 200 N/mm

x = 2 mm, y = 2 mm; load = 200 N/mm

Analytical solution

FEM solution

Analytical solution

−28.45 −33.43 −31.60 0.0000814 0.00222

−28.5 −33.2 −31.4 0.000080 0.00223

−14.57 −15.93 −15.88 0.0000799 0.001846

x = 1 mm, y = 0.01 mm; load = 200 N/mm Analytical solution −15.73 0.628 29.51 −0.0005426 0.001344

FEM solution −15.8 0.65 −30.1 −0.00053 0.00136

FEM solution −14.8 −15.9 −15.65 0.0000785 0.00185

x = 1 mm, y = 0.03 mm; load = 200 N/mm Analytical solution −25.96 0.587 −1.18 −0.0002 0.0022

Numerical results for bi-layered coating system: h1 = 0.02 mm = 20 ␮m; h2 = 0.02 mm = 20 ␮m.

FEM solution −25.7 0.600 −1.25 −0.000215 0.0023

219

j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 1 9 6 ( 2 0 0 8 ) 213–221

Table 5 – Comparison of results for tangential load Position Stress (MPa) (a)  xx  yy  xy u v

Position Stress (MPa) (b)  xx  yy  xy u v

x = 1 mm, y = 1 mm; load = 200 N/mm Analytical solution 27.07 28.72 −27.69 0.0002132 −0.003228

x = 2 mm, y = 2 mm; load = 200 N/mm

FEM solution

Analytical solution

28.5 30.45 −28.2 0.000215 −0.00345

14.59 15.08 −14.29 0.0002163 −0.002621

x = 1 mm, y = 0.01 mm; load = 200 N/mm Analytical solution 236.2 0.0198 −2.96 0.00228 −0.00021

FEM solution

FEM solution 14.65 15.60 −14.45 0.000218 −0.00265

x = 1 mm, y = 0.03 mm; load = 200 N/mm Analytical solution

235.3 0.021 −2.6 0.00219 −0.000221

338.5 0.187 −8.15 0.002211 −0.0002375

FEM solution 338.5 .20 −8.25 0.00223 −0.00024

Numerical results for bi-layered coating system: h1 = 0.02 mm = 20 ␮m; h2 = 0.02 mm = 20 ␮m.

Fig. 8 – Comparison of results from analytical solution and finite element method (normal load). (a) Comparison of results for normal load,  yy for y = 2 mm. (b) Comparison of results for normal load,  xy for y = 2 mm.

ing stresses and displacements in coatings and substrate. A suitable mesh density around the contact area was found to be roughly half the coating thickness (Figs. 4 and 5). “Structural solid Quad 4node 42” elements were used in ANSYS finite element software. The coatings and substrate behavior was characterized as elastic and isotropic. The sliding tip was modeled as completely rigid. The kinetic formulation was determined applying plane strain behavior and loads are applied at nodes. In order to find the numerical results the material properties represented in Table 1 are used (Harry et al., 1999).

The system is modeled by ANSYS finite element software for uncoated, single and double-coated substrate and the stresses and displacement are obtained. Typical results of distributions and displacement are shown in Figs. 6a and b and 7a and b. The figures are topographical stress field and displacement field maps where each color corresponds to a certain stress and displacement level range in coatings and substrate. For normal load and double-coated system the stress  yy and displacement in y direction, v, are as follows and for tangential load and double-coated system the stress ␴yy and displacement in x direction, u, are as follows.

Fig. 9 – Comparison of results from analytical solution and finite element method (normal load). (a) Comparison of results for normal load, v for y = 0.01 mm. (b) Comparison of results for normal load,  zz for y = 0.01 mm.

220

j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 1 9 6 ( 2 0 0 8 ) 213–221

Fig. 10 – Comparison of results from analytical solution and finite element method (normal load). (a) Comparison of results for normal load,  xx for y = 0.03 mm. (b) Comparison of results for normal load v for y = 0.03 mm.

Fig. 11 – Comparison of results from analytical solution and finite element method (tangential load). (a) Comparison of results for tangential load,  xx for y = 2 mm. (b) Comparison of results for tangential load,  xx for y = 0.01 mm.

Fig. 12 – Comparison of results from analytical solution and finite element method (tangential load). (a) Comparison of results for tangential load, u for y = 0.01 mm. (b) Comparison of results for tangential load,  xx for y = 0.03 mm.

8.

Results and comparison

A mathematical model was developed to calculate the distribution of stresses and displacement in thin coatings and

substrate by an indenter under normal and tangential loading. These equations include the effect of material properties of coatings and substrate, the indenter geometry and normal or tangential load. In order to check these equations, we calculated the results again using the ANSYS finite element

Table 6 – Comparison of results between single-coated system and bi-layered-coated system putting h2 = 0 for normal load Position

x = 1 mm, y = 1 mm; load = 200 N/mm

x = 1 mm, y = 0.01 mm; load = 200 N/mm

Stress (MPa)

Analytical solution

Analytical solution

FEM solution

FEM solution

(a)  xx  yy  xy u v

−29.45 −32.76 −31.74 0.0000791 0.002226

−30.1 −33.04 −31.23 −0.000078 0.0023

−12.86 0.635 30.38 −0.000552 0.001345

−12.65 0.73 29.8 −0.00054 0.0014

(b)  xx  yy  xy u v

29.61 30.67 −29.59 0.0002229 −0.00327

31.1 30.01 −30.5 0.00223 −0.00033

245.1 0.0236 −1.9 0.00229 −0.0002171

246.3 0.03 −1.4 0.0022 −0.000229

Numerical results for single-coated system: h1 = 0.02 mm = 20 ␮m; h2 = 0.

j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 1 9 6 ( 2 0 0 8 ) 213–221

software. The numerical results for both uncoated and coated systems, from analytical solution and finite element method, are compared. The results are very close to each other. The numerical results for uncoated half-space, from analytical solution and finite element method, in arbitrary points, are obtained and compared. They are also compared with the results of Timoshenko’s solution Eqs. (23) and (24). Some results are presented in Tables 2 and 3. The numerical results, also, for coated half-space, from analytical solution and finite element method, in arbitrary points, are obtained and compared. The instance results represented in Tables 4 and 5 show that the results are very close to each other. So these equations should be authentic. Comparison between some numerical results from analytical solution procedure and finite element simulation are presented in Figs. 8–12. In order to compare with the results of single-coated system we put h2 = 0 in bi-layered-coated system formulation. In this case, the Ai (w) variables obtained are completely equal to variables obtained from single-coated system, so the numerical results in two cases are equal (Table 6). These equations should be authentic and useful for ranking the coating–substrate adhesion of different coated systems. It also can be used for estimating the mean coating stress for interfacial failure from the critical load.

9.

Conclusions

Fundamental solutions for concentrated normal and tangential forces acting on the surface of a coated half-plane are determined. Solutions of the Airy stress functions for the two cases are obtained and are used to derive the expressions for the elastic fields in coatings and substrate. These problems are also modeled by ANSYS software and the results of finite element method are obtained. The results of analytical method are compared with FEM data. They are very close to each other. These equations are useful for ranking the coating– substrate adhesion of different coated systems, as well as estimating the critical mean stress for interfacial failure. The conclusions obtained in this paper apply to contact of a system coated by tow layers and for concentrated normal and tangential load. For the contact of multi-coated system and

221

other type of loads, fundamental solutions can be re-derived using the general procedures given in this paper.

references

Anderson, I.A., Colllins, I.F., 1995. Plane strain stress distributions in discrete and blended coated solids under normal and sliding contact. Wear 185, 23–33. Aslantas, A., Tasgetrian, S., 2002. Debonding between coating and substrate due to rolling–sliding contact. J. Mater. Des. 23, 571–576. Bentall, R.H., Johnson, K.L., 1986. An elastic strip in plane rolling contact. Int. J. Mech. Sci. 10, 637–663. Blees, M.H., Wenkelman, G.B., 2000. The effect of friction on scratch adhesion testing. Thin Solid Films 359, 1–13. Elsharkawy, A.A., 1999. Effect of friction on subsurface stresses in sliding line contact of multilayered elastic solids. Int. J. Solid Struct. 36, 3903–3915. Harry, E., Rouzaud, A., Juliet, P., Pauleau, Y., 1999. Adhesion and failure mechanisms of tungsten-carbon containing multilayerd and graded coatings subjected to scratch tests. Thin Solid Films 342, 207–213. Ihara, T., Shaw, M.C., Bhushan, B., 1996. A finite element analysis of contact stress and strain in elastic film on a rigid substrate. Trans. ASME, J. Tribol. 108, 527–539. Ma, L.F., Korsunsky, A.M., 2004. Fundamental formulation for frictional problem. J. Solid Struct. 41, 2837–2854. Muskhelishvili, N.I., Radok, J.R., 1975. Some Basic Problem of Mathematical Theory of Elasticity, fourth ed. Noordhoff International Publishing, Leyden. Porter, M.I., Hills, D.A., 2002. Note of complete contact between a flat rigid punch and an elastic layer attached to a dissimilar substrate. Int. J. Mech. Sci. 44, 509–520. Quesnel, E., Pauleau, Y., Monge-Cadet, P., Brun, M., 1993. Tungsten and tungsten-carbon PVD multilayered structures as erosion-resistant coatings. J. Surf. Coat. Technol. 62, 474–481. Sokolnikoff, I.S., 1976. Mathematical Theory of Elasticity, second ed. Mc Graw Hill International Book Company, New York. Sull, K.R., 2002. Contact mechanics and adhesion of soft solids. J. Mater. Sci. Eng. R36, 1–45. Timoshenko, S.P., Goodier, J.N., 1982. Theory of Elasticity, third ed. Mc Graw Hill International Book Company, New York. Wu, T.S., Chiu, Y.P., 1967. On the contact problem of layered elastic bodies. Quart. Appl. Math. XXV, 233–242. Xie, Y., Hawthorne, H.M., 2002. Effect of contact geometry on the failure modes of thin coatings friction on scratch adhesion testing. J. Surf. Coat. Technol. 155, 121–129.