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Correlation factors for the 2D coated Hertzian contact problem
ELSEVIER
Wear -’ f ’_
( 1997 )
Technical
‘65’67_ _
note
Correlation factors for the 2D coated Hertzian contact problem A.V. Olver *
Abstract Elastic conract between
smooth. parallel.
coated cylinders
is considered.
Results of earlier analyses are correlated
the contact width. This enabIes the con&cCtdimension and pressure to he determined is
easily extended to the case in which
K~ww~s:
both bodies are coated.
0
1997 Elsevier
This procedure
Science S.A.
Hertz: Cornact width; Contact stress: Coating5
1. Introductiun
coating has Young’s modulus E, and We shall later extend the analysis to the case where both bodies are coated. The procedure implied by Gupta and Walowit [ I ] is as follows: Calculate the half-width, aL, for the contact of the indenter with a uniform body having the elastic constants of the coating. using Hertzian theory, i.e. son’s ratio Poisson’s
The problem of elastic contact between non-conforming coated surfaces has received much attention in recent years owing to the widespread use of coatings in engineering components, for example in wear protection. The twodimensional problem, which is the coated analogue of the Hertzian line contact,
using scaling factors for
without restrictions on the elastic constants.
can be solved using layered
body stress functions
transform method [ I .2]. This has recently been extinded to rough surface contact [ 21, to plastic deformation [ 3 ] and to fracture i 4 J of coated solids. Howcvcr, the complexities of the solution will deter many and it is therefore useful to have the results of the analysis in a form which can be applied directly to any Hertz-type line contact. For example Gupta and Walowit [ I] give results for a rigid and for an elastic indenter but restrict the analysis to v = 0.25. On the other hand, Cole and Sayles [ 2 ] give results only for Y = 0.30 for coating, substrate and indenter. In this
v2_ The
ratio v,.
and the Fourier
note, a method-based
entirely
on the published
plots given
by earlier workers -is given for finding the contact dimension (the semi-width, a) and the maximum pressure for any values of the elastic constants of the contacting materials.
2. Analysis
Smooth, isotropic, non-conforming bodies are assumed and we first consider the case in which an urrcua~ed indenter (Young’s modulus A??~, Poisson’s ratio v3) is pressed against a body whose substrate has Young’s modulus & and Pois* Corresponding author. 0043- 1648/97/$17.00 0 1997 Elsevier Science S.A. All rights reserved PIISOO43-1648I97)00151-8
PR
a1. = 2(&
I”
)
where Gx=
1-v;
-+6
1-v; E.7
1 -I
K is the radius of relative curvature and P is the load per unit length. Correct for the effect of the substrate. If the elastic constants are exactly as assumed by, for example, Cole and Sayles [ 21 ( & = E3 and vl = vz = V~=0.3) we may then correct this value by multiplying by the factor a/q_, plotted as a function of nLIJr in their figure 4; h is the coating thickness. However. for the present purposes, we require to genera& the results of Cole and Sayles [ 21 in two respects: ! a) for general coated bodies. i.e. where vI # v-, etc.. and ( b) for general indenters, i.e. where E2 f E3 and I+ # Y?. In
order to do this we first follow Oliveira and Bower
141
and describe the elastic mismatch between the coating and the substrate using the Dundurs parameters: (1) and
266
(6) Thus we may the vertical axis
a-03 a - O;r
(-2Y \I’*-
am0.2
\1+rl
a
1+
Cole and Sayles’
, -
( 1( 9I’*_ -OL
1
Y+
PPh but interpret
(7)
1
I+8
Fig.1.Tbcsemi-width of the contact between a rigid cylinder and a coated
solid: a is tb SICUUII cantact semi-width Md q_ is the contaer semi-width for acylinder (indenter) in contact.wirh ahnogenews solid withthcpropchcs of the c&ing: It is tk coa!ing thickness. From Oliveira and Bower 143
(Fig.2).
=r(Kz -I)-(Kl-1) B r(K2+l)+(~1+l)
(2)
where r=E,( 1 + Q)/&( I+ Y,) and ~=3-4~. For most practically important materials, 0.6 > ar> - 0.6 and O< p< a/3. Olivcira and Bower give normalised contact widths f<# this range of materials for a rigid indenter. Their plot is reproduced in Fig. 1. In or& to apply this to a general i.e. elastic indenter, we first recognize [ 1] that the asymptotic values of ala, for the
elastic indenter are: (3)
lim (uIczL) = 1
h+=
which of course reduces to a/a,_ for e= y, as required. Howand Sayles’ plots remain valid only for vt = v, -0.3, i.e. for /3==a/3.5. Plots are given by Gupta and Walowit for both i?j=y and
ever, Co!z
for &= 0. Insertion of trial values confirms that theassumption of uniform scaling is accurate. In addition, Oliveira and Bower’s plots for s-0 and /3= a/3 ( vI = v,=O.25) agree well with those of Gupta and Walowit for the same conditions. However, Oliveira and Bower’s treatment of the elasric indenter is incorrect, owing to their neglect of the necessary scale factor given in IQ. (5). The pressure distribution is Hertzian to a good approximation with the restrictions on a and j3 imposed by [ 4 J, so we may find the maximum pressure from
Altertlatively we may use the maximum pressure correction factors given in [ 1] and 121 with an appropriate scaling
fator for the asymptote.
and 3, Exte~totwocoa&dbUes l-v?
where E$ = 2+Ed
l-p?
Ej
1 -I
(4)
We may then scale Oliveira and Bower’s graph according to the requiredasymptotic limits assuming that th42shape is otherwise unaffected. This is equivalent to using Oliveiraand Bower’s graphs (which have h +O asymptotes of 7*“) directly, but interpreting the vertical axis as It
( 1 t-1
If2
y ,’ Y+ ( 1+5‘) -’
Finally, it is noted that the procedure can easily be extended to the case in which both bodies are coated. First the problem is analysed as above, assuming the indenter to be homogeneous with the properties of its coating (Young’s modulus E.,, Poisson’s ratio vq). We then determine the correction factor from the appropriate curve for the indenter coating thiclcnessh,andproperties a2,&, butscaletheh,-,Oasymptote to the appropriate value for the indenter substrate pressing on the (composite) lower body. This will itself require use of the full layered body procedure described above. 4. Example calculation
(3 Note that for a rigid indenter t= 0 and Eq. ( 5) reduces to a/u‘ in accordance with Oliveira and Bower’s interpretation of their own pph. For the Cole and Sayles assumption of identical indenter and substrate, E= Y and the asymptote reduces to
As an example, consider the experiment described by OIver et ai. [ 51 in which a high alloy steel wire (Young’s modulus &= 169 GPa, Poisson’s ratio y2=0.3) coated with 3 pm thick TiN (El =465 GPa, vl = 0.25) was pressed against a diamond flat ( E3 = 1050 GPa, v3= 0.2). The wire had radius 0.1232mmandthenormal1oadwas0.01515MNm”.From Hertzian theory we have uL = 2.64 pm and hence uJh =
A.V. Olver/ Wear 212 (1997; 265-267 0.880.
To make the necessary correction for the substrate, we use Oliveira and Bower’s plof using cu=O.455, p = a/3.89. Using the interpretation given by Eq. (5) we have l/Z
y t= ( )(y+q;2y
I+“-1 QL
1.20
v+51
-
with y=2,67 and 6=0.453, from which u/aL= 1.147 and a= 3.03 pm. This compares with the accurate numerical value calculated using the method given in [ 21 of 3.00 pm, well within the accuracy of the chart,
267
References I I 1 P.K. Gupta. J.A. Walowit, Contact streswzs between an elastic cylinder and a layered elastic solid. Trans. ASME. J. Rubric. Tech. % ( 1974) 250-257. I2 I S.J. Cole, R.S. Sayks. A nurrwica! method for the contxt of layered elastic bodies with real rough surfaces, J, Tribd. 114( 1991) 729. [ 3 1S.K. Wang, A. Kapoor, Effect of hard ad stiff overlay coatings on the smngth of surfaces in rcpcad sliding, Tribal. ht. 29 ( 1996) 695-702. [ 41 S.A.G. Oliveira, A.F. Bower, An analysis of ftactwc and delti~on in thin coatings subjected to conhct loading, Wear 198 ( 19%) 15-32. IS] A.V.Olver,P.M.E.Cann,J.C.Loric.Aninvesr.iga&bnin~~theproptrtics of a thin solid coating using an optical method. Prw. 22nd I&&-Lyon Symposium on Tribology,Third Bodks in Tribology. Lyon. Septcmbtr 1995, Elsevier. 19%.
5. Conclusitm A procedure is given which enables the contact dimensions and pressure to be estimated for the coated line contact problem with no restrictions on the elastic constants of the indenter, coating or substrate. The procedure. can easiIy be extended to the case in which both bodies are coated.
Andrew Olver is a graduate of Exeter University and obtained a Ph.D. from imperial College in 1986. He worked in the aircraft industry in materials and mechanical engineering before becoming a lecturer at Imperial College in 1992,
Wear -’ f ’_
( 1997 )
Technical
‘65’67_ _
note
Correlation factors for the 2D coated Hertzian contact problem A.V. Olver *
Abstract Elastic conract between
smooth. parallel.
coated cylinders
is considered.
Results of earlier analyses are correlated
the contact width. This enabIes the con&cCtdimension and pressure to he determined is
easily extended to the case in which
K~ww~s:
both bodies are coated.
0
1997 Elsevier
This procedure
Science S.A.
Hertz: Cornact width; Contact stress: Coating5
1. Introductiun
coating has Young’s modulus E, and We shall later extend the analysis to the case where both bodies are coated. The procedure implied by Gupta and Walowit [ I ] is as follows: Calculate the half-width, aL, for the contact of the indenter with a uniform body having the elastic constants of the coating. using Hertzian theory, i.e. son’s ratio Poisson’s
The problem of elastic contact between non-conforming coated surfaces has received much attention in recent years owing to the widespread use of coatings in engineering components, for example in wear protection. The twodimensional problem, which is the coated analogue of the Hertzian line contact,
using scaling factors for
without restrictions on the elastic constants.
can be solved using layered
body stress functions
transform method [ I .2]. This has recently been extinded to rough surface contact [ 21, to plastic deformation [ 3 ] and to fracture i 4 J of coated solids. Howcvcr, the complexities of the solution will deter many and it is therefore useful to have the results of the analysis in a form which can be applied directly to any Hertz-type line contact. For example Gupta and Walowit [ I] give results for a rigid and for an elastic indenter but restrict the analysis to v = 0.25. On the other hand, Cole and Sayles [ 2 ] give results only for Y = 0.30 for coating, substrate and indenter. In this
v2_ The
ratio v,.
and the Fourier
note, a method-based
entirely
on the published
plots given
by earlier workers -is given for finding the contact dimension (the semi-width, a) and the maximum pressure for any values of the elastic constants of the contacting materials.
2. Analysis
Smooth, isotropic, non-conforming bodies are assumed and we first consider the case in which an urrcua~ed indenter (Young’s modulus A??~, Poisson’s ratio v3) is pressed against a body whose substrate has Young’s modulus & and Pois* Corresponding author. 0043- 1648/97/$17.00 0 1997 Elsevier Science S.A. All rights reserved PIISOO43-1648I97)00151-8
PR
a1. = 2(&
I”
)
where Gx=
1-v;
-+6
1-v; E.7
1 -I
K is the radius of relative curvature and P is the load per unit length. Correct for the effect of the substrate. If the elastic constants are exactly as assumed by, for example, Cole and Sayles [ 21 ( & = E3 and vl = vz = V~=0.3) we may then correct this value by multiplying by the factor a/q_, plotted as a function of nLIJr in their figure 4; h is the coating thickness. However. for the present purposes, we require to genera& the results of Cole and Sayles [ 21 in two respects: ! a) for general coated bodies. i.e. where vI # v-, etc.. and ( b) for general indenters, i.e. where E2 f E3 and I+ # Y?. In
order to do this we first follow Oliveira and Bower
141
and describe the elastic mismatch between the coating and the substrate using the Dundurs parameters: (1) and
266
(6) Thus we may the vertical axis
a-03 a - O;r
(-2Y \I’*-
am0.2
\1+rl
a
1+
Cole and Sayles’
, -
( 1( 9I’*_ -OL
1
Y+
PPh but interpret
(7)
1
I+8
Fig.1.Tbcsemi-width of the contact between a rigid cylinder and a coated
solid: a is tb SICUUII cantact semi-width Md q_ is the contaer semi-width for acylinder (indenter) in contact.wirh ahnogenews solid withthcpropchcs of the c&ing: It is tk coa!ing thickness. From Oliveira and Bower 143
(Fig.2).
=r(Kz -I)-(Kl-1) B r(K2+l)+(~1+l)
(2)
where r=E,( 1 + Q)/&( I+ Y,) and ~=3-4~. For most practically important materials, 0.6 > ar> - 0.6 and O< p< a/3. Olivcira and Bower give normalised contact widths f<# this range of materials for a rigid indenter. Their plot is reproduced in Fig. 1. In or& to apply this to a general i.e. elastic indenter, we first recognize [ 1] that the asymptotic values of ala, for the
elastic indenter are: (3)
lim (uIczL) = 1
h+=
which of course reduces to a/a,_ for e= y, as required. Howand Sayles’ plots remain valid only for vt = v, -0.3, i.e. for /3==a/3.5. Plots are given by Gupta and Walowit for both i?j=y and
ever, Co!z
for &= 0. Insertion of trial values confirms that theassumption of uniform scaling is accurate. In addition, Oliveira and Bower’s plots for s-0 and /3= a/3 ( vI = v,=O.25) agree well with those of Gupta and Walowit for the same conditions. However, Oliveira and Bower’s treatment of the elasric indenter is incorrect, owing to their neglect of the necessary scale factor given in IQ. (5). The pressure distribution is Hertzian to a good approximation with the restrictions on a and j3 imposed by [ 4 J, so we may find the maximum pressure from
Altertlatively we may use the maximum pressure correction factors given in [ 1] and 121 with an appropriate scaling
fator for the asymptote.
and 3, Exte~totwocoa&dbUes l-v?
where E$ = 2+Ed
l-p?
Ej
1 -I
(4)
We may then scale Oliveira and Bower’s graph according to the requiredasymptotic limits assuming that th42shape is otherwise unaffected. This is equivalent to using Oliveiraand Bower’s graphs (which have h +O asymptotes of 7*“) directly, but interpreting the vertical axis as It
( 1 t-1
If2
y ,’ Y+ ( 1+5‘) -’
Finally, it is noted that the procedure can easily be extended to the case in which both bodies are coated. First the problem is analysed as above, assuming the indenter to be homogeneous with the properties of its coating (Young’s modulus E.,, Poisson’s ratio vq). We then determine the correction factor from the appropriate curve for the indenter coating thiclcnessh,andproperties a2,&, butscaletheh,-,Oasymptote to the appropriate value for the indenter substrate pressing on the (composite) lower body. This will itself require use of the full layered body procedure described above. 4. Example calculation
(3 Note that for a rigid indenter t= 0 and Eq. ( 5) reduces to a/u‘ in accordance with Oliveira and Bower’s interpretation of their own pph. For the Cole and Sayles assumption of identical indenter and substrate, E= Y and the asymptote reduces to
As an example, consider the experiment described by OIver et ai. [ 51 in which a high alloy steel wire (Young’s modulus &= 169 GPa, Poisson’s ratio y2=0.3) coated with 3 pm thick TiN (El =465 GPa, vl = 0.25) was pressed against a diamond flat ( E3 = 1050 GPa, v3= 0.2). The wire had radius 0.1232mmandthenormal1oadwas0.01515MNm”.From Hertzian theory we have uL = 2.64 pm and hence uJh =
A.V. Olver/ Wear 212 (1997; 265-267 0.880.
To make the necessary correction for the substrate, we use Oliveira and Bower’s plof using cu=O.455, p = a/3.89. Using the interpretation given by Eq. (5) we have l/Z
y t= ( )(y+q;2y
I+“-1 QL
1.20
v+51
-
with y=2,67 and 6=0.453, from which u/aL= 1.147 and a= 3.03 pm. This compares with the accurate numerical value calculated using the method given in [ 21 of 3.00 pm, well within the accuracy of the chart,
267
References I I 1 P.K. Gupta. J.A. Walowit, Contact streswzs between an elastic cylinder and a layered elastic solid. Trans. ASME. J. Rubric. Tech. % ( 1974) 250-257. I2 I S.J. Cole, R.S. Sayks. A nurrwica! method for the contxt of layered elastic bodies with real rough surfaces, J, Tribd. 114( 1991) 729. [ 3 1S.K. Wang, A. Kapoor, Effect of hard ad stiff overlay coatings on the smngth of surfaces in rcpcad sliding, Tribal. ht. 29 ( 1996) 695-702. [ 41 S.A.G. Oliveira, A.F. Bower, An analysis of ftactwc and delti~on in thin coatings subjected to conhct loading, Wear 198 ( 19%) 15-32. IS] A.V.Olver,P.M.E.Cann,J.C.Loric.Aninvesr.iga&bnin~~theproptrtics of a thin solid coating using an optical method. Prw. 22nd I&&-Lyon Symposium on Tribology,Third Bodks in Tribology. Lyon. Septcmbtr 1995, Elsevier. 19%.
5. Conclusitm A procedure is given which enables the contact dimensions and pressure to be estimated for the coated line contact problem with no restrictions on the elastic constants of the indenter, coating or substrate. The procedure. can easiIy be extended to the case in which both bodies are coated.
Andrew Olver is a graduate of Exeter University and obtained a Ph.D. from imperial College in 1986. He worked in the aircraft industry in materials and mechanical engineering before becoming a lecturer at Imperial College in 1992,