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Effect of initial surface curvature on frictionally excited thermoelastic phenomena
Wear, 27 (1974) 195-207 ci> Elsevier Sequoia S.A., Lausanne
195 - Printed
in The Netherlands
EFFECT OF INITIAL SURFACE CURVATURE EXCITED THERMOELASTIC PHENOMENA
R. A. BURTON
ON FRICTIONALLY
and V. NERLIKAR
The Technological Institute, North Western University, Evanston, III. 60201 (U.S.A.) (Received
August
24, 1973)
SUMMARY
Contact in a seal-like configuration is considered where initial imperfections lead to the lips meeting as a periodic sequence of Hertzian contacts. For small ratios of initial contact width to spacing, the effect of sliding is to reduce the contact width monotonically. For the initial conditions such that there is full contact, equivalent to overlap of the Hertzian patches, the effect of speed is more abrupt, giving the appearance of an instability above which the initial pressure distribution is drastically distorted. Tables and equations are offered to facilitate numerical calculation of contact patch widths and pressures for a range of selected operating conditions.
INTRODUCTION
When two semi-infinite plates meet edge to edge and sliding takes place parallel to the common edge, frictional heating can distort this edge so as to produce an uneven pressure distribution. If the plates are initially flat this phenomenon appears as an instability which leads to amplification of any disturbance of the surface above a certain sliding speed i. If, however, the edges of the plates are not flat, but are wavy and meet to form a series of Hertzian contacts, the behavior is different. The Hertzian contacts change continuously in width with sliding speed, and stresses as well as temperatures rise smoothly as speed is increased. That the contacting zone may also be microscopically divided into a large number of asperity contacts is of no particular interest in the analysis of these Hertzian pressure patches, so long as the asperities are small and are large in number. For this initial investigation the most severe case of a conductive body sliding on an insulating body will be examined. This type of contacting pair gives rise to instability more easily than does a thermal conductor sliding on thermal conductor. In addition, the analysis of disturbance behavior is simplified, since the disturbance tends to be stationary in the conducting body. Arguments have been made2 that even thin oxide films may cause metallic contacts to behave in this manner.
196
R. A. BURTON.
V. NERLIKAR
Korovchinsky3 has dealt with the present problem for the case of plane strain, rather than plane stress which would correspond to the present geometry: but the difference between these two cases is small. His results do not extend to the case of initially flat surfaces as a limiting configuration, and he does not deal with the influence of neighboring contact patches as encountered when the surfaces are supported by a series of Hertzian contacts. The present analysis differs from Korovchinsky in that the relationship shown in eqns. (19) and (12) below lead to a much simplified treatment, without loss of generality. The results here are of special interest in that they include the questions of instability as a special case, and show the limits of a large-disturbance configuration for initially flat contact surfaces. They also show when thermal effects significantly distort an initially Hertzian contact. are close to those in ref. 4, but are repeated here Equations (l)-(6) because of the omission of the quantity $ in the earlier publication. RELATIONSHIP
BETWEEN
SEMI-INFINITE
HEAT
FLOW
AND
CURVATURE
OF
FREE
SURFACE
ON
A
SOLID
When the surface temperature distribution can be expressed as a Fourier series of non moving waves, the solution for the heat flow equation in a semiinfinite body is
T = c
T,,e-““y sin nwx
(1)
When the thermoelastic equations are solved for this temperature the boundary at J=O allowed to displace freely.
distribution,
~‘1~~ 0 = -X (cc!nw) T, sin YAWN + c’
with
(2)
It follows that I;2~j?x2~,=, Returning
= C cmo T, sin nwx
now to eqn. (1) it is possible
?Tjiyl,,,
(3) to write
= -JZ IZOI(, sin MLX
(4)
Hence ?2V/(:.X2I,= 0 = - a( (“T/?y) And since the heat flow per unit area q; due to the pressure q=
(5) perturbation
-K(iT/ip)=C(p’V
At the center
of the contact
(6) patch (7)
Pb=P0-Pi pressure
Since pi is the mean contact pressure for the entire on the contact patch is j = p0 4. we may write p0 51 = pi 111
p’. is
edge, and
the mean
(8)
FRICTIONALLY
EXCITED
THERMOELASTIC
197
PHENOMENA
and eqn. (6) becomes
t-4 Inserting this into eqn. (5) we find
ah a.2
= y=o
wP,(wq
(l-5;>
(10)
bearing in mind that y is positive down into the surface. One may use eqn. (1) even in situations where the Fourier series expansion is not used, so long as the pressure distribution is such that it could be expanded as a Fourier series. In order for the net pressure to be zero between contact spots, the perturbation pressure must in that region be -pi. ELASTIC
DISPLACEMENTS
OF SURFACES
DUE TO THE PRESSURE
DISTRIBUTION
For a load distributed along that edge of a plate (see Fig. 1) the displacement of that edge is u = & jlI” Mloglx-
(11)
Differentiation under the integral sign permits evaluation of the derivatives of v, such that d”v 2(-l)“(n-l)! b [P/(X - t;)“] dx (14 dx”= TCE 3 II
H (Dimensionless) Fig. 1. Computer solution compared with straight dimensionless quantities G and H.
line approximation
for relationship
between
the
R. A. BURTON,
198
V. NERLIKAR
If conditions are restricted to a loading symmetrical about x=0, ranging from - I to + 1, the quantity d”v/dx” can now be evaluated at x= 0, from which origin 5 becomes numerically identical with x, and one may write (see Appendix 1): d”u -= dx”
’ ,i _jplWx
2(-l)“(n-l)! 7CE
This neglects the influence secondary influence. SURFACE
BOUNDARY
(13)
of adjacent
patches
on elastic
curvature
at x=0,
a
CONDITION
If the surfaces of the thermal conductor and the thermal insulator must conform over the region of contact from -I to + 1; then, since the point x=0 is within that region, the expansion-induced curvatures from eqn. (10) and the elastic pressure-induced curvatures in the two bodies must be equal and opposite to one another. The same will be true also of the higher derivatives of r. Noting that the same pressure acts on both bodies producing elastic displacement in each, the derivatives of this net displacement will be the sum of the displacements for the two bodies or: 2(-l)“(n-l)!
d”v dx” X=0,ne,=
-I
rcE*
Pdx (14)
.! -I X”
where E* = E, E,/‘(E, + EJ
(15)
Note that for two identical materials, E*= E/2. At this point let us restrict consideration to the second derivative or curvature only, adding the initial curvature l/R to the curvature of the thermoelastic bulge, eqn. (lo), and equating this sum to the relative curvature of the elastic deformation as follows:
POWER SERIES SOLUTION
To obtain according to :
FOR CONTACT
a solution
SPOT PRESSURE
for eqn. (16)
DISTRIBUTION
p may be expanded
in a power
P=Po[1+a,(X/1)2+M2(X/1)4+a3(X/1)6+...] Here only even powers may be rewritten as: P=Po
F+sjIl
are retained
(17)
in the interest
of symmetry.
This equation
(18)
%(;ys]
The right hand term of eqn. (16) may eqn. (IS), and we find:
series
now
be integrated
after
substitution
of
(19)
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EXCITED
TWERMOELASTIC
PHENOMENA
199
Defining n.E” 1/4Rp, E II and
eqn. (20) becomes li+G=
[I-Ca,/(25-1)f
(21)
If the same procedure is carried out for higher derivatives a general equation may be written as CR, =
“’ a,(2R+ 1)
~-3s=l
2s- (2r+ I)1 t= 1,2,..n’
(22)
The term involving R is made to disappear with the assumption that the curved surfaces have a relative parabofic shape with the central radius R, in which case higher derivatives disappear. The procedure used to obtain simultaneous solution to this set was to select a number of terms in the pressure equation s=n, and generate n equations from eqn. (22), for Y from 1 to n. An approximate value for G is inserted in equations r= 1 through r= n, and these are solved for GIN,CX~,..CX,,, The appropriate cti quantities are inserted into eqn. (2f), a value for H is sefected, and it is then solved for G. This value of G is now used in place of the approximate G in a repeat of the procedure, and successive iterations are carried out, rapidly converging to stationary values of G and c+ for the selected H. EFFECT
OF SLIDING
SPEED
ON CONTACT
SIZE
Typical results from such a calculation for a 10 term series are shown in Fig. 1. The straight line may be represented by the equation G-i-f.I8H-1.8--O
Upon reinsertion of the appropriate
(23) quantities into eqn, (23) it becomes: (24)
At this point let us introduce R = I/B& Lk= s,/nt
(25)
where d is a measure of contact width/spacing ratio and a,, refers to this ratio for the Hertzian case. The quantity l,, may be found from eqn. (24) by letting V=O, in which case
R. A. BURTON.
V. NERLIKAR
01
0
4
8
12
V/V,
16
20
2,4
(Dimensionless)
Fig. 2. Relationship between dimensionless contact width and dimensionless speed. Zero speed intercept gives fraction of surface in contact without frictional heating. When i,,,> I physical meaning disappear5 but eqn. (26) remains valid definition for I,,.
(26) The term in parentheses classical Hertz solution. growth is V,= 2Kculc+E”
is quite close to unity as would be the case for the Recalling that the critical velocity for small disturbance = 2KnlmpE”cq then
(l-r;,[pJ C
(271
Regrouping
(28) Solving
the quadratic
for A
-~f~[~~~,‘+7.2(5~-5~~)~~ i=
1
(291 ~“-I!!~ [ Jbh” f This is made meaningful by reference to Table II and Fig. 2, where values of i are shown for selected magnitudes of &, and a range of V/VC. Typical magnitudes for 4 are shown in Table I. For this calculation, however, r was assigned the rounded off value of 0.75. These results become physically meaningless for j-> 1, since this would correspond to overlapping of the contact patches. Referring specifically to Fig. 2. note that for small i,, (initial concentration of contact load), the curves drop off smoothly with increase of V/V,, and the contact patch decreases in size. For larger initial contact widths drop off is more drastic with speed. Somewhere above an 2
FRICTIONALLY
EXCITED THERMOELASTIC
201
PHENOMENA
TABLE I MAGNITUDES .._________-
FOR THE CONTACT
G -.
H
1.624 1.472 1.324 1.176 0.742 0.457 0.177 0.0375 0
0.125 0.262 0.375 0.521 0.875 1.156 1.4075 1.5 1.534
-_____
PATCH PARAMETER -_____
c
s/so
0.7467 0.7541 0.7601 0.7661 0.7811 0.7901 0.7998 0.802 0.8033
0.9295 0.93875 0.946 0.9537 0.9723 0.984 0.9956 0.998
..._.- ___.
.~___
_.....
._~
1
TABLE II EFFECT OF DIMENSIONLESS SLIDING SPEED PATCH WIDTH 1 FOR SELECTED VALUE OF E.,
V/v, ON
DIMENSIONLESS
CONTACT
(Only positive, real solutions retained) -_~-~-____-. F/K
J+
--
0.4
0.2 0.4 0.6 0.8 1.0
0.10 o.to 0.09 0.09 0.09
0.37 0.34 0.3 I 0.29 0.26
1.2 1.4 1.6 1.8 2.0 1.0942 1.0120 0.8604
0.09 0.08 0.08 0.08 0.08
0.24 0.22 0.20 0.19 0.17
.-
-
.._.~
0.1 _. _~_ i
_
0.9
1.6 .__~_.
2.5
cc ___._.~
__ 0.83 0.72
0.61,84.4 0.51. 4.59 0.42, 2.x4
2.3, 18.3 -1.09. 0.55
_-.-
u.35, 0.30, 0.26, 0.23, 0.20,
1.17. 0.41 1.21, 0.33 1.23, 0.28 1.25, 0.24 I .26, 0.21
0.99, 0.44 1.07, 0.34 1.1 I, 0.29 1.15, 0.25 1.17.0.22
2.28 2.01 1.85 1.76 1.69
0.87, 0.47 0.98, 0.36 1.04, 0.29 1.08, 0.25 1.12,0.22 0.6666*
0.7201* 0.8475*
* Corresponding to lowest value of V/C{ for which real solution exists.
initial Hertzian width of unity there is a breakover and the curves become double valued for higher magnitudes of V/V,. Put differentIy, the curves for lower values of a,, may be thought of as the lower leg of such a double valued curve, where the remainder has moved off the scale. Arguments have boen offered’ to show that the upper leg of such a curve is unstable. If the system finds itself to the right of the appropriate curve it will move downward toward smaller 2; but if it is to the left it will move upward. This means that if it is above the upper leg it is also to the left of it and will move upward toward flatter contact. These ideas are in accord with the earlier introduced instability concepts. For example, if the two surfaces were initially flat (&,-+x)), and if a disturbance
202
R. A. BURTON. V. NERLIKAR
with ;1==I could be impressed on them, this disturbance would not grow at low sliding speed but would grow above a certain critical speed. This “growth” would correspond to a shrinking of 1. The crossover is not exactly at V/V,= 1, but this is not surprising in view of the approximations used. Although 1.> 1 is physically meaningless, A,,> 1 remains meaningful. One simply refers to eqn. (26) as defining I,, and ignores the physical interpretation of an actual contact patch width. There is no limit on the magnitudes that 4RPpzE* may have. For example, for R+ccr_., Ih+ccc, and the surface is flat at V=O. COMPARKSON
OF CONTACT
WIDTH WITH INITIAL HERTZIAN
WIDTH
It is instructive to return to eqn. (23) and rederive the final equation so as to display the contact width I in ratio to the Hertzian width I,. Letting H’I= 1.18 H, this becomes G-l-H’=
1.8
(30)
If we let g-G/l;
h=H’/l’;
1’~ 1.8
gls ht2 = I’
(31)
and (32)
(33)
Defining y as (Gl,/l)/Zs, holding in mind that this calls for, Y=
paVnE* l,
(34)
4Kv
it follows from eqn. (32) that f,/lh= -y+(y2+
1)”
(35)
For large 7, E/lh-+iy and behavior is thermoelastic. Equations (34) and (35) represent essentially a solution to the problem, but may be made more useful when it is recalled that 1, = (4PR/zzE*)+
(36)
Where P = 2p,@z
and
5 =#pO
(37)
Substituting eqn. (36) into eqn. (34) and simplifying
Here it is seen that y is determined completely by operating conditions and material properties. once its magnitude is known eqn. (39, Table III or Fig. 3, will give the corresponding l/l,. Knowing 1, from eqn. (36) one can obtain contact width. Equation (37), and the total load will give maximum contact pressure. For this
FRICTIONALLY
EXCITED THERMOELASTIC
PHENOMENA
203
Fig, 3. Effect of operating variable 1 on contact width. TABLE III GLORIFICATION OF HERTZIAN PATCH WIDTH BY THE SLIDING PARAMETER y
0.1 0.2 0.4 0.8 I.2 1.6 2.0 2.5 3.0 3.5 4.0
0.904 0.8198 0.677 0.480 0,363 0.280 0.236 0.193 0.162 0.140 0.123
purpcm 5 may be appraximated or may be found more precisely from Table I as a function of G. By such reasoning we find
14
and approximating P
max=”
p.
=
za
5 as 0.75
!! PE”* 1
(-ix”>
To provide a numerical example to show a typical magnitude for y use the appropriate properties for iron : p
=0.05
c1 =6x 10e6/oF K = 5.6 Ib/sec “F
R =2 in. l/m =O
E* = 9 x lo6 Ib,‘in.2
z =0.05 in. I/ = 500 in/s P = 50 lb,
R. A. BURTON. V. NERLIKAR
204
The sliding speed and stresses are high but might be encountered in gearing. For these conditions ~=0.87. This would correspond to considerable modification of the contact stress. Even at half the sliding speed chosen. the contact width would be halved and the stress doubled, relative to static contact. CONCLUSION
Earlier studies of thermoelastic effects of frictionally heated sliding contact were concerned with initially flat surfaces. The present study takes account of an idealized form of initial waviness, and includes the initially flat contact as a limiting case. Behavior of an initially patch-like contact is a monotonic decrease of contact width with increase of sliding speed, showing no sudden change near the critical speed for small disturbance growth on flat surfaces. For contacts where the hypothetical Hertzian width !,, is greater than the spacing measure m the solution becomes double valued; and arguments may be advanced for an instability-like behavior with waviness not being amplified at small sliding speed and with waviness growing rapidly to an isolated contact patch configuration at a certain of speed. The tables along with eqns. (38) (39) and (40) permit the estimates actual contact width and pressures. COMMENTS
One may well ask how these results may be reconciled with the idea of cosine wave instability at V/V,= 1. It would appear that the wave would grow and break contact at points of low pressure; but the large disturbance solution would apparently call for suppression of this new interrupted disturbance below V/k’,+ 1.4. One should remember, however, that the present large disturbance solution is based upon a series of parabolic disturbances. If at the point of breaking contact the cosine wave would be represented by ?’ = J’J 1-cos QS) The series expansion
Extending
(41)
near x0 would give a parabolic
this parabolic
approximation
I=(x)Y=“=2$JJ
approximation
to y=O to find the equivalent
of the wave as
1, one finds (43)
Since m=71jco l/m= 2171= 0.636 = /.
(44)
Thus at the point of break-away the cosine wave may be thought of as a series of parabolas with d=O.636. Referring to Fig. 2 this corresponds closely to the magnitude of 3. at the lowest V/V, for which the double valued solution can exist. So, within the approximations used, it is not inconsistent for a cosine wave to grow just above V/V,= 1 and move to an intermittent contact solution.
FRICTIONALLY APPENDIX
EXCITED
THERMOELASTIC
PHENOMENA
205
1
The integral dx,
CPM=jt$
for
m an even integer
has unequivocal validity for m < 1. For m> 1, there is a question as to what happens in the integration process when x passes through zero. One must remember that in the present context we are dealing with an influence function valid at points exterior to the zone of application of an elemental load Pdx. Beneath the load the function is something else. To allow for this we may write
441)= i - $dx
+ if
+dx+&)
. +sx
. -I
where V(E) is a special case of cp(l), that is, E is simply a special magnitude for I. Integrating -2 ~(~) = -m-l
1 --A [ r”-’
1
t-44
The equation is satisfied when
Hence, formal integration gives
and the proper result is generated. If one seeks further reassurance, one may take the example of a uniform load for which the well known solution is u =z[(i+x)lny
Differentiating
and at x=0
twice
+ (I-x)ln($d]
206
R. A. BURTON,
One may continue in the paper.
onward
and for each case will confirm
V. NERLIKAR
the relationships
used
NOMENCLATURE
specific heat Young’s Modulus composite Young’s Modulus for two bodies E,E,/(E, +E,) dimensionless quantity see eqn. (21) dimensionless quantity see eqn. (21) mechanical equivalent of heat conductivity of material diffusivity of material (K/p”,,) length of slider half width of contact spot I for zero sliding speed half distance between centers of contact spots m/I load pressure on contact surface designator index in equations index for summation temperature time displacement of surface in J‘ direction sliding velocity critical sliding velocity for growth of small disturbances in contact pressures wear coefficient coordinate in direction of sliding coordinate normal to direction of sliding, measured from contact surface width of slider coefficient of thermal expansion (i = 1,2.3.. .) coefficient in equation dimensionless quantity, see eqn. (34) strain l/m= l/M ,I for zero sliding speed coefficient of friction Poisson’s ratio exponential coefficient of time wave number = 2n n/L p/p,, ratio of mean to maximum contact pressure. ACKNOWLEDGEMENTS
The research reported here was sponsored Naval Research, Contract NOO014-67-A-0356-0013.
by the United States Office of The work reported in references
FRICTIONALLY
EXCITED THERMOELASTIC
PHENOMENA
1, 2, 4, 5 was also performed under the same contract. acknowledgement was omitted in these publications.
207 The authors regret that
REFERENCES 1 R. A. Burton, V. Nerlikar and S. R. Kilaparti, Thermoelastic instability in a seal like configuration, Wear, 24 (1973) 1777188. 2 R. A. Burton, The role of insulating surface films in frictionally excited thermoelastic instabilities, Wear, 24 (1973) 189-198. 3 M. V. Korovchinski, The plane contact problem of thermoelasticity during quasi stationary heat generation on the contact surfaces, J. Basic Eng., (1965) 811-817. 4 R. A. Burton, S. R. Kilaparti and V. Nerlikar, A limiting stationary configuration with partially contacting surfaces, Wear, 24 (1973) 1999206. 5 R. A. Burton and V. Nerlikar, Large disturbance soiutions for initially Rat, frictionally heated thermoelastically deformed surfaces. accepted for publication in Trms. ASME.
195 - Printed
in The Netherlands
EFFECT OF INITIAL SURFACE CURVATURE EXCITED THERMOELASTIC PHENOMENA
R. A. BURTON
ON FRICTIONALLY
and V. NERLIKAR
The Technological Institute, North Western University, Evanston, III. 60201 (U.S.A.) (Received
August
24, 1973)
SUMMARY
Contact in a seal-like configuration is considered where initial imperfections lead to the lips meeting as a periodic sequence of Hertzian contacts. For small ratios of initial contact width to spacing, the effect of sliding is to reduce the contact width monotonically. For the initial conditions such that there is full contact, equivalent to overlap of the Hertzian patches, the effect of speed is more abrupt, giving the appearance of an instability above which the initial pressure distribution is drastically distorted. Tables and equations are offered to facilitate numerical calculation of contact patch widths and pressures for a range of selected operating conditions.
INTRODUCTION
When two semi-infinite plates meet edge to edge and sliding takes place parallel to the common edge, frictional heating can distort this edge so as to produce an uneven pressure distribution. If the plates are initially flat this phenomenon appears as an instability which leads to amplification of any disturbance of the surface above a certain sliding speed i. If, however, the edges of the plates are not flat, but are wavy and meet to form a series of Hertzian contacts, the behavior is different. The Hertzian contacts change continuously in width with sliding speed, and stresses as well as temperatures rise smoothly as speed is increased. That the contacting zone may also be microscopically divided into a large number of asperity contacts is of no particular interest in the analysis of these Hertzian pressure patches, so long as the asperities are small and are large in number. For this initial investigation the most severe case of a conductive body sliding on an insulating body will be examined. This type of contacting pair gives rise to instability more easily than does a thermal conductor sliding on thermal conductor. In addition, the analysis of disturbance behavior is simplified, since the disturbance tends to be stationary in the conducting body. Arguments have been made2 that even thin oxide films may cause metallic contacts to behave in this manner.
196
R. A. BURTON.
V. NERLIKAR
Korovchinsky3 has dealt with the present problem for the case of plane strain, rather than plane stress which would correspond to the present geometry: but the difference between these two cases is small. His results do not extend to the case of initially flat surfaces as a limiting configuration, and he does not deal with the influence of neighboring contact patches as encountered when the surfaces are supported by a series of Hertzian contacts. The present analysis differs from Korovchinsky in that the relationship shown in eqns. (19) and (12) below lead to a much simplified treatment, without loss of generality. The results here are of special interest in that they include the questions of instability as a special case, and show the limits of a large-disturbance configuration for initially flat contact surfaces. They also show when thermal effects significantly distort an initially Hertzian contact. are close to those in ref. 4, but are repeated here Equations (l)-(6) because of the omission of the quantity $ in the earlier publication. RELATIONSHIP
BETWEEN
SEMI-INFINITE
HEAT
FLOW
AND
CURVATURE
OF
FREE
SURFACE
ON
A
SOLID
When the surface temperature distribution can be expressed as a Fourier series of non moving waves, the solution for the heat flow equation in a semiinfinite body is
T = c
T,,e-““y sin nwx
(1)
When the thermoelastic equations are solved for this temperature the boundary at J=O allowed to displace freely.
distribution,
~‘1~~ 0 = -X (cc!nw) T, sin YAWN + c’
with
(2)
It follows that I;2~j?x2~,=, Returning
= C cmo T, sin nwx
now to eqn. (1) it is possible
?Tjiyl,,,
(3) to write
= -JZ IZOI(, sin MLX
(4)
Hence ?2V/(:.X2I,= 0 = - a( (“T/?y) And since the heat flow per unit area q; due to the pressure q=
(5) perturbation
-K(iT/ip)=C(p’V
At the center
of the contact
(6) patch (7)
Pb=P0-Pi pressure
Since pi is the mean contact pressure for the entire on the contact patch is j = p0 4. we may write p0 51 = pi 111
p’. is
edge, and
the mean
(8)
FRICTIONALLY
EXCITED
THERMOELASTIC
197
PHENOMENA
and eqn. (6) becomes
t-4 Inserting this into eqn. (5) we find
ah a.2
= y=o
wP,(wq
(l-5;>
(10)
bearing in mind that y is positive down into the surface. One may use eqn. (1) even in situations where the Fourier series expansion is not used, so long as the pressure distribution is such that it could be expanded as a Fourier series. In order for the net pressure to be zero between contact spots, the perturbation pressure must in that region be -pi. ELASTIC
DISPLACEMENTS
OF SURFACES
DUE TO THE PRESSURE
DISTRIBUTION
For a load distributed along that edge of a plate (see Fig. 1) the displacement of that edge is u = & jlI” Mloglx-
(11)
Differentiation under the integral sign permits evaluation of the derivatives of v, such that d”v 2(-l)“(n-l)! b [P/(X - t;)“] dx (14 dx”= TCE 3 II
H (Dimensionless) Fig. 1. Computer solution compared with straight dimensionless quantities G and H.
line approximation
for relationship
between
the
R. A. BURTON,
198
V. NERLIKAR
If conditions are restricted to a loading symmetrical about x=0, ranging from - I to + 1, the quantity d”v/dx” can now be evaluated at x= 0, from which origin 5 becomes numerically identical with x, and one may write (see Appendix 1): d”u -= dx”
’ ,i _jplWx
2(-l)“(n-l)! 7CE
This neglects the influence secondary influence. SURFACE
BOUNDARY
(13)
of adjacent
patches
on elastic
curvature
at x=0,
a
CONDITION
If the surfaces of the thermal conductor and the thermal insulator must conform over the region of contact from -I to + 1; then, since the point x=0 is within that region, the expansion-induced curvatures from eqn. (10) and the elastic pressure-induced curvatures in the two bodies must be equal and opposite to one another. The same will be true also of the higher derivatives of r. Noting that the same pressure acts on both bodies producing elastic displacement in each, the derivatives of this net displacement will be the sum of the displacements for the two bodies or: 2(-l)“(n-l)!
d”v dx” X=0,ne,=
-I
rcE*
Pdx (14)
.! -I X”
where E* = E, E,/‘(E, + EJ
(15)
Note that for two identical materials, E*= E/2. At this point let us restrict consideration to the second derivative or curvature only, adding the initial curvature l/R to the curvature of the thermoelastic bulge, eqn. (lo), and equating this sum to the relative curvature of the elastic deformation as follows:
POWER SERIES SOLUTION
To obtain according to :
FOR CONTACT
a solution
SPOT PRESSURE
for eqn. (16)
DISTRIBUTION
p may be expanded
in a power
P=Po[1+a,(X/1)2+M2(X/1)4+a3(X/1)6+...] Here only even powers may be rewritten as: P=Po
F+sjIl
are retained
(17)
in the interest
of symmetry.
This equation
(18)
%(;ys]
The right hand term of eqn. (16) may eqn. (IS), and we find:
series
now
be integrated
after
substitution
of
(19)
FRICTIONALLY
EXCITED
TWERMOELASTIC
PHENOMENA
199
Defining n.E” 1/4Rp, E II and
eqn. (20) becomes li+G=
[I-Ca,/(25-1)f
(21)
If the same procedure is carried out for higher derivatives a general equation may be written as CR, =
“’ a,(2R+ 1)
~-3s=l
2s- (2r+ I)1 t= 1,2,..n’
(22)
The term involving R is made to disappear with the assumption that the curved surfaces have a relative parabofic shape with the central radius R, in which case higher derivatives disappear. The procedure used to obtain simultaneous solution to this set was to select a number of terms in the pressure equation s=n, and generate n equations from eqn. (22), for Y from 1 to n. An approximate value for G is inserted in equations r= 1 through r= n, and these are solved for GIN,CX~,..CX,,, The appropriate cti quantities are inserted into eqn. (2f), a value for H is sefected, and it is then solved for G. This value of G is now used in place of the approximate G in a repeat of the procedure, and successive iterations are carried out, rapidly converging to stationary values of G and c+ for the selected H. EFFECT
OF SLIDING
SPEED
ON CONTACT
SIZE
Typical results from such a calculation for a 10 term series are shown in Fig. 1. The straight line may be represented by the equation G-i-f.I8H-1.8--O
Upon reinsertion of the appropriate
(23) quantities into eqn, (23) it becomes: (24)
At this point let us introduce R = I/B& Lk= s,/nt
(25)
where d is a measure of contact width/spacing ratio and a,, refers to this ratio for the Hertzian case. The quantity l,, may be found from eqn. (24) by letting V=O, in which case
R. A. BURTON.
V. NERLIKAR
01
0
4
8
12
V/V,
16
20
2,4
(Dimensionless)
Fig. 2. Relationship between dimensionless contact width and dimensionless speed. Zero speed intercept gives fraction of surface in contact without frictional heating. When i,,,> I physical meaning disappear5 but eqn. (26) remains valid definition for I,,.
(26) The term in parentheses classical Hertz solution. growth is V,= 2Kculc+E”
is quite close to unity as would be the case for the Recalling that the critical velocity for small disturbance = 2KnlmpE”cq then
(l-r;,[pJ C
(271
Regrouping
(28) Solving
the quadratic
for A
-~f~[~~~,‘+7.2(5~-5~~)~~ i=
1
(291 ~“-I!!~ [ Jbh” f This is made meaningful by reference to Table II and Fig. 2, where values of i are shown for selected magnitudes of &, and a range of V/VC. Typical magnitudes for 4 are shown in Table I. For this calculation, however, r was assigned the rounded off value of 0.75. These results become physically meaningless for j-> 1, since this would correspond to overlapping of the contact patches. Referring specifically to Fig. 2. note that for small i,, (initial concentration of contact load), the curves drop off smoothly with increase of V/V,, and the contact patch decreases in size. For larger initial contact widths drop off is more drastic with speed. Somewhere above an 2
FRICTIONALLY
EXCITED THERMOELASTIC
201
PHENOMENA
TABLE I MAGNITUDES .._________-
FOR THE CONTACT
G -.
H
1.624 1.472 1.324 1.176 0.742 0.457 0.177 0.0375 0
0.125 0.262 0.375 0.521 0.875 1.156 1.4075 1.5 1.534
-_____
PATCH PARAMETER -_____
c
s/so
0.7467 0.7541 0.7601 0.7661 0.7811 0.7901 0.7998 0.802 0.8033
0.9295 0.93875 0.946 0.9537 0.9723 0.984 0.9956 0.998
..._.- ___.
.~___
_.....
._~
1
TABLE II EFFECT OF DIMENSIONLESS SLIDING SPEED PATCH WIDTH 1 FOR SELECTED VALUE OF E.,
V/v, ON
DIMENSIONLESS
CONTACT
(Only positive, real solutions retained) -_~-~-____-. F/K
J+
--
0.4
0.2 0.4 0.6 0.8 1.0
0.10 o.to 0.09 0.09 0.09
0.37 0.34 0.3 I 0.29 0.26
1.2 1.4 1.6 1.8 2.0 1.0942 1.0120 0.8604
0.09 0.08 0.08 0.08 0.08
0.24 0.22 0.20 0.19 0.17
.-
-
.._.~
0.1 _. _~_ i
_
0.9
1.6 .__~_.
2.5
cc ___._.~
__ 0.83 0.72
0.61,84.4 0.51. 4.59 0.42, 2.x4
2.3, 18.3 -1.09. 0.55
_-.-
u.35, 0.30, 0.26, 0.23, 0.20,
1.17. 0.41 1.21, 0.33 1.23, 0.28 1.25, 0.24 I .26, 0.21
0.99, 0.44 1.07, 0.34 1.1 I, 0.29 1.15, 0.25 1.17.0.22
2.28 2.01 1.85 1.76 1.69
0.87, 0.47 0.98, 0.36 1.04, 0.29 1.08, 0.25 1.12,0.22 0.6666*
0.7201* 0.8475*
* Corresponding to lowest value of V/C{ for which real solution exists.
initial Hertzian width of unity there is a breakover and the curves become double valued for higher magnitudes of V/V,. Put differentIy, the curves for lower values of a,, may be thought of as the lower leg of such a double valued curve, where the remainder has moved off the scale. Arguments have boen offered’ to show that the upper leg of such a curve is unstable. If the system finds itself to the right of the appropriate curve it will move downward toward smaller 2; but if it is to the left it will move upward. This means that if it is above the upper leg it is also to the left of it and will move upward toward flatter contact. These ideas are in accord with the earlier introduced instability concepts. For example, if the two surfaces were initially flat (&,-+x)), and if a disturbance
202
R. A. BURTON. V. NERLIKAR
with ;1==I could be impressed on them, this disturbance would not grow at low sliding speed but would grow above a certain critical speed. This “growth” would correspond to a shrinking of 1. The crossover is not exactly at V/V,= 1, but this is not surprising in view of the approximations used. Although 1.> 1 is physically meaningless, A,,> 1 remains meaningful. One simply refers to eqn. (26) as defining I,, and ignores the physical interpretation of an actual contact patch width. There is no limit on the magnitudes that 4RPpzE* may have. For example, for R+ccr_., Ih+ccc, and the surface is flat at V=O. COMPARKSON
OF CONTACT
WIDTH WITH INITIAL HERTZIAN
WIDTH
It is instructive to return to eqn. (23) and rederive the final equation so as to display the contact width I in ratio to the Hertzian width I,. Letting H’I= 1.18 H, this becomes G-l-H’=
1.8
(30)
If we let g-G/l;
h=H’/l’;
1’~ 1.8
gls ht2 = I’
(31)
and (32)
(33)
Defining y as (Gl,/l)/Zs, holding in mind that this calls for, Y=
paVnE* l,
(34)
4Kv
it follows from eqn. (32) that f,/lh= -y+(y2+
1)”
(35)
For large 7, E/lh-+iy and behavior is thermoelastic. Equations (34) and (35) represent essentially a solution to the problem, but may be made more useful when it is recalled that 1, = (4PR/zzE*)+
(36)
Where P = 2p,@z
and
5 =#pO
(37)
Substituting eqn. (36) into eqn. (34) and simplifying
Here it is seen that y is determined completely by operating conditions and material properties. once its magnitude is known eqn. (39, Table III or Fig. 3, will give the corresponding l/l,. Knowing 1, from eqn. (36) one can obtain contact width. Equation (37), and the total load will give maximum contact pressure. For this
FRICTIONALLY
EXCITED THERMOELASTIC
PHENOMENA
203
Fig, 3. Effect of operating variable 1 on contact width. TABLE III GLORIFICATION OF HERTZIAN PATCH WIDTH BY THE SLIDING PARAMETER y
0.1 0.2 0.4 0.8 I.2 1.6 2.0 2.5 3.0 3.5 4.0
0.904 0.8198 0.677 0.480 0,363 0.280 0.236 0.193 0.162 0.140 0.123
purpcm 5 may be appraximated or may be found more precisely from Table I as a function of G. By such reasoning we find
14
and approximating P
max=”
p.
=
za
5 as 0.75
!! PE”* 1
(-ix”>
To provide a numerical example to show a typical magnitude for y use the appropriate properties for iron : p
=0.05
c1 =6x 10e6/oF K = 5.6 Ib/sec “F
R =2 in. l/m =O
E* = 9 x lo6 Ib,‘in.2
z =0.05 in. I/ = 500 in/s P = 50 lb,
R. A. BURTON. V. NERLIKAR
204
The sliding speed and stresses are high but might be encountered in gearing. For these conditions ~=0.87. This would correspond to considerable modification of the contact stress. Even at half the sliding speed chosen. the contact width would be halved and the stress doubled, relative to static contact. CONCLUSION
Earlier studies of thermoelastic effects of frictionally heated sliding contact were concerned with initially flat surfaces. The present study takes account of an idealized form of initial waviness, and includes the initially flat contact as a limiting case. Behavior of an initially patch-like contact is a monotonic decrease of contact width with increase of sliding speed, showing no sudden change near the critical speed for small disturbance growth on flat surfaces. For contacts where the hypothetical Hertzian width !,, is greater than the spacing measure m the solution becomes double valued; and arguments may be advanced for an instability-like behavior with waviness not being amplified at small sliding speed and with waviness growing rapidly to an isolated contact patch configuration at a certain of speed. The tables along with eqns. (38) (39) and (40) permit the estimates actual contact width and pressures. COMMENTS
One may well ask how these results may be reconciled with the idea of cosine wave instability at V/V,= 1. It would appear that the wave would grow and break contact at points of low pressure; but the large disturbance solution would apparently call for suppression of this new interrupted disturbance below V/k’,+ 1.4. One should remember, however, that the present large disturbance solution is based upon a series of parabolic disturbances. If at the point of breaking contact the cosine wave would be represented by ?’ = J’J 1-cos QS) The series expansion
Extending
(41)
near x0 would give a parabolic
this parabolic
approximation
I=(x)Y=“=2$JJ
approximation
to y=O to find the equivalent
of the wave as
1, one finds (43)
Since m=71jco l/m= 2171= 0.636 = /.
(44)
Thus at the point of break-away the cosine wave may be thought of as a series of parabolas with d=O.636. Referring to Fig. 2 this corresponds closely to the magnitude of 3. at the lowest V/V, for which the double valued solution can exist. So, within the approximations used, it is not inconsistent for a cosine wave to grow just above V/V,= 1 and move to an intermittent contact solution.
FRICTIONALLY APPENDIX
EXCITED
THERMOELASTIC
PHENOMENA
205
1
The integral dx,
CPM=jt$
for
m an even integer
has unequivocal validity for m < 1. For m> 1, there is a question as to what happens in the integration process when x passes through zero. One must remember that in the present context we are dealing with an influence function valid at points exterior to the zone of application of an elemental load Pdx. Beneath the load the function is something else. To allow for this we may write
441)= i - $dx
+ if
+dx+&)
. +sx
. -I
where V(E) is a special case of cp(l), that is, E is simply a special magnitude for I. Integrating -2 ~(~) = -m-l
1 --A [ r”-’
1
t-44
The equation is satisfied when
Hence, formal integration gives
and the proper result is generated. If one seeks further reassurance, one may take the example of a uniform load for which the well known solution is u =z[(i+x)lny
Differentiating
and at x=0
twice
+ (I-x)ln($d]
206
R. A. BURTON,
One may continue in the paper.
onward
and for each case will confirm
V. NERLIKAR
the relationships
used
NOMENCLATURE
specific heat Young’s Modulus composite Young’s Modulus for two bodies E,E,/(E, +E,) dimensionless quantity see eqn. (21) dimensionless quantity see eqn. (21) mechanical equivalent of heat conductivity of material diffusivity of material (K/p”,,) length of slider half width of contact spot I for zero sliding speed half distance between centers of contact spots m/I load pressure on contact surface designator index in equations index for summation temperature time displacement of surface in J‘ direction sliding velocity critical sliding velocity for growth of small disturbances in contact pressures wear coefficient coordinate in direction of sliding coordinate normal to direction of sliding, measured from contact surface width of slider coefficient of thermal expansion (i = 1,2.3.. .) coefficient in equation dimensionless quantity, see eqn. (34) strain l/m= l/M ,I for zero sliding speed coefficient of friction Poisson’s ratio exponential coefficient of time wave number = 2n n/L p/p,, ratio of mean to maximum contact pressure. ACKNOWLEDGEMENTS
The research reported here was sponsored Naval Research, Contract NOO014-67-A-0356-0013.
by the United States Office of The work reported in references
FRICTIONALLY
EXCITED THERMOELASTIC
PHENOMENA
1, 2, 4, 5 was also performed under the same contract. acknowledgement was omitted in these publications.
207 The authors regret that
REFERENCES 1 R. A. Burton, V. Nerlikar and S. R. Kilaparti, Thermoelastic instability in a seal like configuration, Wear, 24 (1973) 1777188. 2 R. A. Burton, The role of insulating surface films in frictionally excited thermoelastic instabilities, Wear, 24 (1973) 189-198. 3 M. V. Korovchinski, The plane contact problem of thermoelasticity during quasi stationary heat generation on the contact surfaces, J. Basic Eng., (1965) 811-817. 4 R. A. Burton, S. R. Kilaparti and V. Nerlikar, A limiting stationary configuration with partially contacting surfaces, Wear, 24 (1973) 1999206. 5 R. A. Burton and V. Nerlikar, Large disturbance soiutions for initially Rat, frictionally heated thermoelastically deformed surfaces. accepted for publication in Trms. ASME.