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The influence of residual stresses on contact-load-bearing capacity
Wear, 75 (1982)
221-
221
240
THE INFLUENCE OF RESIDUAL BEARING CAPACITY *
STRESSES
ON CONTACT-LOAD-
D. A. HILLS and D. W. ASHELBY Department of Mechanical and Production {Gt. Britain) (Received July 2,1981;
Engineering,
Dent Polytechnic,
Nottingham
in revised farm August 8,1981)
Summary The elastic and shakedown limits for several new contact geometries are determined and compared. A new limit is supplied giving the maximum load which may be elastically supported by bodies containing a cylindrical residual stress system, the optimum value for which is found. This concept is important for the opt~~~ation of the residual stresses in shot-peened, burnished, ion-implied and similarly treated surfaces. The contact &onfi~rations considered are a circular contact patch with imposed traction, an elliptical contact patch under pure rolling loads and the repetitive normal loading of two second-order bodies. Examples of the use of the results obtained are discussed.
1. Concept
of shakedown
Hertz [l] studied the stress field created when two frictionless secondorder bodies were pressed together. His results are valid in the elastic regime and are useful in studying rolling contact. More recent work by others has enabled the effect of a frictional force on the stress field to be found, though only for slightly less general contact arrangements. In particular, Poritsky [2] and Smith and Liu [3] examined the two-dimensional case of two rollers pressed together and loaded by a traction and Hamilton and Goodman [4 ] treated the circular contact with friction geometry. In all
*Paper presented at the International Conference on Wear of Materials 1981, San Francisco, CA, U.S.A., March 30 - April X,1981. 0043-lS48/82/0000-0000/$02.50
@ Elsevier Sequoia/Printed in The Netherlands
222
these cases, if the yield criterion is nowhere exceeded, it is clear that there will be no permanent deformation of the bodies and that no residual stresses will remain after each loading cycle. However, if the yield criterion is exceeded, three things will occur which may permit the system to reach a state of elastic equilibrium quickly. (a) Residual stresses will be generated which, together with the applied stresses, may produce a state of stress which subsequently lies within the yield envelope. (b) Work hardening will occur, increasing the yield stress. (c) In the case of some geometries, such as a ball rolling on a flat block, plastic flow of the block will occur to produce a shallow groove. The contact patch will then be much larger, so that the mean contact pressure is lower, reducing the resultant stresses. All three phenomena contribute to the “sh~edown” of the system, i.e. the condition under which the system, having undergone some initial plastic deformation, continues in a totally elastic manner. However, work hardening and a reduction in pressure are not vital aspects of shakedown and, throughout this paper, models will be presented based on an ideally elastoplastic material. Shakedown in continuous media occurs because, as a component which has been loaded beyond the elastic Iimit is unloaded, stresses develop which are of the opposite sign to those produced by the load. These stresses are maintained by elastic recovery of the surrounding material and reduce the net value of the stresses produced by a subsequent application of the load. The “elastic limit” is the largest load which may be applied, without exceeding the yield criterion, in the absence of residual stresses. The “shakedown limit” is the largest load which may be applied which, together with the optimum residual stress system, lies within the yield criterion, Use will be made of Melan’s theorem, a simple interpretation of which has been given by Symonds [ 51. Melan stated that, if some residual stress system (even if rather cont~ved) may be found which, together with the applied stress system, lies within the yield surface, the system will shake down, although not necessarily in the way proposed, i.e. the residual stresses may differ from those speculated. Shakedown has two important implications for wear. First, whilst (for economy) tribological components should not be overdesigned, it is desirable that plastic flow should not persist. In rolling components this would not be catastrophic immediately but would accelerate fatigue failure and give rise to the forward flow of surface layers, as observed by Crook [ 61 and later by Hamilton [7] . Secondly, the nature of the self-generated stresses may be very important. Fujita and Yoshida [S, 91 have found that, when two cylinders of different widths were rolled together, a tensile stress developed perpendicul~ to the surface: this would accelerate the growth of any subsurface cracks present. In various studies of residual stresses in railway lines, deleterious tensile stresses perpendicular to the free surface have also been found [lo, 111.
223
2. Two-dimensional
sliding contact
The first contribution to shakedown theory in tribology was made by Johnson [ 121 who showed that the shakedown limit for two cylinders rolled together is given by
5
< 4.00
(1)
70
Johnson obtained this result by an examination of the stress history projected onto the deviatoric (n) plane, but a simpler proof which may jmmediately be extended to a sliding cylinder is as follows. Consider the sliding or rolling cylinder of Fig. 1. The problem is strictly two-dimensional and the residual stresses must themselves satisfy the equilibrium equations when the load has passed. These are
(bf
tY
Fig. 1. (a) Coordinate set used in twodimensional work. The axis of the cylinder lies in the t direction. (b) CoordinMe set used in sliding threedimensional work. The circular contact patch is centred on the z axis and progresses in the positive x direction. (c) Coordinate set used in rolling three-dimensional work with an elliptical contact patch.
(2) ak aryz au,, -+-+-= ay
ax
0
a2
Because stresses cannot aaij
aUij
vary with respect to y or z,
o
_=_____~
az
ay
ThUS a7,2
-iiT
ar,y aaxx
=_=_..-‘_=o
(3)
ax
ax
Since the free surface must remain clear of all tractions stresses do not vary with depth 7x2 =7 XY = a,,
and since
for all values of x
=0
Because there can be no residual orthogonal for these cases may be expressed as
(4)
shear, the shakedown
TXY< 7.0
limit
(5)
For a rolling cylinder 7,
ymax
- 0.25p,
giving eqn. (1). Later, Merwin and Johnson f 13 ] proposed an expiration for the forward flow phenomenon and, at the same time, evaluated the residual stresses by using the Prandtl-Reuss flow rules and assuming that, if the yield criterion were not greatly exceeded, the strains could be approximated by the extended elastic strains. These results were then generalized to the case of a sliding cylinder 1141. It is possible to extend this technique and to examine the “noplasticity” condition when there is a preexisting residual stress system. The essential difference between shakedown and previously built-in stresses is that the former are totally independent (subject to the yield criterion and equiljb~um requirement) whereas the latter are cylindrical, i.e. uyyr= ~a, = (T, (where the added subscript r refers to a residual stress), Consider the second invariant of the stress deviator tensor: J2
= &x
+
-
Q2
+ (@yy
T.,Y2+ Tyz2+ TZX2
-
%,)2
+ (Q*+
--
%d21
+
(6)
225
If the shakedown limit of eqn. (5) is to be attained, within the braces must be zero. Since CJ,,~ = 0,
(Jyyr = uxx
it is clear that the term
I
I - @YY , I
(7)
= (J%ZX 0X.X - Qt*
where a prime indicates stress due to the contact If the residual stresses are not independent, must instead simply be minimized, i.e. &Jxxr r
-2
-
uyy’ -
u,p
+ (uyy’
-- u,,‘)2
+ (uz*’
(uxx’ - uyy’ -- u,) + 2(u,,’ + ur - u,,‘)
U’, = + (2u,,’
- uy Y’ - a*,‘)
The new effective
value of J, is
J2 = +{(-&
-$c$)~
=_; (uz’ - u/)2
load alone. the term within the braces
+ (uZ’ - u~‘)~ + (&’
+ u, -- u,.J2}
= 0
=0
(8)
- u,‘)~} + T,_‘~
+ 7/s
(9)
When the shakedown condition is reached at the surface first, i.e. p > 0.367, the worst stresses occur at the origin and are UXX
=
uyy
=
U%Z TXY = -
2v
After substituting limit is given by
(10)
c--p0
P
these values into eqn. (5) it is clear that the shakedown
PO < f_ (11) 70 I-( A computer program was written, based on Smith and Liu’s solution, to determine the elastic limit, shakedown limit and “load limit”, the last quantity being the largest load which may be applied, in addition to the optimum cylindrical residual stress system, without causing plasticity. For I-(> 0.32, the load limit for two sliding cylinders is achieved at the surface first, and is given by
po < 6 ’ (0.04
1 + /P)1’2
v = 0.3
(12)
so that as P becomes very large the shakedown limit (11) and load limit (12) tend to the same value. The results are summarized in Fig. 2. The elastic limit found by Johnson and based on Tresca’s criterion is omitted for clarity. A load limit based on Tresca’s criterion would coincide with the shakedown
226
Fig. 2. A comparison of the elastic, shakedown and load limits for the case of a twodimensional hertzian contact with traction. The diagram also indicates whether the point of severest stress is at or below the surface and the value of the optimum residual stress needed to achieve the load limit.
limit, since only one degree of freedom is required to maintain the largest principal shear stress equivalent to rxY’. The results found are summarized in Table 1, the optimum residual stress depends on the applied load. Thisis not too serious because the value of the applied stresses will not become critical until near the load limit and, therefore, residual stresses based on this pressure should be used. A graph of the optimum residual stresses is shown in Fig. 3 and an illustration of the contraction of the plastic zones caused by these residual stresses is given in Fig. 4. When P = 0,367 the surface value of the shear stress is equal to the subsurface value, and subsequently higher values of p are substituted into eqns. (11) and (12) respectively to find the shakedown and load limits.
227 TABLE 1 Shakedown results for sliding cylinder y*
P
0 0.1 0.2 0.3 0.367
0.866 0.866 0.864 0.861 0.858
rxY maxlpO
x*
-0.500 -0.452 -0.415 -0.384 -0.366
0.250 0.281 0.312 0.345 0.367
Shakedown limit
Load limit
TOIPO
TO/PO
4.00 3.56 3.20 2.90 2.73
3.95 3.49 3.10 2.78 2.59
(~rlPOhimum -0.174 -0.152 -0.128 -0.102 -0.083
0.05
0
-0.05
-0.10
-0.15
- 0.20
Fig. 3. The values of the residual stresses uxxr and uyyr needed to achieve the shakedown limit for two-dimensional contact. Also shown is the circular residual stress corresponding to the load limit.
3. Circular contact
with friction
A second configuration examined was that of two balls rolling or sliding together, or any arrangement resulting in a circular contact patch and hertzian loading. The elastic limit is clearly achieved on the plane of
228
(b)
Pig. 4. Illustration of the extent of the initial yielding zone when a two-dimensional contact with traction is loaded to the shakedown limit (X, zone with no residual stresses; 0, zone with optimum circular residual stress}: (a)pg/k = 4, I-( = 0; (b) pg/k = 3.56, ,~=OO.l;(c)pofk = 3.2,~’ 0.2;(d)pglk= 2.9,/i= 0.3.
229
symmetry, i.e. y = 0 (Fig. ‘I), and may be found by using the solution for the elastic stresses determined by Hamilton and Goodman [4] . The shakedown condition presents extra problems. It has been shown that, for twodimensional rolling or sliding contact, the shakedown limit is governed by the orthogonal shear (eqn. (5)). It will now be shown that the same limitation applies for rolling contact under a general elliptical contact patch. It is reasonable that the same condition should apply for a sliding threedimensional contact and this has been assumed. The shakedown limits found are entirely consistent with those for other cases treated more rigorously. Figure 1 shows a plan view of one contacting body with a general elliptical contact patch. By careful considerations of symmetry, the following relations .are apparent: T,,
(8) = ?,, (-4) = -7yz(-e)
Gus Txy
= -7%,
(t?)= --Txy
(n -- f3)
= T~~(T -4)
(-3)
=
--Txy
(n -_ e)
Thus, residual stresses’T,, and T,.~ can be of no benefit. Two possible shakedown limits are therefore Txz < -2. To TXY
< ’
(13)
70
The solution for the elastic stresses in polar coordinates for a circular contact patch was given by Huber [ 151. These are (with the dimensions normalized to the contact radius)
+
u
_ tz
=-
PO T, __.._=--
PO
(1+ ~)r.$‘~ tan-l
i
2*3 pi5
)
u
u2 + 2*2
u1t2 r*ze2 u2 + 2*2 1 -I-u
and from symmetry
( ull2 1
(14)
230
Tzo =T,(j
=o
where 112 u = $ [r*a + 2*2 --. I -t {(r*2 + a*2 ____
We can transform
into a Cartesian coordinate
+ 4z*2)1i2~
system by using
0X.XSEu, cos B + oe8 sin20 fiYY = or, sin28 + u8@ cos28 %,? = 02, 7XY =- ; (e,, - aeo) sin(28)
(15)
7xz = 7, cos 0 = rrz sin 0 TYZ
(16)
maximizing
T,~ with respect to 8 gives
aT,Y = (urr -o&j) aa
cos(2B) = 0
i.e.
e =n/4 After substitution
from eqns. (14), the maximum
It transpires 7XY
i-1PO max TXI -
value of 7,Y is given by
that the maxima and their locations =
0.05
= 0.214
are
at z* = O.!S,‘r* = 1.11
at z* = 0.35, r* = 0.848
i PO 1 max
The orthogonal which is given by %’ 4 4.675
shearing stress again provides the shakedown
limit,
(17)
70
In the case of elliptical contact, as b/a + 00 the shear rXYmaX falls continuously to zero, and it is therefore reasonable to assume that the orthogonal shear provides the limiting factor in all practical shakedown conditions under traversing contact loading. Returning to the problem of shakedown with a circular sliding contact patch, we must take great care when evaluating the stresses because of the indeterminate nature of the equations given by Hamilton and Goodman. In their notation, consistent with Fig. 1,
231
y=o p
=x*
R, *2
a*2 +x*2
=z
+z *2 - 1 + 2iz*
=x*2
Let R,*2 = J exp(iA), then J = {(x*~ + 5~2
4z*2)112
22*
(
X = tan-l
__ 1~2 +
x*2 + z*2 - 1 )
If R,* = 4 + i3/ (say) then
$, = J1/2
sin !_ (12
The imaginary parts of the complex functions F-H required are F = +(z*$
- I#I)+ $x*‘I’
+52z*$J-J,(z*2+x*2-1)1
$
x*w 4J3
1
232
where
K = (q3+ 2*)2 + (lj
+
1)2
The stresses due to the tractive force on the surface are given by
Shakedown
Elsstic
Limit
Elastic
0
Limit
Limit
01
0.2
03
0.4
0.5
06
0.7
p
Fig. 5. A comparison of the elastic, shakedown and load limits for the case of a circular hertzian contact with traction. The diagram also indicates whether the point of severest stress is at or below the surface and the value of the optimum residual stress needed to achieve the load limit (cf. Fig. 2).
233
1
uxx -=_ PPO GYY ---
ClPo
x*3 _
1
x*3
I-IV - .;
z*H, + (1 - V)X*H, + f **z*AY,. - ZVX*~F
Hv -;x*&)
---~vx*~F/
(18) 1 xz -=_ 2G + ;Hz ClPo x*2
7
+ z*(F + x*F,) - 2v*F
The stresses due to the normal component of the contact load may be found from eqns. (14) and superimposed on those due to the tractive forces. Figure 5 shows the variation in shakedown, elastic and limit loads found exactly as before, whilst in Fig. 6 the optimum residual stresses are
Fig. 6. The values of the residual stresses uxxr and uyyr needed to achieve the shakedown limit for three-dimensional contact. Also shown is the circular residual stress corresponding to the load limit (cf. Fig. 3).
234 TABLE 2 Shakedown results for a sliding ball X*
P
0.848 0.849 0.847 0.842 0.841
0 0.1 0.2 0.3 0.32
z*
Shakedown
0.350 0.321 0.298 0.278 0.274
Load limit
limit
TOIPO
70 IPO
4.675 4.129 3.683 3.315 Surface
4.632 4.054 3.574 3.170 3.077
((JrlPO)optimum
-. -0.208 -0.205 -0.187 -0.179 -0.163
summarized. The results obtained are in Table 2 also, and Fig, 7 shows how the critical points vary with friction. The results found are generally similar to those obtained with finite elements by Rydholm and Fredriksson [ 161, who also deliberately assumed that the residual stresses need not obey any specific constitutive relation (Ponter’s theorem). Their curves differ for the shakedown limit when this is attained below the surface and show higher values in this region. Because of their totally numerical method it is difficult to see where the discrepancy lies, but their higher values are consistent with fewer constraints being imposed on the nature of the residual stresses. At higher coefficients of friction, the shakedown and load limits both occur at
EL
EL
p 50.5 -s
@352>!_i
L
pro.303
0.2--
p -0.303 SL
i 0.4--p-o
,l=
0.352
p-0
0.6-i
Fig. 7. Representation of a vertical section through the centre-line of contact for a circular hertzian contact with friction, illustrating how the points of severest stress change with the coefficient of friction: EL, elastic limit; SL, shakedown limit.
235
the surface. Their values are easily found from the simple relations Hamilton and Goodman. The shakedown limit is given by
given by
Although the foregoing results are of fundamental interest, their use is rather limited. This is because, as a rolling ball on a flat shakes down, a groove is created so that the contact becomes elliptical. This problem is now considered in detail.
4. Shakedown
with an elliptical
contact
patch
A heavily loaded rolling ball creates a groove, and hence an elliptical contact patch, with repeated rolling passes. The minor axis of the ellipse is aligned with the direction of rolling. For a railway carriage wheel rolling along a length of railway track, the contact again is nearly elliptical, but with the major axis extending along the rolling direction (Fig. 8). The solution of the “elliptical” shakedown problem is therefore of practical interest. Since the shakedown condition has been established as r,, C TV, the problem for elliptical contact is reduced to solving the Hertz problem on the plane y - 0. The usual restrictions of isotropy and homogeneity apply, and with reference to Fig. 8 the larger contact semiaxis should be less than the smallest principal radius of curvature. Fessler and Ollerton [17] have experimentalfy shown that the Hertz conditions are satisfied if b c 0.3Rb&
(191
They also give the solution for rXz ; much of the theory is contained in Love’s book [ 181. For the plane y = 0, where the severest stresses act, the auxiliary variable y , when normalized with respect to the semiminor axis, becomes Y
*..--;
ix*2 + x*2
- 1 e (@*a + x*2 - 1)2 + 4a*a)x/a]
(20)
Then
and the shear stress is given by 7XB= -pokQ and the shakedown -_kQ
PO
(22) limit (eqns. (13)) may be written (23)
236
Fig. 8. Ways in which an elliptical contact patch can be generated in practice. On the left a ball is shown rolling in a groove so that the minor axis of the ellipse lies in the rolling direction. On the right is a typical railway contact patch geometry where the major axis of the ellipse lies in the direction of rolling.
The elastic limit may be found from the work of Thomas and Hoersch [ 191, and always occurs on the z axis. An experimental verification of the achievement of the shakedown limit has been performed and the results were encouraging [ZO] , The results of these calculations are shown in Fig. 9 although, as the ellipse becomes extremely elongated in the direction of rolling, a different criterion dominates the shakedown limit. This is when the residual stresses themselves are so large that they are on the point of plastic flow. The criterion for this “secondary plastic flow” may be written in terms of the applied stresses as
237
x:
2*
1.0
1 o-8
2
0.6
3 04
0.2
0 0.1
10
K
1
Fig. 9. A comparison of elastic, shakedown and load limits for the case of elliptical contact without friction: curves 1 and 3, shakedown limit (equal to load limit); curve 2, elastic limit (x = 0). It should be noted that the upper shakedown and load limit curves on the right apply only when the contact patch is suffering a repeated normal loading and there is no rolling motion.
When two bodies are pressed normally onto one another repetitively, the orthogonal shear stresses do not change sign and hence eqn. (24) represents the shakedown limit for all ellipse eccentricities. Figure 10 shows the location of the severest stress state. When the limitation is one of secondary plastic flow, the load limit equations (8) and (9) no longer apply, Instead it should be noted that of7 = 3fD7e
(25)
and that the load limit is given by Jz
=
ii@,,
+
< ’
+
or,.
-
4’
+
((J,,
-
oyy
-
M2
+
(ox,
-
o,,)~)
+
CTij2
2 To
i.e. $[{u,,
+ 31’%() - a2,}2 + {uzz - uyy - 31’27e}2 + (u,, +
z1Tij2
<
- r&2]
+
TO2
This equation must usually be solved numerically pressure which just produces secondary yielding.
(26)
to find the contact
5. Conclusions The elastic, shakedown and load limits were found for a variety of configurations and should find immediate application in both transferred
238
I 3.5-.---___--___L-_-_-
. . ___.
_~
/_.
:
_i
*
_ ~___.
.___
~_~
/
I
I I
3~0 -+---I,
/
ei 0.1
i
’
I
i
,I1 ~~~
/ I
/
;
1
i
1
Ii,11 k
10
Fig. 10. Location of the points of severest stress for the case of an elliptical contact patch without friction, It should be noted that the severest stresses giving the load limit occur at almost the same point as those producing the shakedown limit (cf. Fig. 7).
motion, e.g. rolling element bearings, and normal loading configurations, relay contacts. These limits appear to be consistent with experimental observations.
Nomenclature
a, b J2 k PO r*,x* x*, & V D 7
70
8 yi,
t*
lengths of semiaxes of contact ellipse second invariant of the stress deviator tensor b/a, oblateness of ellipse peak contact pressure polar coordinates normalized to characteristic contact dimension Cartesian coordinates normalized to characteristic contact dimension coefficient of friction Poisson’s ratio normal stress shear stress yield in pure shear
e.g.
239
References 1 H. Hertz, Gesammelte Werke, Vol. 1, Leipzig, 1895, p. 155. 2 H. Poritsky, Stresses and deflexions of cylindrical bodies in contact with application to contact of gears and of locomotive wheels, J. Appl. Meek., 72 (1950) 191 - 201. 3 J. 0. Smith and C. K. Liu, Stresses due to tangential and normal loads on an elastic solid with application to some contact stress problems, J. Appl. Meek., 75 (1953) 157 - 166. 4 G. M. Hamilton and L. E. Goodman, The stress field created by a circular sliding contact, J. Appl. Meek., 88 (1966) 371 - 378. 5 P. S. Symonds, Shakedown in continuous media,J. Appl. Meek., 18 (1) (1951) 85 - 89. 6 A. W. Crook, Simulated gear tooth contact; some experiments on their lubrication and deformation, Proc., Inst. Meek. Eng., London, 171 (5) (1957) 187 - 214. 7 G. C. Hamilton, Plastic flow in rollers loaded above the yield point, Proc., Inst. Meek. Eng., London, 177 (1963) 667 - 675. 8 K. Fujita and A. Yoshida, Surface fatigue failure of case-hardened nickel-chromium steel rollers under pure rolling and sliding-rolling contact, Wear, 51 (1979) 365 - 374. 9 K. Fujita and A. Yoshida, Surface durability of case-hardened nickel-chromium steel rollers under pure rolling and sliding-rolling sliding contacts, Wear, 52 (1979) 37 - 48. 10 A Preliminary description of stresses in railroad rail, NTIS Rep. PB-272054, November 1976 (U.S. National Technical Information Service). 11 G. C. Martin and W. W. Hay, The influence of wheel-rail contact forces on the formation of rail shells, ASME Preprint 72-WA/RT-8, 1972. 12 K. L. Johnson, A shakedown limit in rolling contact, in Proc. 4th Natl. Conf. on Applied Mechanics, Berkeley, CA, June 1962, pp. 971 - 975. 13 J. E.!‘Merwin and K. L. Johnson, An analysis of plastic deformation in rolling contact, Froc., Znst: Meek. Eng., London, 177 (25) (1963) 676 - 685. 14 K. Ls. Johnson and J. A. Jefferis, Plastic flow and residual stresses in rolling and sliding contact, Proc., Inst. Meek. Eng., 177 (25) (1963) 54 - 65. 15 M. T. Huber, Zur Theorie der Beruhrung Fester Elasticher Korper, Ann. Pkys. (Leipzig), 14 (1904) 152 - 163. 16 G. Rydholm and B. Fredriksson, Shakedown in rolling contact problems, in Proc. Znt. Conf. on Applied Mechanics: Contact Problems and Load Transfer in Mechanical Assemblages, Euromeck Colloq. 110, Link&ping, Sweden, September 27 - 29, 1978. 17 H. Fessler and E. Ollerton, Contact stresses in toroids under radial loads, Br. J. Appl. Pkys., 8 (1957) 387 - 393. 18 A. E. H. Love, The MatkematicaZ Theory of Elasticity, Cambridge University Press, London, 4th edn., 1952, pp. 192 - 198. 19 H. R. Thomas and V. A. Hoersch, Stresses due to the pressure of one elastic solid upon another, Bull. 212, 1930 (University of Illinois Engineering Experimental Station). 20 D. A. Hills and D. W. Ashelby, On the groove created by a heavily loaded rolling b.all, submitted to Proc. R. Sot. London.
221-
221
240
THE INFLUENCE OF RESIDUAL BEARING CAPACITY *
STRESSES
ON CONTACT-LOAD-
D. A. HILLS and D. W. ASHELBY Department of Mechanical and Production {Gt. Britain) (Received July 2,1981;
Engineering,
Dent Polytechnic,
Nottingham
in revised farm August 8,1981)
Summary The elastic and shakedown limits for several new contact geometries are determined and compared. A new limit is supplied giving the maximum load which may be elastically supported by bodies containing a cylindrical residual stress system, the optimum value for which is found. This concept is important for the opt~~~ation of the residual stresses in shot-peened, burnished, ion-implied and similarly treated surfaces. The contact &onfi~rations considered are a circular contact patch with imposed traction, an elliptical contact patch under pure rolling loads and the repetitive normal loading of two second-order bodies. Examples of the use of the results obtained are discussed.
1. Concept
of shakedown
Hertz [l] studied the stress field created when two frictionless secondorder bodies were pressed together. His results are valid in the elastic regime and are useful in studying rolling contact. More recent work by others has enabled the effect of a frictional force on the stress field to be found, though only for slightly less general contact arrangements. In particular, Poritsky [2] and Smith and Liu [3] examined the two-dimensional case of two rollers pressed together and loaded by a traction and Hamilton and Goodman [4 ] treated the circular contact with friction geometry. In all
*Paper presented at the International Conference on Wear of Materials 1981, San Francisco, CA, U.S.A., March 30 - April X,1981. 0043-lS48/82/0000-0000/$02.50
@ Elsevier Sequoia/Printed in The Netherlands
222
these cases, if the yield criterion is nowhere exceeded, it is clear that there will be no permanent deformation of the bodies and that no residual stresses will remain after each loading cycle. However, if the yield criterion is exceeded, three things will occur which may permit the system to reach a state of elastic equilibrium quickly. (a) Residual stresses will be generated which, together with the applied stresses, may produce a state of stress which subsequently lies within the yield envelope. (b) Work hardening will occur, increasing the yield stress. (c) In the case of some geometries, such as a ball rolling on a flat block, plastic flow of the block will occur to produce a shallow groove. The contact patch will then be much larger, so that the mean contact pressure is lower, reducing the resultant stresses. All three phenomena contribute to the “sh~edown” of the system, i.e. the condition under which the system, having undergone some initial plastic deformation, continues in a totally elastic manner. However, work hardening and a reduction in pressure are not vital aspects of shakedown and, throughout this paper, models will be presented based on an ideally elastoplastic material. Shakedown in continuous media occurs because, as a component which has been loaded beyond the elastic Iimit is unloaded, stresses develop which are of the opposite sign to those produced by the load. These stresses are maintained by elastic recovery of the surrounding material and reduce the net value of the stresses produced by a subsequent application of the load. The “elastic limit” is the largest load which may be applied, without exceeding the yield criterion, in the absence of residual stresses. The “shakedown limit” is the largest load which may be applied which, together with the optimum residual stress system, lies within the yield criterion, Use will be made of Melan’s theorem, a simple interpretation of which has been given by Symonds [ 51. Melan stated that, if some residual stress system (even if rather cont~ved) may be found which, together with the applied stress system, lies within the yield surface, the system will shake down, although not necessarily in the way proposed, i.e. the residual stresses may differ from those speculated. Shakedown has two important implications for wear. First, whilst (for economy) tribological components should not be overdesigned, it is desirable that plastic flow should not persist. In rolling components this would not be catastrophic immediately but would accelerate fatigue failure and give rise to the forward flow of surface layers, as observed by Crook [ 61 and later by Hamilton [7] . Secondly, the nature of the self-generated stresses may be very important. Fujita and Yoshida [S, 91 have found that, when two cylinders of different widths were rolled together, a tensile stress developed perpendicul~ to the surface: this would accelerate the growth of any subsurface cracks present. In various studies of residual stresses in railway lines, deleterious tensile stresses perpendicular to the free surface have also been found [lo, 111.
223
2. Two-dimensional
sliding contact
The first contribution to shakedown theory in tribology was made by Johnson [ 121 who showed that the shakedown limit for two cylinders rolled together is given by
5
< 4.00
(1)
70
Johnson obtained this result by an examination of the stress history projected onto the deviatoric (n) plane, but a simpler proof which may jmmediately be extended to a sliding cylinder is as follows. Consider the sliding or rolling cylinder of Fig. 1. The problem is strictly two-dimensional and the residual stresses must themselves satisfy the equilibrium equations when the load has passed. These are
(bf
tY
Fig. 1. (a) Coordinate set used in twodimensional work. The axis of the cylinder lies in the t direction. (b) CoordinMe set used in sliding threedimensional work. The circular contact patch is centred on the z axis and progresses in the positive x direction. (c) Coordinate set used in rolling three-dimensional work with an elliptical contact patch.
(2) ak aryz au,, -+-+-= ay
ax
0
a2
Because stresses cannot aaij
aUij
vary with respect to y or z,
o
_=_____~
az
ay
ThUS a7,2
-iiT
ar,y aaxx
=_=_..-‘_=o
(3)
ax
ax
Since the free surface must remain clear of all tractions stresses do not vary with depth 7x2 =7 XY = a,,
and since
for all values of x
=0
Because there can be no residual orthogonal for these cases may be expressed as
(4)
shear, the shakedown
TXY< 7.0
limit
(5)
For a rolling cylinder 7,
ymax
- 0.25p,
giving eqn. (1). Later, Merwin and Johnson f 13 ] proposed an expiration for the forward flow phenomenon and, at the same time, evaluated the residual stresses by using the Prandtl-Reuss flow rules and assuming that, if the yield criterion were not greatly exceeded, the strains could be approximated by the extended elastic strains. These results were then generalized to the case of a sliding cylinder 1141. It is possible to extend this technique and to examine the “noplasticity” condition when there is a preexisting residual stress system. The essential difference between shakedown and previously built-in stresses is that the former are totally independent (subject to the yield criterion and equiljb~um requirement) whereas the latter are cylindrical, i.e. uyyr= ~a, = (T, (where the added subscript r refers to a residual stress), Consider the second invariant of the stress deviator tensor: J2
= &x
+
-
Q2
+ (@yy
T.,Y2+ Tyz2+ TZX2
-
%,)2
+ (Q*+
--
%d21
+
(6)
225
If the shakedown limit of eqn. (5) is to be attained, within the braces must be zero. Since CJ,,~ = 0,
(Jyyr = uxx
it is clear that the term
I
I - @YY , I
(7)
= (J%ZX 0X.X - Qt*
where a prime indicates stress due to the contact If the residual stresses are not independent, must instead simply be minimized, i.e. &Jxxr r
-2
-
uyy’ -
u,p
+ (uyy’
-- u,,‘)2
+ (uz*’
(uxx’ - uyy’ -- u,) + 2(u,,’ + ur - u,,‘)
U’, = + (2u,,’
- uy Y’ - a*,‘)
The new effective
value of J, is
J2 = +{(-&
-$c$)~
=_; (uz’ - u/)2
load alone. the term within the braces
+ (uZ’ - u~‘)~ + (&’
+ u, -- u,.J2}
= 0
=0
(8)
- u,‘)~} + T,_‘~
+ 7/s
(9)
When the shakedown condition is reached at the surface first, i.e. p > 0.367, the worst stresses occur at the origin and are UXX
=
uyy
=
U%Z TXY = -
2v
After substituting limit is given by
(10)
c--p0
P
these values into eqn. (5) it is clear that the shakedown
PO < f_ (11) 70 I-( A computer program was written, based on Smith and Liu’s solution, to determine the elastic limit, shakedown limit and “load limit”, the last quantity being the largest load which may be applied, in addition to the optimum cylindrical residual stress system, without causing plasticity. For I-(> 0.32, the load limit for two sliding cylinders is achieved at the surface first, and is given by
po < 6 ’ (0.04
1 + /P)1’2
v = 0.3
(12)
so that as P becomes very large the shakedown limit (11) and load limit (12) tend to the same value. The results are summarized in Fig. 2. The elastic limit found by Johnson and based on Tresca’s criterion is omitted for clarity. A load limit based on Tresca’s criterion would coincide with the shakedown
226
Fig. 2. A comparison of the elastic, shakedown and load limits for the case of a twodimensional hertzian contact with traction. The diagram also indicates whether the point of severest stress is at or below the surface and the value of the optimum residual stress needed to achieve the load limit.
limit, since only one degree of freedom is required to maintain the largest principal shear stress equivalent to rxY’. The results found are summarized in Table 1, the optimum residual stress depends on the applied load. Thisis not too serious because the value of the applied stresses will not become critical until near the load limit and, therefore, residual stresses based on this pressure should be used. A graph of the optimum residual stresses is shown in Fig. 3 and an illustration of the contraction of the plastic zones caused by these residual stresses is given in Fig. 4. When P = 0,367 the surface value of the shear stress is equal to the subsurface value, and subsequently higher values of p are substituted into eqns. (11) and (12) respectively to find the shakedown and load limits.
227 TABLE 1 Shakedown results for sliding cylinder y*
P
0 0.1 0.2 0.3 0.367
0.866 0.866 0.864 0.861 0.858
rxY maxlpO
x*
-0.500 -0.452 -0.415 -0.384 -0.366
0.250 0.281 0.312 0.345 0.367
Shakedown limit
Load limit
TOIPO
TO/PO
4.00 3.56 3.20 2.90 2.73
3.95 3.49 3.10 2.78 2.59
(~rlPOhimum -0.174 -0.152 -0.128 -0.102 -0.083
0.05
0
-0.05
-0.10
-0.15
- 0.20
Fig. 3. The values of the residual stresses uxxr and uyyr needed to achieve the shakedown limit for two-dimensional contact. Also shown is the circular residual stress corresponding to the load limit.
3. Circular contact
with friction
A second configuration examined was that of two balls rolling or sliding together, or any arrangement resulting in a circular contact patch and hertzian loading. The elastic limit is clearly achieved on the plane of
228
(b)
Pig. 4. Illustration of the extent of the initial yielding zone when a two-dimensional contact with traction is loaded to the shakedown limit (X, zone with no residual stresses; 0, zone with optimum circular residual stress}: (a)pg/k = 4, I-( = 0; (b) pg/k = 3.56, ,~=OO.l;(c)pofk = 3.2,~’ 0.2;(d)pglk= 2.9,/i= 0.3.
229
symmetry, i.e. y = 0 (Fig. ‘I), and may be found by using the solution for the elastic stresses determined by Hamilton and Goodman [4] . The shakedown condition presents extra problems. It has been shown that, for twodimensional rolling or sliding contact, the shakedown limit is governed by the orthogonal shear (eqn. (5)). It will now be shown that the same limitation applies for rolling contact under a general elliptical contact patch. It is reasonable that the same condition should apply for a sliding threedimensional contact and this has been assumed. The shakedown limits found are entirely consistent with those for other cases treated more rigorously. Figure 1 shows a plan view of one contacting body with a general elliptical contact patch. By careful considerations of symmetry, the following relations .are apparent: T,,
(8) = ?,, (-4) = -7yz(-e)
Gus Txy
= -7%,
(t?)= --Txy
(n -- f3)
= T~~(T -4)
(-3)
=
--Txy
(n -_ e)
Thus, residual stresses’T,, and T,.~ can be of no benefit. Two possible shakedown limits are therefore Txz < -2. To TXY
< ’
(13)
70
The solution for the elastic stresses in polar coordinates for a circular contact patch was given by Huber [ 151. These are (with the dimensions normalized to the contact radius)
+
u
_ tz
=-
PO T, __.._=--
PO
(1+ ~)r.$‘~ tan-l
i
2*3 pi5
)
u
u2 + 2*2
u1t2 r*ze2 u2 + 2*2 1 -I-u
and from symmetry
( ull2 1
(14)
230
Tzo =T,(j
=o
where 112 u = $ [r*a + 2*2 --. I -t {(r*2 + a*2 ____
We can transform
into a Cartesian coordinate
+ 4z*2)1i2~
system by using
0X.XSEu, cos B + oe8 sin20 fiYY = or, sin28 + u8@ cos28 %,? = 02, 7XY =- ; (e,, - aeo) sin(28)
(15)
7xz = 7, cos 0 = rrz sin 0 TYZ
(16)
maximizing
T,~ with respect to 8 gives
aT,Y = (urr -o&j) aa
cos(2B) = 0
i.e.
e =n/4 After substitution
from eqns. (14), the maximum
It transpires 7XY
i-1PO max TXI -
value of 7,Y is given by
that the maxima and their locations =
0.05
= 0.214
are
at z* = O.!S,‘r* = 1.11
at z* = 0.35, r* = 0.848
i PO 1 max
The orthogonal which is given by %’ 4 4.675
shearing stress again provides the shakedown
limit,
(17)
70
In the case of elliptical contact, as b/a + 00 the shear rXYmaX falls continuously to zero, and it is therefore reasonable to assume that the orthogonal shear provides the limiting factor in all practical shakedown conditions under traversing contact loading. Returning to the problem of shakedown with a circular sliding contact patch, we must take great care when evaluating the stresses because of the indeterminate nature of the equations given by Hamilton and Goodman. In their notation, consistent with Fig. 1,
231
y=o p
=x*
R, *2
a*2 +x*2
=z
+z *2 - 1 + 2iz*
=x*2
Let R,*2 = J exp(iA), then J = {(x*~ + 5~2
4z*2)112
22*
(
X = tan-l
__ 1~2 +
x*2 + z*2 - 1 )
If R,* = 4 + i3/ (say) then
$, = J1/2
sin !_ (12
The imaginary parts of the complex functions F-H required are F = +(z*$
- I#I)+ $x*‘I’
+52z*$J-J,(z*2+x*2-1)1
$
x*w 4J3
1
232
where
K = (q3+ 2*)2 + (lj
+
1)2
The stresses due to the tractive force on the surface are given by
Shakedown
Elsstic
Limit
Elastic
0
Limit
Limit
01
0.2
03
0.4
0.5
06
0.7
p
Fig. 5. A comparison of the elastic, shakedown and load limits for the case of a circular hertzian contact with traction. The diagram also indicates whether the point of severest stress is at or below the surface and the value of the optimum residual stress needed to achieve the load limit (cf. Fig. 2).
233
1
uxx -=_ PPO GYY ---
ClPo
x*3 _
1
x*3
I-IV - .;
z*H, + (1 - V)X*H, + f **z*AY,. - ZVX*~F
Hv -;x*&)
---~vx*~F/
(18) 1 xz -=_ 2G + ;Hz ClPo x*2
7
+ z*(F + x*F,) - 2v*F
The stresses due to the normal component of the contact load may be found from eqns. (14) and superimposed on those due to the tractive forces. Figure 5 shows the variation in shakedown, elastic and limit loads found exactly as before, whilst in Fig. 6 the optimum residual stresses are
Fig. 6. The values of the residual stresses uxxr and uyyr needed to achieve the shakedown limit for three-dimensional contact. Also shown is the circular residual stress corresponding to the load limit (cf. Fig. 3).
234 TABLE 2 Shakedown results for a sliding ball X*
P
0.848 0.849 0.847 0.842 0.841
0 0.1 0.2 0.3 0.32
z*
Shakedown
0.350 0.321 0.298 0.278 0.274
Load limit
limit
TOIPO
70 IPO
4.675 4.129 3.683 3.315 Surface
4.632 4.054 3.574 3.170 3.077
((JrlPO)optimum
-. -0.208 -0.205 -0.187 -0.179 -0.163
summarized. The results obtained are in Table 2 also, and Fig, 7 shows how the critical points vary with friction. The results found are generally similar to those obtained with finite elements by Rydholm and Fredriksson [ 161, who also deliberately assumed that the residual stresses need not obey any specific constitutive relation (Ponter’s theorem). Their curves differ for the shakedown limit when this is attained below the surface and show higher values in this region. Because of their totally numerical method it is difficult to see where the discrepancy lies, but their higher values are consistent with fewer constraints being imposed on the nature of the residual stresses. At higher coefficients of friction, the shakedown and load limits both occur at
EL
EL
p 50.5 -s
@352>!_i
L
pro.303
0.2--
p -0.303 SL
i 0.4--p-o
,l=
0.352
p-0
0.6-i
Fig. 7. Representation of a vertical section through the centre-line of contact for a circular hertzian contact with friction, illustrating how the points of severest stress change with the coefficient of friction: EL, elastic limit; SL, shakedown limit.
235
the surface. Their values are easily found from the simple relations Hamilton and Goodman. The shakedown limit is given by
given by
Although the foregoing results are of fundamental interest, their use is rather limited. This is because, as a rolling ball on a flat shakes down, a groove is created so that the contact becomes elliptical. This problem is now considered in detail.
4. Shakedown
with an elliptical
contact
patch
A heavily loaded rolling ball creates a groove, and hence an elliptical contact patch, with repeated rolling passes. The minor axis of the ellipse is aligned with the direction of rolling. For a railway carriage wheel rolling along a length of railway track, the contact again is nearly elliptical, but with the major axis extending along the rolling direction (Fig. 8). The solution of the “elliptical” shakedown problem is therefore of practical interest. Since the shakedown condition has been established as r,, C TV, the problem for elliptical contact is reduced to solving the Hertz problem on the plane y - 0. The usual restrictions of isotropy and homogeneity apply, and with reference to Fig. 8 the larger contact semiaxis should be less than the smallest principal radius of curvature. Fessler and Ollerton [17] have experimentalfy shown that the Hertz conditions are satisfied if b c 0.3Rb&
(191
They also give the solution for rXz ; much of the theory is contained in Love’s book [ 181. For the plane y = 0, where the severest stresses act, the auxiliary variable y , when normalized with respect to the semiminor axis, becomes Y
*..--;
ix*2 + x*2
- 1 e (@*a + x*2 - 1)2 + 4a*a)x/a]
(20)
Then
and the shear stress is given by 7XB= -pokQ and the shakedown -_kQ
PO
(22) limit (eqns. (13)) may be written (23)
236
Fig. 8. Ways in which an elliptical contact patch can be generated in practice. On the left a ball is shown rolling in a groove so that the minor axis of the ellipse lies in the rolling direction. On the right is a typical railway contact patch geometry where the major axis of the ellipse lies in the direction of rolling.
The elastic limit may be found from the work of Thomas and Hoersch [ 191, and always occurs on the z axis. An experimental verification of the achievement of the shakedown limit has been performed and the results were encouraging [ZO] , The results of these calculations are shown in Fig. 9 although, as the ellipse becomes extremely elongated in the direction of rolling, a different criterion dominates the shakedown limit. This is when the residual stresses themselves are so large that they are on the point of plastic flow. The criterion for this “secondary plastic flow” may be written in terms of the applied stresses as
237
x:
2*
1.0
1 o-8
2
0.6
3 04
0.2
0 0.1
10
K
1
Fig. 9. A comparison of elastic, shakedown and load limits for the case of elliptical contact without friction: curves 1 and 3, shakedown limit (equal to load limit); curve 2, elastic limit (x = 0). It should be noted that the upper shakedown and load limit curves on the right apply only when the contact patch is suffering a repeated normal loading and there is no rolling motion.
When two bodies are pressed normally onto one another repetitively, the orthogonal shear stresses do not change sign and hence eqn. (24) represents the shakedown limit for all ellipse eccentricities. Figure 10 shows the location of the severest stress state. When the limitation is one of secondary plastic flow, the load limit equations (8) and (9) no longer apply, Instead it should be noted that of7 = 3fD7e
(25)
and that the load limit is given by Jz
=
ii@,,
+
< ’
+
or,.
-
4’
+
((J,,
-
oyy
-
M2
+
(ox,
-
o,,)~)
+
CTij2
2 To
i.e. $[{u,,
+ 31’%() - a2,}2 + {uzz - uyy - 31’27e}2 + (u,, +
z1Tij2
<
- r&2]
+
TO2
This equation must usually be solved numerically pressure which just produces secondary yielding.
(26)
to find the contact
5. Conclusions The elastic, shakedown and load limits were found for a variety of configurations and should find immediate application in both transferred
238
I 3.5-.---___--___L-_-_-
. . ___.
_~
/_.
:
_i
*
_ ~___.
.___
~_~
/
I
I I
3~0 -+---I,
/
ei 0.1
i
’
I
i
,I1 ~~~
/ I
/
;
1
i
1
Ii,11 k
10
Fig. 10. Location of the points of severest stress for the case of an elliptical contact patch without friction, It should be noted that the severest stresses giving the load limit occur at almost the same point as those producing the shakedown limit (cf. Fig. 7).
motion, e.g. rolling element bearings, and normal loading configurations, relay contacts. These limits appear to be consistent with experimental observations.
Nomenclature
a, b J2 k PO r*,x* x*, & V D 7
70
8 yi,
t*
lengths of semiaxes of contact ellipse second invariant of the stress deviator tensor b/a, oblateness of ellipse peak contact pressure polar coordinates normalized to characteristic contact dimension Cartesian coordinates normalized to characteristic contact dimension coefficient of friction Poisson’s ratio normal stress shear stress yield in pure shear
e.g.
239
References 1 H. Hertz, Gesammelte Werke, Vol. 1, Leipzig, 1895, p. 155. 2 H. Poritsky, Stresses and deflexions of cylindrical bodies in contact with application to contact of gears and of locomotive wheels, J. Appl. Meek., 72 (1950) 191 - 201. 3 J. 0. Smith and C. K. Liu, Stresses due to tangential and normal loads on an elastic solid with application to some contact stress problems, J. Appl. Meek., 75 (1953) 157 - 166. 4 G. M. Hamilton and L. E. Goodman, The stress field created by a circular sliding contact, J. Appl. Meek., 88 (1966) 371 - 378. 5 P. S. Symonds, Shakedown in continuous media,J. Appl. Meek., 18 (1) (1951) 85 - 89. 6 A. W. Crook, Simulated gear tooth contact; some experiments on their lubrication and deformation, Proc., Inst. Meek. Eng., London, 171 (5) (1957) 187 - 214. 7 G. C. Hamilton, Plastic flow in rollers loaded above the yield point, Proc., Inst. Meek. Eng., London, 177 (1963) 667 - 675. 8 K. Fujita and A. Yoshida, Surface fatigue failure of case-hardened nickel-chromium steel rollers under pure rolling and sliding-rolling contact, Wear, 51 (1979) 365 - 374. 9 K. Fujita and A. Yoshida, Surface durability of case-hardened nickel-chromium steel rollers under pure rolling and sliding-rolling sliding contacts, Wear, 52 (1979) 37 - 48. 10 A Preliminary description of stresses in railroad rail, NTIS Rep. PB-272054, November 1976 (U.S. National Technical Information Service). 11 G. C. Martin and W. W. Hay, The influence of wheel-rail contact forces on the formation of rail shells, ASME Preprint 72-WA/RT-8, 1972. 12 K. L. Johnson, A shakedown limit in rolling contact, in Proc. 4th Natl. Conf. on Applied Mechanics, Berkeley, CA, June 1962, pp. 971 - 975. 13 J. E.!‘Merwin and K. L. Johnson, An analysis of plastic deformation in rolling contact, Froc., Znst: Meek. Eng., London, 177 (25) (1963) 676 - 685. 14 K. Ls. Johnson and J. A. Jefferis, Plastic flow and residual stresses in rolling and sliding contact, Proc., Inst. Meek. Eng., 177 (25) (1963) 54 - 65. 15 M. T. Huber, Zur Theorie der Beruhrung Fester Elasticher Korper, Ann. Pkys. (Leipzig), 14 (1904) 152 - 163. 16 G. Rydholm and B. Fredriksson, Shakedown in rolling contact problems, in Proc. Znt. Conf. on Applied Mechanics: Contact Problems and Load Transfer in Mechanical Assemblages, Euromeck Colloq. 110, Link&ping, Sweden, September 27 - 29, 1978. 17 H. Fessler and E. Ollerton, Contact stresses in toroids under radial loads, Br. J. Appl. Pkys., 8 (1957) 387 - 393. 18 A. E. H. Love, The MatkematicaZ Theory of Elasticity, Cambridge University Press, London, 4th edn., 1952, pp. 192 - 198. 19 H. R. Thomas and V. A. Hoersch, Stresses due to the pressure of one elastic solid upon another, Bull. 212, 1930 (University of Illinois Engineering Experimental Station). 20 D. A. Hills and D. W. Ashelby, On the groove created by a heavily loaded rolling b.all, submitted to Proc. R. Sot. London.