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Crack initiation criteria in fretting fatigue
Wear, 136 (1990)
329 - 343
CRACK INITIATION
329
CRITERIA
IN FRETTING
FATIGUE
D. NOWELL and D. A. HILLS Department (U.K.)
of Engineering
Science,
University of Oxford, Parks Road, Oxford, OX1 3PJ
(Received April 19, 1989; revised August 7, 1989; accepted September 1, 1989)
This paper is a description of a series of fretting fatigue tests on an Al/Cu alloy in which the contact size was varied while the other relevant parameters were held constant. The fretting fatigue life was found to be infinite below a certain critical contact width. The configuration has been analysed using elastic stress analysis and stresses and displacements have been calculated. The variation of life with contact size can be explained by a variation in amplitude of micro-slip. Composite parameters proposed by Ruiz et ~2. are shown to characterize the severity of fretting damage and the probability of crack initiation, and hence explain the observed results. The physical basis of these parameters and their use in design calculations is discussed.
1. Introduction Fretting occurs when assemblies of components are subjected to vibration. Slip takes place between adjacent surfaces and this can have two potentially serious consequences. First, fretting wear [l] may take place in which particles are detached from the contacting surfaces leading to a deterioration in surface finish and changes in part dimensions. Secondly, and perhaps more importantly from an engineering point of view, the presence of fretting in conjunction with a bulk stress in the body of a component can lead to a marked reduction in fatigue life, sometimes by a factor as great as 10 [2]. This phenomenon is known as fretting fatigue [ 31. The investigation of fretting and fretting fatigue has a long history. The experimental work described here follows directly from work by Bramhall [4] carried out in the early 1970s. The apparatus used by Bramhall is shown schematically in Fig. 1. A parallel-sided specimen was tested in push-pull fatigue so that the bulk direct stress o varied sinusoidally with time. Two cylindrical fretting pads of radius R were mounted against the sides of the specimen and held against it by a normal force P. As the fatigue specimen was placed under tension it extended and consequently the cylinders tended to move. They were, however, restrained by a pair of springs so that a tan0043-1648/90/$3.50
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330
Fig. 1. Schematic view of experimental apparatus.
gential force Q was developed in phase with the bulk stress. This tangential force gave rise to the fretting effect. The pressure distribution beneath the pads was essentially the elliptical distribution predicted by Hertz [5 J. Bramhall noted that for this configuration the peak normal pressure p. was proportional to (P/R)“* whereas the contact semiwidth a varied as (PR) “* . It was therefore possible to vary the size of the contact and hence the size of the resulting stress field whilst maintaining a constant peak normal pressure. Bramhall carried out a range of such experiments and discovered that the fretting fatigue life of the specimen depended on the contact size as well as on the imposed stress levels. There was a critical contact size at which transition from a long to a shortened fatigue life took place. Bramhall’s experimental configuration is amenable to elastic stress analysis and the initial objective of the current work was to attempt to explain Bramhall’s observations using the techniques of contact mechanics and fracture mechanics.
2. Experimental
work
2.1. Apparatus In order to obtain accurate experimental data for the analysis and to confirm the existence of a size effect it was decided to carry out a series of experiments similar to Bramhall’s. The apparatus and experimental technique have been described in detail elsewhere [6, 71 and only the essential features will be summarized here. Figure 2 shows a photograph of the apparatus, which is similar to Bramhall’s, mounted on a push-pull fatigue testing machine. The normal force P is imposed through compliant springs B which prevent any fall-off in load during ‘bedding in’ at the start of the experiment. The cantilever springs A provide the tangential force Q and their stiffness may be adjusted so as to vary the ratio of Q to P, both varying sinusoidally with time. The specimen and pads were both made from HE 15-TF, an Al-4wt.%Cu alloy similar to that used by Bramhall. Pads were made in a range of radii from 12.5 to 150 mm to enable contact semiwidths from 0.1 to 1.0 mm to be obtained. Testing was carried out at 20 Hz and
331
Fig. 2. Experimental apparatus.
each test lasted for lo7 cycles unless there was a fatigue failure before this. The environment for the tests was laboratory air at normal room temperature. 2.2. Results Five main series of tests were carried out, each series consisting of eight tests covering the range of contact widths stated above but with the other parameters (PO, Q/P, u and, as far as it was under external control, the coefficient of friction f) held constant. Measurements of P, Q and the amplitude of the force in the specimen enabled po, Q/P and u to be determined. The value of f was less easy to determine. The experiments were carried out in the partial slip regime (i.e. Q < fP) and only at the outer edges of the contact did the surfaces slip over each other. It was in these regions that we needed to determine f. An additional series of tests carried out in full slip suggested that the coefficient of friction reached a steady state value of 0.75 after about 200 cycles. This estimate was confirmed by comparing measurements taken of the average value of f in partial slip tests and using these to predict the slip zone value [ 7, 81. The experimental parameters for each series of tests are given in Table 1 and corresponding plots of fatigue life against contact semiwidth are shown in Fig. 3. Each series of tests displayed the same phenomenon: at small contact widths the fatigue life was long and specimens were still intact after lo7 cycles, whereas at large contact widths failures occurred after about lo6 cycles. There was a sharp transition between these two regimes at a critical contact semiwidth actit, which varied between different series of tests. Fatigue failures initiated towards the edge of the contact patch and propagated for a short distance obliquely under the contact before turning on to a
332
TABLE 1 Experimental results Bulk stress amplitude a (MN me2)
Peak normal pressure pa (MN rn-?)
Q/P
1
92.7
2 3 4 5
92.7 92.7 77.2 61.8
157 143 143 143 120
0.45 0.24 0.45 0.45 0.45
Series
0.75 0.75 0.75 0.75 0.75
0.28 0.54 0.18 0.36 0.57
- 0.38 - 0.72 - 0.27 - 0.54 - 0.71
12.0
100
Bra&&
T
A
6.
Series
1
*
Series
2
10.0 1
I
/,
8.0
A &
t
0 Series
3
+ Series
4
4
6 0
80-
% 0
” 6.0
-
u) 0
0 0
x 40
-
-
a ‘c 1
6.0 -
40-
0
z ‘3 2Om
20, Y
a A 00
a
+ P a
(a)
ooo,-f,t 0.50
Contact 12.0
a
*
T1II,“‘Tvm 0.00
1 .OO
T---T-;7 n
7; Jo
u
1.50
semi-width
a
(mm)
(b)
CoTtact
semi-
width
a
(mm)
-
0 loo-
ul
adt (mm)
12.0
a, K
f
000
0
Series
5
O
0
0
60 -
0 4.0 2 ‘3 2.0
0.0. 0.00 (cl
I
Contact
, 0 50
semi-width
1
I 1.00
o
(mm)
Fig. 3. Variation of fatigue life with contact size: (a) series 1 and 2; (b) series 3 and 4; (c) series 5.
plane normal to the surface. A failed specimen in the fatigue machine is depicted in Fig. 4. Such a variation of fatigue life with contact size at a constant stress level appears difficult to explain at first sight. It is, however, apparent that, since
333
Fig. 4. Failed fatigue specimen between fretting pads.
the contacts are geometrically similar, the amplitude of micro-slip in the slip zones increases in proportion to the contact size. Calculated maximum slip amplitudes for these experiments were between 0.4 and 4.0 pm (peak-peak). Several researchers have reported a variation in fretting fatigue life with slip amplitude [9,10] and a further series of experiments (series 6) was devised to investigate the effect of slip amplitude alone. The surface tractions within the contact are similar to those described by Mindlin and Deresiewicz [ll] for contacting spheres in which a normal force is applied followed by an oscillating tangential force. In this case the contact is two-dimensional and there is also a bulk stress in the specimen which is not present in the pads. This shifts the central stick zone to one side [121. Under these conditions the size of the slip zones and the resultant slip amp~tude is dependent on the degree of load reversal. A fully reversing load (+ Q,,, + -Q,,, + + Q,,,) results in a much larger slip amplitude than if Q is not reversed (Q,,, + 0 --f Q,,,). The additional series of experiments was run at a constant contact width but with different degrees of load reversal. Figure 5 shows the resulting fatigue lives: a fully reversing load with a large slip amplitude produced a short life, whereas less load reversal and smaller slip ~plitude produced long lives. This indicates that slip amplitude has a marked effect on fretting fatigue life since the maximum stress state and the mode I stress intensity factor range AK, remained constant throughout the series 6 tests. The minimum value of Kr was always zero (i.e. when the crack closed). Transition from the short to the long fatigue life regime took place at a minimum bulk stress between -52.4 and -39.9 MN me2, corresponding to maximum micro-slip amplitudes of between 1.2 and 0.9 pm (peak to peak) in the slip zones. These values will be compared with those from the other 5 experimental series in Section 4.2.
334
T-:
,/’
J,“,~,_,,~ 0
“II
-80 00
v-7
bJ.00
-EO.OO
Minimum
bulk
-20
stress
00
0.00
(MN/rn2)
p
e-c
-a
!a e*‘
BF v
,.A
./’ 3
-.a
--
L?
Y
ea.
QJ!
TT
Vf
Fig. 5. Variation of fatigue life with degree of load reversal (series 6 experiments). Fig. 6. Model of contact for stress analysis.
3. Stress analysis In order to analyse the results of the ex~riments an elastic stress analysis must be pe~ormed on the experimental confi~ation. Figure 6 shows an idealization of the contact, where both the fretting pad and specimen are represented by elastic half-planes. This has been shown to be a reasonable assumption if the fatigue specimen is more than two contact widths in thickness [13]. The configuration modelled is that of a cylinder on a half-plane, corresponding to the initial contact geometry. Since the fatigue tests were carried out in the partial slip regime, very little wear was experienced in the contact. Such small amounts of wear have been recorded in other partial slip experiments 1141 and allow us to assume that the initial contact geometry is little changed by wear during the test and the model can be taken to apply throughout the experiment. Under these conditions the normal pressure developed is given by Hertz [ 5 f and is the well-known elliptical d~tribution 2
l/2
1 i)t x
P(X)=;Po
l-
;
where the peak pressure p. is given by PI3 PO-
i 2x(1 - vZ)R t
l/2
12)
and E and v are Young’s modulus and Poisson’s ratio respectively for the contacting materials (which are taken to be elastically similar).
335
As stated in Section 2, the tangential those described by Mindlin and Deresiewicz is displaced by an amount e because of the men which is not present in the pads [12]. peak value of tangential load Qmax are
1 oi x
q@)
= --fPo
1-
2
surface tractions are similar to [ll] except that the stick zone bulk strain in the fatigue speciThe tangential tractions at the
l/2
;
+ a’(x)
lx-el>c
lxlda
1x1 G a
(3)
where q’(X) = 0
(4)
and the stick-zone
semiwidth
c and eccentricity
e are given by
(5)
e -amax -= a
4fPo
These results are valid [12] provided
that
A sample distribution is shown in Fig. 7(a). In the experiments of series 1 - 5 the tangential force was cycled between + Q,,, and the surface tractions oscillated between plus and minus the limiting value given above. Q was first increased from zero to Q,,,, giving rise to the shear tractions indicated in eqn. (4) with a stick zone centred on x = e bordered by two slip zones. The shear force then started to reverse. This gave rise to instantaneous stick over the whole contact [ll, 121 before reverse slip zones started to form at the contact edges. These regions of slip grew progressively as the tangential force approached -Q,,,, where the shear tractions were then equal and opposite to those described by eqn. (4). Figure 7(b) shows sample shear traction distributions at various points in this half-cycle for series 3 experiments. The shear tractions developed at all points in the cycle were always the sum of elliptical distributions of differing amplitude [ll]. This means that the results of Smith and Liu [15] and Poritsky [16] may be employed, together with the principle of superposition, to calculate the stresses and displacements respectively in the halfplanes. The expressions are somewhat lengthy and will not be reproduced here.
336
(b) Fig. 7. Shear traction distributions within the contact: (a) comparison of predicted maximum tractions with Mindlin model (series 1 experiments); (b) variation of predicted shear tractions with load during half a cycle (series 3 experiments).
4. ~n~~r~t~tion
of results
The stress analysis can now be employed to assist in the interpretation of the experimental resuIts. Two possible explanations for the size effect must be considered: first, that fretting fatigue cracks do not initiate at the smaller contact widths, and secondly, that they initiate but that their growth is arrested. These will be referred to as the crack initiation and growth criteria respectively. 4.1. Crack growth criterion The first possibility examined was that cracks are unable to grow at small contact widths. This explanation was first proposed by Bramhall [ 41. ~ssent~lly, if the contact stress field is not sufficiently extensive to propagate an embryo crack to a length at which it can grow under the bulk stress alone then crack arrest will take place and the specimen will not fail. This argument seems initially attractive and an extensive series of stress intensity factor calculations were undertaken to provide a quantitative explanation of the phenomenon [ 6,7 1. Unfortunately, although good predictions of critical contact width can be obtained [7], there is little experimental evidence to support such a theory of crack arrest. If embryo cracks do self-arrest we would expect to see such cracks in sections of intact specimens. In fact, no such cracks were found and there was a marked contrast between sections of the failed specimens, in which many micro-cracks were present (Fig. 8) and intact specimens, where none were present. The absence of initiated cracks was
337
,
I Imm
Fig. 8. Section of a failed specimen viewed under an optical microscope.
further confirmed when some specimens were reused in subsequent tests by fretting at a different point. No failures were observed at the original fretting scar, as might have been expected if embryo cracks existed there. The other weakness of the crack arrest theory is that it fails to explain the variation in fatigue life with slip amplitude recorded in series 6 experiments. It is clear that the alternative hypothesis, that cracks do not initiate at the smaller contact widths, is worthy of further investigation. 4.2. Crack initiation criterion As we have seen, the experimental evidence suggests that cracks do not initiate at the smaller contact widths and that the amplitude of slip between the two surfaces has a distinct influence on the initiation process. The difficulty with analysing the suggestion that cracks do not initiate is that the mechanics of crack initiation are not clearly understood and that we might expect crack initiation to be a microscopic process, taking place at an asperity level, whereas the elastic stress analysis performed predicts only ‘bulk’ properties for a smooth contact. Any attempt to quantify the conditions for crack initiation using bulk properties will therefore be, at best, somewhat empirical. Ruiz et al. [17] have proposed two parameters which describe the severity of fretting damage and the likelihood of crack initiation during a given number of loading cycles. The amount of work performed in overcoming friction per unit surface area in the slip zone varies with position
338
and can be characterized by the product of 7, the maximum shear traction attained during one cycle at that point, and 6, the relative amplitude of slip between the surfaces at the same position. This parameter ~6 gives a measure of the amount of fretting damage per cycle at any point on the surface. It will be immediately recognized that 76 falls to zero in the stick zone (where 6 s 0) and outside the contact (where r = 0), where no fretting occurs. For the experimental configuration studied here 7 is simply equal to q(x) and given by eqn. (4), whereas 6 can be calculated using the results in ref. 16. The variation of 78 in the trailing slip zone (i.e. the one experiencing the maximum tension in the specimen) for series 3 experiments is shown in Fig. 9. Ruiz and his coworkers also recognize that a crack is much more likely to initiate in a region of bulk tensile stress than in one where the stress parallel to the surface is compressive. They therefore propose that a measure of the probability of crack initiation can be obtained by multiplying ~8 by the stress parallel to the surface CJto produce a composite initiation parameter ~6. The variation of this parameter in the slip zone for series 3 experiments is also shown in Fig. 9. A maximum value is reached very close to the edge of the contact, where cracks are found to initiate in the experiment. Ruiz et al. [17] and Kuno et al. [18] also found good agreement between the locations of maxima of or& and the sites of crack initiation. In the present series of experiments the maximum value of ~8 reached in the trailing slip zone will be proportional to 6 and hence to the contact semiwidth a. The likelihood of crack initiation will therefore be much higher at the larger contact widths. Maximum values of ur6 for experiments at the critical contact width are presented for each series of experiments in Table 2.
a16 E
__-r6E
p~(l-vVa
pf(l-v*)a
0.35 1 0.30 0.25 -
-0.10, -1.00
I
I
I -0.60
I
I
I -0.60
I -0.40
x/a
Fig. 9. Variation of damage parameter ~6 and initiation parameter 0~6 in trailing slip zone for series 3 experiments.
339 TABLE 2 Maximum values of initiation parameter at critical contact widths Series
Critical contact (mm)
1 2 3 4 5
0.28 0.54 0.18 0.36 0.57
semiwidth
adt
-+ 0.38 + 0.72 --f 0.27 + 0.54 + 0.71
Maximum value of initiation parameter (376 ( Nz mmV3) 4.47 3.26 2.42 3.94 3.47
+ + + + +
6.06 4.35 3.63 5.91 4.32
Minimum bulk stress (MN me2) 6
-39.9
+ -52.4
3.04 + 4.17
0.6 -
-f:
5
2
/
TO.6-
6’
/
L 0 -(J0.4 cu 5 .u
4,
/
_
-,’3
P a 0.2 -
/
/
/
1
L
/
1+---t 1
/ 0.0
, 0.00
I
0.20
I
0.40
Experimental
I
0.60
a (mm)
I
0.60
Fig. 10. Comparison of experimental and predicted critical contact sizes.
Cracks appear to initiate and lead to specimen failure within 10’ cycles once ~~78exceeds a value of about 4 N2 mm- 3. The mean value of 4.18 N2 mmm3 can be used to predict critical contact widths for failure and these are presented in Fig. 10. The results of the series 6 experiments, where the degree of load reversal was varied, may also be analysed in a similar manner. The transition from short to long fatigue life takes place at a value of ur3 between 3.0 and 4.2 N2 mme3. This is in good agreement with the critical values observed in the other 5 series (Table 2). It would appear, therefore, that 078 does provide a possible means of characterizing the probability of crack initiation by fretting within a particular number of cycles. The approach adopted is so far rather empirical, but we shall proceed in the next section to discuss whether there is a physical basis for characterizing crack initiation with this parameter.
340
5. A physical
basis for the initiation
parameter
The initiation parameter or6 under discussion is defined using bulk properties derived from an elastic stress analysis of smooth contact. We would, however, expect crack initiation to take place as a result of plastic deformation at the asperity level. Real contacts all exhibit some degree of surface roughness and this will often result in discontinuous contact [19], whereby small contact areas with high normal pressures are separated by regions in which no contact takes place. Quite modest asperity tip curvature gives rise to plasticity [ZO] and it is therefore likely that crack initiation takes place as a result of repeated passes of asperities in the plastic regime. Bower and Johnson [21] point out that the adoption of an e~stic/perfe~tly plastic model of material behaviour for material deformation under contact loads leads to the prediction of a vanishingly thin layer of inifinite shear strain. Some degree of strain hardening is therefore required in any material model. With such a model, Bower and Johnson show that incremental plastic strain in the surface layer is possible under repeated unidirectional passes of a sliding or tractive rolling contact load. In the fretting problem the motion is oscillatory, so that at first sight it is difficult to see how incremental strain can occur. Each asperity pass in one direction appears to be matched by an equivalent pass in the opposite direction as the fretting load is reversed. If we return to the smooth elastic model, however, we can see that the actual situation is somewhat more complicated. Figure 11 shows how the relative displacement of points in the slip zone varies with applied tangential force Q (and therefore with applied bulk stress a) during one steady state fretting cycle for series 3 experiments. A hysteresis effect is apparent because of the requirement for the tangential traction q(x)
Loo0
Q,/fP
Fig. 11. Hysteresis curves showing variation of relative displacement between contacting points in the slip zone with load for series 3 experiments.
341
to change from +fp(x) to --fp(x) at any point before the direction of slip can reverse there. If the overall deformations are little modified by the presence of surface roughness then we can relate this diagram to the stress state when asperities pass over a point on the surface. Let us consider a point on body 1, and allow an asperity on body 2 to pass over it at a relative displacement 6i on the scale of the overall contact when the applied load is Qi. The relative displacement will clearly be the same at the point later in the cycle when the asperity returns, but the applied load (and hence the applied tensile stress in the specimen) will be equal to Q2 (and not equal and opposite to Q,) (Fig. 11). Thus the point does not “see” a symmetric stress cycle, when the combined effects of asperity and bulk terms are summed and the accumulation of incremental plastic shear strain is possible. If accumulation of plastic shear strain is possible in fretting, it is likely that a ch~a~teristic amount of strain is required to initiate a crack. We can now investigate how the increment of strain in each fretting cycle varies with u, r and S. For a pair of surfaces of given roughness (and therefore a given asperity density) the number of asperity passes per cycle is simply proportional to the relative displacement 5. Elastic contact analysis predicts that the mean shear stress in the slip zones will be T, which is proportional to the contact pressure P(X). For a gaussian distribution of asperities with constant curvature Greenwood and Williamson [22] have shown that the number of asperity contacts per unit area is proportional to the mean contact pressure. Therefore, T gives a measure of the density of contacting asperities and the product 78 represents the number of asperity passes in each fretting cycle. The inclusion by Ruiz et al. of o in the initiation parameter is less easy to justify physically. Multiplication by the maximum stress parallel to the surface is rather arbitrary, but does take account of the observed phenomenon that fatigue cracks are more likely to grow in regions of high tensile stress. It is perhaps helpful to distinguish between the process of crack nucleation by ~cremen~l plastic shear strain, as described above, and that of initial crack growth, which might be expected to be controlled by the stress state just below the surface. It is therefore reasonable to suggest that for a crack to form it must both nucleate (requiring a critical total amount of fretting damage, where the damage per cycle is characterized by r6) and be able to grow (requiring a tensile stress e). The composite parameter ~76 expresses the requirement for both these conditions to be present. Configurations with high or8 values might be expected to undergo only a few cycles before embryo cracks form whereas the initiation of cracks in specimens with low a76 values will require a large number of cycles, either because the incremental plastic strain per cycle is low (rS), or because the inplane stress u does not encourage initial crack growth. 6. Conclusions A series of experiments has been reported in which the fretting fatigue life varies with contact size in the partial slip regime. This is shown to be due
342
to the variation of relative slip amplitude between the contacting surfaces. Cracks were found to initiate at large contact widths when the slip amplitude was large. The initiation parameter suggested by Ruiz et al. has been shown to characterize the probability of crack initiation well and there appears to be a critical value of this parameter for crack initiation to take place within a fixed number of cycles. Although the definition of this parameter seems rather arbitrary, it has some physical basis in characterizing the combined effect of the accumulated plastic strain in each cycle together with the tensile stress parallel to the surface. Ruiz et ~1 [17] have suggested that the (~76 parameter can be used as a basis for design against fretting fatigue. The present experiments certainly support this view. Design can concentrate on one of two aspects. First, on reducing the severity of fretting (~6) by reducing the amplitude of microslip, or the shear stress. It should be noted that in cases of partial slip with an imposed tangential force, reducing the coefficient of friction f (a common palliative measure) will have opposite effects on r and 6 and that the combined effect will have to be considered carefully. Secondly, designers can attempt to reduce u by placing contacting surfaces in regions of low tensile or compressive stress. This will not prevent fretting damage but should prevent cracks growing. In the absence of more rigorous means of quantifying the conditions required for crack initiation, the a& parameter does therefore provide useful guidance for designers. Acknowledgment The experimental work described in this paper was supported Science and Engineering Research Council.
by the
References 1 R. B. Waterhouse, Fretting Corrosion, Pergamon, London, 1972. 2 K. J. Fenner, K. H. R. Wright and J. Y. Mann, Fretting corrosion and its influence on fatigue failure, Proc. fni. Co@ on Fatigue of Metu&, Z~~tut~o~ of mechanical Engineers, London, 1956. 3 R. B. Waterhouse, Fretting Fatigue, Applied Science, London, 1981. 4 R. Bramhall, Studies in fretting fatigue, D.Phil. Thesis, Oxford University, 1973. 5 H. Hertz, Uber die Beruhrung fester elastischer Korper, J. Reine Angew. Math., 92 (1882) 156 - 171. 6 D. Nowell, An analysis of fretting fatigue, D. Phil. Thesis, Oxford University, 1988. 7 D. A. Hills, D. Nowell and J. J. O’Connor, On the mechanics of fretting fatigue, Wear, 125 (1988) 129 - 146. 8 D. A. Hills, D. Nowell and A. Sackfield, Surface fatigue considerations in fretting, in D. Dowson, C. M. Taylor, M. Godet and D. Berthe (eds.), Interface Dynamics, Proc. 14th Leeds-Lyon Symp. on Tribology, Lyon, September, 1987, Elsevier, Amsterdam, 1988. 9 K. Nishioka and K. Hiakawa, Fundamental investigation of fretting fatigue, Part 2, Bull. JSME, 12 (50) (1969) 180 - 187.
343 10 K. J. Nix and T. C. Lindley, The influence of relative slip range and contact material on the fretting fatigue properties of 35NiCrMoV rotor steel, Wear, 125 (1988) 147 - 162. 11 R. D. Mindlin and H. Deresiewicz, Elastic spheres in contact under varying oblique forces, J. Appl. Mech., 75 (1953) 327 - 344. 12 D. Nowell and D. A. Hills, The mechanics of fretting fatigue tests, Znt. J. Mech. Sci., 29 (5) (1987) 355 - 365. 13 D. Nowell and D. A. Hills, Contact problems incorporating elastic layers, Znt. J. Solids, Strucf., 24 (1) (1988) 105 - 115. 14 U. Bryggman and S. Soderberg, Contact conditions and surface degradation mechanisms in low amplitude fretting, Wear, 125 (1988) 39 - 52. 15 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. Mech., 20 (1953) 157. 16 H. Poritsky, Stresses and deformations of cylindrical bodies in contact with application to contact of gears and of locomotive wheels, J. Appl. Mech., 72 (1950) 191 201. 17 C. Ruiz, P. H. B. Boddington and K. C. Chen, An investigation of fatigue and fretting in a dovetail joint, Exp. Mech., 24 (3) (1984) 208 - 217. 18 M. Kuno, R. B. Waterhouse, D. Nowell and D. A. Hills, Initiation and growth of fretting fatigue cracks in the partial slip regime, Fatigue Fract. Eng. Mater. Strut., 12 (5) (1989) 387 - 398. 19 J. A. Greenwood and J. H. Tripp, The elastic contact of rough spheres, J. Appl. Mech., 34 (1967) 153 - 159. 20 D. Nowell and D. A. Hills, Hertzian contact of ground surfaces, J. Tribol., 111 (1) (1989) 175 - 179. 21 A. F. Bower and K. L. Johnson, The influence of strain hardening on cumulative plastic deformation in rolling and sliding line contact, J. Mech. Phys. Solids, 37 (4) (1989) 471 - 494. 22 J. A. Greenwood and J. B. P. Williamson, Contact of nominally flat surfaces, Proc. R. Sot. London, Ser. A, 295 (1966) 300.
329 - 343
CRACK INITIATION
329
CRITERIA
IN FRETTING
FATIGUE
D. NOWELL and D. A. HILLS Department (U.K.)
of Engineering
Science,
University of Oxford, Parks Road, Oxford, OX1 3PJ
(Received April 19, 1989; revised August 7, 1989; accepted September 1, 1989)
This paper is a description of a series of fretting fatigue tests on an Al/Cu alloy in which the contact size was varied while the other relevant parameters were held constant. The fretting fatigue life was found to be infinite below a certain critical contact width. The configuration has been analysed using elastic stress analysis and stresses and displacements have been calculated. The variation of life with contact size can be explained by a variation in amplitude of micro-slip. Composite parameters proposed by Ruiz et ~2. are shown to characterize the severity of fretting damage and the probability of crack initiation, and hence explain the observed results. The physical basis of these parameters and their use in design calculations is discussed.
1. Introduction Fretting occurs when assemblies of components are subjected to vibration. Slip takes place between adjacent surfaces and this can have two potentially serious consequences. First, fretting wear [l] may take place in which particles are detached from the contacting surfaces leading to a deterioration in surface finish and changes in part dimensions. Secondly, and perhaps more importantly from an engineering point of view, the presence of fretting in conjunction with a bulk stress in the body of a component can lead to a marked reduction in fatigue life, sometimes by a factor as great as 10 [2]. This phenomenon is known as fretting fatigue [ 31. The investigation of fretting and fretting fatigue has a long history. The experimental work described here follows directly from work by Bramhall [4] carried out in the early 1970s. The apparatus used by Bramhall is shown schematically in Fig. 1. A parallel-sided specimen was tested in push-pull fatigue so that the bulk direct stress o varied sinusoidally with time. Two cylindrical fretting pads of radius R were mounted against the sides of the specimen and held against it by a normal force P. As the fatigue specimen was placed under tension it extended and consequently the cylinders tended to move. They were, however, restrained by a pair of springs so that a tan0043-1648/90/$3.50
@ Elsevier Sequoia/Printed
in The Netherlands
330
Fig. 1. Schematic view of experimental apparatus.
gential force Q was developed in phase with the bulk stress. This tangential force gave rise to the fretting effect. The pressure distribution beneath the pads was essentially the elliptical distribution predicted by Hertz [5 J. Bramhall noted that for this configuration the peak normal pressure p. was proportional to (P/R)“* whereas the contact semiwidth a varied as (PR) “* . It was therefore possible to vary the size of the contact and hence the size of the resulting stress field whilst maintaining a constant peak normal pressure. Bramhall carried out a range of such experiments and discovered that the fretting fatigue life of the specimen depended on the contact size as well as on the imposed stress levels. There was a critical contact size at which transition from a long to a shortened fatigue life took place. Bramhall’s experimental configuration is amenable to elastic stress analysis and the initial objective of the current work was to attempt to explain Bramhall’s observations using the techniques of contact mechanics and fracture mechanics.
2. Experimental
work
2.1. Apparatus In order to obtain accurate experimental data for the analysis and to confirm the existence of a size effect it was decided to carry out a series of experiments similar to Bramhall’s. The apparatus and experimental technique have been described in detail elsewhere [6, 71 and only the essential features will be summarized here. Figure 2 shows a photograph of the apparatus, which is similar to Bramhall’s, mounted on a push-pull fatigue testing machine. The normal force P is imposed through compliant springs B which prevent any fall-off in load during ‘bedding in’ at the start of the experiment. The cantilever springs A provide the tangential force Q and their stiffness may be adjusted so as to vary the ratio of Q to P, both varying sinusoidally with time. The specimen and pads were both made from HE 15-TF, an Al-4wt.%Cu alloy similar to that used by Bramhall. Pads were made in a range of radii from 12.5 to 150 mm to enable contact semiwidths from 0.1 to 1.0 mm to be obtained. Testing was carried out at 20 Hz and
331
Fig. 2. Experimental apparatus.
each test lasted for lo7 cycles unless there was a fatigue failure before this. The environment for the tests was laboratory air at normal room temperature. 2.2. Results Five main series of tests were carried out, each series consisting of eight tests covering the range of contact widths stated above but with the other parameters (PO, Q/P, u and, as far as it was under external control, the coefficient of friction f) held constant. Measurements of P, Q and the amplitude of the force in the specimen enabled po, Q/P and u to be determined. The value of f was less easy to determine. The experiments were carried out in the partial slip regime (i.e. Q < fP) and only at the outer edges of the contact did the surfaces slip over each other. It was in these regions that we needed to determine f. An additional series of tests carried out in full slip suggested that the coefficient of friction reached a steady state value of 0.75 after about 200 cycles. This estimate was confirmed by comparing measurements taken of the average value of f in partial slip tests and using these to predict the slip zone value [ 7, 81. The experimental parameters for each series of tests are given in Table 1 and corresponding plots of fatigue life against contact semiwidth are shown in Fig. 3. Each series of tests displayed the same phenomenon: at small contact widths the fatigue life was long and specimens were still intact after lo7 cycles, whereas at large contact widths failures occurred after about lo6 cycles. There was a sharp transition between these two regimes at a critical contact semiwidth actit, which varied between different series of tests. Fatigue failures initiated towards the edge of the contact patch and propagated for a short distance obliquely under the contact before turning on to a
332
TABLE 1 Experimental results Bulk stress amplitude a (MN me2)
Peak normal pressure pa (MN rn-?)
Q/P
1
92.7
2 3 4 5
92.7 92.7 77.2 61.8
157 143 143 143 120
0.45 0.24 0.45 0.45 0.45
Series
0.75 0.75 0.75 0.75 0.75
0.28 0.54 0.18 0.36 0.57
- 0.38 - 0.72 - 0.27 - 0.54 - 0.71
12.0
100
Bra&&
T
A
6.
Series
1
*
Series
2
10.0 1
I
/,
8.0
A &
t
0 Series
3
+ Series
4
4
6 0
80-
% 0
” 6.0
-
u) 0
0 0
x 40
-
-
a ‘c 1
6.0 -
40-
0
z ‘3 2Om
20, Y
a A 00
a
+ P a
(a)
ooo,-f,t 0.50
Contact 12.0
a
*
T1II,“‘Tvm 0.00
1 .OO
T---T-;7 n
7; Jo
u
1.50
semi-width
a
(mm)
(b)
CoTtact
semi-
width
a
(mm)
-
0 loo-
ul
adt (mm)
12.0
a, K
f
000
0
Series
5
O
0
0
60 -
0 4.0 2 ‘3 2.0
0.0. 0.00 (cl
I
Contact
, 0 50
semi-width
1
I 1.00
o
(mm)
Fig. 3. Variation of fatigue life with contact size: (a) series 1 and 2; (b) series 3 and 4; (c) series 5.
plane normal to the surface. A failed specimen in the fatigue machine is depicted in Fig. 4. Such a variation of fatigue life with contact size at a constant stress level appears difficult to explain at first sight. It is, however, apparent that, since
333
Fig. 4. Failed fatigue specimen between fretting pads.
the contacts are geometrically similar, the amplitude of micro-slip in the slip zones increases in proportion to the contact size. Calculated maximum slip amplitudes for these experiments were between 0.4 and 4.0 pm (peak-peak). Several researchers have reported a variation in fretting fatigue life with slip amplitude [9,10] and a further series of experiments (series 6) was devised to investigate the effect of slip amplitude alone. The surface tractions within the contact are similar to those described by Mindlin and Deresiewicz [ll] for contacting spheres in which a normal force is applied followed by an oscillating tangential force. In this case the contact is two-dimensional and there is also a bulk stress in the specimen which is not present in the pads. This shifts the central stick zone to one side [121. Under these conditions the size of the slip zones and the resultant slip amp~tude is dependent on the degree of load reversal. A fully reversing load (+ Q,,, + -Q,,, + + Q,,,) results in a much larger slip amplitude than if Q is not reversed (Q,,, + 0 --f Q,,,). The additional series of experiments was run at a constant contact width but with different degrees of load reversal. Figure 5 shows the resulting fatigue lives: a fully reversing load with a large slip amplitude produced a short life, whereas less load reversal and smaller slip ~plitude produced long lives. This indicates that slip amplitude has a marked effect on fretting fatigue life since the maximum stress state and the mode I stress intensity factor range AK, remained constant throughout the series 6 tests. The minimum value of Kr was always zero (i.e. when the crack closed). Transition from the short to the long fatigue life regime took place at a minimum bulk stress between -52.4 and -39.9 MN me2, corresponding to maximum micro-slip amplitudes of between 1.2 and 0.9 pm (peak to peak) in the slip zones. These values will be compared with those from the other 5 experimental series in Section 4.2.
334
T-:
,/’
J,“,~,_,,~ 0
“II
-80 00
v-7
bJ.00
-EO.OO
Minimum
bulk
-20
stress
00
0.00
(MN/rn2)
p
e-c
-a
!a e*‘
BF v
,.A
./’ 3
-.a
--
L?
Y
ea.
QJ!
TT
Vf
Fig. 5. Variation of fatigue life with degree of load reversal (series 6 experiments). Fig. 6. Model of contact for stress analysis.
3. Stress analysis In order to analyse the results of the ex~riments an elastic stress analysis must be pe~ormed on the experimental confi~ation. Figure 6 shows an idealization of the contact, where both the fretting pad and specimen are represented by elastic half-planes. This has been shown to be a reasonable assumption if the fatigue specimen is more than two contact widths in thickness [13]. The configuration modelled is that of a cylinder on a half-plane, corresponding to the initial contact geometry. Since the fatigue tests were carried out in the partial slip regime, very little wear was experienced in the contact. Such small amounts of wear have been recorded in other partial slip experiments 1141 and allow us to assume that the initial contact geometry is little changed by wear during the test and the model can be taken to apply throughout the experiment. Under these conditions the normal pressure developed is given by Hertz [ 5 f and is the well-known elliptical d~tribution 2
l/2
1 i)t x
P(X)=;Po
l-
;
where the peak pressure p. is given by PI3 PO-
i 2x(1 - vZ)R t
l/2
12)
and E and v are Young’s modulus and Poisson’s ratio respectively for the contacting materials (which are taken to be elastically similar).
335
As stated in Section 2, the tangential those described by Mindlin and Deresiewicz is displaced by an amount e because of the men which is not present in the pads [12]. peak value of tangential load Qmax are
1 oi x
q@)
= --fPo
1-
2
surface tractions are similar to [ll] except that the stick zone bulk strain in the fatigue speciThe tangential tractions at the
l/2
;
+ a’(x)
lx-el>c
lxlda
1x1 G a
(3)
where q’(X) = 0
(4)
and the stick-zone
semiwidth
c and eccentricity
e are given by
(5)
e -amax -= a
4fPo
These results are valid [12] provided
that
A sample distribution is shown in Fig. 7(a). In the experiments of series 1 - 5 the tangential force was cycled between + Q,,, and the surface tractions oscillated between plus and minus the limiting value given above. Q was first increased from zero to Q,,,, giving rise to the shear tractions indicated in eqn. (4) with a stick zone centred on x = e bordered by two slip zones. The shear force then started to reverse. This gave rise to instantaneous stick over the whole contact [ll, 121 before reverse slip zones started to form at the contact edges. These regions of slip grew progressively as the tangential force approached -Q,,,, where the shear tractions were then equal and opposite to those described by eqn. (4). Figure 7(b) shows sample shear traction distributions at various points in this half-cycle for series 3 experiments. The shear tractions developed at all points in the cycle were always the sum of elliptical distributions of differing amplitude [ll]. This means that the results of Smith and Liu [15] and Poritsky [16] may be employed, together with the principle of superposition, to calculate the stresses and displacements respectively in the halfplanes. The expressions are somewhat lengthy and will not be reproduced here.
336
(b) Fig. 7. Shear traction distributions within the contact: (a) comparison of predicted maximum tractions with Mindlin model (series 1 experiments); (b) variation of predicted shear tractions with load during half a cycle (series 3 experiments).
4. ~n~~r~t~tion
of results
The stress analysis can now be employed to assist in the interpretation of the experimental resuIts. Two possible explanations for the size effect must be considered: first, that fretting fatigue cracks do not initiate at the smaller contact widths, and secondly, that they initiate but that their growth is arrested. These will be referred to as the crack initiation and growth criteria respectively. 4.1. Crack growth criterion The first possibility examined was that cracks are unable to grow at small contact widths. This explanation was first proposed by Bramhall [ 41. ~ssent~lly, if the contact stress field is not sufficiently extensive to propagate an embryo crack to a length at which it can grow under the bulk stress alone then crack arrest will take place and the specimen will not fail. This argument seems initially attractive and an extensive series of stress intensity factor calculations were undertaken to provide a quantitative explanation of the phenomenon [ 6,7 1. Unfortunately, although good predictions of critical contact width can be obtained [7], there is little experimental evidence to support such a theory of crack arrest. If embryo cracks do self-arrest we would expect to see such cracks in sections of intact specimens. In fact, no such cracks were found and there was a marked contrast between sections of the failed specimens, in which many micro-cracks were present (Fig. 8) and intact specimens, where none were present. The absence of initiated cracks was
337
,
I Imm
Fig. 8. Section of a failed specimen viewed under an optical microscope.
further confirmed when some specimens were reused in subsequent tests by fretting at a different point. No failures were observed at the original fretting scar, as might have been expected if embryo cracks existed there. The other weakness of the crack arrest theory is that it fails to explain the variation in fatigue life with slip amplitude recorded in series 6 experiments. It is clear that the alternative hypothesis, that cracks do not initiate at the smaller contact widths, is worthy of further investigation. 4.2. Crack initiation criterion As we have seen, the experimental evidence suggests that cracks do not initiate at the smaller contact widths and that the amplitude of slip between the two surfaces has a distinct influence on the initiation process. The difficulty with analysing the suggestion that cracks do not initiate is that the mechanics of crack initiation are not clearly understood and that we might expect crack initiation to be a microscopic process, taking place at an asperity level, whereas the elastic stress analysis performed predicts only ‘bulk’ properties for a smooth contact. Any attempt to quantify the conditions for crack initiation using bulk properties will therefore be, at best, somewhat empirical. Ruiz et al. [17] have proposed two parameters which describe the severity of fretting damage and the likelihood of crack initiation during a given number of loading cycles. The amount of work performed in overcoming friction per unit surface area in the slip zone varies with position
338
and can be characterized by the product of 7, the maximum shear traction attained during one cycle at that point, and 6, the relative amplitude of slip between the surfaces at the same position. This parameter ~6 gives a measure of the amount of fretting damage per cycle at any point on the surface. It will be immediately recognized that 76 falls to zero in the stick zone (where 6 s 0) and outside the contact (where r = 0), where no fretting occurs. For the experimental configuration studied here 7 is simply equal to q(x) and given by eqn. (4), whereas 6 can be calculated using the results in ref. 16. The variation of 78 in the trailing slip zone (i.e. the one experiencing the maximum tension in the specimen) for series 3 experiments is shown in Fig. 9. Ruiz and his coworkers also recognize that a crack is much more likely to initiate in a region of bulk tensile stress than in one where the stress parallel to the surface is compressive. They therefore propose that a measure of the probability of crack initiation can be obtained by multiplying ~8 by the stress parallel to the surface CJto produce a composite initiation parameter ~6. The variation of this parameter in the slip zone for series 3 experiments is also shown in Fig. 9. A maximum value is reached very close to the edge of the contact, where cracks are found to initiate in the experiment. Ruiz et al. [17] and Kuno et al. [18] also found good agreement between the locations of maxima of or& and the sites of crack initiation. In the present series of experiments the maximum value of ~8 reached in the trailing slip zone will be proportional to 6 and hence to the contact semiwidth a. The likelihood of crack initiation will therefore be much higher at the larger contact widths. Maximum values of ur6 for experiments at the critical contact width are presented for each series of experiments in Table 2.
a16 E
__-r6E
p~(l-vVa
pf(l-v*)a
0.35 1 0.30 0.25 -
-0.10, -1.00
I
I
I -0.60
I
I
I -0.60
I -0.40
x/a
Fig. 9. Variation of damage parameter ~6 and initiation parameter 0~6 in trailing slip zone for series 3 experiments.
339 TABLE 2 Maximum values of initiation parameter at critical contact widths Series
Critical contact (mm)
1 2 3 4 5
0.28 0.54 0.18 0.36 0.57
semiwidth
adt
-+ 0.38 + 0.72 --f 0.27 + 0.54 + 0.71
Maximum value of initiation parameter (376 ( Nz mmV3) 4.47 3.26 2.42 3.94 3.47
+ + + + +
6.06 4.35 3.63 5.91 4.32
Minimum bulk stress (MN me2) 6
-39.9
+ -52.4
3.04 + 4.17
0.6 -
-f:
5
2
/
TO.6-
6’
/
L 0 -(J0.4 cu 5 .u
4,
/
_
-,’3
P a 0.2 -
/
/
/
1
L
/
1+---t 1
/ 0.0
, 0.00
I
0.20
I
0.40
Experimental
I
0.60
a (mm)
I
0.60
Fig. 10. Comparison of experimental and predicted critical contact sizes.
Cracks appear to initiate and lead to specimen failure within 10’ cycles once ~~78exceeds a value of about 4 N2 mm- 3. The mean value of 4.18 N2 mmm3 can be used to predict critical contact widths for failure and these are presented in Fig. 10. The results of the series 6 experiments, where the degree of load reversal was varied, may also be analysed in a similar manner. The transition from short to long fatigue life takes place at a value of ur3 between 3.0 and 4.2 N2 mme3. This is in good agreement with the critical values observed in the other 5 series (Table 2). It would appear, therefore, that 078 does provide a possible means of characterizing the probability of crack initiation by fretting within a particular number of cycles. The approach adopted is so far rather empirical, but we shall proceed in the next section to discuss whether there is a physical basis for characterizing crack initiation with this parameter.
340
5. A physical
basis for the initiation
parameter
The initiation parameter or6 under discussion is defined using bulk properties derived from an elastic stress analysis of smooth contact. We would, however, expect crack initiation to take place as a result of plastic deformation at the asperity level. Real contacts all exhibit some degree of surface roughness and this will often result in discontinuous contact [19], whereby small contact areas with high normal pressures are separated by regions in which no contact takes place. Quite modest asperity tip curvature gives rise to plasticity [ZO] and it is therefore likely that crack initiation takes place as a result of repeated passes of asperities in the plastic regime. Bower and Johnson [21] point out that the adoption of an e~stic/perfe~tly plastic model of material behaviour for material deformation under contact loads leads to the prediction of a vanishingly thin layer of inifinite shear strain. Some degree of strain hardening is therefore required in any material model. With such a model, Bower and Johnson show that incremental plastic strain in the surface layer is possible under repeated unidirectional passes of a sliding or tractive rolling contact load. In the fretting problem the motion is oscillatory, so that at first sight it is difficult to see how incremental strain can occur. Each asperity pass in one direction appears to be matched by an equivalent pass in the opposite direction as the fretting load is reversed. If we return to the smooth elastic model, however, we can see that the actual situation is somewhat more complicated. Figure 11 shows how the relative displacement of points in the slip zone varies with applied tangential force Q (and therefore with applied bulk stress a) during one steady state fretting cycle for series 3 experiments. A hysteresis effect is apparent because of the requirement for the tangential traction q(x)
Loo0
Q,/fP
Fig. 11. Hysteresis curves showing variation of relative displacement between contacting points in the slip zone with load for series 3 experiments.
341
to change from +fp(x) to --fp(x) at any point before the direction of slip can reverse there. If the overall deformations are little modified by the presence of surface roughness then we can relate this diagram to the stress state when asperities pass over a point on the surface. Let us consider a point on body 1, and allow an asperity on body 2 to pass over it at a relative displacement 6i on the scale of the overall contact when the applied load is Qi. The relative displacement will clearly be the same at the point later in the cycle when the asperity returns, but the applied load (and hence the applied tensile stress in the specimen) will be equal to Q2 (and not equal and opposite to Q,) (Fig. 11). Thus the point does not “see” a symmetric stress cycle, when the combined effects of asperity and bulk terms are summed and the accumulation of incremental plastic shear strain is possible. If accumulation of plastic shear strain is possible in fretting, it is likely that a ch~a~teristic amount of strain is required to initiate a crack. We can now investigate how the increment of strain in each fretting cycle varies with u, r and S. For a pair of surfaces of given roughness (and therefore a given asperity density) the number of asperity passes per cycle is simply proportional to the relative displacement 5. Elastic contact analysis predicts that the mean shear stress in the slip zones will be T, which is proportional to the contact pressure P(X). For a gaussian distribution of asperities with constant curvature Greenwood and Williamson [22] have shown that the number of asperity contacts per unit area is proportional to the mean contact pressure. Therefore, T gives a measure of the density of contacting asperities and the product 78 represents the number of asperity passes in each fretting cycle. The inclusion by Ruiz et al. of o in the initiation parameter is less easy to justify physically. Multiplication by the maximum stress parallel to the surface is rather arbitrary, but does take account of the observed phenomenon that fatigue cracks are more likely to grow in regions of high tensile stress. It is perhaps helpful to distinguish between the process of crack nucleation by ~cremen~l plastic shear strain, as described above, and that of initial crack growth, which might be expected to be controlled by the stress state just below the surface. It is therefore reasonable to suggest that for a crack to form it must both nucleate (requiring a critical total amount of fretting damage, where the damage per cycle is characterized by r6) and be able to grow (requiring a tensile stress e). The composite parameter ~76 expresses the requirement for both these conditions to be present. Configurations with high or8 values might be expected to undergo only a few cycles before embryo cracks form whereas the initiation of cracks in specimens with low a76 values will require a large number of cycles, either because the incremental plastic strain per cycle is low (rS), or because the inplane stress u does not encourage initial crack growth. 6. Conclusions A series of experiments has been reported in which the fretting fatigue life varies with contact size in the partial slip regime. This is shown to be due
342
to the variation of relative slip amplitude between the contacting surfaces. Cracks were found to initiate at large contact widths when the slip amplitude was large. The initiation parameter suggested by Ruiz et al. has been shown to characterize the probability of crack initiation well and there appears to be a critical value of this parameter for crack initiation to take place within a fixed number of cycles. Although the definition of this parameter seems rather arbitrary, it has some physical basis in characterizing the combined effect of the accumulated plastic strain in each cycle together with the tensile stress parallel to the surface. Ruiz et ~1 [17] have suggested that the (~76 parameter can be used as a basis for design against fretting fatigue. The present experiments certainly support this view. Design can concentrate on one of two aspects. First, on reducing the severity of fretting (~6) by reducing the amplitude of microslip, or the shear stress. It should be noted that in cases of partial slip with an imposed tangential force, reducing the coefficient of friction f (a common palliative measure) will have opposite effects on r and 6 and that the combined effect will have to be considered carefully. Secondly, designers can attempt to reduce u by placing contacting surfaces in regions of low tensile or compressive stress. This will not prevent fretting damage but should prevent cracks growing. In the absence of more rigorous means of quantifying the conditions required for crack initiation, the a& parameter does therefore provide useful guidance for designers. Acknowledgment The experimental work described in this paper was supported Science and Engineering Research Council.
by the
References 1 R. B. Waterhouse, Fretting Corrosion, Pergamon, London, 1972. 2 K. J. Fenner, K. H. R. Wright and J. Y. Mann, Fretting corrosion and its influence on fatigue failure, Proc. fni. Co@ on Fatigue of Metu&, Z~~tut~o~ of mechanical Engineers, London, 1956. 3 R. B. Waterhouse, Fretting Fatigue, Applied Science, London, 1981. 4 R. Bramhall, Studies in fretting fatigue, D.Phil. Thesis, Oxford University, 1973. 5 H. Hertz, Uber die Beruhrung fester elastischer Korper, J. Reine Angew. Math., 92 (1882) 156 - 171. 6 D. Nowell, An analysis of fretting fatigue, D. Phil. Thesis, Oxford University, 1988. 7 D. A. Hills, D. Nowell and J. J. O’Connor, On the mechanics of fretting fatigue, Wear, 125 (1988) 129 - 146. 8 D. A. Hills, D. Nowell and A. Sackfield, Surface fatigue considerations in fretting, in D. Dowson, C. M. Taylor, M. Godet and D. Berthe (eds.), Interface Dynamics, Proc. 14th Leeds-Lyon Symp. on Tribology, Lyon, September, 1987, Elsevier, Amsterdam, 1988. 9 K. Nishioka and K. Hiakawa, Fundamental investigation of fretting fatigue, Part 2, Bull. JSME, 12 (50) (1969) 180 - 187.
343 10 K. J. Nix and T. C. Lindley, The influence of relative slip range and contact material on the fretting fatigue properties of 35NiCrMoV rotor steel, Wear, 125 (1988) 147 - 162. 11 R. D. Mindlin and H. Deresiewicz, Elastic spheres in contact under varying oblique forces, J. Appl. Mech., 75 (1953) 327 - 344. 12 D. Nowell and D. A. Hills, The mechanics of fretting fatigue tests, Znt. J. Mech. Sci., 29 (5) (1987) 355 - 365. 13 D. Nowell and D. A. Hills, Contact problems incorporating elastic layers, Znt. J. Solids, Strucf., 24 (1) (1988) 105 - 115. 14 U. Bryggman and S. Soderberg, Contact conditions and surface degradation mechanisms in low amplitude fretting, Wear, 125 (1988) 39 - 52. 15 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. Mech., 20 (1953) 157. 16 H. Poritsky, Stresses and deformations of cylindrical bodies in contact with application to contact of gears and of locomotive wheels, J. Appl. Mech., 72 (1950) 191 201. 17 C. Ruiz, P. H. B. Boddington and K. C. Chen, An investigation of fatigue and fretting in a dovetail joint, Exp. Mech., 24 (3) (1984) 208 - 217. 18 M. Kuno, R. B. Waterhouse, D. Nowell and D. A. Hills, Initiation and growth of fretting fatigue cracks in the partial slip regime, Fatigue Fract. Eng. Mater. Strut., 12 (5) (1989) 387 - 398. 19 J. A. Greenwood and J. H. Tripp, The elastic contact of rough spheres, J. Appl. Mech., 34 (1967) 153 - 159. 20 D. Nowell and D. A. Hills, Hertzian contact of ground surfaces, J. Tribol., 111 (1) (1989) 175 - 179. 21 A. F. Bower and K. L. Johnson, The influence of strain hardening on cumulative plastic deformation in rolling and sliding line contact, J. Mech. Phys. Solids, 37 (4) (1989) 471 - 494. 22 J. A. Greenwood and J. B. P. Williamson, Contact of nominally flat surfaces, Proc. R. Sot. London, Ser. A, 295 (1966) 300.