An elastic-plastic model for fretting contact

435

Wear, 157 (1992) 435-444

An elastic-plastic

model for fretting contact

Mikael ijdfalk Saab Scania AB, Scania Divkion, S-151 87 SiidetiiiQe (Sweden)

Olof Vingsbo Depa&nent of Materials Science, Uppsala University School of Engineering Box 534, S-751 21 Uppsala (Sweden), and Department of Mechanical Engineering, Universiry of Houston, Houston, TX 77204 (USA) (Received April 1, 1992)

Abstract The type of surface damage that occurs in fretting contact depends on the magnitude of the surface tractions, and the amount of microslip on each point of the contact surface. The constitutive behaviour of asperity contacts, therefore, is of great importance for understanding the friction and wear mechanisms in fretting. In existing fretting models the relative displacement is assumed to be accommodated mainly through microslip in the contact surface and elastic deformation in the bulk of the contacting bodies. The present

paper argues that plastic deformation in the contact zone may contribute significantly to the relative displacement during fretting of metals. Friction mechanisms for fretting contacts are discussed, and an elastic-plastic fretting model is suggested. 1. Introduction

An elastic model for the behaviour of convex contacts between solid surfaces, subjected to oscillatory tangential relative displacements of low amplitude (fretting contacts) has been worked out by Cattaneo [l] and, independently, by Mindlin [2]. Fretting experiments on hardened steel under moderate loads have been carried out by Johnson [3], and found to be in good agreement with the elastic model. However, measurements of the amount of energy dissipated per fretting cycle indicated a larger energy loss than predicted. This deviation from theoretical values was suggested [3] to be caused by an additional energy consumption through plastic deformation localized to asperity junctions at the contact surface. Similar experiments involving comparatively soft materials under loads high enough to initiate plastic yield in the bulk of the material, have been carried out by Bryggman and Soderberg [4], and by the present authors [5]. In these investigations it was found that the maximum displacement amplitude that could be accommodated without incipient gross slip (the critical displacement amplitude) was higher than predicted by the elastic theory, and was influenced by the oscillation frequency. The existing elastic models [l, 21 do not take frequency dependence into account. In the present paper the previous models are developed to treat plastic deformation and frequency effects. 2. The idealized

elastic

model

It is convenient and judged representative of a general case of mechanical contact between engineering surfaces [S] to study the contact conditions of a sphere being

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436

pressed against the plane surface of a halfspace. It is further assumed that the contacting surfaces are perfectly smooth, and that the bodies are of the same isotropic and ideally elastic material. From Hertzian theory it is known that the contact surface will then be circular, of radius

(1) where FN is the applied E and v are respectively

normal load, R is the radius of curvature of the sphere, and the elastic modulus and Poisson’s ratio of the material (same material in both bodies). The normal pressure distribution over the contact surface is given by l/2

I-w=

( 1 l-

2

r2

(2)

a2

where r is the distance from the centre of the contact surface. A simultaneously applied tangential force FT will generate a tangential traction f=(r) in the contact surface. rr(r) has the same direction everywhere, and points of equal traction are located on a circle. A first assumption that no slip occurs anywhere in the contact surface, requires the traction distribution [l, 21

t*(r) =

FT 2n-a(a2-rZ)lR

It is seen from eqn. (3) that t=(r) has a singularity at the outer boundary of the contact surface @=a). However, it may be assumed that the tangential component of the surface traction cannot exceed the friction, represented by the product of the friction coefficient /I and the normal component of the traction, i.e. tr(T) Q w(‘)

(4)

This implies that slip will occur at some parts of the contact to refs. 1 and 2, slip takes place outside a circle of radius r=a’, l/3 a’=a

( FT ) I_-

surface. According given by

(5)

IJ;N

For a given FT < /.LFN(p is the static coefficient of friction) there is a corresponding a’(F,) such that the contact surface is characterized by a central stick circle of radius a’, surrounded by a slip annulus, as shown in Fig. 1. The tangential traction in the slip annulus

takes the maximum 1R

value

(6)

The requirements of equilibrium in the tangential direction and continuity of the tangential traction component are satisfied by the tangential distribution in the stick circle, given by tT

(r) =

rga’

w5[(1-

$Y gl_ _c)‘“]

(7)

437

SOLID SPHERE

SOLID BLOCK

8(T)

FN

8

-Fs-

Fig. 1. The sphere-flat fretting contact geometry. Fig. 2. F,(6) diagram corresponding to the first halfcycle of a fretting test. The elastic deformation in the sphere and the halfspace will result in a tangential displacement 6 of the centre of the sphere relative to a fixed reference point in the halfspace, far from the contact zone. Mindlin has calculated the displacement as

1 where k is given as a function k_

(1+4(2-v) 2

2 3(1 -v’) [ I

of

(8)

v by

I/3

(9)

It is seen from eqn. (5) that u’ approaches zero when the applied tangential force the friction force flN. FT= p.FN is the condition for incipient gross slip over the whole contact area, with the critical displacement sin,= G(flN), and eqns. (S)-(8) are defined for 0~ FT< pFN. With the tangential traction specified by eqn. (3), the corresponding displacement 6 becomes

FT approaches

When the tangential force is increased from zero, microslip starts at the rim of the contact circle @=a), and penetrates inwards to the radius a’ given by eqn. (5). The corresponding relative displacement 6 will increase according to eqn. (8). During successive unloading, the tangential force is gradually decreased. Slip in the reverse direction will then start at the rim, and penetrate inwards. The process is illustrated in Fig. 2, and it can be seen from the graph that the displacement deviates from an elastic line. At each point the horizontal difference corresponds to slip, and the total displacement may be decomposed into two components according to s=s,+s,

(11)

438

where 6, is the (reversible) elastic component For the case of no slip, S = S,, and with k

and 8, is the (irreversible)

slip component.

(FNE~R>“~

=

e

(12)

k

the elastic component,

as already

described

by eqn. (lo), is uniquely

defined

by (13)

The slip component 6,, however, depends on the loading history of the contact. If, as may be the case in fretting, FT is cycled between two values, + T and - T, where T
= 4 - 6(T) +

IWWI

(14)

where

[l-(1ik)“]

$5

qq=

e

IAMB=

(15)

y e [I-(I-E)a]

(16)

IAF,l=cuF,+T

ff=

I

I

(17)

1 ifs.0

(18) -1

if:

Conversely, ~+(S)=(Y[-

<0 the tangential

force FT is solved as a function

of 6 by

T+~T(~s)/]

(19)

8(T)

T-

T/k,

-T/k.

-8(T)

, -h(T)

Fig.

3. Fretting

Fig. 4. Variations

0

hysteresis

fi(T)

loop,

of 6, 8, and

taking 6, with

elastic time

displacement f during

and

a complete

microslip

into

fretting

cycle.

account.

439

lAT(s)l=2p&

[ l-

( l-

L$.%)“]

(20)

lA6l=cd+6(T)

(21)

Figure 4 shows the variations of 6, and 8, with time during a fretting cycle, where the total displacement 6 is externally applied as a perfect sine function. The area inside the loop of Fig. 3 represents the energy, dissipated per fretting cycle. The energy loss is given by [6] &=

36P2FN2 -[l-(1-

3. Deviations

A)‘“-

from the elastic

S{l-(I-

A)“}]

(22)

model

The experiments carried out by Johnson [3], with a hard steel ball rubbing against the flat end of a hard steel roller, showed good agreement with the elastic model as regards the displacement at incipient slip. However, although the measured energy loss per cycle was of the same order of magnitude as the theoretical value predicted by eqn. (22) there were significant deviations. In experiments on metals of varying hardness in a crossed-cylinders configuration, carried out by Bryggman and Sijderberg [4] and by the present authors [5], 6i”, was found to be considerably higher than predicted by the elastic model. It is suggested that these discrepancies can be explained as elastic-plastic behaviour in the contact zone as follows. There are essentially two ways in which plasticity effects may appear in the present fretting contact situation: by deformation of asperity junctions, and by bulk deformation of the contact zone. In Johnson’s investigation [3], standard bearing elements were used. The normal and tangential traction distributions given by eqns. (2) (6) and (7) refer to Hertzian, elastic calculations for ideally smooth surfaces, whereas the asperity junctions of the real bearing surfaces represent small surface elements, suffering localized stress concentrations, which highly exceed the nominal stress values, and may well exceed the elastic limit. The elastic model is further based on Amonton’s law, which presumes that a bulk contact is made up of a large number of asperity junctions, collectively contributing to (macro)friction. Implicity, Amonton’s law assumes that the junctions are rigid under the load, and undergo sudden fracture, without previous elastic or plastic deformation, when an increasing local traction t exceeds the corresponding local product w at a junction (see eqn. (4)). Obviously, as Johnson pointed out, this simplified picture may be the reason for the discrepancy between model and experimental results. Cheng and Kikuchi [7] have suggested that the concepts of plasticity theory be applied to asperity contacts by substituting Amonton’s law by a yield criterion, a flow rule and a work hardening rule. In the case of fretting a kinematic hardening rule is necessary for taking the Bauschinger effect into account. Qualitatively, consideration of elastic-plastic deformation behaviour of the asperity junctions implies that the contact surface is subdivided into three zones. The contact conditions of Fig. 1 will be correspondingly modified as shown in Fig. 5. Under the traction the asperities are deformed elastically in a central stick zone, surrounded by a yield annulus, in which the asperities have yielded

440

,Slip

Annulus

Elastic

condition

10-I 0.0

FN

0.4

FT-

Fig. 5. Modification

1.2

0.6

1.6

2.0

M

of Fig. 1 to include the yield annulus.

Fig. 6. FNy as function

of M for austenitic

stainless

steel (+)

and a Cu-3%Si

alloy (III).

plastically, but not fractured. The yield annulus, in turn, is surrounded by a slip annulus, where the asperities are subjected to shear fracture in the same sense as in the elastic model. The f&) distribution curve in the lower part of Fig. 5 demonstrates the yield annulus by the rounded transition between stick and slip, as compared with the sharp transition in Fig. 1. Calculations of the traction distribution for the elastic-plastic surface interaction can be used to calculate the elastic deformation and the resulting tangential displacement 6(r) with a similar approach as in the purely elastic model. It should be pointed out that no additional displacement is required to shear off the asperities, and that the elastic-plastic deformation will have no measurable effect on S(flN). At the instance of incipient gross slip, the microslip has just progressed to cover the entire contact surface, and the distribution of tangential traction will be the same as in the elastic model. To explain the much higher zinc values reported for softer metals in refs. 4 and 5, it is necessary to consider plastic deformation in the bulk of the contact zone. A quantitative analysis of the stress field during fretting in a number of different materials [8], reveals that plastic deformation will occur even for modest normal loads in most metals. Under the combined action of normal and tangential forces, characteristic of a fretting contact, plastic yield is an effect of both force components. In particular, the normal yield force FNy will decrease when the ratio M=FTJFN increases. In Fig. 6, FNy is plotted vs. M as a “yield curve” for a stainless steel and a copper-silicon alloy, for a given contact geometry as described in ref. 8. The exact determination of the yield curves is difficult owing to the complexity of the contact situation, with reciprocal interaction between dynamic effects and temperature effects. It is demonstrated, however, that FNy is reduced by more than two orders of magnitude, when M is increased from 0 to 2. The fretting yield force may also be determined experimentally by comparing measured 6(F,) with values calculated from the elastic model, and identifying FrFN combinations for which significant disagreement occurs. The yield force will also be influenced by fretting frequency, precycling history and contact geometry, in addition to materials parameters such as Young’s modulus, or strain rate sensitivity. Once the yield force is determined, the resulting plastic deformation can be calculated continuum mechanically, using relevant constitutive models. This means, however, a very rigorous treatment of a less well controlled system, and therefore a simpler model of elastic-plastic fretting might well render equally valuable information.

441

4. A simple

elastic-plastic

model

A simple elastic-plastic model for the contact conditions in fretting can be based on the elastic model described previously, if the bodies are not assumed to be ideally elastic. Instead, the assumption will be made that there does exist a plastic relative displacement component S,, such that

6=6,-t&+6,

(23)

The plastic behaviour of a fretting contact is illustrated by the schematic plastic hysteresis loop of Fig. 7, in which Fp represents a “fretting yield point”. FTydepends on the geometry of the contact (R), the material (a,,, E and u) and on the normal load FN [8]. For a given geometry and material there are, in principle, three different yield conditions: (1) pr,,I = 0 for FN > FN,,;(2) 0 < IFnl < pFN; (3) pm] > @N; i.e. plastic deformation may be initiated only under conditions 1 and 2. The area of the hysteresis loop of Fig. 7 corresponds to the energy dissipated through plastic work. A linear work hardening is assumed, with a plasticity (work hardening) coefficient

It is further assumed that the fretting contact has experienced a large number of load cycles, so that equilibrium conditions are reached. No further structural or temperature changes will then take place, i.e. no further work hardening or thermal softening can occur, and FTywill remain unchanged as plastic flow is initiated alternately in the positive and negative direction during continued cycling. The hysteresis loop of Fig. 7 is further simplified in two aspects. (1) !cP is treated as constant, although it varies within each halfcycle with the strain rate, which, in turn, varies with the applied displacement rate. (2) A mean kP value (the steepness of the work hardening curve in Fig. 7) will vary with the mean strain rate and displacement rate, which will be a function of the frequency of the fretting oscillation. Thus, kP represents a “fretting hardening coefficient”, and is a parameter of great interest in a model aiming at describing the frequency effects in fretting. The total displacement is still described by the 6(FT) function of eqn. (14), but 6(T) and AS(FT) have to be modified by the addition of plasticity terms, according to S(T)=

e e [l-(1- -$]+/3y/--

Fig. 7. Schematic I$.($,) hysteresis loop for plastic displacement in fretting.

(25)

Fq =O;

‘k, ~0.8

F~y=0.5T;

ke

k,=0.8

Fig. 8. Variations

FT~ =O;

ke

kp =0.4

FT~ =0.5T;

ke

kp =0.4

ke

of 6, 6,, 6, and 6, with time I during a complete

fretting

cycle.

(26) where P=

1 if T> FTy (27)

0 if T
1 if lFTl>FTy and

if $FT>O

Y= 0 otherwise The

plastic

displacement

FT - ~~~ kp

T-FTY -7&i-

and the slip contribution

contribution

1

can now be solved

as (29)

is obtained

from

eqn.

(29)

instead

of eqn.

(ll),

6,=6-&-6,

as (30)

Figure 8 is a schematic illustration of the variation of the three displacement components with time, for combinations of two different values of yield point and work hardening coefficient (see Fig. 4). The elastic loading-unloading parts can be seen as straight segments truncating the 6, curve. The corresponding three 6(FT) functions are plotted together in Fig. 9, which illustrates that energy is lost through slip and plastic displacement (hysteresis loops), but not through elastic displacement (a straight line). (The present model disregards the fact that some energy will be dissipated as heat by phonon generation also in the elastic case.) A schematic representation of the complete FT(8,,) loop for FT
with experimental

data

by the present gross slip displacement zinc was measured (AISI 304) at four different fretting frequencies between

authors for 10 and 800

443

II

-8(T)

I

0

J

8(T)

8 Fig. 9. FT(Be), FT(&) and &(a& Fig. 10. The complete TABLE

hysteresis

F=(8) fretting

loops plotted

hysteresis

together.

loop, corresponding

to Fig. 9.

FN= 11.4 N, k,=16

MN m-‘,

1

Theoretical values &=0.55 N

of k, for stainless

steel:

a,=770

MPa,

F (Hz)

P

& + 6, (wd

6 (@NJ

kp (MN m-‘)

10 50 300 800

0.95 1.6 1.5 1.7

1.05 1.77 1.66 1.88

2.3 2.1 1.8 2.9

1.8 18 39 6.4

Hz [5]. Sine was found to be significantly higher than predicted by the elastic model. The FNy yield force can be calculated from the maximum contact pressure p. and the yield stress uY [6, 81. In the present case, the maximum contact pressure is known from eqn. (2) as

p. = 3FN/2n-a2

(31)

As to determining I+, a common approximation is that the Vicker’s hardness HV corresponds to 3u,, where cry is the yield stress in compression. Direct measurements on the test specimens showed a hardness H=235 HVIo,, which indicates a yield stress u,=H/3 =770 MPa. The present combination of p and a,. gives a fretting yield point FTy = 0.55F, = 0.55 x 11.4N = 6.5 N. Values of k, calculated for different frequencies are presented in Table 1. It is seen that the strong influence of frequency on sin= results in a great variation in k, values. A comparison with the elastic coefficient shows a variation O.llk,
444

experimentally in order to obtain to displacement in fretting.

6. Summary

a more complete

picture

of the plastic contribution

and conclusions

The suggested model is based on micromechanisms in the contact zone of fretting contacts of ductile materials. The formulation is, however, mainly continuum mechanical, and the relation to material parameters (II, a,, E) and the introduction of materialrelated concepts, such as “fretting yield” and “fretting hardening”, provide a framework for interpreting data from bulk measurements. It is of particular interest that the coefficient of fretting hardening in the plastic part of the model (k,) may be a tie to describing the frequency dependence.

Acknowledgement

This work has been financially under contract MSM-851 6963.

supported

by the National

Science

Foundation

References 1 C. Cattaneo, Sul contatto di due corpi elastici, Accademia dei Lincei, Rendiconti, Ser. 6, XXVII (1938) 342-348, 434-436, 474-478. 2 R. D. Mindlin, Compliance of elastic bodies in contact, L Appl. Me&, 71 (1949) 259-268. 3 K. L. Johnson, Surface interaction between elastically loaded bodies under tangential forces, Proc. R. Sot. London, Ser. A, 230 (1955) 531-548. 4 U. Bryggman and S. Sbderberg, Contact conditions in fretting, Wear, II0 (1986) l-17. 5 M. ijdfalk and 0. Vingsbo, Influence of normal force and frequency in fretting, STLEIASME Joint Ttibologv Conf, Fort Lauderdale, December 1989, STLE Preprint No. 89-TC4E-1, 1989. 6 K. L. Johnson, Contact Mechanics, Cambridge University Press, Cambridge, 1985. 7 J.-H. Cheng and N. Kikucbi, An incremental constitutive relation of unilateral contact friction for large deformation analysis, J. A&. Me&., 52 (1985) 639-648. 8 0. Vingsbo and M. Odfalk, Conditions for elastic contact in fretting, Proc. Japan ht. Tribolog~ Co& Nagoya, October 1990, Jpn. Sot. Tribologists, Tokyo, 1990, pp. 833-838.