Wear, 65 (1980) 125 - 129 0 Elsevier Sequoia S.A., Lausanne -- Printed in the Netherlands

125

A NOTE ON SHAKEDOWN

D. A. HILLS and D. W. ASHELBY

Department of Mechanical and Production Engineering, Trent Polytechnic, Nottingham (Gt. Britain)

(Received February 11,198O)

Summary

The residual stresses developed beneath a rigid ball pressed into an ideal- ly elastoplastic half-space are found on the centre-line of contact. Using the theoretical results of Johnson a shakedown limit based on the criterion that residual stresses are maintained at less than the yield criterion is found, and experimental results confirm that this limit obtains.

1. Introduction

When components are loaded in a cyclic manner, the highest usable working load may not be determined by the onset of plasticity. This is be- cause, after a few cycles of loading, residual stresses build up which tend to alleviate the applied loads. Hence the steady state condition reached after only a few cycles may be elastic, even though the applied loads alone would violate the yield criterion. This phenomenon is known as shakedown and the largest load which may be supported is the shakedown limit. The concept has been applied to frameworks, pressure vessels and, quite recently, to con- tact loads. Johnson [ 11 found the shakedown limit for two rolling cylinders and was later able to predict the form of plastic flow occurring [ 21. Experi- mental evidence for the existence of the necessary residual stresses was at- tempted [3] and went some way to showing that the initial theory was cor- rect. Nevertheless, because the contact patch does not change as shakedown proceeds, it is very difficult to establish the exact value of the shakedown limit. In order to make significant progress with this type of geometry, a greatly simplified system was chosen for this investigation, namely a rigid frictionless ball pressing vertically into an ideally elastoplastic surface.

2. Theory

Let us consider Fig. 1. In this coordinate system the deviatoric stresses on the centre-line (r = 0) are [4]

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Fig. 1. The coordinate set used.

8-r = see = PO i - ; (1 + u)(l - 2” cot-’ z*) f ; -&I

f2 1 sz = PO i3(l + V)(l --z* cot-” .P) - F

1 + z*z i

(1)

Since these are principal deviatoric stresses, the elastic limit by the von Mises criterion may be immediately found from

Jz= $(S,,2+See2+S~~2)Gk2

If J2 is maximized with respect to the depth, the severest stress is located at a depth Z* = 0.481 and the corresponding elastic limit is

Polk Q 2.771 (2)

Beyond the elastic limit, care is needed to determine the subsequent stress state but Johnson [ 51 has pointed out that on the centre-line the ratio of the three extended elastic deviatoric stresses remains constant over both load and depth, i.e.

S,, = See = 2Szz

This is important because it means that on the cenhe-line the loading is pro- portional, Another way to look at this would be to consider the deviatoric plane (Fig. 2). As the ball is loaded, the stress trajectory extends along a ra- dial line until point A is reached which corresponds to the onset of plasticity. A subsequent increase in pressure does not cause any alteration of the loca- tion of the tip of the trajectory.

As the load is removed, elastic recovery occurs and hence the stresses remain in the same ratio. Thus the tip of the stress trajectory recedes along the same radius, the length of the vector being dependent on the pressure

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Fig. 2. Stress history projected onto the deviatoric plane.

which was achieved during the loading phase. The residual stresses are there- fore given by

l= k

% %e l =-se, + - 3112

I=_ 2k @zz &z - - 3112

Under cyclic loading, when a steady state is attained, it is possible that during every cycle the plastic regime is entered [6]. However, if it is assumed that to obtain a true steady state all plasticity is eliminated and the loading cycle becomes reversible, then the residual stresses must themselves lie with- in the yield criterion. Thus

J2 = $ {(Q - o&2 + (aeel - u&2 + (a& - CJ$)~ }< k2

or

2J2 < k2

The corresponding maximum load is therefore

p,lk < 5.54

This is therefore a possible shakedown limit.

3. Experimental

(4)

To test eqn. (4) as a possible shakedown limit, a polished steel ball (of diameter 10 mm) was compressed cyclically into a steel block under constant maximum load conditions. It was assumed that a steady state had been at-

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TABLE 1

Experimental results

station P

WY

Impression Mici-o- diameter hardness

(mm) (HV)

PO -2 (kNmm ) Polk

0.823 0.870 213 2.078 5.14

1.357 1.090 213 2.175 5.38 2.346 1.342 233 2.488 5.63 2.624 1.484 221 2.276 5.43 3.899 1.777 221 2.342 5.59 4.537 1.965 215 2.224 5.45 5.026 2.068 213 2.221 5.50 5.871 2.203 215 2.285 5.60

Mean 5.46 Standard deviation 0.15

tained after 100 cycles, and the applied loads were chosen to be well in ex- cess of the elastic limit. Because the final state is assumed to be elastic, all Hertz’s relations apply. The diameter of the indentation formed was mea- sured using a Universal measuring machine (to + 0.1 pm) and the current value of the yield stress was estimated by taking microhardness indentations using a 50 g load in the centre of each impression. Thus work hardening was correctly allowed for. The results are shown in Table 1, the peak contact pressure being given by

p. = 3P/2na2

4. Conclusion

The shakedown limit of eqn. (4) therefore seems to be confirmed. It is interesting to compare this value with the nominal average pressure of an in- dentation test, as estimated by Tabor [ 71. This he gives as

PII3 = 2.9800

If the pressure ordinates were hertzian, this would give a peak pressure 1.5 times greater than the mean, i.e.

PO/k = 7.74

Johnson [S] has found that the unloading phase of a hardness test, which is equivalent to the first cycle of loading, is always accompanied by secondary plastic flow. It would therefore be expected that the peak con- tact pressure for the first loading cycle would exceed the shakedown limit. However, it must be remembered that during this first loading cycle the pres- sure distribution is in reality probably far from hertzian, with a much more

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even distribution of pressure because the ball is supported almost entirely on plastic material.

Nomenclature

PI, k

PlIl PO P r, Z, 0 z* V

*0

radius of contact second invariant of the stress deviator tensor yield in pure shear mean pressure maximum contact pressure applied load cylindrical coordinates Z/a, depth normalized by the contact radius Poisson’s ratio (= 0.3) yield under a uniaxial load

References

1 K. L. Johnson, A shakedown limit in rolling contact, Proc. 4th Natl. Conf. on Applied Mechanics, Berkeley, California, June 1962, pp. 971 - 975.

2 J. E. Merwin and K. L. Johnson, An analysis of plastic deformation in rolling contact, Proc., Inst. Mech. Eng., London, I77 (25) (1963) 676 - 685.

3 R. J. Pomeroy and K. L. Johnson, Residual stresses in rolling contact, J. Strain Anal., 4 (3) (1969) 208 - 218.

4 M. T. Huber, Zur Theorie der Beriihrung fester elastischer Korper, Ann. Phys. (Leip- zig), I4 (1904) 152 - 163.

5 K. L. Johnson, personal communication, November 1978. 6 D. Tabor, Discussion. In J. E. Merwin and K. L. Johnson, Proc., Inst. Mech. Eng., Lon-

don, 177 (25) (1963) 688. 7 D. Tabor, A simple theory of static and dynamic hardness, Proc. R. SOC,. London, Ser.

A, 192 (1948) 247 - 274. 8 K. L. Johnson, Reversed plastic flow during the unloading of a spherical indentor, Na-

ture (London), 199 (4900) (1963) 1282.