Shakedown in Dry and Lubricated Surfaces

Lubrication at the Frontier / D. Dowson et al. (Editors) © 1999 Elsevier Science B.V. All rights reserved.

255

Shakedown in Dry and Lubricated Surfaces G.E. Morales-Espejel a , A. Kapoor b and S. Rodriguez-Sfinchez a Instituto Tecnol6gico de Monterrey, ITESM, Monterrey, N.L., M6xico* b Department of Mechanical Engineering, Mappin Street; Sheffield S1 3JD, U.K. a

In this paper the shakedown process of both dry (with different friction coefficients) and rough elastohydrodynamically lubricated contacts is analysed. For lubricated contacts the EHL problem is solved for contacts with "real" stationary roughness and the calculated pressures are used to obtain upper bounds to the shakedown limit. The results show that for incompressible lubricants, the shakedown mechanism is likely to be controlled by the roughness instead of by the overall Hertzian stresses. The roughness produces a reduction of the required yield load. Fatigue calculations are also performed to show the effects of roughness in lubricated contacts. The results obtained with stationary roughness show that roughness modifies shakedown and fatigue life.

1. I N T R O D U C T I O N Contacting surfaces are present in many industrial mechanisms, e.g. roller bearings, gears, cam-follower systems, wheel-rail systems, etc. It is well known that both dry and lubricated contacts produce damage and therefore, a limited life of the component. For industry, it is important to understand and manipulate the mechanisms which determine life in contacting surfaces so that the design of mechanical elements might be improved. Tribological components are loaded repeatedly and if the load in the first cycle is above the elastic limit, the material undergoes plastic flow. Incompatibility in strain with the surrounding material gives rise to residual

stresses, which are protective in nature and may eliminate plastic flow in subsequent cycles. This is the process of shakedown and the limiting contact pressure below which it is possible is known as the shakedown limit. This limit is the rational criterion for tribological components as a load below this limit will lead to only elastic deformation of the material and consequently a longer life. The exact value of the shakedown limit is obtained by simulating rolling / sliding on a computer, but the procedure is computationally expensive and results non-parametric. In a different method, plasticity theorems are used to establish upper (Melan [1]) and lower (Koiter [2]) bounds to the exact shakedown limit. Johnson and co-workers have used Melan's technique to obtain a lower bound to the shakedown limit for line and point contacts

Present address: ITESM-C. Quer6taro, Departamento de Mec~inicae Industrial. Jesfis Oviedo A. No. 10, Parques Industriales, 76130, Quer6taro, Qro., M6xico.

256

[3-6]. Belyakov [7] applied Koiter's technique to estimate an upper bound to the shakedown limit for a line contact. Recently, Kapoor and Williams [8] have studied upper bounds for the shakedown limits for case-hardened and coated surfaces. However, not enough has been done to understand the effect of roughness and elastohydrodynamic lubrication on the shakedown limits, this is what is attempted in this paper. The results show that for incompressible lubricants, the shakedown limit is controlled by the roughness; the roughness produces a reduction of the required yield load. The effect of roughness on life is also studied, the results show that rough surfaces in lubricated contacts do reduce life. 2. T H E O R Y

2.1 Shakedown Analysis

to the maximum load to be supported purely elastically [2-5]. Many residual stress fields can be considered and the one leading to highest lower bound is chosen to provide the closest estimate to the shakedown limit. The upper bound analysis relies on determining a kinematically possible collapse mechanism, which can be activated with the minimum energy requirement. This energy is supplied by movement of the applied load. Belyakov [7] showed that in line contacts a collapse mechanism involving shearing of material parallel to the surface leads to the lowest upper bound. In fact that bound is identical with the lower bound calculated by Johnson, establishing the exact shakedown limit for the line contacts. The same approach was used by Kapoor and Williams [8]. For details of the technique the reader is referred to that publication and the equation used to determine an upper bound to the shakedown limit, Ps, is reproduced below, p~. =

1

k

(1)

max(~x~)

where

k

is

shear

yield

strength

of

the

material, and Z'xz the unit orthogonal shear stress, i.e. the orthogonal shear stress for a maximum contact pressure of unity. v

A

{~z

B

D

B

Figure 1. Loading conditions in line contacts, an element of material experiences the cycle of reversed shear and compression A-B-C-D-E (as given by Johnson [6]).

2.2. Increase in Load-Carrying Capacity The process of shakedown leads to a load higher than the initial yield limit to be supported purely elastically in the steady state. The extent of this benefit can be calculated by an increase in the load carrying capacity of the junction. For a line contact the load supported is proportional to the square of the maximum contact pressure [6] and so ct , the increase in load carrying capacity can be given as, a

Figure 1 shows a cylinder rolling / sliding on a half-space. The resulting contact pressure and stress distributions are calculated. The lower bound technique relies on determining a residual stress field in self-equilibrium, which together with the applied contact stresses leads

(ps)2 =~ (py)2

here

py

is the initial yield limit.

(2)

257

2.3. Estimation of Orthogonal Shear Stress The orthogonal stress component needed to calculate shakedown limits was obtained by dividing the contact pressure into rectangular elements and superposing the analytical solution of each of the rectangular elements. For both, dry and lubricated contacts the contact pressures were divided in 5334 elements. The meshes performed for the subsurface region and for the pressures were tested for convergence to reduce numerical errors, mesh in x direction had 3523 points and in z direction 50, concentrated near the surface.

1'i- /

0.5

0

=

P [cos 281 - cos 20 z

]

0.5

1

1.5

2

2.5

z/a (a)

The orthogonal shear stresses due to a rectangular strip of normal pressure p and a uniform traction q is given by eq. (3) and (4) respectively (from Johnson [6]).

Z'x. z

.10~

-0.178

1.5

1

•

--

-~

-0 5

(3)

0.0706

-1

Txz = --~q {2(81-82)

+(sin281 -sin282) }

(4)

-1.5 -9

In order to obtain the unit stress, the magnitudes of p and q were set equal to 1 and p was multiplied by the appropriate value of friction coefficient to give q .

0

0.5

1

1.5

2

2.5

z/a (b) Figure 2. The distribution of unit orthogonal

3. DRY C O N T A C T S Shakedown limits for dry contacts have been studied in the past. However, in order to facilitate comparisons with the more realistic lubricated contacts some dry contact results are included. These also serve as a check on the accuracy of the shakedown calculations done for lubricated contacts. For cases which include friction, complete sliding is assumed so that traction q is related to normal pressures p by q = ~ p , where ,u is the friction coefficient. Also, no effect of friction over the normal traction is assumed, as is commonly done in these calculations.

^

shear stress (Z'xz) for (a)

/~

fl-0.0and

(b)

calculated

unit

- 0.3.

Figure orthogonal different

2

shows

the

shear stress fields coefficients

of

"~xz for two

friction,

/z = 0.0

and//=0.3. From these calculations, the maximum value w a s p i c k e d up and used in eq. (1) to obtain upper bounds to the shakedown limit. These are given in Table 1. The initial yield limits were estimated by calculating the von-Mises yield parameter J2 and equating it the shear yield strength of the material, k. Substituting these values in eq. (2) provides the increase in load carrying capacity.

258

In line with the previous findings, it can be seen that the load carrying capacity drops from a value of 1.66 to 1.18 as the coefficient of friction increases from a value of 0 to 0.2. Then if the friction coefficient is increased to 0.3, the load carrying capacity increases to 1.24. This can be explained by the quick reduction of the yield load, since now yield takes place at the surface (see Johnson [6] ).

Pressure and film thickness distributions were obtained for the following data, typical of a cam-follower system; max. Hertzian pressure

Table 1 Results for the dry contact example.

..~ .......

po=l.O41xlO9pa,

fl

max( r xz )

ps/k

py/k

a

0.0 0.1 0.2 0.3

0.25 0.28 0.31 0.34

4.00 3.57 3.20 2.90

3.11 3.06 2.94 2.60

1.66 1.36 1.18 1.24

= .......

: ...........

=:....................................

::::::::::::::::::::::::::::::::::::::::::::::::::::::

.............

:..:...:......:

........

:...:

........

the present analysis four cases were considered: smooth surfaces lubricated with compressible and incompressible fluids, and rough surfaces lubricated with compressible and incompressible fluids. It is widely accepted that lubricants are compressible at high loads, however, incompressible examples are useful to study the effect of higher pressure gradients on shakedown upper bounds.

:-:::-:

surfaces

average

velocity

m/s,

equivalent

~ = 22.178

modulus

E'= 227 x 109 Pa,

viscosity rio =0.048

It is an interesting exercise to simplify the calculation of the maximum unit orthogonal shear stress value for dry contacts. There are well known analytical and numerical procedures for this (see e.g. Johnson [6]). However, it was found simpler to use a single curve-fitting equation. Sufficient numerical simulations were performed and with the use of the least square error method the following good approximation was found,

the

Young

defined

2 / E'= (1 - vl 2) / E 1 + (1 - v22) / E 2 ,

...........

of

as,

lubricant

Pa s , lubricant viscosity

index a = 9.6 x 10 -8 Pa -1 reduced radius R=0.02724 m, and Hertzian semi-width a = 0.5x10 -3 m. x10 -7 5

max(G: / Po ) = max(~x:) = 0.249 + 0.374p -0.715/.t 2 + 1.791/.t 3

(5)

Similarly, the upper bound to the shakedown limit may be approximated by its reciprocal,

| -5

-4

-2

-1

0

1

.

2

x, meters Figure

p., / k = [0.249+0.374p-0.715p 2 + 1.791p3] -l (6)

-3

3.

Initial

undeformed

_ t _ _

3

_ a

.

.

.

.

.

.

4

5

xl 0 -4 roughness

used, with r.m.s. = 1.7 x 10-Tm.

which are valid between 0.0 _
4. L U B R I C A T E D C O N T A C T S For lubricated contacts, the pressure distribution differs from that in the dry case. In

The smooth surface calculations, were done by numerically solving the Reynolds, film thickness and load equations, using a NewtonRaphson method, as described in [11].

259

The rough surface examples were calculated assuming the initial surface roughness shown in Figure 3 on the upper surface and this surface to be stationary. The superposition scheme described in Greenwood and Morales-Espejel [10] was used. This scheme insulates the central zone of the contact, disregarding the contact inlet and outlet, as described by Greenwood and Johnson [12]. Then by applying FFT, the "real" roughness is divided into its sinusoidal components, which are taken as initial displacements in the film thickness equation, this film thickness is substituted in the Reynolds equation to find modified pressures. The pressures are divided into sinusoidal components by FFT and used to calculate elastic displacements. The process is repeated until convergence. After this, the numerical smooth surface solution is superposed to include the overall deformation and pressures. This scheme is highly convergent for "real" roughness cases. Figure 4 shows the pressure and film thickness distributions for the four examples

dry Hertzian contact is estimated by using the following equation,

r] = (p~)2 /

(ps)Zhertz= (p~)2 /

(4.0)2

(7)

for /,/= 0. These values are also included in the table.

Table 2 Results for the lubricated contact examples with zero friction. .

-

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

...

. .........

Surf'/"fluid

psi k

py/ k

Ct

rl

Rough/inc. Smooth/inc. Rough/comp. Smooth/comp

2.53 3.6 3.83 3.88

1.65 3.09 3.28 3.18

2.34 1.36 1.36 1.49

0.4 0.81 0.81 0.94

Smooth, no lubrication

4.00

3.11

1.66

1.00

:::

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.=:::.=::_=:;;.:....

solved, for /./= 0. The calculated mean film thicknesses are; 2.11 xl0-6mfor compressible

5. FATIGUE C A L C U L A T I O N S

lubricant and 2.49 ×10-6m for incompressible lubricant. Figure 5 shows the corresponding unit orthogonal shear stresses for all the cases. The variation of stress at a material point as the contact passes over can be traced by drawing a line in Fig. 5 at that depth. The maximum stress experienced is the maximum value of stress on this line. Its variation with depth is plotted in Fig. 6.

It has been shown that due to the protective effect of the residual stresses it is possible to increase load above the elastic limit, and yet keep an entirely elastic behaviour in the structure. If the elastic shakedown limit is not exceeded, then failure would be expected to be by high-cycle fatigue. Lundberg and Palmgren [13], originally applied fatigue to model life of rolling element bearings. A newer approach due to Ioannides and Harris [9] is used in this study to calculate life cycles for the dry and lubricated contacts. This model improves Lundberg and Palmgren theory and incorporates into the analysis the ability to relate elemental surface rolling contact fatigue with the more general structural fatigue. B e s i d e s , the model also

The upper bounds to the shakedown limits were calculated by picking up maxima from Fig. 6 and substituting in eq. (1). Likewise, the initial yield pressures were calculated by estimating the maximum von-Mises yield criterion J2 and equating it to the shear yield strength of the material. These results are given in Table 2. For comparison the case of a smooth contact without lubrication is also included. The improvement in load carrying capacity with respect to this ideal frictionless

introduces the concept of endurance limit r,,, which opens the possibility for infinite life.

260

2.5~

2.5 _

,

,

,

,

,

'

1.5

1.5

p H

0.5

0.5

.

j 0

. -~'.5

-2 w

.

-~

.

-

.

.

-o'.s.

.

.

j

o.5

6

1

1.5

0

-2

-1.5

-1

43.5

0

1.5

x/a

x/a

(b)

(a)

2.5

2.5

,I

~

I

1.5

1.5

H

P

0.5

0.5

0--2

1

0.5

-1.5

-1

-0.5

0

0.5

x/a

(c)

1

1.5

0~ -2

-1.5

-1

-0.5

0

0.5

1

1.5

xla

(d)

Figure 4. EHL Film thickness and pressure distribution for (a) smooth surfaces lubricated with a compressible fluid, (b) rough surfaces lubricated with a compressible fluid, (c) smooth surfaces lubricated with an incompressible fluid, and (d) rough surfaces lubricated with an incompressible fluid, where P = p ~ Po and H = h R / b 2 .

261

1.1: ~°I

°s1 0

.01

-0.5

-0.

-li

'

"

~

0 196

-1.5 -2

,

0

>-..

,,JJ

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0

0.1

0.2

0.3

0.4

z/a

(a)

~ ~I

-

-

0.5

0.6

0.7

0.8

0.9 - - -

z/a

(b)

~-o

2s4

-

1

0.5

01 o

~o

-1.5~ 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0

: 0.1

0.2

0.3

0.4

0.5

z/a

z/a

(c)

(d)

0.6

0.7

0.8

0.9

Figure 5. Dimensionless shear stresses Z'xz, (a) smooth surfaces lubricated with a compressible fluid, (b) rough surfaces lubricated with a compressible fluid, (c) smooth surfaces lubricated with an incompressible fluid, and (d) rough surfaces lubricated with an incompressible fluid.

262

0.4

0.3

0.25 0.3

0.2

<~,,~0.2

~ 0.1

°1I]

rc - rou gh compressible

0.0

.

0

0.1

0.2

0.3

0.4

O.

"

0.5

0.6

0.7

0.8

0

0.9

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

z/a

z/a (a)

(b)

Figure 6. Maximum unit orthogonal shear stress distribution with depth for smooth a n d rough surfaces lubricated with a compressible fluid, (a), and an incompressible fluid, (b).

A complete background of the approach can be seen in the said paper. For the current simplified analysis, however, only the fundamental aspects of the Ioannides and Harris model are considered, which give,

Table 3 Values of the fatigue ( h = 7 / 3 , c = 3 1 / 3 , S = 0.9). ~U

^ )c / z,~ }iA

(8)

A

where No is a new parameter proportional to life in cycles N e, Z' is z'/aand represents the stress weighted depth; it is given by,

rough/inc. sm o oth/in c. Rough/comp. Smooth/comp. smooth, no lubrication

~v.v

2.0E+5 3.7E + 5 6.7E+5 6.8E+5 5.9E+5

parameter

g, l l

Nc

- - V . l

2.04E+7 6.84 E + 7 19.1E+7 21.8E+7 20.4E+7

(9) The analysis has been done by assuming where z,,, is the maximum depth of analysis

h,c,e are experimental constants, ~,, = r,,/po is the orthogonal dimensionless endurance limit of material, and S is the failure probability. Typical values of S are 0.9 for Ll0 lives or 0.5 for Ls0 lives (here S =0.9). For the other constants, typical values have been taken from the ref. [9].

that the stress cycle for each (x, z) location is given by the stress pattern along the whole x direction. This means that the rough surface results have been obtained assuming a repeated roughness pattern. Even though this is not precisely true, in typical cases it is n o t expected to be far from reality. The results for lubricated and dry contacts are given in Table 3 for two values of the endurance limit.

263

DISCUSSIONS Tribological components are loaded repeatedly. The contact pressure limit that causes plastic flow in the first application of the load is known as the elastic limit. If the load is above this limit, plastic flow takes place and residual stresses are developed. These are protective in nature and may result in an elastic behaviour in the steady state. The limit below which this process is possible is known as the shakedown limit and is a rational design criterion. For a dry smooth contact the limit is 4k. However, real surfaces are not smooth and are almost always lubricated. In the present work these effects are analysed and the appropriate upper bounds are calculated.

the

Figure 6 shows the variation with depth of maximum of unit orthogonal shear

stress

~xz

for various lubricated cases. Since

there are no shear traction ( / t = 0), the value of ~xz at the surface ( z / a - 0 )

would be

zero. This is indeed the case. However, for the rough surface lubricated with an incompressible

fluid, Fig. 5 (d),

Z'xz = 0 . 4

very close to the surface, this is due to the stress concentration near this point produced by the high pressure peaks associated with roughness (see Figs. 4 (d) and 5 (d)). The shakedown limit would be expected to reduce due to this effect, the upper bound obtained for this case is only 2.53 k (a reduction of 37%). In the compressible lubricant solution, Fig. 5 (b), the contact pressures do not have the spikes as in the incompressible case. The resulting contact stresses near the surface are not so severe, Figs. 4(b) and 5(b), and the shakedown behaviour is similar to that for a dry smooth surface. The upper bound to the shakedown limit is now 3.83 k , much closer to the value 4.0 k for a dry smooth contact. Figures 4(a) and (c) show the contact pressure for smooth surfaces lubricated with a compressible and an incompressible fluid, respectively. The contact pressure has the

characteristic pressure peak in the exit region. The distribution of unit orthogonal shear stress is shown in Figs. 5(a) and (c) and the variation of its maxima with depth in Figs. 6(a) and (b). The resulting upper bounds on the shakedown limits are now 3.6 and 3.88 k respectively. This rather counter-intuitive result shows that lubricating an ideal smooth surface decreases its shakedown limit. This is due to presence of the pressure peak. It should, however, be remembered here that a dry smooth surface has a friction coefficient higher than zero and consequently a lower shakedown limit, Table 1. Table 2 shows improvements in the load carrying capacity of lubricated contacts. The values of ct provide the improvement in the load carrying capacity due to the effect of residual stresses or shakedown. It can be seen that the incompressible rough surface case has the largest value, therefore this case most profits from the protective effects of the residual stresses. Of course, for this case, the initial yield limit is very low and so even with this large improvement upper bound to the shakedown limit remains lowest of all the cases analysed. In all other cases the improvement in the load carrying capacity is about 35 to 50%. A practically more important parameter is 7/ , this is the ratio of load carrying capacity of the actual contact analysed and an ideal dry smooth contact. This shows how effective is lubrication in keeping the load carrying capacity close to the ideal case. For the case of rough surfaces lubricated with incompressible fluid the load carrying capacity is only 40% of that of the ideal case. Switching to a compressible fluid doubles it to 81%. This shows that with careful choice of lubricant, the detrimental effect of roughness can be substantially reduced. Compressible lubricants prove better even in case of smooth surfaces, r/ is 94% as compared to 81% for incompressible lubricant. In practice, it means that because of the high stress concentration near the surface due

264

to roughness, the load required to reach yield is drastically reduced.

5. CONCLUSIONS

Table 3, shows the effects of roughness and the endurance limit in life for the lubricated contacts. One can see that there are cases (smooth compressible) where predicted life is larger that the Hertzian value. In general, it is accepted that lubrication redistributes the pressures into a widely spread out curve, which reduces the maximum bulk shear stress and pushes it further into the material; these two aspects may increase life. Table 3 also shows that by comparing only the lubricated cases, the predicted life follows the tendency of the shakedown upper bound values, it is to say that longer lives are expected according to the following order; smooth compressible case, rough compressible, smooth incompressible and rough incompressible. The

The following conclusions can be outlined: 1. For dry contacts, shakedown limits and load- carrying capacity are rapidly reduced by friction. 2. For lubricated contacts with high pressure gradients due to roughness (incompressible fluids) the shakedown mechanism is controlled by the roughness, resulting-in a dramatic decrease in the shakedown limit (for the example shown here, 44 %). 3. In general, for lubricated contacts, rough surfaces produce lower shakedown limits and fatigue lives. 4. High pressure gradients tend to reduce the shakedown limit, therefore compressibility in the lubricant helps to sustain this limit.

A

variation of z"u hardly changes the tendency

ACKNOWLEDGEMENTS

for the two analysed values. Moving roughness involves pressures made of two components [10,11], the moving steady state and the excitation generated at the inlet by the entering undeformed roughness, both components travel with different velocities, therefore the final pressures are constantly changing phase angle in time. This, of course, is even more complex when the two surfaces are rough and both move. The amplitude of those waves depends on several factors and is still being investigated. However, it is already known that the wavelength of the roughness, the load and the relative velocity of the surfaces have a strong influence on the amplitudes. Therefore, one can expect different results for the shakedown limits and fatigue life in conditions of pure rolling and rolling plus sliding, based only on the differences of pressure and stress fields. It is difficult to say what the real tendency will be in practice, since failure is also affected by other factors such as the minimum film thickness, friction, sliding direction, etc. Some recent studies carried out along this line are e.g. N61ias et al. [14] and X. Ai [15].

The work was supported by CONACyTITESM. Contract No. 500100-5-1527PA, ITESM, Campus Quer6taro, M6xico and a grant from Gear Research Foundation of the British Gear Association, U.K. REFERENCES

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265

5. A. K a p o o r and K.L. Johnson. Effects of changes in contact geometry on shakedown of surface in rolling/sliding contact. Int. J. Mech. Sci., 34 (3), pp.223-239, 1992. 6. K.L. Johnson. Contact Mechanics, Cambridge University Press, 1985. 7. A.R. Belyakov. see Gokhfeld, D.A. and Cherniavski, O.F. (1980). Limit Analysis of Structures at Thermal Cycling, (England trans.) Sijthoff and Noordhoff Groningen. 8. A. Kapoor and J.A. Williams. Shakedown limits in sliding contacts on a surfacehardened half space, Wear, 162, pp. 197206, 1994. 9 T.A. |oannides and T. Harris. A new fatigue model for rolling bearings, ASME, J. of Trib., vol 107, pp. 367-379, 1984. 10. J.A. Greenwood and G.E. MoralesEspejel. The behaviour of transverse roughness in EHL contacts. Proc. Intstn. Mech° Engrs. Part J, J. of Eng. Trib., vol 208, pp. 121-132, 1993. 11. G.E. Morales-Espejel and A. F~lix Quifionez. Kinematics of two-sided surface features in EHL. To appear in Instn. Mech. Engrs. J. of Eng. Trib. , 1998. 12. J.A. Greenwood and K.L. Johnson. The behaviour of transverse roughness in sliding elastohydrodynamic lubricated contacts. Wear, vol 153, pp 107, 1992. 13. G. Lunderberg and A. Paimgren. Dynamic capacity of rolling bearings. Acta Polytechnica, Mechanical Engineering Series, Royal Swedish Academy of Engineering Sciences, vol 1, No. e, 7, 1947. 14. D. N~lias, M.L. Dumont, G. Dudragne, L. Flamand. Experimental and theoretical investigation on rolling contact fatigue of 52100 and M50 steels under EHL or MicroEHL conditions. Trans. ASME, J. of Trib., vol. 120, pp. 184-190, 1998. 15. Xioalan Ai. Effect of three-dimensional random surface roughness on fatigue life of a lubricated contact. Trans. ASME, J. of Trib., vol. 120, pp. 159-164, 1998.