I 

Final Report On 

(NAS8-36651) 

, -- . 

MECHANISMS OF ROLLING CONTACT SPALLING 

Prepared for 

George C. Yarshall Space Flight Center 

Alabama, 35812 

(NASA-CB-179243) HECHAIPISHS OF ROLLXlG 188-14380 - 

COBTACT sPALLIIG Final Report (Vanderbilt 

Univ.) 109 p CSCL 131 

Unclas 

63/37 Olt8llS 

A. M. Kumr, S. 3. Kulkami, V. Bhargava, 

C. T. Hahn and C. A. Rubin 

CENTER FOR MATERIALS TRIBOLOGY 

Y. 0. Box 1592, Station B 

Vanderbilt University 

Nashville, TN 37235 

. 

'\ 

May 30, 1987 
TABLE OF CONTENTS 

Page 

ABSTRACT.......................................................... iii 

1. INTRODUCTION...................................,...,,..,...... 1 

2. SUMMARY..................................,,..,................ 2 

2 

2 

2.1 Objectives............................................... 

2.2 Tasks.......................................,......,,.... 

3, CYCLIC FLOW PROPERTIES OF 440C STEEL.......................... 4 

4 

8 

13 

3.1 Introduction............................................. 

3.2 Experimental Procedures .................................. 3.3 Results.................................................. 

4. CYCLIC CRACK GROWTH ............................................ 26 

26 

26 

27 

4.1 Introduction............................................. 

4.2 Experimental Procedures .................................. 4.3 Results.................................................. 

5. ROLLING CONTACT STUDIES ....................................... 33 

33 

33 

35 

5.1 Introduction............................................. 

5.2 Experimental Procedures. ................................. 5.3 Results..............................,.,................. 

6. ELASTO-PLASTIC FINITE ELEMENT ANALYSES OF ROLLING-PLUSSLIDING....................................................... 47 

47 

47 

51 

6.1 Introduction...............................,............. 

6.2 Analytical Procedures.................................... 

6.3 Results. ................................................. 

7. FINITE ELEMENT MODEL OF FRICTIONAL HEATING UNDER ROLLING 

AND SLIDING CONTACT........................................... 61 

61 

62 

68 

7.1 Introduction............................................. 

7.2 Description of Model..................................... 

7.3 Results.................................................. 

8. DISCUSSION.................................................,.. 93 

8.1 Cyclic Flow Properties of 440C Steel..................... 

8.2 Cyclic Crack Growth...................................... 

8.3 Rolling Contact Studies..........,....................,... 

8.4 Finite Element Analyses .................................. 8.5 Finite Element Model of Frictional Heating ............... 

93 

94 

94 

96 

97 

II 

9. CONCLUSIONS................................,..............,.., 100 

102 

103 

ACKNOWLEDGEMENTS.................................................. 

REFEKENCES. ....................................................... 
ABSTRACT 

This report describes the results of a 15-month study aimed at analyzing 

the mechanical-ma terial-interactions responsible for rolling contact spalling 

of the 440C steel, high pressure oxygen turbopump (HPOTP) bearings, A coupled 

temperature-displacement finite element analysis of the effects of frictional 

heating under the contact is presented. 

heat generating, 2-dimensional indent in an elastic-perfectly-plas tic half-space 

with heat fluxes up to 8.6 lo4 KW/m2 comparable to those generated in the 

bearing. Local temperatures in excess of 1000°C are treated. The calculations 

reveal high levels of residual tension after the contact is unloaded and cools. 

Efforts to promote Mode II/Mode I11 fatigue crack growth under cyclic torsion in 

hardened 440C steel are described. Spalls produced on 440C steel by a 3 ball/rod 

rolling contact testing machine were studied with scanning microscopy. The 

classical, V-shaped spalls form with a loose flap that breaks off periodically. 

Surface initiation and subsurface initiation of the spalls is encountered in 

uncoated and copper coa ted samples, respectively. 

The contact is modelled as a stationary, 

The shapes of the cyclic, stress-strain hysteresis loops displayed by 

hardened 440C steel in cyclic torsion at room temperature are defined for the 

plastic 6 train amplitudes encountered in rolling/sliding contact. The loop shapes 

and behavior are well represented by the elastic-linear-kinematic hardeningplastic (ELKP), idealization. Finally, elasto-plastic finite element analyses of 

repeated, 2-dimensional rolling-plus-sliding contact embodyfng the ELKP-properties 

of 440C is presented. The analyses treat Hertzian pressures in the range po 

= 2.5 GPa, to po = 3.64 GPa and full slip with a friction coefficient ,, = 0.2 

in the absence of local heating. The rim distortion, cyclic plastic strains, 

residual stresses, and plastic energy dissipation are evaluated and compared 

with results for pure rolling and perfect plasticity. 

iii 
-1- 

1. INTRODUCTION 

The service life of the HPOTP (high pressure oxygen turbopump) bearings 

is limited by premature spalling of the bearing races and balls. These 

bearings operate at 28,000 rpm and nominal load of 7100 14, and are intended 

to survive for 27,000s with lubrication provided by MoS2 films and liquid 

oxygen co~lIng(~'~). 

in the range 60 RW to 130 KW(2) with inadequate lubrication and loss of 

clearances. Consistant with this, evidence of bearing surface temperatures 

reaching 500°C-1000°C have been reported(294). While the predicted life of 

the bearings is 360,000s with full film lubrication, severe spalling and 

prominent wear are reported after 1090s at full power(l). The spalls are 

largely surface initiated, proceed in both balls and races, and propagate 

inwards at relatively small angles. Subsurface damage in inner races is more 

severe than observed on the surface. 

These conditions can produce high heat generation rates 

Efforts to improve the performance of the HPOTP bearings will benefit 

from analyses of the thermal and mechanical contact stresses and the attending 

plastic deforma tion and spalling generated under operating condi tions . Such 

analyses have been intractible in the past, but are now within reach. The 

present authors have developed elasto-plas tic finite element analyses of 

repeated rolling and sliding contact embodying realistic constitutive 

 relation^(^'^^). 

rolling contact spalllng p~oblem(ll-~~). 

of a 15-month study aimed at applying these capabilities to the HPTOP bearing 

problem. 

They have also extended fracture mechanics methods to the 

This report describes the results 
-2- 

2. SUMMARY 

2.1 Objectives 

The general objective of the research was to analyze the mechanicalmaterial interactions responsible for rolling contact spalling with the aim 

of identifying ways of improving the performance of 440C steel HPOTP bearings. 

2.2 Tasks 

Task 1. Frictional Heating. Previous analyses of the frictional heating 

of the HPOTP bearing were reviewed. A coupled temperature-displacement, 

elas tic-plastic fin1 te element analysis of the effects of frictional heating 

under the contact was performed. The contact is modeled as a stationary, 

heat generating, 2-dimensional indent in an elastic-perfectly-plastic halfspace, with heat fluxes as high as 8.6 * lo4 KW/m2 which correspond with the 

heat generation by the HPOTP bearing, and a film coefficient selected to 

produce temperatures in excess of 1000°C. The calculations reveal that the 

local heating produces high levels of residual tension: 0: 350 MPa, 

1200 MPa, after the contact is unloaded and cooled. The superposition OR : 

z 

of compression by the translating contact can then produce a fluctuating 

tension and a potential Mode I crack driving force that can promote crack 

initiation and crack growth. 

Task 2. Spa11 Initiation and Cyclic Crack Growth. Efforts have been 

made to promote Mode II/Mode 111 fatigue crack growth In torsion test samples 

as a first step in evaluating the Mode I1 and Mode I11 cyclic crack growth 

resistance (i.e., da/dN vs AKII curve). 

wl th circumferential defects were subjected to cyclic torsion with progressively 

increasing stress amplitudes. In all cases, cracks emerging from the defects 

extended in directions that produce Mode I loading. This is attributed to a 

Test bars of the hardened 440C steel 
-3- 

low Mode I resistance relative to those for Mode I1 and Mode 111 growth. 

This problem requires further work. 

Spalls and wear tracks produced with a 3 ball/rod rolling contact 

testing machine were studied with scanning microscopy. 

copper coated hardened 440C steel rods displayed the classical V-shaped 

spalls. The spalls form with a loose flap that breaks off periodically. It 

appears that the spalls in the uncoated samples initiate at the running 

surface, at surface imperfections, while the spalls in the coated samples 

initiate belou the surface. It is not clear whether the copper coated samples 

possessed better quality surfaces to begin with or whether the copper coating 

rendered the surface defects less harmful. 

Both the uncoated and 

Task 3. Cyclic Flow Properties of 440C Steel. The shapes of the cyclic, 

stress-strain hysteresis loops displayed by hardened, HRC-59, 440C steel in 

cyclic torsion have been measured at room temperature and a frequency of 0.5 

tiz. The cons tan t plas tic s train arnpli tude tes ts reproduced the strain 

ampli tudes, 0.0005 < < 0.002, encountered in rolling/sliding contact. 

The loop shapes and behavior are well represented by a 3 parameter, elasticlinear-kinematic hardening-plastic (ELKP) idealization. The torsion tests 

show that the hardened 440C steel undergoes substantial cyclic hardening 

before fracturing after N : 300 to 600 cycles. The near end of life values 

of the ELKP-material parameters of 440C steel: OK = 606 MPa, and M = 188 

GPa, are similar to those obtained for other hardened bearing steels and are 

useful approximations of the properties that govern mechanical response to 

repeated rolling-sliding contact at room temperature. 

Task 4. Elasto-Plastic Finite Element Analyses of Rolling-Plus-Sliding 

Contact. Elasto-plastic finite element anlayses of repeated, 2-dimensional 
-4- 

rolling-plus-sliding contact have been performed for the ELKP-properties of 

440C steel evaluated by Task 3. The calculations treat rolling with full 

slip, Hertzian pressures in the range po = 2.5 GPA to po = 3.64 GPA, a 

tangential-to-normal-force-ratio T/N = 0.2 and a coefficient of friction 

P = 0.2. The rim distortion, the cyclic plastic strains, the residual 

s tresses, and the plastically dissipated energies are evaluated. The results 

show that the contacts produce peak cyclic strains and residual stresses in 

the range: 0.00025 I ~3)/2 s 0.001, and -20 MPa 5 aR s -35 MPa that are 

similar to the values calculated for pure rolling but much smaller than the 

strains and stresses calculated for rolling-plus-sliding with elastic-perfectlyplastic material behavior. The analysis locates the peak plastic strain 

activity at a relative depth of y/w = 0.65 (2w is the contact width) where 

spa11 initiation has been observed. In the cases treated, energy dissipated 

as heat by plastic deformation is from 3% to 15% of the energy dissipated at 

the surface by friction. 

3. CYCLIC FLOW PROPERTIES OF 440C STEEL 

3.1 Introduction 

There Is evidence from many sources that rolling contact failure -- the 

formation of spalls and shells -- is closely connected with cyclic plastic 

def~rmation(~). 

of the Hertzian relative peak contact pressure beyond which repeated contacts 

Such failures begin to occur close to the "shakedown" value 

produce continuing cyclic plasticity*. The material' 8 resistance to cyclic 

*The shakedown value of the relative peak pressure, po/k (where k is the 

shear yeild strength) depends on the shape of the contact ellipse and the 

normal-force-to-shear-traction-ratio P/N. For pure rolling, the shakedown 

values are po/k = 4 and po/k = 4.7 €or line and circular contacts respectively. For rolling with full sliding with a friction coefficient 9 = 0.2 

= T/N, po/k = 3.2 for line contacts, or po 5 1.9 GPa for a bearing steel 

(k : 600 MPa). 
-5- 

plasticity is ther f re on of vera1 ritical material pr rties that 

determine the rim's contact pressure capacity and life*. 

The resistance to cyclic flow is determined by both the value Of k and 

the characteristics of the material's stress-strain hysteresis 

Earlier analyses of the cyclic deformation of the rim were performed for the 

isotropic, elastic-perfectly-plastic (EPP) material beha~iOr(~9~) illustrated 

in Figure 1 (A). In this case, material subjected to repeated Contacts above 

shakedown experiences the non-fully-reversed strain cycle shown schema tically 

in Figure 1(B). 

have measured the hysteresis loops of 42100 bearing steel and a 1073 rail 

steel for the range of plastic strain amplitudes encountered in rolling 

contact above shakedown. They find that the cyclic behavior of these steels 

is anything but elas tic-perfectly-plastic but tends to be kinematic(lO). 

They employ the bilinear, 3 parameter, elastic-linear-kinema tic hardening 

plastic (ELKP) representation of the hysteresis loop shown in Figure 2. 

Finite element calculations reveal that the ELKP material experiences fully 

reversed strain cycles with a mean stress above shakedown [see Figure l(C)] 

with much smaller strain amplitudes. 

More recently, Bhargava, Hahn and Rubin and their co~orkers(6,~) 

The characteristics of the hysteresis loop of hardened 440C steel have 

not previously been measured. 

relevant to rolling contact at room temperature are described in this section. 

The measurements define the ELKP-parameters of 440C steel. Elasto-plastic 

finite element analyses of rolling and sliding, that employ these parameters, 

are described in Section 6. 

Such measurements for the strain amplitudes 

*The other properties that are important are the features of second phase 

particles that act as local stress raisers, the stability of the microstructure, and the resistance to crack initiation 
- 6- 

b 
0 

C 0 

L 

-7- 

0.02% 

0.02% 

Figure 2. 

(ELKP) hysteresis loop, indicating the terminology. 

Schema tic representation of an elastic-linear-kinema tic-plas tic 
. .- ~. . -8- 

3.2 Experimental Procedures 

Cyclic torsion tests have been conducted on type 440C steel which is a 

hardenable grade of martensitic stainless steel containing about 1% C and 

18% Cr. 

After machining, the specimens were given the following heat treatment: 

a. Austenitize for 1 hour at 1930OF. 

b. Quench in liquid nitrogen. 

c. Temper at 325OF for 1 hour. 

d. Air cool at 70°F. 

e. Cold soak in liquid nitrogen for 30 min. 

f. Temper at 325OF for 1 hour. 

This produced a hardness of HRC-59. Typical microstructure of the steel 

after heat treatment is shown in Figures 3 and 4. The monotonic, equivalent 

stress-equivalent strain curve derived from a torsion test is shown in Figure 

5. The monotonic yield stress values of the material are: 

Offset, % Equivalent Tensile Yield Stress, MPA (ksi) 

0.02 1200 (174) 

0.2 1760 (255) 

Torsion Testing. Cyclic torsion tests were carried out on hollow, 

cylindrical specimens. The specimen dimensions, as shown in Figures 6(a) and 

6(b), are not altered by the cyclic deformation. The tests were carried out at 

room temperature with a frequency f = 0.5 Hz on an axial-torsion, servohydraulic 

testing machine. 

controlled and manually adjusted to maintain the desired constant 

plastic strain amplitudes. It was frequently difficult to control strain 

amplitude accurately during the first N = 25 cycles because of the rapid cyclic 

hardening, but it presented no problems during the subsequent cyclic life. 

The stress ampli tude and mean stress were automa tically 
-9- 

DBGINAL PAGE IS 

WR QUALITY * 

. .' 

Figure 3. Microstruchre of 440C Steel after heat-treatment. 
-10- L ,& 

figure 4. Microstructure of 440C steel showing some inClUSiOns. 
202: 

44,2 

-1 1- 

v 

I I I 

c 8.k- 0.S I 1 .@ 1.5 2.8 L 

STRAIN ( sg 1 

Figure 5. Monotonic equivalent stress-strain curve for 440C showing 0.027; 

offset. The curve is derived from a torsion test. 

.U)K1GINAL PAGE IS 

'm DR QUALITY 
7- -- -12- .b9 ? 'r 

ORIGINAL PAGE IS 

DE POOR QUALITY 

Figure 6. 

and after the test. 

(a) Torsion test specimen dimensions. (b) Specimen geometry before 
-13- 

The first cycle was recorded with manual load control at about f= 0.1 Hz to 

define the elastic loading (and unloading) line. Subsequent cycles were 

Derformed at f= 0.5 Hz, and the torque-rotation loop was recorded with a data 

acquisition computer. The computer first installs the elastic shear stressstrain lines. It then converts 400 separate sets of torque and rotation values 

in each cycle to shear stress-strain values consistent with an elastic slope 

of G = 79.6 GPa; and plots the loop (see Figure 7). The system then determines 

the 0.035% shear strain offset (0.02% equivalent strain offset), shear yield 

strength (horizontal marks) and other 2 ELKP parameters described in Figure 2, 

and their tensile equivalents : 

(i) kK and CK = 43k~, the kinematic shear and tension yield strength, and 

M, and M = 3 M,, the plastic shear and tension modulus (linear (ii) 

strain hardening coefficients). 

Two alternative ways of defininp the parameters: uK(l), ?l1 and UK(2), M2, 

illustrated in Figure 2, were examined. In addition the system automatically 

computes U, the loop area, which represents the plastic work per unit volume 

and corresponds closely with heat evolution. The values of U are compared with 

U' = Zc?%,the area of the ELKP-loop. 

3.3 Results 

Examples of the hysteresis loops displayed by the 440C steel samples 

are reproduced in Figure 8. After the first few straincycles the departures 

from linear elastic behavior are observed at very modest stresses. The 

* A more accurate analysis of the torque-rotation values that accounts for the 

plastic contribution of the fillet regions was devised after this work was 

completed. This analysis leads to-50% larger plastic strain range values 

and~20% lower stress values and--% lower M-values than reported here. The 

revised results will be presented in a paper currently in preparation. 
-14- 

Figure 7. 

real 440C s tress-s train loop indicating the terminology. 

An elas tic-linear-kinema tic-hardening-plas tic represents tion of a 
-15- 

s 

4 

P + 

tn 

I 

4 

W 

I 

tn 

a 

XQJ 

DDC 

nK?i 
-16- 

tt 

C 

0000000000 0000000 00 

~NOCDLO-cm I I 

Ln 

N 

-K 

c *- 

ic 

KC 

wxc 

L c- w3 
ensuing non-linear plastic deformation is accompanied by rapid hardening. The 

departure from elastic deformation occurs at negative stresses for the ~t? = 

0.002 strain range and for the LZP = 0.001 strain range sample with a 200 MPa 

mean stress; i.e., plastic compression is obtained even before the stress 

reversal while the stress is tensile [see Figure 8(b)]. The nonreversibility 

of the strain cycle produced by the 200 MPa mean stress rapidly decays (see 

Figure 9). All of these features are symptomatic of the kinematic hardening 

behavior. Figure 10 illustrates that the loop areas defined by the 2 ELKPidealizations agree very well with the actual loop areas for small values of 

strain range. The c~(2), M2 characterization appears to be slightly better 

at the highest strain. 

The variation of the loop parameters OK, oc, oa and M are shown in 

Figures 11-13. The samples displayed noticeable cyclic hardening throughout 

their relatively modest lives: 300 cycles 5 N 5 600 cycles. Both cC, CK and 

ca increase with N, while the plastic modulus remains more or less constant. 

Consequently, the steady state is not well defined by these measurements. 

The near end of life values summarized in Table 1 come closest to describing 

the ELKP properties for even larger numbers of contacts. The variation of 

the cyclic life with strain amplitude is presented logarithmically in Figure 

14 and is approximated by the following Coffin-Manson type expression 

where A = 0.52 and n = -0.853. Figure 15 Illustrates that the ELKP 

parameters OK and M are relatively insensitive to variations of strain 

amplitude and mean stress, while the conventional yield parameter measured 

from zero stress, cc, is a sensitive function of these variables. 
’-‘ 

:- ., 

- 

P 

t47 

- I .t- 

-2.3- 

\ 

\T 

i 

\ 

\ 

\ 

\ ”\ 

? 

\ *-I 

I 

Figure 9. 

(with an imposed mean stress of 200 MPa in shear). 

The exponential decay of non-reversibility with number of cycles 
-19- 

c7’ 

N’ 

3 

? 

\ 

.I 

cc) 

aJ 

L 

4 

0 

0 

2 

n 

0, 

Ln I I I 

I 

d 

2’ /= 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

r Actual Area 

o Theoretical Area based on % (1) 

Theoretical Area based on Ok (2) 

1 

.y 

/O 

- 

/ 

2 .o 2 

Figure 10. Comparison of the actual area of the hysteresis loops with the 

areas calculated based on Uk(1) and ,k(2)9 as a function of plastic strain 

range. 
-20- 

L 

W 

523 

i 

y r 0 

9 0 

x 

\ 

1, 

\ 

I I I I 

- 5m0 1 2 3 4 5 

NUMEER OF CYCLES. LOG( N> 

Figure 11 

Oa, with number of cycles. 

Variation of cyclic yield stress, uoc (0.02%) and stress amplitude, 
-21- 

Acp = 0.001, 7m = 200 MPa 1 

I 

Figure 12. Variation of kinematic yield parameter, Qkl with number of cycles. 
-22- 

ORIGINAL PAGE IS 

OF POOR QUAUCY 

M1 

0 ACp 0.0005, Tm O 

I- 

! X ACp 0 0.0005, Tm = O 

i 1 0CL 1 

Figure 13. 

number of cycles. 

Variation of the plastic modulus parameters, M1 and M2, with 
-23- 

OI 

m 

m 

4 

c? 

9 

2 

0 

U 

c. 

4 

N 

cc! 

0 

4 

0 

c? 

4 

4 

U 

N 

m 

0 

0 

m 

a3 

0 

9 

0 

m 

0 

0 

0 

0 

yc n 

N 

c 

r4 

O! 

N 

h 

w 

ir, 

0 

0 

hl 

N 

h 

U 

U 

02 

U 

U 

I 

0 

m 

N 

In 

0 

m 

0 

N 

0 

C 

9 

B 
-24- 

c 

- L. E.?; \ I I 1 

\ 

\ 

\ 

\ 

\ D 

\ 

\ 

\ 

\- 

\I 

Figure 14. Variation of plastic strain range with number of cycles to failure. 
-25- 

I I 1 1 

- '%. 0 0.8 1.4 2.4 3.2 4.0 

1051 

352 

- 35c 

0 

0 

c 

A 

+ 

Y 

0 

4 

F: 

N = 10 

OC 

OC (1, = 200 MPa) 

N = 250 

OC 

Ok 

M1 

M2 

Figure 15. Variation of the cyclic parameters with plastic strain range, 
4. CYCLIC CRACK GROWTH 

4.1 Introduction 

There is evidence that cyclic crack growth proceeds more slowly in a rim 

subjected to rolling contact than in other structures.*(ll) As a result, the 

largest part of the life of the rim can reside in the crack growth stage. 

For example, Shao et a1.(12) observed crack initiation in a case hardened rim 

after N = 1.4 lo6 contacts. 

proceeded in the interval 1.4 lo6 contacts s N s 3.9 106 contacts, and 

occupied 64% of the life. In order to better understand crack growth under 

the contact, efforts are being made to apply fracture mechanics to this 

problem. This involves: 

In this case, crack growth (and spa11 formation) 

(i) Calculation of the Mode I1 and Mode 111 cyclic crack growth driving 

force, LKII and LKIII(13-15), 

(ii) Measurement of the Mode 11 and Mode 111 cyclic crack growth 

resistance; the da/dN-AIiII and da/dN-AKIII curves for the material. 

In the absence of these properties, the Mode I, da/dN-AKI curve 

has been employed as a first approximation(ll). 

The measurements described in this section were undertaken in an effort 

to evaluate the da/dN-AKII and -AKIII curves for 440C steel at room temperature. 

4.2 Experimental Procedure 

Crack growth tests were performed with cyclic torsion loading and the 

same hollow torsion samples described in Section 3 (see Figure 6). 

authors have previously produced stable crack growth and fatigue striations 

The 

*The compressive nature of the loading and friction at the crack faces 

restrict the crack face sliding that is reponsible for crack growth. 
-27- 

in HRC 4140 steel samples of this design subject to cyclic loading (see 

Figure 16). In order to reduce the stress and time for initiation, first a 

single microhardness impression and then a cluster of impressions were placed 

on the surface to act as stress raisers. These specimens were subjected to 

large numbers of stress cycles with progressively Increasing amplitudes and 

eventually fractured, but without signs of stable cyclic crack growth. 

provide a more potent stress raiser, the sample was fitted with a circumferential, 

3 mm-long, 0.5 mm deep crack-like slit introduced by spark machining. The 

specimen was subjected to cyclic torsion with progressively increasing shear 

stress amplitudes (see Table 2). 

To 

4.3 Results 

The specimen fractured suddenly with no obvious signs of cyclic crack 

growth at an applied torque amplitude of 0.24 kN-m (2.12 kip-in) and shear 

stress amplitude of A712 = 292 MPa. 

to the specimen axis (see Figure 17). Apparently, the crack preferred to 

grow in Mode I rather than in Mode 11 or Mode 111. Fractographs in Figures 

18-21, show signs of cyclic growth in Mode I. 

Crack extension proceeded at about 45O 

Figure 18 shows the edge of the notch from which the fatigue crack 

started at 45O to the plane of the notch, The regions A, B and C identified 

in Figure 18 are magnified and shown in Figures 19, 20 and 21 respectively. 

Regions A and B show the typical fatigue markings which gradually change to a 

ductile mode away from the edge of the notch, as shown by the region C. 
-28- 

4J 

m 

u 

U 

U 

H 

\ 

U 

W 
-29- 

TABLE 2 

CYCLIC TORSION TEST DATA ON 440C STEEL 

HOLLOW CYLINDRICAL TORSION SPECIMEN 

Applied Torque Shear Stress Estimated Applied Number of Cycles 

Ampli tude, T/2 Ampli tude, AT /2 Stress In tens i ty 

(kN-m) ( MPa ) ( MPa dm) 

0.06 

0.09 

0.12 

0.15 

0.18 

0.21 

0.24 

73 

110 

146 

183 

2 19 

25 6 

292 

10 

15 

20 

25 

30 

35 

40 

25,000 

25,000 

10,000 

5,000 

5,000 

5,000 

1,526 
-30- 

Figure 17. 

specimen fractured. 

are presented in Figures 18 through 21. 

Schematic view of prenotched torsion specimen just after the 

Fractographs of regions A, B, C and D Identifled here 
-31- 

'I 

I! 

I 

.. 

ORIGINAL PAGE IS 

QE EOOR QUALITY 
e, 

L 

‘3 

W 

co 

d 

b3 

9) 

d 

(w 

0 

3 

e, 

d 

> 

Q 

9) 

4 

u 

d 

c 

CD 

e, 

L 

al 

C 

0 

.rl 

M 

al 

d 

(w 

0 

W 
~ 

-33- 

5. ROLLING CONTACT STUDIES 

5.1 Introduction 

Spalls are initiated in plastically deforming portions of the rim, 

usually with the involvement of second phase particles or inclusions. This 

occurs within a few micrometers of the running surface in the presence of 

surface tractions(16) (rolling-plus-sliding) or at much larger depths, e.g., 

0.6~ corresponding with peak cyclic strain activity, for well lubricated 

r~lling(~~*~~). 

the running surface in the early life(12,16) or parallel to the surface 

(shelling) in the presence of high circumferential residual stresses that 

retard radial crack growth(17). 

in 440C HTOP bearings under laboratory conditions with large axial loads 

initiate within about 2 pm of the running surface with inclinations of from 

The as-initiated cracks can be inclined from 20° to 300 to 

The cracks observed by Bhat and Dolan(l,2) 

loo to 25O. They are similar in appearance to the near surface cracks 

observed by Soda and Yamamoto in 1045 steel cylinders transmitting shear 

tractions(16). 

the surface contribute to the spalling obtained by Bhat and Dolan. In 

addition, high surface tempera tures, thermal s tresses and attending stress 

fluctuations may also be important in the HPOTP environment. 

It appears that poor lubrication and plastic deformation of 

This section describes direct observations of damaged 440C steel rims. 

The work was undertaken to learn more about the mechanisms of spalling of 

440C steel under different conditions. 

5.2 Experimental Procedure 

Spalls produced by R. Thom of the NASA-MSFC on cylindrical 440C specimens 

in the 3-ball-rod rolling contact fatigue tester have been examined in the 

scanning electron microscope. The specimen geometry, and the design and 
-34- 

opera tion of this rolling machine have been described elsewhere(47). 

of spalls were examined. 

Two sets 

The first set was observed on a 440C test cylinder subjected to lubricated 

rolling contacts at a peak Hertzian pressure of 4.04 GPa (586 ksi), using 

roughened balls. 

and lo7 cycles. 

with copper to a thickness of about 0.5 pm. This specimen was tested at two 

contact stress levels: 4.04 GPa and 5.42 GPa. While no failures were observed 

for the tests carried out at a stress level of 4.04 GPa, the life varied between 

39~10~ and 65~10~ for the tests at 5.42 GPa. 

coating improved the life significantly. In addition, observations were made 

of wear tracks on larger 440C cylinders subject to rolling-plus-sliding 

deform tion experiments produced with the Vanderbilt rolling contact machine. 

The rolling machine is of the cantilever type and is designed to produce 

rolling, sliding and combinations of rolling and sliding under different 

contact loads. Both line contact (2-D) and point contact (3-D) can be 

simulated depending on the geometries of the test specimen and the contacting 

roller. The design and the operation of the machine are described in reference 

48. The contacting roller is made of heat treated 52100 bearing steel (HRC 

62.5) and possesses a double curvature radii of 51 mm and 18.26 mm, such that it 

produces a circular contact patch when in contact with a cylindrical radius of 

57 mm. The 440C test cylinders were 57 mm in diameter and 25 mm wide. A 

contact force of 2344 N was applied during the test, which converts to a 

Hertzian pressure of 2600 MPa and a relative peak pressure: po/kk of 4.2. 

tests were conducted at this contact pressure of 2600 MPa and a relative 

slip of 2Z. Total number of contacts for the two tests was 15,500, each 

under fully lubricated and unlubricated conditions, respectively. Surface 

The number of contacts to failure varied between lo6 cycles 

The second set was observed on a test cylinder ion plated 

Thus, the ion plated copper 

Two 
-35- 

damage after the unlubricated test was very severe, while there was very 

little damage after the lubricated test. A photomicrograph of the specimen 

after the unlubricated test is shown in Figure 22(A). A magnified veiw of 

the wear track Is shown in Figure 22(B), which shows microcracks on the 

surface that may eventually lead to a spall. 

5.2 Results 

Both the uncoated and copper coated 440C samples displayed classical, Vshaped spalls that probably originate near the surface close to the V-apex 

(see Figures 23-29). The dimensions of the spalls are typically: 0.3 mm - 

0.6 mm in the rolling direction and 0.6 mm transverse to the rolling direction, 

with the latter corresponding to 2U, the diameter of the elliptical contact* 

and wear track. The depths of the spall craters were not determined but 

appear to be a small fraction of the other dimensions. The spalls display 

the following features, which are shown schematically In Figure 23. 

(i) The V-apex [see Figure 23(A)] points in the rolling 

( forward ) di re c t ion. 

(ii) The cracks grow laterally and circumferentially on a surface 

making a small angle to the running surface. Thus the cracks 

grow deeper as it extends in the direction opposite to the 

rolling direction. 

(iii) At an early stage, the crack breaks through to the surface 

at the shallow end, producing a cantilevered flap supported 

*The following are the dimensions of the Hertzian contact ellipse produced 

in the 3 balllrod tester, where 2U and 2w are the diameters of the ellipse 

in the transverse and rolling direction respectively: 

Po , GPa 2u,m 2w,mm 

4.04 0.6 0.34 

5.42 0.8 0.46 
-36- 

ORIGINAL PAGE IS 

OE POOR QUALITY 
-37- 

SURFACE 

INITIATED Z-Z 

SUBSURFACE 

IN/ TIA TED 2-2 

Figure 23. 

surface of the wear track; (B) longitudinal section 2-2 of the wear track 

illustrating a surface initiated spall; and (C) a similar section illustrating 

a subsurface initiated spall. A cantilevered flap on the "deeper" side is 

identified as region 1 in (B) and (C). 

Schematic diagram of a V-shaped spall: (A) V-shaped apex on the 
-38- 

by unbroken material on the "deeper" side. Such a flap will exist 

from the very beginning for cracks initiating at the surface [see 

Figure 24(A)). 

(iv) Portions of the flap behind the crack front break off periodically 

producing wear fragments. 

unsupported flap forms and breaks off behind the growing crack. 

Additional fragments are produced as more 

The loose flaps or flap stubs at the deeper end of the spall are evident 

in virtually all of the spalls of the uncoated and coated specimens (see Figures 

24(A), 24(B), 25(A), 28 and 29). Two differences are noticeable. The spall 

surfaces of the uncoated samples are flat and continuous (though not necessarily 

smooth) at the V-apex where they reach the surface. Other surface imperfections 

are frequently visible near this location. In contrast the spalls of the 

coated samples are discontinuous close to the apex, revealing undercutting, 

i.e., a flap stub. In these cases, surface defects were not in evidence near 

the spall apex. The implication is that the spalls in the uncoated samples 

In1 tia te at the running surface, at surface imperfections while the spalls in 

the coated samples initiate below the surface and tend to grow parallel to 

the rim as they approach the running surface. 

Scanning electron micrographs of the running surface produced by lubricated 

and unlubricated rolling and sliding are reproduced in Figures 30 and 31. The 

surface of the specimens was artifically damaged by means of Rockwell C hardness 

impressions. Figure 30 shows a hardness impression after the lubricated test. 

In the case of the unlubricated test (N = 15,500), the surface damage was very 

severe and the "ploughing" was deep enough to completely erase the hardness 

impressions. As shown in Figure 31, the wear track on the specimen after the 

unlubrica ted test show early signs of "V-shaped" features which could eventually 

lead to the formation of flakes. 
L 

Y 

3 

! 

.. 

r 

n 

W m 

ORIGINAL PAGE IS 

CUE EUQR QUALITY 
-40- 

ORIGINAL PAGE IS 

QE EWR QUALIm 

n 

3 

n 

s 

C 

0 

V 

9 

0 
-41- 

n 

W m 

.. 

C 

03 

a 

r 

n 

rn 

W 

bD 

0 

d 
ORIGTNATJ PAGE TS 

QE EOOR QUALITY 

n 

u a0 

1 

n 

s 

v) 

n 

U 
. -43- 

-... . .. 

i’ 

, 

n 

P 

v 
C 

0 

4 

0 

d 

ORJGINAL PAGE IS 

DJ3 EooR QUALITY 

n 

V 

U 

n 

0 

v 

C 

0 

a2 0 Q, 

a +a 

00 M 
-45- 

E 

0 

Figure 30. 

impression (N = 15,000). 

Wear track after the lubricated test showing the hardness 
U 

mc 

a0 
-47- 

6. ELASTO-PLASTIC FINITE ELEYENT ANALYSES OF ROLLING-PLUS-SLIDING 

6.1 Introduction 

In the past, elasto-plastic analyses of repeated rolling and rollingplus-sliding contact have been confined to isotropic, elastic-perfectlyplas tic (EPP) material behavior. Furthermore, the analyses were either 

limited to shakedown or not reliable.* As a result, such features of the 

Contact as the magnitude of the cyclic strain amplitudes and residual stresses 

and their variation with depth remain lll-defined. Recently, Bhargava, Hahn 

and K~bin(l~,~~) have devised a finite element model of repeated rolling 

contact that is free of arbitrary assumptions. The model can treat EPP, 

isotropic hardening and elastic-linear-kinemtic-hardening-plastic (ELKP) 

9 7). It can simulate pure r~lling(~,~ 9 19) as well as rolling-plussllding(6,20) and 2-~(6,7,19*20) and 3-D contacts(*l). The results of such 

calculations reveal that the strain amplitudes and residual stresses are 

markedly affected when ELKP-behavior is substituted for perfect 

Finite element analyses of repeated, 2-D pure rolling contact have been 

carried out for high strength bearing steel with ELKP-material behavior and 

This section describes initial efforts to extend the ELKP 

calcula tions to repeated rolling-plus-sliding with materials parameters 

appropriate for the 440C steel. 

6.2 Analytical Procedure 

The finite element mesh employed is shown In Figure 32. It consists of 

288, 8aoded, isoparametric plane strain elements and is roughly 16 w long 

*The classical analyses of Merwin and Johnson(8) and Johnson and Jef ferls(9) 

do not enforce equilibrium or continuity directly and require the arbitrary 

and over-constraining assumption that the total strain (elastic plus 

plastic) is equal to the Hertzian elastic value. 
-4 8- 

Figure 32. Deformed finite element mesh showing displacements under a normal 

Hertzian pressure po = 2500 MPa and the corresponding traction with a 

coefficient of friction v ~0.2. Displacements are magnified 24x. 
-49- 

and 6 w high (where 2w is the contact width). This mesh is made to mimic 

a semi-inf ini te half space by appropriately displacing the boundaries for 

each loading condi tion. Repeated rolling-sliding is simula ted by translating 

a Hertzian pressure distribution and the appropriate surface tractions across 

the top of the mesh. A steady state is obtained after only two translations, 

hence only two translations need to be made. 

the adequacy of the mesh have been verified previo~sly(1~~~~). 

are performed using the multipurpose finite element package "ABAQUS" and 

consume, typically, 2.10 cpu hrs on a VAX 8800. 

This and other features including 

The calculations 

The calculations employed ELKP behavior and material properties for 440C 

steel derived from the measurements in Section 3. Effects of surface heating 

on the ELKP parameters were neglected. Three rolling conditions were examined: 

Case 1. Rolling-with-(ful1)sliding under a peak Hertzian pressure po = 2.5 GPa 

and a coefficient of friction U= 0.2. 

Same as Case 1 with the value of UK reduced by 8% to reflect 

softening due to rim heating . 

Same as Case 1 with an elevated peak Hertzian pressure of po = 

3.64 GPa and a coefficient of friction u= 0.2. 

Case 2. 

Case 3. 

The material parameters are used in the analysis and the rolling conditions 

simulated by the model for each of the three cases are summarized in Tables 3 

and 4. 
-50- 

TABLE 3 

MATERIAL PARAMETERS USED IN FINITE ELEMENT ANALYSIS 

Young's Modulus, E (MPa) 

Poi s son ' s Ra t io , 

Plastic Modulus, M (GPa) 

Kinematic Yield Stress, Uk (MPa) 

Kinema tic Shear Yield Strength, kk (MPa) 

Case 1 Case 2 Case 3 

2.07 105 2.07 105 2.07 105 

0.3 0.3 0.3 

188 188 188 

1050 962.3 1050 

606.2 555.6 606.2 

TABLE 4 

ROLLING CONDITIONS 

Radius of Half Space, R1 

Radius of Rigid Contact 

Cylinder, R2 (m) 

Hertzian Peak Pressure, po (GPa) 

Hertzian Contact Width, 2w (mm) 

Ratio of Tangential to Normal 

Force, T/N 

Coefficient of Friction, p 

Ratio of Hertzian Peak Pressure 

' to Kinematic Shear Yield 

Strength, Po /kk 

0.05 0.05 0.05 

2.500 2.500 3.64 

2.2 2.2 3.18 

0.2 0.2 0.2 

0.2 0.2 0.2 

4.124 4 .s 6.0 
-51- 

6.3 Results 

The appearance of the mesh after the first and second contact at Polkk 

= 6 is illustrated in Figure 33. Note that the displacements are magnified 

820X to make them readily visible. At this magnification a substantial amount , 

of deformation is visible after the first contact but there are virtually no 

further changes in the displacements after the second contact. The implication 

Is that the plasticity is not reversed during the first contact, but is fully 

reversed during the second and subsequent contacts as a result of kinematic 

hardening. This is confirmed by the stress-s train histories experienced by 

the material as shown in Figure 34. 

The magnitude of the continuing cyclic plasticity experienced during the 

second and subsequent contacts is illustrated In Figures 35-37. These show 

the variations with depth below the surface of AcP, the radial plastic 

strain range, and ATP the equivalent strain range, which peak at a relative 

depth of y/w .. 0.65. The magnitudes of the peak equivalent strain range 

0.0005 s &‘(max) s 0.002 for 4.5 < Polkk < 6, correspond with the values of 

A@ for which the ELKP properties were evaluated in Section 3. 

Y 

The circumferential and axial residual stresses are described in Figures 

38 and 39. These are related to the net amounts of plastic deformation 

produced during the first contact. The peak compressive values, 20 MPa 5 

o max s 35 MPa are quite modest. The small circumferential tensile residual 

stress values obtained close to the surface at the highest contact pressura; 

are not observed in the absence of shear tractions. They are interesting 

because they produce a fluctuating tensile stress, and Mode I driving force 

for radial cracks, but appear to be too small: ax(max) = 17 MPa to enforce 

cyclic crack growth. 

35, 36, 38 and 39) with the same ELKP properties show that the shear tractions 

R 

R 

Comparisons with results for pure rolling (see Figures 
-52- 

(a> After I Contact 

(b) After I1 Contacts 

Figure 33. Deformed mesh for Case 3: po/kk = 6.0, after the load is removed. 

Displacements are magnified 820X. 
-53- 

.*. _. . . -. 

e.?- 

Figure 34. The stress-strain history produced at a depth of y = 0.4~ for 

the first and second contacts, derived from the model for 

3 different relative pressures: (a) po/kk = 4.124, (b) po/kk = 

4.5, and (c) po/kk = 6.0. 
u 

‘ e- 

.’ - 

- ,_ 

- p,;/ 

,. . 

,,’ 

I 

Figure 35. Variation of the radial plastic rtrain range, A$, with 

normalized depth. 
-55- 

ORIGINAL PAGE l.3 

OF POOR QUAl;rc?L 

SECOK3 CONTACT 

I I I I 1 

-'.&:e e. 5 1 .e 

Half equivalent plastic rtrain range X 103 

Figure 36. 

with normalized depth. 

Variation of the half equivalent plastic strain range, A3/2, 
-56- 

Figure 37. Variation of peak values of A3 and A 5 p with Polkk. 
-57- 

-e. 

-e. 

h 

W 

0 

Figure 38. 

normalized depth. 

Variation of the normalized circumferential residual stress with 

ORIGINAL PAGE IS 

OE FQOR QUALITY 
-58- 

-e. 

r. 

-.-. 

SCCG'C CD'iTACT 

-- -, - - .' 

Figure 39. 

depth. 

Variation of the normalized axial residual stress with normalized 
-59- 

have hardly any effect on either the plastic strain ranges or the residual 

stresses. This is not found to be the case for EPP-behavior which leads to 

much larger strains and stresses in the presence of shear tractions.(20) 

Table 5 Illustrates the variation of Ui, the plastically dissipated 

energy per unit translation of a unit length of line contact, with normalized 

depth. 

dissipated in the plastic zone during contact, most of which is converted to 

heat. These values are from 3% to 15% of the energy dissipation of friction*, 

Uf, for S = 2% slip. While the adiabatic temperature rise produced by the 

plastic deformation is significant, e.g., 0.6OC per contact, It is a small 

fraction of the rise produced by friction at the running surface. The heat 

generation rate U = 41.9 KJ/m2 corresponds to U = 51.9 KW.** 

The averages of these values, Up , are the total plastic energy 

*The energy dissipation of friction Is Uf = O.~~wpos where s is the slip. 

**For an average (inner/outer) race diameter of 65 mm and 13 balls rotating 

at 28,000 rpm with an effective contact ellipse dimension v = 1 mm. 
TABLE 5 

VARIATIOE; OF THE PLASTIC ENERGY DENSITY IN 367 pm 

LAYERS OF THE FINITE ELEMENT MODEL WITH RELATIVE DEPTH 

Plastic Energy, Ui (MJ/m 3 1, Density Relative Depth, y/w 

Case 1 Case 2 Case 3 

0.167 

0.5 

0.833 

1.166 

1.667 

0.227 0.320 1.450 

0.602 0.874 3.283 

0.434 0.657 3.077 

0.217 0.326 2.185 

0.039 

0.039 

0.084 

0.084 

0.723 

0.723 

2.334 0 0 0.168 

0.168 

2.867 0 0 0.009 

0.009 

3.0 

4 .O 

5.33 

0 

0 

0 

1.56 3.87 11.8 

0.476 1.18 5.41 P 

T ,oC/contac t 0.08 0.18 0.60 

17.3 17.3 36.5 

Ui ,MJ/m 3 

U ,KJ/m 2 

2 Uf ,KJ/m 

U,KJ/m2 17.8 18.5 41.9 

Ui - Plastic energy density in a layer 

Up - Plastic energy dissipated per unit of contact translation per unit 

thickness: Up = yo Z Ut/", where yo is the total depth of plastic 

zone and n is the number of layers composing it. 

Uf - Energy dissipated by friction at the surface per unit of contact 

translation per unit thickness: Uf = 0.5 pwpos, where 2w is the contact 

width, po is the peak pressure, u = 0.2 is the coefficient of friction 

and s = 0.02 is the slip. 

thickness: U = Up + Uf. 

Uj- Total energy dissipated as heat per unit contact translation per unit 

7 - Adiabatic temperature rise plastic zone arising from Plastic 
-61- 

7. FINITE ELEMENT MODEL OF FRICTLOliAL HEATING UNDER 

ROLLING AND SLIDING COXTACT 

7.1 Introduction 

Poorly lubricated bearings exposed to rolling and sliding contact under 

load may also be subject to high temperatures and thermal stresses arising 

from the frictional heating. The bulk of the literature in this area deals 

with theoretical thermo-elastic treatrnents[22-351. Mow and Chen[36] obtained 

an anlaytical solution to the stresses induced by an arbitrarily distributed, 

fast moving heat source on a convective elastic-half plane. There are a few 

elastic and elastic-plastic finite element analyses, but they are concerned 

with the effects of thermal and thermal-mechanical loading for a rail wheel in 

regions far removed from the contact[ 37-401. Theoretical and experimental 

work has been done to study the reduction in bearing service life due to 

localized overheating observed for HP3TP bearings used in the Space Shuttle 

Main Engine[42$431. A more recent, elastic-plastic, thermomechanical model 

of tk region near a shallow weld in a thick plate developed by Ne~man[~~] 

reflects the growing sophistication of the analyses. The main limitation of 

these models in dealing with the non-stationary temperature field is that 

only the effect of thermal loading is considered. Hence, there is a need for 

- 

coupled tempera ture-displacement analysis of rolling and sliding contact in 

order to deal with the problem more accurately. This section describes 

preliminary steps towards a coupled tempera ture-displacement finite element 

analysis of rolling and sliding. The intent of the study was to determine 

whether the thermal stresses are large enough to affect the local stress 

state produced by elastic plastic contact. 

that radically alter the residual stress state for large heat fluxes. 

The results show large effects 
-62- 

7.2 Description of the Model 

A finite sized mesh with appropriate boundary conditions has been used 

to perform a coupled temperature displacement elastic-plastic finite element 

analysis of a long, stationary, heat-generating Hertzian cylinder indenting a 

semi-infinite half plane. All the heat generated In the cylinder is assumed 

to be conducted into the semi-infinite half plane. The model approximates 

the steady state temperature distribution produced by a rapidly rotating heat 

source, and the stresses and deformations directly under the contact. However, 

it sheds no light on the cyclic temperature, thermal and applied stress 

fluctuations produced by a moving contact. A 3-dimensional model of contact 

that can treat aspects of the moving contact problem is being developed 

separately. 

The finite element mesh (Figures 40 and 41) is made up of 8 noded planestrain elements (biquadratic in displacement and linear in temperature). 

Taking advantage of symmetry, x-displacements along side BC are cons trained, 

sides AD and DC are maintained at OC. For the purpose of calculating the 

displacements along sides AD and DC, the semi-elliptical Hertzian distribution 

is represented by 100 concentrated normal forces indenting an elastic half 

plane. The effect of shear tractions is represented by the heat fluxes, but 

their mechanical effect has not been included. The displacements at the 

nodes along AD and DC are calculated for each of the concentrated forces and 

then superposed. Nonunlformly-dis tributed mechanical and thermal loading is 

applied over a half contact width, w, on side AB starting from point B. The 

remainder of side AB is assumed to be convective. 

Calculations are performed for a Polk = 4.0, (where po is the peak 

pressure and k = a,/ 3, is the cyclic shear resistance), thermal conductivity, 

K = 60 w/m OK, a uniform film coefficient, h = 500 w/m-OK, Young's modulus, 
-63- 

A B 

f 

.I 

C 

Figure 40. 

analysis. The contact half width w = 0.6 mm. 

Finite element mesh used in the coupled tempera ture-displacement 
. 

-64- 

B7 

F- L 

Figure 41. Detailed mesh Inside the region BEF (refer to Figure 40). 
-65- 

E = 207 GPa, Poisson' ra io, P = 0.3, and a temperature inde, ndent yield 

I stress, u0 = 393.98 MPa. The film coefficient was selected to give a peak 

temperature of about 1000°C at the highest heat flux consistent with experimental evidence(2). 

Figure 42. 

perfectly plastic model. Nine calculations were made, one of them being a 

pure displacement analysis and the rest being coupled tempera ture-displacemen t 

analyses (see Table 6). Different values of heat flow were considered for 

the coupled calculations and steady state conditions were assumed to exist. 

The heat flux (9) and the Hertzian (contact) pressure distribution (p) are 

given by the following expressions: 

The calculated temperature gradient is illustrated in 

The absolute value of uo has little significance for the elastic2 where: po = peak pressure, N/m 

li = coefficient of friction 

V = rolling velocity, m/a 

w = half contact width; w = 0.6 mm. 

Calculations were performed for the range of heat fluxes shown in Table 6. 

The highest value corresponds closely with the values of 

for HPOTP bearings(l,*). 

avoid convergence problems. The finite element modeling was done on the 

multipurpose finite element package "ABAQUS" on a VAX 8800 computer. 

CPU time for a calculation was found to be 30 minutes. 

105 KW/m2 reported 

The model was loaded and unloaded in increments to 

Typical 
1 fnP. 

t. 0. VRLUE 

1 *5.00E*U2 

2 *5. 77E.02 

3 *t.S5E*U2 

4 *7.33E*U2 

6 *6.6dE*02 

7 *9.tEE-O2 

8 *1.04E-03 

9 *1. 12E.03 

10 *1.20E*03 

5’ *8. A 1E*u2 

Figure 42. Temperature contours for a heat flux Input of 8.6 x lo4 KW/m*. 
-67- 

TABLE 6 

HEAT FLUX INPUTS USED IN THE ANALYSIS IDENTIFIED 

BY NUMBERS IN FIGURES 12 THROUGH 28 

Anal y s i s Iden t I f ie r Heat Flux kw/m2 

1 

2 

3A 

3 

3B 

- 

8.6 x lo4 

5.7 x 104 

4.3 x 104 

2.9 104 

2.1 104 

4 1.4 104 

5 0.6 x 104 

6 0.3 104 

S 0 

- 
-68- 

7.3 Results 

As mentioned earlier, eight different heat input values were used in the 

coupled temperature displacement analyses, Results are presented for the 

loaded was well as the residual state for each case analyzed. The stresses 

are normalized with respect to K, the strains with respect to K/G and the 

depth with respect to w. 

Figures 43 and 44 show the normalized residual axial stress for the 

various cases, and Figures 45 and 46 present the normalized residual circumferential stress. The presence of thermal loading does seem to have an effect 

on the residual stress state, when the heat rate input exceeds 1.67 x lo4 KW/m2 

for the parameters used here. 

For the pure mechanical loading case, the residual axial stresses compare 

well (qualitatively) with those found in existing lIterat~re[~~,~~]. 

the heat input is low, the residual stress distributions do not change qualitatively but a slight change is observed quantitatively. However, for heat 

flux values exceeding 1.67 x lo4 KW/m2, drastic changes are observed in the 

residual axial and circumferential stresses, qualitatively as well as quantitatively. In the present analyses, for the pure mechanical loading case, the 

residual circumferential stress attains a high compress1,ve value at a depth 

of 0.65 w below the surface and then tapers off to a low tensile value. On 

the other hand the axial stress first becomes tensile then, with depth, 

reverses to attain a high compressive valuk and then tapers of;’. With the 

addition of heat input, the residual axial and circumferential stresses 

change character and for high heat rates, they become totally tensile. For 

the parameters considered here, the axial stress becomes tensile for heat 

transfer flow values above 5.83 x lo4 KW/rn2 whereas the Circumferential stress 

becomes trensile for a heat flux value exceeding 1.33 x lo4 KW/m2. 

When 
-69- 

Figure 43. 

normalized depth for different heat Inputs (see Table 6). 

Variation of the normalized residual axial stress, SXR, with 
-7 0- 

D sx3m 

0 SX4R 

A SX5R 

+ SX6R 

x SXSR 

Figure 44. 

normalized depth for different heat inputs (see Table 6). 

Variation of the normalized residual axial stress, SXR, with 
-71- 

2. leIR 

SZ? I? 

e. 

7EI I -_-- 1 J 

e. 51 rz. 99 1.47 1.94 2.42 -e.w e. e4 

[EPTH / A 

0 

0 

A 

+ 

X 

SZX 

SZ32'f 

Figure 45. Variation of the normalized residual circumferential I tress, SZR, 

with normalized depth for different heat inputs (see Table 6). 
-72- 

ORIGINAL PAGE rs 

OF POOR QUALITY 

_- 

D sz3m 

0 SZG? 

A SZ5R 

+ SZ6R 

x SZSR 

Figure 46. 

with normalized depth for different heat inputs (see Table 6). 

Variation of the normalized residual circumferential stress, SZR, 
-73- 

For a heat flux input of 8.6 x lo4 KW/m2, both the axial and circumferential 

residual stresses are found to be entirely tensile. Figures 47 and 48 plot 

the normalized residual radial and shear stresses against normalized depth. 

As expected, thermal loading has no effect on these residual stresses. 

Figures 49, 50 and 51 show the residual stresses for a heat flux input of 8.6 

x lo4 KW/m2, for a heat flux Input of 2.14 x lo4 KW/d and for zero heat flux 

input, respectively. Figures 52, 53, 54 and 55 show the stress distribution 

for the loaded conditions. All the above figures indicate that for combined 

mechanical thermal loading, the most significant effect of the heat input is 

on the circumferential stress. The thermal input also effects the residual 

strains (Figures 56, 57, 58 and 59) and displacements (Figures 60, 61, 62, 63, 

64, and 65). 
-74- 

ORIGINAL PAGE IS 

DE POOR QUALITY 

RESIDUAL RADIAL STRESS (SIGUAYR) 

DSYIR 

G S'im 

A SY3Bq 

6. 7 

I 

4.? 

[Y 

t 

I -2.ei 

I I 

-4E%4 L (2.93 1.47 1.94 2.42 

+ SY3R 

x SY3AR 

Figure 47. 

normalized depth for different heat inputs (see Table 6). 

Variation of the normalized residual radial stress, SYR, with 
~ 

-75- 

RESIDUAL SHEAR STRESS (TAUXYR) 

I 8.6 

L---- 4- 

e. 53 e. c99 I .47 1.94 2.42 -1 8 B. 04 

DEIPTH / A 

Figure 48. Variation of the normalized residual shear stress, SXYR, with 

normalized depth for different heat inputs (see Table 6). 
-76- 

Y 

\ 

Y; 

m 

w 

RESIDUAL STRESS / K VS. DEPTH / A 

2.1er 0 SX'I? 

(A c? Sy,? 

--+- 

I-- 

Figure 49. Variation of the normalized residual stresses, SIR, with normalized 

depth for a heat flux Input of 8.6 x lo4 KW/m2 (see Table 6). 
-77- 

RESIDUAL STRESS / K VS. DEPTH / A 

4.0r 

-4 I I -1 I J 

e% e. 52 1 .EA ‘I. 56 2.08 2.60 

DEPTH / A 

SX3E 

0 SY3Ff 

A sz393;; 

+ SXY3Ef 

Figure 50. Variation of the normalized residual stresses, S3BR, with normalized 

depth for a heat flux input of 2.14 x lo4 KW/m2 (see Table 6). 
-78- 

4 

5. 

\ 

ORIGINAL PAGE IS 

DE POOR QUALITY 

.%r 

I. 

Figure 51. 

depth for zero heat flux input (see Table 6). 

Variation of the normalized residual stresses, SSR, with normalized 
L' 

\ 

- 

- L. . 

-L. 

A 

Figure 52. Variation of the normalized circumferential stress, SLZ, with 

normalized depth for different heat inputs (see Table 6). 
-80- 

@.e- l 

2. 

Y 

I 

SlZL 

SL 25 

SL Zt 

SLZS 

Figure 53. Variation of the normalized circumferential stress, SLZ, with 

normalized depth for different heat inputs (see Table 6). 
-81- 

-2 

. 

-2 

v 

\ 

-7. 

I 

. $- 

I 

I f.? 

! 

LI 

I. 

IFigure 54. 

depth for different heat inputs (see Table 6). 

Variation of the normalized axial stress, SLX, with normalized 
-82- 

e. 22- 

I 

! 

I 

u< - I *<E- 

=I 

c 

f5 

4 

X 

Figure 55. 

depth for different heat inputs (see Table 6). 

Variation of the normalized axial stress SLX, with normalized 
-83- 

WXR VS. DEPTkl / A 

Figure 56. 

normalized depth for different heat inputs (see Table 6). 

Variation of the normalized residual axial strain, EPXR, with 
EPXR VS. DEFTH / A 

A.€- 

f: 

6* 'i 

Figure 57. 

normalized depth for different heat inputs (see Table 6). 

Variation of the normalized residual axial strain, EPXR, With 
-85- 

-e. 

i 5: I 

r: 

I 

l 

7 

- 5 .E- X 

Figure 58. 

normalized depth for different heat Inputs (see Table 6). 

Variation of the normalized residual radial strains, EPYR, with 
... 

-86- 

-2.4 

-i. X 

EPYR VS. DEPTH / A 

I 

-4.4 

I I 1 I 1 

e. WZ e. 52 1 .ra4 1.56 2.0~ 2.63 

-6 

Figure 59. Variation of the normalized residual radial strains, EPYR, with 

normalized depth for different heat Inputs (see Table 6). 

1’ ’ 
-87- 

Figure 60. 

KW/m* (see Table 6:. 

mesh. Mag. factor = 0.62. 

Displaced mesh (under load) for a heat flux input of.8I.6 lo4 

Solid lines - displaced mesh, dashed lines - original 
-88- 

Figure 61. 

8.6 x lo4 KW/m2 (see Table 6). 

original mesh. Mag. factor = 26.0. 

Displaced mesh (residual state) for a heat flux input of 

Solid lines - displaced mesh, dashed lines - 
Figure 62. 

KW/m2 (see Table 6). 

mesh. Mag. factor = 2.00. 

Displaced mesh (under load) for a heat flux input of 2.9 x 104 

Solid lines - displaced mesh, dashed lines - original 
Fi ure 63. 

mesh. Hag. factor = 280. 

Displaced mesh (residual stste) for a heat flux input of 2.9 x 

10 8 KW/m2 (see Table 6). Solid lines - displaced mesh, dashed lines - original 
-91- 

_- 

-. s=- . _... -.e--- -.”, --- -T I -5 ’E-2’ 

. Figure 64. 

KW/m2 (see Table 6). 

mesh. Mag. factor = 51.0. 

Displaced mesh (under load) for a heat flux input of 0.3 x 104 

Solid lines - displaced mesh, dashed lines - original 
-92- 

Fi ure 65. 

10 KU/m2 (see Table6 1. 

mesh. Hag. factor - 380. 

Displaced mesh (residual state) for a heat flux input of 0.3 x f Solid lines - displaced mesh, dashed lines - original 
-93- 

8. DISCUSSION 

8.1 Cyclic Flow Properties of 440C Steel 

The measurements reported here reveal that the elastic-linear-kinemtichardening plastic (ELKP) representation offers a good description of the 

cyclic stress-strain behavior of 440C steel at room temperature and for the 

relatively long deformation times, i.e., 2s and the low strain rates, i.e., 

10-3s-1, employed in the cyclic torsion tests. It should also be clear 

that the values of OK and M appropriate for rolling contact service conditions, 

which encompass a wide range of tenperatures(1,2) and much shorter deformation 

times, i.e., may be substantially different from the values used in 

the present calculations. 

The torsion tests show that the 440C steel undergoes significant cyclic 

hardening, amounting to a 50% increase of UK in the first N = 600 cycles. 

The hardening is attributed to the cyclic-strain-induced-transform tion of 

retained aus teni te. The prema ture failure of the torsion tes t pieces prevents 

the measurement of UK and M for the much larger number of contacts encountered 

in service. However, it is possible that the transformation proceeds more 

rapidly in the torsion test than under the compressively loaded contact since 

it produces a volume expansion. Consequently, it is not clear whether the 

ELKP-properties derived from short lived torsion tests apply to equally small 

or much larger numbers of rolling contacts. This Issue deserves to be examined 

more closely since it affects the interpretation of the torsion test results. 

It Is possible that the relatively short cyclic lives encountered in cyclic 

torsion - lo4 times shorter than under rolling contacts producing similar 

strain amplitudes - are related to the absence of the high hydrostatic 

pressures produced by rolling contact. 

. 
-94- 

The near end of life values of the ELKP material parameters of 440C 

steel: OK = 606 MPa, and M = 188 GPa are similar to those obtained for other 

hardened bearing steels(’). They are regarded as the best estimate of the 

cyclic strength properties that govern the mechanical response to repeated 

rolling-sliding contact at room tempertaure. 

8.2 Cyclic Crack Growth 

The failure to produce stable, Mode I1 and Mode 111 cyclic crack growth 

in the hardened 440C torsion test pieces is attributed to a relatively low 

Mode I resistance. This makes it easier for the crack to extend in Mode I 

in spite of the less intense Mode I stress state. The fact that Mode II/Mode 

:-I ,&I1 growth was obtained in the same sample and for the same loading in a 

softer 4140 test piece, indicates that the Mode I resistance may degrade more 

with hardening than either the Mode I1 or Mode 111. Alternatively, retained 

austenite decomposition at the tip of the crack identified earlier with cyclic 

hardening, and accompanying residual compression stresses, may interfere more 

with Mode II/III growth than with Mode I. This problem requires further work. 

-.- 

8.3 Rolling Contact Studies 

The process of spall formation proposed here is very similar to the 

process observed by Shao et a1.(l2) and illustrated in Figure 66. 

observations indicate that the increase in life obtained with the copper 

coato-d samples is associated with a shift defect assisted from surface-tosubsurface-initiation. The sensitivity of bearing life to surface imperfections 

has been emphasized before(49). 

samples possessed better quality surfaces to begin with or whether the copper 

coating rendered the surface defects less harmful. Other questions about the 

The 

It is not clear whether the copper coated 

spall process remain unanswered. The identity and location of the subsurface 
-95- 

I 'I B 

?- / 

I B f 

I 

T- I, 

e 

I B 

# 

'# 

I 

I 

8 

f: El 

"J 

r 

Ti. I 

I 

I ORIGINAL PAGE IS 

Q! IeOOR QUALITY I 

j I 

I 

z 

L 

d 

'23 

.I= 

U 

3 

c 

L 

bt 

Y 

U 

(5 

L 

V 

U 

.r( 

L: 

x 

U 

U 

V 

rc 

U 

c 

. 
... ” *(.a 

-9 6- 

4 

initiation sites are unclear in this case. The conditions for crack initiation, 

the life remaining after initiation and the effects of the steep temperature 

and strength gradients arising from surface heating are not understood. 

8.4 Finite Element Analyses 

The finite element analyses reveal that the hardened 440C steel ELKP 

material properties greatly reduce the plastic strains generated by rollingplus-sliding contact. The peak, half equivalent strain range obtained here: 

@/2 = 0.00025 (for Po/kk = 4.5, T/N = 0.2, 

the value: ASP/2 = 0.0032 reported by Ham et a1.(20) for elastic-perfectlyplastic (EPP) behavior for the same contact condition. The relatively large 

radial plastic strain range, reported by Ham et al. at the surface: 

ACE = 0.0022, is absent for ELKP behavior. 

flow of Ah = 20 pm/contact displayed by the EPP model is completely absent 

in the present case. The large changes are attributed to the unidirectional 

character of the strains produced by sliding which is obviated by kinematic 

hardening and by the large value of the plastic modulus. Thus, the results 

for rolling-plus-sliding are close to those for pure rolling for kinematic 

hardening. The calculations assume that the material retains its kinematic 

character in the presence of large mean stresses -- a point that requires 

further study. 

ELKP-behavior has been conflraed for pure rolling. 

arises from the fact that the residual stresses are determined by the first 

one or two contacts and depend on the shape of the hysteresis loop at the 

onset of cycling when the material progressively hardens. 

here probably understate the actual values since they are derived from the 

end-of-life ELKP properties. This issue is discussed more fully in Reference 7. 

= 0.21, is less than 1/10 

The large continuing forward 

It should be noted that the elimination of forward flow by 

Another complication 

The values obtained 
-97- 

The cyclic strain peaks are obtained at relative depths of yfw = 0.65 

which correspond with the depths of spa11 initiation in 2 cases(50). 

the cyclic plastic strains produced by contact are small in the absolute 

sense, they are relatively large considering that comparable equivalent strain 

ranges, 0.0005 - < ASP12 - < 0.002, produce very short cyclic lives, e.g., 

300 N 600 cycles in cyclic torsion. The fact that rims subjected to the 

same cyclic strains under contact conditions survive N = lo6 to N = lo7 

contact cycles (see Section 4) suggests that crack initiation and growth are 

inhibited by the high hydrostatic pressures accompanying the contact. The 

finite element model described in Section 6 fails to treat two consequences of 

the high heat generation rates encountered in service: (1) high thermal 

stresses and (2) the degrading of the cyclic strength in the high temperature 

regions. These effects are dealt with in a preliminary way by the steady 

While 

state finite element model described in Section 7. However, the thermal 

effects need to be incorporated into the transient model. 

Finally, it should be noted that 2 deformed layers are produced by 

rolling-plus-sliding contact. These are shown schematically in Figure 67. 

One involves the macro contact width and extends to a depth 2wm - lo3 pm; the 

other involves the asperity contact width and extends to a depth 2wa - 10 Lm. 

Strains will be more intense in the latter case if the asperity peak pressure 

is greater than po, the macrovalue. 

in the mse of the 440C steel where cracks have been observed to initiate 

very close to the rim surface. 

The shallow asperity layer may be Important 

8.5 Finite Element Model of Frictional Heating 

The tempera ture-displacemen t-coupled finite element model of the 

frictionally heated contact has many deficiencies. It is more properly viewed 

as a model of a stationary, heat generating indent. However, the finding that 
-98- 

0 

w 

0 

0 

Residual Ci tcumferential Stress 

Figure 67. Schema tic representation of the contributions of the asperity and 

macro contact to the residual stress distribution in the rim. The quantities 

2wa and 2wm are the asperity and the macro contact widths, respectively. 
-99- 

8 

very large residual tensile stresses: 

are generated at the surface is undoubtedly valid €or the translating contact as 

well. The residual tensile stresses are a consequence of the steep temperature 

gradient and attending compressive thermal stresses under the contact that 

produce plastic compression in the x- and z-directions. On cooling, elastic, 

and possibly plastic, tension compensate for the thermal contraction. The 

cyclic superposi tion of compression by the transla ting contact can then 

produce a fluctuating tension and potential Mode I crack driving force that 

accelerate crack initiation and cyclic crack growth. The effects may be 

accentuated by the softening of the material at the elevated temperatures. 

More realistic calculations of this type employing the translating finite 

element contact model of Section 6 are possible and may be helpful In setting 

limits on the allowable levels of heat generation. 

ox - 345 MPa and aZ = 1200 MPa W 
-100- 

a 9. CONCLUSIONS 

(i) The room temperature, cyclic stress-strain hysteresis loops of hardened 

440C steel subjected to constant plastic strain amplitude cyclic 

torsion have been examined €or the plastic strain amplitudes: 

0.0005 - < ASP/2 - c 0.002 produced by rolling-plus-sliding-contact. 

characteristics and shapes of the loops are well represented by the 

elastic-linear-kinematic-hardening-plastic (ELKP) idealization. 

The 

(it) The 440C steel undergoes substantial cyclic hardening under these 

conditions and fractures after from N : 300 cycles to N : 600 cycles. 

The values of the 3 ELKP material parameters that describe the near 

end-of-life cyclic stress-s train behavior are: (a) the elas tic tension 

modulus (E = 207 GPa), (b) the kinematic yield strength Ok = 1050 MPa 

(or shear yield strength kk = Ck/ 3), and the plastic modulus M = 

188 GPa. 

(iii) Elas to-plas tic finite element analyses of repeated rolling-plussliding contact have been performed for the room temperature ELKPproperties of 440C s tee1 . The calculations trea t Hertzian pressures 

in the range 2.5 GPa - < po - < 3.64 GPa and full slip with a coefficient 

of friction LI = 0.2. The results show that the contacts produce peak 

cyclic strain ranges and residual stresses of 0.00025 s 0.001 

and -20 MPa S uR S -35 MPa that are similar to the values calculated 

for pure rolling, but much smaller than the strains and stresses 

calculated for rolling-plus-sliding with elastic-perfectly-plastic 

(EPP) material behavior. 

P 
-101- 

(iv) The analysis locates the peak strain activity at the relative depth, B 

ylw = 0.65 (2w is the contact width), where spa11 initiation has been 

observed. 

(v) In the cases treated, the energy dissipated as heat by plastic deformation is from 3% to 15% of the energy dissipated at the surface by 

friction. 

(vi) A coupled, tempera ture-displacement, elas to-plas tic finite element 

analysis of the effects of frictional heating under the contact was 

performed. The calculations reveal that the local heating produces 

high levels of residual tension: 

after the contact is unloaded and cooled. The superposition of 

0: : 350 MPa and IY~ : 1200 MPa 

compression by the translating contact can then produce a fluctuating 

tension and Mode I crack driving force that can promote crack initiation 

r 

and crack growth. 

(vii) The spalls generated in 440C steel bearing surfaces in a 3 balllrod 

testing machine display the classical, V-shaped, loose flap produced 

by a small inclination, near surface crack growing into the rim. 

(viii) It appears that the spalls observed in uncoated samples are initiated 

at the surface by surface imperfections. The spalls in longer-lived 

copper coated samples initiate below the surface. 

(ix) The task of measuring the Mode II/Mode I11 cyclic crack growth 

resistance of hardened 440C steel is complicated by a relatively low 

Hode I crack growth resistance. Cracks from defects subjected to 

Mode IIIMode 111 in cyclic torsion prefer to extend in Mode I in spite 

of the low Mode I driving force. 

i 
w ACKNOWLEDGEMENTS 

The authors wish to thank B. Bhat and R. Parr of the NASA Marshall 

Space Flight Center for their support and encouragement. 

thank Mr. R. Thorn of the NASA Marshall Space Flight Center for providing the 

spalled samples for examination. They also wish to acknowledge the assistance 

of Q. Chen and K. Kim with the torsion tests. They are grateful for the 

permission to use the finite element code ABAQUS provided by Hibbitt, Karlsson 

and Sorensen, Inc. Finally they wish to thank C. Thompson for her contributions 

to the manuscript. 

They also wish to 
-103- 

REFERENCES 

1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 

10. 

11. 

12. 

13. 

14. 

B. N. Bhat and F. J. Dolan, "Past Performance Analysis of HPOTP Beatings," 

NASA TM-82470, March 1982. 

B. N. Bhat and F. J. Dolan, "Analysis of Cryogenic Turbopump Bearing 

Service Life," Proc. Structures, Structural Dynamics and Materials 

Conference, May 1983. 

R. E. Maurer and R. A. Palllni, "Computer-Aided Selection of Materials 

for Cryogenic Turbopump Bearings," ASLE Preprint '85 AM4E-1. 

D. K. Chaudhuri and Ravi Verma, "A Study of Wear Processes in 440C Ball 

Bearings Used in the HPOTP of the SSME," Oct. 31, 1986. 

H. K. Lorosch, "The Life of the Rolling Bearing under Varying Loads and 

Environmental Conditions ," Ball and Roller Bearing Eng' g , 1981, Vol. 1, 

p. 17. 

V. Bhargava, G. T. Hahn, G. Ham, S. Kulkarnl, and C. A. Rubin: "Influence 

of Kinematic Hardening on Rolling Contact Deformation," Proc. 2nd Int. 

Symp. on Contact Mechanics and Rail/Wheel Systems, Kingston, RI, July, 

1986. 

G. T. Hahn, V. Bhargava, C. A. Rubin, Q. Chen and K. Kim: "Analysis of 

the Rolling Contact Residual Stresses and Cyclic Plastic Deformation of 

an SA€ 52100 Steel Ball Bearing," ASME J. of Tribology, (to be published). 

J. E. Merwin and K. L. Johnson, "An Analysis of Plastic Deformation in 

Rolling Contact," Proc. Inst. Mech. Engs., Vol. 177, 1963, p. 667. 

K. L. Johnson and J. A. Jefferis, "Plastic Flow and Residual Stresses in 

Rolling and Sliding Contact," Proc. Symp. on Fatigue in Rolling Contact, 

Inst. Mech. Engr., London, 1963, papers. 

H. Zlegler, "A Modification of Prager' s Hardening Rule ," Quarterly App. 

- Math, 1959, Vol. 17, p. 55. 

V. Bhargava, G. T. Hahn and C. A. Rubin, "Analysis of Cyclic Crack Growth 

In High Strength Roller Bearings," Theory and App. Fracture Mechanics, 

Vol, 5, NO. 1, 1986, pp. 31-38. 

E. Shao, X. Huang, C. Wang, Y. Zhu and Q. Chen, "A Method of Detecting 

Rolling Contact Crack Initiation and the Establishment of Crack Propagation 

Curves ," ASLE Preprint, 87-AM-4E-2, 1987. 

S. D. O'Regan, G. T. Hahn, and C. A. Rubin, "The Driving Force for Mode 

I1 Crack Growth Under Rolling Contact," - Uear, 101 (1985) pp. 333-346. 

T. S. Lei, V. Bhargava, G. T. Hahn and C. A. Rubin, "Stress Intensity 

Values for Small Cracks in the Rim of Disks and Rings Subjected to Rolling 

Contact," ASME J. of Tribology, Vol. 108, pp. 540-544, Oct., 1986. 
?.*,". . 

- 104- 

A 

I 15. Q. Chen, G. T. Hahn, C. A. Rubin and V. Bhargava, "Influence of Residual 

Stresses on the Rolling Contact KII Crack Growth Driving Force for High 

Strength Bearing Steel," (to be published). 

16. N. Soda and T. Yamamoto, "Effect of Tangential Traction and Roughness on 

Crack Ini tiationlpropaga tion During Rolling Contact ," ASLE Trans., 1981 

Vol. 25, pp. 198-206. 

17. A. P. Voskamp, "Material Response to Rolling Contact," J. Tribology, 

1988, Vol. 107, pp. 359-366. 

18. V. Bhargava, G. T. Hahn, and C. A. Rubin, "An Elastic-Plastic Finite 

Element Model of Rolling Contact, Part I: 

ASME J. Applied Mechanics, Vol. 52, 1985, p. 67. 

Analysis of Single Contacts," 

19. V. Bhargava, G. T. Hahn, and C. A. Rubin, "An Elastic-Plastic Finite 

Element Model of Rolling Contact, Part 11: Analysis of Repeated Contacts," 

ASME J. Applied Mechanics, Vol. 41, 1985, p. 75. 

20. G. Ham, G. T. Hahn, and C. A. Rubin, "Finite Element Analysis of Repeated 

Two Dimensional Rolling-Sliding Contacts," J. of Tribology, to be published. 

21. S. Kulkarni, G. Hahn, C. Rubin and V. Bhargava, (unpublished research). 

22. J. R. Barber, "Some Thermoelastic Contact Problems Involving Frictional 

Heating," The Quarterly Journal of Mechanics and Applied Ma thema tics, 

1976, p. 7. - 

23. J. R. Barber, "The Solution of Heated Punch Problems by Point Source 

Methods," Int. J. Engng. Sci., Vol. 9, 1971, pp. 1165-1170. 

24. B. B. Mikic, "Thermal Contact Conductance: Theoretical Considerations," 

Int. J. Heat Mass Transfer, Vol. 17, 1974, pp. 205-214. 

25. J. R. Barber, "The Effect of Thermal Distortion on Constriction 

Resistance," Int. J. Heat Mass Transfer, Vol. 14, 1969, pp. 751-766. 

26. J . R. Barber, "Thermoelas tic Ins tabili ties in the Sliding of Conforming 

Solids," Proc. Roy. SOC., A-312, 1969, pp. 321-394. 

27. J. R. Barber, "Distortion of the Semi-Infinite Solid due to Transient 

Surface Heating," Int. J. Mech. Sci., Vol. 14, 1972, pp. 377-393. 

28. C. Panek and J. Dundurs, "Thermoelastic Contact Between Bodies with 

Wavy Surfaces," Transactions of the ASME, Vol. 46, Dec., 1979, p. 854. 

29. J. R. Barber, "Some Implications of Dundurs' Theorem for Thermoelastic 

Contact and Crack Problems," Journal Mechanical Engineering Science, 

Vol. 22, No. 5, 1980, p. 229. 

30. M. Comninou and J. Dundurs, "On the Barber Boundary Conditions for 

Thermoelastic Contact," J. of App. Mech., Vol. 46, Dec., 1979, p. 849. 
-105- 

31. M. Comninou, M. Dundurs, and J. R. Barber, "Planar Hertz Contact with 

Heat Conduction," J. of App. Mech., Vol. 48, Sept. 1981, p. 549. v 

32. J. R. Barber, "Indentation of the Semi-Infinite Elastic Solid by a 

Hot Sphere," Int. J. Mech. Sci., Vol. 15, 1973, p. 813-819. 

33. F. F. Ling and V. C. Mow, "Surface Displacement of a Convective Elastic 

Half-space Under an Arbitrarily Distributed Fast-Moving Heat Source," - J. 

of Basic Engineering, Sept. 1965, p. 729. 

34. D. A. Hills and J. R. Barber, "Steady Sliding of A Circular Cylinder 

over a Dissimilar Thermally Conducting Half-Plane," Int. J. Mech. Sci., 

VO~. 28, NO. 9, 1986, p. 613-622. 

35. H. V. Korovchinski, "Plane-Contact Problem of Thermoelasticity During 

Quasi-Stationary Heat Generation on the Contact Surfaces ," J. of Basic 

Engineering, Dec., 1965, p. 811. 

36. V. C. Mow and H. S. Cheng, "Thermal Stresses in an Elastic Half Space 

Associated with an Arbitrarily Distributed Moving Heat Source ," 

Z.A.M.P., Vol. 18, 1967, p. 501. 

37. M. R. Johnson, R. E. Welch, and K. S. Yeung, "Analysis of Thermal 

Stresses and Residual Stress Changes in Railroad Wheels caused by Severe 

Drag Braking," J. of Engineering for Industry, Feb. 1977, p. 19. 

38. H. 3. Wetenkamp and R. M. Kipp, "Thermal Damage and Railraod Stresses in 

a 33-Inch Railroad Car Wheel," J. of Engineering for Industry, August, 

1978, Vol. 100, p. 363. 

39. A. T. Hopper, S. G. Sampath, and R. B. Stonesiper, "The Elastic Finite 

Element Analysis of a CH-36 Railcar Wheel Under Mechanical and Thermal 

Loads," Track/Traln Dynamics and Design, Pergamon Press, Eds. G. J. Moyar, 

W. D. Pilkey, and B. F. Pilkey, 1978. 

40. H. R. Wetenkamp and R. M. Kipp, "Safe Thermal Loads," 

41. H. R. Larson, J. M. Coughlin, M. R. Halsy and T. M. Rusin, "Experimental 

Verif ica tion of Computer-Predicted Tempera tures and Elas tic Thermal 

Strains in Railroad Wheels," 

42. B. N. Bhat and P. J. Dolan, "Analysis of Cryogenic Turbopump Bearing 

Service Life," 24th Structures, Structural Dynamics and Materials 

Conference, May 1983, Lake Tahoe, Nevada. 

43. R. P. Mavris and R. A. Pallini, "Computer-Aided Selection of Materials 

for Cryogenic Turbopump Bearings," 40th ASLE Annual Meeting, May 

1985, Las Vega, Nevada. 

44. S. 2. Newman, "FEM Model of 3-D Transient Temperature and Stress Fields 

in Welded Plates," Ph.D. Dissertation, Carnegie-Mellon university, April, 

1986. 
-106- 

- 

U 45. G. Dumas and C. N. Baronet, "Elastic-plsatic Indentation of a Half-space 

by an Infinitely Long Rigid Circular Cylinder," Lnt. J. Mech. Sci., 

Vol. 13, 1971, pp. 519-530. 

46. G. Rydholm, "On Inequalities and Shakedown in Contact Problems ," Linkoping 

Studies in Science and Technology, 1981. 

47. Douglas Grover, "A Ball-Rod Rolling Contact Fatigue Tester," ASTM STP 

711, 1982, pp. 107-124. 

48. S. M. Kulkarni, M.S. Dissertation, Vanderbilt University, 1986. 

49. H. K. Lorosch, "Influence of Load on the Magnitude of the Life Exponent 

€or Roller Bearings," ASTM STP 771, 1982, pp. 275-292. 

50. Q. Chen, G. T. Hahn, C. A. Rubin and V. Bhargava, "The Driving Force for 

Rolling Contact Induced, Mode 11, Cyclic Crack Growth in High Strength 

Bearing Steels, '* to be published.