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Materials Science and Engineering A 483484 (2008) 440443

Prediction of fatigue limits of surface-hardened steel with regardto mean stresses and residual stressesA comparison

Frank Jablonski , Johanna Varvarikes

Fachbereich Produktionstechnik, Fachgebiet Technische Mechanik-Strukturmechanik,

Universitat Bremen, Am Biologischen Garten 2, D-28359 Bremen, Germany

Received 6 June 2006; received in revised form 29 September 2006; accepted 29 September 2006

Abstract

Hardened parts are used to minimize abrasion at contact surfaces and to extend the lifetime of machine components. Different risks to failure,and therefore, different fatigue limits result from imperfections, notches and surface properties. Two methods for calculating fatigue limits of

surface-hardened parts are presented and compared. The first method is based on Weibulls weakest-link concept which allows to compute survival

probabilities of parts with inhomogeneous material properties, the second method is the concept of local fatigue strength. The essential quantities

in both models are the distribution of Vickers hardness, the residual stress state, the surface roughness and if necessary the surface oxidation

depth. The model parameters have to be determined from reference specimens. Both models are successfully validated by comparing experimental

and calculated results established with both smooth and notched specimen under different loading conditions.

2007 Elsevier B.V. All rights reserved.

Keywords: Endurance limit; Weakest-link concept; Local fatigue limit; Surface hardening

1. Introduction

Surface-hardened steels (for example case-hardened or laser-

hardened)are widelyusedin thefieldofmechanical engineering,

especially when low abrasion is required. The experimental

determination of the strength of surface-hardened parts under

cyclic loading is a procedure with a large amount of costs due to

the expenditure of material and testing time. In order to reduce

the number of experiments considerably, calculation methods

are needed which allow the consideration of different influences

like load type, stress multiaxiality, stress gradient and notch

effects. Moreover, the influence of residual stresses and mean

stresses are to be considered.

Fatigue failure of machine parts with inhomogeneous mate-

rial properties under inhomogeneous stress conditions may

occur either at the materials surface or inside the material.

Therefore, a separated view on the surface and the volume of

the material is necessary.

Corresponding author. Tel.: +49 421 218 4452; fax: +49 421 218 7478.

E-mail address:fj@mechanik.uni-bremen.de(F. Jablonski).

2. Advanced weakest-link concept

Thepresented model is based on Weibulls weakest-link con-

cept[1],which allows to compute the survival probability of

specimens and parts which are loaded close to the fatigue limit.

It can be applied in the transition region of the stress-cycle dia-

gram and it assigns to each element of volume Vand each

element of surface Sa survival probabilityPS. The survival

probability is a function of the local equivalent stress amplitude

a,eq, the local fatigue limit Wand the element size[2,3]:

PS(S) = 2S((a,eq(x,y,z))/(WS(x,y,z)))

mS(1)

PS(V) = 2V((a,eq(x,y,z))/(WV(x,y,z)))

mV(2)

In these equationsmSandmVare the exponents of the distribu-

tion which have to be determined experimentally.

The local fatigue limits WSand WVdepend on the local

Vickers hardness (HV) and can be expressed by a relation fol-

lowing Murakami for specimens tested without mean stresses

and residual stresses[4]:

WS0 = FS(HV + 120)YSOYR (3)

WV0 = FV(HV + 120) (4)

0921-5093/$ see front matter 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.msea.2006.09.129

mailto:fj@mechanik.uni-bremen.dehttp://localhost/var/www/apps/conversion/tmp/scratch_3/dx.doi.org/10.1016/j.msea.2006.09.129http://localhost/var/www/apps/conversion/tmp/scratch_3/dx.doi.org/10.1016/j.msea.2006.09.129mailto:fj@mechanik.uni-bremen.de

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F. Jablonski, J. Varvarikes / Materials Science and Engineering A 483484 (2008) 440443 441

In Eq.(3)two influence factors are used of whichYRis the

surface roughness coefficient according to German DIN 3990

[5]andYSOis a diminution factor to describe the decrease of

strength due to surface oxidation[6]:

YR = 1.49 0.471(Rz + 1)0.1 (5)

YSO =

x0x0 + xSO

(6)

whereRz is the surface roughness,xSO the surface oxidation

depth andx0is a fit parameter.

In case of multiaxial stresses caused by notches or combined

load cases, the calculation of a local equivalent stress amplitude

a,eqis necessary. Subsequently, the principal-normal-stress cri-

terion(NH), thecriterionafter von-Mises (VM) andthecriterion

after Dang Van et al. (DV) are used [7].The effects of mean

stresses and residual stresses are considered in the materials

strength for NH and VM hypothesis:

WS/V = WS/V0 HVH0S/V

867

lm +

3i=1

RSi

(7)

where HV is the local Vickers hardness, lmthe local mean

stress, RSi the local residual stresses andH0S/V are material

parameters for the surface and the volume[2,3].

For the DV hypothesis, the effects of mean and residual

stresses are considered in the local equivalent stress amplitude

[2,3]:

a,DV=

a,max + HV3HS/V

3

i=1(ia + im) +

3

i=1RSi

12 +

13

HVHS/V

(8)

where iais the local principal stress amplitude, imthe local

meanstress,RSi thelocalresidual stressandHS/Vare fit param-

eters for the surface and the volume.

Thesurvivalprobabilityof theentire part iscalculated by inte-

grating the survival probabilities of the surface and the volume,

which allows to compute the fatigue limit:

PS(S) = 2

S

((a,eq(x,y,z))/(WS(x,y,z)))mSdS

(9)

PS(V) = 2V((

a,eq(x,y,z))/(WV(x,y,z)))mV

dV (10)

PS,tot = PS(S)PS(A) (11)

To calculate the local and global survival probabilities, the

knowledge of the stress states resulting from external loads and

residual stresses is necessary. Therefore, for complex parts the

use of a finite element program is required.

3. Concept of local fatigue strengths

The second model presented in this work is based on a local

stress/strength-ratio where the influence of mean stresses and

residual stresses is considered using a relation according to

Haibach[8]:

W = W0

1

M

3i=1

im + RSi

W0

(12)

in whichMis a material dependentmeanstress sensitivity factor.

The local fatigue limit W0can be calculated using Eq.(3)for

the components surface and Eq.(4)for the volume.

Taking the effect of local stress gradients into account by

using a relation according to Huck, Thrainer and Schutz[9],the

local stress/strength-ratio can be written as follows:

a,eq(x,y,z)

D(x,y,z) =

a,eq(x,y,z)

W(1 + H). (13)

where=(1/1,max)(d1/dx) is the local specific stressgradient,

a,eqthe localequivalentstress amplitudeandHand aremate-rial parameters. Forsteel thevalues ofH= 0.45 and = 0.30 are

given [9]. Thelocalequivalent stressamplitude canbe expressed

by the principal-normal-stress criterion, the criterion after von-

Mises and the criterion after Dang Van. In this model, the mean

stresses and residual stresses are considered on the materials

strength side, which leads to the following expression for Dang

Vans criterion:

a,DV =

a,max + HV3HS/V

3i=1

(ia)

1

2

+1

3

HV

HS/V

(14)

Failure of the component occurs when the local equivalent

stress amplitude exceeds the local strength.

4. Specimen properties

For the verification of both models, smooth and notched

axisymmetric specimensare investigated whichare loaded under

alternating torsion (AT), rotating bending (RB), repeated ten-

sion (RT) and tension-compression (TC). The diameter of the

net cross-sectional area is 6 mm and the radii of the investigated

notches are 1 and 0.2 mm. The material is 16MnCrS5 according

to European Norm DIN EN 10132-2 which is similar to a SAE5115.

The residual stress state is determined by radiographic mea-

surements and, therefore, a removal of material is necessary to

achieve the residual stresses in dependence on the surface dis-

tance. This leads to a rearrangement of the stress state and the

necessity to carry out equilibrium iterations which is done using

FEM. A detailed description of these numerical equilibrium

iterations is given in ref.[3].

The required experiments have been performed by Stiftung

Institut fur Werkstofftechnik (IWT)in Bremen,Germanyand the

details of the experimental procedure may be found in ref.[10].

InTable 1,all tested geometries and load cases are arranged.

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442 F. Jablonski, J. Varvarikes / Materials Science and Engineering A 483484 (2008) 440443

Table 1

Tested geometries and load cases

Load case Test length (smooth specimen)

(mm)

Notch ground radius (notched specimen)

(mm)

Stress ratio

Rotating bending (RB) 5 1

Rotating bending (RB) 50 1

Rotating bending (RB) 1 1

Rotating bending (RB) 0.2 1Alternating torsion (AT) 30 1

Alternating torsion (AT) 1 1

Alternating torsion (AT) 0.2 1

Tension-compression (TC) 1 1

Tension-compression (TC) 0.2 1

Repeated tension (RT) 1 0.1

Repeated tension (RT) 1 0.5

Repeated tension (RT) 0.2 0.1

Fig.1. Calculated fatigue limitin comparison withthe experimentaldeterminedfatigue limit (advanced weakest-link concept).

5. Calculation results

Both presented models contain material dependent parame-

ters which have to be determined experimentally. Therefore a

set of reference specimens is necessary for each model. Based

on this set of reference parameters the fatigue limits of other

specimens can be predicted.

Fig. 1shows the calculation results of the advanced weakest-

link concept using the Dang Van criterion. The results obtained

with theconceptof local fatigue strengthsare illustratedinFig.2.

Both models are able to predict the fatigue limit with a satis-factory accuracy. The relative error is less than 10% using Dang

Vans equivalent stress criterion.

6. Summary and comparison

Two models for calculating fatigue limits of components

loaded in the transition region of the stress-cycle diagram are

compared. The advanced weakest-link concept allows to com-

pute local and global failure probabilities as well as the global

endurance limit. This is done with a significant effort of numer-

ical investigations. A considerable simpler model is the concept

of local fatigue strengths which also allows to predict the global

Fig.2. Calculated fatigue limitin comparison withthe experimental determinedfatigue limit (concept of local fatigue strengths).

endurance limit of a component as well as the likely position of

failure. But in contrast to the weakest-link concept one cannot

give local failure probabilities, which is a loss of information.

Both models offer a satisfactory accuracy.

The essential quantities in the presented models are the dis-

tribution of Vickers hardness, the exponents of the Weibull

distribution of the surface and the volume and the residual stress

state. The surface roughness and surface oxidation depth are

included by means of diminution factors. Local strengths are

characterized with Murakamis relation, the multiaxiality of the

stress state is expressed by equivalent stresses using, for exam-

ple, von Mises or Dang Vans criterion.

Acknowledgements

The authors would like to thank Dr.-Ing. H. Bomas and his

group for performing the experiments to measure the residual

stresses, hardness distributions and fatigue limits.

References

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[2] H. Bomas, T. Linkewitz, P. Mayr, Fatigue Fract. Eng. Mater. Struct. 22

(1999) 733741.

[3] F. Jablonski, R. Kienzler, Comp. Mater. Sci. 13 (1998) 6166.

[4] Y. Murakami, JSME Ser. I 32 (1989) 167180.

[5] DIN 3990, Teil 3, Tragfahigkeitsberechnung von Stirnradern, Berechnung

der Zahnfutragfahigkeit, Beuth, Berlin, 1987.

[6] T.H. Topper, M.H. El Haddad, Can. Metall. Q. 18 (1979) 207213.

[7] K. Dang Van, B. Griveau, O. Message, in: M.W. Brown, K.J. Miller (Eds.),

Biaxial and Multiaxial Fatigue, EGF 3, Mechanical Engineering Publica-tions, London, 1989, pp. 479498.

[8] E. Haibach, Betriebsfestigkeit, Verfahren und Daten zur Bauteilberech-

nung, VDI-Verlag, Dusseldorf, 1989.

[9] D. Radaj, ErmudungsfestigkeitGrundlagen fur Leichtbau, Maschinen-

und Stahlbau, Springer Verlag, Berlin, 1995.

[10] H. Bomas, M. Schleicher, Dauerfeste Dimensionierung schwingend

beanspruchter Bauteile aus einsatzgeharteten Stahlen, Abschlubericht

zum AiF-Forschungsvorhaben 10867N, 1998.