Numerical modeling and simulation of wheel radial fatigue tests

Engineering Failure Analysis 16 (2009) 1533–1541

Contents lists available at ScienceDirect

Engineering Failure Analysis

journal homepage: www.elsevier .com/locate /engfai lanal

Numerical modeling and simulation of wheel radial fatigue tests

Mehmet Firat *, Recep Kozan, Murat Ozsoy, O. Hamdi MeteThe University of Sakarya, Dept. of Mech. Engineering, 54187 Adapazari, Turkey

a r t i c l e i n f o

Article history:Received 6 May 2008Received in revised form 27 September 2008Accepted 16 October 2008Available online 29 October 2008

Keywords:FatigueLocal strain approachCritical planeWheel fatigue testsFEA

1350-6307/$ - see front matter � 2008 Elsevier Ltddoi:10.1016/j.engfailanal.2008.10.005

* Corresponding author. Tel.: +90 264 295 5451;E-mail addresses: [email protected], firat@saka

a b s t r a c t

A computational methodology is proposed for fatigue damage assessment of metallic auto-motive components and its application is presented with numerical simulations of wheelradial fatigue tests. The technique is based on the local strain approach in conjunction withlinear elastic FE stress analyses. The stress–strain response at a material point is computedwith a cyclic plasticity model coupled with a notch stress–strain approximation scheme.Critical plane damage parameters are used in the characterization of fatigue damage undermultiaxial loading conditions. All computational modules are implemented into a softwaretool and used in the simulation of radial fatigue tests of a disk-type truck wheel. In numer-ical models, the wheel rotation is included with a nonproportional cyclic loading history,and dynamic effects due to wheel–tire interaction are neglected. The fatigue lives andpotential crack locations are predicted using effective strain, Smith–Watson–Topper andFatemi–Socie parameters using computed stress–strain histories. Three-different test con-ditions are simulated, and both number of test cycles and crack initiation sites are esti-mated. Comparisons with the actual tests proved the applicability of the proposedapproach.

� 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Durability assessment of mechanical components early in the design phase plays a key role in the automotive industries.Traditionally, this has been performed mainly with prototype tests in the actual service conditions or by using simulativetests with digitally controlled servo-hydraulic equipment [1]. However, the understanding of fatigue failure mechanisms un-der multi-axial loading conditions is still a practical need in order to propose design and material modifications against ac-tive damage process [2,3]. Therefore, the computer modeling and simulation of multiaxial fatigue process is still a cost-effective technique in order to reduce iteration cycles during product development and refinement processes [4,5].

The simulation modeling for fatigue failure of metallic structures may follow different methodologies depending on thetype of the application and available experimental data characterizing the fatigue damage process [6]. In various studies inground vehicle industries, it has been shown that the local state of stress or strain influence the fatigue strength of a mechan-ical design, and the local strain approach is a practical engineering approach as long as the crack initiation plays a dominantrole in durability assessment of metallic components [7–9]. In this perspective, a fatigue failure simulation model can bebuilt on a fatigue failure hypothesis using the local material stress–strain response together with an accumulation rule[10]. Since the local strain histories at a material point are usually not proportional and monotonic, a multiaxial elastic–plas-tic constitutive model is required in this setting. Moreover, a notch-correction algorithm is integrated to the material modelbecause of the multiaxial stress and strain at geometric irregularities found in most of the automotive parts [11].

. All rights reserved.

fax: +90 264 295 5450.rya.edu.tr (M. Firat).

1534 M. Firat et al. / Engineering Failure Analysis 16 (2009) 1533–1541

In this paper, a computational methodology based on the local strain concept and linear elastic FE analysis is pre-sented for fatigue damage analysis of metallic components under proportional and nonproportional loading conditions.The fatigue damage is estimated using the local material response computed with a cyclic plasticity model coupled witha notch stress–strain approximation scheme. All computational modules are implemented into a software tool called asMete, and its application is presented with numerical simulation of radial fatigue tests of a disk-type truck wheel. Thefatigue test cycles and critical crack initiation locations are calculated using effective strain and Smith–Watson–Topper(SWT) and Fatemi–Socie damage criteria. The number of wheel rotations is estimated for three different test cycles dur-ing fatigue tests, and estimated crack initiation sites are also given. The model predictions are compared with the testresults.

2. Fatigue damage modeling

A computer simulation model for the fatigue damage analysis may be composed of three main computational modules.Firstly, a material model is required to calculate the local stress–strain response at a structural point under a given cyclicloading history. A notch correction algorithm should be integrated with the material constitutive model to approximate localdeformation process accounting the geometric discontinuities on the component. A multiaxial damage parameter is used toassess fatigue damage for individual loading cycles and to transform in to an equivalent fatigue process usually characterizedby laboratory tests of smooth specimens. In this part of the paper, mathematical models employed in numerical fatigue dam-age modeling are briefly presented.

2.1. Multiaxial stress–strain analysis

Rate-independent plasticity models are sufficiently accurate in the simulation of local deformation response of metals atroom temperature [12]. Since stress–strain histories at a material point are usually not proportional and monotonic, a kine-matic hardening rule is necessary to describe in the material deformations in both elastic and elasto-plastic regime [13–16].In this setting, an anisotropic plasticity model is used in accord with small deformation content, and additive decompositionof total strain as elastic and plastic parts is assumed. Hill’s orthotropic yield function, f, is used to describe the plastic yieldingconditions

2f ¼ ðGþ HÞðrx � axÞ2 � 2Hðrx � axÞðry � ayÞ þ ðF þ HÞðry � ayÞ2 þ 2Nðrxy � axyÞ2 � 1 ð1Þ

In the above expression, rx, ry and rxy represent the surface stress components, and a is the backstress tensor representingthe translation of the yield surface in the stress space. G, H, F and N are the material anisotropy parameters that may bedetermined by simple tension tests along the material orthotropy directions. The plastic anisotropy is described with theevaluation of an additive backstress form of a nonlinear kinematic hardening model [13,16]. In the kinematic model the totalbackstress increment is expressed as a series expansion of several backstress components

da ¼Xm

i¼1

daðiÞ ð2Þ

daðiÞ ¼ cðiÞrðiÞðn� LðiÞÞdp ði ¼ 1;2; . . . ;mÞ ð3Þ

where, c(i) and r(i) are two sets of material parameters computed from the material stress–strain curve, and L(i) is defined asthe unit tensor for the ith backstress component. n denotes the unit vector normal to the yield surface at the current stresspoint and expressed by

bnc ¼nx

ny

nxy

264

375 ¼ 1

f

ðGþ HÞðrx � axÞ � Hðry � ayÞðF þ HÞðry � ayÞ � Hðrx � axÞ2Nðrxy � axyÞ

264

375 ð4Þ

The mean strain is purely elastic. Normality condition holds, and plastic strain rate is expressed by

e _p ¼ _pofor¼ _pn ð5Þ

Plastic strain rate can be calculated with the help of unit normal, and the constitutive relation between the stress rate andelastic strain is given by

d _re ¼ ½C�d _eee ¼ ½C�ðd _ee � _pdneÞ ð6Þ

where [C] is plane stress isotropic elasticity matrix. Using the consistency condition

bncd _re � bncd _ae ¼ 0 ð7Þ

the backstress evaluation equation in rate form may be expressed as

M. Firat et al. / Engineering Failure Analysis 16 (2009) 1533–1541 1535

_dae ¼ _pXm

i¼1

cðiÞðdne � rðiÞdLeðiÞÞ ð8Þ

Substituting the Cauchy stress rate and backstress rate expressions into the consistency conditions and rearranging theexpression, the plastic-rate multiplier is obtained

_p ¼ bnc½C� _deebnc½C�dne þ

Pmi¼1ðcðiÞðbncdne � rðiÞbncdLeðiÞÞÞ

ð9Þ

After the calculation of the plastic multiplier, the elastic and plastic strain increments can be calculated and the elastic–plastic stress tensor components are determined with the following set of equations:

b _rc ¼ ½C� � ½C�dnebnc½C�bnc½C�dne þ

Pmi¼1ðcðiÞðbncdne � rðiÞbncdLeðiÞÞÞ

!_bec ð10Þ

2.2. Notch analysis

The geometrical discontinuities are critical design locations at which fatigue cracks mostly initiate [10]. The stress andstrain components at such material points can be calculated effectively using numerical techniques like FEM, but conductinga time-history analysis in the elastic–plastic regime is frequently infeasible considering typical automotive applications.Therefore, an approximation scheme based on the superposition of a series of elasticity solutions is necessary for the elas-to-plastic stress–strain state at a notch. Since Peterson’s compilation of elastic stress concentration factors [17], variousapproximate analysis methods were proposed to estimate notch stress and strain components. In this study, pseudo-stressapproach is employed for this purpose [18]. Accordingly, local fatigue loads are modeled as pseudo-stresses, which are noth-ing but stress solution computed with linear elastic FE analysis of a metallic component under the same boundary condi-tions. Then the notch strain components are estimated with the notch stress–strain curve determined with the Neuber’srule [9], and the pseudo-stress tensor is used to describe the local fatigue loads on the component. Considering a set of Mdifferent external loads, Lm acting on a given component, the pseudo-stress tensor erij is the superposition of a set of M stresstensors equal to the elastic stress tensor calculated for each fatigue load acting on the component individually

erij ¼XM

m¼1

ðCijÞmLm ð11Þ

In this expression, (Cij)m are pseudo-stress tensor calculated for each single external load Lm with unit magnitude. In order toaccount the local loads to the elastic–plastic response at a structural point, a pseudo-stress–notch strain or load–notch straincurve is employed and stress tensor components act as scaling coefficients for the structural yield surface [5,9].

2.3. Fatigue damage criteria

A fatigue damage parameter is used in the damage assessment of individual loading cycles, and then transformed in to anequivalent fatigue process that was determined by laboratory tests of smooth specimens [6–10]. In this study, fatigue dam-age processes are defined with critical material planes at which a damage parameter attains its maximum for a given loadinghistory. In critical plane approaches, the fatigue damage events are generally uses parametric stress–strain functions involv-ing normal and shear components resolved on a material plane [19]. ASME Salt and Seqa effective range measures consti-tutive fundamental damage parameters in these settings and employ local strain amplitudes in the generalized shear-type damage parameters [20,21]. In the most general case, these two damage criteria can be expressed with the followingexpressions:

SALTe ¼ max12j12

��������; 1

2j23

��������; 1

2j23

��������

� �ð12Þ

SEQAe ¼1ffiffiffi

2pð1þ #Þ

½ðj12Þ2 þ ðj23Þ2 þ ðj13Þ2�12 ð13Þ

where ji represents the principal strains. Smith et al [22] introduced a stress–strain function, SWT parameter, to predict fa-tigue damage for materials whose damage development was tensile (normal) strain dominated. The proposed stress–strainfunction is given by

SWT ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðra þ amÞeaE

pð14Þ

where ra and ea are the amplitude of stress and strain, respectively acting normal direction of the critical plane, and rm rep-resent the mean normal stress. Fatigue cracks may also initiate in the plane of maximum shear strain amplitude. Fatemi andSocie [23] suggested a shear–strain based damage parameter which accounts the effect of mean stress. The critical plane for

1536 M. Firat et al. / Engineering Failure Analysis 16 (2009) 1533–1541

this damage model is identified as the plane experiencing the maximum shear strain amplitude and the fatigue life are esti-mated based on the accumulated damage on this plane

c 1þ kðraÞmax

ry

� �¼

s0f2Gð2Nf Þb þ c0f ð2Nf Þc ð15Þ

In this model, k and ry are material parameter, and (ra)max represents the amplitude of normal tensile stress acting on agiven critical material plane. The fatigue cycles corresponding to the computed strain amplitude is calculated by solvingstrain-life equation numerically and the fatigue damage increment associated with a loading cycle is calculated using thelinear damage accumulation hypothesis. The critical planes according to Fatemi–Socie damage parameter are defined asthe material planes on which the amplitude of normal strain and shear strain becomes a maximum. The material planes thatare candidate for the maximum damage is determined by computing the transformation matrices for a given set orientationangles. Total damage for all material planes are calculated by calculating multiaxial damage parameters for all materialplanes, and the maximum damage is selected as the largest total damage among all material planes at the material pointfor the given stress–strain history [5].

3. Computer modeling of wheel radial fatigue tests

The fatigue damage modeling approach presented in the previous section were implemented into Mete software and ap-plied in the numerical simulation of radial fatigue tests of a truck wheel. The wheel radial fatigue test is one of sign-off testscommonly used by wheel manufactures in the case of major design changes or for brand-new designs. In these fatigue tests,a tire–wheel–hub assembly is loaded against a rotating rigid-drum under a prescribed static force by means of a hydraulicactuator and the tire–drum contact is established at a fixed inclination (Fig. 1). The test load is intended for vehicle deadweight acting on the wheel assembly during straight line driving, and a fixed inclination angle provides a lateral force sim-ulating the cornering maneuvers so that the actual tire and road interaction during the service is simulated in the test rig.

An analysis of fatigue performance is required for geometric design parameters such as the disk thickness or welding sizeand to ensure a durable design before submitting into radial fatigue tests. Consequently, an engineering analysis indicatingwheel failure locations and estimating number of test cycles is a practical need during design studies. 20-inc Disk-typewheels made of high strength steel blank of thickness 3.5 mm were tested under a constant vertical load for three test con-ditions (Table 1). The wheel assembly is mounted to the test machine using a base plate via 10 bolts, with a 115 Nm assem-bly moment, to the flange connection of a rotation shaft. At the start of tests, vertical load for a given camber angle are set tothe test level and kept static throughout the test, then the rotation of the drum starts and the rotational speed reaches a con-stant value of 250 rpm approximately in 300–400 cycles. For all loading conditions, three wheels were tested and numbers ofwheel rotations are determined.

In order to calculate local fatigue loading on the wheel, linear elastic FE analyses were done using Ansys program,and a FE mesh composed of 55,672 solid elements was generated for the wheel–hub assembly (Fig. 2). The mechanicalstresses on the wheel are considered in three groups. Firstly, there are manufacturing stresses left on the wheel due toprocesses such as the blank stamping forming and welding. In the second group, pre-stresses exit on part of the wheeldue to the assembly with the other mechanical elements, mainly on the disk region due to bolt pretension and on therim due to tire pressure. In addition, there are dynamic loading stresses caused by the vertical wheel force, corneringforce with the wheel alignment and the centrifugal forces due to the rotation of assembly. Due to the complexities asso-ciated with the description of manufacturing stresses, no attempt is done to describe their contribution to the total

Fig. 1. Schematics showing fatigue test conditions.

Table 1Fatigue test conditions and corresponding test cycles.

Test condition Camber angle (�) Vertical force (kg) Number of test cycles (arithmetic mean)

1 0 8900 ± 50 6.17E+052 5 8900 ± 50 5.03E+053 �5 8900 ± 50 4.55E+05

Fig. 2. Wheel FE mesh.

M. Firat et al. / Engineering Failure Analysis 16 (2009) 1533–1541 1537

stress state at a material point on the wheel. Furthermore, spring elements were used to apply vertical and horizontaltire loads, and the dynamic forces due to tire–drum interaction is neglected. Transient effects during start-up are alsoignored, and centrifugal force acting on the wheel is modeled with distributed body forces at a constant rotationalspeed. As a result the total stress at any material point of the wheel is assumed to be the sum of the stress due tothe bolt pretension, the stress due to constant centrifugal force and the stresses due to the vertical and lateral forcesfor a given camber angle. Initially, two linear elastic stress analyses were conducted to simulate assembly processand tire loading. In the same way, linear elastic FE analyses were employed to calculate the scaling constants for thevertical and horizontal test loads of unit magnitude. So that the total pseudo-stress tensor history erijðtÞ at a materialpoint is given by the following expression:

Table 2Mechan

Young’sCyclic sCyclic yFatigueFatigueFatigueFatigue

erijðtÞ ¼ ðCijÞ0bolt-pretension þ ðCijÞ0centrifugal þ ðCijÞlateralFL þ ðCijÞout-of-phasevertical Fv f ðtÞ þ ðCijÞin-phase

vertical FvgðtÞ ð16Þ

where the set of scaling constants Cij corresponding to a set of individual pseudo-stress distributions corresponding toeach load set to be included in the fatigue analysis and computed with FE method. Two harmonic functions, f(t) andg(t), introduced to describe wheel rotation and have phase angles 0� and 90� phase angles, respectively. Regarding theother set of coefficients in pseudo-stress tensor computations for each material point, three additional linear elasticanalyses were performed. The steel properties together with Coffin–Manson strain-life parameters are given in Table2. The pseudo-stress–notch strain curve is employed for the calculation of actual elasto-plastic stresses on the wheelafter the determination of the time-history variation of pseudo-stress tensor components.

ical properties and Coffin–Manson life curve parameters.

modulus (GPa) 202trength coefficient (MPa) 1238ield stress (MPa) 170strength coefficient (Mpa) 980strength exponent �0.098ductility coefficient 0.21ductility exponent �0.27

1538 M. Firat et al. / Engineering Failure Analysis 16 (2009) 1533–1541

4. A performance assessment

The fatigue tests simulations are conducted in two steps. First, a global analysis is performed in that all material points onthe surface of wheel are analyzed for a single test cycle following a pre-loading step including the bolt pre-tension and cen-trifugal forces. The fatigue damage distribution is predicted, and the test cycles for all nodes on the wheel surface are cal-culated with critical plane parameters described in pervious sections. Next, a local analysis is performed considering themost-critical damage locations determined in the global analysis. In this step, the local stress–strain response is calculatedup to 100 cycles using the same computational settings as in global analyses.

A compilation of predicted test cycles with global analyses are presented graphically in Fig. 3. The correlations of test cy-cle predictions with ASME fatigue parameters are determined to be fairly conservative for all camber angles up to a safetyfactor 20. Compared to Seqa parameter, Salt parameter performs better in all test conditions simulated. In addition, the fa-tigue damage distributions with effective parameters are observed to be fairly identical regardless the test condition. Thecritical fatigue locations are found to be on wheel rim-to-disk transition region close to cooling holes (Fig. 4). Fatigue anal-yses based on stress–strain functions lead significantly better cycle predictions. Smith–Watson–Topper (SWT) parameterpredicts all test cycles up to a safety factor 5, and the most critical sides are found on the rim-well welding and hat(Fig. 5). Fatemi–Socie parameter, on the other hand, shows a somewhat improved correlation with the test results, and pro-vides predictions for all test conditions within a band of 3 for all camber angles. For zero-camber angle, the damage distri-bution shows a similar pattern as of SWT parameter, however, for positive and negative camber angles, the damage

1.E+04

1.E+05

1.E+06

-5 0 5camber angle(degrees)

nu. o

f whe

el ro

tatio

ns

testSaltSeqaSWTFatemi-Socie

Fig. 3. A comparison of average test cycles with model predictions in global analyses.

Fig. 4. Fatigue test cycle distribution predicted with ASME salt parameter at zero camber angle.

M. Firat et al. / Engineering Failure Analysis 16 (2009) 1533–1541 1539

distribution changes moderately and for these two tests the most critical locations are predicted around the cooling holeclose to rim-well (Fig. 6).

Regarding the fatigue failure sites identified during the radial fatigue tests, four wheels failed due to cracks initiated alongthe welding of the rim-well, and in all tests, relatively small fatigue cracks were observed on the edges of cooling holes(Fig. 7). It is observed that both locations were predicted with critical plane parameters. In addition, two wheels in thezero-camber angle tests retained its service performance even though fatigue cracks were determined around the coolingholes. For all wheels tested, a secondary fatigue cracks are also observed on the backside of rim welding region at closestside to closing hole. A comparison of estimated fatigue test failure sites shows that none of the parameters is successfulfor all tests. However, the critical plane parameters involving mean stress correction terms perform significantly better pre-dictions under nonproportional cyclic loadings, and Fatemi–Socie and Smith–Watson–Topper parameters in conjunctionwith critical plane concept provide practical estimates for both test cycles and damage critical locations. It should be alsonoted that Fatemi–Socie parameter correlates best considering all testing conditions simulated.

In the second step, the local analyses are conducted using Fatemi–Socie parameter in order to evaluate the local stress–strain response as well as the variation of fatigue damage with loading cycles. For this purpose, a single node close to thefailure location in actual tests is selected at which the highest damage was also predicted in global analysis (Fig. 6). Thestress–strain history is computed up to 100 cycles, and the variation of fatigue damage per cycle is determined at thismaterial point. The model parameters and analysis conditions are retained with those employed in the global analyses.

Fig. 5. The fatigue test cycle distribution predicted with critical plane SWT damage parameter at zero camber angle.

Fig. 6. The fatigue test cycle distribution predicted with critical plane Fatemi–Socie damage parameter at a camber angle of 5�.

1540 M. Firat et al. / Engineering Failure Analysis 16 (2009) 1533–1541

Comparing zero-camber angle and �5� camber angle cases; a significant increase in the in-plane strain components is ob-served (Fig. 8). Moreover, the tensile mean stress for �5� camber angle case causes a slight shift of the stress loop with eachloading cycle. A similar trend is observed for the zero-camber angle case, however, with a significantly reduced rate.

Fig. 7. Failure locations around rim-well welding and cooling holes (test condition 1).

Fig. 8. In-plane strain response at node 11089.

-100

-50

0

50

100

150

-200 -100 0 100 200

in-plane stress-x(MPa)

in-p

lane

stre

ss-y

(MPa

)

Fig. 9. In-plane stress response at node 11089.

M. Firat et al. / Engineering Failure Analysis 16 (2009) 1533–1541 1541

Examining the in-plane stress component histories, a limited amount of stress relaxation like behavior is observed at thispoint up to the 100th cycle (Fig. 9). There is a slight movement of the stress loop due to mean stress components in localstress history. Furthermore, variations in in-plane stress and strain components cause a fairly small decrease in the damagevalues per cycle, and consequently the change of fatigue damage appears to be insignificant.

5. Conclusion

In this paper, a computational methodology is proposed for fatigue life and failure prediction of automotive componentsand its application is presented with numerical simulations of radial fatigue tests of a disk-type truck wheel. Following ashort review of theoretical models, the computer modeling of wheel radial fatigue tests were described in conjunction withlinear elastic FE stress analyses. In simulation models, the wheel rotation is included with a nonproportional cyclic loadinghistory, and wheel–tire interaction is neglected. The fatigue test cycles and failure locations for three test conditions werepredicted using effective strain, Smith–Watson–Topper and Fatemi–Socie parameters using computed stress–strain histo-ries. In the first step, the critical locations are estimated, and nodal points on cooling hole and the rim-well welding are iden-tified as the fatigue critical locations. Best correlations are obtained with the critical plane parameters involving mean stresscorrection terms when compared with strain range parameters. Fatigue test cycles predicted using Fatemi–Socie damageparameter is observed to be conservative and considerably close to actual test cycles for all camber angles. Furthermore,variations in cyclic damage values appeared to be insignificant.

Acknowledgements

The authors thank to G. Topgoz of Jantsa Wheels (Turkey) for providing the technical material and supporting the re-search in this study. Also the help of technical staff of Jantsa Design and Tooling Dept. are gratefully acknowledged.

References

[1] Wright DH. Testing automotive materials and components. Warrendale (PA): SAE Publication; 1993.[2] Richard C, Rice M. SAE fatigue design handbook. Warrendale (PA): SAE Publication; 1988.[3] Grubisic V, Fischer G. Procedure for optimal light-weight design and durability testing of wheels. Int J Vehicle Design 1984;5(6):659–71.[4] Chu C, Conle FA, Bonnen JJF. Multiaxial stress–strain modeling and fatigue life prediction of sae axle shafts. In: McDowell DL, Ellis R, editors. Advances

in multiaxial fatigue. ASTM STP 1191. Philadelphia: American Society for Testing and Materials; 1993. p. 37–54.[5] Firat M, Kocabıcak U. Analytical durability modeling and evaluation–complementary techniques for physical testing of automotive components. Eng

Failure Anal 2004;11(4):655–74.[6] Chu C. Multiaxial fatigue life prediction methods in the ground vehicle industry. Int J Fatigue 1997;19(1):325–30.[7] You B, Lee S. A critical review on the multiaxial fatigue assessment of metals. Int J Fatigue 1996;18(4):235–44.[8] Pan W, Hung C, Chen L. Fatigue life estimation under multiaxial loadings. Int J Fatigue 1999;21:3–10.[9] Kocabicak U, Firat M. Numerical analysis of wheel cornering fatigue tests. Eng Fail Anal 2001;8:339–54.

[10] Ellyin F. Fatigue damage, crack growth and life prediction. London: Chapman & Hall; 1997.[11] Tipton SM, Nelson DV. Advances in multiaxial fatigue life prediction for components with stress concentrations. Int J Fatigue 1997;19(6).[12] Lemaitre J, Chaboche J-L. Mechanics of solid materials. Cambridge: Cambridge University Press; 1990.[13] McDowell DL, Socie DF, Lamda SH. Multiaxial nonproportional cyclic deformation. In: Amzallag C, Leis BN, Rabbe P, editors. Low-cycle fatigue and life

prediction. ASTM STP 770. Philadelphia: American Society for Testing and Materials; 1980. p. 500–18.[14] Chaboche JL. Time-Independent constitutive theories for cyclic plasticity. Int J Plast 1986;2(2).[15] Chaboche JL. Constitutive equations for cyclic plasticity and cyclic viscoplasticity. Int J Plast 1989;5:247–302.[16] Wang H, Barkey ME. A strain space nonlinear kinematic hardening/softening plasticity model. Int J Plast 1999;15:755–77.[17] Jiang Y. Cyclic plasticity with an emphasis on ratcheting. PhD dissertation, University of Illinois at Urbana-Champaign; 1993.[18] Peterson RE. Stress concentration factors. New York: John Wiley & Sons; 1977.[19] Barkey ME. Calculation of notch strains under multiaxial nominal loading. PhD dissertation, University of Illinois at Urbana-Champaign; 1993.[20] Socie DF. Critical plane approaches for multiaxial fatigue damage assessment. In: McDowell DL, Ellis R, editors. Advances in multiaxial fatigue. ASTM

STP 1191. Philadelphia: American Society for Testing and Materials; 1993. p. 7–36.[21] ASME, Boiler and Pressure Vessel Code, Section III, Division 1, Class 1 Components, Subsection NB. Rules for construction of nuclear power plant

components. New York: ASME; 1988.[22] Smith RN, Watson P, Topper TH. A stress–strain function for the fatigue of metals. J Mater 1970;5(4).[23] Fatemi A, Socie D. A critical plane approach to multiaxial fatigue damage including out-of-phase loading. J Fatigue Fract Eng Mater Struct

1988;11(3):149–65.