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International Journal of Fatigue

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Determination of Chaboche combined hardening parameters with dualbackstress for ratcheting evaluation of AISI 52100 bearing steelSungyong Koo, Jungmoo Han, Karuppasamy Pandian Marimuthu, Hyungyil Lee⁎

Sogang University, Department of Mechanical Engineering, Seoul 04107, Republic of Korea

A R T I C L E I N F O

Keywords:Chaboche modelRatcheting evaluationFinite element analysisUniaxial ratchetingBearing steel

A B S T R A C T

This research proposes a method to obtain parameters of Chaboche combined hardening model for evaluatingratcheting phenomenon in metallic materials when cyclic loadings are applied. Finite element (FE) model is usedfor simulating ratcheting by changing Chaboche model parameters. Although Chaboche model with single termcan describe the ratcheting characteristics, simulated stress - strain (σ-ε) relation deviates from that of metallicmaterials when the applied alternative stress σa is smaller than the yield strength σo. Whereas, Chaboche modelwith dual backstress can simulate not only the ratcheting characteristics but also σ-ε relation even when σa issmaller than σo. Finally, tensile and ratcheting experiments are conducted with bearing steel specimens (AISI52100) to validate the proposed method for evaluation of ratcheting parameters and σ-ε relations.

1. Introduction

Application of cyclic loadings under working condition significantlyaffects the fatigue life of most engineering structures [1,2]. Ratchetingphenomenon, which is the accumulation of plastic strain resulting fromcyclic loading due to cyclic mechanical or thermal stress, also shortensthe fatigue life due to the plastic deformation under cyclic loading[3,4]. Engineering structures under cyclic loading such as bearings,rails, turbine blades, atomic reactors need to be designed with ratch-eting in consideration, since excessive plastic deformation by ratchetingcan be one of the primary reasons for structural failures [5–11]. Overthe last decades, understanding the ratcheting phenomenon has beenthe particular interest of many researchers. As a result, quantitative andqualitative studies have been conducted with ratcheting under cyclicuniaxial and multiaxial loadings [12–20], cyclic indentation [21,22]including thermal ratcheting [23–25] and time dependent ratcheting[26–29].

Numerous kinematic hardening rules have been developed to de-scribe the cyclic behavior including ratcheting phenomenon in metals[30–35]. Prager [34] and Ziegler [35] proposed linear kinematichardening models to account for the Bauschinger effect. Followed by, amulti-surface model of Mroz [31], and a nonlinear kinematic hardeningmodel of Armstrong and Frederick [32] (A-F model), in which a re-laxation term is added to the linear kinematic hardening model, werealso developed. Chaboche [30] and Ohno and Wang [33] improved theA-F model with multiple backstresses (Chaboche model and O-W

model). These models became as most representative models for de-scribing the cyclic behaviors of metals. Since, compared to O-W model,Chaboche model is highly accessible as it is implemented in Abaqus[36] and it shows a notable improvement in the ratcheting prediction,we intensively address the Chaboche model in this paper. Reviews onkinematic hardening models with cyclic plasticity are well documentedby Chaboche [37,38] and Kang [15].

Several researchers investigated the ratcheting phenomenon withChaboche model [16,22,39–41]. It is also important to predict the ac-curate parameters of Chaboche model for realistic simulation ofratcheting phenomenon in metallic materials. Lemaitre and Chaboche[40] proposed a method for obtaining the parameters of Chabochemodel by using stabilized hysteresis loops under strain controlled cyclicstress test. Using a single backstress Chaboche model without isotropichardening, which is the same model as A-F model [32], Xu and Yue[22] and Goo [41] simulated the ratcheting phenomenon, however, it isdifficult to simulate entire ratcheting strain by using this model. AbaqusUser’s manual [36] also describes the method of Lemaitre and Chaboche[40] mainly for cyclic plasticity parameter prediction; but the method isnot suitable for precise ratcheting simulation. Alternatively, trial anderror method can be used to determine the parameters by matchingsimulation results with experimental ratcheting data. Bari and Hassan[39] proposed a parameter acquisition method using a triple backstressChaboche model. Koo et al. [16] demonstrated that a dual backstressChaboche model can simulate ratcheting, and proposed a parameteracquisition method. Bari and Hassan [39] and Koo et al. [16], albeit for

https://doi.org/10.1016/j.ijfatigue.2019.01.009Received 8 November 2018; Received in revised form 24 December 2018; Accepted 19 January 2019

⁎ Corresponding author.E-mail address: hylee@sogang.ac.kr (H. Lee).

International Journal of Fatigue 122 (2019) 152–163

Available online 22 January 20190142-1123/ © 2019 Elsevier Ltd. All rights reserved.

T

different reasons, argued that isotropic hardening should be excluded,which is refuted in this work.

Prediction of stress - strain (σ-ε) relation along with ratchetingphenomenon is also one of the key role while applying the Chabochemodel parameters in the structural analyses. However, previous studieson ratcheting only approximate the ratcheting characteristics (gradualdecrease of ratcheting strain as loading cycle increases, the value ofnon-zero ratcheting strain at steady state) in experiments but not thestress - strain (σ-ε) relation. In that case, the parameters should beavoided from being applied to structural analyses regardless of howwell the ratcheting characteristics are approximated.

Although increasing the number of backstress can improve the si-mulation of ratcheting [42], it makes the parameter decision process tobe complex; adding n backstress requires 2n number of variables, andthe evolution of each backstress affects each other [39]. Therefore, theaim of this study is to present a simple but effective method to obtainthe Chaboche combined hardening model parameters with dualbackstress to evaluate both the ratcheting characteristics and the σ-εrelation at the same time. The importance of including isotropic hard-ening in the Chaboche model is also addressed. By using finite element(FE) model developed for ratcheting simulation, parametric studies arecarried out to understand the effect of each parameter on stress -strainrelation and ratcheting strain. Uniaxial ratcheting experiments areconducted with bearing steel (AISI 52100) specimen to validate theproposed method.

It is important to recognize that most of the engineering parts aresubject to multiaxial stress state; however, this work is limited to uni-axial case. Since, the ratcheting strain in uniaxial loading is larger thanthose in multiaxial stress state [48], one of the main purposes of thisresearch for predicting the parameters from uniaxial ratcheting test canbe helpful to define the upper limit of ratcheting strain.

2. Background

2.1. Chaboche’s combined hardening model

Chaboche’s multiple backstress nonlinear kinematic hardening de-veloped from Ziegler’s linear kinematic hardening can be expressed asfollows [30].

= Cd 1 ( )d di if

eqpl

i i eqpl

total(1)

where the subscript i is used to determine the ith backstress, σ is a stresstensor, α is a backstress tensor, αtotal is the sum of all the backstresstensors, eq

pl is the equivalent plastic strain, σf is the size of subsequentyield surface, C is the translation rate of yield surface, and γ is therelaxation rate of yield surface translation as plastic deformation ac-cumulates. Isotropic hardening can be expressed as follows

=Q Q b[1 exp( )]eqpl

max (2)

where Q is the size change of yield surface, Qmax is the maximum sizechange of yield surface and b is the rate of change of yield surface sizeas plastic deformation accumulates. Therefore, the size of the sub-sequent yield surface σf and the initial yield surface σo can be expressedas follows.

= + Qf o (3)

If the Eqs. (1) and (2) are used together, the size of yield surfacechanges and the center of yield surface translates together in the de-viatoric space as shown in Fig. 1; this is called as combined hardening.In this paper, Chaboche model refers to Chaboche combined hardeningmodel; for the case without isotropic hardening, it will be additionallyspecified.

2.2. Ratcheting behavior of metals

Metals subject to stress-controlled cyclic loading exhibit dissimilarbehaviors from those under monotonic loading. During cyclic loadings,a median value of maximum stress σmax and minimum stress σmin isdefined as mean stress σm, and a half difference between σmax and σmin isdefined as an alternative stress σa as illustrated in Fig. 2. If σm≠0 andσmax < σo, entire deformation is completely restored by purely elasticbehavior (Fig. 3a). On the other hand, plastic deformation accumulatesas the number of cycles N increases for the condition σmax > σo; in-crement of the plastic strain gradually decreases as N increases. Thebehavior in which the increment eventually converges to zero is calledshakedown. Elastic shakedown occurs when σmax < elastic shakedownlimit σel (Fig. 3b); the final shakedown behavior is purely elastic. Ifσmax < plastic shakedown limit σpl, plastic shakedown occurs as shownin Fig. 3c; whereas, for σmax > σpl, the increment does not converge tozero, which is called as ratcheting (Fig. 3d) and the increment of plasticstrain is called as ratcheting strain δε. Initially, the ratcheting strain islarge and then reaches a steady state, at which the ratcheting strain isdenoted as δεsteady as shown in Fig. 4. The ratcheting occurs underunsymmetrical cyclic loading, but the direction of ratcheting does notalways coincides with the mean stress direction; the ratcheting direc-tion is also influenced by the previous loading history [47].

3. Finite element analyses of ratcheting phenomenon

3.1. Finite element modeling

Since there are a lot of similarities between low cycle fatigue andratcheting in metals, specimen dimensions from ASTM standard E606are used for ratcheting simulation. A quarter axisymmetric FE model isconstructed with 11,526 nodes and 11,250 4-node bilinear axisym-metric elements (CAX4) as shown in Fig. 5. Reduced section of thespecimen is refined with smaller elements. The large deformation in the

'3

'1 '2

succeedingyield surface

23 f

0 '

'o

23

initialyield surface

Fig. 1. Evolution of yield surface in deviatoric space.

time0

max

m

min

a

Fig. 2. Schematic representation of stress-controlled cyclic loading.

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FE model is accounted by using nonlinear geometric change (NLGEOM)option in Abaqus/standard [36]. Uniformly distributed stress is im-posed on the top of the specimen. Material behavior for the FE model isassigned based on the Chaboche combined hardening model. Forparametric studies, the reference material constants are chosen with asingle backstress (A-F model) to analyze the relationship between ki-nematic and isotropic hardening parameters in ratcheting simulation aslisted in Table 1; where E is the elastic modulus and ν is the Poisson’sratio of the specimen.

3.2. Isotropic hardening vs. Chaboche model

Most of the structural analyses use the isotropic hardening modelconsidering only the size change of yield surface when the structure isunder monotonic loading. However, it is inappropriate to simulate thematerial behaviors under cyclic loading as it cannot explain severalcyclic plastic behaviors including the Bauschinger effect [35]. On theother hand, Chaboche model can demonstrate the cyclic plastic beha-viors and particularly simulate ratcheting in metals. Fig. 6 compares thenumerically obtained σ-ε curves with isotropic hardening alone and theChaboche model under cyclic loadings. Material shows purely elasticbehavior for isotropic hardening after the first loading, even though itshows the same σ-ε relation up to the initial σmax. When the Chabochemodel is used, the strain accumulates in the form of open hysteresisloop. It is also applicable for the shakedown phenomenon if the

alternative stress σa satisfies σo < σa < σo+Qmax. The detailed de-scriptions of these dynamic phases with Chaboche model are covered inthe subsequent Section 3.3.

3.3. Parametric study with single backstress

3.3.1. Conditions for occurrence of ratcheting strainFE analyses confirm that the Chaboche model is essential for

ratcheting simulation; however, δε does not always occurs even withChaboche model. Therefore, a range of applied stress conditions is in-vestigated by changing only σm and σa while the reference materialparameters in Table 1 are kept constant. FE analyses are performed forthree cases of (σm, σa) MPa= (150, 301), (200, 300), (250, 301)without isotropic hardening (=w/o IH) to prevent the size change ofyield surface. As a result, δε occurs for (σm, σa) MPa= (150, 301), (250,301) as shown in Fig. 7; a common feature of both cases is that σo(=300MPa) < σa (=301MPa). In the case of (σm, σa)= (200, 300)MPa, the cyclic behavior is fully elastic under cyclic stress since theapplied alternative stress is smaller than the size of yield surface. Thistrend alone is not sufficient to reveal whether the occurrence of δεdepends on the size of the initial or subsequent yield surface. Therefore,additional FE analyses including isotropic hardening are carried out. Ifisotropic hardening is included, the δε occurs in the case of σf < σa,and δε converges to zero as σf approaches the value of σa with in-creasing the number of cycles N as shown in Fig. 8. Since the non-zero

Fig. 3. Typical cyclic behaviors of metals under cyclic loading.

initial

steady

total

Fig. 4. Cyclic stress-strain relation and nomenclature of ratcheting behavior.

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δε means the increase of eqpl, the σf increases with N as Eqs. (2) and (3).

Fig. 9 additionally represents the change of δε and σf with the numberof cycles in case of (σm, σa) MPa= (150, 320) in Fig. 8. Consequently,the condition σf < σa should be satisfied to simulate the occurrence ofδε, and δε decreases as σa – σf decreases.

3.3.2. Effects of hardening parameter on stress-strain relationUnder the monotonic loading condition, the effect of kinematic

hardening parameters (C, γ) and isotropic hardening parameters (Qmax,b) on the σ-ε relation can be predicted by using Eqs. (1) and (2) evenwithout conducting FE analyses. The increment of backstress increasesas C increases and γ decreases. Since the total hardening is sum of the

kinematic and isotropic hardening, the increase of dα/dε implies theincrease of dσ/dε, namely the hardening amount for the same incre-ment of strain. As Qmax increases, dσ/dε increases. Additionally, as bincreases, the hardening rate increases. The positive and negative Qmax

can simulate the hardening and softening behaviors, respectively;however, this is when isotropic hardening is solely used. When it iscombined with kinematic hardening, the sum appears to be hardenedwhen the hardening amount by kinematic hardening is larger than thesoftening by isotropic hardening and vice versa. Therefore, it is desirableto understand that the hardening and softening by isotropic hardeningare just the size increase and decrease of yield surface.

3.3.3. Effects of hardening parameter on ratcheting characteristicsThe effects of Chaboche model parameters on the ratcheting are

investigated through FE analyses with cyclic loading conditions as thedirect prediction from the equations is quite difficult. First, the effectsof kinematic hardening parameters C and γ are evaluated by excludingisotropic hardening; while the stress condition σm=150, σa=350MPais fixed in all conditions. FE analyses are performed for C=10, 12,14 GPa with fixed= 40 and for γ=30, 40, 50 with fixed C=12GPa.FE results confirm that δε increases as C decreases and γ increases andremains identical with increasing N as shown in Fig. 10.

Similarly, the effect of isotropic hardening parameters on ratchetinginvestigated by changing the values of Qmax and b. From Eqs. (2) and(3), one can understand that the size of the final yield surface is equal toσo±Qmax. In the previous studies, a positive Qmax is referred to cyclichardening and negative Qmax is referred to cyclic softening by con-vention. Since, this research focuses on simulating both σ-ε relation andratcheting characteristics, we consider the positive and negative Q max

3.5

5

12.514

70

(mm)

Fig. 5. FE model for ratcheting simulation; dimensions are selected accordingto ASTM E606.

Table 1Reference material constants for parametric study.

E (GPa) ν σo (MPa) C (GPa) γ Q max (MPa) b

200 0.3 300 12 40 30 50

Fig. 6. σ-ε curves for isotropic hardening alone and Chaboche model.

(%)0 1 2 3 4 5 6

(MPa

)

-200

0

200

400

600

800

(150

, 301

)

(200

, 300

)

(250

, 301

)

E, , o, C, from Table 1

(m

, a)

MPa

Fig. 7. σ-ε curves for various values of (σm, σa) w/o IH.

Fig. 8. σ-ε curves for various values of σa when σm=150MPa.

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to increase and decrease the size of yield surface, respectively. When±Q max is used, conditions such as σf < σo+Qmax and σo+Qmax < σfcan be established.

(i) For positive Qmax, if σf < σa < σo+Qmax, δε gradually decreaseswith increasing N and converges to zero. Therefore, conditionσo+Qmax < σa should be satisfied to obtain a non-zero steadystate ratcheting strain (δε steady≠ 0).

(ii) For negative Q max, as the condition (σf < σa) satisfies the condition(σo+Q max < σa) definitely, δε never converges to zero. However,

N0 5 10 15 20

(%)

0.00

0.01

0.02

0.03

0.04

0.05E, , o, C, , Qmax, bfrom Table 1( m , a) MPa = (150, 320)

N0 5 10 15 20

f (M

Pa)

317

318

319

320

321

o max 1 exp plqef Q b

Fig. 9. The change of δε (a) and σf (b) with N under (σm, σa) MPa= (150, 320).

N0 2 4 6 8 10

(%)

0.0

0.5

1.0

1.5

2.0

2.5

3.0E, , o from Table 1

m = 150, a = 350 MPa

= 40C (GPa)

10

12

14

N0 2 4 6 8 10

(%)

0.0

0.5

1.0

1.5

2.0

2.5

3.0E, , o from Table 1

m = 150, a = 350 MPa

C = 12 GPa

5040

30

(a) (b)

Fig. 10. Effects of kinematic hardening parameters (a) C and (b) γ on δε -N data.

N0 10 20 30 40 50 60

(%)

0.0

0.5

1.0

1.5

2.0

2.5E, , o, C, , b from Table 1 Qmax (MPa)

- 30

30

N0 10 20 30 40 50

(%)

0.0

0.2

0.4

0.6

0.8

1.0E, , o, C, , Qmax from Table 1b

10

50

100

(a) (b)

Fig. 11. Effects of isotropic hardening parameters (a) ±Qmax and (b) b on δε -N data.

Table 2Kinematic hardening parameters for dual backstress Chaboche model.

case C1 (GPa) C2 (GPa) γ 1 γ 2

0 (single backstress) 12 – 40 –1 6 6 40 402 11 1 40 403 6 6 70 104 6 6 80 05 11 1 70 106 11 1 10 70

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since σa – σf increases as plastic deformation accumulates, δε ratherincreases with N and converges to a non-zero value. Whereas, theparameter b just determines the convergence rate to δε steady

(Fig. 11).

3.4. Parametric study with dual backstresses

Parametric study with single backstress suggests to use the positiveQ max and σo+Q max < σa to obtain decrease in δε with increasing Nand δε steady≠ 0, respectively. Hence, the single backstress Chabochemodel can be used only when σo < σa is satisfied. To consider ratch-eting in engineering structures, very diverse stress conditions areneeded to be accounted and there are many engineering structureswhere σa < σo. Therefore, the single backstress Chaboche model isinsufficient to simulate the ratcheting phenomenon; increasing thenumber of backstresses in the Chaboche model can improve the pre-diction.

In the stress space, the backstress tensor α represents the center ofyield surface. If multiple backstresses are considered, the center of yieldsurface can be represented as α total, which is a sum of all the backstresstensors. This confirms that there is still only one center of yield surfacelike a single backstress.

=i

itotal(4)

Hence, in the Chaboche model with dual backstresses, αtotal is definedby α1+α2 and the parameters C1, γ 1 and C2, γ 2 can be used to de-termine α1 and α2, respectively. Since αtotal can be regarded as a singlebackstress, C and γ can be used to determine αtotal. Based on Eq. (1), it isexpected that the Chaboche model using single and dual backstresshave the same result for C1= C2= 0.5C and γ 1= γ 2= γ. Therefore,relations between dual backstress parameters are established as follows

= + = +C C C , 21 2 1 2 (5)

The parametric study of a dual backstress Chaboche model is

Fig. 12. (a) σ-ε relations for various cases of Table 2 and (b) the enlargement.

N0 20 40 60 80 100 120

(%)

0.0

0.5

1.0

1.5

2.0

2.5

3.0E, , o from Table 1C1, C2, 1, 2 from Table 2

m = 150, a = 350 MPa case5

0, 1, 2

364

Fig. 13. δε -N data for various cases of Table 2.

customized jigs

Fig. 14. Experimental setup for both tensile test and ratchetting experiments.

(mm)0 2 4 6 8 10

P (k

N)

0

5

10

15

20

specimen 1specimen 2specimen 3

SUJ2

Fig. 15. P-δ data from three tensile tests.

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performed for various set of C1, C2, γ 1 and γ 2, that satisfying the re-lation of Eq. (5) as listed in Table 2.

3.4.1. Stress-strain relation and ratcheting phenomenon with dualbackstresses

The effects of dual backstress parameters on the σ-ε relation areinvestigated through FE analyses of monotonic tensile test withσmax= 500MPa as shown Fig. 12. If γ 1= γ 2= γ is satisfied, σ-ε curvesfor cases 0, 1 and 2 are completely identical regardless of C1 and C2values. The σ-ε curves for cases 1, 3 and 4, in which C1= C2 and valuesof γ 1 – γ 2 varies, are located in ascending order, and the gaps amongthem widen along with increasing ε. This indicates that, when C1= C2

is satisfied, dσ/dε at the same strain increases as γ 1 – γ 2 increases. Theσ-ε relation of case 6 is located above that of case 5, and the gap be-tween them widens with increasing ε. From the fact that γ 1 and γ 2 ofcases 5 and 6 are inverted, it can be said that dσ/dε at the same strainincreases due to interaction of larger γ i with smaller Ci (here γ 2 in-teracts with C2 in case 6).

Similarly, effects of dual backstress parameters on ratcheting phe-nomenon are investigated through cyclic loading FE analyses with thestress condition σm=150 and σa=350MPa as shown Fig. 13. Identicalδε -N curve for the cases 0, 1 and 2 confirms that not only the σ-εrelations but also the ratcheting phenomenon are the same regardless ofC1 and C2 values if γ 1= γ 2= γ is satisfied. δε never changes with

(mm)0 2 4 6 8 10

P (k

N)

0

5

10

15

20

exp.FEA

SUJ2

eqpl

0.0 0.2 0.4 0.6 0.8 1.0

f (M

Pa)

0

200

400

600

800

1000

1200

1400Eqn. (6) with w = 0.6SUJ2

t

0.00 0.05 0.10 0.15 0.20 0.25 0.30

t (M

Pa)

0

200

400

600

800

1000

necking pointd = 777.0 MPa

d = 0.1643

e

0.0 0.1 0.2 0.3 0.4

e (M

Pa)

0

200

400

600

800TS = 659 MPa necking point

)b()a(

(c)(d)Fig. 16. (a) σe-εe and (b) σt-εt data converted from P-δ data (c) σf-εeq pl data from the weighted- average method (c) P-δ data from an experiment and FEA with plasticproperties.

(%)0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

(MPa

)

0

100

200

300

400

500

600( initial (%), max (MPa))

(2.36, 540)(2.55, 550)

(3.02, 560)

N0 20 40 60 80 100

(%)

0.000.020.040.060.080.100.120.140.160.180.20

a (MPa)

m = 300 MPa

260250

240

steady (%)a (MPa)260 0.019250 0.014240 0.012

Fig. 17. Experimental results of (a) σ-ε data and (b) δε -N data for various values of σa.

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increasing N, and the δε values themselves are equal. The δε-N curve ofcase 4 shows the δεsteady becomes zero if any γ i is equal to zero. In viceversa, δε decreases with increasing N and eventually converges to a non-zero δε steady for the cases 3, 5 and 6 in which γ 1 and γ 2 both are notequal to zero and different. This is a typical ratcheting behavior. The δε-N data of cases 5 and 6 show that δε steady decreases when larger γ i

interacts with smaller Ci (here γ 2 interacts with C2 in case 6). Theplastic strain at the same stress level is inversely proportional to dσ/dε(Fig. 12); as a result, except for case 4 (γ 2= 0) where δε steady does notoccur, the δε steady in all cases tends to be inversely proportional to dσ/dε.

4. Experiments

For both tensile test and ratchetting experiments, the most com-monly used bearing steel AISI 52100 specimens were fabricated ac-cording to the standards (i) ASTM E8 [43] for tensile tests and (ii)ASTM E606 [44] for ratcheting tests. As round specimens are suitablefor cyclic loading tests especially with compressive loads inherent in arisk of buckling, the round specimens were selected and then tested inInstron universal testing machine (3367). Customized jigs were manu-factured to conduct both the tests in Instron 3367 as shown Fig. 14. Theload data was obtained with Instron load cell, which can measure theload up to ± 30 kN. The deformation within the specimen’s gaugelength was measured with the extensometer (base distance=25mm,maximum displacement=±5mm). For tensile test, the round speci-mens with gauge length of 24mm and the displacement rate of 2mm/min were considered. The calculated strain rate is about 1× 10−3 s−1.Whereas for ratcheting experiments, the round specimens with gaugelength of 25mm and the loading rate of 5 kN/min were considered. Thecalculated stress rate is about 2.12MPa/s. It is beneficial to understandthat this low loading rate level is more related to thermal cyclic loadingrather than actual rotating rate of bearing steel.

4.1. Tensile test

Fig. 15 shows almost identical experimental load-displacementcurves from three tensile specimens; in confirms the reproducibility ofexperimental results. Therefore, a single representative P-δ data isconverted into engineering stress-strain (σe-εe) and true stress - strain (σt- εt) curves as shown in Fig. 16a and b, respectively, by using themathematical definition of stress and strain [45]. Since, for AISI 52100steel, Lüders band exists in the stress-strain curves, an average betweenupper yield point (402MPa) and lower yield point (386MPa) is con-sidered as the yield strength (σo= 394MPa) of the material. The tensile

strength σTS and elongation e of AISI 52100 steel are identified as659MPa and 35%, respectively. To obtain the σt - εt data after diffusionnecking point, the weighted-average method referred by Hyun et al.(2014) [45] is used to calculate the true stress-strain data at the dif-fusion necking, where stress σd and strain εd are 777MPa and 0.1643,respectively. Based on these values, σt - εt data after the necking point isobtained by varying the weight w from 0 to 1 as follows

= + +w w(1 ) (1 )t d t dt

d

d

d (6)

The obtained σt -εt data for w=0.6, are then converted to σf- eqpldata

as shown in Fig. 16c to input as the plastic properties of AISI 52,100steel in the FE analyses. Numerical P-δ curve up to fracture are in goodagreement with the experimental results as shown in Fig. 16d.

4.2. Ratcheting experiments

With occurrence of Lüders band in AISI 52100 steel, a large de-formation occurs at the same load as shown in Fig. 15. If the ratchetingtest is performed with load-control, it may cause fatal damages to theexperimental instruments. Therefore, the specimen is stretched until thestress level increases above the Lüders band under the displacement-control, and then ratcheting experiments are carried out by changing tothe load-control. Experimental results for σa=240, 250, 260MPa withconstant σm=300MPa are shown in Fig. 17. As the initial σ-ε curves upto σmax are the same for all three cases, only the σ-ε data of σmax= 560MPa is drawn, and the (εinitial, σmax) for three cases are in-dicated by gray circles (Fig. 17a). The δεsteady is displayed on the δε-Ndata (Fig. 17b).

5. Results

5.1. Parametric study results

Suitably varying the Chaboche model parameters, the parametricstudies reveal the effect of each parameter on the σ-ε relation and theratcheting phenomenon as summarized in Table 3. dδε/dN in the tablerepresents the variation of the ratcheting strain with N. If this value isnegative and positive, δε decreases and increases with N, respectively.The parametric conditions of Chaboche model for ratcheting simulationcan be found in the Table 3:

(i) single backstress Chaboche model without isotropic hardening (w/oIH): it is possible to simulate ratcheting only when σo < σa is sa-tisfied; however, rather rough ratcheting simulations are possible

Table 3Result summary of parametric study on Chaboche model.

Parametric study on Chaboche model Features

σ-ε relation Ratcheting characteristics

σa > σo (w/o IH), σa > σf (w/IH) – δε occurssingle backstress

C, γC ↑ dσ/dε ↑ δε ↓

dδε/d N=0γ ↓σo+Q max < σa Q max > 0 dσ/dε ↑ dδε/d N < 0

δε steady≠ 0Q max < 0 dσ/dε ↓ dδε/d N > 0

δε steady≠ 0dual backstress

C1+ C2= Cγ 1+ γ 2= 2γ

γ 1= γ 2 regardless of C1, C2 identical dδε/d N=0identical δε

γ 1≠ γ 2 γ 1= 0 or γ 2= 0 – dδε/d N < 0δε steady= 0

γ 1≠ 0andγ 2≠ 0

γ 1 – γ 2 ↑when C1= C

dσ/dε ↑ dδε/d N < 0δε steady ↓

C1 – C2 ↑when C1≠ C2, γ 1 > γ 2

dσ/dε ↓ dδε/d N < 0δε steady ↑

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asδε never varies with increasing N.(ii) single backstress Chaboche model with isotropic hardening (w/IH): the

variation of δε with increasing N can be simulated, therefore, de-tailed ratcheting simulation is possible. A non-zero δε steady andreduction in δε with increasing N can be achieved with positive Qmax and, however, not with the condition σa < σo. If ratchetingcharacteristics are focused with smaller σo than σa, the gap be-tween the σ-ε curves of FEA and experiments becomes quite larger.On the other hand, for a negative Q max, the gap between σ-ε curvesbecomes smaller even when σa < σo; however, δε increases withincreasing N. For these reasons, it is concluded that the Chabochemodel with a single backstress is insufficient to fully simulate

ratcheting.(iii) dual backstress Chaboche model without isotropic hardening (w /o IH):

the non-zero δε steady and reduction in δε with increasing N can beachieved. However, when σa < σo, the simulated σ-ε curve has alarge difference from experiments. It shows that including isotropichardening is essential to reduce the difference of σ-ε relations be-tween FEA and experiments. In conclusion, the dual backstressChaboche model with isotropic hardening (w/IH) is best suited tosimulate ratcheting behavior by satisfying both the σ-ε relation andthe ratcheting characteristics.

identification of material parameters (E, , o) and stress conditions ( m, o)

is a < o ?

Yes No

Qmax can be ignored,set b satisfying Q Qmax at initial| FEA

set Qmax satisfying o + Q max < aand b satisfying Q Q max at initial| FEA

find C and satisfying initial| FEA initial| Exp.

set the initial values of C1, C2, 1 and 2satisfying C1 = C2 = 0.5C, 1 = 2 =

(Eq. (5) should be satisfied from this point on)

increase 1 – 2 to decrease steady | FEA( initial| FEA automatically decreases)

increase C1 – C2 to increase initial| FEA( steady | FEA automatically decreases)

areinitial| FEA = initial| Exp.,steady | FEA = steady | Exp.

satisfied?

No

Yes

output C1, C2, 1, 2, Q max and b

Fig. 18. Procedure for obtaining the parameters of the Chaboche model.

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5.2. Proposed method of parameter acquisition

Based on the results of parametric analyses, a simple but effectivemethod to obtain the parameters of the Chaboche model for ratchetingsimulation is suggested as follows (Fig. 18).

Portier et al. [46] conducted ratcheting tests with 316 austeniticsteel (AISI 316L) and approximated the results with the Chabochemodel. Koo et al. [16] also approximated experimental results of Portieret al. [46] with the Chaboche model. Therefore, the validity of theacquisition method presented in this paper can be verified by com-paring with literature. Since the work of Portier et al. [46] contains no

δε-N data, δε-N data are extracted from εtotal-N data. The σ-ε curve up toσmax= 240MPa and the δεN data up to N=100 are compared betweenall the classes as shown in Fig. 19. The simulation results of Portier et al.[46] are very far from the experimental results in both the σ-ε relationand the ratcheting characteristics; whereas, the simulation results ofKoo et al. [16] show that the σ-ε relation is largely different, but theratcheting characteristics are very consistent with the experimentalresult. Contrary to these previous researches, FE analyses based on theproposed acquisition method give the σ-ε relation and the ratchetingcharacteristics are very close to the experimental results. Hence, theparameter acquisition method proposed in this research enables more

(%)0.0 0.2 0.4 0.6 0.8 1.0 1.2

(MPa

)

0

50

100

150

200

250

Exp. by Portier et al. (2000)

Portier et al. (2000)Koo et al. (2017)present work

FE results by

N0 20 40 60 80 100

(%)

0.00

0.01

0.02

0.03

0.04

0.05Exp. by Portier et al. (2000)

Portier et al. (2000)Koo et al. (2017)present work

FE results by

(a) (b)Fig. 19. Comparison of simulation results of Portier et al. [46], Koo et al. [16] and present work.

(%)0.0 0.5 1.0 1.5 2.0 2.5 3.0

(MPa

)

0

100

200

300

400

500

600

Exp.FEA

E = 200 GPa = 0.3o = 400 MPa

Qmax = - 200 MPab = 120

C1 = 62.1 GPaC2 = 8.1 GPa

1 = 398.68

2 = 1.32

N0 20 40 60 80 100

(%)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

FEA

steady|FEA = steady|Exp. = 0.014 %

Exp.

)b()a(

Fig. 20. Experimental and FE results of (a) σ-ε data and (b) δε-N data for σm=300, σa=250MPa.

(%)0.0 0.5 1.0 1.5 2.0 2.5

(MPa

)

0

100

200

300

400

500

600

Exp.FEA

E = 200 GPa = 0.3o = 400 MPa

Qmax = - 200 MPab = 120

C1 = 62.1 GPaC2 = 8.1 GPa

1 = 398.68

2 = 1.32

N0 20 40 60 80 100

(%)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

FEA

steady|FEA = 0.011 %

Exp.

steady|Exp. = 0.012 %error = 8.3 %

(a) (b)

Fig. 21. Experimental and FE results of (a) σ-ε data and (b) δε-N data for σm=300, σa=240MPa using the same parameters as Fig. 20.

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precise ratcheting simulation with the combined hardening Chabochemodel.

5.3. Experimental verification of the proposed method

By using the experimental results for the case of σm=300 andσa=250MPa, the Chaboche model parameters for AISI 52100 steel areobtained by the proposed method. The simulated σ-ε relation and δε-Ndata are then compared with experimental results as shown in Fig. 20.Similarly, σ-ε relation and δε-N data for σa=240, 260MPa are com-pared with the experimental results (Fig. 21 and Fig. 22) by using thesame Chaboche model parameters. Although a wavy curve appears inthe σ-ε relations when σf converges to σo+Q max, σ-ε curves after 1%strain are well approximated and the error of δεsteady is about only 10%or less for all three cases. It should be emphasized that, in cyclic plas-ticity, the negative Qmax does not mean the cyclic softening in nature[47]. In this work, Qmax is mainly used to improve the prediction of σ-εcurves. The comparison results confirm that the Chaboche modelparameters obtained from the proposed method using dual backstresscan even simulate the cases of σa values slightly different from the σavalues used in the parameter acquisition.

Since the ratcheting strain of initial cycles is greater than the steadystate value, it can be an important factor in early structural integrity. Inthis study, the initial ratcheting strain are overestimated as it focuses onthe ratcheting strain in steady state. Therefore, the prediction of overallratcheting strain will be discussed in the future investigation.

6. Conclusions

In this work, we investigated the effect of combined hardeningChaboche model parameters on stress-strain relation and ratchetingphenomenon in metals. Tensile test and ratcheting experiments wereconducted with the bearing steel material AISI 52100. An effectiveparameter acquisition method was then proposed and verified throughthe comparison with the work from literature. Based on the proposedmethod, the obtained parameters were Qmax=−200MPa, b=120,(C1, C2) GPa= (62.1, 8.1) and (γ 1,γ 2,)= (398.68, 1.32). With thesevalues, the σ-ε curve after 1% strain was well approximated andratcheting strain in steady state (δε steady) of the FE result was equal tothe experimental result in case of (σm, σa) MPa= (300, 250). Similarly,under the other two conditions of (σm, σa) MPa= (300, 240) and (300,260), the differences of δε steady between FE and experimental resultswere 8.3% and 10.5%, respectively. Therefore, these results sufficientlydemonstrate the validity of the proposed method for effectively pre-dicting both σ-ε relation and ratcheting characteristics of metals byusing only two backstress. Although it is not possible to deal with theentire alternative stress range with one parametric case, the proposed

method is highly applicable as it can handle the nearby alternativestresses. Therefore, it is simple but effective method for suggestingacceptable approximations of ratcheting behavior.

Acknowledgment

This research was supported by the Basic Scien ce Research Programthrough the National Research Foundation of Korea (NRF-2017R1A2B3009706).

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(%)0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

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