1-s2.0-s0749641903001359-main

Modified kinematic hardening rule for multiaxialratcheting prediction

X. Chen*, R. Jiao

School of Chemical Engineering & Technology, Tianjin University, Tianjin 300072, PR China

Received in final revised form 8 May 2003

Abstract

A modified kinematic hardening rule is proposed in which one biaxial loading dependent

parameter �0 connecting the radial evanescence term [(�:n)ndp] in the Burlet–Cailletaud modelwith the dynamic recovery term of Ohno–Wang kinematic hardening rule is introduced intothe framework of the Ohno–Wang model. Compared with multiaxial ratcheting experimental

data obtained on 1Cr18Ni9Ti stainless steel in the paper and CS1026 steel conducted byHassan et al. [Int. J. Plasticity 8 (1992) 117], simulation results by modified model are quitewell in all loading paths. The simulations of initial nonlinear part in ratcheting curves can be

improved greatly while the evolutional parameter �0 related to plastic strain accumulation isadded into the modified model.# 2003 Elsevier Ltd. All rights reserved.

Keywords:Kinematic hardening rule; Ratcheting; Cyclic plasticity; Multiaxial loading; Constitutive model

1. Introduction

Ratcheting, accumulation of secondary deformation proceeding cycle by cycleunder stress-controlled conditions, is an important factor in designing structurecomponents. The ratcheting deformation could accumulate continuously with theincreasing number of cycles applied, and it may not cease until fracture. Ratchetingdeformation contributes tomaterial damage and reduces fatigue life (Rider et al., 1995).Before 1990, all cyclic plasticity models cannot give good simulation of ratcheting.

Later on, a number of papers review the state of the art of modeling the ratchetingbehavior (Chaboche, 1994; McDowell, 1994; Ohno and Wang, 1993a,b; Ohno, 1998;Bari and Hassan, 2000, 2001, 2002). Ratcheting experiments have been conducted

International Journal of Plasticity 20 (2004) 871–898

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E-mail address: [email protected] (X. Chen).

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on different materials under various loading conditions (Hassan and Kyriakides,1992a, 1994a,b; Hassan et al., 1992; Jiang and Sehitoglu, 1994a; Portier et al., 2000;Bocher et al., 2001; Igaria et al., 2002). Kinematic hardening rules is very importantin cyclic plasticity simulation (Chun et al., 2002; Yoshida and Uemori, 2002; Genget al., 2002; Yaguchi et al., 2002). Several kinematic hardening rules have been pro-posed for predicting of ratcheting under multiaxial loading. The nonlinear kinematichardening rule by Armstrong and Frederick (1966) was found to over-predictratcheting strain significantly under multiaxial loading paths. Several authors gottheir models by modifying the dynamic recovery term in Armstrong and Frederickmodel (Bower, 1989; Chaboche and Nouailhas, 1989a,b; Chaboche, 1991; Yoshida,2000; Ohno and Wang, 1993a,b; Jiang and Sehitoglu, 1994a,b; McDowell, 1995;

Nomenclature

� Deviatoric backstress tensord� Incremental deviatoric backstress tensordp Magnitude of the plastic strain increment tensorn Unit normal to the yield surface at current stress points Deviatoric stressE Young’s modulus for elasticityG Shear modulusHp Plastic modulusN Number of loading cycles" Strain tensor"x Axial strain"xc Amplitude of axial strain cycle"xp Maximum axial strain in a cycle"� Circumference strain"�p Maximum circumference strain in a cycled"e Elastic strain increment tensord"p Plastic strain increment tensorf Yield surface function�� Axial stress rangen Poisson’s ratio� Stress tensord� Incremental stress tensor�0 Size of yield surface�x Total axial stress�xa Amptude of axial stress cycle�� Circumferential stress��a Amplitude of circumferential stress cycle��m Mean of circumferential stress cycle�mean Mean of axial stress cycle

872 X. Chen, R. Jiao / International Journal of Plasticity 20 (2004) 871–898

Abdel-Karim and Ohno, 2000). Chaboche (1991) showed that a threshold fordynamic recovery of back stress is effective for controlling ratcheting in simulation.Ohno and Wang (1993a,b) introduced a critical state of dynamic recovery, and theyshowed that the critical state expresses no or little ratcheting under uniaxial cyclicloading within the framework of the strain hardening and dynamic recovery format.Nonlinear forms of the dynamic recovery term were then discussed for simulatingratcheting appropriately (Ohno and Wang, 1993a,b; Chaboche, 1994; McDowell,1995) while decaying ratcheting was thus discussed in detail from the view of con-stitutive modeling by Jiang and Sehitoglu (1996) and by Jiang and Kurath (1996).In these coupled models based on the Armstrong and Frederick nonlinear

kinematic hardening rule, the plastic modulus (Hp) is calculated according to thekinematic hardening rule and the consistency condition. Usually the parametersare calculated from the hysteresis loops and uniaxial loading responses. Theseparameters are, in effect, calibrated to produce a better representation of thehysteresis loop and uniaxial ratcheting, however they fail to predict multiaxialratcheting responses. In order to solve the problem of over-prediction by the existingmodels on multiaxial ratcheting responses, many researchers (McDowell, 1995;Jiang and Sehitoglu, 1996; Voyiadjis and Basurychoedhury, 1998) have attempted toadd multiaxial terms and parameters into the Chaboche or Ohno–Wang model.However, these modified models do not improve the simulation of the biaxialratcheting responses compared with the Ohno–Wang model (Bari and Hassan,2002). Thus Bari and Hassan proposed a modified kinematic hardening rule basedon the idea of Delobelle et al (1995) in the framework of the Chaboche model. Sincethe Ohno–Wang model is regarded as the best model to predict ratcheting by theresearchers (Igaria et al., 2002), it is reasonable to do some modification in theframework of the Ohno–Wang model. This study is just to propose an improvedkinematic hardening rule by introducing one multiaxial parameter �0 to the Ohno–Wang model aiming at investigating the modified model for its validity and applic-ability of predicting ratcheting under several different multiaxial loading paths.Although rate-dependent constitutive model has been make great progress (Krempland Khan, 2003; Ho and Krempl, 2002), but in order to simplify the problem, arate-independent model is considered in the paper.

2. Ratcheting experiments

The material used in the study was 1Cr18Ni9Ti stainless steel in the form of roundbar with a diameter of 32 mm after being oil-quenched at 1100 �C for 30 min. Thechemical composition of the material is (wt.%): C 0.065, Mn 1.34, Si 0.95, P 0.03, S0.007, Ni 8.74, Cr 17.54, Ti 0.41. The mechanical properties of 1Cr18Ni9Ti stainlesssteel are shown in Table 1.The specimen used in this study, given in Fig. 1, has a tubular geometry with

outside and inside diameters of 22 and 18 mm, respectively in the gage section. Thetests were conducted on an Instron tension–torsion machine with an MTS axial-torsional extensometer mounted on the outside of the specimen gage section. Strain

X. Chen, R. Jiao / International Journal of Plasticity 20 (2004) 871–898 873

and stress was recorded in the personal computer using an automated data acquisitionsystem. All tests were conducted at room temperature under stress control for axialloading and under strain control for torsional loading. The frequency of cyclicloading was 0.5 Hz.The loading paths in the axial stress–shear strain plane (� � �=

ffiffiffi3

pplane) used in

ratcheting tests are illustrated schematically in Fig. 2. The controlled parameters aregiven in Table 2. These tests consist of a constant-amplitude shear strain cyclingunder a constant axial stress (case 1) and a circular axial stress–shear strain cyclicloading with mean axial stress (case 2).

Fig. 1. Specimen geometry (mm).

Table 1

Mechanical properties of 1Cr18Ni9Ti stainless steel

�b (MPa)
�s0.2 (MPa) (%) �5 (%) E (GPa) G (GPa) � HB
605
310 75 60 193 65.4 0.47 160
Fig. 2. Loading paths in ratcheting experiments.


For 1Cr18Ni9Ti stainless steel, the ratcheting experiments reveal that the rate ofratcheting continuously decreases as cycling continues, but does not fully shake-down or cease. The observations of the uniaxial cyclic stress–strain curve for first 16cycles reveal very slight cyclic hardening as shown in Fig. 3. In the present study,therefore, we neglect the cyclic hardening for simplicity.Generally speaking, non-proportional additional hardening of materials has some

effects on ratcheting and the effects have been taken into account in constitutivemodels (McDowell, 1995; Jiang and Sehitoglu, 1996; Jiang and Kurath, 1996).1Cr18Ni9Ti stainless steel presents significant non-proportional additionalhardening under controlled circular strain path (Chen et al., 2001).However, the non-proportional additional hardening of 1Cr18Ni9Ti stainless steel

is not obvious in the experiments of the paper because of axial stress is quite lowunder axial stress–shear strain cyclic loading. A comparison of torsional stress–strain curves between pure torsion, case 1 and case 2 cyclic loading shows additionalhardening can be neglected (see Fig. 4).

Fig. 3. The first 16 cyclic stress–strain response of 1Cr18Ni9Ti stainless steel under uniaxial cyclic

loading.

Table 2

List of ratcheting experiments

Spec. no.
Path �"2/2 (%) �mean (%) �� (MPa) �mean (MPa)
M130
Case 1 0.4 0 0 200
M140
Case 2 0.4 0 200 200
M150
Case 2 0.6 0 200 200

3. Description of the constitutive model

It is assumed that the total strain increment is decomposed into an elastic strainpart and a plastic strain part:

d" ¼ d"e þ d"p ð1Þ

and that the elastic part obeys Hooke’s law:

"e ¼1þ �

E��

�

Etr�ð ÞI ð2Þ

The plastic flow rule can be stated as

d"p ¼3

2

1

Hpds :nh in ð3Þ

The material is assumed to follow the von Mises yield criterion, which is given by

f ¼3

2s� �ð Þ : s� �ð Þ � �20 ¼ 0 ð4Þ

where s ¼ �� 13 tr �ð ÞI is the deviatoric stress tensor, � is the back stress, and �0 is

the size of the yield surface, I is a unit tensor.

Fig. 4. Comparison of torsional stress–strain curves of pure torsion, case 1 and case 2 for 1Cr18Ni9Ti

stainless steel.


3.1. Ohno–Wang model

3.1.1. Model formulationIn order to better describe ratcheting behavior, Ohno and Wang formulated

kinematic hardening rules superposed several A–F kinematic hardening rules andassumed that each component of back stress �i has a critical state for its dynamicrecovery term. Only after reaching the critical state can the dynamic recovery termswork fully. According to the way the dynamic recovery term is used before thecritical state, Ohno–Wang models can be divided into two models (Ohno and Wang,1993a,b). The initial Ohno–Wang model is proposed in the following form:

Model Ið Þ � ¼PM1

�i; d�i ¼ �i2

3rid"p �H fið Þ d"p :

�i

i

� �i

� �; fi ¼ i

2

� r2i ð5Þ

where i, is ith component of deviatoric back stress �, i is the magnitude of �i,i ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3=2�i :�i

pand �i, ri, are material constants. H stands for the Heaviside step

function.In the Ohno–Wang model (I), before �i reaches its respective critical value ri, each

decomposed hardening rule simulates a linear hardening with a slope (2/3 � iri) andafter that it does not evolve. Consequently, the model becomes a multilinear modelin uniaxial cases. The Ohno–Wang model (I) produces closed hysteresis loops andhence cannot produce any uniaxial ratcheting. To eliminate this limitation, Ohnoand Wang proposed a slight nonlinearity for each rule by introducing an exponentialrelation and before reaching its critical state the dynamic recovery term is partiallyoperative. The formula is proposed as follows:

Model IIð Þ � ¼PMi¼1

�i; d�i ¼ �i2

3rid"p �

iri

� �mi

�i d"p :�i

i

� �� ð6Þ

where mi is a material constant and when mi ! þ1, the Ohno–Wang model (II) isreduced to the Ohno–Wang model (I). In the Ohno–Wang model (II), the slightnonlinearity is introduced by replacing the Heaviside step function with power of mi

in Eq. (6) and the dynamic recovery term of each decomposed hardening rule alwaysworks in the form of an exponential relation that produces unclosed hysteresis loopsin uniaxial cases, thus allowing uniaxial ratcheting to occur.Compared with the A–F model, each dynamic recovery term of the Ohno–Wang

models has its critical state and is inhibited in a certain range. Therefore, underuniaxial and multiaxial conditions, the Ohno–Wang models can predict smallerratcheting strain and in some degree simulate the nonlinear part of a ratchetingstrain accumulation curve that is similar to experiments. What’s more, Ohno–Wangmodels use the term

d"p :�i

i

� �


in place of dp in the Armstrong and Frederick rule. Although this term is the sameas dp in uniaxial cases, in multiaxial loading cases, d"p and �i=ið Þ have differentdirections and the projection result makes

d"p :�i

i

� �

smaller than dp. Hence, a model embracing the former term predicts less developmentof multiaxial ratcheting strain than a model with the latter term as demonstrated byOhno and Wang (1993b).A large number of decomposed rules should be employed in the Ohno–Wang

model (II) in order to use several essentially linear hardening rules to simulate anonlinear hysteresis curve well. In this study, it is found that eight hardening rulesare sufficient to obtain a good stable uniaxial hysteresis loop simulation for1Cr18Ni9Ti stainless steel.

3.1.2. Parameter determinationModel parameters are determined by the tensile curve from uniaxial loading

(Ohno, 1998). The uniaxial–loading tensile curve is divided into several segments asshown in Fig. 5 and the corresponding parameters � i, ri for each segment can bedetermined from the following equations:

Fig. 5. Definition of parameters in the Ohno–Wang model from uniaxial cyclic stress–strain space.


�i ¼1

"p ið Þ; ri ¼

� ið Þ � � i�1ð Þ

"p ið Þ � "p i�1ð Þ

�� iþ1ð Þ � � ið Þ

"p iþ1ð Þ � "p ið Þ

� �"p ið Þ For i 6¼ 1 ð7aÞ

and finally ri is determined by using

XMi¼1

ri þ �0 ¼ �max ð7bÞ

where, �(i) and "p(i) denote stress and plastic strain at the ith point on the monotonictensile stress versus plastic strain curve and � Mð Þ ¼ � Mþ1ð Þ.In the Ohno–Wang model (II), the power mi is an important parameter control-

ling ratcheting response and may be determined by a uniaxial ratcheting experimentresponse. Predicted ratcheting with different mi is compared in Fig. 6 from which itcan be seen that as the exponent mi increases, the predicted ratcheting decreases,thus the predicted ratcheting by the Ohno–Wang model (I) is always smaller thanthat by the Ohno–Wang model (II). That is to say, under the same condition, theprediction of Ohno–Wang model (I) is the smallest prediction that can be made bythe Ohno–Wang model (II). From Fig. 6, it is clear that although the increase of mi

can give smaller ratcheting simulation, predicted ratcheting by the Ohno–Wangmodel (I) is still much larger than experimental ratcheting strain in the first 15 cycles.So it is concluded that the increase mi cannot overcome the over-prediction of theOhno–Wang model (II). In this paper a larger mi is assumed (mi=10) to simulate themultiaxial ratcheting first and the shortcoming in the simulation will be solved byadding terms into the Ohno–Wang model (see Section 3.2).

Fig. 6. Comparison of experimental ratcheting and predicted ratcheting of the Ohno–Wang model (II)

with different parameter mi.


The parameters in the Ohno–Wang model used in this study for simulations arepresented in Table 3.

3.1.3. Simulations of experimental resultsThe simulation of the experiments by the Ohno–Wang model (II) using the above

set of parameters is shown in Figs. 6–8. As it is known, among many existingmodels, the Ohno–Wang models can describe the uniaxial and torsional hysteresiscurves well but over-predicts multiaxial ratcheting though with a smaller over-prediction than the Chaboche model, modified models by McDowell, and Jiang andSehitoglu (Bari and Hassan, 2002).From the numerical computation, it is known that under the same condition, the

predicted ratcheting by the Ohno–Wang model (I) is always smaller than that of theOhno–Wang model (II). But if experimental ratcheting is even much smaller thanthe predicted ratcheting by Ohno–Wang model (I), the Ohno–Wang models lackother parameters that can be adjusted to decrease the biaxial ratcheting simulationand hence cannot satisfactorily simulate some materials’ ratcheting behavior.Bari and Hassan (2002) modified the Chaboche model by adding the Delobelle

kinematic hardening rule (Delobelle et al., 1995). Enlightened by their work, a newkinematic hardening model in the framework of the Ohno–Wang model can besupposed, in which one parameter �0 connecting the radial evanescence term[(�:n)ndp] of the Burlet–Cailletaud model (1986) with the dynamic recovery term ofthe Ohno–Wang model (1993a,b) is proposed in order to improve the ratchetingsimulations under multiaxial loading paths.

3.2. An improved model

In order to simulate the uniaxial ratcheting experiments, Burlet and Cailletaud(1986) modified the radial evanescence term in the Armstrong and Frederick (1966)hardening rule as follows:

d� ¼2

3Cd"p � � �:nð Þndp; n ¼

@f=@�

@f

@�

¼

ffiffiffi3

2

rs � �ð Þ

�0

ð8Þ

The plastic modulus expression obtained from this hardening rule by satisfying theconsistency condition

�f:¼ 0

is the same as that obtained from the Armstrong and

Table 3

Model parameters for 1Cr18Ni9Ti stainless steel

�0 (MPa)
E (MPa) mi (i=1�M) �0o �0st
235
193 000 10 0.07 0.005 8
�1�8=4800, 2400, 1200, 600, 300, 150, 75, 37.5

r1�8=10, 65, 63, 41, 80, 70, l6, 2 MPa


Frederick hardening rule. And under uniaxial loading conditions, the direction of �is the same as that of n and hence the radial evanescence term [(�:n)ndp] is reducedto the dynamic recovery term of Armstrong and Frederick. In addition, becausesimulations of uniaxial ratcheting responses depend entirely on the calculationscheme of the plastic modulus of a model, these two rules produce the same simu-lation; while for biaxial loading, the radial evanescence term [(�:n)ndp] of the Burlet

Fig. 7. Cyclic stable strain–stress curves under uniaxial and torsional loading: (a) uniaxial, (b) torsional.


and Cailletaud rule essentially yields a tensor along the plastic strain-rate directionand the simulation of biaxial ratcheting is like the result of Prager (1956) linearhardening rule that predicts shakedown ratcheting (Bari and Hassan, 2002).Between over-prediction ratcheting by the Ohno–Wang model and shakedownratcheting of the Burlet and Cailletaud model, this study obtains a modifiedhardening rule incorporating the ideas of both the Burlet–Cailletaud and the Ohno–Wang models with a parameter �0 as follows:

d�i ¼ �i2

3rid"p �

iri

� �mi

�0�i þ 1� �0ð Þ �i :nð Þn½ � d"p :�i

i

� �� ; i ¼ 1; 2; . . .M ð9Þ

where, � i, ri, mi, and i in Eq. (9) is the same as the Ohno–Wang model. When �0=0,the modified hardening rule is reduced to the Burlet–Cailletaud model that predictsthe shakedown ratcheting; while if �0=1, it reverts to the Ohno–Wang model (II)that over-predicts ratcheting under multiaxial loading conditions.Following the consistency condition ( f

.=0), the plastic modulus is expressed as

follows:

Hp ¼XMi¼1

�i ri ¼3

2

iri

� �mi

�i :nð Þ�i

i:n

� �� ð10Þ

In Eq. (10), it can be seen that the plastic modulus expression (Hp) does notinclude �0 and �0 can be determined by a biaxial ratcheting response, so �0 can influ-ence biaxial ratcheting simulations without having any effect on both the calculationof plastic modulus and the simulations of uniaxial ratcheting responses.

Fig. 8. Cyclic stable strain–stress curve simulated by Ohno–Wang model (II) for M130.


Because the plastic modulus expression (Hp) is independent of �0 and is the same

as that obtained from the Ohno–Wang model, all of the parameters of the Ohno–Wang model can be used by the modified hardening rule as presented in Table 3.The comparison of the Figs. 8 and 9 shows that the stress–stain simulations by twomodels have no differences. The simulations by the modified model with different �0

Fig. 9. Cyclic stable strain–stress curve simulated by modified model for M130.

Fig. 10. Comparison of experimental ratcheting and predicted ratcheting by the modified model with a

constant �0 for Ml30.


are presented in Fig. 10 in which we can see that if a larger �0 is assumed, the modifiedmodel can simulate the initial nonlinear part but cannot provide the subsequentratcheting rate trend well, while if a smaller �0 is assumed, the modified model canpredict the ratcheting rate trend well but cannot simulate the initial nonlinear part ofratcheting curve. So we can come to the conclusion that the modified model with aconstant �0 cannot predict a good simulation of the whole ratcheting curve. Hence itis better to give �0 an evolutionary character to improve the simulation.

Fig. 11. The influence of �0st, �0o, and on ratcheting strain; (a) �0st, (b) �

0o, (c) .


Fig. 11. (continued)

Fig. 12. Comparison of experimental ratcheting and predicted ratcheting by the modified model with an

evolutional �0 for M130.


An evolutionary equation for �0 is proposed as follows:

d�0 ¼ �0st � �0

� dp ð11Þ

0 0
where, � st is the saturated value of � and is an evolutionary coefficient. The initialvalue of �0 is denoted by �0o. The computation for different �0st (�
0o, and are kept

constant) shows that the value of �0st allows adjustment of the slope of ratchetingrate trend as shown in Fig. 11 (a). The computation for different �0o (�

0st and are

kept constant) reveals that �0o is closely related to the ratcheting rate of the firstseveral cycles as shown in Fig. 11 (b) and decides the ratcheting evolutive rate asshown in Fig. 11 (c). So by a biaxial ratcheting experiment curve (see Fig. l2), �0o isassumed to give a good simulation of the first ratcheting while �0st is decided by theratcheting rate trend and is evaluated to well simulate the ratcheting evolutionrate. The value of �0st, �

0o, and other parameters in the modified model for

1Cr18Ni9Ti stainless steel are given in Table 3.In this paper, the simulation results by the modified model with constant �0 and

evolving �0 are shown by the curves of modified model-1 and the curves of modifiedmodel-2, respectively.Comparisons of improved ratcheting simulations of M130, M140 and M150 by

the modified model with evolving parameter �0 and other parameters of the Ohno–Wang model (II) with experimental data are presented in Figs. 12–16.In order to explore the validity of the modified model with evolving �0, some

published experiments data for CS 1026 steel (Hassan et al., 1992) are used in thispaper. Three test paths, axial strain cycle with constant pressure (case 1), bow-tiecycle (case 2), and reverse bow-tie cycle (case 3), are shown in Fig. 17. Ratchetingsimulation results of the Ohno–Wang model and material parameters in the Ohno–Wang model can be found in the paper of Bari and Hassan (2000). Compared withthe Ohno–Wang model, the modified model with constant �0=0.6, can give bettersimulation of the experiments (see Figs. 18 and 19 in this paper and Fig. 14 of Bariand Hassan, 2000). The modified model with the constant �0 simulates the ratchetingrate trend well as shown in Fig. 18, but fails to predict the initial nonlinear part ofratcheting curves reasonably (It can be seen from other simulations to experimentsdata in other cases of Fig. 19.) So like the simulations of 1Cr18Ni9Ti stainless steel,in order to simulate the initial nonlinear part of ratcheting curves best (see Fig. 19),an evolving parameter �0 as in Eq. (11) can be introduced. The values of �0st, �

0o,

and other parameters in the modified model for simulating data of Hassan et al(1992) are presented in Table 4.The improved ratcheting simulations of three ratcheting experiments by the

modified model with constant �0 (Modified model-1) and an evolving parameter �0

(Modifiedmodel-2) are presented in Fig. 19 and compared with the experiments as well.

4. Results and discussion

Evaluation of the Chaboche model, the Ohno–Wang model and some modifiedmodels based on Ohno–Wang model by McDowell (1995), Jiang and Sehitoglu


(1996), have demonstrated that most of these models are not robust enough tosimulate the set of biaxial ratcheting responses (Bari and Hassan, 2002). In thisstudy, a similar conclusion on the Ohno–Wang model can be obtained (see Fig. 7).The drawback of these models is believed to be the lack of parameters that cancontrol the biaxial ratcheting, which leads to the failure of describing the yieldsurface normal directions that are decided by the kinematic hardening rule of amodel. In order to simulate the biaxial ratcheting experiments well, it is necessary

Fig. 13. Comparison of experimental and predicted ratcheting for M140: (a) shear stress–strain, (b) axial

ratcheting strain.


for a coupled model to introduce special biaxial terms or parameters �0 in the kine-matic hardening rule. As the actual yield surface deforms during plastic loading,using multiaxial ratcheting responses by calibrating these parameters determined bya muliaxial experiment will compensate the adverse influence of the lack of exactnessintroduced in the modeling through the assumption of invariant yield surface shape(Phillips and Tang, 1972; Phillips and Lee, 1979). These terms and parametersshould be recessive under uniaxial conditions but will play an important role indescribing the yield surface normal and thus produce a different plastic straindirection under multiaxial condition. The parameter �0 in the Delobelle modelintroduced by Bari and Hassan into the framework of the Chaboche model has this

Fig. 14. Comparison of experiments and predictions for M140; (a) loading path, (b) axial stress–strain

response, (c) stress response, (d) strain response.


character and the existence of �0 can improve ratcheting simulations remarkably(Bari and Hassan, 2002).This paper introduces the parameter �0, which connects the radial evanescence

term [(�:n)ndp] in the Burlet–Cailletaud model with the dynamic recovery term ofOhno–Wang kinematic hardening rule, into the framework of the Ohno–Wangmodel. The new parameter �0 is not involved in the plastic modulus calculationscheme, so the plastic modulus expression of the modified rule is the same as that ofthe Ohno–Wang model and all parameters determined completely from a uniaxialexperiment for the Ohno–Wang model can be used by the modified rule.The predicted ratcheting by the modified model was compared with experimental

data of 1Cr18Ni9Ti stainless steel for case 1 in Fig. 10. The new parameter �0 can bedetermined by this tension–torsion experiment and it can adjust the predictedratcheting to the range of over-prediction of Ohno–Wang model and the shakedown



of Burlet–Cailletaud model and predict a smaller biaxial ratcheting compared withthe Ohno–Wang model. �0 is effective in adjusting the model to the predicted ratch-eting when other parameters are kept unchanged. Computations for different valuesof �0 show that the ratcheting decreases with decrease of �0, as shown in Fig. 10. Thisis not surprising since the decrease of �0 implies that the radial evanescence term[(�:n)ndp] in the modified model becomes more influential. The modified model withconstant �0 cannot simulate the initial nonlinear part and the subsequent rate trend

Fig. 15. Comparison of experimental and predicted ratcheting for M150; (a) shear stress–strain, (b) axial

ratcheting strain.


of a ratcheting curve at the same time. So in order to improve the simulation of themodified model, �0 is made to evolve with plastic strain accumulation as given byEq. (11). According to the ratcheting curve, �0

0 (the initial value of �0) is assumed tosimulate ratcheting curve of the first several cycles while �st

0 (the saturated value of�0) can be selected to predict the experimental ratcheting rate trend well and theevolutional coefficient can be selected to give a good simulation of the evolutiveratcheting rate as shown in Fig. 10. Analyzed from the perspective of the modifiedmodel, �0 is a coefficient that denotes the proportional relation between the radialevanescence term [(�:n)ndp] and the dynamic recovery term. The smaller �0, thelarger proportion of the radial evanescence term and the smaller the predictedratcheting by the modified model. With the evolutional function of �0, simulations of

Fig. 16. Comparison of experiments and predictions for M150; loading path, (b) axial stress–strain

response, (c) stress response, (d) strain response.


M140 and M150 by the modified model appear to be in a reasonably betteragreement with experimental data (see Figs. 13–l6).Comparisons of experiments and simulations of stress–strain response are

presented in Figs. 13–16 for case 2 under the circular loading path. It can be seenthat the modified model not only predicts the axial ratcheting strain and the stablestress–strain hysteresis loop with reasonable accuracy, but it also simulates the stressresponse, the strain response and axial stress–strain response well to some degree. Itis seen that the modified model can well simulate the evolving relation between axialstress, axial strain and torsional strain.In order to confirm the validity of the modified model and the introduction of

parameter �0 in the model, some experiments data by Hassan et al. (1992) is used tocompare with the simulations by the modified model as shown in Figs. 18 and 19.



We can also find that the modified model with the constant parameter �0 cannotsimulate the initial nonlinear part of ratcheting curves well. �0 is expressed as anevolutional function of plastic strain accumulation by Eq. (11) in modified model-2.In Fig. 19, the simulations by the modified model with evolutional �0 (curves ofmodified model-2) are obviously better than those by the model with constant �0

(curves of modified model-1) in all cases 1–3. Compared with the Ohno–Wangmodel [see Fig. 14 in the paper of Bari and Hassan (2000)], the simulations ofratcheting response by the modified model are improved. The modified model cangive good predictions on different loading paths such as an axial strain cycle withconstant internal pressure, circumferential strain peaks from ‘‘bow-tie’’ and reverse‘‘bow-tie’’ cycles, which proves the modified model is valid and robust.

Fig. 17. Loading paths in ratcheting experiments of Hassan et al. (1992).

Fig. 18. Ratcheting predictions by the modified model-1 with constant �0, the modified model-2 with

evolutional �0 and Ohno–Wang model. The experiment data is obtained from Hassan et al. (1992).


The prediction of ratcheting strain to a high number of cycles and the simulationsof ratcheting on under changeable cyclic loading path are not found in the literatureand this remains an open issue. McDowell (1995) proposed an equation for decay of

Fig. 19. Comparison of experimental and predicted ratcheting by the modified model-1 and the modified

model-2: (a), (b) circumferential strain peaks from case 1; (c) circumferential strain peaks from case 2;

(d) circumferential strain peaks from case 3. Experiment data from Hassan and Kyriakides (1992), Hassan

et al. (1992) and Corona et al. (1996).



Table 4

Model parameters for CS 1026 of Bari and Hassan (2000)

�0 (ksi)
E (ksi) � mi (i=1�M) �0o �0st
18.8
26 300 0.302 0.45 0.15 0.6 5
�1�12=45 203, 13 944, 7728, 4955, 3692, 2135, 1230, 585, 295, 119, 50, 20

r1�12=0.707, 2.597, 0.326, 0.076, 2.985, 2.132, 2.825, 3.754, 2.905, 2.076, 1.96, 10


ratcheting strain after 25–50 cycles to substitute for the integration of theconstitutive equation over the entire history of loading. Bari and Hassan (2001), andsuggested that it would be necessary to introduce anisotropy into the yield surface toenhance the predictive capability of ratcheting strain beyond the current assumptionof invariant yield surface shape. The distortion model of subsequent yield surfaceswas introduced into nonlinear kinematic constitutive equations to consistent withmultiaxial ratcheting modeling (Vincent et al., 2002; Francois, 2001). More effortsare certainly needed for more reliable prediction methods. Thus a comparativeevaluation of the proposed model with other existing models and data in the litera-tures is desirable. More comprehensive verification of the model and furtherimprovement remain as future work.

5. Conclusions

Ratcheting tests were conducted on 1Cr18Ni9Ti stainless steel for two non-proportional loading paths. A modified kinematic hardening rule that incorporatesthe radial evanescence term [(�:n)ndp] of the Burlet–Cailletaud model with theOhno–Wang kinematic hardening rule is proposed. All parameters except a newparameter �0 of the modified rule are the same as those of the Ohno–Wang modelsand �0 can be determined with a biaxial ratcheting response. The parameter deter-mination scheme for this modified model is simple and systematic. In order toimprove the simulation to all parts of the ratcheting curves, an evolving parameter �0

is introduced into the modified model. The model predicts stable stress–strainbehavior of the test material with reasonable accuracy. Ratcheting simulations ofboth two types of loading paths are reasonably accurate for experimental data atlow numbers of cycles. In order to confirm the validity of the modified model andthe introduction of the parameter �0 in the model, simulations to the experimentsdata of Hassan et al. (1992) by the modified model with constant �0 and evolving �0

have been presented.

Acknowledgements

The authors gratefully acknowledge financial support for this work from NationalNatural Science Foundation of China (project Nos. 19872049, 10272080) andTRAPOYT.

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