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Modified kinematic hardening rule for multiaxial ratcheting prediction
International Journal of Plasticity 20 (2004) 871–898 www.elsevier.com/locate/ijplas
Modified kinematic hardening rule for multiaxial ratcheting 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 model with the dynamic recovery term of Ohno–Wang kinematic hardening rule is introduced into the framework of the Ohno–Wang model. Compared with multiaxial ratcheting experimental data obtained on 1Cr18Ni9Ti stainless steel in the paper and CS1026 steel conducted by Hassan et al. [Int. J. Plasticity 8 (1992) 117], simulation results by modified model are quite well 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 is added 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 cycle under stress-controlled conditions, is an important factor in designing structure components. The ratcheting deformation could accumulate continuously with the increasing number of cycles applied, and it may not cease until fracture. Ratcheting deformation contributes to material 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 ratcheting behavior (Chaboche, 1994; McDowell, 1994; Ohno and Wang, 1993a,b; Ohno, 1998; Bari and Hassan, 2000, 2001, 2002). Ratcheting experiments have been conducted * Corresponding author. Tel./fax: +86-22-8789-3037. E-mail address: [email protected] (X. Chen). 0749-6419/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijplas.2003.05.005
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Nomenclature d dp n s E G Hp N " "x "xc "xp " "p d"e d"p f n d 0 x xa a m mean
Deviatoric backstress tensor Incremental deviatoric backstress tensor Magnitude of the plastic strain increment tensor Unit normal to the yield surface at current stress point Deviatoric stress Young’s modulus for elasticity Shear modulus Plastic modulus Number of loading cycles Strain tensor Axial strain Amplitude of axial strain cycle Maximum axial strain in a cycle Circumference strain Maximum circumference strain in a cycle Elastic strain increment tensor Plastic strain increment tensor Yield surface function Axial stress range Poisson’s ratio Stress tensor Incremental stress tensor Size of yield surface Total axial stress Amptude of axial stress cycle Circumferential stress Amplitude of circumferential stress cycle Mean of circumferential stress cycle Mean of axial stress cycle
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 important in cyclic plasticity simulation (Chun et al., 2002; Yoshida and Uemori, 2002; Geng et al., 2002; Yaguchi et al., 2002). Several kinematic hardening rules have been proposed for predicting of ratcheting under multiaxial loading. The nonlinear kinematic hardening rule by Armstrong and Frederick (1966) was found to over-predict ratcheting strain significantly under multiaxial loading paths. Several authors got their models by modifying the dynamic recovery term in Armstrong and Frederick model (Bower, 1989; Chaboche and Nouailhas, 1989a,b; Chaboche, 1991; Yoshida, 2000; Ohno and Wang, 1993a,b; Jiang and Sehitoglu, 1994a,b; McDowell, 1995;
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Abdel-Karim and Ohno, 2000). Chaboche (1991) showed that a threshold for dynamic recovery of back stress is effective for controlling ratcheting in simulation. Ohno and Wang (1993a,b) introduced a critical state of dynamic recovery, and they showed that the critical state expresses no or little ratcheting under uniaxial cyclic loading within the framework of the strain hardening and dynamic recovery format. Nonlinear forms of the dynamic recovery term were then discussed for simulating ratcheting appropriately (Ohno and Wang, 1993a,b; Chaboche, 1994; McDowell, 1995) while decaying ratcheting was thus discussed in detail from the view of constitutive 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 the kinematic hardening rule and the consistency condition. Usually the parameters are calculated from the hysteresis loops and uniaxial loading responses. These parameters are, in effect, calibrated to produce a better representation of the hysteresis loop and uniaxial ratcheting, however they fail to predict multiaxial ratcheting responses. In order to solve the problem of over-prediction by the existing models on multiaxial ratcheting responses, many researchers (McDowell, 1995; Jiang and Sehitoglu, 1996; Voyiadjis and Basurychoedhury, 1998) have attempted to add multiaxial terms and parameters into the Chaboche or Ohno–Wang model. However, these modified models do not improve the simulation of the biaxial ratcheting responses compared with the Ohno–Wang model (Bari and Hassan, 2002). Thus Bari and Hassan proposed a modified kinematic hardening rule based on the idea of Delobelle et al (1995) in the framework of the Chaboche model. Since the Ohno–Wang model is regarded as the best model to predict ratcheting by the researchers (Igaria et al., 2002), it is reasonable to do some modification in the framework of the Ohno–Wang model. This study is just to propose an improved kinematic hardening rule by introducing one multiaxial parameter 0 to the Ohno– Wang model aiming at investigating the modified model for its validity and applicability of predicting ratcheting under several different multiaxial loading paths. Although rate-dependent constitutive model has been make great progress (Krempl and Khan, 2003; Ho and Krempl, 2002), but in order to simplify the problem, a rate-independent model is considered in the paper.
2. Ratcheting experiments The material used in the study was 1Cr18Ni9Ti stainless steel in the form of round bar with a diameter of 32 mm after being oil-quenched at 1100 C for 30 min. The chemical composition of the material is (wt.%): C 0.065, Mn 1.34, Si 0.95, P 0.03, S 0.007, Ni 8.74, Cr 17.54, Ti 0.41. The mechanical properties of 1Cr18Ni9Ti stainless steel 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. The tests were conducted on an Instron tension–torsion machine with an MTS axialtorsional extensometer mounted on the outside of the specimen gage section. Strain
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and stress was recorded in the personal computer using an automated data acquisition system. All tests were conducted at room temperature under stress control for axial loading and under strain control for torsional loading. The frequency of cyclic loading was 0.5 Hz. pffiffiffi The loading paths in the axial stress–shear strain plane ( = 3 plane) used in ratcheting tests are illustrated schematically in Fig. 2. The controlled parameters are given in Table 2. These tests consist of a constant-amplitude shear strain cycling under a constant axial stress (case 1) and a circular axial stress–shear strain cyclic loading with mean axial stress (case 2). Table 1 Mechanical properties of 1Cr18Ni9Ti stainless steel b (MPa)
s0.2 (MPa)
605
310
(%) 75
5 (%)
E (GPa)
G (GPa)
HB
60
193
65.4
0.47
160
Fig. 1. Specimen geometry (mm).
Fig. 2. Loading paths in ratcheting experiments.
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For 1Cr18Ni9Ti stainless steel, the ratcheting experiments reveal that the rate of ratcheting continuously decreases as cycling continues, but does not fully shakedown or cease. The observations of the uniaxial cyclic stress–strain curve for first 16 cycles 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 constitutive models (McDowell, 1995; Jiang and Sehitoglu, 1996; Jiang and Kurath, 1996). 1Cr18Ni9Ti stainless steel presents significant non-proportional additional hardening 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 low under axial stress–shear strain cyclic loading. A comparison of torsional stress– strain curves between pure torsion, case 1 and case 2 cyclic loading shows additional hardening can be neglected (see Fig. 4).
Table 2 List of ratcheting experiments Spec. no.
Path
"2/2 (%)
mean (%)
(MPa)
mean (MPa)
M130 M140 M150
Case 1 Case 2 Case 2
0.4 0.4 0.6
0 0 0
0 200 200
200 200 200
Fig. 3. The first 16 cyclic stress–strain response of 1Cr18Ni9Ti stainless steel under uniaxial cyclic loading.
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Fig. 4. Comparison of torsional stress–strain curves of pure torsion, case 1 and case 2 for 1Cr18Ni9Ti stainless steel.
3. Description of the constitutive model It is assumed that the total strain increment is decomposed into an elastic strain part and a plastic strain part: d" ¼ d"e þ d"p
ð1Þ
and that the elastic part obeys Hooke’s law: "e ¼
1þ ðtrÞI E E
ð2Þ
The plastic flow rule can be stated as d"p ¼
3 1 hds:nin 2 Hp
ð3Þ
The material is assumed to follow the von Mises yield criterion, which is given by 3 f ¼ ðs Þ : ðs Þ 02 ¼ 0 2
ð4Þ
where s ¼ 13 trðÞI is the deviatoric stress tensor, is the back stress, and 0 is the size of the yield surface, I is a unit tensor.
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3.1. Ohno–Wang model 3.1.1. Model formulation In order to better describe ratcheting behavior, Ohno and Wang formulated kinematic hardening rules superposed several A–F kinematic hardening rules and assumed that each component of back stress i has a critical state for its dynamic recovery term. Only after reaching the critical state can the dynamic recovery terms work fully. According to the way the dynamic recovery term is used before the critical 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: 2 M P 2 i Model ðIÞ ¼ i ; di ¼ i ri d"p Hðfi Þ d"p :
i ; fi ¼ i r2i ð5Þ 3
i 1 wherep
i, is ith component of deviatoric back stress , i is the magnitude of i, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
i ¼ 3=2i :i and 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) and after that it does not evolve. Consequently, the model becomes a multilinear model in uniaxial cases. The Ohno–Wang model (I) produces closed hysteresis loops and hence cannot produce any uniaxial ratcheting. To eliminate this limitation, Ohno and Wang proposed a slight nonlinearity for each rule by introducing an exponential relation and before reaching its critical state the dynamic recovery term is partially operative. The formula is proposed as follows: mi M P 2
i i ð6Þ Model ðIIÞ ¼ i ; di ¼ i ri d"p i d"p : 3 ri
i i¼1 where mi is a material constant and when mi ! þ1, the Ohno–Wang model (II) is reduced to the Ohno–Wang model (I). In the Ohno–Wang model (II), the slight nonlinearity 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 always works in the form of an exponential relation that produces unclosed hysteresis loops in 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, under uniaxial and multiaxial conditions, the Ohno–Wang models can predict smaller ratcheting strain and in some degree simulate the nonlinear part of a ratcheting strain accumulation curve that is similar to experiments. What’s more, Ohno–Wang models use the term i d"p :
i
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in place of dp in the Armstrong and Frederick rule. Although this term is the same as dp in uniaxial cases, in multiaxial loading cases, d"p and ði = i Þ have different directions and the projection result makes i d"p :
i smaller than dp. Hence, a model embracing the former term predicts less development of multiaxial ratcheting strain than a model with the latter term as demonstrated by Ohno 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 a nonlinear hysteresis curve well. In this study, it is found that eight hardening rules are sufficient to obtain a good stable uniaxial hysteresis loop simulation for 1Cr18Ni9Ti stainless steel. 3.1.2. Parameter determination Model parameters are determined by the tensile curve from uniaxial loading (Ohno, 1998). The uniaxial–loading tensile curve is divided into several segments as shown in Fig. 5 and the corresponding parameters i, ri for each segment can be determined from the following equations:
Fig. 5. Definition of parameters in the Ohno–Wang model from uniaxial cyclic stress–strain space.
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ðiÞ ði1Þ ðiþ1Þ ðiÞ i ¼ ; ri ¼ "pðiÞ For i 6¼ 1 "pðiÞ "pðiÞ "pði1Þ "pðiþ1Þ "pðiÞ 1
879
ð7aÞ
and finally ri is determined by using M X ri þ 0 ¼ max
ð7bÞ
i¼1
where, (i) and "p(i) denote stress and plastic strain at the ith point on the monotonic tensile stress versus plastic strain curve and ðMÞ ¼ ðMþ1Þ . In the Ohno–Wang model (II), the power mi is an important parameter controlling ratcheting response and may be determined by a uniaxial ratcheting experiment response. Predicted ratcheting with different mi is compared in Fig. 6 from which it can 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 than that by the Ohno–Wang model (II). That is to say, under the same condition, the prediction of Ohno–Wang model (I) is the smallest prediction that can be made by the 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–Wang model (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 the Ohno–Wang model (II). In this paper a larger mi is assumed (mi=10) to simulate the multiaxial ratcheting first and the shortcoming in the simulation will be solved by adding 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.
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Table 3 Model parameters for 1Cr18Ni9Ti stainless steel 0 o
0 (MPa)
E (MPa)
mi (i=1 M)
235
193 000 10 0.07 18=4800, 2400, 1200, 600, 300, 150, 75, 37.5 r18=10, 65, 63, 41, 80, 70, l6, 2 MPa
0 st
0.005
8
The parameters in the Ohno–Wang model used in this study for simulations are presented in Table 3. 3.1.3. Simulations of experimental results The 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 existing models, the Ohno–Wang models can describe the uniaxial and torsional hysteresis curves well but over-predicts multiaxial ratcheting though with a smaller overprediction than the Chaboche model, modified models by McDowell, and Jiang and Sehitoglu (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 the Ohno–Wang model (II). But if experimental ratcheting is even much smaller than the predicted ratcheting by Ohno–Wang model (I), the Ohno–Wang models lack other parameters that can be adjusted to decrease the biaxial ratcheting simulation and 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 new kinematic hardening model in the framework of the Ohno–Wang model can be supposed, in which one parameter 0 connecting the radial evanescence term [(:n)ndp] of the Burlet–Cailletaud model (1986) with the dynamic recovery term of the Ohno–Wang model (1993a,b) is proposed in order to improve the ratcheting simulations 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: 2 @f=@ d ¼ Cd"p ð:nÞndp; n ¼ rffiffiffi 3
@f
¼ 3 ðs Þ
@
2
ð8Þ
0
The plastic modulus expression :
obtained from this hardening rule by satisfying the consistency condition f ¼ 0 is the same as that obtained from the Armstrong and
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Fig. 7. Cyclic stable strain–stress curves under uniaxial and torsional loading: (a) uniaxial, (b) torsional.
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 reduced to the dynamic recovery term of Armstrong and Frederick. In addition, because simulations of uniaxial ratcheting responses depend entirely on the calculation scheme of the plastic modulus of a model, these two rules produce the same simulation; while for biaxial loading, the radial evanescence term [(:n)ndp] of the Burlet
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Fig. 8. Cyclic stable strain–stress curve simulated by Ohno–Wang model (II) for M130.
and Cailletaud rule essentially yields a tensor along the plastic strain-rate direction and the simulation of biaxial ratcheting is like the result of Prager (1956) linear hardening rule that predicts shakedown ratcheting (Bari and Hassan, 2002). Between over-prediction ratcheting by the Ohno–Wang model and shakedown ratcheting of the Burlet and Cailletaud model, this study obtains a modified hardening rule incorporating the ideas of both the Burlet–Cailletaud and the Ohno– Wang models with a parameter 0 as follows: mi 2
i i 0 0 di ¼ i ri d"p ½ i þ ð1 Þði :nÞn d"p : ; i ¼ 1; 2; . . . M ð9Þ 3 ri
i 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 predicts the 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: M X 3 i mi i Hp ¼ i r i ¼ ði :nÞ :n ð10Þ 2 ri
i i¼1 In Eq. (10), it can be seen that the plastic modulus expression (Hp) does not include 0 and 0 can be determined by a biaxial ratcheting response, so 0 can influence biaxial ratcheting simulations without having any effect on both the calculation of plastic modulus and the simulations of uniaxial ratcheting responses.
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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 two models 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.
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are presented in Fig. 10 in which we can see that if a larger 0 is assumed, the modified model can simulate the initial nonlinear part but cannot provide the subsequent ratcheting rate trend well, while if a smaller 0 is assumed, the modified model can predict the ratcheting rate trend well but cannot simulate the initial nonlinear part of ratcheting curve. So we can come to the conclusion that the modified model with a constant 0 cannot predict a good simulation of the whole ratcheting curve. Hence it is better to give 0 an evolutionary character to improve the simulation.
Fig. 11. The influence of 0 st, 0 o, and on ratcheting strain; (a) 0 st, (b) 0 o, (c) .
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Fig. 11. (continued)
Fig. 12. Comparison of experimental ratcheting and predicted ratcheting by the modified model with an evolutional 0 for M130.
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An evolutionary equation for 0 is proposed as follows:
d0 ¼ 0st 0 dp
ð11Þ
where, 0 st is the saturated value of 0 and is an evolutionary coefficient. The initial value of 0 is denoted by 0 o. The computation for different 0 st (0 o, and are kept constant) shows that the value of 0 st allows adjustment of the slope of ratcheting rate trend as shown in Fig. 11 (a). The computation for different 0 o (0 st and are kept constant) reveals that 0 o is closely related to the ratcheting rate of the first several cycles as shown in Fig. 11 (b) and decides the ratcheting evolutive rate as shown in Fig. 11 (c). So by a biaxial ratcheting experiment curve (see Fig. l2), 0 o is assumed to give a good simulation of the first ratcheting while 0 st is decided by the ratcheting rate trend and is evaluated to well simulate the ratcheting evolution rate. The value of 0 st, 0 o, 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 modified model-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 this paper. Three test paths, axial strain cycle with constant pressure (case 1), bow-tie cycle (case 2), and reverse bow-tie cycle (case 3), are shown in Fig. 17. Ratcheting simulation 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 with the Ohno–Wang model, the modified model with constant 0 =0.6, can give better simulation of the experiments (see Figs. 18 and 19 in this paper and Fig. 14 of Bari and Hassan, 2000). The modified model with the constant 0 simulates the ratcheting rate trend well as shown in Fig. 18, but fails to predict the initial nonlinear part of ratcheting curves reasonably (It can be seen from other simulations to experiments data 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 0 st, 0 o, 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 (Modified model-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 modified models based on Ohno–Wang model by McDowell (1995), Jiang and Sehitoglu
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(1996), have demonstrated that most of these models are not robust enough to simulate the set of biaxial ratcheting responses (Bari and Hassan, 2002). In this study, 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 can control the biaxial ratcheting, which leads to the failure of describing the yield surface normal directions that are decided by the kinematic hardening rule of a model. 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.
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for a coupled model to introduce special biaxial terms or parameters 0 in the kinematic hardening rule. As the actual yield surface deforms during plastic loading, using multiaxial ratcheting responses by calibrating these parameters determined by a muliaxial experiment will compensate the adverse influence of the lack of exactness introduced in the modeling through the assumption of invariant yield surface shape (Phillips and Tang, 1972; Phillips and Lee, 1979). These terms and parameters should be recessive under uniaxial conditions but will play an important role in describing the yield surface normal and thus produce a different plastic strain direction under multiaxial condition. The parameter 0 in the Delobelle model introduced 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.
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Fig. 14. (continued) 0
character and the existence of 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 of Ohno–Wang kinematic hardening rule, into the framework of the Ohno–Wang model. The new parameter 0 is not involved in the plastic modulus calculation scheme, so the plastic modulus expression of the modified rule is the same as that of the Ohno–Wang model and all parameters determined completely from a uniaxial experiment 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 be determined by this tension–torsion experiment and it can adjust the predicted ratcheting to the range of over-prediction of Ohno–Wang model and the shakedown
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Fig. 15. Comparison of experimental and predicted ratcheting for M150; (a) shear stress–strain, (b) axial ratcheting strain.
of Burlet–Cailletaud model and predict a smaller biaxial ratcheting compared with the Ohno–Wang model. 0 is effective in adjusting the model to the predicted ratcheting when other parameters are kept unchanged. Computations for different values of 0 show that the ratcheting decreases with decrease of 0 , as shown in Fig. 10. This is 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 with constant 0 cannot simulate the initial nonlinear part and the subsequent rate trend
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Fig. 16. Comparison of experiments and predictions for M150; loading path, (b) axial stress–strain response, (c) stress response, (d) strain response.
of a ratcheting curve at the same time. So in order to improve the simulation of the modified model, 0 is made to evolve with plastic strain accumulation as given by Eq. (11). According to the ratcheting curve, 00 (the initial value of 0 ) is assumed to simulate ratcheting curve of the first several cycles while st0 (the saturated value of 0 ) can be selected to predict the experimental ratcheting rate trend well and the evolutional coefficient can be selected to give a good simulation of the evolutive ratcheting rate as shown in Fig. 10. Analyzed from the perspective of the modified model, 0 is a coefficient that denotes the proportional relation between the radial evanescence term [(:n)ndp] and the dynamic recovery term. The smaller 0 , the larger proportion of the radial evanescence term and the smaller the predicted ratcheting by the modified model. With the evolutional function of 0 , simulations of
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Fig. 16. (continued)
M140 and M150 by the modified model appear to be in a reasonably better agreement 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 seen that the modified model not only predicts the axial ratcheting strain and the stable stress–strain hysteresis loop with reasonable accuracy, but it also simulates the stress response, the strain response and axial stress–strain response well to some degree. It is seen that the modified model can well simulate the evolving relation between axial stress, 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 to compare with the simulations by the modified model as shown in Figs. 18 and 19.
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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).
We can also find that the modified model with the constant parameter 0 cannot simulate the initial nonlinear part of ratcheting curves well. 0 is expressed as an evolutional 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 of modified 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–Wang model [see Fig. 14 in the paper of Bari and Hassan (2000)], the simulations of ratcheting response by the modified model are improved. The modified model can give good predictions on different loading paths such as an axial strain cycle with constant internal pressure, circumferential strain peaks from ‘‘bow-tie’’ and reverse ‘‘bow-tie’’ cycles, which proves the modified model is valid and robust.
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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).
The prediction of ratcheting strain to a high number of cycles and the simulations of ratcheting on under changeable cyclic loading path are not found in the literature and this remains an open issue. McDowell (1995) proposed an equation for decay of
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Fig. 19. (continued)
Table 4 Model parameters for CS 1026 of Bari and Hassan (2000) 0 (ksi) 18.8
E (ksi)
mi (i=1M)
0 o
0 st
26 300 0.302 0.45 0.15 0.6 112=45 203, 13 944, 7728, 4955, 3692, 2135, 1230, 585, 295, 119, 50, 20 r112=0.707, 2.597, 0.326, 0.076, 2.985, 2.132, 2.825, 3.754, 2.905, 2.076, 1.96, 10
5
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ratcheting strain after 25–50 cycles to substitute for the integration of the constitutive equation over the entire history of loading. Bari and Hassan (2001), and suggested that it would be necessary to introduce anisotropy into the yield surface to enhance the predictive capability of ratcheting strain beyond the current assumption of invariant yield surface shape. The distortion model of subsequent yield surfaces was introduced into nonlinear kinematic constitutive equations to consistent with multiaxial ratcheting modeling (Vincent et al., 2002; Francois, 2001). More efforts are certainly needed for more reliable prediction methods. Thus a comparative evaluation of the proposed model with other existing models and data in the literatures is desirable. More comprehensive verification of the model and further improvement remain as future work.
5. Conclusions Ratcheting tests were conducted on 1Cr18Ni9Ti stainless steel for two nonproportional loading paths. A modified kinematic hardening rule that incorporates the radial evanescence term [(:n)ndp] of the Burlet–Cailletaud model with the Ohno–Wang kinematic hardening rule is proposed. All parameters except a new parameter 0 of the modified rule are the same as those of the Ohno–Wang models and 0 can be determined with a biaxial ratcheting response. The parameter determination scheme for this modified model is simple and systematic. In order to improve the simulation to all parts of the ratcheting curves, an evolving parameter 0 is introduced into the modified model. The model predicts stable stress–strain behavior of the test material with reasonable accuracy. Ratcheting simulations of both two types of loading paths are reasonably accurate for experimental data at low numbers of cycles. In order to confirm the validity of the modified model and the introduction of the parameter 0 in the model, simulations to the experiments data 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 National Natural Science Foundation of China (project Nos. 19872049, 10272080) and TRAPOYT.
References Abdel-Kanim, M., Ohno, N., 2000. Kinematic hardening model suitable for ratcheting with steady-state. International Journal of Plasticity 16, 225–240. Armstrong, P.J., Frederick, C.O., 1966. A Mathematical Representation of the Multiaxial Bauschinger Effect. CEGB Report No. RD/B/N 731.
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Bari, S., Hassan, T., 2000. Anatomy of coupled constitutive model for ratcheting simulation. International Journal of Plasticity 16, 381–409. Bari, S., Hassan, T., 2001. Kinematic hardening rules in uncoupled modeling for multiaxial ratcheting simulation. International Journal of Plasticity 17, 885–905. Bari, S., Hassan, T., 2002. An advancement in cyclic plasticity modeling for multiaxial ratcheting simulation. International Journal of Plasticity 18, 873–894. Bocher, L., Delobelle, P., Robinet, P., Feaugas, X., 2001. Mechanical and microstructural investigations of an austenitic stainless steel under non-proportional loadings in tension–torsion—internal and external pressure. International Journal of Plasticity 17, 1491–1530. Bower, A.F., 1989. Cyclic hardening properties of hard-drawn copper and rail steel. Journal of the Mechanics and Physics of Solids 37, 455–470. Burlet, H., Cailletaud, G., 1986. Numerical techniques for cyclic plasticity at variable temperature. Engineering Computation 3, 143–153. Chaboche, J.L., 1991. On some modifications of kinematic hardening to improve the description of ratcheting effects. International Journal of Plasticity 7, 661–678. Chaboche, J.L., 1994. Modeling of ratchetting: evaluation of various approaches. European Journal of Mechanics, A/Solids 13, 501–518. Chaboche, J.L., Nouailhas, D., 1989a. Constitutive modelling of ratchetting effects. Part I: experimental facts and properties of classical models. ASME Journal of Engineering Materials and Technology 3, 384–390. Chaboche, J.L., Nouailhas, D., 1989b. Constitutive modeling of ratchetting effects. Part II: possibilities of some additional kinematic rules. ASME Journal of Engineering Materials and Technology 3, 409–416. Chen, X., Tian, T., An, K., 2001. Non-proportional cyclic hardening behaviors of 1Cr18Ni9Ti stainless steel. Acta Mechanica Sinica 33, 698–705. Chun, B.K., Jinn, J.T., Lee, J.K., 2002. Modeling the Bauschinger effect for sheet metals, part I: theory. International Journal of Plasticity 18, 571–595. Corona, E., Hassan, T., Kyriakides, S., 1996. On the performances of kinematic hardening rules in predicting a class of biaxial ratcheting histories. International Journal of Plasticity 12, 117–145. Delobelle, P., Robinet, P., Bochen, L., 1995. Experimental study and phenomelogical modelization of ratcheting under uniaxial and biaxial loading on an austenite stainless steel. International Journal of Plasticity 11, 295–330. Franc¸ois, M., 2001. A plasticity model with yield surface distortion for non proportional loading. International Journal of Plasticity 17, 703–717. Geng, L., Shen, Y., Wagoner, R.H., 2002. Anisotnopic hardening equations derived from reverse-bend testing. International Journal of Plasticity 18, 743–767. Hassan, T., Kyriakides, S., 1992. Ratcheting in cyclic plasticity, part I: uniaxial behavior. International Journal of Plasticity 8, 91–116. Hassan, T., Corona, E., Kyriakides, S., 1992. Ratcheting in cyclic plasticity, part II: multiaxial behavior. International Journal of Plasticity 8, 117–146. Hassan, T., Kyriakides, S., 1994a. Ratcheting of cyclically hardening and softening materials, part I: uniaxial behavior. International Journal of Plasticity 10, 149–184. Hassan, T., Kyriakides, S., 1994b. Ratcheting of cyclically hardening and softening materials, part II: multiaxial behavior. International Journal of Plasticity 10, 185–212. Ho, K., Krempl, E., 2002. Extension of the viscoplasticity theory based on overstress (VBO) to capture non-standard rate dependence in solids. International Journal of Plasticity 18, 851–872. Igaria, T., Kobayashi, M., Yoshida, F., Imatani, S., Inoue, T., 2002. Inelastic analysis of new thermal ratcheting due to a moving temperature front. International Journal of Plasticity 18, 1191–1217. Jiang, Y., Sehitoglu, H., 1994a. Cyclic ratcheting of 1070 steel under multiaxial stress state. International Journal of Plasticity 10, 579–608. Jiang, Y., Sehitoglu, H., 1994b. Multiaxial cyclic ratcheting under multiple step loading. International Journal of Plasticity 10, 849–870. Jiang, Y., Sehitoglu, H., 1996. Modeling of cyclic ratcheting plasticity, part 1: development of constitutive relations. ASME Journal of Applied Mechanics 63, 720–725.
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Jiang, Y., Kurath, P., 1996. Characteristics of the Armstrong-Frederick type plasticity models. International Journal of Plasticity 8, 117–146. Krempl, E., Khan, F., 2003. Rate (time)-dependent deformation behavior: an overview of some properties of metals and solid polymers. International Journal of Plasticity 19, 1069–1095. McDowell, D.L., 1994. Description of nonproportional cyclic ratcheting behavior. European Journal of Mechanics, A/Solids 13, 593–604. McDowell, D.L., 1995. Stress state dependence of cyclic ratcheting behavior of two rail steels. International Journal of Plasticity 11, 397–421. Ohno, N., Wang, J.-D., 1993a. Kinematic hardening rules with critical state of dynamic recovery, part I: formulations and basic features for ratcheting behavior. International Journal of Plasticity 9, 375–390. Ohno, N., Wang, J.D., 1993b. Kinematic hardening rules with critical state of dynamic recovery, part II: application to experiments of ratcheting behavior. International Journal of Plasticity 9, 391–403. Ohno, N., 1998. Constitutive modeling of cyclic plasticity with emphasis on ratcheting. International Journal of Mechanical Sciences 40, 251–261. Phillips, A., Tang, J.L., 1972. The effect of loading path on the yield surface at elevated temperatures. International Journal of Solids Structures 8, 463–474. Phillips, A., Lee, C.W., 1979. Yield surfaces and loading surfaces: experiments and recommendations. International Journal of Solids Structures 15, 715–729. Portier, L., Calloch, S., Marquis, D., Geyer, P., 2000. Ratcheting under tension-torsion loadings: experiments and modeling. International Journal of Plasticity 16, 303–335. Prager, W., 1956. A new method of analyzing stress and strain in work hardening plastic solids. Journal of Applied Mechanics 23, 493–496. Rider, R.J., Harvey, S.J., Chandler, H.D., 1995. Fatigue and ratcheting interactions. International Journal of Fatigue 17, 507–511. Vincent, L., Colloch, S., Kurtyka, T., Marquis, D., 2002. An improvement of multiaxial ratcheting modeling via yield surface distortion. ASME Trans. J. Engng Mater. Tech. 124, 402–411. Voyiadjis, G.Z., Basuroychowdhury, I.N., 1998. A plasticity model for multiaxial cyclic loading and ratcheting. Acta Mechanica 126, 19–35. Yaguchi, M., Yamamoto, M., Ogata, T., 2002. A viscoplastic constitutive model for nickel-base superalloy, part 1: kinematic hardening rule of anisotropic dynamic recovery. International Journal of Plasticity 16, 1083–1109. Yoshida, F., 2000. A constitutive model of cyclic plasticity. International Journal of Plasticity 16, 359– 380. Yoshida, F., Uemoni, T., 2002. A model of large-strain cyclic plasticity describing the Bauschinger effect and workhardening stagnation. International Journal of Plasticity 18, 661–686.
Modified kinematic hardening rule for multiaxial ratcheting 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 model with the dynamic recovery term of Ohno–Wang kinematic hardening rule is introduced into the framework of the Ohno–Wang model. Compared with multiaxial ratcheting experimental data obtained on 1Cr18Ni9Ti stainless steel in the paper and CS1026 steel conducted by Hassan et al. [Int. J. Plasticity 8 (1992) 117], simulation results by modified model are quite well 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 is added 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 cycle under stress-controlled conditions, is an important factor in designing structure components. The ratcheting deformation could accumulate continuously with the increasing number of cycles applied, and it may not cease until fracture. Ratcheting deformation contributes to material 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 ratcheting behavior (Chaboche, 1994; McDowell, 1994; Ohno and Wang, 1993a,b; Ohno, 1998; Bari and Hassan, 2000, 2001, 2002). Ratcheting experiments have been conducted * Corresponding author. Tel./fax: +86-22-8789-3037. E-mail address: [email protected] (X. Chen). 0749-6419/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijplas.2003.05.005
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Nomenclature d dp n s E G Hp N " "x "xc "xp " "p d"e d"p f n d 0 x xa a m mean
Deviatoric backstress tensor Incremental deviatoric backstress tensor Magnitude of the plastic strain increment tensor Unit normal to the yield surface at current stress point Deviatoric stress Young’s modulus for elasticity Shear modulus Plastic modulus Number of loading cycles Strain tensor Axial strain Amplitude of axial strain cycle Maximum axial strain in a cycle Circumference strain Maximum circumference strain in a cycle Elastic strain increment tensor Plastic strain increment tensor Yield surface function Axial stress range Poisson’s ratio Stress tensor Incremental stress tensor Size of yield surface Total axial stress Amptude of axial stress cycle Circumferential stress Amplitude of circumferential stress cycle Mean of circumferential stress cycle Mean of axial stress cycle
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 important in cyclic plasticity simulation (Chun et al., 2002; Yoshida and Uemori, 2002; Geng et al., 2002; Yaguchi et al., 2002). Several kinematic hardening rules have been proposed for predicting of ratcheting under multiaxial loading. The nonlinear kinematic hardening rule by Armstrong and Frederick (1966) was found to over-predict ratcheting strain significantly under multiaxial loading paths. Several authors got their models by modifying the dynamic recovery term in Armstrong and Frederick model (Bower, 1989; Chaboche and Nouailhas, 1989a,b; Chaboche, 1991; Yoshida, 2000; Ohno and Wang, 1993a,b; Jiang and Sehitoglu, 1994a,b; McDowell, 1995;
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Abdel-Karim and Ohno, 2000). Chaboche (1991) showed that a threshold for dynamic recovery of back stress is effective for controlling ratcheting in simulation. Ohno and Wang (1993a,b) introduced a critical state of dynamic recovery, and they showed that the critical state expresses no or little ratcheting under uniaxial cyclic loading within the framework of the strain hardening and dynamic recovery format. Nonlinear forms of the dynamic recovery term were then discussed for simulating ratcheting appropriately (Ohno and Wang, 1993a,b; Chaboche, 1994; McDowell, 1995) while decaying ratcheting was thus discussed in detail from the view of constitutive 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 the kinematic hardening rule and the consistency condition. Usually the parameters are calculated from the hysteresis loops and uniaxial loading responses. These parameters are, in effect, calibrated to produce a better representation of the hysteresis loop and uniaxial ratcheting, however they fail to predict multiaxial ratcheting responses. In order to solve the problem of over-prediction by the existing models on multiaxial ratcheting responses, many researchers (McDowell, 1995; Jiang and Sehitoglu, 1996; Voyiadjis and Basurychoedhury, 1998) have attempted to add multiaxial terms and parameters into the Chaboche or Ohno–Wang model. However, these modified models do not improve the simulation of the biaxial ratcheting responses compared with the Ohno–Wang model (Bari and Hassan, 2002). Thus Bari and Hassan proposed a modified kinematic hardening rule based on the idea of Delobelle et al (1995) in the framework of the Chaboche model. Since the Ohno–Wang model is regarded as the best model to predict ratcheting by the researchers (Igaria et al., 2002), it is reasonable to do some modification in the framework of the Ohno–Wang model. This study is just to propose an improved kinematic hardening rule by introducing one multiaxial parameter 0 to the Ohno– Wang model aiming at investigating the modified model for its validity and applicability of predicting ratcheting under several different multiaxial loading paths. Although rate-dependent constitutive model has been make great progress (Krempl and Khan, 2003; Ho and Krempl, 2002), but in order to simplify the problem, a rate-independent model is considered in the paper.
2. Ratcheting experiments The material used in the study was 1Cr18Ni9Ti stainless steel in the form of round bar with a diameter of 32 mm after being oil-quenched at 1100 C for 30 min. The chemical composition of the material is (wt.%): C 0.065, Mn 1.34, Si 0.95, P 0.03, S 0.007, Ni 8.74, Cr 17.54, Ti 0.41. The mechanical properties of 1Cr18Ni9Ti stainless steel 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. The tests were conducted on an Instron tension–torsion machine with an MTS axialtorsional extensometer mounted on the outside of the specimen gage section. Strain
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and stress was recorded in the personal computer using an automated data acquisition system. All tests were conducted at room temperature under stress control for axial loading and under strain control for torsional loading. The frequency of cyclic loading was 0.5 Hz. pffiffiffi The loading paths in the axial stress–shear strain plane ( = 3 plane) used in ratcheting tests are illustrated schematically in Fig. 2. The controlled parameters are given in Table 2. These tests consist of a constant-amplitude shear strain cycling under a constant axial stress (case 1) and a circular axial stress–shear strain cyclic loading with mean axial stress (case 2). Table 1 Mechanical properties of 1Cr18Ni9Ti stainless steel b (MPa)
s0.2 (MPa)
605
310
(%) 75
5 (%)
E (GPa)
G (GPa)
HB
60
193
65.4
0.47
160
Fig. 1. Specimen geometry (mm).
Fig. 2. Loading paths in ratcheting experiments.
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For 1Cr18Ni9Ti stainless steel, the ratcheting experiments reveal that the rate of ratcheting continuously decreases as cycling continues, but does not fully shakedown or cease. The observations of the uniaxial cyclic stress–strain curve for first 16 cycles 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 constitutive models (McDowell, 1995; Jiang and Sehitoglu, 1996; Jiang and Kurath, 1996). 1Cr18Ni9Ti stainless steel presents significant non-proportional additional hardening 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 low under axial stress–shear strain cyclic loading. A comparison of torsional stress– strain curves between pure torsion, case 1 and case 2 cyclic loading shows additional hardening can be neglected (see Fig. 4).
Table 2 List of ratcheting experiments Spec. no.
Path
"2/2 (%)
mean (%)
(MPa)
mean (MPa)
M130 M140 M150
Case 1 Case 2 Case 2
0.4 0.4 0.6
0 0 0
0 200 200
200 200 200
Fig. 3. The first 16 cyclic stress–strain response of 1Cr18Ni9Ti stainless steel under uniaxial cyclic loading.
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Fig. 4. Comparison of torsional stress–strain curves of pure torsion, case 1 and case 2 for 1Cr18Ni9Ti stainless steel.
3. Description of the constitutive model It is assumed that the total strain increment is decomposed into an elastic strain part and a plastic strain part: d" ¼ d"e þ d"p
ð1Þ
and that the elastic part obeys Hooke’s law: "e ¼
1þ ðtrÞI E E
ð2Þ
The plastic flow rule can be stated as d"p ¼
3 1 hds:nin 2 Hp
ð3Þ
The material is assumed to follow the von Mises yield criterion, which is given by 3 f ¼ ðs Þ : ðs Þ 02 ¼ 0 2
ð4Þ
where s ¼ 13 trðÞI is the deviatoric stress tensor, is the back stress, and 0 is the size of the yield surface, I is a unit tensor.
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3.1. Ohno–Wang model 3.1.1. Model formulation In order to better describe ratcheting behavior, Ohno and Wang formulated kinematic hardening rules superposed several A–F kinematic hardening rules and assumed that each component of back stress i has a critical state for its dynamic recovery term. Only after reaching the critical state can the dynamic recovery terms work fully. According to the way the dynamic recovery term is used before the critical 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: 2 M P 2 i Model ðIÞ ¼ i ; di ¼ i ri d"p Hðfi Þ d"p :
i ; fi ¼ i r2i ð5Þ 3
i 1 wherep
i, is ith component of deviatoric back stress , i is the magnitude of i, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
i ¼ 3=2i :i and 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) and after that it does not evolve. Consequently, the model becomes a multilinear model in uniaxial cases. The Ohno–Wang model (I) produces closed hysteresis loops and hence cannot produce any uniaxial ratcheting. To eliminate this limitation, Ohno and Wang proposed a slight nonlinearity for each rule by introducing an exponential relation and before reaching its critical state the dynamic recovery term is partially operative. The formula is proposed as follows: mi M P 2
i i ð6Þ Model ðIIÞ ¼ i ; di ¼ i ri d"p i d"p : 3 ri
i i¼1 where mi is a material constant and when mi ! þ1, the Ohno–Wang model (II) is reduced to the Ohno–Wang model (I). In the Ohno–Wang model (II), the slight nonlinearity 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 always works in the form of an exponential relation that produces unclosed hysteresis loops in 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, under uniaxial and multiaxial conditions, the Ohno–Wang models can predict smaller ratcheting strain and in some degree simulate the nonlinear part of a ratcheting strain accumulation curve that is similar to experiments. What’s more, Ohno–Wang models use the term i d"p :
i
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in place of dp in the Armstrong and Frederick rule. Although this term is the same as dp in uniaxial cases, in multiaxial loading cases, d"p and ði = i Þ have different directions and the projection result makes i d"p :
i smaller than dp. Hence, a model embracing the former term predicts less development of multiaxial ratcheting strain than a model with the latter term as demonstrated by Ohno 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 a nonlinear hysteresis curve well. In this study, it is found that eight hardening rules are sufficient to obtain a good stable uniaxial hysteresis loop simulation for 1Cr18Ni9Ti stainless steel. 3.1.2. Parameter determination Model parameters are determined by the tensile curve from uniaxial loading (Ohno, 1998). The uniaxial–loading tensile curve is divided into several segments as shown in Fig. 5 and the corresponding parameters i, ri for each segment can be determined from the following equations:
Fig. 5. Definition of parameters in the Ohno–Wang model from uniaxial cyclic stress–strain space.
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ðiÞ ði1Þ ðiþ1Þ ðiÞ i ¼ ; ri ¼ "pðiÞ For i 6¼ 1 "pðiÞ "pðiÞ "pði1Þ "pðiþ1Þ "pðiÞ 1
879
ð7aÞ
and finally ri is determined by using M X ri þ 0 ¼ max
ð7bÞ
i¼1
where, (i) and "p(i) denote stress and plastic strain at the ith point on the monotonic tensile stress versus plastic strain curve and ðMÞ ¼ ðMþ1Þ . In the Ohno–Wang model (II), the power mi is an important parameter controlling ratcheting response and may be determined by a uniaxial ratcheting experiment response. Predicted ratcheting with different mi is compared in Fig. 6 from which it can 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 than that by the Ohno–Wang model (II). That is to say, under the same condition, the prediction of Ohno–Wang model (I) is the smallest prediction that can be made by the 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–Wang model (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 the Ohno–Wang model (II). In this paper a larger mi is assumed (mi=10) to simulate the multiaxial ratcheting first and the shortcoming in the simulation will be solved by adding 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.
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Table 3 Model parameters for 1Cr18Ni9Ti stainless steel 0 o
0 (MPa)
E (MPa)
mi (i=1 M)
235
193 000 10 0.07 18=4800, 2400, 1200, 600, 300, 150, 75, 37.5 r18=10, 65, 63, 41, 80, 70, l6, 2 MPa
0 st
0.005
8
The parameters in the Ohno–Wang model used in this study for simulations are presented in Table 3. 3.1.3. Simulations of experimental results The 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 existing models, the Ohno–Wang models can describe the uniaxial and torsional hysteresis curves well but over-predicts multiaxial ratcheting though with a smaller overprediction than the Chaboche model, modified models by McDowell, and Jiang and Sehitoglu (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 the Ohno–Wang model (II). But if experimental ratcheting is even much smaller than the predicted ratcheting by Ohno–Wang model (I), the Ohno–Wang models lack other parameters that can be adjusted to decrease the biaxial ratcheting simulation and 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 new kinematic hardening model in the framework of the Ohno–Wang model can be supposed, in which one parameter 0 connecting the radial evanescence term [(:n)ndp] of the Burlet–Cailletaud model (1986) with the dynamic recovery term of the Ohno–Wang model (1993a,b) is proposed in order to improve the ratcheting simulations 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: 2 @f=@ d ¼ Cd"p ð:nÞndp; n ¼ rffiffiffi 3
@f
¼ 3 ðs Þ
@
2
ð8Þ
0
The plastic modulus expression :
obtained from this hardening rule by satisfying the consistency condition f ¼ 0 is the same as that obtained from the Armstrong and
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Fig. 7. Cyclic stable strain–stress curves under uniaxial and torsional loading: (a) uniaxial, (b) torsional.
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 reduced to the dynamic recovery term of Armstrong and Frederick. In addition, because simulations of uniaxial ratcheting responses depend entirely on the calculation scheme of the plastic modulus of a model, these two rules produce the same simulation; while for biaxial loading, the radial evanescence term [(:n)ndp] of the Burlet
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Fig. 8. Cyclic stable strain–stress curve simulated by Ohno–Wang model (II) for M130.
and Cailletaud rule essentially yields a tensor along the plastic strain-rate direction and the simulation of biaxial ratcheting is like the result of Prager (1956) linear hardening rule that predicts shakedown ratcheting (Bari and Hassan, 2002). Between over-prediction ratcheting by the Ohno–Wang model and shakedown ratcheting of the Burlet and Cailletaud model, this study obtains a modified hardening rule incorporating the ideas of both the Burlet–Cailletaud and the Ohno– Wang models with a parameter 0 as follows: mi 2
i i 0 0 di ¼ i ri d"p ½ i þ ð1 Þði :nÞn d"p : ; i ¼ 1; 2; . . . M ð9Þ 3 ri
i 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 predicts the 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: M X 3 i mi i Hp ¼ i r i ¼ ði :nÞ :n ð10Þ 2 ri
i i¼1 In Eq. (10), it can be seen that the plastic modulus expression (Hp) does not include 0 and 0 can be determined by a biaxial ratcheting response, so 0 can influence biaxial ratcheting simulations without having any effect on both the calculation of plastic modulus and the simulations of uniaxial ratcheting responses.
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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 two models 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.
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are presented in Fig. 10 in which we can see that if a larger 0 is assumed, the modified model can simulate the initial nonlinear part but cannot provide the subsequent ratcheting rate trend well, while if a smaller 0 is assumed, the modified model can predict the ratcheting rate trend well but cannot simulate the initial nonlinear part of ratcheting curve. So we can come to the conclusion that the modified model with a constant 0 cannot predict a good simulation of the whole ratcheting curve. Hence it is better to give 0 an evolutionary character to improve the simulation.
Fig. 11. The influence of 0 st, 0 o, and on ratcheting strain; (a) 0 st, (b) 0 o, (c) .
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Fig. 11. (continued)
Fig. 12. Comparison of experimental ratcheting and predicted ratcheting by the modified model with an evolutional 0 for M130.
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An evolutionary equation for 0 is proposed as follows:
d0 ¼ 0st 0 dp
ð11Þ
where, 0 st is the saturated value of 0 and is an evolutionary coefficient. The initial value of 0 is denoted by 0 o. The computation for different 0 st (0 o, and are kept constant) shows that the value of 0 st allows adjustment of the slope of ratcheting rate trend as shown in Fig. 11 (a). The computation for different 0 o (0 st and are kept constant) reveals that 0 o is closely related to the ratcheting rate of the first several cycles as shown in Fig. 11 (b) and decides the ratcheting evolutive rate as shown in Fig. 11 (c). So by a biaxial ratcheting experiment curve (see Fig. l2), 0 o is assumed to give a good simulation of the first ratcheting while 0 st is decided by the ratcheting rate trend and is evaluated to well simulate the ratcheting evolution rate. The value of 0 st, 0 o, 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 modified model-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 this paper. Three test paths, axial strain cycle with constant pressure (case 1), bow-tie cycle (case 2), and reverse bow-tie cycle (case 3), are shown in Fig. 17. Ratcheting simulation 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 with the Ohno–Wang model, the modified model with constant 0 =0.6, can give better simulation of the experiments (see Figs. 18 and 19 in this paper and Fig. 14 of Bari and Hassan, 2000). The modified model with the constant 0 simulates the ratcheting rate trend well as shown in Fig. 18, but fails to predict the initial nonlinear part of ratcheting curves reasonably (It can be seen from other simulations to experiments data 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 0 st, 0 o, 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 (Modified model-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 modified models based on Ohno–Wang model by McDowell (1995), Jiang and Sehitoglu
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(1996), have demonstrated that most of these models are not robust enough to simulate the set of biaxial ratcheting responses (Bari and Hassan, 2002). In this study, 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 can control the biaxial ratcheting, which leads to the failure of describing the yield surface normal directions that are decided by the kinematic hardening rule of a model. 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.
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for a coupled model to introduce special biaxial terms or parameters 0 in the kinematic hardening rule. As the actual yield surface deforms during plastic loading, using multiaxial ratcheting responses by calibrating these parameters determined by a muliaxial experiment will compensate the adverse influence of the lack of exactness introduced in the modeling through the assumption of invariant yield surface shape (Phillips and Tang, 1972; Phillips and Lee, 1979). These terms and parameters should be recessive under uniaxial conditions but will play an important role in describing the yield surface normal and thus produce a different plastic strain direction under multiaxial condition. The parameter 0 in the Delobelle model introduced 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.
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Fig. 14. (continued) 0
character and the existence of 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 of Ohno–Wang kinematic hardening rule, into the framework of the Ohno–Wang model. The new parameter 0 is not involved in the plastic modulus calculation scheme, so the plastic modulus expression of the modified rule is the same as that of the Ohno–Wang model and all parameters determined completely from a uniaxial experiment 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 be determined by this tension–torsion experiment and it can adjust the predicted ratcheting to the range of over-prediction of Ohno–Wang model and the shakedown
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Fig. 15. Comparison of experimental and predicted ratcheting for M150; (a) shear stress–strain, (b) axial ratcheting strain.
of Burlet–Cailletaud model and predict a smaller biaxial ratcheting compared with the Ohno–Wang model. 0 is effective in adjusting the model to the predicted ratcheting when other parameters are kept unchanged. Computations for different values of 0 show that the ratcheting decreases with decrease of 0 , as shown in Fig. 10. This is 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 with constant 0 cannot simulate the initial nonlinear part and the subsequent rate trend
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Fig. 16. Comparison of experiments and predictions for M150; loading path, (b) axial stress–strain response, (c) stress response, (d) strain response.
of a ratcheting curve at the same time. So in order to improve the simulation of the modified model, 0 is made to evolve with plastic strain accumulation as given by Eq. (11). According to the ratcheting curve, 00 (the initial value of 0 ) is assumed to simulate ratcheting curve of the first several cycles while st0 (the saturated value of 0 ) can be selected to predict the experimental ratcheting rate trend well and the evolutional coefficient can be selected to give a good simulation of the evolutive ratcheting rate as shown in Fig. 10. Analyzed from the perspective of the modified model, 0 is a coefficient that denotes the proportional relation between the radial evanescence term [(:n)ndp] and the dynamic recovery term. The smaller 0 , the larger proportion of the radial evanescence term and the smaller the predicted ratcheting by the modified model. With the evolutional function of 0 , simulations of
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Fig. 16. (continued)
M140 and M150 by the modified model appear to be in a reasonably better agreement 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 seen that the modified model not only predicts the axial ratcheting strain and the stable stress–strain hysteresis loop with reasonable accuracy, but it also simulates the stress response, the strain response and axial stress–strain response well to some degree. It is seen that the modified model can well simulate the evolving relation between axial stress, 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 to compare with the simulations by the modified model as shown in Figs. 18 and 19.
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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).
We can also find that the modified model with the constant parameter 0 cannot simulate the initial nonlinear part of ratcheting curves well. 0 is expressed as an evolutional 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 of modified 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–Wang model [see Fig. 14 in the paper of Bari and Hassan (2000)], the simulations of ratcheting response by the modified model are improved. The modified model can give good predictions on different loading paths such as an axial strain cycle with constant internal pressure, circumferential strain peaks from ‘‘bow-tie’’ and reverse ‘‘bow-tie’’ cycles, which proves the modified model is valid and robust.
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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).
The prediction of ratcheting strain to a high number of cycles and the simulations of ratcheting on under changeable cyclic loading path are not found in the literature and this remains an open issue. McDowell (1995) proposed an equation for decay of
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Fig. 19. (continued)
Table 4 Model parameters for CS 1026 of Bari and Hassan (2000) 0 (ksi) 18.8
E (ksi)
mi (i=1M)
0 o
0 st
26 300 0.302 0.45 0.15 0.6 112=45 203, 13 944, 7728, 4955, 3692, 2135, 1230, 585, 295, 119, 50, 20 r112=0.707, 2.597, 0.326, 0.076, 2.985, 2.132, 2.825, 3.754, 2.905, 2.076, 1.96, 10
5
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ratcheting strain after 25–50 cycles to substitute for the integration of the constitutive equation over the entire history of loading. Bari and Hassan (2001), and suggested that it would be necessary to introduce anisotropy into the yield surface to enhance the predictive capability of ratcheting strain beyond the current assumption of invariant yield surface shape. The distortion model of subsequent yield surfaces was introduced into nonlinear kinematic constitutive equations to consistent with multiaxial ratcheting modeling (Vincent et al., 2002; Francois, 2001). More efforts are certainly needed for more reliable prediction methods. Thus a comparative evaluation of the proposed model with other existing models and data in the literatures is desirable. More comprehensive verification of the model and further improvement remain as future work.
5. Conclusions Ratcheting tests were conducted on 1Cr18Ni9Ti stainless steel for two nonproportional loading paths. A modified kinematic hardening rule that incorporates the radial evanescence term [(:n)ndp] of the Burlet–Cailletaud model with the Ohno–Wang kinematic hardening rule is proposed. All parameters except a new parameter 0 of the modified rule are the same as those of the Ohno–Wang models and 0 can be determined with a biaxial ratcheting response. The parameter determination scheme for this modified model is simple and systematic. In order to improve the simulation to all parts of the ratcheting curves, an evolving parameter 0 is introduced into the modified model. The model predicts stable stress–strain behavior of the test material with reasonable accuracy. Ratcheting simulations of both two types of loading paths are reasonably accurate for experimental data at low numbers of cycles. In order to confirm the validity of the modified model and the introduction of the parameter 0 in the model, simulations to the experiments data 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 National Natural Science Foundation of China (project Nos. 19872049, 10272080) and TRAPOYT.
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