Acta Mechanica 163, 9–23 (2003) DOI 10.1007/s00707-003-1017-2
Acta Mechanica Printed in Austria
Modeling of ratcheting behavior under multiaxial cyclic loading X. Chen, Tianjin, China, and K. S. Kim, Pohang, Korea Received July 29, 2002; revised January 15, 2003 Published online: May 20, 2003 Springer-Verlag 2003
Summary. A two-surface plasticity theory is used to predict ratcheting strain under multiaxial loading. A kinematic hardening rule that combines the Mroz and Ziegler hardening rules is employed along with the plastic modulus given as an exponential function of the distance between the yield surface and the bounding surface. Model results are compared with the experimental data obtained on medium carbon steel under proportional and nonproportional axial-torsional loading. The model predicts reasonably well the experimental ratcheting behavior at relatively low cycles. Predictions overshoot the actual ratcheting strains at high cycles, yet the results look favorable compared with other data found in the literature.
1 Introduction When structural components are cyclically loaded in the plastic regime, progressive plastic deformation can occur by a combination of primary (steady) loading and secondary (cyclic) loading. This deformation behavior is referred to as ratcheting. The ratcheting deformation could accumulate continuously with the applied number of cycles, and it may not cease until fracture. Ratcheting deformation contributes to the damage of material and reduces fatigue life [1]. Although this behavior has been known for some time, it was not until recently that systematic studies have been conducted. A number of papers review the state of the art of modeling the ratcheting behavior [2]–[6]. Ratcheting experiments have been conducted on different materials under various loading conditions [7]–[13]. The studies with existing models indicate that the simulation of uniaxial ratcheting primarily depends on the plastic modulus calculation scheme, whereas in multiaxial loading ratcheting depends essentially on the kinematic hardening rule employed in the model [7]–[9]. Several kinematic hardening rules have been proposed for prediction of ratcheting under multiaxial loading. The conventional linear kinematic hardening rules such as Prager [14] and Ziegler [15] are inadequate for ratcheting modeling since they produce closed hystersis loops. Mroz¢ [16] kinematic hardening rule overpredicted the ratcheting strain [3]. The nonlinear kinematic hardening rule by Armstrong and Frederick [17] was found to overpredict ratcheting strain significantly under multiaxial loading. Several authors modified their model by introducing additional terms [18]–[24]. These new nonlinear kinematic hardening models can describe multiaxial cyclic ratcheting significantly better, but at the expense of increased complexity and model parameters. Also, as indicated by Bari and Hassan [5], [6],
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X. Chen and K. S. Kim
undesirable features in the predicted results of ratcheting strain still persist in most of these models. Ratcheting modeling may be formulated within the framework of the classical plasticity theory or by adopting multisurface plasticity models. In the former, the plastic modulus is coupled with the hardening rule through the consistency condition for plastic flow, while they are not explicitly coupled in the latter approach. The modified versions of the two-surface model proposed by Dafalias and Popov [25] have been most popularly used in the latter approach [9], [10], [26]–[30]. Bari and Hassan [6] evaluated a few of these models, along with other hardening rules used with the classical theory, and concluded that the Chaboche [24] kinematic hardening rule with three decomposed terms and the Ohno and Wang model [21] provided better results than others. In all these studies, however, the comparison of the predicted and experimental ratcheting strains is restricted to a low number of cycles, especially for complex biaxial loading paths. Chen and Abel [30] proposed a kinematic hardening rule that combines the Mroz and Ziegler hardening rules to model nonproportional cyclic stress-strain behavior and multiaxial cyclic ratcheting on the basis of a two-surface plasticity theory. This model has been successfully used to simulate cyclic nonporportional plasticity for type 304 stainless steel [31]. The present work is aimed at investigating the model for its capability of predicting ratcheting strain under multiaxial loading. Ratcheting tests were conducted on tubular specimens of medium carbon steel under four types of nonproportional axial-torsional loading. The model results will be compared with experimental data over a larger number of cycles than in other studies.
2 Ratcheting experiments The material used in the study was medium carbon steel S45C (in Korean materials designation), which was acquired in the form of a round bar with a diameter of 32 mm. The material was held at 850 C for 30 minutes and oil-quenched. It was then tempered at 600 C in the furnace for 40 minutes and air-cooled. The chemical composition of the material is (%wt): C 0.43, Si 0.18, Mn 0.69, P 0.023, S 0.007. The mechanical properties 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 12.5 mm and 10 mm, respectively, in the gage section. The tests were conducted on an Instron tension-torsion machine with an axial-torsional extensometer mounted on the outside of the specimen gage section. Strains and stresses were recorded in the personal computer using an automated data acquisition system. All tests were conducted at room temperature under load control for axial loading and under strain control for torsional loading. The frequency of cyclic loading was 0.5 Hz. Triangular waves were employed in the tests. The loading paths in the axial stress-shear strain plane (r c plane) used in ratcheting tests are illustrated schematically in Fig. 2. The controlled parameters are given in Table 2. These tests consist of (i) a constant amplitude shear strain cycling under a constant axial load (Case 1), (ii) a constant amplitude proportional path with mean axial stress (Case 2), (iii) a 90 out-of-phase Table 1. Mechanical properties of S45C steel ru (MPa)
r0:2 (MPa)
s0:2 (MPa)
RA (%)
d (%)
E (GPa)
G (GPa)
m
798
590
320
39
17
206
79
0.298
11
Modeling of ratcheting behavior under multiaxial cyclic loading 170.0 30.0
40.0
30.0
40.0
30.0
q30.0
12.5 q10.0
R100.0
Fig. 1. Specimen geometry (mm)
Axial stess (MPa)
Axial stess (MPa)
100
-1.0
-0.5
0.0
0.5
0
1.0
-1.0
-0.5
Shear strain (%)
0.0
0.5
1.0
Shear strain (%)
Case 1
Case 2
Axial stess (MPa)
100
Axial stess (MPa)
100
0 -1.0
0 -0.5
0.0
0.5
1.0
-1.0
-0.5
Shear strain (%)
0.0
0.5
1.0
Shear strain (%)
Case 3
Case 4
Fig. 2. Loading paths in ratcheting experiments
Table 2. The list of ratcheting experiments Spec. No.
Path
Test-8 Test-10 Test-7 Test-12
Case Case Case Case
1 2 3 4
Dc=2 (%)
cmean (%)
Dr=2 (MPa)
rmean (MPa)
Ds=2 (MPa)
0.866 0.866 0.866 0.866
0 0 0 0
0 50 50 50
100 50 50 50
270 239 260 274
12
X. Chen and K. S. Kim
path with mean axial stress (Case 3), and (iv) a butterfly path with mean axial stress (Case 4). In all cases the shear strain amplitude was set at 0.866%. In addition to the four axial-torsional tests, completely reversed, uniaxial and torsional baseline fatigue tests were conducted at several strain amplitudes to obtain the cyclic stressstrain curves.
3 Description of the constitutive model Following the work of Mroz [16] on a multi-surface plasticity theory, Dafalias and Popov [25] proposed a two-surface plasticity model. Based on their concept, Tseng and Lee [32], McDowell [26], and Ellyin and Xia [27] proposed different versions of the two-surface model. The total strain increment is decomposed into the elastic and plastic strain increments: e_ ¼ e_e þ e_p :
ð1Þ
The material is assumed to follow the von Mises yield criterion, which is given by f ðs a; RÞ ¼ ½ðs aÞ : ðs aÞ R2 ¼ 0;
ð2Þ
where s ¼ r ðrkkp =3ÞI ffiffiffiffiffiffiffiffiis the deviatoric stress, a is the backstress, and R is the radius of the yield surface (R ¼ 2=3ry ), I is a unit tensor, and the inner product is defined by, e.g. s : t ¼ sij tij : The bounding surface in two-surface plasticity theory is given by fð s a; RÞ ¼ ½ð s aÞ : ð s aÞ R2 ¼ 0:
ð3Þ
Since the focus in this study is on ratcheting under the condition of stable cyclic stress-strain response of the material, the yield surface and bounding surface are assumed to be subjected to no isotropic hardening. Thus, R_ ¼ 0 and R_ ¼ 0, where a dot over a variable implies its time rate. The generic kinematic hardening rules of our need are those introduced by Ziegler and Mroz. The Ziegler hardening rule [15] is given by a_ ¼ l_ z ðs aÞ;
ð4Þ
where l_ z is a scalar multiplier. The Mroz hardening rule [16] in the context of a two-surface model can be written as a_ ¼ l_ m ð sm sÞ;
ð5Þ
where l_ m is a scalar multiplier, sm is the stress point on the bounding surface with the same outward normal vector n of the yield surface as shown in Fig. 3, and can be expressed as sm ¼ aþ
R ðs aÞ: R
ð6Þ
The motion of the bounding surface is given by the hardening rule of Prager [14]: _ ¼ H0 e_p ; a
ð7Þ
where H0 represents the asymptotic plastic modulus in the stable uniaxial cyclic stress-strain curve. It is known that the Ziegler rule causes an arrest of ratcheting and that the Mroz rule overestimates the rate of ratcheting. To improve the situation, a superposition of the two kinematic hardening rules was proposed by Chen and Abel [30]. The new kinematic hardening rule is given by
13
Modeling of ratcheting behavior under multiaxial cyclic loading
s2
sm
n s ds
R
a
nm
ns n
ns
R
a
s3 s1
Fig. 3. Two-surface model with superposed kinematic hardening rule in the stress space
sm sÞ þ Cs ðs aÞ; a_ ¼ l_ s ½ð
ð8Þ
where Cs is a model parameter. For Cs ¼ 0, this represents the Mroz hardening rule. The proportionality factor l_ s can be obtained from the constancy condition f_ ¼ 0: l_ ¼
ðs aÞ : s_ RR_ ; ðs aÞ : ð sm sÞ þ Cs R2
ð9Þ
Cs is a function of plastic strain accumulation. The following evolution equation is proposed for Cs : C_ s ¼ bðCst Cs Þg_ ; where b and Cst are material constants and 2 p p 1=2 : e_ : e_ g_ ¼ 3
ð10Þ
ð11Þ
The direction of movement of the yield surface is expressed as follows: ns ¼
dm nm þ Cs Rn ; kdm nm þ Cs Rnk
ð12Þ
sm s where the unit vectors are defined by n ¼ ksa sak, nm ¼ ksm sk, and
sm sÞ : ð sm sÞ1=2 : dm ¼ ½ð The plastic flow can be stated as e_p ¼
1 hs_ : nin; H
ð13Þ
where H ¼ 23 Ep , Ep being the plastic modulus determined from a stable uniaxial cyclic stress– strain curve, and hs_ : ni ¼ 0 if s_ : n < 0; hs_ : ni ¼ s_ : n if s_ : n 0. The plastic modulus is assumed to be a function of the distance ds from the current stress point s on the yield surface to the stress point s on the bounding surface in the direction of movement of the yield surface. The distance can be expressed as
14
X. Chen and K. S. Kim 800 Ep0
Bound
Stress (MPa)
400 R a
0
-400
-800 -0.02
-0.01
0
0.01
0.02
Plastic strain
Fig. 4. Definition of parameters in uniaxial cyclic stress–strain space
Ep (MPa) 300000
Ep=9269Exp(0.008ds) ∆e/2=0.1% ∆e/2=0.5% 200000
100000
0 0
100
200
300
ds (MPa)
400
500
Fig. 5. Plastic with ds
modulus
h i1=2 ds ¼ s : ns þ ðs : ns Þ2 þ R2 ksk2 :
correlated
ð14Þ
The recommended form of H by Dafalias and Popov [25] can be written as H ¼ H0 þ hðdin Þ
d ; din d
ð15Þ
where H0 is the initial plastic modulus, and d is the distance in the hardening direction, din is the initial distance which can be updated with load reversals in cyclic loading, and h is a positive
15
Modeling of ratcheting behavior under multiaxial cyclic loading
Table 3. Model parameters for S45C steel R (MPa)
R (MPa)
Ep0 (MPa)
k
Cst
b
200
530
9269
0.008
200
1
6
b =1 s=100MPa
Axial strain (%)
Cst=80 Cst=100
4
Cst=150 Cst=200
2
0 0
100
200
a
300
6
500
Cst=100 b=0.5
s=100MPa
Axial strain (%)
400
Number of cycles
b=1
4
b=2 b=5
2
0 0
b
100
200
300
Number of cycles
400
500
Fig. 6. The influence of Cst and b on ratcheting strain; a Cst , b b
function related to the shape of the stress–strain curve. This equation was designed such that H ¼ H0 as the yield surface contacts the bounding surface and H becomes infinite as d approaches din . Chaboche indicated that updating din in axial cyclic loading could make the
16
X. Chen and K. S. Kim 2.5 Case1
Axial strain (%)
2
1.5
1
0.5 Experiment Prediction 0 0
100
200
300
400
Fig. 7. Comparison of predicted and experimental ratcheting strain for Case 1
10
Fig. 8. Evolution of parameter Cs as a function of the effective plastic strain
Number of cycles
200 160
Cs
120 80 40 0 0
2
4
6
h
8
predicted stress–strain curve overshoot the actual curve [33]. In this study the following equation was used for H: H ¼ H0 expðkds Þ;
ð16Þ
where k is a constant that can be determined from the axial cyclic stress–strain curve. The modulus H approaches to H0 as ds reduces to 0, and it becomes large enough, though not infinite. For the material under consideration, two stable stress–strain hysteresis loops obtained under uniaxial cyclic loading (strain range De=2 ¼ 1:0% and De=2 ¼ 0:5%) were used to determine material constants. The definitions of some material constants and model variables are shown in Fig. 4 for uniaxial cyclic loading. The relation between the plastic modulus and the distance of a stress point from the bounding surface is shown in Fig. 5. The constants obtained from the test data are listed in Table 3. The stress–strain response of the Case 1 ratcheting test was used to examine the effects of the values of Cst and b. The axial constant stress in this test was 100MPa and the torsional cyclic strain amplitude was 0.866%. Computations show that the shear stress amplitude does not
Modeling of ratcheting behavior under multiaxial cyclic loading 800
17
S45C
Axial stress (MPa)
400
0
-400 Experiment Prediction -800 -1.5
-1.0
-0.5
a
0.0
0.5
1.0
1.5
Axial strain
400
S45C
Shear stress (MPa)
200
0
-200 Experiment Prediction -400 -2.0
b
-1.0
0.0
Shear strain (%)
1.0
2.0
Fig. 9. Cyclic stable strain-stress curves under uniaxial and torsional loading; a uniaxial, b torsional
change appreciably with Cst and b, and that the shape of the stress–strain loop is only slightly affected by the two constants. However, it was found that the ratcheting strain is very sensitive to these constants. When b ¼ 1, computations for different values of Cst show that the ratcheting rate increases with a decrease of Cst , as shown in Fig. 6a. This is not surprising since the decrease of Cst implies that the Mroz hardening rule becomes more influential. Figure 6b shows that, with Cst ¼ 100, the ratcheting rates at high cycles did not change much with b. These trends imply that Cst is mainly related to ratcheting rates at high cycles, whereas b is related to transient changes of ratcheting strain in the initial phase. Cst and b are determined by a trial method for Case 1. The values of Cst and b used in this study for S45C steel are given in Table 3. For Case 1, there was a slight overprediction in the early stage of the test but a good agreement with experimental data was obtained later in Fig. 7. It is also noted that in numerical
18
X. Chen and K. S. Kim 300 Case 3 200
Shear stess (MPa)
100
0
-100
-200 Experiment Prediction -300 -1.0 -0.8 -0.6 -0.4 -0.2 0.0
a
0.2
0.4 0.6 0.8 1.0
Shear strain (%)
120 Case 3
Axial stess (MPa)
80
40
0 Experiment Prediction -40 -400
b
-200
0
Shear stress (%)
200
400
Fig. 10. Comparison of predicted and experimental loops for Case 3; a shear stress–strain, b axial stress–shear stress
computations of the four loading cases the initial value of Cs was set to zero. Thus, the initial operative hardening rule was the Mroz model. The parameter Cs evolved as the plastic deformation was shown in Fig. 8 for the present material, in which it evolved from zero to the saturated value Cst .
4 Results and discussion The model was able to simulate the stable cyclic stress-strain response under R ¼ 1 strain control tests. The predicted axial and shear stress-strain curves under different strain ranges are
Modeling of ratcheting behavior under multiaxial cyclic loading
19
120 Case 4
Axial stress (MPa)
80
40
0 First 10 cycles Experiment -40 -400
-200
a
0
200
400
Shear stress (MPa)
120 Case 4
Axial stress (MPa)
80
40
0 First 10 cycles Prediction –40 –400
b
–200
0
Shear stress (MPa)
200
400
Fig. 11. Stress response in the first 10 cycles for Case 4; a experiment, b prediction
shown in Fig. 9 in comparison with experimental results. The material showed a yield plateau in the initial loading both under axial and torsional cycling. Under cyclic torsion with a shear strain amplitude of 0.866%, it is found in Table 2 that the shear stress amplitude is approximately at the same level in all nonproportional loading tests, which indicates, in comparison with pure torsion, that there was no apparent additional hardening. This is perhaps due to low applied axial stress. As a consequence, the shear stress– strain loop and the axial stress–shear stress response for Case 3, predicted without isotropic hardening in the model, were found to be in good agreement with experimental results (Fig. 10). For circumstances where the additional hardening effect is sizable, the isotropic hardening rule may be included in the model as in [26].
20
X. Chen and K. S. Kim 2
Axial strain (%)
1.6
1.2
0.8 Experiment Case 2
0.4
Case 3 Case 4
0 0
200
400
600
800
1000
Number of cycles
a 2
Axial strain (%)
1.6
1.2
0.8
Prediction Case 2 Case 3
0.4
Case 4
0 0
b
50
100
150
Number of cycles
200
250
Fig. 12. Ratcheting strain for different loading paths with mean stress of 50 MPa; a experiment, b prediction
For Case 4, the axial stress-shear stress response of the first 10 cycles was predicted well by the model, as depicted in Fig. 11. It is noted that a perfect plastic material model was enforced for the first half cycle in the analysis to enforce the yield plateau behavior. The ratcheting strain was measured up to fatigue failure of the specimen in all tests. The rate of ratcheting decreased continuously as cycling continued, but did not arrive at the state of shakedown. A comparison of the ratcheting strain for the three tests with the same mean stress, Case 2–4, showed that the loading paths effects on the ratcheting strain are rather small; Fig. 12a. It is apparent that the level of mean stress carries more importance in ratcheting, and the loading path has only a secondary influence. Figure 12b shows that the model predicts
Modeling of ratcheting behavior under multiaxial cyclic loading
21
correctly the order of ratcheting strain magnitudes among the three cases under constant mean stress. In Cases 2–4 predictions appear to be in reasonably good agreement with experimental data for a relatively low number of cycles, but the model tends to overpredict the ratcheting strain as the cycle accumulates. The prediction of ratcheting strain to a high number of cycles is not found in the literature, and it still remains a difficult problem. McDowell [34] proposed an equation for the decay of the ratcheting strain after 25–50 cycles to substitute the integration of the constitutive equation over the entire history of loading. Bari and Hassan [6] suggested that it would be needed to introduce anisotropy into the yield surface to enhance the predictive capability of ratcheting strain beyond the current status. More efforts are certainly needed for more reliable prediction methods. The quality of predictions obtained in this study compares favorably with other data available in the literature. A comparative evaluation of the proposed model with other existing models is desirable. The loading conditions used in this study are limited. More comprehensive verification of the model and further improvement remain as future work.
5 Conclusions Ratcheting tests were conducted on S45C steel for four different nonproportional loading paths. A kinematic hardening rule that superposes the Ziegler and Mroz hardening rules was incorporated in a two-surface plasticity theory. An exponential equation was used for the plastic modulus given in terms of the distance of the yield surface from the bounding surface. The number of material parameters employed in the model was relatively small compared with nonlinear kinematic hardening models pursued in recent years. The model predicted stable stress–strain behavior of the test material with a reasonable accuracy. The accuracy of the predicted ratcheting strain varied with the loading conditions and the number of cycles. Predictions were reasonably accurate at a low number of cycles, but they became significantly larger than the actual values as the cycle increased. The results still look encouraging in comparison with other data available in the literature.
Acknowledgements The authors gratefully acknowledge financial support for this work, in part from Brain Korea 21 Program at Pohang University of Science and Technology, and in part from National Natural Science Foundation of China and TRAPOYT.
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[33] Chaboche, J. L: Time independent constitutive theories for cyclic plasticity. Int. J. Plast. 2, 149– 174 (1986). [34] McDowell, D. L.: Stress state dependence of cyclic ratcheting behavior of two rail steels. Int. J. Plast. 11, 397–421 (1995). Authors’ addresses: X. Chen, School of Chemical Engineering and Technics, Tianjin University, Tianjin 300072, P.R. China (E-mail:
[email protected]); K.S. Kim, Department of Mechanical Engineering, Pohang University of Science and Technics, Pohang 790-784, Korea (E-mail:
[email protected])
