Multiaxial cyclic ratchetting under multiple step loading

Pergamon

International Journal of Plasticity, Vol. 10, No. 8, pp. 849-870, 1994 Copyright © 1994 Elsevier ScienceLtd Printed in the USA. All rights reserved 0749-6419/94 $6.00 + .00

0749-6419(94)00029-8

MULTIAXIAL CYCLIC RATCHETTING UNDER MULTIPLE STEP LOADING

YANYAO JIANG a n d HUSEYIN SEHITOGLU University of Illinois at Urbana-Champaign

A b s t r a c t - Strain ratchetting responses of 1070 steel are reported for multiple step cyclic load-

ing histories. The stress amplitude and mean stress are varied between loading steps in multiple step loading. Experimental results reveal that the material exhibits a strong memory of the previous loading history, and such memory plays a discerning role on the subsequent ratchet° ring. The material could ratchet in the opposite direction to the mean stress or could reverse its ratchetting direction with time. The origin of the ratchetting transients has been linked to the variation of the plastic modulus within the loading cycle for proportional loading and the noncoincidence of the plastic strain rate direction and yield surface translation direction for nonproportional loading. Many of the constitutive relations proposed for cyclic loading are not designed to handle the ratchetting evolution. Based on the Armstrong-Frederick hardening algorithm, the model forwarded by Bower can qualitatively predict the ratchetting directions for certain multiple step loading cases, but the predicted ratchetting rates differ from the experimental values. The Ohno-Wang model, which introduces threshold levels of dynamic recovery in nonlinear hardening, can simulate negative ratchetting under positive mean stress, or vice versa, as well as the ratchetting direction reversal during step loadings. This model can provide results that agree with experimental observations for a class of nonproportional cases, where the plastic strain rate direction and yield surface translation direction are noncoincident. Its performance deteriorates for proportional loading.

I. INTRODUCTION

Ratchetting, which denotes cyclic accumulation of deformation, is often induced under asymmetric loading and is commensurate with plastic deformation. Asymmetric loading in uniaxial loading is achieved by subjecting the material to unequal stress levels in tension and compression. Its definition for multiaxial loading is not as simple because a mean shear stress could induce axial ratchetting even under equal tension-compression loading (JL~_nO[1993]; JIANG& SEHITOGLU[1994a]; McDowEI.L [1991]). The ratchetting phenomena could accelerate damage accumulation and, superimposed with fatigue loading, could produce early material failures. The problem is important in rolling-sliding contact where shear deformation accumulated in the rolling direction is synergistically coupled with the reversed shear and normal stresses. Infinitesimal plastic deformation is sufficient for ratchetting to initiate and a small ratchetting rate could accumulate to large values over many cycles. Over the years the modeling of elastic-plastic behavior of metals for complex loading histories has received considerable interest. Early studies were mainly concentrated on monotonic and uniaxial loading, and recent effort has been directed toward cyclic multiaxial plasticity. Many researchers have investigated strain-controlled experiments where mean stress decreased with asymmetric strain cycling. The more complicated cyclic ratchetting under stress control has drawn less attention and remains to be a challenge 849

850

Y. JIANG and H. SEHITOGLU

in modeling. Some conclusions drawn from the early strain-controlled experiments may not be expanded to stress-controlled ratchetting. For example, the MROZ [1967,1969] multiple surface model and the associated two-surface models produce commendable stress response for strain-controlled multiaxial loading (LAMBA [1976]; LAMBA & SIDEBOTTOM [1978]; McDow-ELL [1985a,b,c]). However, the Mroz model predicts fully closed hysteresis loops for any proportional loading and unrealistically high ratchetting rates for nonproportional ratchetting loading (JIANG & SEHITOGLU [1994b]). Because ratchetting rate is often small and subject to transient changes, it is imperative to study ratchetting over many cycles. When ratchetting was studied in the literature, often a few dozen loading cycles were considered. In these cases, the ratchetting rates could appear constant, and some plasticity models can predict this trend. However, the ratchetting rate is generally a nonlinear function of the loading cycles and, depending on the material, may increase or decrease with loading cycles (JIANG [1993]; JIANG & SEHITOGLU [1994a]). Moreover, it has been generally accepted that the ratchetting direction follows the mean stress direction. This assertion was verified for ratchetting under proportional and constant amplitude loading. Constant amplitude case refers to the loading path with constant maximum and minimum stresses over the duration of the test. When the loading path a n d / o r stress amplitude vary, the experiment is classified as a "multiple step" test. The ratchetting directions may not follow the mean stress directions under multiple step loading. This article explores the ratchetting responses of 1070 steel for both uniaxial and multiaxial multiple step loadings. Cyclic ratchetting behaviors are monitored, and intricate material behaviors are documented. Some of the existing plasticity models are discussed in light of the experimental results. I!. EXPERIMENTAL PROCEDURE

Two types of loading stress states were considered: uniaxial and biaxial (axialtorsional) loadings. Uniaxial tension-compression experiments were conducted using solid smooth specimens of 1070 steel. The specimens were machined from hot rolled bars, which were heat treated under 870°C for 4 h followed by air-cooling. The monotonic yield strength (0.2% offset) of the material is 450 MPa, and the reduction of area is 30%. The round smooth specimen has a diameter of 10.5 mm and a gauge length of 12.7 mm. The specimens were polished to about 30 micron surface finish before testing. A microcomputer was employed to control the experiments and collect the data for axial load and deformations. An extensometer of 12.7 mm gauge length was used to measure the axial deformation, and an extensometer of 10.5 mm gage length measured the diametral strain. Because diametral strain was measured, the true stress can be computed using instantaneous cross-sectional area. Tubular specimens were chosen for the axial-torsional loading tests. This specimen choice results in a near-homogeneous shear stress state. The specimen has an internal diameter of 25.4 mm and a wall thickness of 3.80 ram. The gauge length is 25 mm. The tubular specimens were heat treated under the same conditions as the uniaxial smooth specimens. A biaxial extensometer of 25 mm gauge length was used to measure the deformations in the axial and torsional directions. For the selected loading cycle, the computer acquires 200 data points per channel, which were stored for later analysis. During ratchetting experiments precaution was taken to ensure that the deformation was not excessively large and was within + 5 % . The corresponding fatigue lives were large enough that fatigue failure or any measurable damage growth was not expected during the ratchetting experiment.

Ratchetting under multiple step loading

851

IlL EXPERIMENTAL RESULTS

III.1. Uniaxial loading We present in Fig. 1 the stress-strain responses for a "three step" uniaxial loading. In this case, the mean stress is maintained constant at 280 MPa, while the stress amplitudes are 375 MPa, 425 MPa, and 375 MPa, in Step 1, Step 2, and Step 3, respectively. Note that the horizontal axis is broken for the three steps to avoid overlapping of the hysteresis loops. In Step 1 the ratchetting rate decreases with increasing loading cycles, following a power law relation in terms of the number of cycles (JIANG [1993]). The ratchetting rate in Step 2, however, is lower than expected compared to the case of Step 2 in isolation. When the stress range decreases after Step 2, the ratchetting rate is lowered further. A close look at the details of the stress-strain behavior in Step 3 indicates that the axial strain ratchets about 0.03°70 in the first 10 cycles and 0.01 070 in the remaining cycles. The results from this ratchetting experiment confirm that when mean stress is maintained constant and the stress range varies in a multiple step test, the ratchetting rate decreases with increasing number of cycles, but the ratchetting direction remains consistent with the mean stress direction. Figure 2 shows the stress-strain responses of two consecutive loading steps from a "six step" uniaxial ratchetting experiment. The values of the controlled stress ranges and mean stress are all the same in each loading step except that the sign of the mean stress in the consequent loading step is opposite to that of the previous step. In Step 1, the stress range is 764 MPa and the mean stress is 382 MPa. Step 2 has a stress range of 768 MPa and a mean stress of - 3 8 4 MPa. It is noted that the hysteresis loops for Step 1 are identical to those for Step 2, only one is tensile and the other compressive. Ratchetting rate decreases with increasing number of cycles in each loading step. However,

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Fig. 1. Experimental ratchetting for a uniaxial step loading with a constant mean stress.

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the ratchetting rate does not continuously decrease with the entire loading history. Consequently, the commonly used isotropic hardening that relates the hardening to either the accumulated equivalent plastic strain or accumulated plastic strain energy density, both quantities being proportional to the total loading history, cannot explain the ratchetting phenomenon experimentally observed. Previous research has established that the strain ratchetting direction is identical with the mean stress direction. An early study by JIANG and SEHITOGLU[1994a] confirmed this finding for single step (constant-amplitude) proportional loading. However, this assertion does not always hold for ratchetting under multiple step loading. Figure 3 shows ratchetting behavior for a four step uniaxial loading. The stress range is about the same in each loading step, whereas the mean stress in each loading step differs. The central line denotes the mean stress in each plot. The stress-strain response for Step 1 is similar to that shown in Fig. 1. When the mean stress is reduced to 77 MPa in Step 2, the strain ratchets in the negative direction opposite to the mean stress. When the mean stress is changed to - 1 2 5 MPa in Step 3, the ratchetting switches to the negative direction. After Step 3, the mean stress is reduced to - 2 0 MPa in Step 4. From the right side plot o f Fig. 3b it can be ascertained that with a negative mean stress the strain ratchets in the positive direction. DOLAN [1965] reported that for 4340 steel when the mean stress was reduced to zero after an asymmetric loading history, the strain ratchetted in the opposite direction to the mean stress of the previous loading step. He noted that "the plastic stretch of the specimen from the previous cycling was gradually being partially recovered by a cyclicdependent shortening." The backward ratchetting after a tensile mean stress loading history was explained by strain recovery, i.e. the tendency for the material to recover from a non-zero strain state to zero deformation. However, the strain-recovery does not explain the current ratchetting results displayed in the right side plot of Fig. 3b where the axial strains in the four loading steps are all positive. According to the deformationrecovery explanation, the strain should have ratchetted in the negative direction in Step 4. In fact, the result shows an opposite behavior. The deformation may recover to some degree, but clearly such recovery is not significant for the material under investigation.

Ratchetting under multiple step l o a d i n g

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(b) Fig. 3. Experimentalratchetting for a multiple step uniaxial loading. (a) Stress-strain response: Step 1 and Step 2; (b) Stress-strain response: Step 3 and Step 4.

To strengthen our understanding of ratchetting, more multiple step uniaxial ratchetting experiments were conducted, and some of the results are shown in Figs. 4 and 5. The stress amplitudes are 396 MPa in both steps shown in Fig. 4. The mean stresses are 204 MPa and 78 MPa in Step 1 and Step 2, respectively. Step 1 experiences 65 cycles. The stress-strain responses are produced in Fig. 4. In Step 2, referring to the cycle labels in Fig. 4, the strain ratchets in the negative direction opposite to mean stress during the first 250 cycles and then shifts to the positive direction for the remainder of the cycles. Comparing the two step loading history of Fig. 4 with Fig. 3a, we find that the only difference between the two cases is that in the Fig. 3a case the first step experiences 4100 cycles, whereas Step 1 in Fig. 4 experiences only 64 cycles. In other words, if Step 1 in Fig. 4 had experienced a higher number of loading cycles, the strain in Step 2 would ratchet in the negative direction for all the loading cycles. The ratchetting phenomenon

854

Y. JIANG and H. SEHITOGLU

t~_ = 396M]Pa 2 2~-= 396MPa

o ~=~MPa

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Fig. 4. Experimental ratchetting for a two step uniaxial loading.

displayed in Fig. 4 indicates that the duration of the previous loading step has a profound influence on the ratchetting of the current loading step. To illustrate the change in ratchetting direction further, we present in Fig. 5 the results from a two step uniaxial ratchetting experiment where the stress amplitude is 405 MPa and the mean stress is - 2 1 1 MPa in Step l, and the stress amplitude is 437 MPa and

Ao = 437MPa 2 . = -77MPa

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r- 6 0 0

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Fig. 5. Experimental ratchetting for a two step uniaxial loading.

....., .....Loo -0,025

Ratchetting under multiple step loading

855

the mean stress is - 7 7 MPa in Step 2. In the right side plot of Fig. 5 we reveal for Step 2 that the strain ratchets in the positive direction for about 1000 cycles, followed by a shift in ratchetting direction to the negative direction. Referring to the ratchetting behavior in Fig. 3a, it is noted that if the stress ranges are the same in both steps, the ratchetting in Step 2 would be all opposite to the mean stress. This suggests that the relative ratio of the stress ranges between the loading steps plays a role in the ratchetting of the following loading step. To gain further understanding of the previous history effect, the influence of a single overload on the ratchetting behavior is demonstrated in Fig. 6. The maximum stress in the first reversal is 980 MPa, which corresponds to an axial strain of about 4.5°7o. The following loading step has a stress amplitude of 420 MPa and a mean stress of 100 MPa. In Fig. 6 we observed that the strain ratchets in the negative (backward) direction opposite to the mean stress. In a separate test, which is not presented, the overload was lower than that in Fig. 6, the strain ratchetting was initially in the negative direction opposite to the mean stress and then shifted to the positive direction. Ill.2. Multiaxial loading The material memory effects are studied for multiaxial stress states. A two step tension-torsion loading path is shown in Fig. 7a, and the corresponding axial stressstrain and shear stress-strain responses for Steps 1 and 2 are shown in Figs. 7b and 7c, respectively. Step 1 represents proportional compression-torsion, where the axial mean stress is - 3 0 0 MPa and the shear mean stress is -110 MPa. In Step 2 the axial mean stress is reduced to zero, and the other loading parameters remain the same. From the right side plots in the figures we can find that, as a result of the change of axial mean stress, the axial strain ratchets in the positive direction, whereas the ratchetting in the shear direction displays a complicated behavior. The shear strain ratchets in the positive direction opposite to its mean stress during the first 60 cycles, then in the negative direction for the remaining 2000 cycles. Similar phenomenon was observed if, instead of changing axial mean stress from Step 1 to Step 2, the shear stress is reduced to zero. In that case the shear strain ratchets in the opposite direction to its previous direction in Step 1, and the axial ratchetting direction changes during Step 2 loading. In Fig. 8 we consider a nonproportional loading path. In each loading step the axial

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856

Y. JIANGand H. SEHITOGLU

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Fig. 7. Experimental ratchetting for a two step nonproportional axial-torsion loading history. (a) Stresscontrolled loading path; (b) Axial stress-strain response; (c) Shear stress-strain response.

stress is maintained constant, and the shear stress is cycled symmetrically. The axial mean stress in the Step 1 is 300 M P a and the shear stress amplitude is 230 MPa. The ratchetting behavior is illustrated by plotting the axial strain versus the shear strain. In Step 2, the shear stress range is 230 M P a but the axial stress is reduced to 60 MPa. The result o f the axial stress change, as s h o w n in Fig. 8c, is that ratchetting develops in the compressive direction in Step 2. Clearly the axial mean stresses in both steps are positive, and the ratchetting in the negative direction during Step 2 develops as a result o f the previous history effect.

Ratchetting under multiple step loading

400-

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Fig. 8. Experimental ratchetting for a two step nonproportional axial-torsion loading history. (a) Stresscontrolled loading path; (b) Biaxial strain response: Step 1; (c) Biaxial strain response: Step 2.

From the observations o f the experimental results for multiple step loading, it is noted that when the mean stress does not vary between the loading steps the ratchetting direction is unaltered. When the mean stress is reduced in magnitude yet maintains the same sign after the initial loading step, the material displays a tendency to ratchet in the opposite direction to the mean stress. The intensity o f this material memory is dependent on the detailed previous loading history, including the loading amplitudes and number o f cycles, and dissolves with cyclic loading. Farthermore, the 'anomalous' ratchetting phe-

858

Y. JIANG and H. SEHITOGLU

nomena, such as ratchetting in the direction opposite to the mean stress and ratchetting direction change with time, observed on 1070 steel may be strongly related to the ratchetting rate decay of this material. For the materials exhibiting constant or increasing ratchetting rate, this "anomalous" behavior may not be observed.

IV. REVIEW OF CONSTITUTIVE MODELS

The existence o f a yield surface and, when strain accumulation is small, the normality flow rule are often assumed in incremental plasticity. The von Mises type of yield surface for an initially isotropic material is given as, f = ( S i j - o~ij)(Sij - o~ij) - 2k 2 = 0,

(1)

where Sij -- a~j - ~a~jS~j are the components of the deviatoric stress tensor, a~9 represent the components of the deviatoric backstress (center of yield surface), and k is the yield stress in simple shear. The plastic strain rate is, 1 de~j = ~ ( d S k : n k t ) n i j ,

(2)

where (x) = 0.5 (x + Ix[). e~s denote the components of the plastic strain tensor, n~j are the components of the unit exterior normal on the yield surface at the loading point, and h is the plastic modulus function. Under stabilized uniaxial tension-compression loading, h = 2 ( d a / d e p ) ' where a and e p are the axial stress and plastic strain, respectively. While the framework is mostly the same, it is the definition of the plastic modulus function, h, and the translation direction of the yield surface that distinguish one plasticity model from another. The classical kinematic hardening models proposed by PRAGER [1955] and ZIEGLER [1959] are not designed to handle ratchetting. When the strain hardening modulus is expressed as a function of the second stress invariant (DRUCKER *, PALGEN [1981]), a constant ratchetting rate can be predicted for constant amplitude loading. The multiple surface models of MRoz [1967,1969] and GARUD [1981,1982] do not predict ratchetting for any proportional loading. These models can predict ratchetting for general nonproportional loading, and the predicted ratchetting rates are much larger than the experimental observations (JIANG & SEHITOGLU[1994b]). The two-surface models (DAFALIAS & POPOV [1975,1976]; KRIEG [1975]; McDOWELL & MOYAR [1991]; TSENC & LEE [1983]), with nonlinear expressions for the plastic modulus function, can predict ratchetting. However, the two-surface plasticity theory is not able to reproduce the power law ratchetting rate decay observed in experiments (JIANG & SEHITOGLU [1994a]). In addition, none of the aforementioned models can explain the experimental phenomenon of ratchetting opposite to the mean stress direction for uniaxial loading. ARMSTRONG and FREDERICK [1966] introduced a nonlinear kinematic hardening rule, introducing a recovery term linked to the transitory strain memory effect. With the introduction of a nonlinear term, the Armstrong-Frederick rule gives constant strain ratchetting for constant amplitude stress-controlled loading. Generally, this model leads to large overestimations of ratchetting (INOUE e t al. [1985]).

Ratchetting under multiple step loading

859

BOWER [1987,1989] made a modification o f the Armstrong-Frederick rule, (Bower)

dot 0 = cl ( r d e ~ - (oli2 - 13ij)dA )

(3a)

dl3ij = c2(o~ii - 13i2)dA,

(3b)

and

where dA x/-deude e, 13~jare components of an additional internal state variable that has an initial value of zero, and, c~, c2, and r are material constants. The term Cl (oru - / 3 u) in eqn (3a) plays as the driving force for ratchetting in the Bower model. Noting that all3u is proportional to (oru - / 3 u ) , under stress-controlled tension-compression with non-zero mean stress, c~ (c~o- -/3ij) decreases with the loading cycles. As a result, this model is able to predict ratchetting rate decay. However, the Bower model cannot describe the basic stress-strain curve well and can account for the ratchetting rate decay only for a limited number of cycles. The model always delivers ratchetting arrest, which could be contrary to experiments (JIA~G t, SEHITO~LU [1994a]). Nonlinear hardening rules based on the Armstrong-Frederick rule are generally expressed in the form of the decomposition of several backstress components. It was postulated (CI-IAaOCHE et al. [1979]) that the total backstress is decomposed to several components, M

d°LiJ---- Z

(4)

d°t(m),

m=l

where ot,j are the components of the total backstress, or(m) represents a constituent of the total backstress, m = 1,2 . . . . . M, and, M is the number of terms in backstress expansion. Each backstress part obeys an Armstrong-Frederick type rule, da(')

= c(m)(r(m)de~j - w(m)~i(jm)dA)

(m = 1,2 . . . . . M ) ,

(5)

where c (m) and r (m) are material constants, and W (m) is a function o f stress state (MoosBRUC,~ER & McDowELL [1989]; McDowELL [1992]). The plastic modulus function is determined by the consistency condition, which insures the stress state point laying on the yield surface when the elastic-plastic deformation occurs, and hardening rule. The initial model of Cn_nmOCHE et al. [1979] can be represented by W (m) = 1 in eqn (5). Under constant amplitude tension-compression, this model predicts a transient ratchetting response consistent with experiments, but the duration of the transient response is short-lived. Subsequently, the model predicts constant ratchetting rate, which contrasts to the long-term decay of ratchetting rates experimentally observed (JIANG [1993]; JIAI~G & SEHITOGLU[1994a]). In addition, the Chaboche model predicts ratchetring direction consistent with the mean stress direction for uniaxial tension-compression loading, which makes it impossible to predict negative ratchetting under positive mean stress and ratchetting direction change, the phenomena shown in Figs. 4 and 5. A recent model by CHABOCHE et al. [1991] introduced a threshold level for each backstress part, W (m) =

1

~m(m) \)

(m) oLU

oLU

1

( m = 1,2, "" . , M ) ,

(6)

860

Y. JIANG a n d H. SEHITOGLU

where ~m is the threshold level for dynamic recovery of the mth backstress. When the value of a backstress part is below a certain level, W (m) is zero and the kinematic hardening is linear. A backstress exceeding a critical value will result in 0 _ W (m) _< 1 and nonlinear kinematic hardening. The introduction of the threshold makes the model predict less ratchetting than does the model without the threshold. OHNO and WANG [1991,1993a,b] proposed the following expressions for a new hardening algorithm, (Ohno-Wang)

W ~'')

=

(ct(m)] ~''~-\

rm

/

" ( " ' "]

(m = 1,2,.

"rtijL'ij

,M),

..

(7)

where Xm is a constant, g m = ot (m) _

rm <

(8)

1

represents a limiting surface with a radius of rm, and, L!m)

1J

=

Oli(jm) ~ (m;

(9)

and, (~ (m) = ~(.m)o/.(.m) U

(10)

lj

are the unit vector and magnitude of a backstress component, respectively. In the Ohno-Wang model each backstress part ct~m) is either within or on a limiting surface with a radius of rm. When the backstress part is within a limiting surface, W cm) is non-zero; therefore, the hardening is nonlinear. When the backstress reaches the limiting surface, the model yields d a i j n i j = 0. Under this condition, the translation direction of this backstress is in the tangential direction to the limiting surface. The coefficient xm controls both the magnitude and pattern of the ratchetting rate predicted using the Ohno-Wang model (JIANG & SEnrro6Lu [1994a]). McDOWELL [1992] suggested that Xm be further a function of the degree of saturation of short range backstress and on noncoaxiality o f the plastic strain rate and each backstress, Xm = A ( ( M

+ l - m) - B

~(m) ) t/ rl [(m)'~ l+C(l-(6t(m)/rm)) rm

x-ij--ij

.

,

(11)

where A, B, and C are positive constants and are determined from the nonproportional ratchetting experiments. This equation suggests that the coefficient Xm is larger for proportional loading than for nonproportional loading, since for proportional loading ( n i j L i j(m) ) is either unity or zero while for nonproportional loading j, n: -o ~" i( " ) ) is either zero or smaller than unity. From eqn (11), Xm is also dependent on c~(m)/rm, the relative distance o f a backstress part from its origin in deviatoric stress space. V. COMPARISON BETWEEN EXPERIMENTAL AND PREDICTED RESULTS

Predictions using two plasticity models are compared with the experimental results. These models are the Bower model and the Ohno-Wang model. Due to the difficulty

Ratchetting under multiple step loading

0.04

0.04

- -

Exl~riment

"t~0.03-

/

I

~ 0.020.01

~

=

m=3

m=

0.03

eqn(ll]

Ill1

y

0.02

eqn(11)

0.01

F~ 0.00

m=

~.1 ~+3~

//'/ "

~

861

F~

n ~ ';'~"~n 0 ~ ';'C 'u ~ ';'C"

100

Number of Cycles

1000 1u

';'"",

10

~ ';'C 'l

'n ~ 'i'~'"' 10 ~ ',;'C 100 1000

~ ','"'"J

~ ',~'"",

100

Number of Cycles

Number of Cycles

0.00

1000

Fig. 9. Comparison of experimental data and ratchetting strain predicted by plasticity models for a three step uniaxial loading with constant mean stress (Fig. 1).

in determining the material constants, the threshold model by CnAaOCI-IEet al. [1991], eqn (6), is not discussed. The results of the simulations along with the experimental data are shown in Figs. 9 through 12. The material constants used in each model are listed in Table 1. The basic material constants for the models are obtained by fitting the uniaxial stress-strain curve at the strain amplitude A e/2 = 0.8°70 under strain-controlled condition. The material constants in the Bower model are further adjusted to fit some of the ratchetting data from uniaxial experiment. The 1070 steel displays little isotropic hardening under strain-controlled loading (JIAr~O [1993]), therefore the isotropic hardening is ignored in the simulations. Either the accumulated ratchetting strain or the ratchetting rate, the amount of ratchetting per loading cycle can be used in the comparison of the predicted and the experimental results. We found in a previous paper (JIANOR, SrnrrooLu [1994a]) that for 1070

o.os].

z:,ro, I

~ 0"04] '

OhnO'wang'

.....

0.03]

.........

~ /.,..,/ ,/.

,]

I

~lm-3 0,04

~

0.03

y X m = 3

0.02

"... ...........................

~0.02 1 '¢C0.011

-000 ,

,~.~--'~'~ecln(11) . . . . . . .

;'o

. . . . . .

;;0

0.05

. . . . .

Number of Cycles

:;'00.... 10'000

0.01

'

' ''""1

10

'

' ''""1

100

Number

'

'''""1

'

1000

of Cycles

' ......

I

O00

10000

Fig. 10. Comparison of experimental data and ratchetting strain predicted by plasticity models for a two step uniaxial loading (Fig. 3).

862

Y. J ] a N o and H. SEHITOGLU

0.03 -

--E,..,menll

I

...... eowe~ I Ohno-Wang I

=_

i

003

i

...........

/

e~

09 0.02

-

I

'i

!

,f

.=_ ~p

~(m= 1

~=3

/'

if2 0 . 0 1

0 02

i

-

001

q.(11)

x

<

0.00

-

' l 'l'"l

, ; ,l,,'l

10

N u m b e r of

100

I I

'

'

' ' ''"I

'

'

' '''"l

t0

'

'

' '''"I

100

'

'

'

1000

' ''"I

,

,

, ,,N

000

10000

Number of Cycles

Cycles

Fig. 11. Comparison of experimental data and ratchetting strain predicted by plasticity models for a two step uniaxial loading (Fig. 4).

steel under the constant amplitude loading condition, the ratchetting rate decreases with the increasing number of cycles, and the relationship between the ratchetting strain and the number of cycles can be best described by a power law. Due to the sign change of ratchetting rate for multiple step loading conditions, ratchetting rate cannot be shown in logarithmic coordinates. Therefore, in this presentation, the accumulated strain is chosen for comparison. A difficulty with the selection is that the predicted ratchetting strains by the models are generally very different from the experimental results, and the differences accumulate during the step loading, To evaluate the theoretical models in light of the experimental results, the predictions are subjected to a parallel shift along the ordinate, where the ordinate is the ratchetting strain and the abscissa is the number of cycles, to permit same initial ratchetting strain in experiments and predictions. In Figs. 9 through 12, the loading cycle numbering begins at each loading step. The solid lines denote the experimental ratchetting strain, which is the average of the maxi-

005

0 O5

-

:................... . /

........ o,~- ,~1 .[:

0,04-

U') ~0.03

~oo2n" ~0.01

0.00

-

y

/

/ F "°<11, ~Xm= 1

"" I

lO

I

1oo N u m b e r of

Cycles

I

I

1ooo

10000

. . . . . . .

'I

1o

'

'

''''"

1oo N u m b e r of

. . . . . . . . . . . . . . ' ' . . . . . . . . . . . . . .

1ooo

l

0

O0

10000

Cycles

Fig. 12. Comparison of experimental data and ratchetting strain predicted by plasticity models for a two step nonproportional tension-torsion loading consisting o f constant axial stress and alternating Shear (Fig. 8).

Ratchetting under multiple step loading

863

Table 1. Material constants for plasticity models 1070 Steel E = 210000 MPa/z = 0.3 Model Bower Ohno-Wang

Material Constants k = 165MPa, c1=483, c2=20, r=249MPa k = 165 MPa, Xm = 1.0 C1 200, c2 = 550, c3 = 260, c4 = 80, c5 = 25 r~=78MPa, rE=58MPa, r3=43MPa, r4=41MPa, =

r5=360MPa

mum and minimum strains in a cycle. The central lines are the predictions by the Bower model, and the dotted lines denote the predictions by the Ohno-Wang model. In the Ohno-Wang model, the coefficients Xm were 1.0 and 3.0. The modification made by McDow~I.I. [1992] on the coefficient Xm, eqn (11), is also considered. The constants A, B, and C in eqn (11) were 5.0, 1.0, and 4.0, respectively. The selection of those four constants is made based on a best fit of the experimental ratchetting rate for a nonproportional loading path consisting of constant axial stress and varied shear cycling shown in Fig. 8b. Figure 9 shows the comparison between the experimental and predicted results. The experimental results are presented in Fig. 1, and in Fig. 9 they are shown with solid lines. The Bower model predicts declining ratchetting rate for the first 100 cycles and ratchetting arrest thereafter in Step 1. Subsequently, this model predicts no ratchetting in the second and third steps. The results from the Ohno-Wang model demonstrate the dependence of the predictions on the selection of the coefficient Xm. When the coefficient Xm is 1.0, the predicted ratchetting strains exceed greatly the experimental results in Steps 1 through 3. When coefficient Xm takes the expression eqn (11), the predicted ratchetting predictions are lower than the experimental results for Steps 1 and 2 but are in agreement with the experimental result for Step 3. When Xm is 3.0, the predictions are in close agreement with the experimental data for the first few dozen cycles in each loading step, but the deviation between the predicted and the experimental results increases with increasing cycles. Figure 10 reveals simulations for Steps 1 and 2 presented earlier in Fig. 3a. The Bower model predicts ratchetting in the tensile direction for the first 60 cycles in Step 1 before reaching a ratchetting arrest. In Step 2, the Bower model predicts negative ratchetting followed by a ratchetting arrest. Despite the capability of the Bower model in predicting the ratchetting directions, the magnitude of the predicted strain differs noticeably from the experimental data. For constant-amplitude loading, the larger the coefficient Xm in the Ohno-Wang model, the smaller the ratchetting rate predicted by the model (JIAN6 & SEI-IITOOLU[1994a]). We can observe this in the left side plot of Fig. 10. Having a close look at Fig. 10 for Step 2, we find that when Xm is 1.0, the Ohno-Wang model predicts ratchetting in the negative direction for the first 50 cycles, then switching to positive direction. The tendency predicted when Xm is 3.0 is the same as when Xm is 1.0. If eqn (11) is used for xm, the Ohno-Wang model predicts no ratchetting for Step 2. The experimental and predicted ratchetting results displayed in Fig. 11 are for the two step loading shown in Fig. 4. Due to the limited number of loading cycles for Step 1, the predictions by the Ohno-Wang model at Xm = 1.0 and 3.0 are in agreement with the experimental results, but using eqn (11) for Xm results in small ratchetting strain predicted. The Bower model predicts much higher ratchetting than experimentally observed

864

Y. JIANG and H. SEHITOGLU

for Step 1. As we can observe in the right side of the figure, the experimental result shows a ratchetting direction change; the strain ratchets in the negative direction for the first 250 cycles and in positive direction thereafter. Clearly, the Bower model is not able to predict the ratchetting direction change. When Xm = 1.0 and 3.0, the Ohno-Wang model predicts qualitatively correct tendency in the ratchetting directions for Step 2, but the predicted change points of the ratchetting direction disagrees with the experimental observation. When eqn (11) is used for x,n, the Ohno-Wang model predicts ratchetting arrest for Step 2. For this two step loading, the Ohno-Wang model with Xm 3.0 provides reasonable ratchetting prediction. In Fig. 12, we display results for the nonproportional step loading case presented in Fig. 8. With the constant tensile axial stresses, the axial strain ratchets in the positive direction in Step 1 and negative direction in Step 2. The Bower model and the OhnoWang model (Xm = 1.0 and 3.0) predict the ratchetting directions correctly for both loading steps. Once again, the predicted results by the Bower model differ greatly in magnitude from the experimental observations. The Ohno-Wang model on the other hand predicts good quantitative results for both loading steps. This model with X,, being expressed by eqn (11) provides improved predictions for Step 1, but it predicts ratchetting direction shift at about 700th cycle in Step 2, which contrasts to the experimental observation. One notable phenomenon with the Ohno-Wang model is that the model with Xm = 3.0 provides prediction closest to the experimental results for Step 2, while in Step 1 the model with Xm = eqn (11) matches the experiments closely. =

VI. DISCUSSION OF RESULTS

Under the basic framework of plasticity theory of yield surface and yield surface translation in the absence of rate-dependence and isotropic hardening, the plastic deformation is determined by the plastic modulus function, h, and the translation direction of the yield surface. When the deviatoric stress components are proportional all the time (proportional loading), the principal stress directions are unchanged, and the translation direction of the yield surface always coincides with the exterior normal direction. Therefore, for proportional loading the plastic deformation is determined by the variation of the plastic modulus function, h. The proportional loading is then classified as Type I (JL~NG& SEmTO6LU [1994a]). Figure 13 presents a schematic of the stress-strain

Sij 1

si~ s~7-

st 1

Fig. 13. Schematic o f ratchetting due to variation of plastic modulus function for proportional (Type I) loading.

Ratchetting under multiple step loading

865

behavior due to variation of plastic modulus function, where h + is the plastic modulus function at stress point P and h - represents the plastic modulus function at P ' in the opposite loading direction, and points P and P ' are symmetric with respect to the mean stress state S~. Obviously, the difference of h + from h - results in ratchetting. When the ratio h + / h - is less than 1.0 in a loading cycle, the strain will ratchet in the positive direction, and when h + / h - is larger than 1.0, ratchetting will be in the negative direction. Shown in Fig. 14 are the variations of the ratio h + / h - at cycles 4 and 4096, respectively, for Step 1 and Step 2 of the uniaxial ratchetting experiment shown in Fig. 3a. The plastic modulus function, h, is determined from the numerical differentiation of the experimental stress-strain curves. The horizontal axis in Fig. 14 represents the stress away from the mean stress state. Shown in Fig. 14 in solid lines for Step 1, the h + / h - ratio is less than 1.0 at cycle 4, which corresponds to the ratchetting in the positive direction. The h + / h - ratio is approaching 1.0 as the loading history evolves, which is consistent with the ratchetting rate decay. At cycle 4096 of Step 1, h +/h - is approximately 1.0. The stabilized h +/h - ratio is altered by the shift of mean stress in Step 2. Noting the dashed lines in Fig. 14, at cycle 4 of Step 2 the h + / h - ratio is always larger than 1.0, which results in ratchetting in the negative direction. The difference of h + / h - ratio approaches 1.0 as the loading cycle increases in Step 2. At cycle 4096 the h + / h - ratio approaches 1.0 again. In contrast to Type I, the other extreme case is termed as Type II ratchetting in which the plastic modulus function plays no role and the ratchetting is solely contributed by the inconsistency of the translation direction of the yield surface and the normal (plastic deformation) direction. The loading case shown in Figs. 8 or 12, where the axial stress is constant and the shear stress varies symmetrically, is a typical example of Type II ratchetting (JIANG & SEHrroGLu [1994a]). Referring to Fig. 15, denote 0 as the angle between the normal direction and the translation direction of the yield surface. For Type II, the ratchetting rate decay corresponds to the continuous 0 decrease with the loading cycles. From eqn (2), because h and ( d S ~ l n k t ) are all nonnegative, the ratchetting direction is determined by the normal direction n e. For axial-torsional loading, because $11 = 2 ~ x / 3 , where ax is the axial stress, the sign of nil decides the sign of dep, the plastic strain

1070 Steel

1.2-

[ 1.1-

cycle4 cycle 4096 ~

'..C

1.0-

Step11

........... Step 2 I

/

.......................

........................

cycle 4

0.9-

0.8 100

I

I

I

i

200

300

400

500

la,- ~p I Fig. 14. Variation of

h+/h

-

ratio from a two step uniaxial experiment.

866

Y. JIANG and H. SEHITOGLU

d(t 0

¥O Fig. 15. Schematic of yield surface and normal n in deviatoric stress space.

increment in the axial direction, n t ] being positive for a whole cycle results in ratchetting in the positive strain direction, and vice versa. Under the basic framework of plasticity theory of yield surface, normality flow rule, and yield surface translation, the components of the unit exterior normal on the yield surface can be determined from the numerical differentiation of the experimental stress-strain relations (JtANG & SEI-IITOGLU [1994a]). Figure 16 shows the ntl variations with the shear stress for the nonproportional ratchetting experiment displayed in Fig. 8. At cycle 4 of Step 1, nil undertakes a positive value, implying a positive ratchetting. As the loading cycle increases, n~] is approaching zero, which results in near-zero ratchetting. When the axial mean stress is reduced in Step 2, nn becomes negative, which corresponds to ratchetting in the negative direction. With increasing loading cycles in Step 2, nH approaches zero again, commensurate with decreasing ratchetting rate in the axial direction. Due to the introduction of the additional internal state variable /3ij, the Bower model, eqn (3), can predict negative ratchetting at positive mean stress or vice versa. However, this model always predicts a ratchetting arrest after a certain number of loading cycles, and it cannot predict the ratchetting direction change because ~ij neutralizes

o

BA ' ~A " ~ A/ ~ 0.3

B'

B

B' ...~-...A.~-r~:..~. B

1070 Steel

~

B

0.2 0.1

--~

............. __ E

cycle 4 ~

A~2048

...... A . . . . . . . . . .

0.0 ~cycle4

-0.1

/

'

-0.2 -0.3 -300

I -200

I -100

i 0

I 100

I 200

I 300

Shear Stress, MPa Fig. 16. Variation of the normal c o m p o n e n t nll from a nonproportional experiment.

Ratchetting under multiple step loading

867

with increasing loading cycles. By adjusting the material constants in the model, the predictions can be improved to some degree but such adjustment of the material constants cannot change the overall tendency of the model. Despite the inaccuracy of the predictions, the Bower model displays some advantages. The additional internal variable/3ij enables the model to predict decrease, increase, and constant ratchetting. /~ij on the other hand serves to render memory from the previous loading history. From the experimental observations on the ratchetting behavior for both single and multiple step loadings, it is reassuring to have such an internal variable. An improvement can be made on the Bower model by employing the same expansion of the backstress similar to that used in the Chaboche model (CrIAaOCrlE et al. [1979]) and the Ohno-Wang model. The characteristics of the Ohno-Wang model predictions for constant amplitude loading are that it predicts ratchetting rate decay for a number of loading cycles before reaching constant ratchetting rate (Jt~rc & SEmTOGLtr [1994a]). In the region of ratchetting rate decay, the tendency of the ratchetting rate is in general agreement with experimental observations. However, the duration of this region depends both on the coefficient Xm and the loading type. Generally, the larger the Xm, the faster the predicted ratchetting rate and longer the ratchetting decay region. With the same Xm, value this region is larger for Type II than that for Type I loading. When Xm = 0, the model predicts constant ratchetting for the proportional or Type I loading and ratchetting decrease for the Type II loading. When Xm = +o% the model results in full closure of hysteresis loop for Type I loading but predicts transient ratchetting for Type II loading. From the model we can deduce that a material undergoing constant ratchetting for Type I loading will display ratchetting rate decay for Type II loading, which contrasts to the experimental findings of HASSANet al. [1992] and BOWER [1987]. Except for Xm, the other material constants in the Ohno-Wang model are determined from the symmetric strain or stress-controlled uniaxial loading. Based on the comparisons in Figs. 10 through 12, we note that the coefficient Xm controls the ratchetting predictions of the Ohno-Wang model. The Ohno-Wang model at Xm = 3.0 provides results close to the experimental observations for the first 100 cycles in the first loading step and better overall results than the other two Xm selections. One notable outcome is that for Xm = 3.0 the model provides encouragingly good backward ratchetting in Step 2 shown in Fig. 12. In fact, the performance of the model for the case of Fig. 12 with any of the three X,~ selections is better than the other cases of Figs. 9 through 11. Knowing the ratchetting shown in Figs. 9 through 11 belongs to Type I and Fig. 12 belongs to Type II, we conclude that under the multiple step loading condition the OhnoWang model is more pertinent for Type II ratchetting prediction than for the Type I case. Equation (11) places a higher value for the coefficient Xm for proportional loading than for nonproportional loading. As a result, for the proportional loading shown in Figs. 10 and 11, the predicted ratchetting by the Ohno-Wang model using Xm--- eqn (11) is lower than the experimental results. Although in Fig. 12 superiority of eqn (11) is verified in the prediction of Step 1 ratchetting, the ratchetting direction change predicted in Step 2 differs from the experimental observation. Consequently, from the predictions of multiple step ratchetting we conclude no overall preference for eqn (11) over the choice of a constant Xm. We note the possible influences of isotropic hardening and nonlinear elasticity in material ratchetting behaviors. The isotropic hardening for the 1070 steel is not significant. Nonlinear elasticity effects have been observed in 1070 steel, but its contribution to the ratchetting deformation is limited.

868

Y. JIANG and H. SEHITOGLU

VII. CONCLUSIONS

From the observations of the ratchetting behavior of the 1070 steel under multiple step loadings and the evaluations of the existing plasticity models in light of the experimental results, the following conclusions can be drawn, 1. Under multiple step loadings, the material exhibits a strong memory of the previous loading history, and such memory has a great influence on the subsequent ratchetting. The material could ratchet in opposite direction to the mean stress or could reverse ratchetting direction with time. 2. Classical incremental plasticity theories which perform favorably for straincontrolled cases are inappropriate for the stress-controlled ratchetting predictions. A recently developed model based on the Armstrong-Frederick hardening rule by Bower can quantitatively predict the ratchetting directions in some cases. Although the model predictions deviate severely from the experiments in some cases, the concept of including an additional variable in the Armstrong-Frederick rule is promising for ratchetting representation. 3. The Ohno-Wang model can predict negative ratchetting under positive mean stress, or vice versa, and ratchetting direction changes during a step loading. This model can provide results which agree with experimental observations for Type II nonproportional loading, but its performance on the ratchetting prediction for Type I proportional loading is unsatisfactory. 4. Ratchetting transients have been partially explained by the variation of plastic modulus variation in the cycle and the noncoincidence of the plastic strain direction and the yield surface translation direction. Acknowledgements-The research was supported by the Association of American Railroads, Technical Center, Chicago, Illinois, with Dr. Dan Stone as monitor. The cooperation of Roger Steele, Gerald Moyar, and Michael Fec is acknowledged. The experiments were conducted in the Advanced Materials Testing and Evaluation Laboratory (AMTEL) at the University of Illinois at Champaign-Urbana. The assistance of Drl Peter Kurath, director of AMTEL, is greatly appreciated.

REFERENCES

1955 1959 1965 1966 1967 1969 1975 1975 1976 1976 1978

PRAGER,W., "The Theory of Plasticity: A Survey of Recent Achievements," Proceedings, Institute of Mechanical Engineers, London, 169, 41. ZIEGLER,H., "A Modification of Prager's Hardening Rule," Quarterly of Applied Mechanics, 17, 55. Dol~a,~,J.T., "Nonlinear Response Under Cyclic Loading Conditions," Proceedings, Ninth Midwestern Mechanics Conference, University of Wisconsin, Madison, WI, p. 2. ARMSTRONG,P.J., and FREDERICK, C.O., "A Mathematical Representation of the Multiaxial Bauschinger Effect," C.E.G.B., Report R D / B / N 731. MROZ,Z., "On the Description of Anisotropic Workhardening," J. Mech. Phys. Sol., 15, 163. Mgoz, Z., "An Attempt to Describe the Behavior of Metals Under Cyclic Loads Using a More General Workhardening Model," Act Mech., 7, 199. DnvnlJAS, Y.F., and PoPov, E.P., "A Model of Nonlinearity Hardening Materials for Complex Loading," Acta Mech., 21, 173. KRmG, R.D., "A Practical Two Surface Plasticity Theory," ASME J. Appl. Mech., 42, 641. DAFALIAS,Y.F., and POPOV, E.P., "Plastic Internal Variables Formalism of Cyclic Plasticity," ASME J. Appl. Mech., 43, 645. LAMBA,H.S., Nonproportional Cyclic Plasticity, T&AM Report No. 413, Department of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign. LAUmA,H.S., and S1OEBOTTOM,O.M., "Cyclic Plasticity for Nonproportional Paths. Part ii; Comparison with Predictions of Three Incremental Plasticity Models," ASME J. Eng. Mat. Techn., 100, 104.

Ratchetting under multiple step loading

1979 1981 1981 1982 1983 1985 1985a 1985b 1985c 1987 1989 1989 1991 1991 1991 1991 1992 1992 1993 1993a 1993b 1994a 1994b

869

CrtAaocrm, J.L., DAr~GVAN, K., and COI~.DmR,G., "Modelization of the Strain Memory Effect on the Cyclic Hardening of 316 Stainless Steel," Transactions of the Fifth International Conference on Structural Mechanics in Reactor Technology, Div. L, Berlin, L l l / 3 . DRUCKER,D.C., and PAL~EN, L., "On Stress-Strain Relations Suitable for Cyclic and Other Loading," ASME J. Appl. Mech., 48, 479. GAROO,Y.S., "Multiaxial Fatigue of Metals," Ph.D. Dissertation, Department of Mechanical Engineering, Stanford University. GARtrO,Y.S., "Prediction of Stress-Strain Response Under General Multiaxial Loading," in ROnDE and SWEASErCOEN(eds.), Mechanical Testing for Deformation Model Development, ASTM STP 765, American Society for Testing and Materials, p. 223. TSENC,N.T., and LEE, G.C., "Simple Plasticity Model of Two-Surface Type," ASCE J. Eng. Mech., 109, 795. INotm, T., IGAm, T., YOSmDA,F., SUZUKI,A., and M ~ , S., "Inelastic Behaviour of 2¼ Cr-Mo Steel Under Plasticity-Creep Interaction Conditions," Nucl. Eng. Design, 90, 287. McDowELL, D.L., "An Experimental Study on the Structure of Constitutive Equations for Nonproportional Cyclic Plasticity," ASME J. Eng. Mat. Techn., 107, 307. McDowELL, D.L., "A Two Surface Model for Transient Nonproportional Cyclic Plasticity, Part I: Development of Appropriate Equations," ASME J. Appl. Mech., 52, 298. McDoWELL,D.L., "A Two Surface Model for Transient Nonproportional Cyclic Plasticity, Part II: Comparison of Theory with Experiments," ASME J. Appl. Mech., 52, 303. BOWER,A.F., "Some Aspects of Plastic Flow, Residual Stress and Fatigue Due to Rolling and Sliding Contact," Ph.D. Dissertation, Emmanuel College, Department of Engineering, University of Cambridge. BOWER,A.E, "Cyclic Hardening Properties of Hard-Drawn Copper and Rail Steel," J. Mech. Phys. Solids, 37, 455. MOOSBROC, GER, J.C., and McDoWELL, D.L., "On a Class of Kinematic Hardening Rules for Nonproportional Cyclic Plasticity," ASME J. Eng. Mat. Techn., 111, 87. Cn~a3ocrm, J.L., NOUAILI'IAS,D., PACOU,D., and PAULMIER,P., "Modeling of the Cyclic Response and Ratchetting Effects on lnconel-718 Alloy," Eur. J. Mech. Solids, 10, 101. McDoWEtt, D.L., "Nonproportional Cyclic Plasticity of Rail Steels," Final Report for AAR Contract META-91-290. McDoWELL,D.L., and MOYAR,G.J., "Parametric Study of Cyclic Plastic Deformation in Rolling and Sliding Line Contact with Realistic Nonlinear Material Behavior," Wear, 144, 19. Om~o, N., and WAr,C, J.-D., "Nonlinear Kinematic Hardening Rule: Proposition and Application to Ratchetting Problems," Transactions of the 1lth International Conference on Structural Mechanics in Reactor Technology, Tokyo, Japan, Vol. L, p. 481. HASSAt~,T., COROr~A,E., and KYRIAKIDES,S., "Ratchetting in Cyclic Plasticity, Part II: Multiaxial Behavior," Int. J. Plasticity, 8, 117. McDoWELL,D.L., "Description of Nonproportional Cyclic Ratchetting Behavior," Progress Report on AAR Contract No. META-92-195. JIAr~G,Y., "Cyclic Plasticity With an Emphases on Ratchetting," Ph.D. Dissertation, Department of Mechanical Engineering, University of Illinois at Urbana-Champaign. Om~o, N., and WANG, J.-D., "Kinematic Hardening Rules With Critical State of Dynamic Recovery. Part I: Formulation and Basic Features for Ratchetting Behavior," Int. J. Plasticity, 9, 375. OrlNo, N., and WANG, J.-D., "Kinematic Hardening Rules With Critical State of Dynamic Recovery. Part II: Application to Experiments of Ratchetting Behavior," Int. J. Plasticity, 9, 391. JL~qG,Y., and SErnrOGLU, H., "Cyclic Ratchetting of 1070 Steel Under Multiaxial Stress State," Int. J. Plasticity, 10, 579. JIANC,Y., and SEHITOGLU,H., "Comments on the Mroz Multiple Surface Type Plasticity Models," Submitted to International Journal of Solids and Structures.

AMTEL Univ. of Illinois at Urbana-Champaign 104 S. Wright St. Urbana, IL 61801, USA Department of Mechanical Engineering Univ. of Illinois at Urbana-Champaign 1206 West Green Street Urbana, IL 61801, USA ( Received in final revised form 10 February 1994)

870

Y. JIANG a n d H . SEHITOGLU

NOMENCLATURE A , B, C

= material constants

c, c1, c2, Cm = m a t e r i a l c o n s t a n t s in p l a s t i c i t y m o d e l s d

= p r e f i x d e n o t i n g i n f i n i t e s i m a l i n c r e m e n t or d i f f e r e n t i a t i o n

E

= Young's modulus

h

= plastic m o d u l u s f u n c t i o n

k

= yield stress in s i m p l e shear

llij

= c o m p o n e n t s o f the unit e x t e r i o r n o r m a l to the yield surface at the stress state

M

= a positive integer d e n o t i n g the n u m b e r of t e r m s in the backstress e x p a n s i o n for a n ArmstrongF r e d e r i c k type m o d e l

r, r m

= m a t e r i a l c o n s t a n t s in plasticity m o d e l s

S~

= c o m p o n e n t s o f the d e v i a t o r i c stress t e n s o r

ctij

= c o m p o n e n t s o f the d e v i a t o r i c b a c k s t r e s s t e n s o r

(m)

c~ij

= m t h b a c k s t r e s s t e n s o r ( m = 1 , 2 , . . . , M ) for a n A r m s t r o n g - F r e d e r i c k type m o d e l = K r o n e k e r delta

cP

= c o m p o n e n t s o f the plastic s t r a i n t e n s o r

A

= e q u i v a l e n t plastic s t r a i n

Xm

= material constant

= Poisson's ratio