ratcheting in cyclic plasticity, part i: uniaxial behavior

International Journal of Plasticity, Vol. 8, pp. 91-116, 1992 0749-6419/92 $5.00 + .00 Printed in the U.S.A. Copyright © 1992 Pergamon Pre~ Ltd.

R A T C H E T I N G I N C Y C L I C P L A S T I C I T Y , P A R T I: U N I A X I A L B E H A V I O R

T,~,s~ H~ss.~,N and STELIOS KYRIAKIDES

The University of Texas at Austin

(Communicated by David McDowell, Georgia Institute of Technology)

Abstract-This paper is concerned with the phenomenon of cycfic creep, or ratcheting of metals cyclically loaded in the plastic range (ratcheting here describes the cyclic accumulation of deformation). Part I is concerned with time-independent ratcheting under uniaxial loading; Part II addresses various ratcheting mechanisms under biaxial loading. In the uniaxial case, a systematic set of stress-controlled cyclic experiments was conducted on 1020 and 1026 (heat=treated) carbon steels, in which the effect of the cycle amplitude and mean stress on the rate of ratcheting was established. Three rate-independent cyclic plasticity models are critically reviewed with respect to their performance in predicting the rate of ratcheting measured experimentally. A modification of the Dafallas-Popov model is outlined in which the linear bounds are allowed to "translate" in the direction of the cyclic creep at the rate of ratcheting. With this modification, the model is shown to improve the simulation of ratchcting in all ranges of parameters tested experimentally.

1. INTRODUCTION

In many applications, structures and structural components must be designed to with- stand cyclic loads that can lead to occasional excursions into the plastic range of the material. Examples are structures in earthquake-prone areas, pressure vessels and various nuclear reactor components, offshore structures in extreme weather conditions, and many structural components operating at elevated temperatures. A necessary tool for the prediction of the allowable number of load cycles such structures can sustain is a dependable constitutive model.

Motivated by these needs, the modeling of the elastic-plastic behavior of metals under complex loading histories, which includes repeated loading and unloading, has received considerable interest over the last 15 years. The works on the subject can be categorized into those that deal with time-independent phenomena and those that also include time- and temperature-dependent behavior. A nonexhaustive list of major con- tributions to time-independent cyclic plasticity includes the works of Mgoz [1967,1969, 1976], D~XLL~S and PoPov [1975,1976], E~SENnER(~ [1976], LX~B~ and SR)E~OTTOU [1978], Snm~TOPd et al. [1979], DZUCKE~ and PXL~EN [1981], D~XLL~S [1981,1984], OHNO [1982], TSE~ and LEE [1983], T~aq~o, et al. [1985], CIL~BOCH~ [1986], NA~mI and NIrd~I. [1954], McDow~.L [1987], CI-I~nOCm/and NOLr~LI4~S [1989a,b], YOSmDA et al. [1978] and others. A similar list for works on time-dependent cyclic behavior includes BODhmR et al. [1979,1987], IQ~MP~. et al. [1984,1986], V~.~NIS and LEE [1984], M I ~ Z [1987], BE~qXL~.XL and IVL~Qu~s [1987], Cn,~m)cm/[1989], and others.

Extensive studies of the mechanical behavior of metals under uniaxial cyclic loading histories (see Jn~qS~,I~ [1975], L,,,h'DOP.~.F et al. [1969,1970], Moggow [1965]) have re- vealed that, under such loading conditions, metals can harden or soften. Cyclic hard-

91

92 T. HASSAN and S. KYRIAKIDES

" AAAA o. VVVVI/_L'

~-. (~si)

SSS04

-40

Fig. 1. Cyclic hardening behavior of SS-304. (From ALAMEEL [1985]. Reprinted by permission.)

ening or softening can best be demonstrated in a strain-symmetric cyclic history, as shown in the inset of Fig. 1. The cycle-dependent hardening of stainless steel 304 can be seen in the experimental results shown in Fig. 1. By contrast, carbon steel 1020, which was mechanically work hardened during its manufactur ing process, is seen in Fig. 2 to soften under a similar loading history. In both cases, the hysteresis loop tends to stabilize to one that is closed (stable hysteresis loop) after a number of cycles.

I f the strain cycles have a strain offset, the hysteresis exhibits a progressive shift o f its mean stress and stabilizes at the zero mean stress level. This shift is known as cyclic relaxation (see L.~r~r~ogar [19701).

Another interesting phenomenon occurs under unsymmetrical stress cycling. Figure 3a shows the stress-strain response of stainless steel 304 cycled in a stress control fashion, as shown in the inset. The cycles had a positive mean stress. The induced hysteresis loops never close, and, as a result, the recorded strain gradually creeps, or ratchets, in the direction of the mean stress (the rounding of the stress peaks indicates the time-dependence

Ratcheting in cyclic plasticity, Part I 93

" AAAAA o. VVV /l/_L'

CSl020

(ksi)

I -~-~(%)

- 7 5 "

Fig. 2. Cyclic softening behavior of CS 1020.

of this material, even at room temperature). The maximum strain recorded in each cycle (exp) is plotted in Fig. 3b as a function of the number of cycles (N). The rate of ratcheting is faster in the first 10 cycles. As the material hardens, the rate decreases to approximately a constant value. (It is important to note that for this material the rate of ratcheting does not decay to zero as Nincreases, as reported by SArcoOR for SAE 1045 steel [1972, Ch. 6]).

Figure 4a shows the ratcheting behavior of carbon steel 1020, which was shown earlier to be a cyclically softening material. Because the material softens with every cycle, the rate of ratcheting is exponential in nature, and leads to failure in a relatively small number of cycles. This illustrates a dangerous consequence of cyclic softening.

Inelastic material behavior under cyclic loads is indeed complicated. Time-dependence of the material can further complicate the phenomena described. However, a constitutive model that has the capability of simulating the major uniaxial phenomena described above and that can successfully generalize them to a multiaxial setting, while retaining a degree of simplicity, can be a very useful engineering tool in the analysis of the class of difficult structural problems described above (DRucrmR [1981]).

Motivated by the pioneering results of L,~n~, and Sm~.noTTo~ [1978], considerable effort has been directed toward cyclic hardening and, to a lesser degree, cyclic softening and relaxation. As a result, many state-of-the-art models have successfully demon-


(ksi)

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Fig. 3. (a) Cyclic creep (ratcheting) behavior of SS-304. (From AL~C~Et [1985]. Reprinted by permission.) (b) Maximum strain per cycle as a function of the number of cycles.

strated good agreement with experimental results on these phenomena. On the other hand, the more complicated phenomena related to cyclic creep have received less attention and remain a difficulty in modeling.

The main objective of this work is to address the problem of ratcheting. For simplicity, but also for a sharper definition of the problem, the study is limited to time-independent behavior, with the understanding that this is the first step toward a more complete study that includes time-dependent effects. It was recognized that the devel-


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opment of a consistent experimental data base was necessary in such an undertaking. Uniaxial and biaxial sets of experiments involving ratcheting were conducted. A number of models from the literature are critically reviewed by comparison to the experimental results generated, and various suggestions for optimum agreement with the experiments, without compromising simplicity, are made.

For easier reading, we divide the presentation into uniaxial and multiaxial behavior addressed in Parts I and II, respectively. However, the work must be read as an entity. For brevity, references that appear in Part I are not repeated in Part II. Equations ap- pearing in the two parts are numbered consecutively.

!1. RATCHETING UNDER UNIAXIAL LOADING

The phenomenon of strain ratcheting, which occurs in stress-control unsymmetric cycling of metals, was reported as early as 1911 by BAmSTOW. The subject received a new impetus in the 1960s with experimental works by C o r t ~ [1964], B e ~ [1960], MOR- ~tOW [1965], Lm'c~OR~V et al. [1969], and others. More recent works include those of Pn.o et al. [1979], Yosnm~, et al. [1980], AzAu~e~. [1985], and Ruoox.~.s and KI~a ,L [1989]. The prime motivation of these works was not always material modeling as such, and as a result they do not make available sufficient information on the stress-strain


responses to accommodate modeling. In view of this, a systematic experimental program was undertaken in order to generate the data necessary for evaluating the consitutive models.

The test specimens for each material tested were machined from the same material stock and processed in exactly the same way. The experimental work conducted includes sets of uniaxial ratcheting tests and various biaxial ratcheting tests. The biaxial tests involved combined internal pressure and cyclic axial loading. Details on these are reported in Part II.

Uniaxial ratcheting experiments were conducted on AISI 1020 and 1026 carbon steels. All test specimens were tubular, with a test section of 1.0 in. (25.4 mm) outside diameter and 0.050 in. (1.27 mm) wall thickness. The test specimens were machined from long, tubular stock with 1.25 in. outside diameter and 0.25 in. wall thickness. Special care was taken in the machining process to reduce its effect on the material properties of the specimens. The CS 1020 test specimens were tested as machined, whereas the CS 1026 specimens were heat treated prior to testing. The heat treatment involved heating at 1300°F (705°C) for 2 h (in air) followed by furnace cooling. The purpose of the heat treatment was to reduce the effect of mechanical work from the manufacturing process on the measured properties.

The experiments were conducted in a closed-loop servo-hydraulic test facility with an axial load capacity of 55 kips (250 kN). The facility provides the capability of testing under load, displacement, or strain control. The load was monitored through a cali- brated load cell, and the strain was monitored through strain gages bonded to the test specimens, and extensometers. The command signals were provided by a function gen- erator or through a computer-based data acquisition and control system. The load, displacement, and strains were continuously monitored and recorded through this data acquisition system. Data were acquired at intervals of strain of 0.02°70, and, in addition, all the load peaks were recorded. The axial stress-strain response was also recorded on analog plotters as a back-up to the digital data recorded in the computer.

The results presented in Figs. 1-4 demonstrate that cyclic hardening and softening can have a strong influence on the rate of ratcheting in unsymmetric stress cycling. In an effort to reduce the coupling of the two phenomena, all test specimens were cyclically stabilized by axial strain symmetric cycling in the range of _+ 1070. The results from this process on CS 1020 and CS 1026 are shown in Figs. 2 and 5, respectively. Twelve cycles were found to lead to a reasonably stable hysteresis. The rate of loading during this part of the history was 4 min/cycle. CS 1020 is seen in Fig. 2 to exhibit significant softening. In the case of CS 1026, the heat treatment reduced the yield stress of the material and introduced the elastic-perfectly plastic behavior, characteristic of mild steels, in the initial part of the response. At the same time, the heat-treated material is seen (Fig. 5) to be quite stable to strain symmetric histories. This indicates that the cyclic softening exhibited by CS 1020 may be attributed to the cold work in the manufacturing process.

We note that the cyclic stabilization adopted does not stabilize the material in the sense described by LAr~])~RAF et al. [1969]. Indeed, quite possibly, an incremental step test of the type they described is necessary for better stabilization. However, it will be shown that the influence of cyclic softening is diminished to such a degree by this prehistory that it is reasonable to neglect this phenomenon in the modeling.

Following the strain symmetric cycling, the specimens were unloaded to approximately zero axial stress and strain, as shown in Figs. 2 and 5. The test machine was then


(ksi ) . P ~

;

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_ _ ~ " " ~ . . . . . . . . . . . ~ ( % )

Fig, 5, Strain-symmetric stress-strain response of CS 1026 (heat treated).

switched to the load control mode, an initial load offset (mean stress, o,:,~) was manu- ally prescribed, a stress amplitude (OxQ) was selected, and stress control cyclic loading was commenced. All stress cycles prescribed had a period of 2 rain.

HI. EXPERIMENTAL RESULTS

Figures 6a and 7a show typical stress-strain responses obtained from the stress control experiments on CS 1020 and CS 1026, respectively. In each case, the mean stress and amplitude stress of the cycle are normalized by the yield stress oo,, of the monotonic part of the response (0.2070 strain offset yield stress, for CS 1020 and the "plateau" stress for CS 1026) and are given in the figures. The stable hysteresis loop obtained in the prehistory is included in the figures and is indicated by a dashed line.

The test specimens were cycled until the rate of ratcheting reached approximately a constant value, or until an axial strain of 3-4070 was accumulated (whichever came first). Thus, in all cases, the cycling was interrupted well before failure of the specimens. In a number of exper'tments, following the ratcheting test, a hysteresis loop with strain amplitude of 1 070 was traced (under strain control). Such loops are included with dashed lines in Figs. 6 and 7. In all cases, these hysteresis loops were found to be almost iden- tical to the stable hysteresis loops developed in the prehistory. It was thus concluded that, as a result of the ratcheting, such hystereses tend to translate along the strain axis at the ratcbeting rate. This observation was found to be crucial in modeling the phenomenon (see next section).

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Fig. 7. Axial strain ratcheting of CS 1026; comparison between (a) experiment and (b) prediction.

Two sets of experiments were conducted for each material group. In the first set, the mean stress was kept constant and the amplitude of the cycle was varied. In the second set, the amplitude was kept constant and the mean stress was varied. The stress-strain curves obtained are similar in nature to those presented in Figs. 6 and 7, and they are not included here. A summary of these results is presented in Figs. 8 and 9, where the maximum strain recorded in each cycle is plotted as a function of the number of cycles. It is clear from these results that the two stress cycle variables chosen significantly af- fect the rate of ratcheting in both sets of results presented. Increase in either of these quantities leads to a faster rate of strain accumulation.

In the case of CS 1020, and for the range of parameters tested, the rate of ratcheting experiences a reduction in approximately the first 10 cycles, Following this initial transient, the strain accumulated is seen to be directly proportional to the number of cycles applied. This is also reflected in the stress-strain results, a representative sample of which is shown in Fig. 6a, where, following the first few cycles, the loops traced un- derwent only small changes in shape.

For the range of parameters tested for CS 1026, the initial transient behavior is ex- tended to approximately 20 cycles. Further cycling occurs at approximately a constant rate (see Fig. 9). At the same time, as the cyclic history progresses, a gradual change in the shape of the stress-strain loops traced is seen to occur in Fig. 7a. This is due to some reduction in the elastic modulus, some reduction in the size of the elastic region, and


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Fi~. 8. M ~ i ~ u m strain as a fu,ct ion of number of cycles in u n i ~ i a l ratehetin~ experiments on CS !020. (a) Constant mean stress; (b) constant ampStude stress.

a gradual change of the shape of the stress-strain curves. The net result is a hysteresis that encloses a larger area. These changes get progressively larger with the number of cycles. In the range of parameters tested, the changes in the hysteresis loops cause a gradual reduction in the ratcheting rate. This is important because what is conventionally accepted as cyclic softening was shown in Fig. 4 to lead to an acceleration in the rate of ratcheting. It is thus difficult at this stage to categorize the induced changes in terms of the conventional terms.

One experiment in which the mean stress of the stress cycle was negative was conducted in order to establish any differences in the material response from the results presented above. The results are shown in Fig. 10 (note that the termination of the strain


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symmetric prehistory was altered to maintain similarity with the other experiments). The ratcheting behavior recorded has essentially the same characteristics as those observed in the experiments with positive mean stress.

IV. MODELING OF U N I A X I A L RATCHETING

The modeling of the ratcheting behavior observed in the cyclic experiments presented will be carried out within the fundamental assumptions of classical cyclic plasticity theory. A guiding constraint in this effort will be the simplicity of the models considered. In view of the approach adopted and the observations made in the experimental part of the study, we will restrict our attention to cyclically stable material behavior. The

102 T. Hnss~ and S. KYRIAKIDES

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m o d e l s considered incorporate the s impl i fy ing assumpt ions o f J2-type plasticity with various k inemat ic type hardening rules. The yield surfaces are represented by:

f(o - a ) = [ ~ ( s - a ) - ( s - a ) ] ~/2 = Oo O )

where o is the stress tensor, s is the deviatoric stress tensor, ot is the current center o f the yield surface in the stress space, and a is the center in the deviatoric stress space. For a cycl ical ly stable material , Oo is a constant . Al l other surfaces required by the mode l s used will a lso be as sumed to be Mises surfaces.


The plastic strain increment is given by

1 (~f.d~)~f d'~ = T~ \ ~. a-;" (2)

Under the assumptions adopted, for uniaxial loading H in (2) is given by

H = de. ~. (3)

The model is completed by incorporating a suitable hardening rule to the flow rule, which prescribes the translation of the yield surface in stress space during the loading history. Various hardening rules will be discussed in detail in Part II of this paper.

Three different ways for evaluating the plastic tangent modulus H were found to be promising for the problems considered and are summarized below.

IV. 1. Model I: The Drucker-Palgen model (1981)

The plastic modulus H is assumed to be strictly a function of the second invariant of the deviatoric stress tensor J2 of the following form:

H = (Aj~V) -'. (4) A and N are material constants evaluated from a segment A E in Fig. 11 of the stable hysteresis loop obtained from an experiment conducted in a suitable strain range. For the purposes of this paper, A E was fitted with the three-parameter Ramberg-Osgood expression as follows:

o- - Experiment

,//---- .......... /

-, / ~ Loop

~&. i i. ~xpedmen~l by~{e~ese~ ~d p~fio~ f~om model I.

104 T. HASSAN and S. KYRL~KrOEs

° - ,

. . . . t5) 7 E \~v/

where E, o~,, and n are material constants. From (3)-(5)

H= [ ~ ~E ~,--a~-~ { 3J2 ~'n-"/2 ] (6)

IV.2. Model II: The Dafalias-Popov model (1976)

The basic idea of this model is that the current tangent modulus depends on the "distance" in stress space of the current stress state, and of that representing the immediately previous elastic stress state, from a bounding surface. In the uniaxial setting in its sim- plest form, the bound is linear, as shown in Fig. 12. The two measures o f distance men- tioned above are 6 and ~in, respectively. The following relationship was proposed for H

(7a)

where

a h = (7b)

Eo P is the plastic modulus of the bound; the constants a, b, and m are evaluated from two uniaxial stress-strain curves.

l - / i ~ , o ~

~ i ,~

Fig. 12. Definition of parameters for model II.


IV.3. Model III: The Tseng-Lee model (1983)

This model is very similar to the model outlined above, except that the bounding surface is replaced by a memory surface. This surface is centered at the origin and grows isotropically every time its stress level is exceeded. Thus, it represents the biggest state of stress developed in the loading history (j~a~).

In the uniaxial setting, the memory surface is represented by two horizontal lines at the maximum stress level I O'm [ in history as shown in Fig. 13. ~ and 6in have the same meaning as in model II, but are measured from the memory lines. H can be evaluated from (7) with Eo ~ being the plastic modulus on the memory surface and ~b being replaced by Ore.

IV.4. Performance o f the models

The performance of the three models with respect to cyclic creep was evaluated by direct comparison to the experimental results obtained for CS 1020. In each case, the model parameters were selected in a way that would optimize the performance of the model. The parameters were selected from the experiments with 8xa = 0.709 and 8xm = 0.127(t~ -ffi ~/oo,).

In the case of Model I, the parameters were obtained from the upper half of the stable hysteresis loop. They are presented in Table 1.

In the case of Model II, E, t~o, E~, and ob were obtained from the stable hysteresis loop. The constants a, b, and m were obtained by fitting the two branches of the first loop in the stress-controlled loading history through eqn (7). The measured variables are given in Table 2. It is of interest to note that this fit yields a very good prediction of the stable hysteresis loop.

Model III was shown by TS~.N~ and L~.E [1983] to lead to ratcheting which exhibits a constant rate. In the experiments, the rate of ratcheting was not found to be constant. Thus, in order to optimize the performance of the model, parameters a, b, and m were

,i

O"

0

Memory Line

. . . . . ~ ~ , ~

i, I ~ I

I ;

~P

Fig. 13. Definition of parameters for model Ill.

106 T. HASSAr~ and S. KYRIAKIDES

Table 1. Parameters for model 1

Material E, ksi tro, ksi ay, ksi type (GPa) (MPa) (MPa)

CS 1020 25,125 35.0 27.0 3.9 (173.2) (241) (186)

CS 1026 26,320 19.0 17.8 5 (181.5) (131) (123)

obtained from the stress-strain loop traced during the 21st cycle when a steady state of ratcheting had been reached in the experiment. Parameters E, ao, Eft, and am were again picked from the stable hysteresis loop. The parameters chosen are given in Table 3.

The predictions from the three models of the experiment from which the material parameters were extracted are summarized in Fig. 14. The results presented involve the maximum strain predicted in each cycle exp plotted against N. The same sets of parameters were used to predict the response for a second case, with 8x,~ = 0.127 and 8xa = 0.627, which exhibits slower ratcheting. The predictions for this case are also presented in the same figure. In each case, the corresponding experimental results are included for comparison.

All three models exhibit ratcheting, although the rate of ratcheting predicted from each one differs considerably. Model I grossly overestimates the experimental rate of ratcheting for both cases (indeed, for all cases t e s t ed - s ee also A~AU~Er~ [1985]). The main reason for this performance can be attributed to the very simple nature of the flow rule adopted, in which the tangent modulus is assumed to depend only on J2 (or the value of stress in the uniaxial case). For example, in Fig. 11, B'D" is the response predicted for the B'D part of the experimental response; it is obtained by horizontal translation of segment BC of the stable hysteresis loop. Clearly, the prediction is overly soft and will lead to an accelerated rate of ratcheting. (Note that the remainder of the loop shown was predicted very well by the model.)

This model was primarily developed for predicting cyclic hardening, softening, and relaxation. Its performance in those phenomena is very good. Thus, in spite of its in- adequate performance in the ratcheting experiments discussed here, the basic simplicity of this model makes it attractive for use in many applications involving complex loading histories. (A modification of this model, suggested by ShAw and KYRIAKIDES

Table 2. Parameters for model lI

Material E, ksi Eo ~, ksi ~o, ksi a, ksi type (GPa) (GPa) ( M P a ) (GPa) b m

tr b varies with each experiment.

CS 1020 25,125 466.8 25.0 146,200 40 3 (173.2) (3.22) (172) (1008)

CS 1026 26,320 200.0 19.0 7!,100 27 2 (181.5) (1.38) (131) (490.2)


Table 3. Parameters for model IlI

Material E, ksi Eo ~, ksi Oo, ksi a, ksi type (GPa) (MPa) (MPa) (GPa) b m

CS 1020 25,125 466.8 25.0 304,200 141 3 (173.2) (3.22) (172) (2097)

CS 1026 26,320 200.0 19.0 281,900 214 4 (181.5) 0.38) (131) 0944)

om varies with each experiment.

[1985] for primarily strain symmetric cyclic Loading histories, suffers from the same de- ficiency as it essentially adopts the strict dependence of H on J2.)

Model II, with linear bounds, leads tO a reasonably good prediction of the racheting rate in the first few cycles for all experiments carded out. However, as the strain accu- mulates; the distances ~ and 6in, which are measured from the inclined bounds, increase for the loading part of the loop and decrease for the reverse loading part. This leads to smaller accumulation of plastic deformation in each cycle (reduction in the rate of ratcheting). Persistent cycling will always lead to arrest of the ratcheting (shakedown) for all

2

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-I

•

/ I . . " • •

~I • " / / / cs-,o2o . . . .

/ /

/~ ~ •° ° ~ ~ ~.--.127

I; ~ / . . . . . ~ " Exp. j~ ~ . : . ~ o 9 ~ ; ; ~ .

I ; ~ ~ ~27{ °E~ - .

~ o o o o o o o ~ 2 2 ~ 2

~ o o ~ 2 2 2 _ % ~ % ~ . . . . . . . ~ ~ ~ & ~ ~

~ ~ ~ . ' , b - - - ~ - - _ ~ _ _ - - _ ' ~ 4'o~ ~

~ ~ ~ ~

Fig. 14. Maximum strain per cycle as a function of number of cycles from two uniaxial ratcheting experiments and corresponding predictions from three models (o ~b = 61.0 (420), om= 64.5 (445); O ~b = 59.5 (410), o m = 63.0 (434) ksi (MPa)).


ranges of (6x,~, #xQ) if Eft > 0. This tendency is clearly seen in the predictions tor the experiment with fi~ = 0.709. On the other hand, the predictions for 6~ = 0.627 are seen to be much closer to the experiment. In this case, the rate of ratcheting is small, and as a result, many more cycles than shown are required for arrest of ratcheting to occur in the predictions.

The strengths of this model in predicting many complex aspects of inelastic material behavior have been demonstrated in previous publications. The special weakness demonstrated here is a direct result of the assumption that the bound is linear as shown in Fig. 12. A modification of this assumption for alleviating this problem will be suggested in the following section.

In model III, the bound is replaced with a memory line that, for our stress-controlled uniaxial experiments, is horizontal at a fixed value of stress tl m (developed in the strain symmetric prehistory). This, coupled with the assumption that the material is cyclically stable, leads to a constant rate of ratcheting.

In the case of the experiment with 6x,, = .709, and because of the special way the material parameters were selected, the predicted ratcheting rate is in very good agreement with the experiment. Indeed, good agreement was also obtained between the predictions and the experiments for 6~a = .683 and .711. However, for small values of 6xa, the predictions progressively deviate from the experiments. Worst disagreement occurs for fixQ = .627, where the model predicts negative ratcheting, which is clearly not the case in the experiment. This is a direct result of the way the parameters a, b, and m were selected. Some improvement in the performance of the model can be achieved by modifying the function h in eqn (7b). However, it was not possible for us to find a set of parameters that would lead to acceptable predictions for all the ratcheting experiments conducted.

IV.5. Modification of model H for ratcheting

As pointed out above, the performance of model II in predicting the cyclic creep experiments conducted, was found to be good in the first few cycles, but got progressively poorer for larger numbers of cycles. This inadequacy of the model can clearly be attributed to the assumption that the bound is linear. A linear bound with Eo P > 0 will lead to an eventual arrest of the cyclic creep process. A clue about how this might be cor- rected comes from the experimental observation (see Figs. 6 and 7) that the strain loops with 1 °70 amplitude translate along the strain axis at the ratcheting rate. In view of this, and with the definition of the bound in mind (DA~ALIAS ~ POPOV [1976]), it seems that a possible modification of the model, for improving its performance in ratcheting, might be to allow the bound to translate in the strain direction at the rate of ratcheting.

Two practical questions arise when implementing this concept for modifying the bound. The first is, when should the translation of the bound start? The second is, how do we include the bound translation without altering the basic working characteristics of the model, whose performance has been proven for various complex uniaxial loading histories?

The first question is answered by comparing the predictions of the original model to the experiments. The model as is initially yields ratcheting rates that are in good agreement to the experimental ones. We thus delay the onset of the translation of the bound until the predicted ratcheting rate starts to deviate f rom the experimental one. For example, in the case of CS 1020 the transition value of eft = 0.9070 was used for all pre-


dictions carried out. For each case the corresponding value of stress, ~z, was calculated from ~b and Eo P (note that [ ~z [ > [ ~ I).

The second question is resolved as follows: Let the point in the stress-strain space at which the bound starts to shift be point B shown in Fig. 15. In the prediction of the fn'st half of the first loop shown, the variable ~ will be measured from AB if e P < (~L - ~,)/Eo P. At B the bound starts to shift in the direction of the strain at an amount equal to the strain increment. Thus, for the same part of the response, ~ will be measured from BC if eP > (aL - ~b)/Eo P. The slope of the bound will be assumed to remain constant at Eo P at all times.

The lower bound has also shifted along eP an equal amount to the shift of the top one (the bound surface center does not change during the shift). Thus, on unloading and reloading, the distance 6 is now measured from the shifted bound, as shown in the figure (dashed line). On the loading branch of the second loop shown, the distance 6 is measured from A'B' until the strain reaches the value corresponding to B'. Beyond B' the bound starts to shift again in the direction of ~P and ~ is again measured from B'C. The process repeats itself and yields a constant ratcheting rate.

If the mean stress of the stress cycles is negative, the bounds can be made to shift in a similar fashion in the direction of -~P. A simple scheme for generalizing this concept to multiaxial setting will be discussed in Part II of this paper.

V. COMPARISON BETWEEN EXPERIMENTAL AND PREDICTED RESULTS

The modification described was used to predict the ratcheting responses of the two sets of experiments carried out. The same set of material parameters was used for all predictions made in each material group (an exception is ~b which was evaluated indi- vidually from each stable hysteresis loop). The material parameters used in each group

cr

B B'

~fnE

"~'~~~'-'~ Modified Bound

~P

Fig. 15. Modification of the Dafalias-Popov bounding line for ratcheting.

i 10 T. HASSAN and S. KYRIAKIDES

P 0.9% for CS are given in Table 2. The value of strain at the transition points was ex = 1020 and eft -- 2.350/o for CS 1026. (Note: The maximum stress in the stable hysteresis loop obtained following the strain controlled prehistory varied by a small amount in each test specimen. As a result, the value of trb calculated varied. For example, in the CS 1026 tests reported in the two papers the mean value of trb was 39.55 ksi and one standard deviation was 0.537 ksi. In the 11 CS 1020 tests the mean value of o~ was 60.15 ksi and the standard deviation was 0.762 ksi. In order to obtain the best possible predictions we opted to use the actual measured value of ob from each experiment.)

The stress-strain response predicted for one of the CS 1020 experiments is shown in Fig. 6b, together with the corresponding experimental one (the comparison between the experimental and predicted result starts at point A in the stress-strain curve shown). The predicted stress-strain loops change somewhat in shape up to the transition strain, but retain their shape after that point. In general, the stress-strain response predicted is in good agreement with the experiment. The rate of ratcheting in the simulation is also in good agreement with the one in the experiment. The initial and final strain loops predicted are included in the figure. They are in very good agreement with the experimental ones. Overall, the performance of the model is very good in this case.

Similar calculations were carried out for all experiments conducted. They are simb lar in nature to the ones shown in Fig. 6, and will not be shown here. A summary of the quality of the predictions is given in Fig. 16, which includes the maximum predicted strain in each loop as a function of the number of cycles applied. Results for a total of 11 cases are shown together with the corresponding experimental results. In all cases, the rate of ratcheting initially decreases with N, but is seen to stabilize, after a number of cycles, to a constant rate. Overall, the agreement between experiment and analysis is very good, especially in view of the fact that the values of ox~ and o ~ were varied quite significantly.

A key variable for the modified model is the selection of the value of strain at which the bound starts to shift. For the CS 1020 runs, the particular value chosen came from the experiment with ~ = 0.709 and ~ , , = 0.13. However, for this group of experb ments, the value of this transition strain did not vary significantly from one experiment to another, and as a result, the sensitivity of the predicted results to this parameter was not very large. Thus the predictions would not vary significantly if the transition strain is selected from another experiment.

Similar calculations were made for the CS 1026 experiments. The material parameters used are listed in Table 2. One of the predicted stress-strain responses is shown in Fig. 7b. The same observations made with regard to the results in Fig. 6b hold for these results. A summary of the results is shown in Fig. 17, where the maximum strains predicted in each cycle are plotted against the number of cycles applied for six experiments with various values of Oxa and o~,~. The corresponding experimental results are also included in the same figures. The predictions are similar in nature to the experiments. The agreement with the experimental results is reasonably good, but not as good as that demonstrated in Fig. 16 for CS 1020. The reason for this dissimilar performance in the two materials is due to the bigger changes exhibited in the hysteresis loops of CS 1026 during the cycling process in spite of the initial cyclic stabilization of the material. Such changes cannot be captured by the current model since it is based on the assumption of a cyclically stable material. A broadening of the model to include such cycle-dependent changes in the hysteresis loops must address the interaction of cyclic hardening, soft~ ening, and, perhaps, damage of the material with the cyclic creep process.


(xp (%) o

/ 'j CS1020 o ° o ~,=.13 o Experiment / . 7 1 1 ~

- - P r e d i c t i o n ~ ' / y

~ ~ , ~ * ~

~ * ~ ~

I ~ . . . . . . o . 6 ~

o ~b ' ~b ' ~ ' ~ ' N (a)

(xp

(i °l,,

~.

/ o 2 1 8 o CSl020 / o O , ,. o ~,,,,.185

~ = 6 6 / o o ° ~ "=° " . / o ° o ° / o ° 1 6 9 o Experiment / . o o o / o o ,

Prediction / _ ~ o ~ o o = / o ° o o o o ° oo

o ,147 ° o ~ o o o o o

° oo° ooo o * ° ooOO ° ~ ° ooooOO ° ~ ° ° .127 ° o o o = o o ~ o =

ooO o o oo

0 ' I~ ' 2~ ' ~ ' 4 ~ _ ~ (b)

Fig. 16. Maximum strain per cycle as a function of number of cycles for CS 1020 ratcheting experiments. Comparison between experiments and predictions.

The performance of the model in predicting ratcheting to a larger number of cycles was also tested in an experiment that was cycled to approximately 90 cycles. The results are shown in Fig. 18. The performance of the model is again seen to be quite good.

The parameters given in Table 2 were also used to reproduce the experiment with compressive mean stress of ~x= = -0.157 and 8m = 0.770. The results are compared to the experimental ones in Fig. 10. The predictions are in good agreement with the experiment.

A final test of the modified model was conducted by considering a stress history in which the mean stress of the cycles was kept constant (8~,m = .161), but the amplitude


~p (%) o ° o.815 CS 1026 o o

~,--.16 o o ° ° ° Experiment ,o o ~ o ° oo

- - Prediction o o o ° 7 8 7 o "

o ~ , ~

2 ~ ~ ~ ~ ~ ~ ~ ~ ,~ o o o ~ o o

~ ~o ° ° ~

,~ ~

. . . . . . . . . ~ ~ *

~ ~ ~ ~ , ~

o ,b ~o ' ~o ~o ~ (a)

~,. ~ o 2 2 5 (%) ~ ~ [ ~ 3 CS 1026

E,=79 ~ o o o

~ Experiment ~ o ~ ~ ~ o 160

~ Predictio~ ~ o ~ _ o o o o o

~ ° ° - _o ° ° ° ° ° ° °

~ ~ o ~ * ~* 2 o o o o o.104

o..o .... ~ ~ ~ * ~

~ * ~

0 I0 20 30 40 ~ N (b)

Fig . 17. M a x i m u m s t ra in per cycle as a f u n c t i o n o f n u m b e r o f cycles fo r C S 1026 r a t c h e t i n g e x p e r i m e n t s . C o m p a r i s o n b e t w e e n e x p e r i m e n t s a n d p r e d i c t i o n s .

was increased in four steps from ~x, = 0.622 to 0.690, 756, and 0.827. Ten cycles were conducted at each stress amplitude. The experimental and predicted results are compared in Fig. 19. All major features of the experiments are reproduced. The predicted rate of ratcheting is very similar to the experimental one. Jumps in strain, which occur during the first cycle immediately following the stress amplitude change, are somewhat under- estimated.

VI. CONCLUSIONS

This paper discusses the ratcheting behavior of metals under stress-controlled cyclic loading. A series of uniaxial experiments were conducted on 1020 and 1026 (heat-treated)

Ratchcting in cyclic plasticity, Part I 113

~xp ( % )

_

.

i .

0

t~O~ ~ 000000 CS 1 0 2 6 o o o o - ~

~,o =.790 o o o o - ~ - - 0 0 o-,,,=.195 o o o o _ ~ o ~o

oO ooO°

0° 0~

K~eriment

Pred i c t i on

' 2'o ' 4 0 ' ' Co ' ~ N

Fig. 18. Ratcheting strain as a function of N in an experiment with large N. Experiment and prediction.

carbon steels, in which the mean stress and amplitude of the stress cycles were varied. The test specimens were cyclically stabilized in a _+ 1% strain symmetric loading applied prior to the stress-controlled cycling. This was done in order to reduce the effect o f cyclic softening on the ratcheting rate.

The Drucker-Palgen (I), Dafal ias-Popov (II), and Tseng-Lee (III) models were used to simulate the experiments. Model I was found to predict a ratcheting rate that is much higher than that measured experimentally. This is a direct result o f the very simple nature of the flow rule adopted. Model II was found to yield good predictions of the rate of ratcheting at the earlier part o f the cyclic history. However, this model always leads to eventual arrest o f ratcheting if a linear bound is used. Model I I I predicts a constant ratcheting rate if the material is cyclically stable. The rate of ratcheting predicted by this model can be quite good for stress cycle parameters that are close to those of the experiment fitted. However, the predictions were not satisfactory for the whole range of experimental parameters considered.

In the experiments it was observed that large-amplitude strain-controlled loops, induced following significant strain ratcheting, can be obtained by a simple shift o f the stable hysteresis loop by a strain equal to the ratcheting strain. In view of this, a simple scheme for shifting the linear bound in Model II is proposed. In this modification, the bound is allowed to translate in the direction of ratcheting at the rate of ratcheting. Using this modification, the predicted rate of ratcheting was found to be in good agreement with the experimental one for all ranges of loading considered.

Acknowledgements-The test facility and data acquisition and control systems used to conduct the experiments were developed partly with the financial support of the Office of Naval Research under the equipment grarlt No. N00014-86-G-0155. The work was conducted with the support of the National Science Foundation under grant MSM-8352370. The work of George Alameel [1985] on the subject was very helpful in establish- ing the goals of this study. The authors wish to thank Edmundo Corona for many critical comments he made in the course of this study.

114 T. 14_ASSAN and S. KYRIAKIDES

or= (~)

~ - _!__;:~ ..................... ~ ~ _ _ _ _ _

//,/¢./~ ~

~__l . . . . . . . . . . . . . . . . ~;~. t_ _ _-- ~

(a)

(b)

E=p CSI026 ~m='161 o °

oo ° .8,27o ~ /

, ~o/ oo~ ~ .75~o o o / o ~ n t _ o /

~ 6 ~ . . . . . ~ i Pr~ict~n

o ~o ~o ~o #o ~ N

(c)

Fi~. ]9. Ratchets8 results o b ~ n e d by i n c r ¢ ~ i ~ the stress cycle amplitude in steps. (a) E x ~ d m e n t ; (b) predictions; (c) m ~ i m u m strain versus N.

REFERENCES

1911 BAIRSTOW, L., "The Elastic Limits of Iron and Steel Under Cyclical Variations of Stress," Philosophical Transactions of the Royal Society of London, Series A, 210, 35.

1960 BE~HA~, P.P., "Axial-Load and Strain-Cycling Fatigue of Copper at Low Endurance," Journal of the Institute of Metals, 89, 328.


1964 Co~m, L.F., J~t., "The Influence of Mean Stress on the Mechanical Hysteresis Loop Shift of 1100 Aluminium," ASME Journal of Basic Engineering, ~6, 673.

1965 Morrow, J.D., "Cyclic Plastic Strain Energy and Fatigue of Metals," ASTM STP, 3~ , 45. 1967 MltOZ, Z., "On the Description of Anisotropic Workhardenin~" Journal of the Mechanics and Phys-

ics of Sofids, 15, 163. 1969 ~ , R.W., Mo~gow, J.D., and ESDO, T., "Determination of Cyclic Stress-Strain Curve,~ Jour-

nal of Materials, 4, 176. 1969 l~to~, Z., "An Attempt to Describe the Behavior of Metals Under Cycfic Loads Using a More General

Workhardening Model," Acta Mechanica, 7, 199. 1970 LAm~O~.A~, R.W., ~The Resistance of Metal to Cyclic Deformation," ASTM STP, 467, 3. 1972 SAmJOn, B.I., Fundamentals of Cyclic Stress and Strain, The University of Wisconsin Press. 1975 DA~,UAS, Y.F., and POpOV, E.P., "A Model of Nonlincarly Hardening Materials for Complex Load-

ing," Acta Mechanica, 21, 173. 1975 JHAiCSA~, H.R., "A New Parameter for the Hysteresis Stress-Strain Behavior of Metals," ASME Jour-

nal of Engineering Materials and Technology, 97, 33. 1976 DA~AtD, S, Y.F., and Popov, E.P., "Plastic Internal Variables Formalism of Cyclic Plasticity," ASME

Journal of Applied Mechanics, 43, 645. 1976 E~SBNEERG, M.A., "A Generalization of Plastic Flow Theory with Application to Cyclic Hardening

Softening Phenomena," ASME Journal of Engineering Materials and Technology, 9~, 97. 1976 M~toz, Z., SHI~VASTAV̂ , H.P., and DUEEY, R.N., "A Non-Linear Hardening Model and Its Appli-

cation to Cyclic Loading," Acta Mechanica, 2~, 51. 1978 LAtmA, H.S., and SIDE~Yrrou, O.M., "Cyclic Plasticity for Nonproportional Paths: Part 1 and 2,"

ASME Journal of Engineering Materials and Technology, 100, 96. 1978 YOSHnJA, F., TAr~A, N., I~JAm, K., and StimATOm, E., "Plastic Theory of the Mechanical Ratch-

eting," Bulletin of the JSME, 21, 389. 1979 BOD~4~R, S.R., PA~TOU, I., and PA~tTOU, Y., "Uniaxial Cyclic Loading of Elastic-Viscoplastic Ma-

terials," ASME Journal of Applied Mechanics, 46, 805. 1979 Pn.o, D., REXK, W., MA.V~, p., and MACH~I~AUCH, R., "Cycfic Induced Creep of a Plain Carbon Steel

at Room Temperature," Fatigue of Engineering Materials and Structures, 1, 287. 1979 SHntATOm, E., Ir~GAm, K., and Yosam^, F., "Analysis of Stress-Strain Relations by Use of an An-

isotropic Hardening Plastic Potential," Journal of the Mechanics and Physics of Solids, 27, 213. 1980 YOSmDA, F., MUI~XA, K., and SH~ATOm, E., "Constitutive Equation of Cyclic Creep under Increas-

ing Stress Condition," Bulletin of the JSME, 23, 337. 1981 DA~A~AS, Y.F., "A Novel Bounding Surface Constitutive Law for the Monotonic and Cyclic Hard-

ening Response of Metals," 6th SMIRT Conference, L, L3/4. 1981 D]tucg.E]t, D.C., and PA~O~N, L., "On Stress-Strain Relations Suitable for Cyclic and Other Load-

ing," ASME Journal of Applied Mechanics, 48, 479. 1982 OHio, N., "A Constitutive Model of Cyclic Plasticity with a Nonhardening Strain Region," ASME

Journal of Applied Mechanics, 49, 721. 1983 Ts]~o, N.T., and LEE, G.C., "Simple Plasticity Model of Two-Surface Type," ASCE Journal of En-

gineering Mechanics, 109, 795. 1984 DA~AUAS, Y.F., "Modelling Cyclic Plasticity: Simplicity Versus Sophistication," in Mechanics of En-

gineering Materials, D~SAI, C.S., and GA~LAOHER, R.H. (eds.), John Wiley and Sons, New York, p. 153.

1984 KS~M~L, E., and Lu, H., "The Hardening and Rate Dependent Behavior of Fully Annealed AISI Type 304 Stainless Steel Under Biaxial In-Phase and Out-of-Phase Strain Cycling at Room Temper- ature," ASME Journal of Engineering Materials and Technology, 106, 376.

1984 N^GH~I, P.M., and NW~K~, D.J., J]t., "Calculations for Uniaxial Stress and Strain Cycling in Plas- ticity," ASME Journal of Applied Mechanics, $1, 481.

1984 VAJ.AmS, K.C., and L~, C.F., "Endochronic Plasticity: Physical Basis and Applications," in Mechan- ics of Engineering Materials, DBSA~, C.S., and GAXJ.A~JH~]t, R.H. (eds.), John Wiley and Sons, New York, p. 591.

1985 AL~Mr~J., G.M., "Cyclic Loading of Inelastic Materials: Experiments and Predictions," M.S. The- sis, The University of Texas at Austin, EMRL Report No. 85/2.

1985 SHAW, P.K., and KYlttA~m~S, S., "Inelastic Analysis of Thin-Walled Tubes under Cyclic Bending," International Journal of Solids and Structures, 21, 1073.

1985 TA~AXA, E., M ~ , S., and OOKA, M., "Effects of Strain Path Shapes on Non-Proportional Cyclic Plasticity," Journal of the Mechancis of Physics and Solids, 33, 559.

1986 C'I4AmJCH~, J.L., "Time-Independent Constitutive Theories for Cycfic Plasticity," International Jour- nal of Plasticity, 2, 149~

1986 IO~MPL, E., McMAHo~, J.J., and YAO, D., "Viscoplasticity Based on Overstress with a Differential Growth Law for the Equilibrium Stress," Mechanics of Materials, $, 35.

1987 Bi~NAX~J., A., and MAgQU~S, D., "Constitutive Equations for Nonproportional Cyclic Elasto-Vis- coplasticity," ASME Journal of Engineering Materials and Technology, 109, 326.

116 T. Hassnn and S. KYRIAKIDES

1987

1987

1987

1989

1989a

1989b

1989

BODNER, S.R., "Review of a Unified Elastic-Viscoplastic Theory," In Unified Constitutive Equations for Creep and Plasticity, MILLER, A.K. (ed,), Elsevier, New York, p. 273. McDowELL, D.L., "Simple Experimentally Motivated Cyclic Plasticity Model," ASCE Journal of Engineering Mechanics, 113, 378. MILLER, A.K., "The MATMOD Equations," In Unified Constitutive Equations for Creep and Plas- ticity, Miller, A.K. (ed.), Elsevier, New York, p. 139. CHABOCHE, J.L., "Constitutive Equations for Cyclic Plasticity and Cyclic Viscoplasticity," Interna- tional Journal of Plasticity, 5, 247. Cltn~ocrm, J.L., and NOI~AILrlAS, D., "Constitutive Modeling of Ratchetting Effec ts -Par t 1: Ex- perimental Facts and Properties of the Classical Models," ASME Journal of Engineering Material and Technology, 111, 384. CI-IABOCrm, J.L., and NOtlAILI-IAS, D., "Constitutive Modeling of Ratchetting Effects-Par t ll: Pos- sibilities of Some Additional Kinematic Rules," ASME Journal of Engineering Material and Tech- nology, 111,409. RUGGLES, M.B., and KREMPL, E., "The Influence of Test Temperature on the Ratchetting Behavior of Type 304 Stainless Steel," ASME Journal of Engineering Materials and Technology, l l l , 378.

Engineering Mechanics Research Laboratory Department of Aerospace Engineering and Engineering Mechanics The University of Texas at Austin Austin, Texas 78712

(Received 9 August 1990; in final revised form 10 January 1991)

a

E

fro ~

H

N

n

s

t

~, ~in

~, ~x

~xp

~P~ P ~x

d~ P

O, 0 x .~-

~ .~-

( l x a , 17xm -~-

#xa, #xm =

0 o ~

o o , ~

~ b ~

0 L ~

0 m ~

ay =

NOMENCLATURE

= current position of yield surface center in deviatoric stress space

= Young's modulus

= plastic modulus of bound

= generalized plastic modulus

~ ~S 'S

= number of loading cycles

= Ramberg-Osgood hardening parameter

--- deviatoric stress tensor

= time

= distance of stress points from bounds

--- axial strain

= maximum strain in each cycle

--- plastic axial strain

= plastic strain increment tensor

axial stress

stress tensor

amplitude and mean value of stress cycle

Crxa /Oo , , O x r a / O o,

size of yield surface

yield stress (0.2°70 strain offset (1020) or plateau stress (1026))

bounding stress

ratcheting limit stress

memory stress

Ramberg-Osgood yield parameter

ratcheting in cyclic plasticity, part i: uniaxial behavior

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