ratcheting of cyclically hardening and softening materials: i. uniaxial behavior

36
Pergamon International Journal of Plasticity, Vol. 10, No. 2, pp. 149-184, 1994 Copyright © 1994 Elsevier Science Ltd Printed in the USA. All rights reserved 0749-6419/94 $6.00 + .00 0749-6419(93)E0003..J RATCHETING OF CYCLICALLY HARDENING AND SOFTENING MATERIALS: I. UNIAXIAL BEHAVIOR TASNIM HASSAN and STELIOS KYRIAKIDES The University of Texas at Austin Abstract-The phenomenon of ratcheting of materials cyclically loaded in the plastic range is studied through combined experimental and analytical efforts (ratcheting here describes the cyclic accumulation of deformation). In particular the work seeks to illustrate how cyclic hardening and softening influence ratcheting. To this end, systematic sets of experiments were performed on stainless steel 304 and carbon steel 1018 which, respectively, exhibit cyclic hardening and soft- ening. Due to the wide variety of behavior observed, and to better illustrate the modelling chal- lenges, the results are divided into uniaxial and multiaxial behavior, and are presented in Parts I and II, respectively. In Part I, the results from a series of uniaxial stress-controlled experi- ments are presented, which illustrate the parametric dependence of ratcheting in the two mate- rials examined. Results from a set of auxiliary strain-controlled experiments required for quantifying the cyclic hardening and softening characteristics of the materials are also presented. In a preceding publication, the authors demonstrated that ratcheting in cyclically stable mate- rials could be simulated with consistent accuracy by allowing the bounds of the two-surface model of Dafalias-Popov to translate in the direction of ratcheting at the rate of ratcheting. This mod- ified model, coupled with previously developed schemes for simulating cyclic hardening in strain- controlled cycling, are used to simulate the experimental results developed. Strengths, weaknesses and plausible alternatives are critically presented. The results are quite promising. I. INTRODUCTION Ratcheting or cyclic creep in the context of mechanical behavior of materials describe a phenomenon where, under cyclic loading, the material experiences progressive accu- mulation of deformation, which can result in failure. The dangerous nature of the phe- nomenon is demonstrated in Fig. la, which shows a uniaxial stress-strain response of carbon steel (CS) 1020 in a cyclic history in which the stress was prescribed. The cycles had amplitude of trx~ = 57.47 ksi (396.3 MPa), and a mean value of a~m = 20.15 ksi (139 MPa) (see inset in figure). The strain is seen to grow at an increasing rate with each applied stress cycle. In this case, the specimen failed after ten cycles. The phenomenon is well known and has been reported among others by BAmSTOW[1911], BE~ [1960], Corr~ [1964], Dot~.N [1965], LArzt~,r [1970], P~o et al. [1979], YosmoA et al. [1980], AtOlL [1985], ~ o c n ~ and Nou.~m.rIAS [1989], RUC,6LES and Kt~m,L [1990], and HASSAN aria K ~ I D E S [1992]. Under multiaxial loading, ratcheting can occur if at least one component of stress is prescribed in a multiaxial cyclic loading history involving some plastic deformation (see H~s~ et al. [1992]). In such cases ratcheting will, of course, be in the direction of the prescribed stress(es). The phenomenon falls into the broader subject of cyclic plasticity, which has received considerable attention over the last twenty years. This has included micromechanical as 149

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Page 1: Ratcheting of cyclically hardening and softening materials: I. Uniaxial behavior

Pergamon International Journal of Plasticity, Vol. 10, No. 2, pp. 149-184, 1994

Copyright © 1994 Elsevier Science Ltd Printed in the USA. All rights reserved

0749-6419/94 $6.00 + .00

0749-6419(93)E0003..J

R A T C H E T I N G O F C Y C L I C A L L Y H A R D E N I N G

A N D S O F T E N I N G M A T E R I A L S :

I. U N I A X I A L B E H A V I O R

TASNIM HASSAN a n d STELIOS KYRIAKIDES

The University of Texas at Austin

Abstract-The phenomenon of ratcheting of materials cyclically loaded in the plastic range is studied through combined experimental and analytical efforts (ratcheting here describes the cyclic accumulation of deformation). In particular the work seeks to illustrate how cyclic hardening and softening influence ratcheting. To this end, systematic sets of experiments were performed on stainless steel 304 and carbon steel 1018 which, respectively, exhibit cyclic hardening and soft- ening. Due to the wide variety of behavior observed, and to better illustrate the modelling chal- lenges, the results are divided into uniaxial and multiaxial behavior, and are presented in Parts I and II, respectively. In Part I, the results from a series of uniaxial stress-controlled experi- ments are presented, which illustrate the parametric dependence of ratcheting in the two mate- rials examined. Results from a set of auxiliary strain-controlled experiments required for quantifying the cyclic hardening and softening characteristics of the materials are also presented. In a preceding publication, the authors demonstrated that ratcheting in cyclically stable mate- rials could be simulated with consistent accuracy by allowing the bounds of the two-surface model of Dafalias-Popov to translate in the direction of ratcheting at the rate of ratcheting. This mod- ified model, coupled with previously developed schemes for simulating cyclic hardening in strain- controlled cycling, are used to simulate the experimental results developed. Strengths, weaknesses and plausible alternatives are critically presented. The results are quite promising.

I. INTRODUCTION

Ratcheting or cyclic creep in the context of mechanical behavior of materials describe a phenomenon where, under cyclic loading, the material experiences progressive accu- mulation of deformation, which can result in failure. The dangerous nature of the phe- nomenon is demonstrated in Fig. la, which shows a uniaxial stress-strain response of carbon steel (CS) 1020 in a cyclic history in which the stress was prescribed. The cycles had amplitude of trx~ = 57.47 ksi (396.3 MPa), and a mean value of a~m = 20.15 ksi (139 MPa) (see inset in figure). The strain is seen to grow at an increasing rate with each applied stress cycle. In this case, the specimen failed after ten cycles. The phenomenon is well known and has been reported among others by BAmSTOW [1911], B E ~ [1960], Corr~ [1964], Dot~.N [1965], LArz t~ , r [1970], P~o et al. [1979], YosmoA et al. [1980], A t O l L [1985], ~ o c n ~ and Nou.~m.rIAS [1989], RUC,6LES and Kt~m,L [1990], and HASSAN aria K ~ I D E S [1992].

Under multiaxial loading, ratcheting can occur if at least one component of stress is prescribed in a multiaxial cyclic loading history involving some plastic deformation (see H ~ s ~ et al. [1992]). In such cases ratcheting will, of course, be in the direction of the prescribed stress(es).

The phenomenon falls into the broader subject of cyclic plasticity, which has received considerable attention over the last twenty years. This has included micromechanical as

149

Page 2: Ratcheting of cyclically hardening and softening materials: I. Uniaxial behavior

150 T. HASS^N and S. KYRIAKIDES

Ox (k~i) 100- CS 1020 Virgin

o

oj . . . . . -50 - - - - " ' - ~× (%) (a)

(kasi) 100 t CS 1020 Stabilized

t 50 I / ~ ] ~ ~-+*a = 52.13

0 ~ ~ ~ . ~ . ~ , . / 3 ! 0 ~++-o,~. = 9.24

- 5 0 ~ ~ +~(%)

4-

Stable Hysteresis (b) Loop

0 O • •

o o 0 o eo e°

o e • o

CS 1020 o Virgin • Stabilized

Exp (%) t

Ox

Gxm

V V V V V J_-'

0 ' 1'0 20 ' (c) ~ N

Fig. 1. Cyclic creep (ratcheting) behavior of CS 1020. (a) Virgin material, (b) stabilized material, and (c) max- imum strain per cycle as a function of N.

well as macromechanical investigations, which have led to the development of a signif- icant number of models of elastic-plastic material behavior under complex, cyclic load- ings (for a review of the subject, see O n t o [1990]). Other major phenomena which influence the mechanical behavior of materials under cyclic loads are cyclic hardening, softening, and cyclic relaxation. These are best demonstrated in strain-controlled exper- iments. Figure 2a illustrates cyclic hardening as it occurs in uniaxial, strain-symmetric cycling of stainless steel (SS) 304. By contrast, CS 1018, which was cold worked dur- ing its manufacturing, is seen in Fig. 3a to soften under the same loading history. In both cases, after a number of cycles the hysteresis loops tend to close, or stabilize (stable hys- teresis loop), as shown in plots of the stress amplitude (Ox,,) of each cycle vs the num- ber of the cycle (N) in Figs. 2c and 3c.

Cyclic mean stress relaxation is best demonstrated if the strain cycles have a strain off- set (exm) as shown in Fig. 4a. In this case, the hysteresis loops tend to translate with each cycle so as to minimize the absolute mean stress (ax,,) of the cycle as illustrated in a plot of axm vs N in Fig. 4c (see MoRRow and SI~CLAm [1958]; JHANSALE and TOPPER [1973]),

Cyclic hardening, softening, and relaxation result in additional challenges at the multi-

Page 3: Ratcheting of cyclically hardening and softening materials: I. Uniaxial behavior

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Page 4: Ratcheting of cyclically hardening and softening materials: I. Uniaxial behavior

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Page 5: Ratcheting of cyclically hardening and softening materials: I. Uniaxial behavior

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153

Page 6: Ratcheting of cyclically hardening and softening materials: I. Uniaxial behavior

154 T. HASSAN and S. KYRIAKIDES

axial setting. One of the complications first demonstrated by LAMBA and SmEBOTTOM [1978], is that there is not a one-to-one correspondence between the hardening in a uni- axial test and that in multiaxial cyclic loading. However, considerable work conducted over the past fifteen years has resulted in significant progress on our understanding of mechanical behavior under strain controlled cyclic loading. By contrast, cyclic loading involving at least one prescribed stress has received less attention. The mechanical behav- ior under such loading histories is not well understood and remains a difficulty in mod- eling (see CrtABOCHE and NOUAILHAS [1989]). The main objective of this work is to fill this gap.

In two preceding papers we first demonstrated the parametric dependence of ratch- eting in a systematic set of uniaxial (HAsSAN and KYRIAKIDES [1992]) and multiaxial (HASSAN et al. [1992]) experiments. Due to the complexity of the problems involved, the investigation was purposely limited to time independent and cyclically stable materials. This was achieved by using structural carbon steels which exhibited limited time depen- dence, and by cyclically stabilizing all test specimens used in an 1 °70 strain-symmetric, cyclic, uniaxial prehistory. The results of a uniaxial ratcheting experiment conducted on such a stabilized material are shown in Fig. lb. In this case, following an initial transient extending a few cycles (in this case less than ten), the material ratchets at approximately a constant rate. This was the case at least up to 307o of strain when this experiment was terminated. Similar results were recorded for a wide range of cycle parameters.

Even though the cycle parameters in the two experiments shown in Fig. la and lb are different, it is interesting to compare the rate at which ratcheting occurs in the two exper- iments (see Fig. lc, where the peak strain exp, in each cycle is plotted against the num- ber of cycles applied N). Quite clearly, cyclic softening causes an acceleration in the rate of ratcheting. Conversely, cyclic hardening can be expected to cause some reduction in the rate at which a material ratchets in a similar experiment.

Interaction of softening and relaxation is illustrated in Fig. 4a. Relaxation causes the loop to shift in the direction that reduces the mean stress (axm) of each cycle, as shown in Fig. 4c. Softening reduces the amplitude of each cycle (axa) as also shown in Fig. 4c. As a result, the cycle peaks with positive stress move down at an accelerated rate, whereas the peaks with negative stress remain approximately stationary. The opposite effect can be expected in the case of a similar experiment on a cyclically hardening mate- rial (see ALAMEEL [1985]).

In the present study, we continue to limit our attention to time independent material behavior, but seek to understand and quantify how ratcheting in particular, and stress controlled cycling in general, are influenced by cyclic hardening and softening. It was recognized that a consistent set of experimental data is necessary for such an undertak- ing. Two different sets of experiments involving a cyclically softening and a cyclically hardening metal were conducted. Each set involves uniaxial cyclic experiments, includ- ing several ratcheting experiments and several types of biaxial cyclic experiments. In each set we provide experimental data to accommodate modeling and its evaluation.

HASSAN and KYRIAKIDES [1992] and HASSA~q et al. [1992] used the models of DAFA- LL~S and PoPov [1975, 1976], DRUCKER and PALGEN [1981], and TSENG and LEE [1983] to illustrate strengths and weaknesses in the state of the art in modeling ratcheting phe- nomena. A number of improvements to these models were suggested and various defi- ciencies were identified for future development. For example, ratcheting in a uniaxial experiment on a cyclically stable material was shown to be predicted, with consistent accuracy, if the bounds in the Dafalias-Popov model were allowed to translate in the direction of ratcheting at the rate of ratcheting. The improved models, enhanced by

Page 7: Ratcheting of cyclically hardening and softening materials: I. Uniaxial behavior

Ratcheting: Part I 155

many recent developments by other investigators in the modeling of cyclic hardening, will be used to evaluate the modeling of ratcheting in cyclically hardening and soften- ing materials.

For easier reading, we divide the presentation of the work into uniaxial and multi- axial behavior, addressed in Parts I and II, respectively. References which appear in Part I are not repeated in Part II, and equations are numbered consecutively.

II. U N I A X I A L EXPERIMENTS

The experiments were conducted on cold finished CS 1018 and on SS 304. The first is a cyclically softening material and the second a cyclically hardening one. All CS 1018 specimens used 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 tubu- lar stock with 1.25 in outer diameter and 0.25 in wall thickness.

Stainless steel uniaxial experiments were conducted on round, solid bar test specimens with test section diameters of 0.5 in (12.7 ram) machined from solid stock with diam- eter of 0.625 in (15.9 mm). Tubular specimens, like the ones described above, were used for biaxial experiments. Prior to testing, all stainless steel specimens were heat treated to 1960°F (1070°C) for 40 minutes followed by furnace cooling. The purpose of the heat treatment was to reduce the effect of mechanical work from the manufacturing process (cold finishing). It resulted in increased cyclic hardening of the material.

The experiments were conducted in a closed loop servo-hydraulic axial-torsional test facility suitably extended to include a pressure loading capability also. The command signals were provided either by function generators or by a computer. The data were recorded in a data acquisition system for later processing and on x-y plotters for real time monitoring of the experiments. In strain-controlled cycles the strain rate used in the experiments was 1.2070 per minute. In stress-controlled cycles the stress rates were 87-91 ksi (600-628 MPa) per minute for SS 304, and 167-189 ksi (1152-1303 MPa) per minute for CS 1018. The experimental setup used and procedure followed are the same as those reported previously by the authors (1992) and will not be repeated here.

II. 1. Ratcheting experiments

A series of uniaxial ratcheting experiments were conducted for each of the two mate- rials used. They involved stress cycles with constant amplitude (axa) and mean stress (axm) as shown in the inset in Fig. 1. Figure 5a shows a typical result obtained from a CS 1018 specimen. The cycle parameters are given in the figure (#xm -- axm/a6 and #xa "~- tTxa//aO, where a6 is the yield stress [stress at 0.2°70 strain offset] of the initial monotonic part of the material response). The complex interaction between cyclic soft- ening and ratcheting is quite obvious. Initially, the rate of ratcheting is relatively slow. As the material softens with cycling, the rate of ratcheting increases while the hystere- sis loops change shape. These changes are more easily observed in Fig. 6, where discrete loops recorded in a similar ratcheting experiment for the 1st, 30th, 58th, 90th, 120th, and 140th cycles are shown. In addition to the obvious progressive "fattening" of the loops, a gradual decrease in the elastic range and some decrease in the elastic modulus are observed as the cyclic history progresses.

In the experiment shown in Fig. 5a, stress cycling was terminated after 85 cycles. We then switched to strain control and performed 10 cycles with strain amplitude of 0.5070, and 10 additional cycles with amplitude of 1.0070 (the mean strain of these cycles was

Page 8: Ratcheting of cyclically hardening and softening materials: I. Uniaxial behavior

Ox (ksi) 120-

CS 1018 Oxr~ = 0.276

~xa = 0.756

-80

cx 120- (ksi)

-0 .25 0

1

(a)

x m

3.0

" ~x (%)

o~ 120- (ksi)

-0,25 ' ~ ex (o~) 0

.0 2~E- - i

-8C {c)

156 T. HASSAN and S. KYRIAKIDES

Fig. 5. Axial strain ratcheting of CS 1018. (a) Experiment, (b) prediction, Model I, (c) prediction, Model II, (d) loop width as a function of N, (e) plastic work as a function of N, (f) maximum strain in cycle as a func- tion of N comparison of predictions with experiment, and (g) response of strain controlled cycles following ratcheting in (a). (Figure continues.)

2 .690 ) . The results are shown in Fig. 5g. The smaller ampli tude loops are seen to exhibit pr imari ly relaxation. The larger ampl i tude loops are seen to exhibit re laxat ion as well as softening.

Two sets o f experiments were conducted for each material . In the first set, the mean stress of the cycles was kept cons tant and the ampl i tude of the cycle was varied. In the

Page 9: Ratcheting of cyclically hardening and softening materials: I. Uniaxial behavior

Ratcheting: Part I 157

0.45 e

(%)

0.6 W p (ksi)

120- (Tx

(ksi)

-80

CS 1018

axm = 0.276

-~- Expe__ Moderli~len,

i t i

50 100

" ~ ' - - ' ~ - N (d)

CS 1018 / - /

Gxrn = 0.276 /

~ Experiment - - Model II

0 i i 5 100

~ N ( e )

CS 1018

exm = 2.69%

'£xp (%)

t

(g)

Fig. 5 continued.

3- CS 1018 / ~ / ~xm = 0.276 / / / ~xa = 0.756 / ~ '

, Y - - o - Experiment

f --- Original Model

- - - Model I

- - - Model II

5'0

O)

' 16o ------~- N

=0.5% f ~ , ~ = 1 . 0 %

3.0 J J " 3.5

,~ ex (%)

second set, the amplitude of the cycles was kept constant and the mean stress was var- ied. In the case of CS 1018, three experiments were conducted in which the mean stress was 5xra = 0.207. Three additional experiments were conducted with #xa = 0.751. The stress-strain responses obtained in these experiments are similar in nature to that pre- sented in Fig. 5a, and they are not included here. A summary of the results is presented in Fig. 7a and 7b, where the maximum strain recorded in each cycle (exn) is plotted as a function of the number of cycles (N). It is seen that except for the experiment with

Page 10: Ratcheting of cyclically hardening and softening materials: I. Uniaxial behavior

158 T. HASSAN and S. KYRIAKIDES

CS lO18

~ ,~ = 0 .249 Ex tended

Mono ton i c Cu rve Mono ton i c Cu rve o~ ~*~ = 0 .749 j / /

1 . . . . . . . . . . . o . . . . . . . . . . J JJ J JJ i

80 ~ Cycle no - - - - - -

Fig. 6. Evolution of hysteresis loops and bound in a ratcheting experiment on CS 1018.

#xa = 0.811 and #x,~ = 0.207, the initial rates of ratcheting do not differ much from each other. However, in all cases the rate of ratcheting increases as the number of cycles increases. As expected, higher cycle mean stress and higher cycle amplitude result in higher rates of ratcheting.

This dangerous type of behavior was also reported by CI-L~ and LAIRD [1987| in ratch- eting experiments on the same material. The amplitudes and mean stresses of the cycles they applied were in general smaller than those used in our experiments. Consequently, in some of their results acceleration of ratcheting was exhibited after 100 to 1000 cycles.

Two methods were used to monitor the changes introduced to the stress-strain loops during ratcheting. The first method involves monitoring the width of each loop, e, defined in the inset in Fig. 7c (see also MoRRow [1965]; CnAI and LAIRD [1987]). The second method involves monitoring the area of each hysteresis loop during cycling (plas- tic work W p) as shown in the same inset. Figures 7c and 7d show how these quantities vary with the number of applied cycles for the three ratcheting experiments shown in Fig. 7b. Both quantities increase monotonically with N, and they do so in approximately the same fashion. Increase in mean stress is seen to result in increased rate of growth of both quantities. The corresponding result for the three experiments in Fig. 7a had the same trends and are not included here.

A typical stress-strain response from a ratcheting experiment on SS 304 is shown in Fig. 8a. As demonstrated in Fig. 2a, this material exhibits cyclic hardening. This causes a reduction in the rate of ratcheting as shown in Fig. 8a. Plots of Cxp vs N from five such experiments, in which the cycle mean stresses was #xm = 0.157 and the cycle ampli- tude was varied, as shown in Fig. 9a. Results from three additional experiments in which the cycle amplitude was ~x, = 1.014 and the mean stress was varied are shown in Fig. 9b. In all experiments the material exhibited negative ratcheting during the first cycle. In Fig. 9a the amount of negative ratcheting is seen to increase as am increases. In Fig. 9b this initial reduction in strain decreases as axm increases. In subsequent cycles, strain ratcheting was positive. In all experiments the rate of ratcheting is observed to reduce as N increases. However, given that this material exhibits significant cyclic hardening it can be concluded that the interaction between ratcheting and hardening in this case is relatively weak. Another observation is that for the cycle parameters con- sidered, the rate of ratcheting was not significantly affected by either the amplitude or

Page 11: Ratcheting of cyclically hardening and softening materials: I. Uniaxial behavior

. " \

~ " \'\, E

I i

'/

" i/

E d

m ~ (..) i~

z

E

o

- ,~ ,

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~ E \ ~ \

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8

0

~ ; . ge

.~ .~

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159

Page 12: Ratcheting of cyclically hardening and softening materials: I. Uniaxial behavior

160 T. HASSAN and S. KYRIAKIDES

xlj (kTi) 40

° l 1to

SS 304

~,m = 0.201 ~a = 1.015

Extended Monotonic Curve

(a)

(ksi) 40-

f /

/ / /

0

-40

o15

(b) E, (%)

e (%)

0.6 SS 304 W P Gxm = 0.201 ~'xa = 1.015 (ksi)

~ - e - Exper iment l - - Model II \

\ \

0.3 SS 304 ( l x r n = 0 .201

axa = 1.015 , -=- Experiment ~ - - Model II

\ N

0 0 100 0 50 1 0 m N

- - ~ . - N (c) (d)

Fig. 8. Axial strain ratcheting of SS 304. (a) Experiment, (b) prediction, Model I1, (c) loop width as a func- tion of N, (d) plastic work per cycle as a function of N, and (e) response of strain controlled cycles follow- ing ratcheting in (a). (Figure continues.)

the mean value of the stress cycles applied. Finally, we observe in Fig. 8a that the loops remained relatively unchanged as ratcheting progressed. This was true for all experiments in this group. This observation is further supported by the results in Figs. 9c and 9d, where the variation of both e and W P with N is small. It is also worth noting that for this material both e and W e decrease for cycles with higher mean stress.

11.2. Strain-control led exper iments

A number of additional experiments were performed to obtain the cyclic hardening and softening characteristics of the two materials required for modeling purposes. For example, a strain-symmetric cyclic experiment with a strain amplitude of 1 07o was con-

Page 13: Ratcheting of cyclically hardening and softening materials: I. Uniaxial behavior

Ratchet ing: Pa r t I 161

(ksi)

t 60

-60

SS 304

exrn = 2 . 3 %

exc = 1.0%

I I 2.0

(e)

Fig. 8 con t inued .

-~ Ex (%)

ducted for each material. The stress-strain responses recorded for SS 304 are shown in Fig. 2a. The variation of the stress amplitude of each cycle with N is shown in Fig. 2c. The amplitude is seen to increase by approximately 4007o and to stabilize in approxi- mately 25 cycles. The corresponding results for CS 1018 are shown in Figs. 3a and 3c, respectively. In this case, the stress amplitude of the cycles decreases by approximately 17°70 in 25 cycles. The material did not quite stabilize after 25 cycles, but the rate of addi- tional softening was rather small. In the same figures, we also observe that for CS 1018 the cycles have a positive mean stress, which can be attributed to the cold finish of the tubular stock from which the test specimens were machined. The mean stress of the cycles is seen to relax during cycling. As a result, the compressive stress peaks soften at a smaller rate than the tensile stress peaks. It is also interesting to observe that the mean stress relaxes to a nonzero positive value. Another observation from Fig. 3a is that the modulus of the elastic range is gradually reduced by cycling. Similarly, some reduc- tion in the linear range of the loops is observed as the cycling progresses.

A cyclic, multistep strain-symmetric test was conducted for each of the two materi- als (see LANDGRAF et al. [1969]) to establish the amount of hardening or softening at different strain amplitudes. In the case of CS 1018, we started by cycling the material at a strain amplitude of 0.307o. After stabilization the strain amplitude was increased to 0.5070. The material was cycled at this amplitude until it stabilized again. This was repeated at strain amplitudes of 0.75070 and 1.007o. A total 130 cycles were applied. The stress-strain results are shown in Fig. 10a.

The final stabilized loop (exc = 1.0070), produced in the multistep experiment shown in Fig. 10a, was found to be almost congruent to the one produced by direct cycling at this same strain amplitude (shown in Fig. 2a). Strain cycling with 1.007o amplitude was also performed after the completion of the ratcheting experiment shown in Fig. 5a. That is, the cycling was performed at a mean strain of 2.69°7o. The loops obtained are shown in Fig. 5g. The stabilized loop is again almost indistinguishable from the two other sta-

Page 14: Ratcheting of cyclically hardening and softening materials: I. Uniaxial behavior

162 T. HASSAN a n d S. KYRIAKIDES

E.xp (%7

t

Exp (%)

f i

4- SS 304

1,151 o~,a

c>--

1.076

2" ~ . . . . . . . . . _o. __o_..~o ..... 1 0 1 4

. ~ _ - ~ ~3o-- _ . . . . . . . . . . . . 0 .948

l - ' L I ~ J "

- ~ 0 . 8 4 3

/

3 -

1

J

(a) - ~ N

SS 304 ~×a= 1 ,014

0.201 ~z. _~>_-~>

~o.~ ~4,.--_~- ~ - . . . . . 0 .158 . . . . .

. . . . . . . 0 . 0 8 1 ....

\ -o.-. Experiment G,~,

- - - - Model II )

(b) ~ N

e 0 . 6 - SS 304 Wp 0 . 3 -

~xa= 1.o14 (ksi)

t °-°0'

0.158 I~r. ~ = _ ~ _~ ,=-il-~-~ 0,201

5 0 1()0 ~ 0

(c) ~ N

SS 304

(~:~a = 1,0t4-

, ~ 0.081

~ : =__.==_-=.-=,o.158 i~,m

50 100

(d) - - - - ~ N

Fig. 9. M a x i m u m strain per cyc le as a funct ion o f N for SS 304; c o m p a r i s o n be tween exper iments and pre- dict ions . (a) C o n s t a n t m e a n stress, (b) cons tant ampl i tude stress, (c) l oop width as a func t ion o f N , and (d) plastic w o r k per cycle as a func t ion o f N.

Page 15: Ratcheting of cyclically hardening and softening materials: I. Uniaxial behavior

Ratcheting: Part I 163

E,

T': o vvvvvvvvvi'

(Ix (ksi) 120

CS 1018 ~ ]

ax 100- (ksi)

l 80 -

6 0 - <

20-

i ~ .

0 . 3 "~ CS 1018 ¢,c(%) ~ a,a

Exper iment -o - (~xm

- - Mode l II

,-?:.:.--;:- i '" - . . . . . . . . . . . . .

i 50 I00 150

(c) " N

W p (a) (ksi)

ox

c,,,,I ~° T

1,5-

1.0

0.5

CS 1018 1.0

r . . . . . i

i 0.75

~ 5 " ~ Exper imenl

I :.c(%) [ - Model II

' ~o 1;o (b) (d) ~ N

1 150

Fig. 10. Multistep strain-symmetric cycling on CS 1018. (a) Experiment, Co) prediction, Model II, (c) amplitude and mean stress per cycle as a function of N, and (d) plastic work per cycle as a function of N.

ble loops with the same amplitude mentioned above. We thus concluded that for this material the stable loops, and perhaps the cyclic stress-strain curve, that is, the locus o f points joining the peaks o f the stabilized loops, do not depend on history (FELTm~R and LAmP [1967], reported that this is usually the case for materials which exhibit wavy slip mode) .

Figure 10c shows how the amplitudes and mean stresses o f the cycles vary with N, and Fig. 10d shows how the area o f the loops varies with N in this experiment. These results will be used to evaluate the performance o f models in the next section.

Results from a similar experiment on SS 304 are shown in Fig. 11. In this case the strain amplitudes o f the four steps were 0.25%, 0.5%, 0.75% and 1.0% and a total o f 75 cycles were applied. The stress-strain results are shown in Fig. 1 la. The same com- parison o f stabilized strain loops described above for CS 1018 was also performed for this material (see Figs. 2a, Be, and 1 la). The three stabilized loops obtained by strain cycling at amplitude o f 1.0% but with different prehistories were found to be almost congruent. Thus, it is concluded that the cyclic stress-strain curve, in the range o f parameters o f our experiments, can be considered to be independent o f the history, for this material also.

Page 16: Ratcheting of cyclically hardening and softening materials: I. Uniaxial behavior

164 T. HASSAN and S. KYRIAKIDES

-1 ]

~x

(ki i)

50-

40-

3O

0

-10 •

S SS 304

, ~ 2 5 / " ~ ~:~1 Experiment

• • - - Model II "', ~:x: ( % )

100

(c) - - ~ N

(a)

-1 0

(b)

WP 1.2- (ksi)

t 0.8

04

SS 304 ~

l

- - 1

0 " ~ + Experimenl

- Model II J

0'.25 "

_ _ . ~ - - - ~ 715

N (d)

Fig. 11. Multistep strain-symmetric cycling on SS 304. (a) Experiment, (b) prediction, Model I1, (c) ampli- tude and mean stress per cycle as a function of N, (d) plastic work per cycle as a function of N, and (e) satu- rated bounds at different strain ranges (e) and fit. (Figure continues.)

JHANSALE and TOPPER [1973] suggested that ratcheting and relaxation could be viewed as different manifestations of the same process, resulting from control conditions. To investigate the validity of this observation, we conducted some additional relaxation experiments. The stress-strain results from a relaxation experiment performed on CS 1018 were discussed earlier in relation to Fig. 4. Figure 12 shows results from an addi- tional four-step relaxation experiment performed on the same material. The strain cycles prescribed had amplitudes of 0.5°70. Forty cycles were first performed about a mean strain of 0.5070. This was followed by 30 cycles at each of the following values of mean strain: 1.0070, 1.5070, and 2.0070. Figure 12c shows how the cycles' mean stresses and amplitudes varied with N. The stress amplitudes are seen to reduce monotonically, indi- cating that softening takes place almost in a continuous fashion. During the first 40 cycles, the cycle mean stress is seen to relax. The mean stress experiences a step increase every time the cycle mean strain is increased, due to the induced strain hardening. The mean stress relaxes again during the 30 cycles performed at exm = 1.0070. This sequence of events is repeated during steps three and four of the experiment. The relaxation of

Page 17: Ratcheting of cyclically hardening and softening materials: I. Uniaxial behavior

Ratcheting: Part I 165

(ksi)

t 40

Obol

2 0 -

Monotonic Curve

SS 304

0 ' 014 ' 0'.8 ' 112

- - - - . ~ (O/o) (e)

Fig. 11 continued.

the mean stress of the cycles is approximately the same in each of the last three sets of 30 cycles. Figure 12d shows that the areas of the loops traversed suffered a rather small reduction during the 130 cycles performed. This, in spite of the significant softening of the material.

Figure 13 shows the results from a similar four-step relaxation experiment performed on SS 304. The strain amplitudes of the loops were again 0.5°7o. The mean strains of the four sets of loops were again 0.5070, 1.0°70, 1.5070, and 2.0070. The number of cycles performed was 40 for the first set and 30 for each of the three subsequent sets. The mate- rial is seen to harden significantly during the first 40 cycles, during which the mean stress showed almost no change (see Fig. 13c). During the three subsequent sets of cycles the material experiences small amounts of cyclic softening and relaxation. This type of rever- sal in cyclic behavior has been observed before (LAMB [1977]). It seems to be at least affected by the size of the loops but other characteristics of materials exhibiting this behavior are not known. W p is seen to experience a gradual and continuous reduction as N increases (Fig. 13d. See also MORROW [1965]).

IlL MODELING OF UNIAXIAL CYCLIC BEHAVIOR

In our preceding work (1992), it was demonstrated that the prediction of ratcheting in cyclically stable materials under uniaxial loading is a challenging task. It was shown that the models of D~AtL~S-POPOV [1975,1976], DRUCr~R-PAZGFN [1981], and TSENO- LEE [1983], in spite of their demonstrated capability to predict other complex phenom- ena associated with cyclic loading histories, could not yield consistently accurate results for ratcheting. Some reasons for this inconsistent performance were also given. For example, it was demonstrated experimentally that, in a ratcheting experiment, the

Page 18: Ratcheting of cyclically hardening and softening materials: I. Uniaxial behavior

F

i

i

i

,/ + + ' ,

~ g 2

E ~2

e4

u~

~2 o

i

~ Z

t

r

O w

t: E

~÷~ ~o: i I,, ~/-~ o o 0

E

c,l

Z ~

.o U e~

E

I

0

=-

e~

e~

0

o

x

Rg

u

166

Page 19: Ratcheting of cyclically hardening and softening materials: I. Uniaxial behavior

g

O9

i

o co

- r_

" ¢ + I ~t3

2 o <

d ' - N m

o

g

o

E

0 o

E ----

+ ;

u~

O

rd

.2

\

O

o

t~ t~

Z t.~

o

0

v

o

t~

jj v

0

. o

o

g

r ~

o

x

~ R g o ~ . , ~

°~

m y .

167

Page 20: Ratcheting of cyclically hardening and softening materials: I. Uniaxial behavior

168 T. HASSAN and S. KYRIAKIDES

bounds in the Dafalias-Popov model translate in the direction of the accumulated strain at the rate of ratcheting. A simple method for incorporating this observation into the original model was proposed. The modified model was then shown to be capable of pre- dicting ratcheting in cyclically stabilized materials with good accuracy for a wide range of cycle parameters.

In the present work, we use this model with the modification as the starting point in an effort to evaluate its performance in predicting ratcheting in the cyclically harden- ing and softening materials used in our experiments. We first briefly review the model and the modification as they apply to cyclically stable materials.

III. 1. Cyclically stable behavior

A guiding constraint in this effort has been the simplicity and ease of implementa- tion of the model. In the spirit of this constraint, we adopt the simplifying assumptions of J2-type plasticity with various kinematic-type hardening rules. The yield surfaces are represented by:

f ( o " - - nt ) = [ ~ ( S - - a ) - ( s - - a ) ] 1/2 = a o , (1)

where a is the stress tensor, s is the deviatoric stress tensor, a is the current center of the yield surface and a is the current center of the yield surface in the deviatoric space. For a cyclically stable material, a0 is a constant.

The plastic strain increment is given by

1 (Of .do) Of d P= -d (2)

Under the assumptions adopted, for uniaxial loading, H in the equation above is given by

do H - (3) deP"

H will be evaluated according to the Dafalias-Popov model. The model is completed by incorporating a suitable hardening rule to the flow rule. The hardening rule 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 work.

A main innovation of the DArAZtAs-PoPov [1975,1976] model is that the current tan- gent 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 sur- face. In the uniaxial setting, in its simplest form, the bound is linear. The two measures of stress mentioned above are t5 and tS;n, respectively (see Fig. 14). The following rela- tionship was proposed for H

(4a)

Page 21: Ratcheting of cyclically hardening and softening materials: I. Uniaxial behavior

Ratcheting: Part I 169

o B E~

J

| l . -

f / , t

J

J S Eo p

Bound

Fig. 14. Definition of parameters of Dafalias-Popov model.

where

a

h = b[Sin~ m. (4b)

I + \2Ob]

Eo "° is the plastic modulus of the bound; the constants a, b, and m are evaluated from two uniaxial stress-strain curves. An alternate expression for h, suggested in SE~D- RANYBARI [1986], will also be used and is

6i, ] h = ho Ce + (I - Ce) 2(ab= ao) ' (4c)

where the constants h0 and Ct are evaluated from two different stress-strain curves. In the multiaxial setting, the bounding surface will be assumed to have the follow-

ing form

F(# -/~) = [ 2a-(~ - b)" (,~ - b)] I/2 -- trb, (5)

where # is the image of ¢ on the bounding surface and/~ is the center of the bounding surface, ~ and b are the corresponding variables in the deviatoric stress space, and ab is the radius of the bounding surface. For cyclically stable materials ~b is a constant. The evolution of/3 will be discussed in Part II. The distance 8 is generalized as follows

t5 = [(~ - g)- (# - g)] I/2. (6)

Page 22: Ratcheting of cyclically hardening and softening materials: I. Uniaxial behavior

170 T. HASSAN and S. KYRIAKIDES

Figure 15a shows the results of a ratcheting experiment on CS 1020 following cyclic stabilization through 1.0070 strain-symmetric cycling (from HASSAN and KYRIAKIDES [1992]). The stabilized hysteresis loop, which resulted from the cyclic prehistory, is also included. The bound at the commencement of ratcheting can be obtained by extrapo- lation from the stable hysteresis loop as shown in the figure. After 50 stress-controlled cycles with the parameters given in the figure, a ratcheting strain of approximately 2.8°7o has resulted. Another strain-controlled loop, also shown in the figure, was prescribed at this time from which the current bound is established. The position of the current bound is seen to be significantly different from the initial one. Indeed, use of the ini- tial bounds for simulating this type of experiment will result in a gradual deceleration and eventual arrest of the predicted ratcheting, as shown in HASSAN and KYRIAKIDES [1992].

This observation led us to the conclusion that prediction of such experimental results requires that the bounds in the original Dafalias-Popov model, be allowed to ratchet in the direction of accumulated strain at the rate of ratcheting. With this modification the model predicts constant rates of ratcheting. The particular materials used by HASSAN and KYRIAKIDES [1992] to conduct uniaxial ratcheting experiments, exhibited an initial transient which lasted a few cycles. Following the transient, ratcheting progressed at a constant rate. This is clearly illustrated in Fig. 16, which shows 11 sets of cycle peak

CS 1020 Gx Extended Current initial bound bound

5,m : 0.159 (ksi) 75 , A __ _ . . . . . . ~ L . - _____----\~.~..p~_.._'~l.

//J/

_4 / ' °I~Y- ' /// ..... ~oop~"" " ...... /- ............... -~'--'~H;steresis ......... :;i Ia,

Hysteresis Loop

~3x

<7 > 75- r __. / ._~-- . . . . . . . . . . . . . . .

-I ,'" O ~ ' . "" 3

~x

'-" ........... -721 ~ ............. - ,(ooi (o)

Fig. 15. Axial strain raicheling of stabilized CS I070. (a) Experiment, (b) prediction, Model I, and (c) pre- diction, Model If.

Page 23: Ratcheting of cyclically hardening and softening materials: I. Uniaxial behavior

Ratcheting: Part I

~xp -]

(%) / c s ~020

3 / o ~.,~=o.~3o o°

/ - - M o d e , , "'" "'" " °

" ~ . . . . . -o-o-" -

"t cs0o (%) l ~xa - 0 . 660 o o

3 o Experiment o ° M o d e l I o o

o g~ ---Model II o o o o '

o ~ ~o 0 o ~ . , ~ o o o o ~

o / 2 O / t

9 ~

o 0 .218 o

o 0 . 1 8 5 0

/ ~ o o

o . - o 0 . 1 6 9 o o ~ o o

- ~ o o f ~ 0 . 1 5 9

o O . . .

- ~ J . ~ ~'~"o ~% ~_~

171

0

Fig 16 M^- (b) 40 (a" "~ " ~u~lmum strain ~-r • ~ N

J uonstant mean stre.~¢ ~ c y c m a s a function o f N " - • . .

~ = - u m ) constant a m p l i t u d e s ; r n ~ F m ~ a x i a l ~a~cheting exv,~,~ . . . . . . . . . . ~ o m p a r J s o n b e t w - ~ " u ' ' c m s on stabilized C S fn '~n

~ u experiments a n d pred 7 .v, .v. ~ctions.

strain results as a function of the number of stress cycles applied, obtained from ratch- eting experiments on CS 1020. Because o f this transient, the bounds were alowed to start shifting after an initial plastic strain of 0.9070 was reached. The predictions of the eleven sets of experimental results obtained with the modification are shown in Fig. 16 (/den-

Page 24: Ratcheting of cyclically hardening and softening materials: I. Uniaxial behavior

172 T. HASSAN and S. KYRIAKIDES

tiffed as Model I). The predictions are in good agreement with the experiments for all cases.

Figure 17a shows schematically the implementation of this modification in the model. B~ B2 represent the bound from which 6 is measured during the stress-strain history 1-2. After 2, the bound shifts to the right. The bounds corresponding to point 3 are shown with a dashed line (B3B 6 and B4Bs). The new bounds remain fixed during the history 3-4-5-6-7. The bounds move again to the right during 7-8 (not shown in the figure). bl b2b3babsb6b7b8 show how the center of the bounds moves in the o - e P space dur- ing this history. The rightward parallel shift of the bounds has also resulted in a simul- taneous downward shift. In the two surface theory of plasticity, a downward shift of the bounds is the mechanism through which relaxation is achieved, as clearly demon- strated by SEYED-RANJBARI [1986].

Let us now reconsider the suggestion made by JI-IANSAZE and TOPPER [1973], that ratcheting and relaxation are different manifestations of the same process, resulting from different loading control conditions used. Let us consider again the stress-strain history shown in Fig. 17a, and for a moment let us disregard under what mode of loading it was produced. The loop 3-4-5-6-7 has resulted in (r7 < a3. In other words, the loop has shifted down, causing a reduction in the mean stress of the loop; that is, mean stress relaxation has taken place. If this loop was obtained by strain-controlled cycling of con- stant amplitude, then it is completed at 7. I f stress-controlled cycling (constant ampli- tude) is used, then it will continue up to point 8, resulting in a net ratcheting strain of (e8 P - e7P). We thus conclude that indeed ratcheting and relaxation are related (follow- ing a different line of argument, CHABOCHE and NOUAILHAS [1989] reached a similar conclusion).

As a result of these observations, we consider an alternative, but hopefully equiva- lent, way of modeling ratcheting through two-surface plasticity theory. In the new approach we use relaxation, or the downward shift of the bounds, to simulate the cor-

0

2 0 b

i

_ L

. ~ " ,Be f

Oin /~.3 I I /

5, . . . .

B2 B37

/ 7

~ b3_~_47 bs 4

2 0 b -

__r

J

7

/ / - i - I °

/ ~ - j [34

.l=,.-

,£P

(a) Model l (b) Model II

Fig. 17. Modifications of Dafalias-Popov bounds for ratcheting of stable materials. (a) Bounds shift with ratcheting at the rate of ratcheting (Model I). (b) Bounds relax during reverse loading (Model II).

Page 25: Ratcheting of cyclically hardening and softening materials: I. Uniaxial behavior

Ratcheting: Part I 173

rect rate of ratcheting. The instantaneous shift of the bounds causes a change in the instantaneous "modulus" of the bound which becomes

E~" = E g + C~[1 - sign(/3da)]]/3], (7)

where 13 is the current center of the bounds (see Fig. 14) and do is the current stress incre- ment. This is in essence the form of E~ suggested in SEXrED-RANYaARI [1986], where Cr is called the material relaxation coefficient and is determined from a relaxation experiment.

The net effect of eqn (7) is that, for a positive/3 and negative do (e.g. history 2-3-4 in Fig. 17b), the bounds experience an accelerated rate of translation and their center moves along b364. Thus, at 4, the bounds have moved so that they pass through points B4 and Bs. E P in (7) is the instantaneous slope of the locus of points formed by join- ing the images of the current stress points on the current bound, i.e. BaB4, shown with a dashed line. The bound center will move according to d/3 = E~de p, which implies that b3b4 in Fig. 17b will be parallel to B3B4. If both/3 and do are positive as for path 4-5-6-7, then E~ = E~'. In this case, the bound center will move along a straight line (b4b5b6b7). Note that, if Cr = 0 then, for the case shown in Fig. 17b, blbEb3 and b4bsb667 would be co-linear.

If we compare Figs. 17a and 17b, we observe that, although the mechanism adopted by the two models for shifting the bounds is different, the net result after one cycle is the same. Thus, instead of Cr being a relaxation coefficient, we name it a ratcheting coefficient and evaluate it from a ratcheting experiment instead. All other features of the original model are retained. The modified model is henceforth identified as Model II. We note that Cr can remain in the model for all loading histories, including strain- symmetric ones. In such cases, its effect will be to introduce small changes in the shapes of the predicted loops. Otherwise the predictions will remain unaffected, because the relaxation introduced during the two halves of each loop will cancel each other.

For the cyclically stabilized CS 1020 ratcheting experiments in Fig. 16, Cr was eval- uated by matching the predicted ratcheting rate to that of the experiment with 6xa = 0.711 and 6xm = 0.130. All other experiments in the set were then predicted (material constants used are given in Table 1). The predicted ratcheting strains are shown by dashed lines in Fig. 16. The model is seen to simulate well the initial transient seen in the experimental results, and to yield a nearly constant rate of ratcheting after that. Overall, the predictions are somewhat different from those of Model I, but they are in quite good agreement with the experimental ratcheting rates.

The stress-strain response predicted for one of the experiments in this set, is shown in Fig. 15c. The predictions are similar to those of Model I, shown in Fig. 15b, and are in reasonably good agreement with the corresponding experimental results shown in Fig. 15a.

As already mentioned, success in predicting ratcheting by both models discussed, is based on the successful modeling of the shift of the bounds, which is accomplished in different ways in the two models. In Fig. 18a and 18b, we compare the evolution of with plastic strain for Models I and II, respectively. In Model I, the shift of the bounds starts after e~ = 0.9°70. In Model II, shifting commences from the first cycle. However, following the first few cycles, the two histories are seen to have a similar trend. It's inter- esting to compare the evolutions of/3 yielded by the two models with that yielded by the original model; that is, no shifting of the bounds is allowed. In that case,/3 evolves linearly with ex p as shown by dashed lines in the two figures.

Page 26: Ratcheting of cyclically hardening and softening materials: I. Uniaxial behavior

174 T. HASSAN and S. KYRIAKIDES

Table la. Model I and II material parameters

E ksi Eo P ksi Ot, o ksi oas (eOl) ksi Material (GPa) ~ R (GPa) (MPa) (MPa) Ch Cb ~ C r

CS 1020 25,125 0.33 0 .418 466.9 5 9 . 7 4 . . . . 30 (stabilized) (173.2) (3.22) (411.9)

CS1018 28,500 0.33 0 .364 291.0 105.8 72.0 150 5.56 0.2 2.3 (196.5) (2.01) (729.5) (496)

SS 304 27,800 0.33 t 440.0 29.8 65.0 82 8.86 0.2 2.8 (191.7) (3.03) (205.5) (448)

?fro -- 16.0 ksi (110.3 MPa).

Table lb. Shape function parameters

a ksi ho ksi Materials (GPa) b in (GPa) Ce

CS 1020 146,200 40 3 - - (Model I) (1008.0)

CS 1020 191,700 43 3 - - (Model II) (1321.7)

CS 1018 - - - 19,000 9.0 (131.0)

SS 304 - - - 20,000 3.7 (137.9)

Final ly, let us define Al3r as the net shift in/3 after each ratchet ing cycle (see Figs. 18a and 18b). The values of A~r predicted for each cycle by Models I and II are plotted as a func t ion of N in Fig. 18c. Fol lowing differences in the first few cycles, the two mod- els yield approximately the same/t /3r . This results in constant rates of ratcheting, which are approximate ly the same.

1II.2. Cyclically hardening and sof tening behavior

Cyclic hardening and sof tening will be modeled by in t roduc ing his tory-dependent s imul taneous changes in the b o u n d i n g and yield surface sizes ob and Oo. These exten- sions to the original model were developed in DAFAtIAS [1981] and DArALIAS and SEYED- RANJBARI [1982]; they are also discussed in detail by SEYED-RANJBARI [1986]. For completeness, we summarize the major features of these extensions and point out some minor modif ica t ions we have in t roduced for improved performance .

The extent to which a material cyclically hardens or softens depends on the strain ampli tude, as illustrated in Figs. 10 and 11. Thus, a memory plastic strain surface (PSS) is in t roduced as follows (CHABOCI-IE et al. [1979]; OHNO [1982])

S = [ ] ( e P - v ) " ( e e - v ) ] = ¢o 2, (8 )

where e e is the plastic strain, ~, is the plastic strain at the center of the surface, and eo its current radius. The surface evolves as follows:

Page 27: Ratcheting of cyclically hardening and softening materials: I. Uniaxial behavior

Ratcheting: Part I 175

OS ~/(n-n*)de~, S = O, ~ e P .de p > 0

deo = OS

O, S < 0 or ~ .de p < O,

(9)

where n is the unit normal to the yield surface at a, n* is the unit normal to the PSS,

dePe = [ ] d c P . d e p ] , / 2 ,

and ~ is a material constant (0 < ~/< 0.5, is chosen for best performance of the model). The first of eqn (9) can also be expressed as follows:

The PSS translates in the Ziegler-Prager direction, thus

d7 = dA(e p - ~'), (10)

where dA is evaluated from the consistency condition for this surface. The size of the bounding surface, % in (5), evolves as follows:

dab = Cb ( %s -- oo) de e, (11)

where % is the current radius and %s = oos(e0) is the stabilized (saturated) radius for the current size of the PSS. Thus, the current size of the PSS decides the size of the sta- bilized bounding surface. For any value of Co, the rate at which obs is approached is set by Cb, a material constant. The value of oos(eo) is decided from the empirical relationship

Obs(~-.O) -~ O ' b s ( ~ . 0 l ) - - [ 0 r b s ( ~ . O l ) - - %o]e-Ch% (12)

where obs(eo,) is the size of the stabilized bound at a large value of Co, %0 is the size of the bound of the monotonic stress-strain curve, and Ch is a material constant deter- mined experimentally.

The determination of the constants in (12) for SS 304 is illustrated in Fig. l l e . Included in the figure are the monotonic stress-plastic strain curve, the initial bound, and the sizes of the stabilized bounds at strain amplitudes of 0.25070, 0.5070, 0.75070, and 1.0070 obtained from the cyclic multistep test shown in Fig. 1 la. In this case, %s(eOl ) = 65 ksi (448 MPa), %o = 29.8 ksi (206 MPa), and Ch = 82. With these constants, a~(eo) is also plotted in the same figure (assumed that eo ~ exP¢). The fit is seen to be in good agreement with the experimental results identified by (.). (For this material, cyclic hard- ening for strain amplitudes less than approximately 0.307o was much less than that exhib- ited for larger strain amplitudes; see also LANImRAF et al. [1969]).

The parameter Cb, which governs the rate of hardening (or softening), was deter- mined from the strain-symmetric experiment shown in Fig. 2a. An approximate value

Page 28: Ratcheting of cyclically hardening and softening materials: I. Uniaxial behavior

176 T. HASSAN and S. KYRIAKIDES

(ks i )

t

/ CS 1020 / / /

(~xm= 0 " 1 3 0 / / / / ~ / ~ r ~ ~ / / / / ~ x a = 0.711

_~_ M:~ :~ : , Model

I 11o ' 2'o

(a) ~ ex p (%)

P (ksi)

2 ¸

0

/ CS 1020 .,

/

~xm = 0.130 / ~ , ,, I . , / J . ~ / / ' . ~ ' Z -

~xa = 0.71], . . . . .

/ / , - Mo o,,,

- - Original Model

i 0 i 1. 210 P (%)

( b ) ~ E:x

(ksi)

t 0 . 4 ¸

0 . 2

i t " . . . . . . . . . . . . . . . . . . . . . .

J

CS 1020

(~xm = 0.130 m ( ~ x a = 0 . 7 1 1

- Modet I f / - - M o d e l II

/ I

I 2' ' 1 0 0 30

(c) - - - - ~ N

Fig. 18. Evolution of the center of bounds in three models for uniaxial ratcheting of stabilized CS 1020. (a) Comparison between original model and Model I, (b) comparison between original model and Model II, and (c) comparison between Models I and II.

of the constant is first calculated by substituting the bound sizes for the first two cycles in an expression obtained by integrating (11) (see SE~D-RA~m,~aU [19861). This value is then improved iteratively by comparing the predicted stress-strain response with the experimental one. Cb = 8.86 was found to yield optimum results for this case.

Page 29: Ratcheting of cyclically hardening and softening materials: I. Uniaxial behavior

Ratcheting: Part I 177

The evolution of the size of the yield surface was found to be different in the two materials used in the experiments. For SS 304, the yield surface size was found to remain very nearly constant during all uniaxial cyclic experiments performed. Thus, o0 was taken to be constant for this material. By contrast, in CS 1018, the suggestion of SEX'ED- Rx~m~au [1986] that the size of the yield surface evolves so that its size remains in direct proportion to the bounding surface, was found to represent well the observed behav- ior. In this case, we assumed that if initially

oo=Ooo then doo=Odoo 0_<0_<1. (13)

The value of 0 = 0.364 was found to represent well the experimental results. All material constants for SS 304 are given in Table 1. These constants were used to

simulate the experiment shown in Fig. 2a. The results are shown in Fig. 2b. The cycle stress amplitudes as a function of N from this simulation are compared to the corre- sponding experimental values in Fig. 2c. Some difference between the simulation and the experiment is observed in the first few cycles. Subsequently, the results from the sim- ulation are seen to be in good agreement with the experiment. The loop areas W p from the simulation are compared with the measured values in Fig. 2d. In the simulation, W p stabilized at a somewhat different value than in the experiments, indicating that some difference exists between the shape of the predicted loops and those traced in the experiment.

The numerical simulation of the cyclic multistep test for SS 304 is shown in Fig. 1 lb. The predicted loops are seen to be in good agreement with the corresponding experimen- tal ones appearing in Fig. 1 la. This conclusion is reinforced by the good agreement between the predicted and measured cycle amplitudes, mean stress and W p demon- strated in Figs. 1 l c and 1 l d.

A procedure similar to the one described above for SS 304, was used to establish the material constants for CS 1018, using the experimental results in Figs. 10 and 3. The calculated constants are listed in Table 1. Figure 3b shows the simulation of the exper- iment in Fig. 3a. The loop amplitude and mean stresses of the simulation are in very good agreement with the experimental results as demonstrated in Fig. 3c. However, the discrepancy between the calculated and measured loop areas is seen in Fig. 3d to be somewhat worse for this material. This is at least partly due to changes in elastic mod- ulus, which are not included in the model. (As a result of cold work performed in the manufacturing process, this material exhibited some initial anisotropy. This, for instance, results in the asymmetry between tension and compression observed in the loops in Fig. 3a. This effect was approximated in the model as an initial shift of the bounding surface of 7 ksi [48.3 MPa] in the positive stress direction. One of the effects of this initial anisotropy is the relaxation of the mean stress of the loops quantified in Fig. 3c. In this case, the model reproduced the relaxation quite well.)

The numerical simulation of the multistep test for this material is shown in Fig. 10b. The simulation is seen to be in good agreement with the experimental results which appear in Fig. 10a. Quantitative comparison between the simulated results and the exper- iments are shown in Figs. 10c and 10d. Overall, the comparison is favorable, although some differences are observed in Fig. 10c in the rate of softening for cycles with smaller strain amplitudes, and in Fig. 10d between the simulated and measured values of W P.

The main conclusion from these comparisons is that, as demonstrated before by DAFALL~S [1981] and DArALL~S and S~Y~.D-RA~YnA~ [1982], the model is capable of reproducing cyclic hardening and softening, in strain-symmetric uniaxial experiments, with engineering accuracy.

Page 30: Ratcheting of cyclically hardening and softening materials: I. Uniaxial behavior

178 T. HASSAN and S. KYRIAKIDES

III.3. Ratche t ing in cyclically hardening or so f ten ing materials

We are now ready to return to modeling ratcheting in cyclic hardening and softening materials. We first revisit one of the ratcheting experiments on CS 1018, shown in Fig. 6. Included in this figure is the monotonic stress-strain curve from which the ini- tial bound is established by construction and extrapolation. After 140 stress cycles, a ratcheting strain of 3.0% has accumulated. Following this, a larger loop is developed under strain control, part of which is shown in the figure. The upper half of the loop is seen to be significantly lower than the monotonic curve, illustrating that the bound has shifted. Of course, in this case the shift observed is partly due to cyclic softening and partly due to ratcheting. This observation was also found to be true in the case of ratcheting experiments involving cyclic hardening materials (e.g. see Fig. 8a).

To model this behavior we combine the two schemes for simulating ratcheting in cycli- cally stable materials, discussed in Section III. 1, with the model extensions for simulat- ing hardening or softening summarized in Section II1.2.

II1.3.1. M o d e l IL The procedure followed for combining the two sets of changes for the uniaxial case is as follows. Assume that we are at a state of loading where all prob- lem variables are known. Let the next step differ from this state by do. We first evalu- ate the new value of di. Using (5 and ~in w e evaluate H from eqns (4). The strain increments de and de P are evaluated next from (3) and the corresponding elastic rela- tionship. We then evaluate dab from (11) and (12) and doo from (13). In the uniaxial case, da = do - doo, dl3 = EPde P - d~rb, deo is evaluated from (9) and d7 = ( 1 - ~)de P. Finally all variables are updated.

111.3.2. M o d e l L In the case of Model I, the procedure followed is essentially the same. However, the presence of the strain surface, introduced for modeling cyclic hard- ening, can now be exploited to implement the shifting of the bounds in the direction of ratcheting. This is achieved as follows. Consider path 1-2-3 in Fig. 19. Let the PSS cor- responding to 1 be centered at 71 and its size to be eOl. In going from 1 to 2 the PSS does not evolve and, as a result, the bounds remain stationary. The PSS evolves dur- ing 2-3. During any increment that the PSS evolves, the bounds shift by the amount d e P. At point 3, the bounds have the position shown by dashed lines in the figure and the PSS has moved to 3'3 and has size of e03. During path 3-4-5, the PSS does not evolve and as a result the bounds remain stationary. The same is true for path 5-6-7. The PSS evolves again during 7-8 when the bounds experience a new shift.

IV. COMPARISON BETWEEN EXPERIMENTAL AND PREDICTED RESULTS

The performance of the two models described above in simulating ratcheting will be evaluated by direct comparison to the experimental results from CS 1018 and SS 304. A set of model parameters were determined for each material, by using results from the following three experiments: (1) the 1%0 strain-symmetric cyclic experiments (Figs. 2 and 3); (2) the multistep strain-symmetric cyclic experiments (Figs. 10 and 11); and (3) a ratcheting experiment.

The parameters Obo, oa~(eol), Ch, and Cb in eqns (11) and (12) were determined using experiments (1) and (2) (as discussed in Section III.2) and the values are given in Table 1. The constant 71, which determines the rates of growth of the PSS (see eqn [9]), was set at 0.2, as this value was found to reproduce both cyclic hardening and soften- ing quite well in experiments (1) and (2). In eqn (13) 0 = 0.364 was found to represent

Page 31: Ratcheting of cyclically hardening and softening materials: I. Uniaxial behavior

Ratcheting: Part I 179

B2 B37

I~A.-'////Z)(////A k\\\"~\\\\\\\\~l

,. / ~'a / , I

2(~b+t'~b) 2c £P

Model I

Fig. 19. Implementation of Model I using plastic strain surface (PSS) for cyclic hardening and softening materials.

the experimental results for CS 1018 well. As discussed earlier, in the case of SS 304 the size of the yield surface did not vary significantly in the experimental results; it was thus assumed to remain constant (a0 = 16 ksi).

For the two materials under consideration, the linear shape function h in (4c) was found to yield better predictions than the alternate one given in (4b), and it was pre- ferred. The value of ho for each material was evaluated from experiment (1). Ce in (4c) was evaluated by fitting a stress-strain loop with smaller amplitude from experiments (3). The bound modulus E0 p was obtained from a monotonic stress-strain curve. Finally, the ratcheting coefficient Cr in (7) was selected by matching the predicted ratcheting rate to that of an experiment. The ratcheting experiment with ~x,~ = 0.158, Ox~ = 1.014 was used for SS 304, and that with #xm : 0.249 and #x~ = 0.751 for CS 1018 (experiments [3]). The full set of parameters used for each material appear in Table 1.

The asymmetry in the hysteresis loops for CS 1018, observed in strain-controlled cycling (e.g. Fig. 2a), was included in all calculations by giving an initial shift of 7 ksi to the bounding surface in the positive stress direction. In addition, the first monotonic stress-strain curves were modelled separately, as they are distinctly different from all other hysteresis curves. The monotonic stress-strain curve parameters for CS 1018 are Oo = 74.5 ksi, h = 15, 120 ksi, and the values of the remaining constants are the same as those given in Table 1. The monotonic stress-strain curve for SS 304 has o0 = 20-24 ksi, and the remaining constants are as given in Table 1.

IV.1. Ratcheting behavior

These model variables were used to simulate all ratcheting experiments. The stress- strain responses predicted by the two models for one of the CS 1018 experiments are

Page 32: Ratcheting of cyclically hardening and softening materials: I. Uniaxial behavior

180 T. HASSAN and S. KYR1AKIDES

shown in Fig. 5, together with the corresponding experimental response. Both predic- tions are seen to be in good agreement with the experiment. The shape characteristics of the predicted loops with N, as represented by the evolution of e and W P, are com- pared with the experimental results in Figs. 5d and 5e. We observe that the evolution of the predicted loop "shapes," at least according to these parameters, differ to some degree from the experimental results. Initially, the evolution of both of these quanti- ties is slower than was measured. After approximately 60 cycles the evolution of the pre- dictions is much faster than was measured. The ratcheting strains predicted by the two models for this experiment, as a function of N, are compared with the experimental results in Fig. 5f. Both models are seen to predict rates of ratcheting which are in good agreement with the experiment. In the same figure we also include predictions from the original Dafalias-Popov model extended to include cyclic hardening/softening. Initially, the three models perform equally well. However, the rate of ratcheting predicted by the original model is seen not to follow the acceleration seen in the experimental results after approximately 50 cycles.

The predicted stress-strain responses for the remaining ratcheting experiments on CS 1018 are similar to the ones shown in Fig. 5 and will not be included. The predictions from all the experiments in the set are summarized in the exp - N plots shown in Figs. 7a and 7b. The agreement between experiments and predictions from both models is good for all cases. Overall, the predictions from Model II are somewhat better than those from Model I. However, we remind the reader that Model I has one less parameter than Model II. The rather poor performance by the original Dafalias-Popov model demon- strated in Fig. 5f is consistently repeated for all cases shown in Figs. 7a and 7b. The improved performance exhibited by Models I and II is directly related to the modifica- tions (bound shift or relaxation) that were implemented.

Similar simulations were performed for the ratcheting experiments on SS 304. The pre- dicted ratcheting response for one of these experiments is compared to the experimen- tal results in Fig. 8. The predicted loops are somewhat different from the ones measured, but overall, the results can be said to be in qualitatively good agreement with the exper- iment. In addition, the rate of ratcheting predicted is seen in Fig. 9b to be in good agree- ment with the experimental values. However, the loop changes predicted are significantly different from those measured, as evidenced in Figs. 8b, 8c, and 8d.

The predictions from Model II of all SS 304 ratcheting experiments are summarized in the e ff - N plots shown in Figs. 9a and 9b. The predicted ratcheting rates, for most cases, are in good agreement with corresponding experiments. However, differences between the predicted and measured ratcheting strains during the first few cycles result in an approximately constant difference between the two sets of results. This difference is a direct result of the inability of the model to reproduce the appropriate hardening induced during the first few cycles of the simulation, and illustrated in Figs. 8c and 8d. In this material there seems to be some difference between hardening during large mono- tonic excursions in strain and that induced during cycling involving small loops. The nature of this difference is not well understood at this time and was not modelled. This contributed to the quality of the predictions (D~AuAS [1981] briefly addressed this mat- ter). The mechanical response of SS 304 exhibits some time dependence even at room temperature. This was neglected in the models used. Although this is expected to play a relatively small role in the quality of the predictions shown in this paper, a quantita- tive comparison was not performed.

The predictions of these experiments from Model I were found not to be of very good quality. This was also related to the inability of this model to reproduce the material

Page 33: Ratcheting of cyclically hardening and softening materials: I. Uniaxial behavior

Ratcheting: Part I 181

cyclic hardening with accuracy in the first two cycles. More specifically, this is directly related to the inability of this model to reproduce accurately the negative ratcheting exhibited by this material during the first two cycles (see Fig. 9).

IV.2. Strain-controlled experiments

Model II was also used to predict all the strain controlled experiments reported in this paper, some of which have already been discussed. The predictions for a single step relaxation experiment on CS 1018 are compared with the experiment in Fig. 4. Over- all, the predicted stress-strain response is seen to be in good agreement with the one mea- sured. The evolution of the amplitude and mean stresses of the loops are seen in Fig. 4c to also be in good agreement with the experimental results. However, the mean stresses predicted are seen to be relaxing at a somewhat faster rate than in the experiment. Indeed, with continued cycling, the model will yield relaxation to zero mean stress. Due to initial anisotropy this material relaxes to a nonzero value. This anisotropy was not properly accounted for in the modeling; thus, the difference between the predicted and measured relaxation. The corresponding W p - N predictions are compared to the experimental values in Fig. 4d. They are numerically reasonable but qualitatively some- what different from the experiment.

The predictions for the multistep relaxation experiments performed on CS 1018 and SS 304 are shown in Figs. 12 and 13, respectively, together with the experimental results. In both cases, the loop shapes are well reproduced. The W p - N predictions are also quite representative of the experimental results. The rate of softening of CS 1018 is reproduced well for the last three steps, but not for the first step where the amount of softening induced is underpredicted. The mean stress relaxation predicted for CS 1018 follows the pattern of the experimental results, except for a downward shift of all results due to excessive relaxation in the first few cycles. In the case of SS 304, the growth of the loop amplitude resulting from cyclic hardening is reproduced well in the first relax- ation step. In the later steps, the model predicts hardening while the experimental results show some cyclic softening. The reasons for this reversal in behavior are not well under- stood, and thus this feature was not included in the model. The mean stress relaxation in the same experiment is well reproduced for all steps except the first, where the exper- iment exhibited no relaxation.

v. CONCLUSIONS

A set of uniaxial experiments have been presented demonstrating ratcheting in cycli- cally hardening (SS 304) and softening (CS 1018) materials. A number of auxiliary uni- axial experiments were also performed on each material to establish the cyclic hardening and softening characteristics of the materials, required for modeling.

Cyclic softening and hardening were found to influence significantly the rate of ratch- eting. In the case of CS 1018, ratcheting initially occurs at an approximately constant rate. However, following some accumulation of strain, the rate of ratcheting experiences an exponential growth which quickly results in failure. Cyclic hardening was found to reduce the rate of ratcheting, but in the range of parameters considered, ratcheting was never arrested.

The experiments performed were simulated numerically using the Dafalias-Popov model, with a set of additional features required for modeling cyclic hardening in strain- controlled loading histories. The performance of the modified model in predicting cyclic

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182 T. HASSAN and S. KYRIAKIDES

hardening and softening in strain-controlled experiments was evaluated by compar ing predictions with a number o f different strain-controlled experiments. Overall, the per- formance of the model was good. The following special features o f the material behav- ior recorded resulted in modeling inaccuracies.

1. SS 304 does not exhibit significant cyclic hardening for strain cycles with ampli- tudes less than 0.3°70. This feature was not included in the model .

2. Hardening induced to SS 304 by large monotonic excursions in strain seems to dif- fer f rom that induced during cycling involving small loops. This difference was not included in the model .

3. Cycling o f CS 1018 in the plastic range results in some reduction of the elastic mod- ulus, but also some changes in the shapes o f the hysteresis loops. This indicates that some fo rm o f damage accumulat ion takes place.

4. The CS 1018 used exhibited an initial anisotropy, which was not properly modeled.

The modeling o f ratcheting by the original model was shown to be deficient. Two schemes for correcting this deficiency have been proposed, together with all other mod- ifications required for compatibility with the original model. The modified models were first shown to be successful in predicting ratcheting in a cyclically stable material. Their pe r formance in predicting ratcheting in cyclically softening materials was good for all experiments performed. However , some differences in the evolution o f loop shapes between predictions and measurements were found to occur. These differences can be at t r ibuted to the reasons given above.

The per formance of bo th models in predicting ratcheting in the cyclically hardening material was not as good. Model II yielded rates o f ratcheting which were accurate, but the predictions differed f rom the experiments by a constant factor for all cases. The pre- dictions f rom Model I were not o f good quality due to the inability o f this model to reproduce the negative ratcheting, which was exhibited in the first one to two ratchet- ing cycles. These difficulties indicate that a more tho rough unders tanding o f the phys- ical phenomena (at the micro level) observed in this material and their causes is required before the models can be improved.

Acknowledgements-The test facility and data acquisition and control systems used to conduct the experi- ments were developed partly with the financial support of the Office of Naval Research under the equipment grant No. N00014-86-G-0155. The work was conducted with partial support from the National Science Foun- dation under the PYI award MSM-8352370. The authors would like to thank Y. F. Dafalias for bringing the work of Seyed-Ranjbari to their attention.

REFERENCES

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the Institute of Metals, London, 89, 328. 1964 COFFIr~, L.F., "The Influence of Mean Stress on the Mechanical Hysteresis Loop Shift of 1100 Alu-

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Midwestern Mechanics Conference, Madison, WI, 3. 1965 MORROW, J.-D., "Cyclic Plastic Strain Energy and F~igue of Metals," ASTM STP, 378, 45. 1967 FEtTNER, C.E. and LAmD, C., "Cyclic Stress-Strain Response of FCC Metals and Alloys-I. Phe-

nomenological Experiments," Acta Metallurgica, IS, 1621.

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Ratcheting: Part I 183

1969 LANDOgAr, R.W., MORROW, J., and ENDO, T., "Determination of Cyclic Stress-Strain Curve," Jour- nal of Materials, 4, 176.

1970 LANDGgar, R.W., "The Resistance of Metal to Cyclic Deformation," ASTM STP, 467, 3. 1973 JrthNsAt~, H.R. and TOPPER, T.H., "Engineering Analysis of the Inelastic Stress Response of a Struc-

tural Metal Under Variable Cyclic Strains," ASTM STP, 519, 246. 1975 DAtums, Y.F. and Poeov, E.P., "A Model of Nonlinearly Hardening Materials for Complex Load-

ing," Acta Mechanica, 21, 173. 1976 DAFAI.IAS, Y.F. and PoPov, E.P., "Plastic Internal Variables Formalism of Cyclic Plasticity," ASME

Journal of Applied Mechanics, 43, 645. 1977 LAIRD, C., "The General Cyclic Stress-Strain Response of Aluminum Alloys," ASTM STP, 637, 3. 1978 LAMBA, H.S. and SIOEBOTTO~, O.M., "Cyclic Plasticity for Nonproportional Paths: Part 1 and 2,"

ASME Journal of Engineering Materials and Technology, 100, 96. 1979 Cm, taocrm, J.L, DANG VAN, K., and COROIER, G., "Modelization of the Strain Memory Effect on

the Cyclic Hardening of 316 Stainless Steel," 5th SMiRT Conference, Berlin, August 1979, L, L 11/3. 1979 P~o, D., KEn(, W., MAYa, P., and MACHERAUCI-I, E., "Cyclic Induced Creep of a Plain Carbon Steel

at Room Temperature," Fatigue of Engineering Materials and Structures, 1,287. 1980 YOSHmA, F., MtraXrA, K., and SrImA~ORY, E., "Constitutive Equation of Cyclic Creep under Increas-

ing Stress Condition," Bulletin of the JSME, 23, 337. 1981 DRUCKER, D.C. and PAtGErL L., "On Stress-Strain Relations Suitable for Cyclic and other Load-

ing," ASME Journal of Applied Mechanics, 48, 479. 1981 DAFALUtS, Y.F., "A Novel Bounding Surface Constitutive Law for Monotonic and Cyclic Harden-

ing Response of Metals," Transactions, 6th SMiRT Conference, Paris, France, August 1981, L, L3/4. 1982 DAFALIAS, Y.F. and SAYED-RANJBARI, M., "Constitutive Modeling in Cyclic Plasticity," in Current

Advances in Mechanical Design and Production, Proceedings 2nd Cairo University MDP Confer- ence, 429.

1982 OHNO, N., "A Constitutive Model of Cyclic Plasticity With a Nonhardening Strain Region," ASME J. Appl. Mech., 49, 721.

1983 TSEN6, N.T. and LZE, G.C., "Simple Plasticity Model of Two-Surface Type," ASCE Journal of Engi- neering Mechanics, 109, 795.

1985 At^MEEI., 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.

1986 SEYEo-RANJnAIU, M., "Further Development, Multiaxial Formulation, and Implementation of the Bounding Surface Plasticity Model for Metals," Ph.D. dissertation, University of California, Davis.

1987 Cn~, H.-F. and LAMP, C. "Mechanisms of Cyclic Softening and Cyclic Creep in Low Carbon Steel," Materials Science and Engineering, 93, 159.

1989 Cnhaocrm, J.L. and Nou~rL~S, D., "Constitutive Modeling of Ratchetting Effects-Par t I: Exper- imental Facts and Properties of the Classical Models," ASME Journal of Engineering Material and Technology, 111, 384.

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1990 RUC_,OLES, M.B. and KRE~L, E., "The Interaction of Cyclic Hardening and Ratcheting for AISI Type 304 Stainless Steel at Room Temperature- I. Experiments," Journal of the Mechanics and Physics of Solids, 38, 575.

1992 HASSAN, T. and KYRIAKIDr~S, S., "Ratcheting in Cyclic Plasticity, Part I: Uniaxial Behavior," Int'l Journal of Plasticity, 8, 91.

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

(Received in final revised form 10 May 1993)

a

b

c~ ch c,

NOMENCLATURE

= current position of yield surface center in deviatoric stress space

= current position of bounding surface center in deviatoric stress space

= rate at which the bounding surface size changes

= constant governing rate at which the stable bounding surface grows

- ratcheting coefficient

Page 36: Ratcheting of cyclically hardening and softening materials: I. Uniaxial behavior

184 T. HASSA~ and S. KYRIAKIDES

E = Young ' s m o d u l u s

Eo p = a sympto t i c plast ic m o d u l u s of bounds

E ~ = in s t an taneous m o d u l u s of bounds

H = genera l ized plast ic m o d u l u s

n = un i t n o r m a l to the yield surface

n* = un i t n o r m a l to the plast ic s t ra in surface

N = n u m b e r of load ing cycles

s = dev ia tor ic stress tensor

t = t ime

W P = plast ic work or area of a hysteresis loop

a = center o f the yield surface

Ag = center o f the b o u n d i n g sur face

AJ3r = r e l axa t ion of bounds in the one ra tche t ing cycle

t5 = d i s tance o f cur ren t stress po in t f rom b o u n d

~in = dis tance o f last y ie ld ing poin t f rom b o u n d

e, cx = axia l s t ra in

exc, exm = a m p l i t u d e and m e a n values of a s t ra in cycle

eo = size of the plast ic s t ra in surface

Cxp = m a x i m u m axia l s t ra in in a stress cycle

e P, e~ = plast ic axia l s t ra in

dee v = equ iva len t plast ic s t ra in inc rement

de P = plas t ic s t ra in inc rement tensor

1' = plast ic s t ra in of the center o f the plast ic s t ra in surface

~7 = rate a t which the plast ic s t ra in surface size changes

0 = O0/Ob

a, Ox = axia l stress

a = stress tensor

# = image of a on the b o u n d i n g surface

aa = cur ren t size of b o u n d i n g surface

obo = in i t ia l b o u n d i n g surface size

Obs(eo) = m a x i m u m b o u n d i n g sur face size for the cur rent va lue of eo

abs(eOl) = m a x i m u m b o u n d i n g surface size for a large value of eo

oxa, ox,,, = a m p l i t u d e and m e a n value of a stress cycle

#x,,, #xm = oxaAr6, ~xm/O6

o0 = size of yield surface

o~ = yield stress of m o n o t o n i c s t ress - s t ra in response (0.2°70 s t ra in offset)