ratcheting in cyclic plasticity, part ii: multiaxial behavior

International Journal of Plasticity, VoI. 8, pp. 117-146, 1992 0749-6419/92 $5.00 + .00 Printed in the U.S.A. Copyright © 1992 Pergamon Press Ltd.

RATCHETING IN CYCLIC PLASTICITY, PART II: MULTIAXIAL BEHAVIOR

TASNIM HASSAN, EDMUNDO CORONA, a n d STELIOS KYRIAKIDES

The University of Texas at Austin

(Communicated by David McDowell, Georgia Institute of Technology)

Abstrac t -Par t II is concerned with time-independent ratcheting of materials under biaxial, cyclic loading histories. Carbon steel 1026 (heat-treated) tubes were tested under constant internal pressure and two axial loading histories. In the first set of experiments, the tubes were axially cycled in a strain symmetric fashion. For this loading, the material exhibits ratcheting in the circumferential direction. The second set involved stress controlled axial cycling in the presence of internal pressure. This loading results in strain ratcheting in the axial as well as the circumferential directions. Also, even though the material was cyclically stabilized prior to the ratcheting experiments, the biaxial loading was found to induce significant additional changes to the shapes of the hysteresis loops traced. The three cyclic plasticity models discussed in Part I are used to simulate the two sets of experiments using the material parameters selected for the modeling of the uniaxial ratcheting experiments. Prediction of the correct rate of ratcheting in the first set of experiments was found to be very sensitive to the hardening rule incorporated in the models. An explanation of this sensitivity is given. A hardening rule proposed by Armstrong and Frederick, with suitably selected parameters, was found to provide reasonably good predictions of the six experiments conducted. Prediction of the second set of experiments was found to require modeling of the changes induced to the shape of the basic hysteresis loop by this cyclic history.

1. RATCHETING UNDER BIAXIAL LOADING: EXPERIMENTS

Strain ratcheting under biaxial loading has traditionally been studied through two types of experiments. The first, and most widely used, experiment involves strain symmetric torsional cycling of thin-walled circular tubes in the presence of a constant axial load. The interaction of the constant axial and cyclic shear stresses leads to a progressive elon- gation of the tube (axial strain ratcheting). WOOD and BENDLER [1962] gave one of the earlier reports on the subject from experiments on copper tubes. They also made various micromechanical observations on the slip systems associated with the phenomenon. Similar experiments were conducted by MOYA~ and St~cL~aR [1963], BENrtA~ [1965], and Ft~trDENrm~ and RONAY [1966]. Since then, the experiment has been widely used by many other researchers.

The second experiment involves axial strain-symmetric cycling of a thin-walled tube in the presence of constant, internal pressure. The interaction of the axial and circumferential stresses leads to a progressive increase in the diameter of the tube (circumferential strain ratcheting). One of the earlier references to the phenomenon is due to Ruiz [1967] in work motivated by progressive collapse of pressure vessels (see also EDmmDS and B~.R [1961]). This type of experiment subsequently has been widely used in both low cycle fatigue studies as well as cyclic plasticity studies. SI-Ino.roai et al. [1978~] ~ and

1[ ]I indicates references listed in Part I.

117

118 T. HASSAN et al.

YOSHIDA et al. [1979 ~] used both experiments in key studies of cyclic creep and its modeling.

In both experiments, the rate of ratcheting of a given material has been shown repeat- edly to depend on the value of the constant load applied, as well as on the amplitude of the strain cycles prescribed.

One of the goals of this study was to develop a consistent set of experimental results on cyclic creep behavior of metals that can be used as a basis for evaluating models in a systematic fashion. Thus, in addition to the uniaxial ratcheting experiments presented in Part I, a complementary set of biaxial experiments was conducted. The test specimens were made from 1026 carbon steel with the same geometry and heat treatment as those used in the uniaxial experiments. The experiments conducted involved internal pressure and axial loading.

The test facility, data acquisition system, and test procedure used are the same as described in Part I. The test specimens were subjected to the same strain-symmetric (stabilization) prehistory as the uniaxial specimens. Following cyclic stabilization, they were unloaded to zero stress and axial strain. (In most cases, a small residual circumferential strain was registered at this stage, which was found to be related to the way Luder bands developed during the cyclic prehistory.) Internal pressure was applied next, using a hydraulic pump. The pressure and axial load were coupled through feedback so that the specimen was under pure lateral pressure loading (zero axial stress). In the experiments presented, the pressure was kept constant by using an accumulator in the pressure line. With some periodic manual adjustments, it was possible to keep the pressure within 3070 of the nominal value throughout the course of the experiment. Two major types of cyclic experiments were conducted, which are described below.

I. 1. S t ra in-symmetr ic cycling at constant pressure

A set of experiments was conducted in which, following pressurization, the test specimen was axially cycled in a strain-symmetric fashion. A set of results from such an experiment is shown in Fig. 1. The axial stress and strain are identified as (a,~, ex), and the circumferential values are identified as (a0, co). The amplitude of the strain cycles is identified by e,~c, and the constant circumferential stress by #0 (= go/go,). In the particular case shown, ex,- = 0. 5070 and ~0 = 0.245. The specimen was cycled 25 times.

Figure la shows the axial stress-strain loops recorded. The stable hysteresis loop from the prehistory is also included. The presence of the constant go causes a shift of the loops in the positive Ox direction.

Figure lb shows a plot of the ex-eo response obtained. The ratcheting in e0 can be clearly observed. Initially, the ratcheting rate is somewhat faster, but following the first few cycles, it becomes nearly constant (see also Fig. 2a).

A series of six such experiments was conducted in which the values of the constant circumferential stress and the strain cycle amplitude were varied. The stress-strain responses from these experiments are similar to those shown in Fig. 1 and are not included here. The results obtained are summarized in Fig. 2, which shows plots of the maximum circumferential strain recorded in each cycle (e0~,) as a function of the number of cycles applied. In the experiments reported in Fig. 2 (a & b), the specimens were cycled until a steady ratcheting was reached, or until a circumferential strain of 2070 was accumulated, whichever came first.

Ratcheting in cyclic plasticity, Par t II 119

(ksi) 45~ ........... :

~,=.z,~5 - ; - - ; ,~1" Li / / ,xo--.~o% .-'"" ~ ~ /~ I ;

I ~ I

,] ~

-~ / " I ~ ; ~ ,' I ~ ' ~ ~ (%)

// ~ "'/ fl ~ ~

i / . . . . . . . . . . . . . . - - ~ - H~ter~ is

. . . . . . . . . . . _45~ Lo~ (o)

- . _ _ ~ . ~ x ~ o / , )

~b)

Fig, 1. Axial strain-symmetric cycling of a thin-walled tube at a constant internal pressure. (a) Axial stress- strain response; (b) ratcheting response of ee.

In all cases, a constant (approximately) rate of ratcheting was achieved within the first few cycles. In one case (Se = 0.178, exc = 0.5°70), shown in Fig. 2c, the specimen was cycled 69 times to an ee of 2.27~0 in order to see if any change occurred in the rate of ratcheting. Following the first few cycles, the ratcheting rate was found to remain con-


stant (the experiment was interrupted well before failure). The results in Fig. 2 clearly demonstrate that an increase in exc or a0 increases the rate of ratcheting.

1.2 Stress control axial cycling at constant pressure

Two types of ratcheting have been discussed so far. In the first, unsymmetric stress cycles lead to ratcheting in the direction of the mean stress. In the second, strain symmetric cycling in the axial direction in the presence of a constant stress in the circumferential direction results in strain ratcheting in the circumferential direction. The major difficulty in the modeling of the first type of ratcheting was in the accuracy of the plastic modulus calculation as discussed in Part I of this study. It will be shown in the next

~ep (%)

2 -

C S 1 0 2 6

~ = 2 4

°.65 ° o o¸50

o o o o o

o o o

o ° o°

o o

o o o o o o ~40 o o o

o o o ° ° ° ° " ~ o o ° o o o

o o o o° o o ~,~ (°Io) o o o

o o o o ° o o o

o o O o

o Ib 20 ~'o 4'0 - - -N

(~)

~ep [%) C S 1 0 2 6

~ ,e= .5%

° 3 5 7 2 4 5 o o o , o o o

o o ° o o ° o 178

o o ° o o o ° " o o ° o o o o ° ° ° °

o o ° °°° o oO o°° ...... 122

o o °°

o o o o o ° ° ° ° ° ° ° ° ~ ~ o o o o o ° ° ° ° ° °

° o o o o o o o o . . . .

o ' ib ' 7o -------l,- N

(b) Fig. 2. Maximum circumferential strain as a function o f number of cycles from strain symmetric cycling at constant 80. (a) #0 = 0.24; (b) exc = 0.5%. (Figure continued on facing page.)

Ratcheting in cyclic plasticity, Part II 121

(ep (%)

3-

I

o

o

o

CS 1026 ~o = .178 ~,,c=5% o ° °

o o

o

o

o

o o

o fo ~'o ' eb ~'o - ~ N

(c)

Fig. 2 continued. (c) Strain accumulation for larger number of cycles.

section that the difficulties in the modeling of the second type of ratcheting are primarily rooted in the hardening rule adopted and will be addressed independently. In addition to the above, a third series of experiments, in which both types of ratcheting occur, was conducted in order to establish any interaction between the two mechanisms and, at the same time, provide a more independent way of verifying the validity of the models.

The experiments involved stress control axial cycling in the presence of a constant, internal pressure. The same prehistory was applied to the test specimens as before. Fol- lowing pressurization, stress controlled axial cycling was prescribed. The selection of suit- able mean value and amplitude for these stress cycles, for a given internal pressure, is much more complicated than at zero pressure.

A set of experimental results obtained from such an experiment is shown in Fig. 3. The applied internal pressure resulted in a circumferential stress of ~o = 0.120 and the stress cycles had an amplitude of ~x~ = 0.800 and mean value of #x,, = 0.130. The axial stress-strain response obtained is shown in Fig. 3a. The axial ratcheting is quite clearly seen. It is also interesting to observe that, although the material was cyclically stabilized, the hysteresis loops traced show significant change as the cyclic history progresses. The change in the hysteresis loops is much larger than that observed in the uniaxial experiments. The corresponding ~x-e0 response is shown in Fig. 3b. Both ~x and eo are seen to ratchet. The e0 ratcheting is negative and is primarily due to the Poisson effect. The effect of the pressure on the ratcheting rates can be established by comparing the results in Fig. 3b with those in Fig. 4b, which were obtained from an experiment with the same axial stress cycles but zero pressure. The internal pressure reduces both the axial (positive) as well as the circumferential (negative) rates of ratcheting.

Further increase in the internal pressure (keeping exa and ~xm the same) leads to ratcheting in the positive e0 direction and further reduction in the rate of axial strain

122 T. HAssAr~ et al.

(ksi) 45~

I _ -- ~ ~- - - ~

-30

i .~ ~,(%)

I$ (a) Loop

.075

lO .~3 .6 9 • ~ ~

= =, (%)

(%)

l _j.,

CS 1026 Ko =.120 ~,.=,~oo e,,~.13o

-.45 ± (b)

Fig. 3. Stress-controlled axial cycling with #0 = 0.120. (a) Axial stress-strain response; (b) axial-circumferential strain response.

ratcheting. A set of results for #0 = 0.249 is shown in Fig. 5. A significant transient is observed in the first 13 cycles followed by a more gradual change. The hysteresis loops traced in Fig. 5a are again seen to undergo significant change during the cyclic history.

When the pressure is increased even further, the rate at which e0 ratchets increases further and starts to dominate the behavior in the axial direction. A set of results for #0 = 0.353 is shown in Fig. 6. The material is seen to ratchet in the positive e0 direction and the negative ex direction, even though the prescribed axial stress cycle has the same positive mean stress value of Oxm = 0.130 as the previous experiments.


¢r~ ( ks i )

l 45.

_ _ ~ ~

2 -.3 ~ .

-30' /

~ ~ ~ Hysteresis . . . . . . ~ Loop

- - ( a )

~ ~,,(O/o)

-.3 0 .5 .6 .9 ~ ,~ ~ I I I

-.~, c~ -.,5 ,o~,O, ~ _ _ I First two cycles

=45' ~

(b)

Fi~. 4. Uni~ial ratchetin~ experiment with the same ~ia] stress cycle as the experiment in FiB. $.

A total of six experiments were conducted with the same axial stress cycle parameters as the experiments discussed above, but with different values of internal pressure. The results are summarized in Fig. 7, in which we plot the mean value of ex(ex,~) versus the mean value of e0(e0,~) achieved in each cycle (the mean here is the average of the two peak values in each cycle). In order to illustrate all features of the interaction between the two ratcheting mechanisms, we show the strains achieved during 45 stress cycles for all experiments. The maximum rate of axial ratcheting occurs for o0 = 0. In this case,


~r, (ksi) 45-

. . . . . . t _ _ ~ ~ ......... y . . . . . . . . . . ,:

N _!:i . . . . . : : : : ~ I - - - ~ . . . . . . . ~ "* -~'~-- ~oY~! eeresis

~3

~" .45. (%]

t CS 1026 ~ ~. =.249 &'~'%.~. ~,o =.8oo ~ --JSO

.~.-

~'-'~'-%~'~ ' ~"~" 6

(b~

I .9

- - - ~ cx C%)

Fig. 5. Stress-controlled axial cycling with 00 = 0.249. (a) Axial stress-strain response; (b) axial-circumferential strain response.

60m grows negatively at approximately one-half the rate of ex,~ (as expected, in view of incompressibility of plastic deformations). The presence of internal pressure leads to a positive growth in e0. At lower pressures, this causes a reduction in the negative rate of growth in eOm and a corresponding reduction in the positive rate of growth of exm. Fur-


(a)

-. I

-I.2

(b)

¸.9

.6

.5

0

-.I 5

ksi)

---~,(%)

resis Loop

~o (%)

CS 1026 ~ =.55:.3 ~,o =.80O ~,,,=.~50

I

.6 - - - ~ ( % )

Fig. 6. Stress-controlled axial cycling with 80 = 0.353. (a) Axial stress-strain response; (b) axial-circumferential strain response.

ther increase in pressure leads to a positive rate of growth of eo,~, but the rates of ratcheting of both strains are significantly reduced. Clearly, in the positive e0-ex quadrant there is a combinat ion of tr0 and the axial stress cycle parameters at which the rates of ratcheting are relatively small. At higher pressures, the rate of ratcheting in e0m increases significantly and dominates the behavior in the axial direction through the Pois-

126 T. HASSAr~ et al.

-.5

~ e m (%)

~ , : ~ . 2 1 0 \

I ~ ~ I

~ .,5 I

~ . 1 2 0 ""c~.....

=5 " - - - . ~ .

CS 1026 ~o : .800 ~,,, :.130 Nm~:45

I 1.5

~ . ~ ( % )

~ ' ~ . . . o 0

Fig. 7. Mean axial versus circumferential strain recorded in each cycle in stress-controlled axial cycling experiments at values of #~ (o --- values at every fifth cycle).

son effect. At some value of pressure, the material does not ratchet in the ~-xm direction. Further increase in pressure leads to ratcheting in the negative exm direction. Continued increase in pressure will lead to a much faster rate of ratcheting in EOm and the negative exm direction.

Another experiment from this family is shown in Fig. 8. The stress parameters of this case were: ~o = 0.246, ~:,~ = 0.800, and #xm = 0. The pressure dominates over the axial behavior, and as a result, the material ratchets in the negative ex direction.

A common feature for all the biaxial ratcheting experiments is the significantly larger change induced to the tr,~-ex hysteresis loops during cycling as compared to the changes observed in the uniaxial ratcheting experiments described in Part I. The reason for this difference is not well understood. At the macromechanical level, LASmA and SIDEBOTTOM [1978 ~] observed that under biaxial cyclic (90 ° out of phase axial/torsional) loading materials exhibit significantly more hardening than that exhibited under uniaxial (strain symmetric) cycling. However, it is not clear whether the behavior observed in our experiments is related to their observations. A similar observation was made by SLiwowsra [1979] in different types of experiments. He observed that equivalent stress-strain loops in stress-controlled experiments become "thinner" dur ing cycling. The "thinning" can be attributed to cyclic hardening of the material used. Biaxial loading leads to more thinning or more hardening. This behavior seems to be similar to the one observed in our experiments but with the opposite effect on the hysteresis loops (fattening). WOOD and BE~DLER [1962] observed that, under biaxial cyclic loading (tension/cyclic torsion), the material experiences an accelerated enlargement of slip zones previously activated by monotonic loading, which, qualitatively, could explain the difference. McDow~t~ et al. [1988] conducted a microstructural examination of the


-I.5

I

(a)

45 (kii) _ _ ~ - . ~

, ~ ¢ - _ ~ ~ - - ~ . . . . -

l ; l

t I I t

i t

/ 0 ~ /! ,~

,,'--~,(°/ol /

~

-~0 % • ~ ~ - ~ ~ ~ Stable

Hyste res is

Loop

I I

(b)

-.9 (%)

.6

.3

0

-.15

CS 1026 ~ =.246 ~o :.800 ~ r . - - O

I I I

.6

----~ ~,(°/o)

Fig. 8. Stress-controlled axial cycling with 6xm = 0. (a) Axial stress-strain response; (b) axial-circumferential strain response.

Lamba-Sidebot tom observation, described above, for 304 stainless steel. They observed differences in the microstructure resulting f rom uniaxial cycling and that resulting f rom biaxial strain cycling. The main conclusion f rom these studies, affecting macromechanical modeling, is that fundamental material features are altered by biaxial cyclic load-


ing, and, as a result, multiaxial experiments are necessary for a more complete representation of the material behavior. The results presented in this paper support this point of view.

11. MODELING OF BIAXIAL RATCHETING

The biaxial ratcheting behavior observed in the experiments will be simulated using the three models outlined in Part I of this study. Each model can be completed by pre- scribing an appropriate hardening rule. In the modeling, it is assumed that, for cyclically stable materials, yield surfaces retain their size and shape. Under these assumptions, the hardening rule defines the way yield surfaces translate in stress space. It is well known that actual yield surfaces expand/contract and deform during loading (see Pnn~- LIeS & LEE [1979]). In view of this, the performance of models that do not account for such changes, but depend strictly on yield surface translation to represent the evolution of yield surfaces, can be expected to be sensitive to the hardening rule adopted (see McDOWELL [1987], SrIAW & KYRIAKIDES [1985t]). This will be shown to be the case for the models and ratcheting histories considered here.

The Drucker-Palgen (I) and Dafalias-Popov (lI) models belong to the class of models that emphasize the determination of the plastic modulus (see DAFALIAS [1984t]). Their flow and hardening rules are uncoupled. Typically, in these models, given an increment in stress (strain), the current moduli and strain (stress) increments are evaluated first, from the flow rule. Once these are determined, the hardening rule and the consistency condition are used to evaluate the movement of the yield surface. Although this procedure can involve iteration, the order of these operations is retained, which provides the uncoupling of the hardening and flow rules.

This characteristic provides essentially an extra "degree of freedom" to these models, as compared to the competitor class of models which direclty couple the tangent moduli calculations to the hardening rule (e.g., ARMSTRONG & FREDERICK [1966]; BOWER [1989]; CHABOCHE et al. [1979]; MROZ [1976~]). In the former group of models different hardening rules can be implemented as deemed appropriate for the application at hand. This flexibility will be exploited by implementing a number of hardening rules in the calculations performed. Their performance will be evaluated by direct comparison to the experimental results.

II. 1. Hardening rules

In eqn (1) 2 ot represents the current center of a yield surface in stress space, dot is the current increment of ot. The following expressions for dot (hardening rules) will be considered:

1. Prager-Ziegler rule (Pe, AaER [1956], ZIEGLER [1959]).

dot = d/z(~r - ot) (8)

where d/z is to be evaluated from the consistency condition.

2Equations (1) to (7) are in Part I.


2. Armstrong-Frederick rule (Ae~tsrRo~vG ~ FR~DI~RICK [1966]). This rule was proposed as a nonlinear extension of Prager's linear kinematic hardening model. Indeed, it represents the heart of various models which couple the hardening and flow rules. It is usually written as

dt~ = c~ de p - c2~ dee ~, de~ P = x/~ d ~ " d ~ ~'.

In order to be incorporated in the class of models used here, it will be written as

d~t = d t ~ [ ( 1 - k ) ( ~ - ot) - k o t ] ( 9 )

where k is either a constant or more generally

k = k ~

and d~ is determined from the consistency condition.

3. Mroz rule (MRoz [19671]). This geometric rule was proposed by Mroz for his multisurface model. It relates the movement of two nested surfaces in the fashion shown in Fig. 4 of the reference. It can be expressed as follows:

dot = d/z(# - a) (10)

where # is the image point of a on the outer surface. For model II # is determined from

(# _ / ~ ) = at, ( a - ot) ( 1 1 ) •o

where ~ is the current center of the bounding surface and ab is its size. Model I, in its original form, does not involve a second surface. A modification proposed by SHAw and KVRIA~IDV.S [1985 ~] has added a loading surface to the model. The loading surface grows isotropically and represents the largest state of stress achieved in history. In this case ~ is determined from

t l a x

# = - - (o - ot). (12) Oo

4. Stress increment rule. Experiments have consistently shown that subsequent yield surfaces do, under some circumstances, translate in the stress increment direction (see Pni~Ln, s ~ LEE [1979I]. For completeness we also consider the following rule:

dot = da. (13)

5. Tseng-Lee rule (Ts~vG ,~ L~z [1983~]). This rule was proposed as a way of implementing Phillips's experimental observations on movement of yield surfaces in a prac-

130 T. HASSAN el al.

tical problem. It was developed as part of the Tseng-Lee model, but can also be applied to other two-surface models. The hardening rule is given by

d~¢ = d#~, (14)

where the direction of the unit vector ~ is defined in eqn (8) in Ts~NG and LEE [1983~]. In addition to a hardening rule, model II requires a rule for the movement of the

bounding surface. We will directly adopt the method recommended by the originators of the model [1976~]. In the multiaxial case, the stress quantity/i in eqn 7 was taken as in eqn (21) of the same reference.

II.2. Performance o f the models

II.2.a. Strain symmetr ic cycling at constant pressure. The performance of the three models and various hardening rules outlined above, with respect to their ability to predict strain ratcheting under biaxial loading, is evaluated by direct comparison with the experimental results shown in Fig. 1. The model parameters used are the same as those given in Tables 1 and 2 of Part I, with tr b = 38.79 ksi (267 MPa) in model II and o,n --- 39.80 ksi (274 MPa) in model III.

The results obtained by model II with the Prager-Ziegler (1) and Mroz (3) hardening rules are shown in Fig. 9. The predicted axial stress-strain responses, shown in Fig. 9a, are seen to be in good agreement with the corresponding experimental results (Fig. 1). The shapes of the predicted loops, as well as the vertical shift of the loops, are quite well reproduced by both hardening rules.

Figure 9b shows the corresponding ~x-~o response predicted. In the case of the Prager-Ziegler hardening rule, the predicted response stops ratcheting after approximately four reversals in the axial strain (this was also observed by HA~rCEr~L and HARVEY [1979], KA~EKO [1981], and others). This, clearly, is in disagreement with the experimental results. The cause of this behavior will be explained below.

The e0-ex response predicted with the Mroz hardening rule and the same model is seen to lead to ratcheting in e0. However, the rate of ratcheting is much faster than in the experiment. This is shown in Fig. 10 where the maximum circumferential strain in each cycle is plotted against the number of cycles (N). The corresponding experimental results are also included in the figure. (The accelerated rate of ratcheting observed in the predictions was also observed by YOSHIDA et al. [1978 ~] in the predictions of a similar experiment using the original multisurface model of Mroz.)

The same calculations were conducted with model I, using the same two hardening rules. The predictions were found to be very similar, in all respects, to those of model II and are summarized in Fig. 10. It can be observed that the absolute value of cop predicted by the two models, as well as the rate at which it accumulates per cycle, are very similar when the same hardening rule is used.

The calculations were repeated using models I and II with the stress-increment hardening rule (4) and the Tseng-Lee hardening rule (5). A summary of the results is included in Fig. 10. Both hardening rules yield a rate of ratcheting that is much faster than the experiment, although the results from the stress-increment hardening rule are somewhat better than the other set. Again, the corresponding predictions from the two models are quite similar.

Predictions made by the Tseng-Lee model (III) and hardening rule (5) are also shown.


(ksi) 4 5 .

~.~'.~°,~ I ..~- --~, ,,o:.~o% -- - - ,Y-~ / / i

/ " / I/ / /

- ' / / I / .,L:.,., / / t / / - - ; / _ ~ .-~s,~,.

/ ~ [ ~ ~ / Hysteresis ~ . . . . . . ~ ~ ~Op

. . . . . . . P m g e r - Z i e g l e r - 4 5 ' (~) ~ Mroz

:5 ~£.5 -.2, ' ._~x(O/o]

(b)

Fig. 9. Predictions of experiment in Fig. 1 by model II with the Prager-Ziegler and Mroz hardening rules.

The results are seen to be very close to those of models I and II with the same hardening rule. (Note: The corresponding results from model I, with the hardening rule found by ShAw and KY~TAK~eS [1985 ~] to work in a cyclically loaded structural problem, lie between those from the same model and hardening rules 3 and 4. As a result, they are not included here.)

132 T. HAss~ et al.

___________ °° ?? ~'"'"'/'~®®

o Ib 2'0

CS 1026 ~,; =.245 o Exp. { (xe=.,~)%

{ - - Model I Predi. I . . . . Model f I

l - -Model "fIZ

3'O 4'O ~ N

Fig. 10. Maximum circumferential strain in each cycle as a function of N. Predictions of experiment in Fig. 1 from three models and five hardening rules (1-5 identify the hardening rules).

The results presented above can be summarized as follows: (a) the particular loading history addressed here seems to be relatively insensitive to the flow rule adopted (at least within the family of models considered); and (b) the rate of ratcheting predicted by the models used is very sensitive to the hardening rule used.

• The Prager-Ziegler hardening rule leads to a quick arrest of ratcheting. • Hardening rules 3, 4, and 5 yield rates of ratcheting that are higher than the exper-

imental ones.

The role of the hardening rule in this particular cyclic history can be better understood by considering the paths followed by a and at in two of the calculations using model II. Figure 1 la shows the results for hardening rule 1 The bolder line OABCD represents the stress path and oabcd the path followed by the center of the yield surface (Y.S.). (Be- cause of the prehistory imposed, the initial position of the Y.S. is not at the origin. This has no significant impact on the discussion that follows.) Y.S.~, Y.S.2, and Y.S.3 are segments of the yield surface at three instances during the loading history, and ate, at2, and at3 are the respective centers of these Y.S.'s.

Paths OA and oa correspond to the initial pressurization. This resulted in some plastic deformation indicated by the movement of the Y.S. from o to a. During the first quarter cycle of the axial strain, the stress moves along A B and the Y.S. center moves along ab. data shows the direction of translation of the Y.S. when its center is at ate. During the second and third quarters of the axial cycle, the stress moves along B C and the Y.S. center along bc. The Y.S. is seen to continue to move in the positive o0 direction, but at a decreasing rate (compare orientation of dat2 with that of dat~). During the next half of an axial cycle, the stress point moves along CD and the Y.S. center along cd. During this part of the loading history the Y.S. center reaches (nearly) the level of


(ksi)

d~t~,. _ _ . ~ | I~ ~"[f f_ _ _ _ d ~ s

, , , I _ , ~ * , / / ~ . i 0 . ~ . ~ ,~

I -15 x

~)

(ksi)

B.S.~ y~ :~0

,

( . . . .

-15

(b)

,¢d,h ~ ~

~ ' ~ b , ( ] ~ - ; ~ ~ ~ /

B.S.I

r / / /~ ~fll

~(ksi)

Fig. 11. Stress and yield surface center paths followed when simulating the experiment in Fig. 1 by Model II and (a) the Prager-Ziegler hardening rule; (b) the Mroz hardening rule.

the stress path. Subsequently, d~t remains oriented along BC and ~ moves between d and e.

Figure 1 lb shows a similar plot of the corresponding paths followed when the Mroz hardening rule is used. In this case, in addition to two Y.S. segments the corresponding bounding surface (B.S.) segments are also shown. (The movement of the center of the bounding surface is small for this stress path, and is not included in the plot.) OABC and oabc represent, again, the stress path and the path followed by the Y.S. center, d ~ is the instantaneous direction of translation of Y.S. 1 and, according to hardening rule 3, it is oriented along P~ Q~. Similarly, d~t 2 is oriented along Ja2Q 2.

The Y.S. is seen to move very quickly in the positive a0 direction. Movement in this direction continues until the loading point and its image on the bounding surface are both at the a0 level of the stress path (BC). Correspondingly, the Y.S. center reaches the level represented by bc, which is different from that of BC. Subsequently, the Y.S. moves strictly along bc.

In summary, in the predictions from the two hardening rules, the Y.S. centers have

134 T. HASSA~ et al.

(~0 I ~15

~ ~ - h ~ \ ~ ., ~ / > ~ _ . . . . , , , / ~ - - \ ~ ~ '~

., ~ ¢ - ~' - ~ JiB' Z ' ~ ~ ~" ~

~ ..... / ' ~ ~ Z c- . / i ' . -

~ ~ (kS l )

-30 [

Fig. 12. Steady-state motion of yield surfaces and direction of plastic strain increments by the Prager-Ziegler and the Mroz hardening rules.

stabilized at different levels along the a0 axis. The influence of the level of stabilization of the Y.S. in stress space is illustrated in Fig. 12, which shows the Y.S.'s from the two hardening rules, at the two extreme points of the stress paths, after stabilization. The normals to the surfaces, which represent the instantaneous directions of the plastic strain increments, are also included. In the case of the Prager-Ziegler hardening rule, the Y.S. moves along the same line as the stress point. As a result, during the loading and reverse loading paths of the cycle, the normals to the Y.S. at the loading point are paral- lel (opposite signs). This, combined with the symmetry of the cycle, results in plastic strains which, during the loading and reverse loading parts of each cycle, are equal and opposite in sign, and cancel each other. As a result, this hardening rule does not predict ratcheting if the material is cyclically stable. Cyclic histories with less symmetry in the stress paths can result in some ratcheting in %.

In the case of the Mroz hardening rule, the Y.S. stabilized at a value lower than a0. As a result, the normal to the Y.S. during the loading part of the cycle is different from that during the reverse loading part. The positive e~" developed during the tension-to- compression part of the cycle is much larger than the negative ~0 p developed during the compression-to-tension part of the cycle. As a result, a net positive e0 is accumulated after each cycle. This is qualitatively similar to what happens in the experiments, but the rate of accumulation of e0 predicted is too high.

From the above discussion, it can be concluded that within the framework of our assumptions, the key to successful prediction of this type of experiment is a hardening rule that will cause the Y.S. to stabilize at an appropriate stress level (t~0). This value must lie between the values at which the Ziegler and Mroz hardening rules stabilized.

II.2.b. Performance of the Armstrong-Frederick hardening rule. This hardening rule, when combined with the class of plasticity models considered in this paper, allows the user the freedom of choice of k in eqn (9) (k determines the strength of the influence of the two contributions to da ) . The simplest choice is for k to be a constant (ko). For


the particular experiment shown in Fig. 1, the following values of k0 were found to yield the best agreement with the experimental results:

Model I: ko = 0.091

Model II: ko = 0.097.

CS 1026 (ksi) 45 ~. --.245 I " ~ ~ . . . . . . . . 7 ,,o:.~oo, o . _ - / - } / / /

" / t / / ,,;" ///

-I t ~ 1111 I / ~ ~ ~ = ( % )

/, ~ ~ - ~ t o n e

/ - _ ~ - - - ~ ~ ~ H y s t e r e s i s

L . . . . . . . . . . . . . _--/45 ~ ~Op

(~)

Go

.'-,~ = ~(%) (b)

Fig. 13. Predictions of experiment in Fig. 1 by Model II and the Armstrong-Frederick hardening rule.

136 T. [-IASSAN et al,

Figure 13 shows the stress-strain responses predicted with model II and this hardening rule. All features of the predicted results are seen to be in very good agreement with the experimental results shown in Fig. 1. The corresponding cop versus N results are shown in Fig. 10. It can be observed that, with k constant, the model predicts a constant rate of ratcheting for all N. The predicted rate is, however, in good agreement with the experimental one.

In the case of model I the amplitude of the predicted axial stress was somewhat underestimated for all cases (by about 5%). The predicted rate of ratcheting is seen in Fig. 10 to be in very good agreement with the experimental results.

Figure 14 shows the stress and yield surface center paths followed in the calculations conducted with model II and this hardening rule. We observe that, after approximately one axial cycle, the yield surface center stabilizes at a level of o0 which lies between those of the Mroz (bc) and Prager-Ziegler (b 'c ' ) hardening rules shown earlier in Fig. 12.

The values for ko given above were used to simulate the other five experiments in this group (i.e., cases with different values of o0 and exo). The predictions from the two models are summarized in Fig. 15. The rates of ratcheting predicted by the two models are very similar, although the absolute values of the peak strains differ by a constant value (this can be mainly traced to differences in the predictions from the two models of the first quarter cycle). The predicted rates of ratcheting are, in the main, in good agreement with the experimental rates.

As observed above, if k = ko, the predicted cop is linearly related to the number of axial strain cycles. However, the corresponding experimental results exhibit some non- linearity in the first few cycles. The predictions can be improved by making k a function of the accumulated plastic strain as proposed in eqn (9). The following form was found to yield good results:

~ d k = ko + k~e -vr, r = ~ e~d t (15)

(ksi) I oT

t / ~ \, ,i . ,, - ) . . . . ,

~ _ _ ~ / . . . . . ~ , _ ~ , ,- , _ ~

-~5 ~ -,5~ ~o o ~~::(,s0

Fig. 14. Stress and yield surface center paths followed when simulating experiment in Fig. 1 by Model I! and hardening rule 3.


Eep (%) Experiment o

CS 1026 I Model I ~ =.24 Prediction I Model ] ' I - - -

°.65 o

o o . . ~ . ~ 5 0

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

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

~ / ~ ~ . / i %o

/ ~ ~

~ ~ /

, i

l'o '

(a) "~-'~" N

~ep (%)

t CS 1026 ~,c =.5%

° 357 o o ~ . 2 4 5 ~ . ~ . ~ .~ ~ .

o

o / °~ °~ / - ~ ~ - ~

'

' ' ' 2 " 0 ' ' I0 30

(b)

i i

4O ~,-N

Fig. 15. Maximum circumferential strain as a function of N. Comparison between experiments and predictions with k = k 0. (a) O0 = 0.24; (b) ~xc = 0.5%.

where ko, kl and "r are constants selected f rom one of the biaxial experiments. The following values were found to yield the best predictions: k0 = 0.059, k~ = 0.069 and ~/= 3.1. The improved predictions are shown in Fig. 16. The improved performance is seen to also hold for larger values of N as shown in Fig. 16c.

It has been demonstrated above that the key to successful prediction of this particular class of experiments is a hardening rule that causes the yield surface center to stabilize at an appropriate level along the o0 axis. Guided by strictly mechanical reasoning, it is easy to generate additional hardening rules which lead to good predictions. For example, a hardening rule that involves a combination of the Prager-Ziegler and the Mroz rules has been found to be quite promising. A hardening rule which combines the stress increment and Ziegler rules has been found to yields results that are of the same qual- ity as those in Fig. 15. In each case, the two hardening rules can be "mixed" in a direct

138 T. HAssAr~ et al.

way, as in the Armstrong-Frederick rule (eqn (9)), or in a less direct way, as proposed by SI-~W and Kvgua(mEs [1985~]. The best "mix" parameters can be evaluated in the way used to find k in the Armstrong-Frederick rule above.

The nonuniqueness of the solution to the problem is intellectually unsettling. Some of the rules found to be successful in this particular loading history may be eliminated if it can be demonstrated that they fail in other loading histories. However, at the same time, the Prager-Ziegler and the other hardening rules which have been shown to fail here cannot be universally rejected without further work.

This state of affairs is the price paid for the simplicity we have required from the mod-

(ep (%)

CS t028 o Experimenl

~ =.24 - - Prediction

°.65 o -

o o ¸50

o o

o o o oooo ° ° ° o - 4 0

° ° o o o o ° ° °

,°(%)

0 2'0 (c~)

~ N

~e~

(%) CS 1026

~,c =.5 % 2

°.557 o o ~ . 2 4 5

o°/ °° o~°°°°° 1 o o oo o o o°

o ooO ...... 122

o o o ooooooO°°° ~ o

' ' ' 2 " 0 ' ' 0 I0 30

(b) ~ 'N

Fig. 16. Maximum circumferential strain as a function of N. Comparison between experiments and predictions with k as in eqn (15). (a) #0 = 0.24; (b) exc = 0.5%. (Figure continued on facing page.)


Eop (%)

_

CS 1026 ,z. =.~78 / ~,,c=.5% / o o

o o oO oO ~ o

° o °

° o °

ooO o°° i l p l r i m e n t o - ..... Prediction

o z'o 4'o ' ~b ~'o

(c)

Fig. 16 continued. (c) Larger number of cycles.

- ~ - N

els used. Indeed, it has already been demonstrated that more general models which allow deformation of the yield surface can successfully predict the major features of the particular loading histories considered here (e.g., Snn~xoRi et al. [1979~], HARVEY et al. [1985]). It is easy to see from Fig. 12 that even moderate deformation of the yield surface at the stress point can alleviate the problem caused by the Prager-Ziegler rule.

II.2.c. Stress controlled axial cycling at constant pressure. The modeling schemes, proven to adequately simulate the uniaxial ratcheting in Part I and the biaxial ratcheting discussed in the preceding section, were combined and used to simulate the five biaxial ratcheting experiments presented earlier. The modification of the bounds of model II, found to improve the performance of the model in predicting uniaxial ratcheting, was generalized to the multiaxial setting in the following simple manner. The plastic equivalent strain achieved during the loading history is continuously monitored. When the current value exceeds a critical value, the position of the bounding surface in stress space is kept fixed. The bounding surface resumes its motion only if the value of the equivalent plastic strain starts decreasing. The first critical value is obtained from a uniaxial ratcheting experiment as described in Part I (point B in Fig. 15). During the loading history this value is updated to correspond to the largest value of equivalent plastic strain.

The material parameters used in the predictions are those given in Table 2 of Part I, combined with the Armstrong-Frederick hardening rule as given in eqns (9) and (15) with ko = 0.059, k~ = 0.069 and .g = 3.1.

Due to the number of cycles involved, this generalization was exposed to only lim- ited use when simulating the five experiments in this group (i.e., for most of the history,

140 T. ~IASSAN et al.

the original form of model II was adequate). As a result, the generalization was not adequately tested. More testing of the concept will be necessary before more general use in other cyclic loading histories.

The predictions of the biaxial ratcheting experiments for the two extreme values of pressure tested (~0 = 0.120 and ~0 = 0.353) are shown in Figs. 17 and 18. They are seen to be generally in good agreement with the corresponding experimental results shown in Figs. 3 and 6, respectively. A more quantitative comparison between experiments and predictions is shown in Fig. 19, in which the mean values of ~x and ~0, recorded every fifth load cycle, are plotted. It can be observed that for ~0 = 0.120 the predicted "direction" of ratcheting is in reasonably good agreement with that of the experiment, but the rate at which ratcheting occurs is underestimated. The same can be said about the results for 60 = 0.353 but in this case the rate of ratcheting is overestimated.

A distinct difference between the experimental and predicted results is that in the pre-

(ksi) 4 5

t

2 -•~ ! " ' .6

(0)

•075-

.9 ~ ~ (%)

-----~ ~,(%)

.3 .6 .9

CS 1026

We =.120

~,o =.~oo -.5 ~x~. =.1:50

(b)

Fig. 17. Predictions of stress controlled axial cycling with 00 = 0.120. (a) Axial stress-strain response; (b) axial-circumferential strain response.


I

-1.2

Ii o (a)

(ksi)

.6

~ ~,(%)

-.9 % (%)

.6 CS1026 ~ ~353

~o=.800 ~:.~o

.3

-I.2 -.6

(b)

\ _ _ _ . ~ ~,(*/.)

.15

Fig. 18. Predictions of stress-controlled axial cycling with ~0 = 0.353. (a) Axial stress-strain response; (b) axial-circumferential strain response.

dictions, the trx-ex hysteresis loops retain their shape, whereas in the experiments, they undergo significant changes during cycling. Clearly, the assumption that the material is cyclically stable, made throughout this study, is inappropriate for this set of experiments. The changes in the shape of the basic hysteresis loops affect both the direction

142 T. HASSAN et ak

-I

~ ~em

~ ., (i°) it 'y ',~

"'~ , i ,i i

~ % ~

I x ' Z ~ I -.5 "~

~ E xperiments - * - Predict ions

CS 1026 ~ o = 8 0 0

~,.~ : .150 Nr.~.= 45

i i i ~= . . . . ... -- .5 I 15 ~ . _ _ ~ . . . . . . . . ~ 12(3

Fig. 19. Mean axial versus mean circumferential strain recorded in each cycle in stress-controlled axial cycling experiments at different values of 00 (o,o --- values at every fifth cycle). Experiments and predictions.

as well as the rate at which ratcheting occurs. The effect of these stress-strain loop shape changes is even stronger in the other three experiments with ~0 = 0.210, 0.249 and 0.318. In these cases, the two mean stresses (fi0 and ~x,~) are such that they have a more pronounced "cancelling" effect on the ratcheting induced in both the ex as well as the e0 directions. The rate of ratcheting is significantly reduced in these experiments. As a consequence of this, the changes induced to the hysteresis loops play a more dominant role, demonstrated by the more significant changes induced to the direction of ratcheting shown in Fig. 19. In view of this, the predicted directions of ratcheting are significantly different from the experimental ones for these three cases. The cause of the changes undergone by the hysteresis loops must first be understood and followed by appropriate changes in the modeling before these predictions can be improved.

Figure 20 shows the predictions of the ratcheting experiment conducted under zero axial mean stress, 8xa = 0.800 and ~0 = 0.246. In this case, the pressure causes ratcheting in the negative ex direction. Also in this case, the effect of the changes undergone by the ax-ex hysteresis loops on the ratcheting is relatively small. As a result, this experiment was qualitatively and quantitatively (see Fig. 21) reproduced very well by the model used.

!11o CONCLUSIONS

This paper addresses the ratcheting behavior of 1026 carbon steel under two biaxial, cyclic loading histories. The first history involved axial, strain symmetric cycling of a


.45 (ka)

/ o ~ .~

---~+~(%)

-50

(~)

! I I I -L2 -.6

(b)

~'e '.9 (%)

.6

CS 1026 ~ =.246 ~,,, =.800

.3 ~ =0

I I 0 .6

~ __.__~ ~(%)

-.15

Fig. 20. Predictions of stress-controlled axial cycling with ~xm = 0. (a) Axial stress-strain response; (b) axial-circumferential strain response.

th in-wal led tube in the presence o f cons tan t in terna l pressure . The second h is tory involved stress con t ro l l ed axia l cycl ing and cons tan t in te rna l pressure .

In the first case, the presence o f in terna l pressure causes ra tche t ing in the c i rcumfer - ent ial d i rect ion. Fo l lowing an init ial t ransient , the mater ia l was found to exhibi t a con-


I -2

~ CS 1026 ~ ~0 =.z46

"~,,.,~, ~o:.800 ~ -, ~,,~0

,,, N~.~= 45

~ ~ x ~ +

-* - Pre6icfion ~ .

I I I -I.5 -I =5

---~xm(%)

.I.5

0

Eom (%)

,

Fig. 21. Mean axial versus mean circumferential strain recorded in each cycle for case in Fig. 20 (o ,o -~ values at every fifth cycle). Experiment and prediction.

stant rate of ratcheting. The rate of ratcheting is directly related to the value of internal pressure and the amplitude of the strain cycles applied. The three cyclic plasticity models outlined in Part I were used to simulate the experiments. The models were found to re- produce the major features of the material response, but yielded wrong rates of ratcheting. The rate of ratcheting was found to be very sensitive to the hardening rule adopted in the models. For example, the Prager-Ziegler hardening rule leads to arrest of ratcheting in the first couple of cycles. The Mroz hardening rule yields a rate of ratcheting that is much higher than those recorded in the experiments. These discrepancies are at least partly due to the simplicity built into the models adopted by requiring that the yield surface retain its shape and size throughout the loading history. Under these assumptions, the Armstrong-Frederick hardening rule, with suitably selected material parameters, was found to lead to reasonably good predictions of the rate of ratcheting.

The cyclic history used in the second set of experiments combines the features of the uniaxial ratcheting experiments and those of the biaxial ratcheting experiments described above. It was found to lead to ratcheting in both the axial and circumferential directions. The two ratcheting mechanisms were found to interact strongly through the Pois- son effect. The axial hysteresis loops traced in these experiments were found to exhibit significant changes in shape during cycling. These changes were found to be more pronounced than those observed in the uniaxial ratcheting experiments presented in Part I. The Dafalias-Popov two surface model with the same material parameters used in the other cyclic histories studied was used to simulate these experiments. The main features of the material responses were found to be adequately reproduced, but for some cases


the rate and direction of ratcheting were found to be influenced by the changes in the shape of the stress-strain loops observed during cycling.

The modeling used throughout this study was based on the assumption that the material is cyclically stable. As a result, the predictions of the ratcheting rate and direction were not satisfactory in all cases in the second set of biaxial experiments. Improved performance will require that the models be extended to include changes to the hysteresis loops introduced by the cyclic loading.

Acknowledgement-The test facility and data acquisition and control systems used in the experiments 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 the support of the National Science Foundation under grant MSM-8352370.

REFERENCES

1956 PRAGER, W., "A New Method of Analyzing Stresses and Strains in Work Hardening Plastic Solids," ASME Journal of Applied Mechanics, 23, 493.

1959 Zm~U/R, H., "A Modication of Prager's Hardening Rule," Quarterly of Applied Mathematics, 17, 55. 1961 ED~otqDs, H.G., and BEER, EJ., "Notes on Incremental Collapse of Pressure Vessels," Journal Me-

chanical Engineering Science, 3, 187. 1962 WooD, W.A., and BENDLER, H.M., "Effect of Superimposed Static Tension on the Fatigue Process

in Copper Subjected to Alternating Torsion," Transactions of the Metallurgical Society AIME, 224, 18.

1963 MOYAR, G.J., and Sn~ctMR, G.M., "Cyclic Strain Accumulation Under Complex Multiaxial Load- ing," Proceedings of the Joint International Conference on Creep, Institution of Mechanical Engi- neers, London, 2-47.

1965 BEmi~r~, P.P., "Some Observations on the Cyclic Strain-Induced Creep in Mild Steel at Room Tem- perature," International Journal of Mechanical Sciences, 7, 81.

1966 ARMSTgOSG, P.J., and FREDERICg, C.O., "A Mathematical Representation of the Multiaxial Bauschinger Effect," Berkeley Nuclear Laboratories, R & D Department, Report No. RD/B/N/731.

1966 Fg~trOESTrIAt, A.M., and RONAV, M., "Second Order Effects in Dissipative Media," Proceedings of the Royal Society, A 294, 14.

1967 RuIz, C., "H. igh-Strain Fatigue of Stainless-Steel Cylinders: Experimental Results and Their Appli- cation to Pressure-Vessel Design," Journal of Strain Analysis, 2, 290.

1979 CnAaocm~, J.L., DANG VAN, K., and CORDIER, G., "Modelization of the Strain Memory Effect on the Cyclic Hardening of 316 Stainless Steel," 5th SMIRT Conference, L, L11/3.

1979 HANCELL, P.J., and HARVEY, S.J., "The Use of Kinematic Hardening Models in Multiaxial Cyclic Plasticity," Fatigue of Engineering Materials and Structures, I, 271.

1979 PHILLIPS, A., and LEE, C.Wo, "Yield Surfaces and Loading Surfaces: Experiments and Recommen- dations," International Journal of Solids and Structures, 15, 715.

1979 SLIWOWSKI, M., "Behaviour of Stress-Strain Diagrams for Complex Cyclic Loadings," Bulletin de l'Academie Polonaise Des Sciences, Serie des Sciences Techniques, XXVII, 115.

1981 KAt~go, K., "Proposition of New Translation Rule in Kinematic Hardening," Bulletin of the JSME, 24, 9.

1985 HARVEY, S.J., TOOR, A.P., and ADKINS, P., "The Use of Anisotropic Yield Surfaces in Cyclic Plas- ticity," in Multiaxial Fatigue, ASTM STP 853, MILLER, K.J. and BRows, M.W. (eds.), p. 49.

1987 McDow~tL, D.L., "An Evaluation of Recent Developments in Hardening and Flow Rules for Rate- Independent, Nonproportional Cyclic Plasticity," ASME Journal of Applied Mechanics, 54, 323.

1988 McDowEzz, D.L., "Biaxial Path Dependence of Deformation Substructure of Type 304 Stainless Steel," Metallurgical Transactions, 19A, 1277.

1989 BOWER, A.F., "Cyclic Hardening Properties of Hard-drawn Copper and Rail Steel," Journal of the Mechanics and Physics of Solids, 37, 455.

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 )


n ~

~ x c -~"

~ x m ~

~0 ~--"

~.Om ~

eOp -~

~ =

~ 0 -~

~O -~

NOMENCLATURE a

normal to yield surface

center of yield surface in stress space

center of bounding surface in stress space

amplitude of strain cycles

mean of peak values of axial strain in cycle

circumferential strain

mean of peak values of circumferential strain in cycle

peak value of ~0

(-~ s.s) ~/~ circumferential stress

o0/o o,

3Additional variables defined in Part I.

ratcheting in cyclic plasticity, part ii: multiaxial behavior

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