Elastic-plastic memory and kinematic-type hardening

ELASTIC-PLASTIC

MEMORY

AND R.

J.

KINEMATIC-TYPE

HARDENING*

ASAROt

In this article it is proposed that the classical plasticity concept of kinematic hardening is much more general and descriptive of plastic deformation in metals than has heretofore been acknowledged. Furthermore it is shown how microstructure1 ideas involving dislocation micro-mechanisms do, when properly coupled to the classical continuum theories, prescribe not, one but several forms of kinematic hardening and that these forms are suggestive of strain hardening mechanisms on the micro as well as on the macroscale. Experimental illustration of two forms of this hardening are given. Finally, it is shown that kinematic type hardening involves a sort of plastic memory and that to describe such a memory state from dislocation theory a more complete description of dislocation densities than is normally used is required. MEMOIRE ELASTIQUE-PLASTIQUE ET DURCISSEMENT DE TYPE CINEMATIQUE On sugg&re dans cet article que le concept du durcissement ci&matique en plasticit classique est plus g&n&al et plus repr&entatif de la deformation plastique des m6taux qu’on ne l’a jusqu’8 p&sent reconnu. On montre de plus comment des consid&ations SW la microstructure impliquant des micromCcanismes de dislocations peuvent, lorsqu’elles sont associ6es convenablement aux theories classiques des milieux continus, faire intervenir non pas une, mais plusieurs formes de durcissement ci&matique, et que ces formes de durcissement font penser h des m6canismes d’brouissage 8.1’8chelle microscopique comme 8, 1’6chelle macroscopique. On donne une illustration exp&imentale de deux de ces formes de durcissement. On montre enfin que le durcissement de type cinbmatique met en jeu une sorte de m&moire plastiquc et que pour d&r& un tel &at B partir de la th6orie des dislocations, on a besoin d’une description des densites de dislocations plus complBte que celle qu’on utilise habituellement. ELASTISCH-PLASTISCHES

ERINNERUNGSVERMC)CEN UND KINEMATISCHER VERFESTIGTJN’GSTYP In dieter Arbcit wird dargelegt, daO das klsssische Konzept der kinemstischen Verfestigung in der PlastizitiLt vie1 allgemeiner und geeigneter zur Beeobreibung der plastischen Verformung in Metallen ist als bislang angenommen. Dariiberhinaus wird geseigt, wie Vorstelhmgen iiber die Mikrostruktur einschliel3lich der Versetzungs-Mikromechanismen bei richtiger Ankopphmg an die klassische Kontinuumstheorie nicht nur eine, sondern inchrera Erscheimmgsformen kinematischer Verfestigung vorschreiben, und daD dieee Formen Verfestigungsmechanismen sowohl im Mikro- als such im Makrobereich andeuten. Beispiele fti zwei Formen dieses Verfestigungstyps werden angefilhrt. SohlieBlich wird gezeigt, daB die kinematische Verfestigung eine Art plastischen Erinnerungsvermiigens umfa&. Urn dieeen Zuatand versetzunptheoretisoh zu beschreiben, miissen die Versetzungsdiohten vollstiindiger bekannt sein ab iiblicherweise benutzt.

1. INTRODUCTION

The direct relationship between the details of microst,ructure and the mechanical state of crystalline solids has been the subject of a great deal of discussion in the recent literature on plasticity. In particular, it is generally believed that aside from lattice frictional stresses and the like, the mechanical state of a crystalline body is determined by ita microstructure, including dislocation, precipitate and second phase distributions as well as by the details of the micromechanical processes involved with plastic deformation ; the latter in turn depend quite specifically upon microstructure. It is &so known that with plastic straining t,he substructure (in particular the dislocation content), and hence the mechanical state, changes. In this way ‘microscopic’ theories are theories for the flow stress (i.e. the stress to impose a given uni-axial strain rate) and are often theories for strain hardening (i.e. t,he increase in flow stress wit.h plastic strain at a prescribed strain rate). Just what these dislocation substructures are, on the other hand, is not necessarily * Received January 21, 1975; revised April 8, 1976 t Scientific Research Staff, Ford Motor Company, Dearborn, Michigan, U.S.A. ACTA

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well known and so the microstructural theories have proceeded on the basis of models for the microstructure with very little direct help from the mechanical data..$ The dislocation ‘forest’ is one such dislocation model which attempts to account for dislocation-dislocation intersections and hardening; the major experimental guide for quantitative analysis is electron microscopy which measures a scalar p, the dislocation density. p is actually a measure of dislocation line density which prescribes an isotropic kind of hardening and is unsuited for describing anisotropic effects some of which are discussed in this article. At present no such suitable structural quantity exists for describing anisotropic effects. Furthermore, we note that the mechanical data aligned with the microstructural theories is obtained almost exclusively from uni-directional stress-strain tests from which it is impossible to separate isotropic and anisotropic contributions to strain hardening. For these remans we conclude that there presently exists no adequate theoretical description of the dislocation substructure and hence for the mechanical state. t: Some latent hardening studies of, for example, Sharp and Makin’l’ or Basinski and Jackson’z’ are notable exceptions.

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Continuum plasticity theories on the other hand, while they have generally paid less attention to, or even ignored, microstructural details are mathematically more complete in their descriptions of t,he mechanical state, They contain the idea of strain hardening in the form of, for example, Hill’s isotropic hardening 1aw,t3) F’rager’s kinematic law@’ or more recently, Mroz’s ‘field of work hardening moduli.‘(5)t They are also concerned with describing much more complex multi-axial loading and straining histories including both isotropic and kinematic effects. The notion of a mechanical state determined and fixed by strain history is, in these ‘macroscopic’ theories called ‘memory’ which brings us to an interesting point that forms the basis of this article. Memory in the microscopic theories has, in the past, implied reversibility as in ‘shape-memory’ phenomena discussed by Bakercab and by Nagasawa et uZ.(~) In plasticity two very striking forms of this kind of memory appear in the reversible twinning of Fe-Be alloys discussed by Bolling and Richman or in the anelastic shape recovery of polymeric materials during fatigue studied by Rabinowitz and Beardmore.ce) We thus link memory with reversibilit,y; elastic behavior is a perfect memory. The existence of memory in solids that plastically strain by dislocation motion is then associated with dislocation substructures that are reversible, at least in their strain hardening import. It is then the purpose of this article to discuss plastic memory from the standpoint of micro-mechanics and to show that, when properly coupled to the microscopic theory, the continuum theories embody concepts of plastic memory in the herein adopted sense. This is important because it provides an expressive theoretical link between the mathematically sophisticated continuum theories and the physically based microstructural theories. In particular, it will appear that the notion of plastic memory prescribes at least three types of kinetic hardening with which we can evaluate strain hardening. Experimental examples of two of these will be presented and discussed in the light of specific micromechanical models. An a.ttempt to establish the generality of the ideas will be made on the basis of dislocation theory and with the aid of other data. Finally, the relationship of memory phenomena to recent phenomenological theories for the plastic equation of state will be discussed. It is the contention here that studies of memory, as detined here, will provide valuable information concerning deformation substructure and the mechanical state. t Mroz’s “field of work hardening moduli” is a general form of kinematic hardening.

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2. KINEMATIC

HARDENING

AND

MEMORY

For the present, the discussion of the continuum theory for plasticity and kinematic hardening will view plastic deformation in the time-independent idealization. It is then possible to introduce the concept of a yield function F(u) where Q is the stress tensor. We will see shortly, however, that, this leads to some difficulty concerning what we call plastic strain. Changes in the yield function with plastic strain, representing changes in mechanical state, involve various combinations of expansion, translation and distortion. We are here most concerned with expansions (isotropic hardening) and translatiohs (kinematic hardening) and so the yield function fits the relation F(o

-

5) -

h(l) = 0,

(1)

where 5 is a tensor describing translation in stress space and h is a measure of expansion; 1 is a scalar function of plastic strain. A simple mechanistic interpretation of kinematic hardening is to imagine that strain hardening is caused by the development of essentially long range residual stressesor, inthenomenclature of the microscopic theories, back stresses. c is then a ‘rest stress’ as discussed by Ric@) or a back stress as discussed by Seeger.(ll) As it translates by the residual stresses, the yield surface retains its size and shape and within the scope of the microscopic theories we may imagine that this means that the various micro-mechanical processes for slip are unchanged and only the net resolved stress on a dislocation segment is changed. Let us now specialize to the case of uni-axial tension-compression so that we can use data obtained by cyclic deformation test,ing. Since in this case kinematic hardening involves the motion of the yield surface symmetrically up and down the stress axis we can say that there exists a sort of memory in that during the reversal of a forward strain we recover the initial state of hardness in the interim before reaching an equivalent strain-hardened state in the reverse direction. This is shown schematically in Fig. 1. To understand the microstruct,ural implications of this we may proceed as follows. The

FIG. 1. Schematic illustration of the relevant segments of a yield surface before and after uni-axial straining.

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strain rate, d, and temperature, T, are fixed and we write that the applied stress, o, required to impose this strain rate is a function of the developed internal structure, i.e.

u = ~(a, j3,y,. . . ,); E, T fixed.

(2)

a, B, y, . . . , are presumed to be some suitable set of internal variables describing the microstructure. E is plastic strain. It is important that we do not write Q = (T(E) since plastic strain, not a state quantity itself, cannot be a measure of a state quantity such as the mechanical state. Presumably we might seek to write an additional relation of the form da, d/A dy, . . . , is a function of de

(3)

and, if it is required, be concerned with whether or not an inverse relation exists. Current knowledge of micro-plasticity would of course have us rewrite equation (3) so that it reads L, 899, - * - , is a function of 8. T, a, 8, y, . . . ,

(4)

where a dot indicates the time derivative and T is the temperature. Some important consequences of relations like equation (4) have been recently discussed by Riceoo*rz) in connection with the plastic potential. Equations (2 and 3) are consistent with the concept of a mechanical equation of state as set forth by Zener and Holloman in terms of plastic strain, i.e. CT= U(E, 8, T).

(5)

Equation (3 or 4) would relate to the integrating factor in Hart’s(l~) very recent adaptation ‘of the equation of state in the following way. Hart’s theory interprets equation (5) as meaning that there exists a unique relation between stress and strain rate at a given value of plastic strain. Plastic strain is, by virtue of equation (3 or 4), equivalent to structure. Now with reference to the way that equation (2) is written we may imagine that the structures form in the sequence a, j3, y, . . . , and that the yield surface translates according to equation (2). If the material has memory, CC,p, y, . . . , are recoverable with reverse plastic straining, and we can show how various forms of kinematic hardening result. We will for convenience consider the three most likely types which we call KI, KII and KIII. Kinerndic type I The first type of kinematic hardening can be found by taking a very old view of polycrystalline deformation due to Masing. We view a solid as being composed of a set of n elastic-plastic elements as shown in Fig. 2(a). Each element is rigid plastic, symmetric in tension and compression, of the same dimensions, 8

FIG. 2. (a) Model structure composed of elastic-rigid phtio elements; (b) Stress-strain behavior for model elements in (a); (0) Composite stress-strain behavior for structure shown in (a).

but each has a different yield point (Fig. 2b). We want to derive the fully reversed strain hardening curve. In Fig. 2(c) we imagine that the structure of Fig. 2(a) is loaded in tension past a stress level of u1 at which point the first and weakest element yields ; the loading continues up to u2 where the second element begins to yield. We can oompute the residual stresses on element 1 upon unloading by noting that the total load on the frame is u, Area while the total load on element 1 is a, - Areal. To unload we apply -~a +Area and since Area = n - Areal, we find the residual stress on 1 equal to -(a, - ul) while the residual stresses on each of the other n - 1 elements (all elastic at this point) is (a, - uJ(n - 1). If we were to continue forward loading to ua the residual stress on element 1 would be -(us - ur), on element 2 would be - (aa - ua), etc. For this case reverse plastic straining would begin at -bR wherelu,l = 01-t 02 - 63. This construction, consistent with an original model discussed by Masing( shows that if the forward curve has a form u = j(e) then the reverse loading curve has the form u = 2j(+). It is easy to see that this same reverse flow law is obtainable from Mroz’s ‘field of work hardening moduli.‘@) In this the yield loci are viewed as a set of concentric surfaces. Initial yielding takes place when the first surface is reached with plastic moduli hr. The inner surface translates until it makes contact with the second at which point the plastic modulus is h, and both surfaces translate with plastic strain until the third surface is met. If we view a, p, y, . . . , the microstructural variables determining the flow strength, as being the plastic strain in the various elements then we have the situation where recovery (memory) has the sequence 2a, 28, 2y, . . . . In other words element 1 deforms l

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plastically twice as much in compression before 2 yields 8s it did originally in tension; likewise for elements 2, 3, etc. Furthermore the above model implies that (&r/aa) is not a function of/l, y, . . . , and likewise for ~~u~~~~,(~~~~), . . . . To give further microst~~~al ~te~retation to the elastic-plastic elements discussed above we may say that they represent the individual grains in a random polycrystaiiine aggregate. Following along the lines of the slip theory of Budiansky and WU(*~)those grains most favorably oriented for slip are compared to the weakest element. The internal stresses that cause strain hardening are produced by the pile-ups of glide dislocations at the boundaries and those grains that ,slip first experience the largest residual stresses For this reason as well 8s their favorable orientation to the stress axis they are the first to deform during unloading end reverse io8ding. An entirely similar situation results if the elements are identified as dislocation cells which 8re known to form in fatigued or uni-directionally strained metals. A theory based on the intern81 stresses formed within the cells would only then require that the cell walls, which emit dislocations, are not all the same and hence the critical stresses necessary to emit dislocations differ. This difference is needed so that the individual cells have stress--strain relations similar to Fig. 2(b). The two models discussed in the above paragraph bear resemblance to the el~tic-pl~tic element model of Fig. 2 since both contain the idea that plastic ffow isinhomogeneous and hence there are regions which deform plastically (viz. weak elements or grain or cell interiors) and regions which deform ela,stically (viz. strong elements or grain or cell boundaries). The elastic regions support stresses which are, upon unloading, transmitted as residual stresses to the plastic regions. The curve CT= 2j(&) contains a Bauschinger effect due to the residual stresses. Such Bauschinger effects are found in single erystalsu” without cells and thus where the hard regions are less obvious. All of this suggests that a more general approach to the internal stresses developed by inhomogeneous plastic flow is required. In particular, Asaro(38) has shown how a dispersion of dislocation glide loops, which delineate slipped from unslipped regions, gives rise to such long-range stresses. The resulting stresses follow directly from the form of stress field caused by a closed loop as opposed to a straight dislocation. By considering closed loops the effects of the body’s free surfaces were included. Models using this idea need specify exactly what structural feature delineates the elastic and plastic regions; i.e. what determines the size, shape and orientation of the loops. Regardless of

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these fine details it should be recognized that any loop density of this type necessarily gives rise to residual stresses and kinematic hardening effects. It is interesting, at this point, to recall the strain hardening model of Marcus et tzZ.(r81 which is based on the Bilby et a&(1*)analysis for pI~ticity near a crack tip. Marcus et csl. assume that plastic strain is developed within sheas bands (pileups) which form against the friction stresses of the slip planes. With reverse loading they arbitrarily fix the existing pileups and form equivalent pileups of the opposite sense. The strain hardening 18w they obtain is non-linear and is of 8 general enough shape to fit a good deai of stress-strain data. Their model indicates that the internal stresses of the pileups sse recovered in the sequence 2a, 2& 2y, . . . ; the first structures to form in the forward direction 8re the first to be recovered in the reverse direction. Examination of their equations (5-l@, especially equation (la), reveals that their reverse loading curve is of the predicted form for KI hardening. That is, we can write for their case, that if in the forward direction E = f-l(u) then, in the reverse direction )E = j-l(&). Finally, we note that type KI hardening is quite general. Figure 3 shows examples of four polycrystalline alloys tested in cyclic deformation. In each c8se both the KI and an isotropic idealization has been drawn to be compared with the experimental curve. The kinematic is computed from the forward curve, o =j(e), by writing $0 =j(+f. Figure 3(a) is for a commercial aluminum alloy, 5086-F, tested in the as received condition. Figure 3(b) is for the superalloy Nimonic 80A which was well annealed for 9 hr at 108O’C and water quenched to surpress precipitation of the y’ phase. For both these materials we would conclude that KI hardening describes the initiation of reverse plastic strain quite well. The overall hardening is, though, somewhat isotropic in that both materials undergo cyclic hardening. Figures 3(c and d) are for a Cu-20 Zn alloy precipitation hardened with Cr. Figure 3(c) shows the initial response which is similar to those just discussed. Figure 3(d) shows the fully reversed behavior of the same brass sample of Fig. 3(c) after 240 strain reversals. The hardening is now almost perfectly KI type. The point here is that a very wide class of alloys show strain hardening that is as much kinematic as it is isotropic and that for cyclically stable materials the strain hardening is nearly pure KT type. ~~~e~~~c ~urde~j~ type II Other iypes of hardening can be envisaged by changing the sequence of recovery operations. For

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FIG. 3. Fully reversed &as-strain response of four polyorystalline materials shown for comparison with KI and isotropic hardening. Curve 3(a) is due to Sherman .(*O) Test frequenoies were 8ll about 0.016 cps.

example, imagine the sequence of structural operations a, 8, Y, . . . , being reversed as a, B, y, . . . . In this case the f&t structures to form are the tirst to relax but do so only until they disappear. The reverse curve is shown schematically in Fig. 4 along with the KI curve. The hysteresis loop thus obtained is now inflected about the origin and the area within the loop (the plastic work) is reduced.

substructural recovery on the microscale and so, aside from the lattice frictional stresses (point A in Fig. 4), such a material behaves much like a non-linear elastic solid. 3. EXPERIMENTAL

A detailed experimental program was carried out on a material chosen with the hope that memory efleots would be greatly enhanced. From simple geometric

Kinematic hardening type III A still more perfect form of recovery or plastic memory can be constructed by adopting the recovery sequence . . . y, /3, a,. Such recovery occurs by the law that the last structures to form are the first to relax and, like KII, the structures relax until they are gone. The total plastic work in a given strain cycle is minimal since, like KII, there is no stored energy in going through half a strain cycle and reverse straining takes optimum advantage of the internal stresses. The latter statement holds true since au/as > in the example shown in Fig. 4. au/a@ > aajay... One other interesting feature of types KII and KIII memory is that at zero plastic strain (point F’ in Fig. 4) the magnitude of the flow stress is that of the initial yield point, A. KIII memory is suggestive of

4 STRESS

FIQ. 4. Fully reversed stre~strain response for hypethetioal materials showing types RI, KII or KIII memory. For chbrity the str8in hardening has been approximated by a series of straight lines.

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considerations one would expect that planar glide configurations and debris would produce pronounced memory effects; since the dislocrttions all lie on their slip-planes they are able to retrace their paths during reverse straining. The original Orowan idea for particle hardening, adopted by many for a description of oxide dispersion hardening provides a model system for plastic memory. The particles provide the required stabilizing influence for planar debris such as Orowan shear loops. To promote planarity further the matrix should have a low stacking fault energy. For these reasons it was decided to study the dispersion hardened superalloy In-753 which has a composition Ni-2OCr-2.5Ti-1.5 Al-0.05C by weight. The matrix is dispersion hardened with spherical Y,O, particles comprising 3.1 ok vol of the material. Quantitative stereology performed on extraction replicas revealed that the average particle edge-on spacing was 5650 A and the mean dia of the particles in the plane was 1500 A. The recrystallized grain structure showed highly elongated grains; the direction of elongation being (100). X-ray analysis verified that the samples had an almost perfect (100) wire texture. Samples were tested in cyclic tension-compression deformation at room temperature. The specimens were hour-glass shaped with treaded grips; the dimensions of the grip portions were & in. long with +20 threads. The gauge section had a roughly 0.475 in. radius and was 0.10 in. dia at the neck. Strain was measured diametrically on most of the specimeds although a few larger samples were tested Both the with longitudinal strain measurement. loading rate and total logitudinal strain rate were controlled, but since no significant difference in behavior was observed, only tests for controlled loading rate will be presented. The test frequency for the dat’a to be shown was 1 cycle/min. Plastic strain amplitudes were fixed by controlling the loading ram (MTS 2.5 kip capacity closed loop) with an analog computer programmed to compute longitudinal elastic and plastic strains from the diametral measurements. For most of the tests the plastic strain amplitudes were symmetric in tension and compression. Other test variables included beginning the tests in tension or compression, and plastic strain amplitude. Typical results for the first half cycle are shown in Figs. 5(a and b) along with attempts to fit the data to t,he theoretical curves based on KI, KII or KIII memory. At these strain amplitudes the material evidently displays a very pronounced memory which, noting the inflections in the curves, is either type KII or KIII. Both theoretical curves display rather poor fits to the data at the end points which may be

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(b) Fully reversed stress-plastic strain response for dispersion hardened Nimonio BOA. Also shown by dotted lines BIB the theoretical memory responses.

FIG.

6.

improved by the following reasoning. Both KII and KIII memory assume that the initial yield stress acts as a friction stress so that reverse plastic flow begins only when the reverse stress reaches point C in Fig. 5(a). However the initial yield stress (point A) is associated with the critical forces necessary to (i) develop mobile dislocation groups on the slip planes and (ii) overcome, by the Orowan process, the oxide dispersoid. Both barriers are reduced to some extent in their effectiveness ; the particles themselves are screened from the glide dislocations by, for example, the Orowan loops. The initial Orowan stress in this material can be estimated to be about 9 x IO3psi. Both the curves KII and KIII should then be lowered by at, least this amount ; this yields KII’ and KIII’. It is also evident from the figures that there is a large amount of recoverable strain upon simple unloading. This anelasticity produces a correction that is simulated by

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the segment ‘An.’ Irreversibility in the 5ow would necessitate increasing the 5ow stress above the curves KII and KIII; this is represented by segment ‘IF.’ Visual inspection also shows that segment DE is the trace of AB. Although both KII and KIII memory show a good qualitative fit to the data, type KIII is better quantitatively, especially at the point of initial reverse yielding. We conclude then that the observed memory is an imperfect KIII type. Tests started in tension produced results quite similar to those shown for the compression first tests shown in Fig. 5. There is a difference in that recovery of the original yield stress at 0 plastic strain is more complete for the compre~ion 5rst tests. This effect, which appears to have some generality with regard to the Bauschinger effecP) is shown in Fig. 6.

renrton FIG. 6. A comparison between reverse and forward straining in compression first and tension first tests.

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Fro. 7. Fully reversed stress-&rein response for dispersion hardened Nimonic 8OA at a plastic strain smpiitude of 0.03. The KIII memory is now suppressed and the behavior is more KI type.

than 0.04 the additional hardening is apparently more isotropic and non-recoverable. With continued cycling the memory also changes. To begin with we should note that even up to plastic strain amplitudes of 0.04 there is relatively little cyclic hardening. The effect is shown in Fig. 9 where various cyclic stress strain curves have been defined. Up to plastic strain amplitudes of 0.015 the maximum hardening occurs after about 3 reversals but the hardening is only cu 5 per cent of the initial yield. Furthermore, the hardening after eaoh half-cycle is never more than 20 per cent of the forward strain hardening increment. After approx 10 cyoles,t this material was observed to then soften as indicated by the 175 cycle curve. Such hardening followed by softening is often attributed to unstable microstruoture that initially raises the strength of the material. In the present case the instability has not been positively identified although there is reason to believe

0

The memory effect changes as a function of strain amplitude and with cycling. Figure 7 shows a fully reversed stress-strain curve for a longitudinal plastic strain amplitude of 0.03. At this amplitude the KIII type memory is less evident; the memory that exists is more like KI although some of the hardening is isotropic. In Fig 8 the composite effect in strain amplitude is shown. We plot the reverse plastic strain, as measured from the end of the forward prestrain, required to raise the reverse stress so that it is equal in magnitude to the initial yield. As discussed with reference to Fig. 4, KIII memory yields a straight line of slope 1. Figure 8 shows that up to 0.02 longitudinal plastic strain amplitude (approx 0.035 shear strain) the predictions of KIII memory are closely followel. For strain amplitudes larger

I- 4t a 6 3g; 82 p”

=o

I2

3

PLASTIC

PIE-STRAIN

4 1%)

‘P

Fm. 8. Reverse plastic strain required to raise the flow stress to the vrslue of the initial yield is plotted a% a function of plsstic pre-strein. The predictions of KID memory &xc followed up to strain amplitudes of cdrnost 2 per cent. t At plastic strain amplitudes of around 0.63 or 0.04 the the hardening is evident for only one reversal before softening OOr,WS.

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Fro. 9. Cyclio stress-strain curves defined for 3 and 175 oycles intended to show the emall extent of ayolic hardening followed by softenin&

that during the solution treatment and quench some y’ precipitation is inevitable.@s,ae) y’ strengthened alloys are known to cyclictllly harden and then soften(“); the explanation offered is the shearing of the precipitates. Figure 10 shows that after 40 cyoles the memory changes from KIII to essentially KI type.

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develop in a very general way whenever there are net plastic strain ~omogeueities. The oxide dispersion in a plastic matrix presented a model experimental system for such elastic-plastic incompatibilities. Even so, there were two types of memory displayed, KI and KIII type. The KU1 memory was attributed to dislocation reversibility on the microscale. The K.I type can ~ntatively be attributed to the long-range stresses developed within the dislocation free regions between dislocation wall structures. In what follows we disouss explicit models for these two types of memory in this system and close by proposing a new approach for the description of the mechanical state. waodetfor KIII beh&or

A. A ~~cr~~~u~~

Our microstructural model for the initial dislocation structures responsible for strain hardening and memory is shown in Fig. 11. Slip is thought to begin by

4. DISCZTSSION

In the present article we have dealt with the question of recovery of strain hardening. We heve shown that both the continuum theories for plasticity as well 8s the ~eros~et~al theories embody the notion of recovery 8nd that there is a simple and natmal link between the two which is b8sed upon the idea that the microstructure determines the mechanical St&e. Studies of memory in complex materials and chiaraoterization of memory should & of extreme value in choosing models for the complex flow proCesae%. For example, we have shown that plastic memory is 8 natural outcome of strain hardening theories that attribute the hardening to long-range stresses. Furthermore, long-range stresses have been shown to

m

‘Jnrlipped

(b)

.

lipped

Q

fd

FIG. 11. Schematic illustrretion of initial slip processes including nucleation and growth of sheer zones end the Orowan bowing process.

FIG. 10. Fully reversed stress-strain response for dispersion hardened Nimonic 8OA at a plastic strain amplitude of I per cent. The memory is shown to shift from KXII to KI type with continued cyaling.

the expansion of shear zones or dislocation loops and by the bowing of dislocation segmeuts 8bout the dispersoid. Thus the initial yield strength is associated with (8nd herein identitied with) the critical forces to begin the bowing process and to create a mobile population of glide loops. The bowing force has already been estimated; there is presently, though, no adequate model for the rest of the “yield force. Once the bowing process has begun, subsequent dislocations gliding past particles will leave shear loops around the parficles as envisioned by the Fisher, Hart aud Pry model.(s) These loops contribute to

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both t.he long range back stresses, di8icussedin Section 2 and to t-helocal stress field whichraises the bowing force. For a quantitative treatment of the hardening associated with an increased bowing force we can refer to the work of Brown and Stobbs(@*’ or to that of Hart.‘@@) In particular Hart? showed, that the critical force followed a relation of the form T = 1 + y(M + &W),

(6)

where y is a constant and the quantity Jf is pxoportional to the plastic strain. Hart’8 analysis was very approximat,e in that he chose simplified form8 for the shear loop distribution and the loop stress field. Brown and Stobbs’ results did not include a detailed account of the bowing process. Since the functional form of equation (6) (at least in strain) is consistent with experimental data it is important to know if similar models which incorporate more detailed loop geometry and loop stress field8 are also conei8tent. Recent ca1cu1ation8~@*~ have, in fact, shown that this is entirely the case ; equation (6) is very general and can account quali~tively for the non-linear strain hardening. We do not, at present, mean for the model to be too quantitative-the reason is that to describe a strain hardening curve for room temperature flow is not conclusive proof of any model. Similar flow curves, consistent with experimental data, have been derived by Ashby[@@)who used the well known dimensional description relating flow stress to the square root of dislocation density. The description of memory is a much more stringent test of a model. Another microstructure1 feature of our model for part of t,he memory effect, that is not dependent on the presence of the particle 8hear loops, are the glide dislocations, or actually glide loops, distributed on the glide planes as suggested by Fig. 11. These pile-up8 exert local back-stresses that oppose further glide and, ideally, behave like a dispersion of small viscous zones. This idea is similar to one proposed by Zenert30) for anelasticity in shear bands. The possibility of significant inelastic strains stored in these zones can be explored by using Eshelby’sc+i) result for elliptical pile-ups. The symmetrical glide strains stored in a single circular pile-up subjected to the resolved stress r is given by (7) where S is the number of pile-up8 per unit volume, c is the dia of a pile-up,? and 17= n(2 - ~)/4( 1 - Y); t The parameter c is taken a8 roughly equal to the mean edge on particle spacing, 6650_& as measured from extraction replicas.

AND

KINEMATIC

HARDENIXG

1263

B is Poisson’s ratio and p is the shear modulus. The isotropic shear modulus is calculated from the longitudinal and diametral data to be 11.4 x lo6 psi. A reasonable ratio of 7111 would then be lo-* and equation (6) then yield8 8 w 1O-16,N. If there is on the average a pile-up for each particle (as there would be if the particle8 were arranged in a square threedimensional array) N w lOI@ and & w 10Vs. In fact since the error here can be a factor of 5 we could argue that nearly one half the reversible strain can be stored in pile-ups. According to this model and equation (7) the strain hardening would be a linear function of strain with a fixed number of dislocation8 on the pile-up. As the arrays collapse during reveme etraining, and if they do 80 ideally, they would give rise to a KU1 type memory. Furthermore, the pile-up8 tend to concentrate the applied force on the leading bowing dislocation segment thus facilitating the Orowan mechanism. For example, with the above numerology we e8timate the number of dislocations, n, in a typical pile-up to be cu 2 and this would inereaae the applied force on the particle8 by essentially the 8ame factor. We should al8o note that with T/P m lOma, r w BY psi which is about the same value as the 9 x 103 psi computed for the Orowan strees. It would appear then that our original picture for the early stages of yielding and flow in this material, depicted in Fig. 11, is entirely reasonable. Pile-ups or dispersions of local shear bands seem capable of storing plastic strains of magnitudes up to several times 10V3 while the rest of the KIII memory and strain hardening is cau8ed by other planar structurt?the Orowan shear loops. It is entirely likely, however, that in the later stage8 of pla8tic straining (after cu 2 per cent longitudinal pla8tic strain) the dislocation structures do become more three-dimensional. This can occur by the crossslip mechanism discussed by Hirsch and Humph. reys(*@) or more recently by Brown and Stobbs.c@@) Hirsch and Humpbreyscaa) have derived a flow curve using their model for prismatic loops formed during the cross-slipping operation. Their curve, like all other model curves, is consistent with their data. It would appear then that at the small plastic strains considered in the pre8ent work the predominant strain hardening comes about from highly anisotropic defect structures such as shear loops that are reversible and that are consistent with the observed memory. Beyond uni-directional plastic strains of ca 0.04 strain hardening is irreversible and is consistent with forest models or other model8 for non-recoverable strain hardening.

1264

ACTA

B. A miwostrwtwal

METALLURGICA,

model for KI behavior

At present there is no complete model for the plastic flow process in cyclically stable materials and such a development is beyond the intent of this article. The memory behavior does, on the other hand, provide a considerable guide for evaluating any tentative proposals. For example, we note that it is generally observed that cyclically stable materials develop dislocation cell struatures comprised of dense clusters of dislocations making walls that enclose relatively dislocation free regions. In fatigued dispersion hardened materials, single crystal studies have shown that the cell structures are replaced with more two-dimensional wall structures which form at or near the particles.@~~~) Dislocation motion is believed to be confined to the regions within the cells or between the walls. The walls then serve as the elastic regions described in Section 2. Hence the essence of a model for plastic flow would be that there exists (for whatever reason) a difference in stress at which flow begins in the various cells. Adjacent cells undergo different plastic strains and the walls themselves undergo little plastic strain. There must then develop residual stresses within those cells that plastically deform the most. These residual stresses constitute the strain hardening. Reverse deformation will not be the exact trace of the forward flow but rather the first cells to deform will be the first to relax-this leads to KI behavior. Any more detailed microstruotural theory would need to contain this idea to be consistent with the data. (C) A dislocation

deeption

for the mechanic& state

We now note some further ~pli~atio~ of the memory phenomena to fundamental theories for the description of the mechanical state. Although it is generally thought that the mechanical state is determined by subst~ct~, e.g. dislocation substructure, there is presently no complete description of dislocation distributions, at least complete enough to describe multi-axial deformation or memory phenomena. Recent microstructural analyses for dispersion hardening(z7) and strain hardening(a) have made use of the scalar quantity, p in describing the dislocation subst~ct~ and these treatments rely on the notion that dislocation-dislocation interactions produce strain hardening. The increase in flow stress derived from these ‘forest’ type models is isotropic at least in a glide plane which cuts the forest. This type of model cannot then describe memory or account for other phenomena such as plastic anisotropy or cyclic stability. The analysis of residual stresses produced by a ~tribution of shear loops, such as previously

VOL.

23,

1975

discussed or as discussed by Brown and Stobb@@F for dispersion hardening, suggests that consideration of dislocations as closed loops must be explicit in a ~nd~en~l ~slocation theory for plasticity. For example it is quite impossible, on dimensional grounds, to deal with free surface effects by treating dislocations as straight lines as in the forest modeis. A proper description of the dislocation distribution would therefore need to include a loop density tensor to describe all the long-range internal stresses. This in turn suggests that a more realistic description of the dislocation distribution be developed from the spatial moments including line, dipole, loops, etc. densities rather than a siugle parameter idealization. Eaoh moment is determined by the order in which the stress field falls off with distance from the defect. A single loop, e.g. gives rise to isotropic hardening in a glide plane cutting through it by vktue of its line moment and to anisotropic hardening in its own plane caused by the loop stress field and the free surface effects. Planar loops, uncomplicated by tangling with other dislocations, give rise to reversible flow and KIII type memory. The classical notion of kinematic hardening seems quite closely associated with the dislocation loop density and the details of this relationship constitutes the subject of continuing experimenta and theoretical research. 5. CONCLUDING

REMARK

It should be noted that all of our discussion about memory and the mechanical state was not in anyway limited to dispersion hardened systems. In fact preliminary work has also established the existence of KIII memory in vacuum cast Fe-9 wt. % Al polycrysstals and electron microscopy has confirmed that these Fe-Al alloys are not internally oxidized nor contain any substantial substructure aside from grain boundaries. ACKNOWLEDGEMENTS

It is with great pleasure that I acknowledge the discussions and assistance of R. W. Landgraf and F. D. Richards, the comments of C. L. Magee, and the communications with L. M. Brown. All were most helpful in the development of the present work. REFERENCES 1. J.V.i%uwandM.J.M~~~~,Cun.J.Ph~8.445,519(1967). 2. 2. S. BASINSKI and P. J. JACKSON, Phya. StatzapSoEidi @, 806 (1985). 3. R. XII&L, The ~~~~~~~c~ Themy of PZ~~~j~y. Oxford University Press, London (1950). 4. UT. PBAOER, Proc. In&. Mech. Engrs 169,41 (1955). 6. 2. MBOZ, J. i%cJb. Phys. solid8 15, 163 (1967). 6. C. BAPER,M~LSC~. J. 5,92 (1971). 7. A. h'aoAsAwA, K. ENAMI, Y. ISHIXO, Y. ABE and S. NEXWO, ScriptrsMet. 8, 1055 (1974).

ASARO 8. ;~$BOLLING

: ELASTIC-PLASTIC

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23. P. BEARDMORE,Mater. Sci. Engng. 5, 350 (1970). 24. C. E. FELTNER and P. BEARDMORE, ASTM, STP 467 (1970). 25. J. C. FISHER, E. W. WART and R. H. PRY, Bctn Net. 1, 336 (1953). 26. E. W. HART, Acta Mel. 20, 275 (1972). 27. M. F. ASHBY, Phil. Msg. 14,1157 (1966). 28. P. B. HIRSCHand F. J. HUMPHREYS,Proc. R. Sot. A318, 45 (1970). 29. L. M. BRO~T and W. M. STOBBS, Phil. Mag. 2& 1185 (1971). 30. C. HEXER,h’.?asticity and Anelasticity of Metals. University of Chicago Press, Chicago (1948). 31. J. D. ESEELBY, Phys. Status Solidi 8, 3057 (1963). 32. W. M. STOBBS,D. F. WATT and L. M. BROWN, Phil. Mag. 25, 1169 (1971). 33. 0. T. Woo, B. RAMASWAMI, 0. A. KUPCIS and J. T. MCGRATH,Acta Met. 22, 385 (1974). 34. M. DONER, H. CEANO and H. COXRAD, Met. Trans. 5, 1383 (1974). 35. R. DEWIT, Solid St. Phye. 19,249(1960); or V. VOLTERRA, Annla. Ecole Norm. Suner. 8. 24. 400 (19071. 36. J. D. ESHELBY, J. appj. P&s. $5,255: (1964). 37. R. J. As11~0, Int. J. Engng Sci. lS, 271 (1975). 38. R. J. ASARO, Rept. SR-75-7, Ford Motor Company (1975).