Work hardening and dynamical recovery of F.C.C. metals in multiple glide

A& meralf. Vol. 37, No. 6, pp. 1521-1546, 1989 Printed in Great Britain. All rights reserved

Copyright 0

OVERVIEW

GQol-6160/89 $3.00+0.00 1989 Pergamon Press plc

NO. 82

WORK HARDENING AND DYNAMICAL RECOVERY F.C.C. METALS IN MULTIPLE GLIDE

OF

F. R. N. NABARRO Condensed Matter Physics Research Group. University of the Witwatersrand, Johannesburg, PG Wits 2050, South Africa and National Institute for Materials Research, CSIR, PO Box 395, Pretoria 0001, South Africa (Received 18 Ju/_v 1988) Abstract-The work-hardening processes in single crystals in symmet~cal orientations, in single crystats in “single slip” in Stage III, and in polycrystals, seem to be similar, and we attempt to characterize these processes. Hart in 1970 defined the conditions under which the internal state could be described by a single “hardness” parameter, and we discuss some problems arising from his analysis. Both Hart and his colleages [Constitutive Equations in Plasticity, M.I.T. Press, Cambridge, Mass. (1975)] and Mughrabi [Strength c&Metals and Alroyf (KS&fA J), Pergamon Press, Oxford (1980)] found one-parameter models too restrictive, and developed two-parameter theories. Hart’s theory is essentially phenomenolo~cal, and emphasizes the importance of the strain-rate sensitivity. It offers only a very simpli~ed atomistic model of sfip zones embedded in an elastic matrix. Mughrabi develops a detailed ceflular dislocation model with hard cell walls and soft cell interiors, but the theoretical discussion is confined to the behaviour at zero temperature and an intermediate strain rate. The present review has three aims: (if to provide an atomistic basis for some of the empirical results of analyses based on Hart’s theory; (ii) to suggest a modification of Mughrabi’s model which has a clearer physical foundation; (iii) to reconcile the apparently divergent approaches of the Hart and Mughrabi schools. The process of dynamical recovery in Hart’s “hightemperature” regime is important in the discussion. It can set in at homologous temperatures T/T, G 0.4, whereas the diffusional processes associated with hip-temperature creep set in only above T/TM iz?0.5. We suggest that it is controlled by the climb of edge disl~ations mediated by the vacancies produced by plastic defo~ation. Climb mediated by vacancies in thermal equiljbrium bound to devotions will have a similar activation energy and stress dependence, but leads to static rather than dynamic recovery, while abundant cross slip may occur in the same range of temperature and stress. R6sn&-Les processus d’&rouissage dans les cristaux uniques en orientations symetriques, dans les cristaux uniques soumis au “glissement simple” au stade III, et dans les polycristaux, paraissent semblables, et nous proposons de caracteriser ces processus. En 1970, Hart a defini les conditions SOUS lesquelles l’etat interne peut Btre decrit par un parametre de “durete” unique, et nous examinons quelques problemes qni resultent de son analyse. Hart, avec ses collegues [Conscitutive Equations in Plasticit_v, M.I.T. Press, Cambridge, Mass. (1975)], et Mughrabi [Srrength of Metals and Alloys (fCSMA J), Pergamon Press, Oxford (1980)] ont trouvi tous deux que les modiles fond&s sur un seul parametre etaient trop limit&s, et ils ont deveioppe des theories ;i deux paramitres. La theorie de Hart est essentiellement ph~nomenologique, et met l’accent sur I’importance de la sensibilim a la vitesse de deformation. Elle n’offre qu’un modele atomistique trbs simplifie de zones de glissage encastres dans une matrice elastique. Mughrabi a developp6 un mod&e detail16 a dislocations en configuration cellulaire dans lequel les joints des cellules sent dures et l’interibur des eellules est molle, mais la discussion theorique se borne au comportement a temperature zero et a une vitesse de deformation moyenne. Les buts de l’article present sont: (i) de dormer une fondation atomistique a quelques-uns des resultats empiriques des analyses fondles sur la theorie de Hart; (ii) de suggerer une modification du mod&e de Mughrabi qui possede une fondation physique plus Claire; (iii) de r&concilier lea approaches apparament divergentes des holes de Hart et de Mughrabi. Le proeessus de revenu dynamique dans le regime “haute-tem~rature” de Hart joue un role important dans la discussion. I1 peut commencer aux temperatures homologues T/Thi < 0.4, tandis que les processus diffusionnels associb au ffuage a haute temperature commencent seulement au dessus de T/T, % 0.5. Nous suggerons qu’if est determine par la monte6 des dislocations coins m&die&par ies lacunes produites par la dbformation plastique. La montei: mediee par les lacunes en equilibre thermique likes aux dislocations aura une &nergie d’activation et une influence de la contrainte semblables, mais elle produit un revenu statique plutdt que dynamique, tandis qu’une deviation abondante peut se montrer dans la meme &endue de temperature et de contrainte. Z~~nf~~-Die Ve~esti~ngsprozes~ in ~inkristauen in symmetrischer Q~entierung, in Einkristallen in ~infac~~leitun~ im Bereieh III und in V;elk~stalien seheinen &hnlich zu sein. Wir versuchen hier diese Prozesse zu charakterisieren. Hart definierte in f970 die Eedingungen, under welchen der innere Zustand durch einen einzigen “HIrte” Parameter beschrieben werden kann. Wir diskutieren einige Probleme, die aus seiner Analyse entstehen. Hart und Mitarbeiter [Constituzive ~Q~UZ~Q~S in ~~ast~ci~y, M.I.T. Press, Cambridge, Mass. (1975)] sowohl als such Mu~r~bi [.S’trength &f Metals and Allo$ (ICSMA 51, Peraamon Press. Oxford (198O)l fanden Ein-parameter Modelle zu einschrlnkend, und kntwickelten ZweLparameter Theorien. Harts Theorie ist im wesentlichen phenomenologisch und hebt die Bedeutung der Abhangigkeit von der Verformungsgeschwindigheit hervor. Sie bietet nur ein sehr vereinfachtes atomische Model1 von Gleitbiindern, die in einem elastichen Medium eingelargert sind. A.M. 37X5---A

1521

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OVERVIEW NO. 81

Mughrabi entwickelte ein ausfilhrliches Model1 von Versetzungszellen mit harten Zellwanden und we&hem Inneren, jedoch seine theoretischen Betrachtungen sind auf das Verhalten beim Nullpunkt und bei einer mittleren Verfo~ungsgeschwindig~t beschrankt. Der hier gegebene ~~rblick hat drei Zeile: (i) Eine atom&he Grundlage filr einige empirische Ergebnisse, welche mit hilfe von Harts Theorie gewonnen werden, zu geben; (ii) Eine Modifikation von Mughrabis Model1 vorzxschlagen, die physikalisch besser begriindet ist; (iii) Die anscheinend abweichenden Auffassungen der Hart und Mughrabi Schulen in Einklang zu bringen. Der Prozess der dynamischen Erholung in Harts “Hochtemperature” Bereich ist wichtig in dieser Beziehung. Br kann bei einer homologen Temperatur T/T, < 0,4 beginnen, obgleich die Diffusionsprozesse, welche das Hochtemperature kriechen begleiten, erst iiber T/TM = 0,s einsetzen. Wir schlagen vor, dass der Prozess der dynamischen Erhohlung durch das Klettern von Versetzung under Hilfe vor Ixerstellen, die wlhrend der plastichen Verformung gebildet werden, kontrolliert wird. Klettern mit Hilfe von Leerstellen, im Tem~raturgleichgewicht und an Versetzungen gebunden, hat eine Bhnliche Akt~vie~ng~ner~e und Spannun~bh~gigkeit, fiihrt aber zu einer stat&hen und nicht zu einer dvnamischen Erhohlun~, wahrend ~uergleitung im I_&4htss im gleichen Tem~ratur- and Spannungsbereich auftreten kann. -

1. INTRODUCTION Historically, theories of crystal plasticity have concentrated on crystals in single-glide orientations. This situation seemed conceptually simple and, in any case, multiple glide could be expected to occur only if the principal axes of stress happened to be in directions having a special relationship to the crystal axes. Admittedly, multiple glide is inevitable in poly crystals. What is more important is that single glide is unstable in cubic crystals. Stage I is indeed a close approximation to single glide, but in Stage II, which has received the most theoretical attention, there is a transition from single glide to multiple glide, even though the geometry of the crystal deformation corresponds closely to that of single glide. The behaviour of a “single-glide” crystal in Stage III is not very different from that of a polycrystal or of a single crystal in “multiple-glide” orientation throughout their deformation histories. This regime of turbulent flow [l] may prove easier to understand than Stage II, if this latter is really a region of transition between the “laminar” flow [l] of Stage I and the turbulent flow of Stage III of polycrystals and of crystals in symmetrical-glide orientations. The mode1 of the Stuttgart school [2,3] takes Stage II to be fundamental, and regards Stage III as a breakdown of Stage II which arises because cross-slip occurs when the stress and the temperature are high enough. The theory is mainly concerned with this region of breakdown, and says little about Stage III except that the rate of work hardening is less than that in Stage II. The extension of the model to include a theory of work hardening in Stage III [4] involves some arbitrary assumptions. Although it leads to satisfactory estimates of stacking-fault energies, the theory of the onset of Stage III is open to severe criticism (cf. 151).A basic problem is that cross-slip is more easily initiated at a pre-existing jog than in a smooth portion of a screw dislocation [6]. It will occur, at least in some regions, by the former process before the stress has risen to the level at which the latter process can become effective. The correlation with stacking-fault energy could also appear if Stage Ii1 involved the climb of dissociated dislocations.

Kuhlmann-Wilsdorf [7,8] has emphasized that parabolic hardening is to be expected in any simple model, such as that of Taylor [9], in which the dislocation structure becomes denser while remaining simitar to itself as defo~ation proceeds, except that the distance between barriers (here the cell size D) remains constant. However, in Stage III a reciprocal relation between D and the tensile stress (Tof the form UCCD-’

Cl)

is commonly observed [lo]. Figure 8 of [ 1 l] indicates that the relation (1) persists throughout Stages I, II and III. In fact [12,4], if the observations are analyzed in terms of a model which does not allow for the annihilation of dislocations, they indicate that the mean free path of dislocations actually increases during Stage III, while the cell size is decreasing. Also [4], the theory does not account for the strong temperature dependence of hardening in Stage III. Kocks [ 131pointed out that two distinct processes of recovery can occur in a cell wall, Either dislocations of opposite sign on opposite sides of the wall may come together and annihilate, or dislocations of the same sign on the same side of the wall may rearrange to form a rudimentary sub-boundary of relatively low energy. The combination of these two processes leads to polygonization. These latter processes cannot occur in the idealized model proposed by Mrghrabi [14-161, which is entirely free from the type of lattice rotation illustrated (e.g,) in Fig. 12 of 1.51.The dynamic recovery mechanism aided both by stress and by thermal activation, which Kocks proposes for the first process, is that suggested by Tangry and Shastri [17]. Suppose (Fig. 1) that on one side of the wall a mobile dislocation with Burgers vector b,,,which is approaching the wall interacts with a “forest” dislocation with Burgers vectors br, to form two nodes joined by a resultant dislocation with Burgers vector b, + b,, while a similar configuration with Burgers vectors of the opposite signs is formed on the other side of the wall. Under the combined action of the external stress and of their mutual attraction, the mobile dislocations will bow out into the positions represented by the dashed lines. The components of the dislocation line tensions which

NABARRQ:

OVERVIEW NO. 81

Fig. 1. One mechanism of dynamic recovery. Mobile dislocations with Burgers vectors t b, combine with forest dislocations * b, to produce resultant dislocations + (b,,, + b,). When the mobile components attract and annihilate, the segments of resultant dislocation shorten.

elongate the resultant disI~t~o~s are reduced, and the resultant dislocation links shorten. They may shorten to such an extent that thermal activation brings a pair of nodes together, and the mobile dislocation moves freely towards the mobile disfocatiun of the opposite sign. The higher the stress, the shorter are the d&cation finks, and the smaller is the a~~va~on energy required to bring the nodes together. The recovery process appears to be “dynamic”, because it becomes less rapid if the applied stress is removed. IIowever, this process of thermallyassisted glide is not a true recovery process which leads to the annihilation of dislocations. If mobile edge dislocations approach the wall from opposite

1523

The theory can be represented by “‘spring-anddashpot” models f2I] but, so far_ there has been little ~nte~retat~o~ on the atomic scale* It became dear that the effects which occur when the direction of deformation is reversed, in addition to some other effects, cannot tW described by a model which contains only a single hardness parameter, and it later proved useful ta develop a two-parameter theory 1221 which had been outlined in [21]. The two parameters are an isotropic “friction stress” and a directional “back stress”’ or ‘~k~ne~~~ hardening”. An alternative two-parameter theory has been developed by Mughrabi [IQ-16] using a dislocation model. We shall present this model, and modify it in a way which renders it more physically plausible and at the same time in better agreement with experiment. The two parameters of the model may be chosen to be the isotropic friction stress and the directional back s&ess within the ceE, in close aaafogy with Hart’s stresses We are thus led to explore the &at&n between Hart’s theory and Mughrabi’s. Wart% theory takes the rate of deformation i as a basic variable, while this quantity does not appear at all in Mughrabi’s theory. This leads us to discuss the nature of dynamic recovery in some detail.

We begin with an early survey by Mitchell [23f, which shows (Pig. 2) the temperature dependence of the flow stresses of single crystals of silver, copper, nickel and aluminium, each one measured at a given state of work hardening. The curves consist of three portions. At low temperatures, the flow stress desides, the; must necessari$ come from &&rent creases rapidly with increasing tem~rat~re, until a sources, and wifr Aardiy ever be moving on the same plateau is re~~~~~at a temperature which we calf rL. atomic plane- When tbey meet, they wilt not annihilAbove rL is a ptateau in which the flow stress is ati, but wilt form a narrow dipole. Essmann and Mughrabi [18] have provided much evidence that a almost independent of temperature, the dependence being no greater than the temperature depndence of mechanism must exist by which narrow dipoles can annihilate, whether they are in screw or in edge the shear modulus. This plateau extends to a temorientation, and we regard this annihilation as being perature TH, above which the flow stress again decreases rapidly with increasing temperature. It seems t.he essential step in dynamic recovery. that there is one tbe~~ly activated process which Cons~de~ble success has been achieved by a twoparameter phenome~olog~~al theory due to Hart ]I9]+ controIs the flow stress below TL, and is so rapid above Tt that the temperature de~~de~~e of its rate The original theory begins [Is] with a rigorous analysis of the conditions which must be met if the plastic above TL is no Xanger important, and another therstate of the polycrystal is to be represented by a single ‘“hardness” parameter. Experiments [20] then show 1.Q that these conditions are met over a wide range af conditions during monotonic deformation. Analysis of the observations shows ]2l] the existence of a scahng law, which leads to the ~~dj~tion of a regime of steady-state creep which does not depend on adventitious factors such as necking or recrystallization. However, this prediction depends on the crossing of pairs of curves which, according to the 0 600 1000 1PixY 400 800 2oa Temperature ---? "K basic theory, summarized at the beginning of the same paper, must not gross. We shall show how this Fig. 2. The temperature dependenoe of flow stress r(T), mmdized by the shear modulus GfT). After Mitchell 1231. apparent contradictian is resolved.

1524

NABARRO:

OVERVIEW

Table 1. Dependence on temperature 7’ of the saturation fatigue stress rr_, of copper [25]. The melting point is TM

TIT, %&@a)

0.003 73.5

0.06 48.4

0.22 25.2

mally activated process which only begins to occur at an effective rate above TH. We would like to identify those processes. These measurements, of the Cottrell-Stokes type [23], determine the temperature dependence of the ffow stress at a constant structure. It is at least equally important to know the temperature dependence of the rate at which the structure alters with increasing strain. Because the stress-strain curve changes its shape as the temperature changes (except in the “athermal” Stage II), this temperature dependence is not easy to define in monotonic deformation. The saturation stress in fatigue is a well-marked quantity. Its dependence on temperature in copper is given [25] in Table 1. Since T,/T, for copper in Fig. 2 is about 0.17, the rapid low-tem~rature decreases in Fig. 2 and in Table 1 occur in the same t~perature range, and we assume that they are produced by the same mechanism. If stress-strain curves in Stage III are extrapolated to higher stresses, the extrapolation suggests that a limiting flow stress 7,,im will be reached. In fact, particularly at high temperatures, the limiting flow stress has a value 7, which is greater than %I,,,, indicating that two different processes of recovery are operating. Siethoff f26] has plotted published observations of rlnm and 7s as functions of temperature. For copper and aluminium, both stresses, and for silver r,, seem to change their temperature dependence at T/T, = 0.5-0.6. For gold, 7, changes its temperature dependence at T/TM w 0.6, while 71,kM changes its dependence at T/TM ;i:0.43. Various observations of steady-state creep in copper may be analyzed 1271 to show that at temperatures above T/T, = 0.4-0.5 a process of deformation sets in which has an activation energy 0.51-0.72 of that for self-diffusion, while creep data for f.c.c. cobalt [28,29] show an activation energy 0.64 times that for self-diffusion below T/TM = 0.48. As has already been mentioned, there are two kinds of softening process. The first allows dislocations to overcome or evade obstacles to their motion; the second leads to the mutual annihilation of dislocations of opposite signs. Dislocations in f.c.c. metals are extended on a single glide plane and contain stacking faults. Glide on this plane may reduce the internal stress and my perhaps soften the crystal, but it is not thermally activated and does not concern us here. Thermally activated processes include the intersection of repulsive forest dislocations and the cross-slip of screw dislocations. Both of these processes are controlled by the stacking-fault energy. We need to distinguish between cross-slip initiated at a random point on a dislocation and cross slip at a

NO. 81

pre-existing constriction such as a jog. The climb of isolated edge dislocations may depend on the presence of vacancies in thermal equilibrium, the concentration of interstitials in equihbrium being negligible. The rate of this process depends on the concentration and mobility of the vacancies, measured by the coefficient of self-diffusion. It necessarily involves some sort of constriction of the extended dislocation, and is therefore hindered if the stacking-fault energy is low and the dislocation is widely extended. Vacancies [30] and possibly interstitials [31] are formed by plastic deformation. We re-interpret the calculations of Mecking and Estrin [32] to suggest that these defo~ation-produced point defects will be sufficiently abundant to produce recovery in the regions of highest dislocation density. We shall consider only the effect of the vacancies. If interstitials are produced during plastic deformation, they will be so mobile at all but the lowest temperatures that their contribution to dislocation climb will be independent of temperature and of strain rate. The temperature dependence of the activation energy for flow provides some info~ation on the rate-controlling process. In some regimes of flow, the rate-controlling processes are the formation and migration of point defects. The activation energies for these exceed kT by a factor of order 30, the logarithm of the number of lattice vibrations occurring in the duration of a typical experiment, and are virtually independent both of the state of work hardening of the material and of the applied stress. These processes of constant activation energy are characterized by horizontal straight lines in a plot of activation energy kT(d In e/a In T) against T. In other regimes of flow the rate-controlling processes are the motion of dislocation segments over potential barriers. In these processes, the activation energy when the stress is low is much greater than kT because they require the simultan~us motion of many atoms in a potential field, and these processes do not occur at a measurable rate under low applied stresses. The activation energies under low applied stresses depend strongly on the state of work hardening, being proportional to the length of the dislocation segements, and thus inversely proportional to the flow stress. These activation energies decrease with the applied stress, and flow occurs at a measurable rate when they are reduced to the order of 30 kT. They thus occur at a constant activation entropy of order 30 k, which under experimental conditions is virtually independent of the state of work hardening. In a plot of experimentally determined activation energy against temperature, they are represented by straight lines passing through the origin. The breaking of attractive junctions between mobile dislocations is a process of the kind which has just been discussed. The process by which a large mobile dislocation segment cuts an immobile forest dislocation is less easily characterized; it could occur at an apparently constant entropy of activation at

NABARRO:

T

OVERVIEW

OK

Fig. 3. Activation energies for creep of pure alurninium as a function of the absolute temperature. --Calculated curve using six discrete activation energies and five relative weights or frequency factors. l Experimental data. (Numbers in parentheses refer to number of determinations made; only one determination made on all other points.) The segments of the full line either are horizontal or pass through the origin (redrawn from Sherby et al. [33]).

low temperatures or at an almost constant energy of activation at high temperatures. Figure 3 shows the activation energy for creep of pure (99.996%) aluminium, measured as a function of temperature by Sherby et al. [33]. The dashed line is an experimental plot using six activation energies and five relative weights, while the full line depends only on the two constant activation energies AB, CD and the two constant activation entropies OA and BC. The regions AB and CD are well fitted. The region OA is not well fitted, and supports the view of Sherby et al. that several processes are involved. The region BC appears to be a separate region of constant entropy of activation, with transition regions about 20K wide into AB and CD, but it could also be a broad direct transition from AB to CD. The activation energy in the region CD, corresponding to a temperature of 17,860 K, agrees with that for selfdiffusion in aluminium, 16,01OK, but that in the region AB, 13,840 K, is far both from this energy and from that to move an existing vacancy, 7540 K. It might represent diffusion along dislocations or grain boundaries. We may try to analyze Fig. 2 in terms of these considerations, remembering that there is a rough

152.5

NO. 81

proportionality between the melting temperature and the activation enthalpy for self-diffusion. Table 2 shows the low-temperature limits TL, the hightemperature limits T,, the melting temperatures TM, the ratios TL/TM and TH/TM, and the quantity 103y/@ (where y is the stacking-fault energy) which measures the closeness of approach of partial dislocations in equilibrium, for the metals appearing in Fig. 2. Consider first the lower limit T,. There is no correlation with the melting temperature: nickel, which has the highest melting point, has a low TL. There is a clear negative correlation with stackingfault energy. The data on the upper limit TH are inadequate, but there does not seem to be a correlation with TM, while there is a possible negative correlation with y /pb, which appears much more clearly in the analyses of the onset of Stage III [4]. Table 3 shows the enthalpies Hfv to form and H,,,, to move a vacancy, taken from Ref. [34] and converted to degrees K by the factor 1 eV = 1.1605 x lo4 K, the Debye temperatures en, the values of &,/TM, and the Debye frequencies v = k&,/h, with k/h = 2.083 x 10” K-l s-‘, for the same metals. Table 4 shows the equilibrium concentration of vacancies C, given by C, = exp( - Hrv/ T) and the jump frequency of vacancies v, given by v, = v exp(- H,,,,/T)

Metal

T,.

Ag

300 230 200 190

CU Ni Al

of plateau (K) TH 500 440

Melting point

Limits of Plateau (homologous)

G?

T,lThn

THIT,

1234 1353 1726 933

0.243 0.170 0.116 0.204

0.370 0.472

Stacking-fault energy

1O’vlrb 2.6 3.8 6.3 18.9

Table 3. Enthalpy to form a vacancy Hfv and enthalpy to move a vacancy H,. (from [34]), expressed in K. Debve temperature 0, and Debye frequency Y

Metal

Enthalpy to form vacancy Hr, (K)

Enthalpy to move vacancy H,v (K)

Ag CU Ni Al

I 1,490 Il.950 15,670 8470

9980 12,300 17,180 7540

(3)

at the lower and upper limits of the plateau. It is clear that the the equilibrium concentration of vacancies at the lower limit of the flow-stress plateau is too low for vacancies in equilibrium to affect the flow process. It is also clear that the jump frequency of vacancies, even if they are formed by plastic deformation, is too low at the lower limit of the plateau for them to contribute to the flow. Thus the process which leads to the low-temperature drop in Fig. 2 cannot be mediated by vacancies. We have no

Table 2. Lower limits TL and upper limits r, of the flow-stress plateau (from [23]), melting points r, and normalized stacking-fault energies Limits

(2)

Debye temperature 0,

WT,

215 315 375 394

0.174 0.233 0.217 0.422

Y = k&,/h (lo’*s-l ) 4.48 6.56 7.81 8.21

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OVERVIEW NO. 81

,

3.9

-S.T LOG

t -6.1

I

-5.7

a -4.7

LOG STRAIN-RATE

I

-3.7

I -21

-s.7

-4,

STRAIN-RATE

-31

-2.1

(SEC-')

Fig. 5. As for Fig. 3, 1100 aluminium alloy, in the hightemperature regime.

(SEC-‘)

Fig. 4. Points of constant work-hardening parameter F indicated by symbols of the same type. The curves are curves of constant hardness u*, in the low-temperature regime. Type 316 stainless steel. After Wire et al. [35].

must he substracted the effective stress

from the applied stress u to yield

tra=a-or. compelling reason to neglect the influence of interstitials formed by deformation, other than the belief

that their influence will be independent of temperature throughout the temperature range under consideration. However, in view of the clear negative correlations of TL with stacking-fault energy shown in Table 1, we believe that this process is either cross-slip or the intersection of mobile and forest dislocations by a process involving their constriction. Although the flow stress Q in this region depends on temperature and plastic strain rate i, the rate of internal hardening of a specimen of given structure [r in equation (67)] is independent of strain rate. If we plot curves of constant structure, termed constant hardness, in the plane of (In i, In a), then (Fig. 4) the curves of constant f coincide with those curves. This means that for monotonic deformation in the region of low temperatures and high strain rates the structure depends only on the total plastic strain cP, which is then an acceptable state variable in this region. We follow Korhonen et al. [35] in believing that the fact that the development of the dislocation structure with strain in this region is independent of strain rate implies that it is also independent of temperature. The paths which the dislocations follow are uniquely determined, but the stress cr required to move them along these paths contains a pure “friction stress” ur which depends on temperature and strain rate and

region there is a process of dynamical recovery which occurs only during plastic deformation and which reduces the hardening during slow deformation but not during rapid deformation. In this region the change in dislocation configuration depends on the

20

L

--t--j ”

I

\

ou

400 Temperature

A8

Equilibrium atomic concentration of vacancies C, TL TH 2.3 x IO-”

Jump frequency of vacancies Y, (s-l) TL TH 1.6 x 1O-2 -

CU Ni Al

2.7 x Wz3 9.4 x 10-35 4.4 x 10-20

3.9 x IO_” 3.9 x IO_‘5 4.8 x 10-s

4.2 x:0@’ 4.4 x 10~9

(K)

Fig. 6. Schematic diagram of the strain-rate sensitivity of aluminium as a function of temperature, showing also the influence of strain rate. After Saimoto and Duesbery [37].

Table 4. Equilibrium atomic concentration of vacancies C, and vacancy jump frequency at the lower and utmer limits of the flow-stress olateau

Metal

(4)

The situation in the region of high temperatures and low strain rates is entirely different. As is shown in Fig. 5, the rate of work hardening increases more rapidly with increasing strain rate than does the flow stress at constant structure. This implies that in this

1.4 x 102 3.0 x 10s

vv

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OVERVIEW NO. 81

(a)

(b)

Fig. 7. (a) Flow stress of annealed 99.99% copper as a function of strain rate at room temperature, measured at constant threshold stress (flow stress extrapolated to 0 K). From Follansbee and Kocks [38]. (b) As for (a) but for samples which have reached a strain of 0.15 at a uniform strain rate.

strain rate and not only on the increment of plastic deformation. Saimoto and Duesbery [37] presented the same features in another way. Recognizing that there is a monotonic, though not necessarily simple, relation between the effects of increasing strain rate and of decreasing temperature, their Fig. 5 (Fig. 6) is essentially the gradient of Fig. 12, which summarizes the work of Hart and his colleagues. In agreement with Figs 4 and 5, it shows that at low (homologous) temperatures, the strain rate does not influence the temperature dependence of the strain-rate sensitivity of the flow stress. Temperature affects the mobility of dislocations, but not the rate of work hardening. At higher temperatures, there is a process of dynamical recovery which is more effective at low strain rates than at high strain rates. The situation has also been examined by Follansbee and Kocks [38], who studied the strain-rate dependence of the flow stress of annealed 99.99% copper at room temperature as a function of strain rate. At a constant mechanical threshhold stress (flow stress extrapolated to 0 K), the stress increased slowly and uniformly with the logarithm of the strain rate [Fig. 7(a)]. When the same material was tested at a strain of 0.15, achieved at a constant strain rate, the flow stress increased considerably more rapidly, but still uniformly with the logarithm of the strain rate [Fig. 7(b)] provided the strain rate was not very high. At very high strain rates, the flow stress increased rapidly. [We note that each curve in Fig. 7(a) shows an indication of a rapid increase in flow stress beginning at a strain rate of 5 x lo3 SC’; the region of very high stresses above 104s-’ in Fig. 7(b) is cut off in Fig. 7(a).] We deduce from the observations that there is a process of recovery in copper at room temperature which is more effective at lower strain rates in the range 10-4-104 ss’, and a process, which may or may not be the same one, which is ineffective at strain rates above lo4 s-‘. It is important to recognize that the process or processes involved in dynamical recovery are com-

pletely different from those involved in static recovery. In his original analysis [39] of what is frequently called the Bailey-Orowan equation, Orowan was concerned to demonstrate that, in order to consider plastic flow as recovery flow, it would be necessary to consider rates of recovery exceeding those actually observed by order of magnitude. In a later paper [40], Orowan demonstrated quantitatively that (a) the flow stress is influenced to an unmeasurably small extent by the (static) recovery occurring during a tensile text, and that (b) even in the lowest-melting technical metals the stress-strain curve is not appreciably influenced by recovery. He emphasized that the temperature dependence of the rate of recovery is much more rapid than that which is observed for the flow stress. Bailey [41] had been led to propose that creep resulted from a balance between work hardening and recovery by noticing that the temperature dependence of the recovery time of a low-carbon steel “(calculated from two points only)” was in “close agreement in the order of magnitude” with the temperature dependence of the creep fracture lives of some other steels. Later observations have often claimed to demonstrate the occurrence of recovery creep, in which the work hardening produced by the increment of strain in a given time interval just compensates the static recovery of the flow stress during the same time interval. Lloyd and McElroy [42] have analyzed seven of these sets of observations. In each case the rate of work hardening which leads to agreement with the model of static recovery is between 0.2 and 1.6 times the shear modulus or Young’s modulus. This is totally incompatible with the rate of athermal linear hardening of a single crystal in Stage II, which, using resolved components of stress and strain, is about 0.0035 of the shear modulus. The corresponding value for a polycrystal is obtained by multiplying by the square of the Taylor factor, and is about 0.03 of the shear modulus. The experiments in question were carried out by the “stress-dip” technique. A sample

1528

NABARRO:

OVERVIEW

is creeping under a constant stress. The applied stress is suddenly reduced by a small amount Aa (Aa > 0) and, after a rapid transient, the creep rate becomes very small. After a time At, creep again sets in at a rate slower than, but comparable with, the initial creep rate. After some time, the stress is increased by Au to its original value. After a transient during which the sample creeps relatively rapidly through a strain Ac, a steady rate of creep close to the initial rate is achieved. The rate of recovery is take to be Au/At, and the rate of work hardening do/At. The experiments show that AC/At is close to the steady-state creep rate, even though Aa/Ac is much greater than the normal rate of work hardening. We would expect the rate of work hardening in a creep test to be less than that in a tensile test conducted at a much higher strain rate. We interpret the discrepancy in terms of the type of model is implied in equation (4), and will be developed in the following sections. The flow stress consists of a major part which opposes dislocation motion in either direction, and a minor “back stress” which opposes dislocation motion in the direction which contributes positively to the strain rate, and aids dislocation motion in the opposite direction. This “back stress” may arise in various ways-from small pile-ups of dislocations against barriers, or from the curvature of dislocation segments. While this component is normally smaller than the “friction stress”, it will increase much more rapidly than the friction stress after a small increment in the applied stress because it can be produced by small glide displacements of the dislocations. This leads to an anomalously high value of Aa/Ae. Since the value of Be/At agrees with the steady-state strain rate, it follows that AC/At must represent a rate of recovery which is anomalously high in two ways. Firstly, the arguments of Orowan and those contained in Ref. [5] show that recovery at the normal rate is totally inadequate to give the observed creep rate when work hardening occurs at the normal rate of static recovery and, secondly, the rate of work hardening which is actually balanced is much greater than the normal. We explain the first factor by noting that in our model of dynamical recovery the rate of recovery decays only slowly after the strain rate is suddenly reduced. We explain the second factor by considering the specific model of the back stress produced by dislocation pile-ups. When the small stress change Aa is suddenly applied, the friction stress does not allow the dislocations in the pile-up to move, the stress concentration at the head of the pile-up is unchanged, and the rate at which leading dislocations at the heads of pile-ups are removed by climb is unchanged. Since plastic flow has ceased, dislocations are not replaced, and the back stress falls rapidly. During transients, both hardening and recovery are concentrated in the rapidly-relaxing “back stress” component, and recovery is still dynamic rather than static. There is in fact direct evidence that the crystal

NO. 81

structure [43] and the flow stress [44,45] recover much more rapidly in a sample under stress than they do in an unstressed sample. For example [44], a sample of polycrystalline high-purity aluminium was strained 15% at 78 K and then annealed without load for 2 h at 530 K. Its flow stress was then 7400 p.s.i. It was allowed to creep under a stress of 1500 p.s.i. at 530 K for 5 h, which produced a strain of 0.0086, and its flow stress fell to 4700 p.s.i. The same flow stress was reached in a sample annealed without external stress only after 1750 h at 530 K or 117 h at 472 K. Similarly [45], a cold-worked pure aluminium sample with a room-temperature yield stress of 22,800 p.s.i. achieved a reduction of yield stress to 19,000 p.s.i. after annealing at 477 K for 63 h without load, but a reduction to 12,000 p.s.i. after annealing for the same time at the same temperature under a load of 6000 p.s.i., which caused a creep strain of 8.1%. Creep at lower temperatures caused less weakening, or even strengthening, work hardening being more important than recovery. This illustrates the combined action of strain and thermal activation in dynamical recovery. Blum and Reppich [46] have provided evidence of a process of recovery which decays over a strain range of order 1% and a time of order a day after a substantial reduction in the load applied during a creep test. The materials reported were pure aluminium at 400 K and A1-4.5 at.% Mg at 573 K, but Blum and Reppich state that “after a large stress reduction the elastic and anelastic response is always followed by a range where the creep rate decreases . . . there is agreement that the decrease of i after a stress reduction is an expression of the dynamic recovery of the dislocation structure. . . after a large stress reduction the dislocation density decreases with time. . . . Obviously the decrease in dislocation density goes along with a decrease in creep rate”. This explanation in terms of a decrease in the dislocation density is not entirely convincing. Blum and Reppich describe these samples of Al and A14.5% Mg at high temperatures as “class M”, i.e. as being effectively pure metals, as contrasted with “class A”, typical alloy materials. As we shall explain in connection with Fig. 14, the glide velocity of dislocations in class M materials is a rapidly increasing function of stress, and a decrease in dislocation density has little influence on the strain rate. It seems more likely that the process of rapid dynamical recovery which is occurring during rapid straining at a high stress continues for some time after the stress and stain rate have been reduced. The specific model we propose is that vacancies are produced abundantly during rapid straining and lead to recovery by the annihilation of edge dislocation dipoles. When the stress and strain rate are suddenly reduced, the vacancy concentration and so also the rate of dynamical recovery decay slowly, leading to a decelerating creep rate.

NABARRO:

The ideas of a two-parameter model and of dynamical recovery are not closely linked. Thus Mecking and Kocks [47] (who also briefly discussed a two-parameter model) developed a rather successful one-parameter model which incorporates both work hardening and dynamical recovery. Even at the upper limit of the plateau, the equilibrium concentration of vacancies is too low for them to produce static thermal recovery at an appreciable rate. However, although the numerical estimates are very rough, it seems that, at the upper limit of the plateau, the jump frequencies of vacancies formed by plastic deformation in copper and in aluminium are of the order of 5 x 10’s_‘. This would imply a diffusion length of the order of 80 interatomic spacings in a second, which suggests that the process which begins at this temperature is dynamical recovery by the migration of vacancies produced by the plastic deformation. This view is supported by a re-analysis of the observations of Atkinson et al. [48] on the relaxation of the Bauschinger effect in copper single crystals hardened by dispersed particles of silica. When the direction of plastic deformation is reversed, two effects are observed, a rounding of the flow stress curve described [49] as “transient softening”, and a depression of the absolute value of flow stress below that which is required to cause flow in the original direction at the same cumulative strain. The latter persists to large reverse strains, and is known as “permanent softening”. The observations of Atkinson et al. suggested that there are two distinct mechanisms of plastic relaxation which lead to permanent softening, and result in the formation of distinct dislocation structures, “that the A structure is formed when the crystal undergoes plastic relaxation with the load off, and that the B structure is

formed when the load is on”. They go on to argue “that the A structure could form only at the expense of straining the specimen against the applied load’, so that the A structure cannot be formed under load. This argument is incomplete: softening under load is almost a hundred times faster than softening in the absence of load, and the essential observation is not that the A structure cannot be formed under load, but that the B structure can only be formed under load. Indeed, in their Fig. 11 they refer to the process as “obtained under dynamic softening conditions”, and we believe that it occurs by the same process of dynamical recovery as do the processes we have just discussed. Atkinson et al. find the activation energy for softening “when the specimen is under load” to be 0.9 + 0.1 eV, in reasonable agreement with the activation energy of 1.06 eV required to move a vacancy in copper. Figure 3 of [5], reproduced here as Fig. 8, shows the normalized stress-strain curves of single crystals of various face-centred cubic metals, interpolated so that they can be displayed at common homologous temperatures T/T,. It was pointed out in [S] that the behaviour of aluminium is anomalous, its reduced flow stress moving systematically from being the highest at low temperatures to being the lowest at high temperatures. We believe that the lowtemperature anomaly is to be explained as follows. The empirical correlation between the melting temperature and the enthalpy for self-diffusion justifies the use of TM as a normalizing factor at higher temperatures where self-diffusion is important. At low temperatures, self-diffusion does not occur, and the appropriate normalizing factor is not the melting temperature. Table 5 shows that the entropies of melting of Ag, Cu and Ni all lie within 0.07 units of 1.17 entropy units, while the entropy of melting of TITm=0,13

TITm=0,07

0

50

2x 1o-3

100 CC%)

1529

OVERVIEW NO. 81

150

0

50

100

150

E(%)

TITm=0,22

2x10-3

-

TITm=0.30

a/II’

.Cu

lW-

0

50

100 et%)

156

0

50

100

150

E(%)

Fig. 8. Work hardening curves of single crystals of f.c.c. metals at selected homologous temperatures. After Nabarro [5].

NABARRO:

1530 Table 5. Entropies Enthalpy of melting, kJ mole’ Metal Ag CU Ni Al

Hr., 15'31

of melting

10.26 11.26 14.35 7.76

Entropy of melting S,IR = H,/RT,

3.1. The Second Law of Thermodynamics

1.10 1.16 1.24 I .38

aluminium is 1.38 units. Regarded as a phase change in the solid, melting is more strongly first-order in Al than in the other metals. This means that the melting point of Al is anomalously low, and homologous temperatures normalized by means of the melting temperature are anomalously high. This explains the apparently high strength of Al at low temperatures; the decrease at high temperatures is explained by the very low stacking-fault energy.

3. HART’S ANALYSIS Curves such as those in conducted at conventional the strain rate C raises the can be no unique relation

Fig. 5 are obtained in tests stain rates. An increase in curves, showing that there of the form

z =?(E, T).

NO. 81

state creep, the problem of the double-valued parameter, and its resolution by comparison with the thermodynamic analogy.

of f.c.c. metals

RTM kJ mol-’

11.27 13.01 17.8 10.7

OVERVIEW

(5)

In the original version of Hart’s theory c represents the plastic strain, the elastic and anelastic components of strain having been subtracted. It is clear that L is not an acceptable coordinate if reverse straining is allowed; a rod extended from 6 = 0 to 6 = c0and then compressed back to L = 0 recovers the intitial values of E and T, but its flow stress has increased. Hart and Solomon [20] showed that E is not an acceptable variable even in monotonic deformation. Using a definition of “hardness” which will be described below, they found that in aluminium an increase of hardness which was produced by a strain of 0.061 at a strain rate of 1.6 x 1O-3 SK’ required a strain of 0.145 at a strain rate of 1.6 x 10m6SK’. Hart therefore chose i as the observable coordinate. His analysis is formally similar to part of Carathtodory’s treatment [51] of the Second Law of Thermodynamics, and the analogy is useful to readers familiar with the Second Law. However, the analogy is purely formal, and questions such as “whether i is an acceptable thermodynamic variable” are not practically important. Hart’s analysis is based on a branch of mathematics which the physicist or the metallurgist meets in connection with the Second Law of Thermodynamics but rarely in other contexts. We therefore begin our discussion with a recapitulation of the arguments associated with the Second Law. We then outline Hart’s phenomenological theory and the experimental evidence for its validity. We proceed to the empirical scaling law, the discussion of steady-

We consider a simple fluid system in which the state can be completely specified by the two parameters, volume V and pressure p. For a mechanical system such as a cube of sponge rubber maintained at ambient temperature, V is a unique function of p, and the state of the system is determined by a single parameter. For a vessel filled with gas, the relation between p and V depends on the temperature 0 of the gas, as determined by the reading 0 of an arbitrary thermometer. The equation ofstute of this mass of gas may be written in the form 6=

B(P,

V)

(6)

and may be represented by a surface in the space of p, V and 6 (Fig. 9). The First Law implies that there is an internal energy function U, with U=

U(P,

V)

(7)

which may be represented by a surface such as that of Fig. 10, in which we have allowed for the increase in specific heat with increasing temperature. In any change of the parameters p and V, the external work done by the system is p dV. The change in internal energy dU is not in general given by dU=

-pdV

(8)

and we define the heat SQ on which has been supplied to the system by SQ = dU -p dV.

(9)

There is no quantity Q(p, V) depending only on the present state of the system and satisfying the linear or Pfaffiian form dQ =($)Vdp

+[($-p]dV

(10)

and we indicate this by writing SQ rather than dQ. The theory of differential forms then tells us that (lo), together with (7), being a differential form in two variables, always has an integrating factor l/t(p, V) leading to an integral s(p, V), such that ds = sQ/t.

(11)

Moreover, any function f(s) is also an integral, which areses from the integrating factor f ‘(s)/t. All of this results from applying purely mathematical arguments to equations derived from the First Law of Thermodynamics. The Second Law allows us to select a specialf(s) which has important properties. We will not reproduce the argument [51,52] here, but draw attention to an essential step in the argument. We consider two simple fluid systerns, specified by (p,, V,) and (p2, V,). When these

NABARRO:

OVERVIEW

pv = 0

1531

NO. 81

u = pv +

O.l(pV)s

gas. .ture.

Fig. 11. The entropy

S(p, V) of a perfect

gas.

NABARRO:

1532

are in thermal equilibrium,

OVERVIEW

an equation of the form

@(PI, V,,Pr, V,)=O

(12)

must be satisfied. Therefore, when the two are in thermal equilibrium, the combined system is fully described by three independent variables. In general, a differential form in three independent variables does not have an integrating factor. The Second Law of Thermodynamics tells us that the form

NO. 81

anelastic components can be recognized and substracted from the total strain, and we consider only Ed, which for brevity will be written as L. The theory takes account explicitly of the workhardening coefficient Y and of the strain-rate sensitivity v of the flow stress 0 by writing dlna

=Y& +vdlni

(16)

which may be written in the alternative 6t =adlno+/IdIni.

form (17)

This is a Pfaffian form analogous to (10) if, and only if tl = Gl(cr,i) subject to (12), does have an integrating factor l/T. This is also the special integrating factor for system 1. This integrating factor, and the corresponding integrals S1 and S, for the combined system, have a number of special properties: (1) The reciprocal of the special integrating factor, T, called the thermodynamic temperature, is a function only of the arbitrary thermometer temperature 8 T(P, V) = TP(P,

VI

(14)

and is therefore independent of the particular thermodynamic system which is under observation. (2) The corresponding integrals S, called the entropies, are extensive, i.e. s=s,+sr.

(15)

The statements that (13), applying to the compound system, has an integrating factor, that there exists an integrating factor (14) for the first subsystem which depends only on the temperature 8 on an arbitrary scale and is independent of the subsystem, and (15) that there exists an integral which is extensive, are essentially equivalent representations of the Second Law. The entropy S(p, V) of a gas is represented by a surface such as that of Fig. 11. Figures 9, 10 and 11 show that if the state variables p and V of a simple fluid system such as a perfect gas are known, then the temperature 0 on an arbitrary scale (and therefore the thermodynamic temperature T), the internal energy U and the entropy S are all uniquely determined. We shall now relate Hart’s formalism for plastic deformation [19] to the formalism of the First and Second Laws. 3.2. Hart’s phenomenologicaI theory Hart’s theory, in its original form [19], is “restricted to deformation histories at a single temperature”, and to “plastic tensile strain” which at least by implication, must by monotonic. The conditions are supposed to exclude grain-boundary sliding, which Hart had already discussed in an earlier paper [53]. The total strain L, is written as the sum of the elastic strain c,, the recoverable anelastic strain ca and the plastic strain eD.It is assumed that the elastic and

(18)

and B = B(a, i) which may be written equivalently

(19) as

Y =r(a,i)

(20)

v = v(a, i).

(21)

and

Whether or not (20) and (21) are true is a matter for experiment to decide, just as the experiments which underlie the First Law show that the coefficients of dp and dV in (10) are functions of p and V alone. If, in addition to (18) and (19), experiment should show that (&x/a In f?), = (~?/?/aIn u)~

(22)

then (17) is directly integrable to give E =E(a,t)

(23)

u = a(c, i).

(24)

or, equivalently

This is not in general the case. We have seen that the flow stress u at a prescribed strain rate i may be higher if the prescribed strain E has been achieved by rapid straining than if it has been achieved by slow straining. If (18) and (19) hold, as is often true to an adequate approximation, but (22) does not, the form (17), being a Pfaffian form in two variables, always has an integrating factor leading to an integral y(a, i). As in the thermodynamic case, there are infinitely many integrating factors and infinitely many integrals. We choose [21] the integrating factor which leads to an integral a*(o, i) which has the dimensions of stress, and call CT*the hardness parameter of the material. There is nothing in Hart’s formal analysis which is analogous to the Second Law. The Second Law relates the behaviour of one thermodynamic system to that of another, while Hart’s analysis is concerned with a single mechanical system. Among the most useful consequences of the Second Law are equations of the form of Maxwell’s thermodynamic relations.

NABARRO:

OVERVIEW

1533

NO. 81

For example, the exact differential relation dU = TdS -p dV

(25)

leads to the useful Maxwell relation

(gs= -(i$

(26)

The analogous relation da*=Cdu+Edi leads to the analogous relation

but not obviously

(27) useful LOG i

(is). =gX_ Nevertheless, there are analogies between the hardness a*(~, i) and the entropy S(p, V). Both are measures of the internal disorder of the system. A strain-rate-sensitivity curve a(c) at constant hardness is analogous to a p(V) curve at constant entropy. Hart calls the former a “hardness curve”, i.e. a curve taken at constant hardness, while the latter is an adiabatic curve, i.e. a curve taken at constant entropy. When the hardness or the entropy is reduced by a reduction in the internal disorder, heat is released. If, to an adequate approximation, a unique hardness a*(o, C) exists, we may invert the relation to obtain the plastic equation of state u = a(a*, i).

(29)

This is analogous to writing the equation of state of a fluid in the form P =p(K

V)

(30)

which is unfamiliar but permissible. Since g* is assumed to be a unique function of Q and C, it is natural to assume [19] that “for the existence of a plastic equation of state, it is necessary that all hardness curves form a unique one parameter family of curves for which no curve intersects another”. “If a a(i) curve is determined for another specimen of the same material after a different deformation history but such that the point (a,, i,) is common to both characteristics”, then “if the two curves coincide we shall have the important conclusion that each state of plastic hardness of the material as a result of deformation is unique and is independent of the deformation path by which it was reached. This relationship can be called a plastic equation of state” [20]. Evidence for the existence of a well-defined hardness curve in some systems, though not in others, is given in Refs [20,21,54]. After large anelastic transients, a(i) curves which pass through a common point continue to coincide as the strain rate decreases over a wide range in a stress relaxation test. This completes the initial formal part of Hart’s programme.

Fig. 12. A schematic plot of a curve of logo against loge’ (strain-rate sensitivity curve) for a material of constant internal structure (“hardness”). The temperature is constant, but the regions H and L are in the hightemperature and low-temperature rCgimes respectively. From Hart et al. [21].

The general form of the graph of In u against In i at constant u * and constant temperature is shown in Fig. 12. It was not possible experimentally to cover the whole range of C in Fig. 12 at constant temperatures; Fig. 12 was obtained by synthesizing portions of the curve obtained at different temperatures using the “temperature--compensated strain rate” of Zener and Hollomon [55], or even on different materials. More recently, at least a partial direct verification has become available [56] for aluminium at room temperature. We may relate Fig. 12 to Mitchell’s graphs [23] of u as a function of T at conventional strain rates t reproduced Fig. 2. Similar experimental results for copper have been reported by Tangri and Shastry [17]. In the absence of any thermal activation, there is a flow stress represented by the top right portion of Fig. 12 or the top left portion of Fig. 2. At finite but relatively low temperatures there is a thermally activated process which assists dislocation motion. The portion of Fig. 12 marked f. shows that at a given temperature this process is too slow to reduce the flow stress greatly at high strain rates, while below a certain strain rate there is ample time for it to occur, and the flow stress reaches a plateau which is independent of strain rate. Similarly, on Fig. 2, at a given conventional strain rate, this thermally activated process reduces the flow stress to lower and lower values as the temperature increases, until a temperature is reached at which the process occurs so fast that further increase in the temperature produces no further decrease in the flow stress. At some low strain rate in Fig. 12, and at some high temperature in Fig. 2, these plateaux end as a second thermally activated process begins to be effective. We have given a preliminary discussion of the physical natures of these two processes in Section 2. 3.3. The scaling relation The experimental observations reported in [20] and [21] led to a new, empirical, scaling relation. Suppose

NABARRO:

1534

OVERVIEW

we take hardness curves for a given material in two different states of hardness, and join points on these curves which have the same strain-rate sensitivity v of the flow stress [equation (16)]. Then, for a given pair of curves, these joining vectors are all equal, and the slopes fi =(13 lncr/a lni),

(31)

are the same for any pair of curves. That is to say, any hardness curve may be derived from any other hardness curve by rigid translation in the direction p. This quantity B, which for aluminium is 0.22 + 0.01, is characteristic of the material. Since the physical processes which are rate~ontrolling in the low and high strain-rate regions of Fig. 9 are quite different, we expect, and observe, [54] that the scaling slopes ~1 in these two regions will be different. If the slopes in the regions of inflection of curves of different hardness are equal, the scaling slope is undefined, and may be interpolated between the regions H and L. It must be emphasized that the existence of this simple linear scaling law is compatible with, but is in no way required by, the existence of a single hardness parameter. When the linear scaling law is valid, the strain-rate sensitivity may be written v =

(8

in

@/ain i), = v[In(a/uo)

P. [ 01

-p ln(l,&)] = v, d

2

(32)

(33)

form of F is

The appropriate

as shown in equation integrated to give

(32), so that (38) may be

ln$=J&:=J& x d[ln(a/a,,) -P

(34)

F = P/(P -v).

Whatever the form of the function v, this implies that

w It is useful to relate the form of equation (40) to that of the equations which lead to the Cotterell-Stokes Law 1241and to linear Haasen plots 1573. We first assume that the flow stress may be expanded as the product of a function t(u*) of the internal hardness variable alone and a function a(& 7’) of strain rate and temperature alone, i.e. u(i, T) = a(& T)t(u*).

which reduces to

(&),= -Pi.

(36)

(42)

If we increase the hardness to at and repeat the experiment we fmd c+,(T,) - c+,(r2) = ]a(& Ti) - a(&

Tdf@b*) (43)

leading to

~0,) - dTd = t(bb*)_ u,V,) dT,) - u,V,) f(u,*) CAT,)

(-9

so that the changes in stresses are proportional to the stresses themselves. This is the Cottrell-Stokes Law. In the Haasen plot, (&r/a In c’)Tis plotted against (T. If equation (41) is obeyed

=(a In a/a in +a

-@dln2,,

(37)

and it then follows from (33), (34) and (37) that -v)]dv.

(38)

Here v and v’ are both functions of the same variable,

(45)

which is a straight line through the origin. If the flow stress u contains, in addition to the term (41), a “friction stress” u,,(&,T,), we have + a&‘, T).

(46)

Then the Cottrell-Stokes Law is obeyed only if a&, 7’) is proportional to a& T). Cottrell and Stokes used a theoretical model in which (41) is replaced, not by (46), but by a(& T) = a(i, T)t(a*)

and this is ensured by equation (32). From equation (32)

+[v/v’(p

(41)

Then if we measure the flow stress of a material in a given hardness state a: at two tem~ratures T, and T2, at a standard strain rate 6 [or, equivalently, if we change the strain rate at a given temperature], we find

a(& T) = a(i, T)t(a*)

dlno*=dlnu

(39)

(aoja lni),=facrja ini@

To verify that (34) is indeed an integrating factor when p is constant, we observe that it is an integrating factor if

dv =v‘dInu

ln(i&)].

a,(T,) - a,(T,) = ]01(&jr,) -N(& T,)]t(e:)

Here &, and a, are universal dimensional constants such as 1 s-r or the Debye frequency of the material for i0 and 1 Pa or the shear modulus of the material for a,. Following [21] and [36] we can explicitly integrate equation (17) to obtain In u *. Let F/a be the appropriate integrating factor. Then (17) becomes d in e* = (F/E) d& = F(d In CT- vd In i).

NO. 81

-t

7(u*)

(47)

and deduced from their experimental observations that “‘the tem~rature-de~ndent part of the flow stress is proportional to the total flow stress”. The Haasen plot is a straight line not passing through the origin if (46) is obeyed, but when (47) is obeyed the Haasen plot is a straight line only if the Cottrell-Stokes Law holds.

NABARRO:

The form (40) appearing in Hart’s analysis is more genera1 than (41)-(47), and the Cottrell-Stokes and Haasen relations hold only iff(z) in (40) has the form f(z) = A + B In z.

(48)

Equation (40) then implies a linear relation between In 0, In Q* and In i. Such a relation holds in the linear region of Fig. 9, but not at very high or very low strain rates. It is also interesting to relate the scaling law, that p in equation (3 1) is constant over a rather wide range of hardness parameters g* and strain rates i, to a very broad class of dislocation-based models of plastic deformation. These models are guided by the principle of similitude [58], according to which, in a given regime of deformation, the dislocation configuration remains similar to itself as work hardening occurs, only reducing its scale. The internal state of the system in this regime is defined by a single parameter, the mean dislocation density p. All internal stresses scale in proportion to the hardness cr*, which is proportional to p Ii’. However, we recognize that a strict application of the principle of similitude leads to linear work hardening, whereas work hardening in the multiple-glide region with which are concerned is roughly parabolic. Where similitude applies, rate-controlling parameters such as the vibrational frequency of a dislocation loop or the time taken for a vacancy to diffuse either randomly or under a concentration gradient from one dislocation to its neighbour depend on the scale length P -‘i2, and possibly on the temperature. The configuration thus has a characteristic limiting strain rate ~*(a*, T), and an actual strain rate i under an applied stress 0 which we expect to be of the form i(c, rJ*, t) = i*(a*, T)f(cr) xexp[-g(a/e*)U(e*)/kT].

= Au*PB(T),

f(fJ) = C(a/Poq

(50) (51)

and g(u/u*)

= (1 - u/ru*)

(52)

where A, B, C, p, q, r, s, are constants and that U(u*) is a constant U,, independent of u*. We then have lni(u,u*,T)=lnA+plnu*+lnB(T) +lnC+qln(u/p)-U,(l-u/ru*)”

and (54) If, at a given temperature, a In i/a In u takes the same value for two hardnesses of and u: when the applied stresses are u, and u2 respectively, then (54) shows that

u,/u: =u,/u:

(53)

(55)

and it follows that lni,-1ni2=pln(u~/u~)+q1n(u,/u2) =(P +q)lna,lu2

and 1 In u, - In u2 =InC, -In;, p +q independent of u, , u2, u f, uf and T in accordance with the scaling law. However, the principle of similitude implies that if U(a) is determined by the energies of dislocation lines, it must be of the form Up/u*, because the energy at each stage of the activation process is proportional to the line energy of the dislocation in its unactivated state, which is proportional to its length and so to l/u*. The last terms in (53) and (54) then become -U(p/u*)(l

- u/ru*)

(57)

and 3-l (58) respectively. Equation (55) is now replaced by

(49)

In the exponential factor, the activation energy under zero applied stress U(a*) depends only on the scale P “2 of the dislocation configuration, which is measured by G*. Because of similitude, this energy is reduced under an applied stress u by a factor of the form g(u/u*) where g(0) = 1. We now make the arbitrary but plausible assumptions that, within any given regime of deformation, in which u*, u and T can vary over fairly wide ranges i*(u*/T)

1535

OVERVIEW NO. 81

(59) and

(60) The last term in (60) does not vanish, and is not of the form of the logarithm of a ratio of two stresses. The scaling law holds only if we happen to be operating in the region where u/u* = r/(r + 1), when we again obtain (56). The total range of u over which it is possible to test the scaling law for a given metal is at most a factor of 2, and it may prove to be a useful approximation rather than an exact law. The essential problem remains that non-linear hardening proves that similitude has broken down, and we do not understand the nature of this breakdown. Following [21], we may now specialize further, and note that the experimental observations in the “hightemperature” region are in practice well fitted by the

curve

mech&nisms will not be easily distinguished by measurir@ the activation energy for creep, since the ac~v~~o~ energies for the ~j~~t~on of an existing vainer in the bulk crystal and f”arthe formation and ~~gra~~ of a vacaneg in the ~~ghb~~r~ood of a d~~~~o~ are Mh about half the ac~~~tio~ energy of ~~f~~~~s~o~ by the formation and migration of vacancies in the bulk. However, the former process will hehnve as a dynamic recovery and the latter as a static recovery.

r = @ tn @*j&g.

(671

Rtperirnentally this quantity appears always to be positive for a sample under~~j~g monotonic deformatizln at a constant tern~~t~r~. We exclude hero the rework-soften~~g~’ ~he~o~e~ which can occur if a sampie is estranged nnder a new stress system, or when the sampb is re-strained under the same stress system at a lower ~M~rat~re~ Ex~~~ta~~y~ for a~~~i~j~~ in th% ~~~gh-~~~~tur~~’ region which is the characteristic form of power-law creep. There is no really satisfactory theory of the exponent 4.5. Accordirq to the model which wilt be suggested in Section 41, the rate of d~a~~ea~ recovery is control&d by the rate of an~~h~~at~onoF edge dipoles by chmb mediated by wicancies formed by the proe~~ssof plastic de~~~~~~~~~. The “natural kw of

WiXp &cm3

ws

is ex~~~j~~ by saying that the ~te-~o~trolli~g pro* eess is the a~~~h~~at~onof edge d~~~~ of height A in the eelI walk+, The lifetime of a dipole is of order h”;iin,, where U, is the coe&ient af ~~~d~~u~o~ by the vacarrcy ~~ban~s~~ Now P is j~~~e~y ~r~~~ t&n& to the d~s~~~~~~ density p, whib &IVis independent of p, Each dip&e a~~h~~a~ion releases ;I number af mobile dislocations proportional to p, each of which @des a distance equal to the cell width f) in a time ~~~~~~The strain rate Z is thus proportionat to p~~~~h~~ which is ~ropo~~o~a~ to p2& or, from equation (I), t0 p”jo. Since rrccp’% equation @%j follows. B* is ~ropQ~~u~~~ b the In the model we s formed by the c~t~~~~ of rate at which vacan forest dislocations of density p, i.e. D,Kp, and f65) is replaced by

r = e(ajC)~.&(fl*/c)‘2~s

@Q

where C is a constant, and it may by s~~ifi~~t

that

t2s - 7.8 = 4.7 wbitr: for the. same rnateriaf l/j4 = 46 f: KS. The observed rate of work harden@

is

As Hart er nf. [2i] point on& the vanishing of I+ appears to provide a d&et and natural exphifollowed nation of a region of ~~~~d~-state~9 by decelerating creep, without having recourse to extraneous considerations such as necking or recrystallization. The I&S of negative work h~~den~n~ during rno~~ iaa* (66) tonic ~tens~o~ at a #~sta~t temperature is unwhich is &.+ses to the observed ~wer faw than is the familiar in metaf plasticity, bet it is familiar in the theory oftbe ~~a$t~c~t~of lithium fhroride f?B, 6If and ‘“n2Stumf” law (Sj. OIif fbf fhp:Same of ~ate~~~s with the diamond structure &2]. He*, L. M. Brown fS9] hm power I;lw (66) is ~~~~~ if we amme that the r&e the ~~~t~~~~e~t~~~of d~s~~tjo~s a~~~~~~ lo pmof didocation climb is cantroiled hy pipe diiR&on of duce the imposed rate of strain reduces the flow stress vacancies along dislocations, since the number of by an amount greater than the work hardening of dislocations dislocation channels is proportional to 6’. The two produced by the a~~~~~t~on

NABARRO: 9r

OVERVIEW NO. 81

GERMANIUM

A

Fig. 13. Stress-strain curve of a single crystal of germanium. Redrawn from [62].

(Fig. 13). Ultimately, the continued a~umuiation of dislocations causes the flow stress to rise again. Yaney et al. [63] have clearly shown the competing influences of dislocation multiplication in increasing the number of carriers of plastic deformation and in increasing the internal stress and decreasing the dislocation free path by observing the transient changes in strain rate after sudden small changes in stress. The effect expected in a pure metal is shown schematically in Fig. 14(a). The steady-state relation between stress and strain rate is given by the full curve. After a sudden increase

‘ure Metal Type

1

LOGCStress)

Uoy

Type

tOGK%tressf

Fig. 14. The steady-state relation between stress and strain rate (full curves) and the transient behaviour during a rapid small change of stress (a) in a pure metal, (b) in a solid solution. From Yaney et al. [63].

1537

in stress, the effective stress on the mobile dislocations increases, their velocity and the strain rate increase rapidly, and the observed strain rate is greater than that in the steady state, as shown by the dashed line. Work hardening then causes the strain rate to approach that given by the full curve. In an alloy, the dislocation velocity and strain rate increase only slowly with the effective stress, and the abnormal strain rate falls below the steady-state curve, as shown in Fig. 14(b). Only after the number of mobile dislocations has increased as a result of the increased stress does the observed strain rate rise to the steadystate value. The observed curves for pure aluminium and for a sohd solution of Al-5.8% Mg are similar to Fig. 14(a) and (b) respectively, demonstrating that the considerations established for alkali halides and the diamond structure also apply to metals and alloys. This is a situation in which dynamic recovery is negligible in the experiment and totally neglected in the theory. Hart’s “high-temperature” region is one in which dynamic recovery is present both in the experiments and in the theory which will be developed. A mechanism for the annihilation of dislocations exists which becomes more effective the higher the dislocation density. The rate of work hardening therefore decreases rapidly with increasing strain, while the density of dislocations continues to increase. It is thus not unreasonable that rr* increases, but the ffow stress e decreases, at a constant strain rate i.

3.5. A double -valued hardness parameter We have seen that Hart’s phenomenological interpretation of steady-state creep, or rather of a minimum in the rate of creep at constant stress, is compatible with a physical model of dynamical recovery. However, it applears to be incompatible with the fundamental requirement for the existence of a hardness parameter, “that all hardness curves form a unique one-parameter family of curves for which no curve intersects another”. Before and after the minimum in the creep rate there exist two states of the material which creep at the same rate i under the same stress CT,but which are clearly different, because the creep rate decelerates in the former state and accelerates in the latter state. They must therefore have different hardness parameters a: and a:. The hardness curves with these two parameters intersect in the point (cr, e). The resolution of this paradox appears when, instead of drawing the family of curves [IS(~)],., we draw the surface a*(~, i). The situation which arises is familiar in the thermodynamic analogy. Suppose we draw the surface T(p, V) representing the equation of state, or the surface S(p, V), representing the entropy, of water in the range O”C-10°C. At constant (atmospheric) pressure, a given volume may correspond to T N 2°C or T u 6”C, and similarly for the entropy. The surface representing the equation of

1538

NABARRO:

OVERVIEW NO. 81

1-c 12 10

8 6

f 3.0 -

4

5 2.9 8 -J 2.8

2

2.7 -

/

*P A

* j .”

0

\ PWTI ’ ’ ’ ’ ’ -7 -6 -5 -4 -3 -2

Fig. 1.5.The surface T(p, V) for cold water.

Log Strain Rate (set-1)

state is sketched in Fig. 15. It is roughly a portion of a circular cylinder with its axis pointing in the direction (-2 x 104atm, 1 unit strain, 2~*C). It has two important properties:

Fig. 17. Two hardness curves measured at 270°C on highpurity abminium, ifiustrating the behaviour sketched in Fig. 12. From Wart [64J.

(i> The surface is single, smooth and regular in the relevant region. (ii) While a pair of values (p,V) defines T, any given pair of values (p, V) in this region either determines two values of Tor determines no values of T. The boundary between these two regions in the (p+ a/-)plane is a curve (approximately a straight line) which is the edge of the projection of the surface T(p, V) on to the (p, V) plane, and does not correspond to a singular line on the surface.

in the region of low and intermediate strain rates, the flow stress at intermediate strain rates increasing from 1 to 2 to 3. According to the scaling law explained below equation (31), these curves are derived from one another by translation parallel to the alfowed Iine. It is clear that each pair of curves has a common point in the (a, C) plane_ A practical exampie of this is shown in Fig. 17. If we consider a creep curve under a constant applied stress CT,the specimen first reaches curve 1 of Fig.16, where its strain rate L is decreasing. As it hardens, it reaches curve 2 where the strain rate reaches a minimum value, and then curve 3, where the strain rate has increased to its original value, the increased hardness being more than compensated by an increase in the number of mobile dislocations. Each point in the diagram below the arrowed line corresponds to two curves of constant hardness, and points above the arrowed line do not lie on any hardness curve. The hardness surface o*(a, <) is generated by displacing each curve perpendicular to the plane of Fig. 16 by a distance proportional to its hardness parameter Q*.

The surface S(p, V) has similar properties. Figure 16 shows the co~esponding situation in the case of hardness curves. Curves I, 2, 3 represent I

1

Fig. 1ft. Three hardness curves in the r@me of high temperatures and low strain rates. The flow stress at intermediate strain rates, and hence the hardness u*, increases from curve 1 to curve 2 to curve 3. At a constant applied simss 8, the strain rate first decreases as the sample hardens, and then increases. Redrawn from Hart et al, PI].

samples

The theory has been extended in a number of ways. Three-dimensional stress systems, plastic anisotropy, grain-bounda~ &ding and the analytic form of the working-harding coefficient have been included, and a block diagram representation of the spring-anddashpot type has been given [22]. The interaction of the strain dependencies of o and tr* [as discussed below equation (71)] with geometrical instabilities in the tensile creep test is anafysed in Ref. {64]. Reference [65] gives a dislocation model which can be governed by the equations of the theory, and consists (Fig. 18) of slip zones in an elastic medium. The

NABARRO:

OVERVIEW NO. 81

1539

4. MUGHRABI’S MODEL

Fig. 18. Hart’s model of slip zones in an elastic medium [65]. motion of the dislocations in the slip zone is governed by a law relating dislocation velocity to the resultant stress acting on the dislocations, and dislocations are assumed to “leak” from the ends of the slip zones. These developments do not affect the basic structure of the theory, which depends on the single internal parameter, the isotropic resistance to dislocation motion u*. The major development in Ref. 1721is that the reversible anelastic strain a and the corresponding directed back stress a,, are no longer treated as complications to be eliminated from the theory, but as essential components of the model. Since a and a, are taken to be related by cr,=Ma

(74)

where llrf is a constant modulus, this introdu#s a second internal parameter. On the one hand, as is recognized in [66], this brings the theory much closer to Mughrabi’s “‘two-parameter model” which we shall discuss in Section 4; on the other hand, it takes the theory outside the formal framework which was developed in Ref. 1191and found in Ref. [20] to be valid over a wide range of parameters in monotonic deformation. Hannula, Korhonen and others [67,68] developed a dislocation model whose plastic properties correspond to this analysis. The ends of Hart’s slip zones are represented by “strong obstacles” on the glide planes, while the mobility of dislocations in the glide zones is controlled by “weak obstacles” (Fig. 19). Both strong and weak obstacles become stronger as work hardening proceeds. The analogy with Mughrabi’s model of strong cell walls and weak cell interiors is obvious.

Fig. 19. A glide plane with strong and weak obstacles dislocation motion. From [68].

to

Mughrabi’s approach differs from that of Hart, Li and their colleagues in several ways. It is not a codification of experimental observations, but a theoretical analysis of a dislocation model suggested by experimental observations. Though the evolution during the course of straining of the dislocation structure which is considered must depend on processes which are thermally activated, the properties of the model are discussed in terms which do not involve thermal activation. It is essentially a zerotemperature system in which the temperature and the strain rate do not appear. Although its most recent development [16] is entitled “A Two Parameter Description of Heterogeneous Dislocation Distributions in Deformed Metals”, which suggests an analogy with Hart’s later model involving a * and a,, we shall see that it is essentially a three-parameter model in which a relation between the three parameters is established by means of a hypothesis concerning the evolution of the dislocation structure during straining. This hypothesis is admittedly speculative; we shall offer an alternative hypothesis which has a clearer physical basis, and which appears to lead to results in better agreement with experiment. 4.1. Mughrabi’s theory The first evidence that a plastically deformed metal contained internal strains which could be interpreted in terms of “soft cells” in which the internal stress opposed the external stress applied during deformation and “hard cell walls” in which the internal stress was in the same direction as the applied stress came from the X-ray studies of Culhty [69,70] in silicon-iron and in nickel. This model was confirmed by transmission electron microscopy, which showed cells of low dislocation density separated by walls of high disl~ation density. Mughrabi [71] developed it quantitatively, and he [72,14-161 and Pedersen et al. 1731proceeded to explore its consequences. The geometry of the specific model [Fig. 20(a-c)] would apply most precisely to a crystal of NaCl compressed or extended along (100) in double glide. The f(1 lO> edge dislocations shown in Fig. 20(a) combine to form (100) dislocations as shown in Fig. 20(b). The screw components moving normal to the plane of the figure have opposite helicities in the two systems, and the strains they produce conform to the strains produced by the (100) edge dislocations which are illustrated in Fig. 20(c). We take the width of the cells in Fig. 20 to be a, of which a fraction f, is occupied by the interior of the cell andf, by the cell wall. The mean dislocation density is p, and the densities in the cell interior and in the walls are pe and pw. The isotropic friction stresses, resolved on the glide planes, for the motion of dislocations in the interiors and in the walls are r, and z,, related to p, and pw by ~~= aGhpJ/’

(75)

f540

NABARRO:

OVERVIEW NO. 81

intern& stress can be reduced by annealing while leaving the flow stress on the glide plane unchanged. ~~ = uGbp tv’2 (76) This correspands to the fact that “transient softening” is resistant to annealing, whife “perwhere b is the Burgers nectar and u is about 0.4. manent softening” is easify distroyed. It is dso The dislocation arrays between the cefts and the consistent with our analysis of the resufts of “s#ress wagis in Fig. Z@(b) prod= tensile stresses AG, and AC, as in Fig. 20(c). These have ~m~onents in the dip” tests assembkd by L;loyd and M&troy {42j, which assumes that the magnitude of the internal glide planes of At, and AT, defined as positive if they stresses can be changed rather easily. In Mugbrabi’s assist the forward motion of glide dislocations. There are 7 paramters, D, S,, f,.z,,z,,AT,and AZ,, case the internal stresses do not affect the flow stress ~~CEUXX plastic flow is occurring in both cell interiors between which certain relations exist. and cell wall; in the dip test oniy the ceI1 interiors Geometry shows that deform plastically, and the changes in Az;. affect the (77) creep rate, while Ihe corresponding changes in 67, f,+.&J= f. are a~omm~at~ efasticaify. Equ~~ib~~~ requires that The Bow stress z in equations (79, 80) is an f, AZ, +f, Azhz, = 0. ml externally controllable parameter, but in equation Since both the cell interiors and the toll walls are (81) we have introduced an additional internal parameter p, We thus have 8 internal parameters, deforming plastically under an applied resolved shear related by the 5 identities (77), (78), (79), (80) and stress 5 @2), leaving 3 internal parameters which may r, + ATC= 7 (79) be reduced to 2 by a physical assumption on the development of the dislocation structure. and Mughrabi’s assumption in ft6j is that a process of ~~,+Az,=z @1>) a~~ib~lation of dislocation pairs occurs, predomiFinally, the mean dislocation density p is given by nantly in the cell wails, so that two dislocatio~s~ which will be of opposite sign because they are 631) moving in opposite directions, annihilate one another f,&f.AvPw=P if they pass within a distance y+ It is assumed that the which, from equations (67 and 68), may be written screw component of the dipole formed by the two &t: +f,5$ = ~~~~~2~. WI dis~~ations is of ~redo~n~t ~~~n~~ and therefore that y is inversely proportionaf to f, or to p jj’. It follows frcm (77), (B), (79) and @O) that No physical mechanism is given for the annihilation, (83) but Essmann and Mughrabi [18] have given strong f,% +fw%v = 2% arguments that such a mechanism must exist both far Although the internal stresses AZ, and AT, play screw and for edge components. The dislocatian an essential part in the physical model, they do not density in the cell walls will then have a quasiappear in the final formula (83) for the flow stress. stationary value if the expectation that a dislocation This is in agreement with the observation [59] that the and

Fig. 20, Mughrabi’s model of cell formation in doubk glider. (a) The motion of two sets af dislocations, (b) Dislocations of the two sets combine, actually or notionally, to form walls of interface dislocations. (c) The stresses and strains produced by the interface dislocations. (From [IS].)

NABARRO:

crossing a cell wall will be annihilated unity, i.e. f,Dp,,,y =f,Dr;y/a*G%* If we accept the assumption proportional to z,, so that

OVERVIEW

in that wall is = 1.

(84)

that y is inversely

y = K’!tw

(85)

f;,.r, = ct2G2b2/K’D.

(86)

then we find

Similarly fctc = ~~G2b~~K~D=fwz,,

(87)

Equation (87), in conjunction with (83), implies that the cell interiors and the walls contribute equally to the resolved flow stress r at all relevant stages of the deformation. The experimental evidence [74] is that, in [OOl]-orientated single crystals, the cell interiors contribute more than twice as much as do the walls at a strain of 0.068, this ratio falling to unity at a strain of 0.52. The theory also seems to be quantitatively unsatisfactory. If we eliminate t, between equations (76) and (861, we obtain p,, = u2GZb2/K”D2w

(88)

where 0, =fwD

(89)

is the thickness of a cell wall. With the numerical values given by ~ughrabi, this becomes pw = 4/D:, which means that, in a section such as that of Fig. 20(a), the number of dislocations in an area of cell wall height equal to its width is just 4, whereas electron micrographs suggest a density considerably higher. 4.2. A rnodz~cot~o~of Mughrabi’s theory These discrepancies probably arise because, as is pointed out in 1181, the annihilation of screw components of dislocation dipoles is relatively easy, and the rate-controlling process in the annihilation of the edge components. It is possible for an edge dipole of length 1 and height h normal to the Burgers vector to convert itself into a circular edge loop of radius (lh,‘z)‘:* with a considerable reduction in the elastic and core energies, by a process of pipe diffusion. Ordinary pipe diffusion has about half the activation energy of bulk diffusion, and will set in at a temperature not very different from that at which existing vacancies become mobile. However, ordinary pipe diffusion involves the interchange of atoms by the migration of trapped vacancies, a process which we may model as the formation of a jog pair at an existing geometrical jog and its subsequent migration. The process we are now considering involves the continued transport of matter by the migration of trapped vacancies, which we may model by the formation of two sets of jog pairs and the migration of one set. The activation energy to form two sets of

1541.

NO. 81

jog pairs and move one set is high, and we suggest is not likely to control the process which removes edge dipoles at temperatures close to Tu, although we have no quantitative argument for excluding this mechanism. We now discuss an alternative mechanism, which we believe to be the rate-controlling process in dynamical recovery. Here we de$ne dynamical recovery to be the process which occurs in the hightemperature and low-strain-rate regions of Figs 2 and 12, but not at an appreciable rate on their plateaux. We notice from Table 4 that, at the temperature at which dynamical recovery begins, the concentration of vacancies (and a fortiori that of interstitials) in thermal equilibrium is negligible, while their jump frequency has just reached values at which they could be expected to influence plastic deformation at conventional strain rates. We couple this with the generally accepted view that dynamical recovery (sometimes defined by other criteria) occurs at a rate proportional to the strain rate, and does not occur simply by the passage of time when the strain rate is zero. The evidence for this, evidence that, except at very high temperatures, edge dipoles can only be removed by their interaction with existing point defects, and evidence that the concentration of mechanically produced point defects in cell walls can become high enough to influence the process of plastic deformation, are summarized in the last two sections of Ref. [5]. The evidence of Table 4 supports this inte~retation. To consider the influence of point defects on the glide process, we concentrate attention on the screw components of the rectangular dislocation loops whose edge components are shown in Fig. 20(a). These loops expand in the interior of the cell, which we shall take to be a cube of side D. The average length of each screw side is D/JZ. We assume that this screw segment penetrates halfway through the screw wall of the cell, and, in doing so, it intersects DD,p, screw dislocations of the other set, producing D&p, jogs, which each drag for an average distance of D,/242. Each screw segment thus produces DDZ,p,,,/2J2b point defects, which, if we follow the argument of Hirsch [30], will mostly be vacancies, and therefore become mobile in the high-temperature region of Fig. 2. At the same time, an edge dislocation of the kind shown lightly drawn in Fig. 20(a) is produced. This edge dislocation is of length I), and, since the dislocation density p, is composed of equal numbers of dislocations of Burgers vectors b, , -b, , b2 and - b2, its distance from the nearest dislocation of opposite Burgers vector is 2~;“~. The number of vacancies required to annihilate this dislocation pair is p; ‘/‘D/b’, and in the quasi-steady state we must have DD;pw/2J2b

= p,“‘D/b’,

or D;/P,“~

= 4,/2.

(90)

1542

NABARRO: QVERVIEW NO. 81

The number ofedge dislocations in an area of edge wall of height equal to the thickness of the wail is Dip,, which from (90) is 4J2/bpzz. Both in its order of magnitude and in the fact that it decreases with increasing pw, that is, with increasing deformation, this result seems more satisfactory than the result 4 given by M~ghrab~s model. Remembering that D, =,f,D, and approximating (81) by p = fWpW we may write this in the form f$‘“D=p;l’=b = 4,/2.

(91)

Mughrabi [16] gives the empirirraI result pB* ~~280

(92)

which implies that a cell wall contains about 70 dislocations at all stages of deformation, If we insert (92) into (91), we obtain

fw= l/24SOpb2 = D2J~~~OOOb2.

(93)

Since D decreases with decreasing strain, this implies that f, decreases with increasing strain. This is compatible with the common view (possibly based on obse~atjo~s of samples strained at high temperatures) that the cell wails become sharper with increasing strain. However, the observations of Mughrabi et al. [74] clearly show that under their experimental conditions fw increases with increasing strain. We believe that under these conditions the dislocation density is increasing rapidly, and the dislocation structure is evoiving rapidly, contrary to our ~sumption of a quasi-stationary state. We suggest that (93) should apply in the iater stages of deformation, and that the limiting cell size will be reached when

f, = b/D.

(94)

Combining (93) and (94), we find that the limiting cell size is given by B/b = X8

(95)

which is too small, but not unreasonable in order of magnitude. Combined with the estimate of 70 dislocations in each cell wall, it would imply large rotations across the subgrain boundaries. These arguments assume that vacancies are generated in the screw walls and absorbed in the edge waifs. They will travel rapidly from their sources to their sinks by pipe diffusion along the disl~at~on loops. This mechanism was suggested by Davies et al. [75] as early as 1965. The rate-controlling process will generally be their migration through the lattice from an edge dislocation to an edge dislocation of opposite sign. The analysis of Mecking et al. [76] is based on the extrapolation of stress-strain curves to very high strains. It shows that the activation energy for dyn~i~1 recovery in this limit is not close to that for ~lf-d~~~s~on, but strongly correlated with the stacking-fauft energy, which controls the effectiveness of dislocations as vacancy sinks. our model is, at least qualitatively, compatible with their analysis.

When dynamical recovery first sets in, its rate is controlled by the rate of production of vacancies and by lihe migration of already-formed vacancies through the lattice, the activation energy being that of the latter process. In deformation to very large strains, vacancies are formed abundantiy in the cell walls, and it is likely that the rate~ontrolli~g process wit1 not be the supply of vacancies to edge dislocations, but their absorption, which is controlled by the stacking-fault energy. Whereas in Mughrabi’s model it is reasonable to assume that the processes occurring in the cell interiors are the same as those occur&g in the cell walls, this would not be reasonable in the present model. It would not even be appropriate to con~ntrate attention on the efimination of edge dipofes from the cell interiors, because these dipoles are readily swept by glide into the cells walls. The experimental observations are themselves confusing. Gil Sevillano et al. [77] review the behaviour of metals at large strains (roughly c > OS) in conditions of “cold work”, T/T, < 0.3, This corresponds to the plateau region in Figs 2 and 12. The principal results are that cells shrink with increasing strain, tending by a rather abrupt transition to a limiting size. “Cell walls sharpen with strain while cell interiors appear progressively cleaner”. The cell wall thickness decreases, while the volume fraction occupied by cell walls may increase with deformation. In the region of high-tem~raiure creep (T/TM S- 0.5, well into the high-tem~rature region in Figs 2 and f2), Bird et uE. [?8] emphasize that “the arrangement of dislocations within subgrains after creep is similar in appearance to the arrangements that are observed in cells after low temperature deformation, An even greater similarity is found between the high and low temperature dislocation substructures when quantitative counts of disclocation densities are made”, Figure 20 of their review shows that the dislocation densities “within subgrains after steady-state creep”” are proportional to the square of the applied stress, while in “cold work” “cell interiors become progressively cleaner”. Indeed, their Fig. 20 suggests that under conditions of steady-state creep the term fez, in equation (83) may account for most of the observed flow stress 2, while Fig, 10 of [74] suggests that this term contributes at most half of 7 after large strains at lower temperatures. The mechanism of dynamical recovery discussed in this section in inhibited in different ways at low temperatures and at high strain rates. At low temperatures, vacancies, even if present, are immobile. At a given high temperature, the mechanism implies that the excess concentration of vacancies is proportional to the rate at which they are produced, and hence to the strain rate. Excess vacancies are removed by dislocation ciimb at a rate propo~~onaI to their concentration. Competing processes which remove excess vacancies, such as combination with interstitials or the nucleation of dislocation loops, occur

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Mughrabi’s model is a closer representation of the actual dislocation distribution observed by electron microscopy, and it replaces the simple picture of slip zones in an elastic matrix (possibly appropriate for Stage I) by a more realistic and more symmetrical model of soft cells encased in hard, but still plastic, cell walls. In a crude approximation one may relate Hart’s stresses B* and u, to Mughrabi’s stresses 7c and -AZ,. Hart’s analysis depends essentially on the strain rate i. The temperature of testing also appears; it does not play an essential part in the structure of the theory, but is introduced by means of empirical rules for scaling C in terms of the temperature. Mughrabi’s analysis does not involve the strain rate or the temperature; it represents a section of Hart’s analysis at the value of i used in a conventional tensile test, and at room temperature. 5.2. Stage III hardening and dynamical recovery In introducing the idea of finding an atomistic interpretation of Figs 2 and 12, we suggested five temperature-dependent mechanisms: cross-slip of dislocations induced by vacancies in thermal equilibrium in the matrix climb of dislocations induced by pipe diffusion, i.e. by vacancies or interstitials trapped on dislocations (effectively jog pairs) and present in thermal equilibrium climb of dislocations induced by point defects produced in the matrix by plastic deformation thermally assisted cutting of repulsive forest dislocations.

I:bclimb of dislocations (ii’)

(iii)

Fig. 2 1. Modification of Hart’s slip zone model (Fig. 18) to allow for the presence of two equally stressed and (almost) orthogonal slip systems. The resemblance to Fig. 20(a) is

at a rate more than linear in the vacancy concentration, and therefore become important at high strain rates. 5. DISCUSSION 5.1. The relation between Hart’s models

and Mughrabi’s

While Mughrabi’s geometrical dislocation model of Fig. 20 is at first sight very different from Hart’s model of Fig. 18, the two are in fact very closely related. If the process indicated in Fig. 18 is occurring in a cubic crystal, there must be another set of glide planes making a large angle with the planes illustrated, and almost equally stressed. It is an obvious simplification to take this second set of glide planes to be perpendicular to the first set, and equally stressed, as in Fig. 21. The close relationship between Fig. 21 and Fig. - 20(a) ._ is obvious.

(iv)

It has become clear that process (ii) will be negligibly slow in the temperature range under consideration. We argued that process (ii’) might occur less readily than process (iii), but that it might not be easy to distinguish between them experimentally. It also seems clear that there are two distinct processes of cross-slip, the one which occurs at low temperatures and at the stresses occuring in Stage I, and the one studied by Bonneville and Escaig [6] which leads to coarse cross-slip lines and occurs at the higher stresses (though, as shown by the observations of Basinski and Basinski [1 l] on copper at 77 K, not necessarily at high temperatures) usually associated with Stage III. We will call the former occasional cross-slip and the latter abundant cross-slip. To distinguish their mechanisms, we remember that Friedel’s criticism [79] of the Stuttgart theory of cross slip included two elements which would make cross slip easier: (a) that cross slip could be initiated at a pre-existing constriction such as would occur at a unit jog or at the node of an attractive junction, and (b) that the release of energy when the dislocation

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OVERVIEW NO. 81

dissociated on the cross slip plane would facilitate cross slip. The factor (b) must always be present, and Bonneville and Escaig have demonstrated its importance in abundant cross slip. We suggest that occasional cross slip occurs when factor (a) is also present, and abundant cross slip when cross slip can occur without the assistance of factor (a), while recognizing that Escaig’s theory of cross slip [SO]was developed for the case of a pre-existing constriction. This interpretation is supported by the observation of Kuhlmann-Willsdorf et al. [81] that, while cross slip in gold is difficult on account of the low stacking-fault energy, the first dislocations to move in gold foils prepared from a sample quenched from 900°C and aged for an hour at lOO”C, and strained in the electron microscope, cross-slipped profusely. These grown-in dislocations are heavily jogged, while the dislocations produced during plastic deformation are extended. The list of likely mechanisms is thus replaced by: (i’) (ii’) (iii) (iv)

cross slip of dislocations at a pre-existing constriction cross slip of dislocations at general points climb of dislocations induced by point defects produced by plastic deformation thermally assisted cutting of repulsive forest dislocations.

If this classification is followed, process (ii’), which is thermally assisted but can occur even at low temperatures where point defects are immobile if the stress is high enough, controls Stage III. Process (iii) is stress assisted, for example because edge dislocations of opposite sign are found closer together when the stress in high, but it is thermally controlled, occurring only at temperatures where vacancies are mobile. It controls dynamic recovery, which is then a quite different process from the onset of Stage III. Process (i’) is responsible for occassional cross-slip. Both (i’) and (iv) are thermally assisted but stress controlled. They depend on the stacking-fault energy. They are indepent of the mobility of point defects, and can therefore occur at low temperatures. The jogs at which process (i’) can occur will normally be formed by process (iv) but a version of (i’) can also occur at attractive junctions which are formed without any energy barrier. The possibility that dislocation climb can occur below T/T, = 0.5 is not usually considered, and the direct experimental evidence for it is much less striking than the evidence for cross slip. This may be a consequence of the observational situation rather than of the processes actually occurring. Cross slip of screw dislocations is observed on the faces of a crystal where edge dislocations enter or leave. both primary and cross-slip steps are well marked, and the two types of slip trace make large angles with one another. Climb of edge dislocations is observed on the

faces of a crystal where screw dislocations enter or leave. The slip steps are in principle unobservable. They are observed in practice because of accidents such as the irregularity of the surface or the rupture of surface oxide films. The unperturbed primary slip traces and the traces of edge dislocations whose motion has a small climb component superposed on a large glide component are inclined at a small angle. The typical signature is a slight fanning-out or a splitting of the traces. Such divergent bundles were observed by Pfaff [82] in crystals of nickel alloyed with 50 or 60% of cobalt, in the former alloy only above 673 K (T/T, m 0.39), and in the latter at 873 K (T/T, z O.SO),and attributed by him to climb. These temperatures conform to the values of TX/TM in Table 2. The alloys used have very low stacking-fault energies. In aluminium, with a very high stacking fault energy, similar if less pronounced splittings of glide bands are to be seen in Figs 9, 13 and 15a of Cahn [83], and in the top sections of Fig. 7(a) and (b) of Miiller and Leibfried [84], both papers dealing with deformation at room temperature (T/T, z 0.31). The work of Pfaff [82] contains some support for our suggestion that two different processes, which we have called dynamic recovery and the onset of Stage III, may be involved in a range of temperatures centred on T/T, z 0.35. He plots the stress till at which “Stage III” begins as a function of temperature, and claims to observe sharp kinks in the curves at T = 525 K (T/TM = 0.30) for the alloy with 50% Co and at T = 640K (TIT, =0.37) for the alloy with 60% Co. If these kinks are real, they are evidence of a change from one dominant mechanism of recovery to another, both below the temperature at which appreciable diffusion can occur in the bulk. In summary, we suggest that process (i’) is responsible for occasional cross-slip. Both (i’) and (iv) are thermally assisted but stress controlled. They depend on the stackin~fault energy. They are independent of the mobility of point defects, and can therefore occur at low temperatures. The jogs at which process (i’) can occur will normally be formed by process (iv), but a version of (i’) can also occur at attractive junctions which are formed without any energy barrier. We have drawn attention to other possible processes; the complexity of the theoretical picture matches that of the experimental observations. Nate added in proof-The evidence of Lloyd and McElroy [42] that the rate of work hardening after a dip in stress during a creep test can exceed Young’s modulus is clearly confirmed by the observations of H. Nakashima and H. Yoshinaga on pure aluminium, in Creep and Fracture of ,??ngineeringMaterials and Structures (edited by B. W&hire and R. W. Evans), p. 15. Inst. Metals, London (1987). A~knowledge~nt~-in the course of this inv~ti~tion I have benefited from discussions with many colleagues, particulary Z. S. Basinski, L. M. Brown, J. L. Crawford, U. Essmann, J. T. Fourie, P. Haasen, E. W. Hart, Che-Yu Li, H. Mughrabi, D. G. Pettifor, S. Saimoto, A. Sceger and D. Stone. None of them can be held responsible for more than

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OVERVIEW NO. 81

a fraction of the tentative suggestions made in this paper. J. L. Crawford programmed Figs 9, 10 and Il.

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