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Work-hardening and work-softening of face-centredcubic metal crystalsA. Seeger a , J. Diehl a , S. Mader a & H. Rebstock aa Institut für theoretische und angwandte Physik der Technischen Hochschule, Stuttgart, andMax-Planck-Institut für Metallforschung, Stuttgert, Germany

Version of record first published: 12 Oct 2010.

To cite this article: A. Seeger, J. Diehl, S. Mader & H. Rebstock (1957): Work-hardening and work-softening of face-centredcubic metal crystals, Philosophical Magazine, 2:15, 323-350

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[ 323 3

Work-Hardening and Work-Softening of Face-Centred Cubic Metal Crystals t

By A. SEXIQEB, J. DIEHL, 6. MADEB and H. REBETOOK Inetitut fiir theomtiache und aqpwandte Phyaik der Teohniaohen Hoohachule Stuttgart, and Mtu-Planck-htitut fiir Metallfomchung, Stuttgert, Qermany

(Reoeived Bugnet 7, 1966)

b E T B A O T

THE preeent paper investigetee by experiment and theory the mechanisms governing work-hardening, work-softening and slipband formation in face-centred cubic metals. The rapid hardening stage (etage 11) of the atreae-strain curve is investigated by flow-streas meaaurements at different tempemturea, combined tensile and torsion experiments, and by the observation of the length of slip linea a8 a function of preatmin. Stage I11 of the atmae-strain o w e and the work-softening phenomena rtesocia.ted with it are investigated with the electron microsoope, showing that the principal surface patterns (slip ban&, fragmentation) am due to cross dip. It is argued that in stage JI the slip distance is decreesed con- t i nudy by the formation of Lomer-Cottrell dislocations. Thk ah30 acoounte for varioua observations on the end of the easy glide region. The temperature dependenoe of work-hardeniug in s twe 111 is caused by screw dialocationa circumventing the seasile Lomer-Cottrell dialocationa by cross dip. The mechanisms through which crost3 elip mums glide band formation and fragmentation are discuesed in some detail.

$1. INTBODUOTION RECENT studies of work-hardening of metale have concentrated on single crystals of faoe-centred cubic metals. It haa proved useful to distinguish three stages of the etreee-etrain curve which may be characterized i~

Stage I is the 80 called ' easy-glide region. The coefficient of work- hardening O=dr/& is rather small, O/Q (@=shear modulus) being of the aame order of magnitude aa in typical hexagonal metale (Zn, Cd) at low temperatures. It is probably justified to consider the work- hardening mechanism in stage I aa being malogone to the work-hardening in hexagonal metals.

The coefficient of work-hardening of stage II is about one order of mag- nitude larger than that of stage I and rather independent of the impurity content ; stage I1 seema to be charecteriefic for all face-centred cubic metah. The observation that the streae-strain curves of hexagonal metale

follows (fig. 1):

t Communicated by the Anthore.

SEB. 8, VOL. 2, NO. Is.-- I957 Z

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3 24 A. Seeger et al. on Work-Hardening and

do not show a stage witb a compondingly large coefficient of work- hardening suggeste that stage I1 is connected with the capability of face- centred cubic metals to slip on intersecting glide-planes.

Staye III is characterized by a decrease of 0 with increming strain. From the temperature dependence of work-hardening it icc obvious that in this stage the work-hardening process ie partially balanoed by a thermally activated process which will be d e d ' dynamical recove y '. It should be noted that this dynamical recovery proceeds during deforma- tion even at temperatures where no static recovery k observed.

The main object of the present paper is an investigation of the mechanisms of uork-hardening and dynamical recovery going on in stage I1 and stage 111 by experiment (work-hardening curves, combined tensile- toreion tests, Rurface observations by ordinary and electron microscope)

Fig. 1

Schematic drawing of the typical shear straw-shear strain curve of a face- centred cubic metal at two different temperatures T, and T,>T,.

and theory. We shall be but little concerned with stage I which is probably best investigated together with hexagonal metals such aa zinc and cadmium. The small coefficient of work-hardening, its dependence on the diameter of the crystal (Pateraon 1965, Suzuki et al. 1956)t and the appearance of very long (and rather faint) slip lines are beat explained by assuming that in stage I the slipdistance of the dislocations ia comparable with the diameter of the specimen (Mott 1962). We mention in pawing that contrary to stage I1 the etress-strein curve in the easy-glide region is rather sensitive to structural irmgularities (Dield 1966 b).

(1966 a). ?For 8 comparieon of the data obtained by different authors see Diehl

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Work-Softening of Fw-Centred C&ic Be&' Cry8lalS

$ 2. T H E MECHANISM OF WOBK-HAE~DENING IN STAGE I1

325

2.1. Survey The extent of the easy-glide region and therefore the strain cI at which

stage I1 begins depends very maxkdy on the orientation of the tensile axis with respect to the crystallographic directions of the crystal. c Z is smallest if the tensile axis is in a symmetric position. Several authors have s ~ g g ~ t e d that the onset of rapid hardening at the beginning of stage I1 ir, due to dip in secondary glide-systemst (Masing and Raffelaieper 1950, Rohm andDiehl1952, Lucke and Lange 1962, Haaaen and Leibfried 1952). A rather striking argument in favour of such an explanation M the temperature dependence of cp (Seeger 1964 a). We shell return to the theory of this observation later (9 4.2).

A more detailed consideration of the mechanism through which slip in secondary glide-syateme is able to increase the coefficient of work- hardening poses the following questions :

(1) IS the increased 8 caused by a continuous admixture of dip in secondary glide-systems, or ie the action of the eeoondary s p t e m ~ confined to the beginning of stage I1 only, aa proposed by Friedel (1966)?

(2) IS the main effect of the secondary glide systeme due to an inorease in density of the dislocation foreet which must be cut by the dislocations moving in the primary glide sp tem (Paxton and Cottrell 1964) or is it of a more indirect nature due to the formation of obstacles which cannot be overcome by the dislocations in the primary glide-system with the aid of thermal energy ?

These problem have been attacked experimentally by a combination of tensile and twisting experiments on &I single crgeth. w e preferred copper to aluminium beceuee copper shows at room temperature all stages of the stresgstrain curve fully developed, and because extensive data on the stress-strah curve of the same material are available (Diehl 1956 a). The main differenca between the technique of Paxton and Cottrell (1964) and o m wae that we avoided complioations due to the elastic core during twisting by using tubular cryetala. The external diameter of the crystals waa 6-6mm; the thickness of the w d was 0-76mm. For details concerning the preparation of the 6peoime1~ reference should be made to Rebetook (1966).

Problem (1) WM inveatigated mainly by meaaurementa of work- hardening in tension after twisting ($2.2). The basic idea waa to intro- duce by twisting ' artScially ' additiond &locations in eecondary glide- s p t e ~ n ~ and to simulate thereby the anticipated effect of glide in secon- dary apterne in &age II.

Problem (2) was investigated by meaaurements of the (reversible) change in flow s t m connected with a change in temperature (0 2.3).

t " h e octahedral glide-system with the lergeef mwlved sheer fitress T is called ' primary slip-~pbm ' ; sll the other one41 are denoted as ' secondary ' dip-systems

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326 A. S e e p et al. on Work-Bwdening and

Both techniques were supplemented by obaervationa of the surface appearance of deformed cryatala (3 2.4).

2.2. Measurements of Work-Hardening Figure 2 gives some aeleated data on shear streagehear strain curvea,

ourve A corresponding to an ordinary tenaile test. Curvee B and C were obtained by t e d e teats on crystale with the same crystallographic orientation aa crystal A. At the strains indicated by arrows the specimens were unloaded, twisted without removal from the apparatus and reloaded in temion. It will be seen that a single twist increeses the flow streas. There is no incresse in the rate of hardening during the subsequent tensile test. Dimegarding the behaviour of the streas-strain curve immediately after reloading in ternion, it may be said that the prinoiple effect of an

Fig. 2

sH61R S T R M 6 Tensile work-hrrrdening c m e i ~ of tubular copper crystah at room temperature :

The origin of c m e a A is shifted by dc=O.l ; B end C were shifted in addition to this in order to agree at c=cl with A. An m w 4 or

y being the d a c e &ear of the twist. indicah an interruption of the t e d e test and a twist of the d*

intermediate twist in stage 11 of the work-hardening curve is a pardel displacement of the stress-strain curve towards higher shear stresses by an amount depending on the shear of the twist (see daehed part of B and C).

Contrary to what should be expected on the b e of Friedel’s hypo- theeia, slip on secondary system occurring only once doee not result in an increase of the rate of hardening. Sinca however thejow 8tre.w is increesed by such a procedure it may be expected that repeated twists and therefore repeated action of seoondary glide-system do give an incresee in the average rate of hardening. This is verified experimentally by o w e Din fig. 2. After B shear strain c = 0 4 in tenaion

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Work-Softeniw of Face-Centred Cubic Metal Cqy8ta€% 327

crgefal D WM m p t d l y twisted by the same moun t y 88 indimted in fig. 2. The streee-etrain o w e obtained by the technique ss discussed above does indeed show an increase of its mean dope after r=0.2. We conclude from these experiments that the inoreseed rate of hardening in stage I1 is due to Coratind slip in seoondary glide-syetems.

2.3. Revereible Change of Flow-Stress with Ternperdure The following discussion wi l l be baeed on the theory of flow-stress given

elsewhere (Seeger 1964 a, b, c, 1966 a). The flow-strese of a pure metal can be divided into two contributions TQ and T~ acoording to

For B given dialooation pattern TQ depends on temperakure only indirectly through the temperature dependence of the shear modulus o! (or other elsstic constants) and hae ita main origin in the long range strees fields of the dislocatione of the primary glide-system. T~ ia the contribution of

7 = 7 Q + 7 s . . . . . . . . . - (1)

Fig. 3

H A R -N€

Division of the etreegatrain curve of a copper crystal deformed in tension at room temperature into TQ and 79.

the dialooation forest threading the primary glide-plane. Leaving seide certain oorreotione, T~ depends linearly on the absolute temperature T. 3 gives the diviaion of T into T~ and TQ 88 8 funotion of shear strain Q for a single-crystal of copper at mom temperatm. These results were obtained (see Rebetock 1966) by memuring the flow-etrees after ohanging the temperature between T,=QOo~ and T , = 2 9 3 ' ~ ,

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328 A. Seeger et d. on Work-Hardening and

and were evaluated using the theory of flow-stress as given for Cu (Seeger 1955a) and the temperature dependence of the shear modulus aa calculated h m single crystal elastic constants (Overton and G a h e y 1956). Figure 3 shows that whereas T~ and 70 are of comparable magni- tude in stage I the relative contribution of T~ to the flow-stress is sub- stantially reduced in stage 11. This difference between stage I1 and the beginning of plastic deformation ie shown even more directly (with hardly

Fig. 4

*m Mrn'

cbange in flow streee accompanying a change in temperature between 90"~ and 293% aa a function of the flow-etreee at 293'6 for copper single crystals. Full circles were ohteined in tension done, empty marks after one or two intermediate twista. The hatched area indicates which experimentel pointe would have been obtained if the increase iu flow Stress after a tWiet companding to e surface shear y=0-02 had been due

any theory intervening) in fig. 4, which gives the difference of the flow- stress T~ at T , = 9 0 " ~ and 7398 at T , = 2 9 3 " ~ aa measured by change-in- temperature tests, plotted against r a p s . The data corresponding to the end of stage II agree with those derived from the experimental results of Adam6 and CottreU (1966). This ie rather satisfactory these meaeuremente relate m d y to stage 111. For small strains we were unable to confirm the aaeertion of Adam6 end Cottrell (1965) that the ratio of the temperature dependent part of the flow stress to the total flow stress is independent of the pmtrain.

fo an inrreaee in 7s only.

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Work-Softening of Fw-Centred Cubic Me&d Crydab 329

-om the reeults of the tests with change of temperature we conclude that the increaaed rate of hardening in stage I1 is not due to a substantial increase in the density of the dislocation foreat. This is not surprising shoe the observation8 on the change in orystal orientation during slip (Taylor and Elam 1926, v. Goler and Saoha 1929, Sack and Weerte 1930) show that only a rather limited amount of slip in aecondary glide systems occura before the onset of double-alip. If slip in seoondary glide systems is at all reeponeible for the increaaed hardening of etage I1 it must caw (indirectly through the formation of obstaoles to dielocation movement which cannot be overcome with the aid of thermal energy) an increase in rQ. Whether thia is so or not can b6 investigated by measuring the temperature dependence of flow-etrees after twisting.

Figure 4 oontaina alao those results that were obtained in temion after tzoisting. They do not differ from those obtained without twisting. Thia impliee that the divieion of the total flow-strees into the contributiona by T~ and rs remaina practically unchanged by an intermediate twist, i.e. that the slip on aecondary system results mainly in an increase of TQ.

This holde even for large twists, since the data at ~ ~ ~ = 3 . 2 k g / m m ~ (indicated by a square) waa obtained after a twist corresponding to a (surface) shear of y=O-1. The other data were obtained with twbta correqonding to a h e m between y=O-OI and y=0.02. The shaded area in fig. 4 indicates which experimental values would have been obtained if the olmerved inoreeee of flow-stress by twista of y=O.O2 mre due to an

We may summarize th’e COllCI‘U8iOnB drawn in QQ 2.2 and 2.3 aa follows ; The rapid hardening in etage I1 and the inoreaae of flow-strees after inter- mediate twists am both due mainly to an inoream Of 7@ This is explained by the oapeoity of alip on secondary glide systems to form obstaclee which hinder dip in the primary slip systems rather effectively. (The nature of them obataolee will be discuased in Q 4.2.) The resulta preeented in 52.2 show that the formation of them obstaclee muet ooour continually during etage 11 (and not only at the beginning of this stage). We expect there- fore a continual reduotion of the slip dbtance during stage II. This otm be oheoked experimentally by e u r f m obeervations.

incresee in 7s only.

2.4 L@ O f r s E i p - L i ~ The d isowon of thia section will be baeed on the asaumption that the

surfwe markings of deformed fsoe-oentred oubic metala oorrelate with the slip and hardening meohhasism in the bulk of .the material. Thia seema to be well justified in view of the reaulta of other investigations in which such a oomhtion could be established (Diehl et d. 1966, Miiller end Leibfried 1986, Seeger 1988 a). The following obeemations mre obtained by dark field illumination with an ordinary miorosoope. Comparison with eleotron microscopy work showed that the ‘dip-linea’ visible under these oonditiona in stage I1 were arrangements of individual lines, the length of which is about half that vieible in the optical microscope.

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330 A. Seeger ed al. an Work-Hardening and

In oder to observe the variation of slip-distance with strain we have to study what we call the ' active length of slip-linea ' 1. This is the average length of slip-linea fonned at a certain strain. In the preeent case where the active length of &p-lines decreaeee with increasing strain it is rather different from the average of the length of d slip-lines that were formed during the prestrain. For thia -on we adopted the method of Blewitt el d . (1956) (in its 8hpl08t form fh t employed by Crussard 1946). It coneiste of poldung electrolytically the crystal after various prestrains 4,

thereby removing all tracea of the previous deformation, and observing the surfam pattern after 8 str8.h increment d c . Figures 6 (a)+) are photograph obtained by this technique from the same orystal (x0=490 ; A,=62" in the notation of Schmid and Bow 1936) after various amounts

Fig. 6

h i p r o d of active length of &p h e e 1 of copper crptab BB 8 fnnOt iOn of (ehea-) pmdrain in t e d e testA at mom tampmature. Additional shear etrains aRer polishing dc=O.O6. The orientationa of the tensile axits of crysfels A , E, and F are 88 indioated in the atemgraphic triangle.

of preatrain in tension. Fqpm 6 (a) shows the very long dip-linea of stage I (prestrain c=O). Figures 6 (b) and 6 ( G ) were taken in stage I1 with preetrains r=0.1 and r=0.2. Figurea 6 (d), 6 (e) and 6(& were obtained in &age III with preetrains of r=0.3 and r=0-6. They show the phenomenon of ' fragmentation ' (slip-line8 appear ta be intenupted, the fragments being dieplaced sideways), to which we shall return later

It can be seen from these photograph that the aotive length of dip- lines doee indeed decresee with increaeing prestrain. Following Blewitt et al. (1966) we have plotted in fQ. 6 1/Z (as obeerved on the top eurfwe of the OryBtal after dr=0.06) versus sheer atrain. In &age I1 thia F88ultB

(0 3-31.

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Work-Softening of Face-Centred Cubk Meid Cryetals 331

in stmight linea, the dope of whioh doe6 not depend markedly on orystal orientation. A t larger prestrains we note 8 somewhat stronger decrease of the active length of dip-lines with strein, the detaih depending on the

Fig. 7

SHEAR SlRAIN 6

Work-brdening curvea of the crystale A, El and F R~QWII in fig. 6.

Fig. 8

Reciprocal of active length of slip lines of a function of preetrain obtained with an additional sfrain of As=O-O&i after one twiet (B and C) or with two twiete (D). The atreee-etrein curvee are given in 6g. 2.

orientation of the cryatah. A comparison with the stre- - curve8 of them or@& (Q. 7) suggeeta that the deviation from the straight line starta at the beginning of stage III. In stage Ill fragmentation of slip

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332

lines OCCUTE (see Q 3.3). In stage I11 we have taken 1 equal to the length of the individual fragments (compm figs. 6 (d) and 6 (e)), following Blemitt et al. (1965). The experimental results given by these authors resemble those of crystal (A), although their crystal had a somewhat different crystallographic orientation (indicated by BCR in the stereographic triangle in fig. 6). The rise of l/Z with strain is not very marked for these orientations. This may explain why Blewitt et al. (1955) described the relationship between 1/Z and as being linear over the whole range of stages I1 and III.

Figure 8 contains the results that were obtained after twisting and refers to the same crystals as the streas-strain curve8 in fig. 2. It will be seen that the effect of intermediate twists on 1/E is analogous to that on the work-hardening curve. A single twbt in stage I1 results in a shift of the graph to larger l/Z values ; repeated twists however (curve D) give an increased slope of the 1/1 venue E plott, We may therefore say that the observation of the surface markings have confirmed the con- clusions drawn above on the basis of mechanical measurements.

A. Seeger et al. on Work-Hardening and

Q 3. DYNAMICAL RECOVERY AND WORK-SOFTENINQ m STAGE III

3.1. Survey

Ae was dearly shown by the results of Blewitt e2 al. (1956) on the varia- tion of the stresgstrain curve of copper single crystah on temperature, and aa may also be derived from the data of Andrade and Henderson (1961) on other face-centred cubic metals if proper allowance is made for the dependence of the shear st-ahear strain curve on orientation, the alightly temperaturedependent stage I1 is followed by stage J I I of the stmgetrain curve which depends strongly on temperature. Increasing the temperature shifte the shear stress T* (fig. l) , at which stage I11 replaoee stage 11, to smaller values.. A straightforward interpretation ie that in 6-0 I1 the dislocated structure of face-centred cubio metals varies with pleetic deformation but is independent of temperature. This has the consequence that T~ irc ah0 independent of temperature (if we neglect the temperature variation of theelastic constants). A temperature dependence of the dope 8 of the s t rewtra in curve may arise through a variation of the density of the dislocation forest with strain (Seeger 1964 a, 1964 b).

The very s m d temperature variation of 8 in stage I1 in copper stale is in full agreement with the experimental result (me Q 2.3) that the inoreaee

t For ~ 2 0 . 2 this curve waa obtained by polishing after the p* indicated in fig. 8, twiding by y=0.006, straining by d c 4 - 0 2 in temion, twieting by y=0.006, straining again by dc-0-02 in tension and obaerving the length of the dip linea in dark field illumination. By a suitable choioe of the conditione of illumination it waa possible to eliminate the effeot of the eliplinea farmed in torsion thereby retaining the dip-linee formed in tension only. For further details 888 Rehstock (1966).

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Work-Sofining of Face-Centred Cubic Metal Crystals 333

1 ~ . density of the dislocation forest is rather s m d t . Since at the trami- tion to stage I11 no abrupt change in the division of flow-stress into T~

and T~ is observed, we conclude that the temperature dependence of stage 111 is due to the change in the dielocated structure with strain (and thereby also d.r,/dr) being temperature dependent. Since unloaded copper crystals do not show meohmical recovery at room temperature, this change in the dislocation pattern during plmtic deformation must be due to the combined effecte of temperature and applied stress. Such a change is of course irreversible and is thus distinguishable from the reversible change of 78 accompanying a change in temperature. The irreversible changes manifest themeelves clearly in the phenomenon of ' work-softening I, studied in some detail on aluminium crystals by Cottrell and Stokes (1966). It seema to be generally agreed that the mechanism responsible for the work-softening phenomenon is the same as that causing the temperature dependence of 70 in stage I11 of the work-hardening curve.

1. Diehl et al. (1966) have propoeed and seconded by observations on the formationof alip bandsin face-centred cubic metala that stage I11 ia characterized by screw-dialocationa surmaunting obstacles in the primary glide eystem by cross dip.

2. Friedel (1966) and Cottrell and Stokes (1965) have propoeed that the temperature dependenoe is caused by dislocations breaking through obstacles on their glide-planes.

The main object of this paragraph is to decide experimentally which of these proposals is the correct one. ' We shall show, mainly by uae of the electron ~ ~ C I V E C O ~ , that the empirical reeults are in agreement with the croaa slip hypothesie but not with the breakdown hypothegia.

There are mainly two proposals for this mechanism :

3.2. Formation of Slip Bands Before attacking the questions outlined in 0 3.1 we shall briefly discuss

the formation of slip bande. h waa first shown by Heidenreich and Shockley (1948) and later Btudied in detail by other authors (Brown 1962, Wilsdorf and Kuhlmann-Wilsdorf 1962 a, b) the btrong ' slip-lines ' visible on the surface of polished aluminium cryatale deformed at room temperature can be resolved by the electron microsoope into a cluster of individual slip-lines. Henceforth we ahall refer to these individual lines, whether clustered into slip bands or occurring aa isolated lines, as slip lines.

For some time it waa believed that these broad slip ban& were typical for deformed face-centred oubic metala. Diehl et d . (1955) wem able to demonstrate that they are typical for stage 111 only. They showed that

t There may be metele (aluminium aa a face-oentred cubic metal with high nttacking-fault. energy being poeeibly O D ~ of them) where because of the small latent hardening (Seeger 1U56 c) the didooation forest contribuh si@cantly to the inoresee of the flow-strese with prestrsin. This would give rise to a temperature dependence of the mork-hardening coefficient in &age II.

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334 A. Seeger et al. an Work-Hardening and

aluminium crystals deformed at the temperature of liquid oxygen, and copper and nickel crystals deformed at or below room temperature do not exhibit clustered slip bands but a pattern of rather evenly spaoed individual slip-linea of varying depth, provided the plestic deformation waa not too extensive (fig. 9 and fig. 10). By polishmg and strain- increment experiments slip band formation WBB shown to be a stage IIL phenomenon (fig. 11 and fig. 12).

The correlation of the first appearance of slip bands with increasing strain and the beginning of stage I11 for various metals and different temperatures waa so close in these experiments that it wm very natural ta conclude that the formation of alip bands is caused by the same prows that is responsible for the temperature dependence of T@ as diacusaed above (Diehl et al. 1956). Cross slip had indeed been proposed earlier w a posaible baaic m e c h h m for dip band formation (Koehler 1952 a, b, Leibfried and Haaaen 1964, Seeger 1955 b).

The reaaon why on aluminium crystals deformed at room temperature (or above) slip bands form at rather early stages of deformation is 88

follows: At a given temperature T,/Q depends on the staaking-fault energy of a metal (see Seeger 1966 c) in such a way t h t it is the smaller the b h e r the stacking-fault energy of the metal under consideration 7. The stacking-fault energy of aluminium is so high that T~ at room temperature is of the same order of magnitude aa T ~ , stage I1 of the work-hardening curve thereby reducing virtually to an idexion point in the streae-strain curve. Aluminium cryetala show therefore typical stage 111 behaviour already after rather small deformations at room temperature.

3.3. Fragmentation of Slip Bands Blewitt et al. (1955) have demonstrated that the slip markmgs visible

on copper crystah in dark field illumination in an optical microscope extensively deformed at mom temperature show ' fragmentation ' (see e.g. figs. 6 (d) and 5 (e) ). By a detailed investigation of this phenomenon we found that extensive fragmentation ocourred only if the deformation was carried right into stage III. In stage II no fragmentation w w observed. The beginning of the fragmentation appeara to be connected with the rice of the 1/E versus E graph (compare fQ. 6). Since the beginning of fragmentation seem to be related to the begin-

ning of stage 111 $ it wee thought worth while to investigate fragmented slip bands by the electron microscope. The electron microscope employed in the experimental part of this paper WBB an Elmiskop I (manufactured by Siemens Berlin, maximum voltage 100 kv). Three typical photographs are shown in figs. 13, 14, and 16. They were obtained from a oopper

t The e t m * curve for Al et -186'0 in fig. 2 of (Djehl et al. 1956)

$ Thia ie in line with the o b e t i o n of Blewitt e4 d. (1966) that wpper has been miedrawn ; in reality stage IIJ begins at ~a=1.4 k g / m L .

cryetsle deformed at 4.2% show neither &age III nor fragmentation.

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Wmk-SOft&ng Of F ~ - C m e d Colbic Metac Cv& 336

crystal that was polished after a prestrain of c-0-6 at mom temperature, strained in addition by an increment in shear strain of d c 4 . 0 6 , and treated with the same replica teohnique aa d by Diehl et al. (1966). The replioaa were shrdowed with palladium in suoh a way that the pm- jeotion of the shadowing direation on the surface of the replioa formed a rather acute angle with the traces of the primary glide aptam.

Figum 13 shows that fragmentation pattern may indeed oontain traces of one of the secondary slip-planee. In thb caw 88 well 88 in all similar onee these h o e s ooinoided with the tram of the omaa dip (111)-plane aa predicted from the oryetallographicorientation of the cryetal. Qure 14 shows that between the Sajacent ends of two frcrgments traces of both the primary and the oroes-dip plane may ooour in a kind of ' staircaee pattern'. Figure 15 waa included to demonstrate that sometimes only the primary dip-plane oas be observed. Comparing this set of photographs there oan be little doubt that these tram are alao connected by c r o ~ slip, the crow slip lines however being so weak that they could not be made visible by our technique. A general idea of the strength of the crow dip linee between two fragments may be obtained by observing that the method of shadowing waa euoh aa to favour the cmm slip t ram. We conolude from this that the individual crow slip linea axe rather weak aa compared with the slip lines of the primary slip-sptem contained in the slip bands. Sometimes one gets the impression that the crow slip is distributed over a certain region rather than concentrated into individual lines (see e.g. fig. 16).

Comparing the hgmentation of the type shown in fig. 16 with the clustering of dip-lines into bands, one gets the impression that there ie an intimate relation between them. The mechanism by which they me formed could well be the same. The electron microscope techniques at present available give little hope that cmaa slip between the slip linea of a slip band oan be made visible. Since however slip bands and frag- ments are of closely related appearance and sinoe both of them begin to occur at the beginning of stage 111 we feel justified in ooncluding that the formation of slip bands is due to croaa slip. This conoluaion had been reached earlier on the basie of somewhat more indirect arguments (Diehl et al. 1966, Seeger 1966 b).

The general impreaaion from both ordinary microsoope and electron microsoope photographs i~ that in stage III the dislocations overcome the obstaclee in the primary glide-plane not by breaking through them aa suppoeed by Friedel (1966) and Cottrell and Stokes (1966) (see the diaousaion in 5 3.1) but by surmounting and oiroumventing them. This argument oan be made quantitative on the bsaie of the reeults presented in fig. 6. As already mentioned the active length of dip lines I in stage 111 was taken equal to the length of those portions of dip bands that follow a unique tram of a dip plane, counting the fragments eeparately. On the baaia of the breakdown mechanism I should start to increase at the begin- ning of stage m. The experimental resulte show that oontrarg to thb

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336

the onset of stage III is marked by an accelerated decrease of 1. Even if it is admitted that the possibility of some of the weaker slip-lines invisible in stage I1 becoming viaible through their sideways growth into slip bands might contribute to thie acceleration it may be said that this observation contradicts flatly the breakdown hypothesis.

A. Seeger e.t al. on Work-Hardening and

3.4. Work-Softening It was already recognized by Cahn (1961) on the basis of investigations

with the optical microscope that crow slip may be a prominent process in the deformation of aluminium single crgetals at elevated temperatures, The surface pat tern called cross elip by Cahn can be made very pro- nounced even at normal temperatures by work-softening of aluminium crystals (Cottrell and Stokes 1955, Kelly 1956). The work-softening procedure coneiets in pmtraining a crystal at a lower temperature, until

Fig. 17

1 I - e

Strea3-strain curves of a face-centred cubic metal crystal showing the phenome- non of work-softening when changing the temperature of the temile teat from T, to a higher temperature T,. The dotted line gives the streas-strain curve at Ts. (Schematic.)

a flow-stress 7 is reached which at a higher temperature would correspond to stage 111. If the crystal is unloaded, warmed up to this higher tempera- ture and strained again a stress-strain curve of the type shown in fig. 17 is observed. The explanation is that the work-hardened state reached at the lower temperature is unstable under the combined action of the higher temperature and the applied stmae 7. We have already mentioned in 8 3.1 that the mechenism by which the unstable part of T~ is reduced in work-softening is thought to be the same aa that responsible for the decreaae of the rate of hardening in stage 111.

Figures 18 and 19, which were obtained on aluminium crystals pre- strained at the temperature T,=90"1( as shown schematically in fig. 17,

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Work-Softenng of Face-Centred Cvbic &fetal C+yetal% 337

polished, and further strained at room temperature (!Pa), give the typical surface appearance of a work-softened crystal. At small m@cations the slip-bands appear wavy in the ordin- microscope (fig. 18). Larger magnrfication (fig. 19) shows that the slip-bands coneist of straight pieces connected with what haa been termed ' prominent orom slip ' by Cahn (1 951)t.

It must be streased that this type of cross slip visible under the optical microscope only rarely follows the trace of the cross slip plane. Sometimes i t doesnot appear to follow at all the trace of a crystallographic plane but to be of a non-crystallographic nature. In some cmea Cahn waa able to resolve such non-crystallographic cross slip into a zig-zag pattern by using high magnification of the ordinary microscope.

Definite results on these problems which can be used to discriminate between varioua theories of work-softening can be obtained by the eleotron- microscope. Figure 20 shows an electron micrograph of 'non- crystallographic ' cross slip. It will be seen that it is resolved into a ' stair-caae ' of slip on the primary slip plane and on one of the secondary planes. The trace of the secondary slip plane w a ~ identified in this (and in all other cases investigated) aa that of the crotw slip plane. This is strikingly demonstrated in fig. 21, which ie an electron micrograph of an aluminium crystal that showed double slip during work-softening. Slip bands were formed both by the primary slip system (dark traces) and by the conjugate slip system (white traces). The primary slip system and the conjugate slip system have their crom slip planes in common. The glide directions of the cross slip systems are different however. Thk is borne out in fig. 21. The cross slip traces of both types of slip lines are parallel. Since the glide directions in them traces and therefore also the nature of the surface steps are different they appear in Merent colours due to the shadowing technique. In the croas slip pattern of aluminium crystab work-softened at room temperature and not suitably oriented for double slip we never observed more than two different slip-planes, corresponding to the primary slip plane and the cross slip plane.

Cross slip can be made visible in the optical microscope also on copper crystals, ae shown in figs. 22 (a), (6). They were obtained by preetraining r=0.6 at the temperature of liquid oxygen, polishing, and work-softening at 300"c with an additional average atrain of Ac=O-OEi. As WBB to be expected the crow slip is less marked than on aluminium crystals work- softened at room temperature. This is confirmed by the electron micro- graph fig. 23, which shows an appearance intermediate between the fragmentation of copper crystah deformed at room temperature (compare

t We think that the so-celled ' intimate c r w slip ' observed by Cahn (1961) on aluminium M eesentielly the same phenomenon 88 the fragmentation dis- cuesed in § 3.3. It WAB raolved by \ilSdorf and Kuhlmann-Wilsdorf (1953 b) nith the electron microscope info individunl lines of croa slip, which were &o rather week compared with the slip lines cluseered into bands.

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338 A. Seeger et d. on Work-Hardening and

fig. 13 to 15) and aluminium crystala work-softened at room temperature (compare fige. 20 and 21).

We were unable to find on work-eoftened face-centred cubic metal cryatah those very long dip ban& (or dip-linee) that can be obaerved by the optical miomcope on plastically deformed single oryatclls of zino and that would be expected if work-softening were due to a bmak down of barriera in the primary glide plane. The aurface pattern was alwaya determined by a wavy or fragmented appearance of slip bands whioh with the aid of the electron microscope could be resolved into a sequence of slip in the primary slip system and in the crow dip aystem. Thia shows that under the conditions of work-softening (i.e. high temperature and high stmaea) it ie rather easy for the dislocations to change over from their primary slip plane to their c r w slip plane and back again to the primary plane. The observationa on work-softened cryetah therefore fit very well into the picture which we have drawn for the processes responsible for the dynamical recovery in stage 111.

Fig. 24

Rectangular dielocation loop. L, and L, are dip distanoes of edge and' somw dialomtiom.

f 4. THEORETICAL DISCUSSION 4.1. General Dakcu.a&m of Work-Hardening

The discussion of our experimental findings given in 5 2 and f 3 was of a somewhat phamomenologicd nature. For example, we have been talking of the effect of glide in secondary glide-systems on the propa- gation of slip in the primary glide system etc. Our present taak is to translate these results into the language of didooation theory and to fit them into a unified picture of the dislocation proceesee during the plastic deformation of h - c e n t r e d cubic metal crystals.

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Work-SofteninS of Facecentred Cubic Metal Cry- 339

We have 888n that the flow-stress r(6) of a metal oryetal oan be divided into two contributions TQ and 7g accordlng to eqn. (1). In genera,l T~

is inversely proportional to the mean di~tance between dislocation Lines penetrating the primary alip plane and therefore proportional to the square- root of the density of these dislocations (Seeger 1964 a, 1966 a). T@ is somewhat more di6cult to interpret quantitatively. It may be said however that it is the larger the smaller the mean dietanoe between dislooations in the primary glide syatem is. The ooefficient of work- hardening due to TQ, whioh will be denoted by $Q&T&, is related to the mean slip dirrtanoe L of dislocations in the primary glide system. The strain increment is proportional to the product of the density of gliding dielooatiow end the slip-dietanoe. The inaream in flow-~tress however is independent of the distance L slipped by the dielocations before they get stuck in the crysfal. Ba ie therefore small for large L and Vice uer8a.

In $ 1 we have briefly mentioned the characteri8tic faturea of S- I. Both the very long and rather weak dip-linee and the small coefficient of work-hardening (in typid ~8888 of pure metah BIG being of the order 2 x 104 to am readily explained by essuming that the mean alip- distance of the Wooatiom in stage I is rether large (of the order of 0.01 om to 0-1 om).

the following discuseion of slip-distanoes and work-hrtrdening we consider the dislocation rings in the glide plane to have the shape of rectangular loop^. The alip-dietances of edges and smws in such a loop am denoted by L, and La. The total number of such loops per unit volume is d d N.

The observations reported in f igs. 0 and 8 give the aotive length of dip lines I on the top surface of the crystal. We put them equal to twice the mean-slip distance L, of sorew dislocations. Accordingly the active length of slip linea on the eide EW~~MXJ will be idenlSed with twice the dip-distance L, of the edge dislocations. The procedure of oorrelating the length of slip linea m observed on the E W ~ ~ W of the crystal with the slip- distances of the dislocations in the bulk of the materid is justified by the consistency of the results we shall obtain on the ba& of this essumption preeently. Our observationa of I gave 1=0-20 cm at the beginning of stage 11

(fig. 6). This is in good agreement with the discuseion of stage I given above. During stage 11112 (for which we now Write La) decreases with increaaing strain aocordmg to the law

To

La=As/(;-e*). . . . . (2)

e* is shghtly smaller than €8 and E ~ O W E a similar dependence on crystal orientation as this quantity. A typical numerical value for a crystal in the oentral part of the orientation triangle is 1 1 8 ~ 4 ~ lo4 om.

The available observations on the length of slip lines on the side surfme are leaa detailed than those on the top ~urfrcce, Since the h t mentioned type of slip-linee is more difiicult to obaerve. It can however be said that

83pB. 8, VOL. 2, NO. I5.--"BoH 1957 Z A

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340 A. Seeger et d. on Work-Hardening and

they were dwaya about twice aa long &B t h m on the top emface. can therefore write

We

. . . . . . . . L,=AJ(€--E*) (3) with A1-2A2.

In copper cry&& deformed at mom temperature, stage I1 of the work-hardening curve can be well represented by a etraight line (Blewitt et d. 1966, Diehl 1966 a). Aomrding to the diecumion in Q 2.3, 8 is not very different from 8, in this cam. It wma to be a reasonable extra- polation from experiment to coneider 8, to be conatant in stage 11. The constancy of 0, and the validity of the empirical law eqna. (2) and (3) can be related and undemtmd with the following model of work- hardening (Seeger 1956 a).

8, ie considered to be due to etreee fielda around ahtietically dietributed groupe of a e m d number of dialocatiom. The amplitude of the straw field generated by the edges of the dielocation loope can be written aa

N L , is the number of edge-dielooations of one aign per unit area and ia therefore inversely proportional to the equare of the mean dbtmce between theee edge dialocationa. a , takee care of the numerical factom involved. It dependa on the d e W of the goometrical arrangement of the dislocations, in particular on the effective number n of dieloca- tiom in the &location groupe. This quantity is characterized by eaying that the far reaching streaa field around such a group is that of a tingle dislocation of strength nb. If 7, is due to a distribution of individual dislocation lines (n=l ) which ia probably a good approximation to the situation in undeformed cryd id (see Seeger 1964 a, c), a, should be of the order 1/5. The infiniteeimal inoreaee dr of the shear-drain due to dN dialocation loope per unit volume moving through the alipdistancea L, and La ia given by

.r,=a,bQ(NL,)u'. . . . . . . . . (4)

In the preeent c888 we expect al= 1/6 nus.

dr=bL1 L , dN. . . . . . . . . (6)

When these loope get stuck in the c r y a t a l they will contribute to the etrese field in the crystal and therefore caw a change in 7,. The work- hardening coefficient of this change is

If we eliminate L , and N from eqn. (6) with the aid of eqna. (2), (4), and (5), we obtain the differential equation

drO a,Wb r-r* dr 2Al ' 70

(7) -- -- -- . . . . . The general eolution of thia equation ia

6 .r42 = - *+ a l a P - ( c - r * ) a . . . . . . (8)

c f-! . 3111

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W~k-Lsoften;ns Of F ~ - C e n t r e d Cubic J f d CTY& 34 1

Since the streee at the beginning of stage 11 must be finite we have to put the constant of in tep t ion C equal to zero. The p h y a i d y sdmieeclble solution of eqn. (8) is a linear sfress-atrein curve with the coefficient of work-hardening

b eq3al . B . (m- . . . . . . .

The same type of treatment holde for sorew-dielooetiona with d x e a 1 and 2 interchanged.

The obeerved mffioient of work-hardening of h-oentred cubic met.& is in stage II of the order 8,=4 x lo4 0. Inserting the Al-value given above we find from eqn. (9)

Oa/cf=al . 3 x loJ. . . . . . . . (10) We conclude that al ia of the order 1 and n therefore of the order 20 to 267. The implication that n doea not vary very much with strain in stage 11 ie in line with the d t of Blewitt et d. (1966) on the electrical reeirrtivity due to dialocationa as a function of flow-strew. The electrical reaietivity ie under very general aaaumptiona proportional to the number of dialocationa per unit areax. The experimental reault ia that it ia proportional to the square of the flow-streas both in stage I1 and in stage 111. Tbh shows directly that eqn. (4) with al fixed ia applicable not only in stage I1 but also in stage 111. A more complicated expreaaion than eqn. (6) will have to be used for the strain-increment in stage nI in order to allow for the contribution of the glide in the c r w alip system.

4.2. Formation of Obstacles to Slip in the Primary Qlide-Plane : Lomer-Cottrd Dielocatione

From studiea of the change of orientation with strain it ia known that slip on aecondaq syeteme contributes very little to the glide in face-centred cubic crystab the orientation of which is in the interior of the stereo- graphic triangle. In stage I1 the number of dialocetiona in secondary glide-systema ia therefore in general smal l compared with the total number. The Iarge effect that those didocations ham on 8, must therefore be due to .a,' catalytic" wtion. We have already (f 2.3) pictured thie aa the formation of obatacles in the primary glide-plane which cannot be over- come with the help of thermal energg' under the strees conditions of stage 11.

inoreaae the coefficient of work-hardening. The only type of obstacle in fihx-centred cubic metala known at present which fulfils them require- mente is a special kind of s d e dielocation, the so called Lomer-Cottrell dislocation (Lomer 1961, CottrelllQ62). h m e r - h t t r d dislocations were

t This numerical 'eetimate ie not foo reliable gince the ratio between the d i v e length of dip lines end the mwn dip-dietenoe may be somewhat different from two.

$ For p808nt diaoumiode of the effeat of dislocetiom on the electrid reeirrtiVity of mowvdent metale ~ 8 8 S e e p end Stehle (1966) and k g e r (1966 b).

Such bbshcl- would dec- the a l ip-d ie tan~ Ll and L , md themby

2 A 2

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342 A. k g e r et d. on Work-Hardening and

used in the discuaaion of work-hardening by Mott aa early aa 1962 (Mott 1952) and later by Leibfried and Haaeen (1954). They play also an essential r61e in the views on work-hardening and work-softening held by Cottmll and Stokes (1965) and Friedel (1966).

Lomer-Cottrell dislocrttiona are formed along the intersection of two (111)-planes and lie therefore along (llO)-directiom. The detailed arrangement of partial dielocktiom and stacking-faulta ia aa shown in fig. 26. (Thie ia the most important type with which we shall aolely be concerned. Further typea which are leaa ' stable ' are discueeed by Haasen (1953) and Friedel (1966).) The eesential feature ie that the stacking-fault ribbon ' bends over ' fiom one octahedral plane to another. The combination of partial dielocationa is confined to both of them planes and can therefore neither glide nor climb away from the line of intersection of theae plan^. Since them d e didocatiom can be removed only with a considerable expenditure of energy they form effective obstacles to the glide motion of dielocations meeting them.

Fig. 25

Lomer-Cottrell d e dielocation, consisting of three partial edge dialocations connected by stacking fault ribbons.

The LomerXottrell dislocation shown in fig. 26 can be formed in two waye, namely either by the reaction of a dielocation of type 1/2 [OTT], (11T)withoneoftype 1/2 [11'0'J,(l~),orbyareactionbetweenl/2[1TO], ( I l r ) and 1/2 [Olr ] , (lrr). The dialocatione of such a pair attract each other through their long-range atreas fielde. The Lomer-Cottrell di~locct- tion formed aa the result of the didocation reaction between the dislocations

~~~ ~~ ~

t W e denote here dielocations by giving their Burgers vect~m and their glide planes. They am u n d e w to run e l e l to the Loner-Cottrell didocation.

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Wmk-So&nning of Fw-Centred Cubic dletal Cy&& 343

of each of them paire ie ' stable ' becam energy ie required to decompoee it into the original dislocations and to separate thew from eaoh other.

The following diecumion will be bamd on the convention that the primary glide ayetern ie 1/2 [lo1 1, (111). This ie the octahedral syetem of maximum shear strese if the tensile direction lice in the interior of the triangle cmm hatohed with two perpendioular seta of linee (fig. 26).

Fig. 26

Combinations of glide-eyeteme capable of yielding LomefcottreU &le The direction of the L o r n 4 t t m U dielooation I indicated didooations.

by the diredion of hatohing.

The range of orientationa in which the shear streee is a maxhum in the other glide-system with glide plane (1lT) are alao c m hatahed. The dislocations of the glide-systems with glide-plane (1 11) can form Lomer- Cottrell dislocations with those glide-system whom trianglm of maximum shear etreee am hatched by the same set of lineat. The direction of the Lomer-Cottrell didocation is uniquely related to the direction of hatching and ie the mme for full and for dotted lines aa indicated in fig. 26.

t Them are 12 odehedral glide apterne and 24 trianglea in the graph of fig. 26. Therefore two trianglea of madmum ehear etreee belong two eaah of them glide-g?&rna. We have hatched both trianglm of euch a pair only in the caw where both are neighbow to the trim& belonging to the primary glide system.

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344 A. Seeger ei d. on Work-Hardening a d

Figure 26 k useful for a qualitative discmion of the relative resolved shear stress in different glide-systems. The smaller the dietance between the point representing the tensile axis and one of the triangles belonging to a certain glide eyetem, the larger is the reeolved shear stress in this glide-syetem. For quantitative g raph id representations see Diehl et d. (1964).

For the generation of an appreciable number of Lomer4ottrell dis- locations of a certain crystallographic direction it is prerequiaite to have enough mobile dislocations of the typea cooperating in their formation. This will be the caae if the ' effective ' resolved shear stress (applied ahear stress minus the latent hardening in the dip ayetern concerned) acting on these dislocations is larger than the critical shear stress. Since not very much is known about the latent hardening in stage I the following dis- cuasion k qualitative only and does not allow for the differences in the latent hardenings of different glide-systems.

Extendmg the idea firat proposed by Rohm and Diehl (1962) we think that the transition from stage I to stage I1 begins when the applied stress becornea large enough to generate for the firat time a considerable number of dislocations in secondary glide-sptems which a m able to form Lomer- Cottrell dislocations lying in the main glide-plane. This stress will in general, e.g. in copper, depend on temperature, Bince at leaat part of these dislocations have to cut the forest of dislocations in the primary glide- systems which has been built up during atage I. The resolved shear stress in the secondary systems correeponding to an appreciable production of dialocations mcreaees with decreasing temperature. Since the slope of the easy glide region of the s t e a i n curve is roughly independent of tempemture this meam that at a given orientation of the tensile axis the easy glide region should extend to larger strains z s at lower temperatures than at hrgher ones (Seeger 1964a). !I'hia k exactly what is observed experimentally (see fig.. 1).

With the present picture it is possible to understand the variation of with the crystallographic orientation of the tensile axh, too. The

orientation dependence of in the c888 of copper single crystale deformed at room temperature is given in fig. 27 taken from Diehl (1966a). Its charactariatic feature is that the largest values occur for orientations in the [llO]-corner of the orientation triangle but excluding the [110]- directibn and ita immediate neighbowhood. This is contrary to the orien- tation dependence of the critical ahear stress which takea its minimum values in the middle of the stemgraphic triangle (see fig. 27). Ow explanation of the variation with crystal orientation ia as follows : In the whole region adjacent to the plane (OlT) (which containe the [loo]- direction and the [Ill]-direction) there k a tendency to form those Lomer-Cottrell dialocations characterized in fig. 26 by the hatching with vertical full liners. They require dislocations in only one of the secondary systems, namely the so-called oonjugate system. In the immediate neighbowhood of the [I 101-direction of the tensile axie there is a tendency,

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Work-Softening of Facecentred Cvbic Metid Cy8tols 346

to form those Lomer-Cottrell dislocations characterizsd by horizontal h a t c h either with full or with dotted hea by the combination of dialoca- tion pairs in e e c o n k glide-systems. In whatever direction the orienta- tion of the tensile ax& moves away from the [IlO]-pole into the interior of the stemgraphic triangle the tendency for the formation of this sort of Lomer4ttrell dielocations will rapidly dimininh ainm the reeolved shear stress in at lead one of the two glide-systems of each of the pairs will decrease. This accounts for the principal features of the experimenhl situation.

Pig. 27

Dependence of e4-/t, and ro/r0 m u on the orientation of the tensile sxie of oopper cryebb deformed st room temperature.

The end of the transition between stage I and stage 11 will preeumably be reached when a sufficient number of Lomer-Cottrell didocations in aIl of the three possible directions are formed. From there on the dope of the streee-etrain curve is mainly determined by the rate of production of Lomer-Cottrell dislocations which must be such aa to bring about the e m p ~ c a l law eqn. (2) for the variation of dipdistance with atrain. Theee procams should be rather insensitive to the degree of perfection of the undeformed crystal, to ita impurity content and to the diameter of the specimen. This egreea with the observations that for a given crystal orientation the work-hardening coefficient is very well reproducible in stage II. Ita variation with crystal orientation is small compared with that of the length of stage I (Diehl1956 a). Thia is understandable aince 80 many secondary glide-system COOp0r8te h determining 0 in Stage 11 that a small dependence on the orientation of the t e d e axis should reeult. We have so far d i s c d the r61e of Lomer-CottreU dialocations in

holdmg up the dialocatione moving in the primmy glide plane and in determining the alip-diabca and thereby ah0 the coefficient of work- hardening in stage 11. As WBB pointed out by Mott (1952), Lomer- Cottrell dislocations may mgn%cantly contribute to the stabilization of the deformed state against backward glide after unloading by forming obstacles in the rear of groups of dielocationa piled up under the action of the applied stress. In our present pioture this impliea that the stability

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346 A. Seeger et ol. on Work-Hardening and

of the deformed state ia leae in the eaay glide region where no Lomer- Cottrell dielocations are available than in etage 11. We have found a striking confirmation of this view in a recent paper by Ruckley and Entwiatle (1966). Theae authore show in their fig. 8 that the rate of increase of the Bauechinger effect (memured in t e rn of the ‘ Bauachinger glide strain ’ which is a me88ure for the instability of the deformed etate under unloading or mvereed loading) with increasing hardening ismuch larger in the easy glide region than in stage I1 and 111 of aluminium single cryetals drained at room temperature.

4.3. The M w h n h of Slip-Band Formatian and Fragmentation

The prinoipal reBult of the diacusaion of experimenta in 5 3 wm that the orom elip of sorew dislocations is the governing prooeaa for the tempera- ture dependence of the work-hardening in etage III, for the phenomenon of work-softening, for the formation of slip-bands, and for the fragmenta- tion of dip-ban&. The objeot of the preeent section is to consider eome of the theoretical problema connected with these phenomena. As pointed out earlier B O ~ W didocations undergo oroaa slip in order to

circumvent and eurpase obetaclea which they are unable to break through. The discmaion in $4.2 suggwta that theae obetdea are Lomer-Cottrell dielocationa. The Lomer-Cottrell dislocations holding up ecrew dieloca- tions are those which cannot be formed by dielocations of the primary glide s p k m but reqnire the cooperation of dislocations in the other glide- egsteme with the primary glide plane. It is therefore essential (Seeger 1956a) not to restriot the diacuegion of Lomer-cOttrell dielocations to thoee formed by the dialocations in the primary system glide 88 waa done by other authore.

Next we will take up the related problems of the detailed mechanism of crow dip and of the &ability of Lomer-Cottrell dialomtione a g h t ‘collapsing’ under the combined action of atreas and temperature. Dialocetions generated during plaetio deformation in face-centred cubic metals are exfended ones. They can undergo omaa elip only if the two partial dislocations of an extended acrew dislocations are brought together over a certain length. The resulting complete eomw-dialomtion may then aepmte into partiala in the ~ O B B alip plane.

Lomer-Cottrell dialocatiom can give way to the preseure of a group of dialomtiom piled up again& them in several w a p which have been diaouseed by Gtmh (1966). For our pment purpoae it suilices to oonaider that way with the 10- energy barrier. Its etep ie to reunite over a certain length the partial &locations which meke up the Lomer-CottreU dielocation. The calculations of the energy barrier (aotivation energy) involved ia very similar to the oalculation of the energy barrier of crosa dip and will have to follow the linee indimted in a simple cam by Schoeck and Seeger (1965), Seeger (1964d). Since more detailed oaloulations of this type are still under way we will confine omelves fo a qualitative

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Work-Softening of Facecentred Cubic M& C ys& 347

disoueeion of the relative magnitudes of the aotivation energies in both c m . The same stress component is involved in bringing together the partial dislocations of the eorew didocations and of the Lomer-Cottrell dialocations'againat whioh they am piled up. The lengths over whioh the recombination to complete didocations haa to be aohieved depend on Merent componente of the streee tensor but will not vary very much under typical conditions as shown by the caloulations of Sohoeck and Seeger (1964). The parameter whioh d m determine the aotivation energies is the width of the steoking-fault ribbon between the partial dislocations involved. "hem widths are very muoh amaller in aluminium than they am in copper (or other low staoking fault energy metale) (Seeger and Sohtiok 1963). Thia implies that the aotivation energies for aluminium are much lower than thoee for oopper under equivalent etrese conditions. In a given material the width of the etaoking-fault varies with the oharacter of the oomplete dislooations. It is a minimum for somwe and a maximum for edges. For Lomer-Cottrell dislocations it is somewhat smaller than €or edges but still considerably larger than for screw didocetions. Thia has the oonsequenoe that the aotivation energy for cross slip is smaller than that for the oollapse of Lomea t t r e l l didocatiom, and that piled up p u p s of sorewdislooations will ' leak ' by cross alip before they am able to break down the Lomer-Cottrell disloca- tiom holding them up in the primary glide plane. The theoretical aitua- tion is therefore m agreement with the faota deduoed from experiment.

The preliminary doulations 80 far available ehow that the temperature variation of T~ (being the reeolved shear stress at which a noticeable rate of cmsa slip is reaohed) oan be amounted for by the theory. The theory also explains fully the quantitative dii€erence in the behaviour of aluminium and oopper whioh is partioularly obvious from the experimental results presented by Diehl et d. (1966). The empirid oorrelation with the magnitude of the s h k i n g fault energy, though at that time baeed on much less oomplete experimental evidenoe than available now, wm in fact the starting point for the work on slip bands preeented here (Seeger 1966 a).

In $6 3.2 and 3.3 we have preeented the empirical evidence that the tnrnefer of slip from one alip-line in a alip band to an adjacent one is due to omas dip. We shall now disouss possible meohanisms in some detail.

It follows from the preceding discueeion that in stage 111 emw disloca- tions are pushed out by cross slip from the head of dislocation p u p e piled up against Lomer+ttrell didocations. The etrees componente at the head of euch a pile up are proportional fo the number of didocations in the pile up (Eehelby et al. 1961). The strong& and lo@ dip-lines will therefore leak first. (In quantitative considerations one must take into account, however, that the number of dislocations effeotive in a pile- up will in general be comiderably smaller than the number of dislocation rings in the slip-line due to the ' sorwning ' effeot of dislocations in the environment of the dip-line.) The didooations in the cross dip plane

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348 A. Seeger et d. on Work-Hardening and

will at first be pushed away from the group with a repuleive force varying roughly invemely proportional to the distance from the group. Later they will either get stuck in the work-hardened material, or they will meet another piled-up group of screw dislocations of the oppoate sign. They will then annihilate with some of these, thereby giving the way free for more dislocations following the aame way. Since the annihilated disloca- tions in the group juet mentioned will in some be replaced from their source a weak slip line (aay of the elementary structure type) may develop into a strong one. Thie mechanism is preaumably stopped by the edges of the dialocation rings spmadmg in the croaa slip plane. There are no dislocations groups with whioh they can Rnnihilate ; they will therefore get stuck in the work-herdened material after a certain dip dietance and create a pile-up the back stressee of which will eventually prevent further croaa dipping. This wcounta for the fact that the cross diplinea a,re rather weak ae diecussed in 8 3.3. Since the streas rnagdcation acting on dislocations which am not quite at the head of a pile-up ia still large, them dislocations may undergo croam dip, too, thereby giving riee to more of these weak croaa slip linea. This e x p l h why one hee frequently the impremion that croaa slip is spread out over an area rather than concen- trated in strong linea.

The screwdielocations which have not annihilated after having moved out of the primary glide plane by c m slip may under the action of shear streaaea and temperature undergo once more croaa slip back to the primary glide plane. Since by the argument given above the edges remaining in the croaa slip plane will be held up somewhere the whole configuration forms B Frank-Read aource in the primary glide plane and may by the Frank-Read mechaniam generate a new slip line. Thie ia (with modifica- tions) the mechanism of " double cross dip '' firat envisaged by Koehler (1962 a, b). It should be mentioned that ah0 a mixture might occur of the two mechaniame which we have diecusaed : the dislocations may annihilate after crow dipping back to the primary glide plane.

Dependmg on the s d e on which these p r o m happen they may either lead to slip band formation or to fragmentation. If the new slip lines are formed within a few hundred Angstroms distance from the original one we observe clustering of dip lines into bands. If the dislocations slip farther away in the C~OBB dip plane before creating a new line or making an existing one @ow one obtdna fragmentation. Both the 'double c r w slip '-and the ' andda t ion ' mechanism allow glide to proceed with a smaller incresee in work-hardening than would comapond to stage 11. They am therefore both able to account for the decresse of the work- hardening coefficient in stage 111. The relative impottance of the two mechanisms will depend on the conditione of streaa and preetrain. If the material ia very little work-hardened aa ia aluminium at the beginning of stage In at high enough temperaturea double crom slip may well be important. In a highly work-hardened material however the annihilation mechanism should be predominant. It accounts in particular for the

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Wmk-So$hiw of F ~ - C m t r d Cubic M d Crydtole 349

drop iu ~ ~ O W - H ~ I W E occurring during work-aoftening which in aome way must be connected with the asuihilation of didooations which had con- tributed to the work-hardening. We conclude with a few remarka on the oonditiom at largestrsine.

AE can be seen h m fig. 24 of Diehl et d. (1955) at high enough straine there ie no dip in the material between dip bands. Slip p r o d by adding more lamellae to dip bmds (and poaaibly also by growth of slip linm within the bands). The qumtion may ark whether a slip-line at the edge of a basd may be EO we& and oontain 80 few dislocations that none of these can be expelled by crow slip, thereby atopping the aidewaye growth of the band. The general i m p d o n is that banda oontinue to grow aide- way^ with increasing strain. Thie is preeumably due to the fact that the ecrew-dislocations in the interior of a band help to pueh out by c r o ~ ~ slip the dielooationa in the outer dip linee.

56. CONCLUDINO RI~IURXE We shall not wmmarim here the concluaiom oonmrning the various

didocation meohaniama in the work-hardening end work-softening phenomena of faoe-oentred oubio metals reaohed in this paper. W e ehould rather like to etreae the nnlfvlng r8h played in theee oomiderationa by the gtacking-,fau& encrgy. Dialocationa generated during oold-work in faoe-centred oubic metala are extended onee, and their behaviour under the aotion of streee and temperature depends to a large meaeure on the width of the ataoking-fault ribbons between the halfdielocatiow. Until it waa realized that aIuminium, being a high etacking-fault energy metel, differs in this reepeot fmm other faoe-oentred cubio metale suoh aa oopper, silver, gold and nickel it waa impoaaible to integrate the eeemingly oon- flicting evidence on the temperaturedependence of flow-streee, on the temperature-dependenoe and the shape of the work-hardening ourve, on surfam IIlIvkinge and dip-band formation into a Mified pioture valid for all faoe-centred cubio metals. We hope to have been able to develop euoh a pioture, whioh of course still requires refinement by future investigations.

ACXXOWIJEDOMENTB It is a p e t plemure for the authora to thank Professor U. Dehlinger

for hie enoouregement and for his kind inter& in thie work. They ale0 aoknowledge p b f u l l y the hano id support of the Deuteche Foreohungegemehchaft .

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350 Work-Hardening and Softening of Face Centred-Cubic Meid Cryat&

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