DISCLINATIONS IN LARGE-STRAIN PLASTIC DEFORMATION … · 2012-02-13 · ones”. The arising excess dislocation boundaries were still penetrable to mobile dislocations. How-ever,

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Corresponding author: M.Seefeldt, e-mail: [email protected]

Rev.Adv.Mater.Sci. 2 (2001) 44-79

© 2001 Advanced Study Center Co. Ltd.

DISCLINATIONS IN LARGE-STRAIN PLASTICDEFORMATION AND WORK-HARDENING

M. Seefeldt

Catholic University of Leuven, Department of Metallurgy and Materials Engineering, Kasteelpark Arenberg 44,B-3001 Heverlee, Belgium

Received: September 01, 2001

Abstract. Large strain plastic deformation of f.c.c. metals at low homologous temperaturesresults in the subdivision of monocrystals or polycrystal grains into mesoscopic fragments anddeformation bands. Stage IV of single crystal work-hardening and the substructural contributionto the mechanical anisotropy emerge at about the same equivalent strain at which the fragmentstructure becomes the dominant substructural feature. Therefore, the latter is likely to be thereason for the new features in the macroscopic mechanical response. The present paper reviewssome experimental and theoretical work on deformation banding and fragmentation as well asa recent model which tackles the fragment structure development as well as its impact on themacroscopic mechanical response with the help of disclinations. Incidental or stress-inducedformation of disclination dipoles and non-conservative propagation of disclinations are consideredas “nucleation and growth” mechanisms for fragment boundaries. Propagating disclinations getimmobilized in fragment boundaries to form new triple junctions with orientation mismatchesand thus immobile disclinations with long-range stress fields. The substructure development isdescribed in terms of dislocation and disclination density evolution equations; the immobiledefect densities are coupled to flow or critical resolved shear stress contributions.

1. INTRODUCTION

The activation of rotational modes of deformation,i.e. the spatially and temporally progressing re-orientation of parts of monocrystals or polycrystalgrains in the course of plastic deformation, and itsimpact on the macroscopic mechanical responsewere intensively studied in Physical Metallurgy sincethe 1930s. The phenomenon was discovered throughX-ray diffraction at deformed metals exhibiting Laueasterism, i.e. a subdivision of the ideal Bragg reflexesinto sets of subreflexes, corresponding to the sub-division of grains into sets of subgrains.

Monocrystal or grain subdivision was studied ina variety of different contexts: In the early years ofPhysical Metallurgy, deformation banding, especiallykink banding, was studied in order to understandkinking as an additional mechanism of plastic de-formation next to dislocation gliding and deforma-tion twinning [1,2] (and next to grain boundary glid-ing and deformation-induced phase transformations which were discovered later). With the new possi-bilities of transmission electron microscopy (TEM)

in the 1960s the attention of metal physicists wasdrawn to individual dislocation behaviour in the earlystages of plastic deformation. However, deformationbanding was rediscovered in texture research wherethe disregard of grain subdivision was understoodto be one reason for the deficiencies in texture pre-diction, especially in local texture prediction.

Already at the beginning of the 1960smonocrystal subdivision on a smaller scale than theone of the deformation bands was discovered in TEM:misorientation bands were found, a band substruc-ture arising during stage III of work-hardening. Ex-tensive TEM studies in the 1970s and 1980srevealed deformation banding on the II. mesoscopicscale and fragmentation on the I. mesoscopic scaleto be the characteristic features of the substructuredevelopment at intermediate and large strains, inthe developed stage of plastic deformation, as Rybinphrased it. At the same time, the stages IV and Vof work-hardening as well as the substructuralcontribution to the mechanical anisotropy were dis-covered. However, only in Russia it was tried fromthe beginning to connect these new features in the

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macroscopic mechanical response to the new fea-tures in the substructure evolution, namely to frag-mentation. Still today, the connection between stageIV and fragmentation is not yet generally accepted,neither on experimental nor on theoretical grounds.

This development was also driven by technologi-cal interest: All the industrially relevant forming pro-cesses, as, for example, flat rolling, deep drawingor wire drawing, involve large strains, high strainrates and multiaxial deformation modes. Recently,the production of ultra-fine grained and bulknanostructured materials by means of severe plas-tic deformation – which is essentially based on ex-tensive fragmentation – found increasing interest alsoin industry.

This section gives a brief overview on deforma-tion banding and fragmentation and large strain work-hardening – the framework within which the presentwork is situated. The second section explains theconcepts of reaction kinetics and disclinations usedin the proposed model – the approach or point ofview taken by the present work. The third sectionpresents a semi-phenomenological application tothe cell and fragment structure development and theirimpact on the flow stress for single crystals withsymmetric orientation. The model is capable ofreproducing stages III and IV of the room tempera-ture work-hardening curve for copper.

1.1. Deformation banding and frag-mentation

Since the 1930s and extensively in the 1950s lightand electron microscopy studies of slip line pat-terns and X-ray studies of Laue asterism were usedto investigate deformation banding during plastic de-formation. In most cases, aluminium single crystalsafter tensile deformation at room temperature werestudied. Barrett and Levenson [3,4] used macro-scopic etching methods to reveal orientation differ-ences and found band-shaped surface structures.They defined a deformation band as a region inwhich the orientation progressively rotates away fromthat in the neighbouring parts in the crystal andsuggested that “deformation bands form as a resultof the operation of different sets of slip systems indifferent band-shaped regions”.

Honeycombe [5,6] distinguished, based on op-tical microscopy surface studies and X-ray aster-ism measurements at aluminium single crystals,two types of bands: a) kink bands are narrow bentregions (typically with double curvature) normal tothe active slip direction which separate slightly dis-

oriented crystal lamellae, and b) bands of second-ary slip are regions nearly free of primary slip tracesand almost parallel to the primary slip planes in theearly stages of plastic deformation. The influence ofthe crystal orientation, the temperature, the defor-mation speed and the crystal purity was studied.The essential feature of the kink bands is the bend-ing which gives rise to a diffuse, streak-like Laueasterism. The bands of secondary slip act as pref-erential sites of activation of unexpected slip sys-tems and thus also exhibit disorientations, but theyare essentially free of curvature.

Kink bands were almost absent in deformedcrystals with highly symmetric initial orientations(deforming through conjugate slip on two or moreslip systems from the beginning) and were mostpronounced in crystals with initial orientations closeto the centre of the standard triangle of the stereo-graphic projection (deforming mainly through primaryslip). Bands of secondary slip were observed for allinitial orientations, but were most pronounced andeven of macroscopic size in multiple-slip orientedcrystals (deforming through conjugate slip e.g. onfour active systems). Kink bands were observed ina wide range of temperatures. However, the spac-ing between the bands turned out to be about 2-3times larger at high temperatures and too small forobservation with optical microscopy at liquid nitro-gen temperature. At temperatures above 450 °C,polygonization of the bent regions was observed,resulting into a sharp, small spot-like Laue aster-ism. The influence of the deformation speed on bothtypes of banding was found to be small.

From recovery experiments [6], it was concludedthat the Laue asterism was due to deformation-in-duced banding. Honeycombe [5] attributed the for-mation of kink bands to the restraints imposed bythe grips on the crystals. In case of multiple slip,the stresses due to these restraints could bereduced by slip on alternative slip systems. The for-mation of bands of secondary slip was ascribed tounpredicted cross-slip at the beginning of plasticdeformation which could hinder slip on primary slipsystems. So as early as 1951 two different types ofdeformation banding were distinguished and tracedback to two different reasons, namely to an imposedone on the macroscopic scale – the restraints ow-ing to the deformation mode – and to an intrinsicone on the microscopic scale – dislocation dynam-ics leading to unexpected cross-slip.

Staubwasser [7] used surface slip line and X-ray asterism studies on aluminium single crystalsafter tensile deformation at room and liquid air tem-

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perature in order to investigate the orientation de-pendence of excess dislocation boundary and kinkband formation and work-hardening. Low-symmet-ric single-slip orientations in the lower central partof the standard triangle showed strong asterism rightfrom the beginning of plastic deformation,corresponding to the formation of excess disloca-tion tilt boundaries perpendicular to the glide direc-tion. With increasing deformation, strong kink-band-ing was found, also perpendicular to the glide direc-tion and with net misorientation across the band.High-symmetric multiple-slip orientations, on thecontrary, exhibited only weak asterism,corresponding to the formation of dislocation wallsperpendicular to all of the active slip planes. No kink-banding at all was observed. Furthermore,Staubwasser suggested a mechanism for kink bandformation. Groups of primary dislocations could getstopped at (unspecified) obstacles, and “such onneighbouring planes could be held fast by the formerones”. The arising excess dislocation boundarieswere still penetrable to mobile dislocations. How-ever, at the end of stage I, the stress fields aroundthe dislocation boundaries would activate a second-ary slip system, so that immobile barriers could beformed through dislocation reactions. Mobile dislo-cations would then pile up at the now intransparentboundaries and induce the characteristic S-shapedlattice curvature of the kink bands. Fully developedkink bands would then be impenetrable obstaclesand have a serious impact on work-hardening.

Mader and Seeger [8] distinguished, based onreplica studies of slip lines patterns on copper singlecrystals deformed at room temperature, two differ-ent types of kink bands: the kink bands of the firsttype arise already at the end of stage I and mainlyin stage II, are of macroscopic size (length betweenseveral 100 µm and the specimen size, width be-tween 10 and 100 µm) and are very straight. Thekink bands of the second type arise at the begin-ning of stage III, are of mesoscopic size (only about100-200 µm long and very narrow) and are lessstraight, a bit wavy and connected to each otherthrough branches. They correspond to themisorientation bands on the mesoscopic scale stud-ied later on in TEM. The present paper focusses onthese misorientation bands on the (first) mesoscopicscale rather than on the kink and deformation bandson the grain-size or (second) mesoscopic or evenmacroscopic scale.

In the early TEM era at the beginning of the1960s, Steeds [9] and Essmann [10] investigatedthe dislocation arrays in single-slip oriented copper

monocrystals after tensile deformation at room tem-perature. At the beginning of stage III, they foundflat, disk-shaped areas parallel to the primary slipplanes which were limited by secondary disloca-tions and included long bowed-out primary disloca-tions, but only few forest dislocations. According toEssmann, the disks were misoriented by 0.2-1.0°with respect to each other and the boundaries hada mixed tilt-twist character. He drew a remarkableconclusion from a comparison of Steeds’ and hismicrographs: while both of them agreed on the diskstructure parallel to the primary slip planes, theirstatements on the height of the disks varied: Steedsfound 1-2 µm, Essmann 0.4 µm. FollowingEssmann, Steeds’ result matched with the periodof the large misorientations which he also observed,but found to be subdivided on a finer scale. In spiteof these early TEM findings of a misorientation bandsubstructure, the attention of metal physicists wasthen mainly drawn to the dislocation structures inthe stages I and II of work-hardening, especially toindividual dislocations, Lomer-Cottrell barriers, pile-ups, dislocation bundles and cell walls (for an over-view see e.g. [11]).

However, in the mid 1960s, deformation bandingand fragmentation were rediscovered in anothercontext, namely in texture research. A long-stand-ing problem in this field was the mechanism for theformation of a <111> and <100> double fibre texturein wire-drawing of f.c.c. metals. Ahlborn [12,13] stud-ied the orientation development of f.c.c. singlecrystals (Ag, Au, Cu, Al, brass) during wire-drawingusing the X-ray rotation crystal method. He foundthe single crystals to split up into deformation bandswith volume fractions of the two components stronglyvarying among the metals. He connected this varia-tion to the variation in stacking fault energy. Lateron, Ahlborn and Sauer [14] used TEM to study de-formation banding on the (second) mesoscopic scaleas well as fragment and cell structure developmenton the (first) mesoscopic and microscopic scalesat copper single crystals during wire-drawing. Theextensive TEM study by Malin and Hatherly [15]also aimed at a better understanding of the couplingbetween fragmentation, deformation banding andtexture evolution.

In line with Essmann’s remark, Likhachev, Rybinand co-workers were the first to distinguish betweenthe cell structure development on the microscopicscale and the fragment structure development onthe (first) mesoscopic scale [16-18]. The cell struc-ture development, as reflected in the average cellsize, cell wall width or misorientation across cell

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walls, was found to be very slow at large strains, sothat Rybin [18] called the cell structure “frozen” 1.

Therefore, large-strain plastic deformation andwork-hardening were ascribed to the fragment struc-ture evolution. Fragmentation was recognised asan almost universal phenomenon which takes placein f.c.c., b.c.c., and h.p. mono- as well as polycrys-talline metals and alloys. Cell sizes, fragment sizesas well as orientations and misorientations of frag-ment boundaries were measured mainly at molyb-denum (as a representative for the b.c.c. materials)mono- and polycrystals but also at nickel (for thef.c.c. materials) and α-titanium (for the h.p. materi-als) [17,18]. The St. Petersburg school of plasticityinterpreted the arising fragment boundary mosaicas a network of partial disclinations in the mosaic’striple junctions [16,20,21] and ascribed fragmenta-tion to the elementary processes of generation,propagation [18,22,23] and immobilization of par-tial disclinations (see below).

Hansen, Juul Jensen, Hughes and co-workersstudied grain subdivision mainly in pure aluminium[24-28] and nickel [25,26,29-31] but also in Ni-Coand Al-Mn alloys [25,29] as well as in copper [32].Cold-rolling [24-28,30,32] as well as torsion[25,29,30] were applied. Recently, single crystal frag-mentation was investigated in pure aluminium afterchannel-die compression [33,34] and cold-rolling[35]. The cell and cell block sizes as well as thewall and boundary orientations and misorientationswere characterized extensively. Whereas the cellsize and the misorientation across cell walls (CW)remain almost constant at large strains, the cellblock size continues to decrease, and themisorientation across the cell block boundaries(CBB) continues to increase. The cells are roughlyequiaxed, while the cell blocks are elongated andshow a strong «directionality». The CW were inter-preted as «incidental dislocation boundaries» (IDB),the CBB as «geometrically necessary boundaries»(GNB). Both of them are considered as low energydislocation structures (LEDS). Especially, the in-terpretation of the CBB as Frank boundaries free oflong-range stresses turned out to correspond with

1 This steady state of the cell structure is a dynamicone: Dislocations would constantly get trapped into

and emitted from the cell walls. However, there is ex-perimental evidence [19] that at least in copper cellrefinement is realised by cell subdivision rather than

by cell wall motion. Strictly speaking, the geometry ofthe cell structure is frozen in the steady state, but notthe cell structure itself.

TEM results [36]. According to the Risø school, frag-mentation is due to the activation of different sets oflocally less than five slip systems (giving lower den-sities of intersecting dislocations and thus of jogs)[25] or due to different slip rates within the sameset of active slip systems [36].

Recently, the orientation dependence of grainsubdivision was investigated. In a TEM study oncold-rolled aluminium polycrystals, Liu et al. [28]classified the subdivision patterns into three types:Type AI grains (mainly near the Goss and S texturecomponents) split up into a quite uniform cell blockstructure with crystallographic boundaries parallelto the active {111} slip planes. Type AII grains (scat-tered) split up into a non-uniform structure with crys-tallographic boundaries. Type B grains (roughlyaround the cube type texture components) split upwith non-crystallographic boundaries and on twodifferent scale levels, on the (first) mesoscopic oneand on the grain-size or (second) mesoscopic one.The latter probably corresponds to deformation band-ing in cube-oriented single crystals. The subdivisionpatterns of channel-die compressed and cold-rolled,respectively, aluminium single crystals with thebrass, the copper and the cube orientation were dis-cussed in [33-35].

To the author’s knowledge, Koneva, Kozlov andco-workers [38-42] were the first· to distinguish two nucleation mechanisms for

fragmentation, namely an intrinsic one and agrain boundary-induced one and

· to distinguish the discrete misorientations acrossthe fragment boundaries and continuousmisorientations bending and twisting the frag-ment interiors (for a scheme, cf. Fig. 1).Their conclusions were based on TEM studies

of compressed f.c.c. Cu-, Ni- and Fe-based alloysin the long-range and short-range ordered solid so-lution states.

According to Koneva, Kozlov and co-workers,the beginning of stage III is characterized by theemergence of terminating as well as continuous dis-location boundaries with discrete misorientations.These boundaries appear in band-like configurationsmade up of parallel boundaries which rotate thelattice inside the band with respect to the matrix(and grow in band direction) as well as in loop-likeconfigurations made up of a closed boundary ring(see and in Subsection 3.2.3. below) which rotatesthe loop interior with respect to the matrix (and growsperpendicular to the loop). The different configura-tions are the result of different nucleation mecha-nisms:

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· grow of band systems from polycrystal grainboundaries, especially from steps in the grainboundaries and

· nucleation and development of closed subboun-dary rings bordered by partial disclination loops.

Which of the mechanisms operates, depends,for instance, on the previous substructure and onthe presence of grain boundaries. Both mechanismslead to a misorientation band substructure with firstone and later two or more intersecting band setswhich gradually cover the whole volume of the speci-men. At the beginning of stage III, the sets wereoriented parallel to non- or only slightly active octa-hedral slip planes and were either of a pure tilt orpure twist character. With increasing deformation,the sets deviated more and more from the octahe-dral slip planes and were of a mixed tilt-twistcharacter. In stage IV, the misorientation band sub-structure is present in the whole specimen volume.Beyond that, continuous misorientations with smallbut increasing orientation gradients develop insidethe misorientation bands due to local bending andtwisting. At the beginning of stage IV, mostly eitherpure bending or pure twisting was found. At larger

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deformations, distortions of a mixed type predomi-nated.

Kozlov, Koneva and co-workers used the follow-ing quantities to characterize the substructure: theredundant1 (or “incidentally stored” or “scalar”) dis-location density ρ

r and the non-redundant or excess

(or “geometrically necessary” or “tensorial”) dislo-cation density ρ

exc, the subboundary spacing d

f , the

radial and azimuthal subboundary misorientationsω

rad and ω

az, the lattice curvature or lattice bend twist

κ. The non-redundant dislocation density ρexc

wascalculated as the density of dislocations accom-modating lattice rotation mismatches through divid-ing the lattice rotation difference by the bendingextinction contour width �

ρκ

excb

= =∆ϕ

�. (1)

While the growth rate of the redundant dislocationdensity was maximum in stage II, the one of thenon-redundant dislocation density was maximum

1 The distinction between redundant and non-redun-dant dislocations was introduced by Weertman [37].

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in stage III. It was observed that extensive genera-tion of non-redundant dislocations starts at a criticalredundant dislocation density of about 1.5 - 3.1014

m-2. Plotting the mean misorientation ω and the av-erage lattice curvature κ over the redundant disloca-tion density ρ

r clearly revealed the transition from

stage III to stage IV. The results can be summa-rized in the scheme presented in Fig. 1 taking pat-tern from Kozlov et al. [40].

Continuous and discrete misorientations werealso distinguished and analysed quantitatively in arecent OIM study at cold-rolled aluminium AA1050by Delannay et al. [43]. The subdivision of each grainwas characterized by measuring the orientationspread within the grains. Two parameters were de-fined: the average misorientation of all grid pointswithin one grain with respect to the mean orientationof that grain (reflecting a large orientation spreadacross the grain), and the average misorientationbetween neighbouring grid points in the orientationmap (reflecting the subdivision into highly misorientedfragments). Both parameters were calculated sepa-rately for each grain. It was concluded that grainswith orientations close to cube have higherorientation spreads than grains along the β-fibre.Furthermore, among the grains with a strong latticedistortion, only those with orientations close tocopper and TD-rotated cube (cube rotated by 22.5 °around the transverse direction) have highneighbouring-points misorientations.

The models which were proposed to describeand explain fragmentation, can roughly be groupedinto two families. The first family is based on tex-ture-type approaches. These models start from ei-ther different sets of active slip systems or differentslip rates within the same set of active systems indifferent crystal parts (see, for instance, [44-47]).However, in general they lack an explanation for theinitiation of the fluctuations in the slip regime. Thesecond family is based on nucleation and growthapproaches. These models discuss the generationof short segments of fragment boundaries and theconditions for their growth. The arising fragmentboundaries then impose the variations in the slipregime. However, the models in this family usuallyfail in describing the transition from a homogeneousto an inhomogeneous slip regime quantitatively, i.e.they fail in treating the fragments as separate tex-ture components starting from a certian criticalmisorientation.

Considering any inhomogeneous slip regime, ithas to be emphasized that compatible deformationof all crystal parts has to be ensured, so that spa-tially varying sets of slip rates may only result in

varying lattice rotation rates but not in varying strainrates, i.e. they have to be different solutions of thesame Taylor problem. The Risø school proposedthe selection of different sets of less than five activeslip systems in different cell blocks because withfewer active slip systems less dislocation jogs haveto be produced and less energy is dissipated [25].The selection of different sets of active slip systemsresults in different lattice rotation rates and thusrequires the formation of geometrically necessaryboundaries to accommodate the lattice rotationmismatch and to separate the evolving cell blocksfrom each other. Dislocation dynamics has to pro-vide the dislocations to construct these boundaries.

In this approach the choice of different sets ofactive slip systems is thus prior to the formation ofthe geometrically necessary boundaries. This wouldrequire spontaneous collective behaviour of dislo-cations in a mesoscopic volume which is not likelyto occur. Supposing that local dislocation dynam-ics is controlled only by the locally present stressfields.

In the author’s opinion, fragmentation should betriggered off by local rather than global slip imbal-ances across an obstacle. This results in the gen-eration of a both-side terminating excess disloca-tion wall which can be completed to form a both-side terminating small-angle grain boundary as anucleus of a misorientation boundary which wouldthen, with the help of its long-range stress fields,impose different slip rates. In this approach, thenucleation and growth of a misorientation boundarywould thus be prior to the choice of different sets ofslip rates. In the framework of the model to be dis-cussed in this paper, the nucleation and growth offragment boundaries is only possible in thecombination of bundling of mobile dislocations inslip bands and of stochastic trapping in a semi-trans-parent obstacle.

1.2. Large strain work-hardening

The coupling between the macroscopic mechani-cal properties of a material and its micro- andmesoscopic structure is the key to the developmentof new materials with high strength and good form-ability. Since all the industrially relevant forming pro-cesses as, for instance, flat rolling, deep drawing orwire drawing, involve intermediate and large strains,a better understanding of the coupling between work-hardening and substructure development is stronglydesirable especially for intermediate and largestrains.

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The substructural background of the transitionfrom stage III to stage IV of work-hardening is still amatter of controversial discussion. What all stageIV models have in common is the use of (at least)two substructural variables, mostly dislocation den-sities, as suggested by Mughrabi [48] for any modelconsidering substructure development and work-hardening at intermediate and large strains. The earlystage IV models [49-52] started from a dislocationcell structure, treated it as a compound made up ofthe cell interiors as a «soft phase» and of the cellwalls as a «hard phase» [48,53], used the cell inte-rior and cell wall dislocation densities as substruc-tural parameters and ascribed the shear resistancesof the two phases to the respective local disloca-tion densities. The ideas on the underlying disloca-tion kinetics varied. It was noticed that the lowhardening rate of stage IV was rather close to theone of stage I of single crystal plasticity and sug-gested that the underlying elementary processesshould also be related to each other (see e.g. [54]).In line with this assumption, Prinz and Argon [49]proposed the formation of hardly mobile dislocationdipoles in the cell interiors and their drift to the cellwalls to be the rate-controlling mechanism of stageIV. Haasen [51] and Zehetbauer [52], on the contrary,were prompted by different features of stage IV inlow and high temperature deformation to focus onthe different storage and recovery mechanisms ofscrew dislocations in the cell interiors and of edgedislocations in the cell walls. The model byZehetbauer [52] took the concentration of plasticvacancies as an additional substructural variable intoaccount. With four fitting parameters (which couldquantitatively be interpreted on physical grounds),it was able to reproduce work-hardening curves fora variety of different temperatures in excellent agree-ment with experiment.

However, the model was based on questionableassumptions concerning the elementary processesand the substructural framework: interactions be-tween screw and edge dislocations were neglected,and experimental evidence of fragmentation (e.g.misorientation bands, cell blocks) as well as of long-range stresses (bow-out of dislocations in the cellinteriors in TEM [11,55], asymmetric XRD line broad-ening [56]) was disregarded. Furthermore, the modelpredicted substructural features which were inconflict with experimental results: considerable dis-location densities in the cell interiors were predicted,whereas TEM studies in the unloaded as well as inthe neutron-pinned load-applied state revealed onlyvery small densities [11,55], and a transition fromthe screw to the edge dislocation density dominat-

ing dislocation kinetics was concluded to cause thetransition to stage IV, while recent X-ray studies in-dicate that such a change in the dominating dislo-cation character takes place already in stage III [57].

With the increasing amount of experimentalresults on grain subdivision it became clear that asubstructure-based model of stage IV work-hardening should take fragmentation into account.Haasen [51] had already assumed that different setsof slip systems were active in different cells andcausing «geometrically necessary» boundaries be-tween the cells, but Argon and Haasen were thefirst to model the coupling between increasingmisorientation and work-hardening [58]. They as-cribed the stage IV work-hardening rate to the factthat dislocations of opposite signs, approaching amisorientation boundary from the two adjacent frag-ments, do not fully compensate each other anymore. Consequently, elastic misfit stresses accu-mulate, or, in defect language, “virtual misfit dislo-cations” with a “Burgers vector” equaling the differ-ence of those of the two dislocations from the twosides (see Fig. 2). These elastic stresses or “virtualmisfit dislocations” were translated into an organizedshear resistance of the cell interiors. Themisorientation was first assumed to accumulate withthe square root of the shear strain, ϕ = B γ . Nabarro[59] proposed the coefficient B to scale with theinverse square root of the fragment size,

ϕ γ=b

df

. (2)

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Based on this scaling law, a work-hardening rate of

d

d

B G

Gb

d

G

c

f

τ

γ νγ

ν

γ

νϕ

=−

=

−=

−

8

3 3 1

8

3 1

8

3 1

2

2

( )

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(3)

for the shear resistance of the cell interiors wasderived.

Argon and Haasen demonstrated that the misfitstresses obtained from their model were compatiblewith the X-ray residual stress measurements byMughrabi and coworkers [56] (as one might alreadyguess from the arrangement of “virtual misfit dislo-cations” in the case of single slip, see Fig. 3, whichresembles the arrangement of “interface dislocations”in the case of multiple slip according to Mughrabi[56]). While Mughrabi and coworkers decomposedthe asymmetric X-ray peak profiles into two sym-metric components and ascribed them to the two“phases” of cell walls and cell interiors (with the peakwidths reflecting the respective local dislocationdensities and the peak shifts reflecting the respec-tive local residual stresses), Argon and Haasendecomposed them into three symmetric compo-nents and ascribed one to the “phase” of cell walls(also with the width reflecting the local dislocationdensity and the shift reflecting the local residualstress) and the remaining pair of peaks to the“phase” of cell interiors (with the two widths reflect-ing the alternating tensile and compressive elastic

mismatch stresses). An important conclusion fromthis reasoning was that the decomposition and sub-structural interpretation of X-ray profiles is notunique. For example, a similar misfit stress distri-bution as proposed by Argon and Haasen on thebasis of “virtual misfit dislocations” can be suggestedon the basis of a regular array of partial disclinationsdescribing the mosaic of fragment boundaries (seeFig.3).

However, the mechanism proposed by Argon andHaasen does not distinguish between the cell andthe fragment structure. If one applies it to the cellstructure, problems arise in case if the cell wall widthis of the same order of magnitude as the cell size,as it is typical for substructures evolving during de-formation at low homologous temperatures, i.e.without significant influence of diffusion. Mobile dis-locations of opposite signs from the two adjacentcells would hardly get into contact with each other,and penetration would more and more replaced bytrapping of one dislocation plus emission of another.The consequences on the accumulation of elasticmisfit stresses would be unclear and would stronglydepend on the cell wall morphology. If one applies itto the fragment structure, the scaling law for themisorientation, Eq. (2) might have to be modifiedand the coupling with the cell stucture evolutionmodel would get lost.

To the author’s knowledge, Koneva, Kozlov andcoworkers [38-42] were the first to propose the ex-cess dislocation density as the second substruc-tural variable required to describe substructure de-

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velopment at intermediate and large strains and tocouple the increasing excess dislocation densityand the increasing misorientation to an additionalflow-stress contribution. They suggested· to connect the transition to stage IV of work-

hardening to misorientation bands and fragmentsbecoming the predominant substructural featuresand

· to consider the excess dislocation density ρexc

as the substructural variable which controls stageIV work-hardening.

In contrast to Argon and Haasen, they did notascribe this connection to elastic misfit stressesdue to the presence of the misorientation bound-aries themselves but to the boundaries acting asglide barriers, to Randspannungen (end stresses)around the edges of terminating boundaries and tothe lattice curvature due to the continuous miso-rientations in the fragment interiors.

The kinetics of band substructure developmentand of the growth of its volume fraction was observedto be connected to the growth of subboundariesthrough the material, in the course of which the ratioof the number of terminating subboundaries and ofthe total number of subboundaries remainedconstant. The terminating subboundaries were iden-tified as defects of disclination type on grounds ofan analysis of the magnitude and functional decayof the long-range stress fields around them. In thesame measure as the non-redundant dislocationdensity, the long-range stress fields grew. Startingfrom the middle of stage III, when the volume frac-tion of the band substructure became significant,the macroscopic flow stress proved to be propor-tional to the average lattice curvature. The origin ofthis flow stress contribution was proposed to beclosely related to the long-range stress fields. Themacroscopic flow stress would then be controllednot only by the local shear resistance due to elas-tic and contact interaction between the dislocations,but also by the barrier resistance due to the newboundaries and by long-range stresses due to thenew excess dislocation configurations. The flowstress was observed to be proportional to· excess dislocation density,· subboundary density, and· misorientation.

Randspannungen (end stresses) were first sug-gested by Seeger and Wilkens [60] around the edgesof terminating boundaries as an additional sourceof long-range stresses in stage II of plasticdefomation, after it had turned out that classical pile-ups were well observed in TEM, but not in sufficient

number to explain the stage II work-hardening rate[11]. Later on, they were often mentioned as a pos-sible source of long-range stresses (see, for instance,[29,48]), but, to the author’s knowledge, not incorpo-rated into a work-hardening theory so far.

2. MODELLING FRAMEWORK

In the theory of crystal plasticity, plastic deforma-tion and work-hardening are understood as a resultof the generation, motion, immobilization andannihilation of point, line and plane crystal defects.These elementary mechanisms are controlled bythe external applied stress field as well as by mate-rial and process parameters as, for example, stack-ing fault energy, vacancy formation enthalpy, tem-perature and deformation speed. A major task ofplasticity modelling is to make a choice of defectpopulations which are relevant under given materialand process conditions. This choice should be basedon a physical consideration of the dependence ofthe elementary mechanisms on the material andprocess parameters (see subsection 2.1). In orderto couple the defect populations to the macroscopicbehaviour and properties of the material, the volumedensities of the point, line and the defects popula-tions are represented by plane defects. A dynamicconsideration of plastic deformation and work-hard-ening then requires a description of the evolution ofthese densities with time or strain (see subsection2.2). In a third step, the impact of the stored de-fects on the local shear resistance and on the long-range internal stresses has to be discussed andthe defect densities have to be coupled to criticalresolved shear stress or flow stress contributions(see subsection 2.3).

2.1. Materials and processes

In the present paper, the quasistatic deformation off.c.c. metals with intermediate and high stackingfault energies at room temperature is discussed. Inthe model presented in section 3, uniaxialcompression of a multiple-slip oriented pure coppermonocrystal is considered. In the following, someconclusions are drawn from the material parameterson the relevance of certain defect populations in thedeformation process. Especially, the respectivecontributions to dynamic recovery throughannihilation of screw dislocations after cross-slip andthrough annihilation of edge dislocations after climbare estimated. On this basis, it can be decided,whether the two dislocation characters have to betreated separately, and whether vacancies have to

��

be taken into account as a relevant defect popula-tion.

Materials with intermediate and high stackingfault energies are chosen in order to avoid deforma-tion twinning. Pure metals were used because theydo form cells. The annihilation rate of immobile screwdislocations cancelling out after cross slip withmobile screw dislocations of the opposite sign pass-ing by scales with an effective annihilation lengthy

s,eff. If the mobile dislocation passes by within this

length, an annihilation event takes place (see alsosubsubsection 3.2.2). Pantleon [61] used the cross-slip model by Escaig [62] – in a version slightlymodified for large strains – to calculate the effectiveannihilation length according to

y yb

P e ys eff s

SF

cs s, ,� .= + − +−�

��

�1

11

0χχ� � (4)

where the stacking fault width lSF

is connected tothe stacking fault energy γ

SF through

�SF

SF

Gb=

2

16πγ. (5)

The cross-slip probability Pcs

, in addition, scaleswith the cross-slip energy

EGb

bCS SF

SF=2

37 32 3

.ln�

� (6)

and, especially, with the local shear resistance dueto the local dislocation density in the cell walls ac-cording to

PE

k T

b

E

k T

CS

CS

B SF

CS

B

w SF w

= − + =

− +

��

��

��

��

��

��

−

−

exp

exp .

1 3

1 48

1

1

τ

γ

πα ρ� (7)

The quantity χ in Eq. (4) expresses a total probabil-ity of cross-slip events during penetration of a mo-bile dislocation through a cell wall, but is irrelevantin the present context because the term in largebrackets in Eq. (4) approximately equals 1 forquasistatic deformation of copper as well asaluminium at room temperature. For more details,the reader is referred to [61]. Table 1 displays theeffective annihilation lengths for copper and, as apoint of reference, also for aluminium at room tem-perature together with their stacking fault energies[63] and homologous temperatures for three differ-ent local cell wall dislocation densities correspondingto three different cell sizes (see below).

One can conclude that – within the framework ofthe cross-slip model by Pantleon – for cold deforma-tion of copper up to intermediate strains with cell sizesof more than 0.1 µm, the screw dislocation annihila-tion length for cross-slip is very similar to the edgedislocation annihilation length for dipole disintegra-tion, so that the both characters do not need to bedistinguished. For aluminium, on the contrary, theannihilation length of screws is much larger than theone of edges from the beginning of cross-slip, so thatone can assume, as a first approximation, that thecell structure consists mainly of edge dislocations.Annihilation of edge dislocations after vacancy-as-sisted climb can be neglected because low homolo-gous temperatures are considered: bulk diffusion ofplastic vacancies is too slow due to the low mobilityof the vacancies, and core or pipe diffusion wouldrequire significant dislocation densities in the cell in-teriors (where plastic vacancies are generated by mov-ing jogs in the mobile dislocations) to transport thevacancies to the cell walls (where they would beneeded to assist climb and subsequent annihilationof cell wall with mobile dislocations).

2.2. Disclinations

If the formation of fragment boundaries is consid-ered as a nucleation and growth process, then it isnatural to model fragmentation in terms of disclina-

Table 1. Effective screw dislocation annihilation length for copper and aluminium at room temperature fordifferent local cell wall dislocation densities.

Cu Alγ

SF/J m-2 55.10-3 200.10-2

RT/Tm/1 0.2 0.3

ys,eff

(ρw = 1014 m-2 or d

c = 2.4 µm)/b 6 584

ys,eff

(ρw = 1015 m-2 or d

c = 0.75 µm)/b 6 831

ys,eff

(ρw = 1016 m-2 or d

c = 0.24 µm)/b 36 1591

&� ��

tion dynamics. In the present paper, it is assumedthat the nucleus of a fragment boundary is a both-side terminating excess dislocation wall. Such a wallcan be formed by collective stopping of mobile dis-locations of the same sign on neighbouring parallelslip planes at an obstacle. If the excess disloca-tions in the wall are arranged at equal and minimumpossible distances, i.e. if they form an ideal low-angle grain boundary (LAGB) segment, then the bor-der lines of the both-side terminating wall are partialdisclinations. This is, for instance, the case for singleslip and an obstacle perpendicular to the slip planes(see Fig. 4, left). If it is not the case, e.g. for singleslip and an obstacle inclined to the slip planes (seeFig. 4, middle), the long-range stress fields aroundthe non-ideal dislocation configuration would triggerErgänzungsgleitung [64,65] (completion slip) onadditional slip systems to complete the non-idealboth-side terminating excess dislocation wall to anideal low-angle grain boundary segment (see Fig.4, right).

Disclinations are the rotational “twin-sisters” ofthe translational dislocations1: disclinations are linesof discontinuity of misorientation, they separateregions where crystal parts are (already) misorientedwith respect to each other from regions where thisis not (yet) the case, they accommodate latticerotation mismatches, while dislocations are linesof discontinuity of shear, they separate regionswhere crystal parts are (already) sheared with respectto each other from regions where this is not (yet)the case, they accomodate strain mismatches.Disclinations were, together with dislocations, in-troduced into the theory of elasticity by Volterra as

early as in 1907 [68]. In a Volterra process,disclinations are generated by introducing a cut intoan elastically isotropic hollow cylinder, rotating thetwo shores with respect to each other, and insertingthe missing or withdrawing the surplus matter. Dis-locations, on the contrary, are produced by shiftingthe two shores with respect to each other. In a syn-thetic definition, a Frank vector representing axisand angle of the rotation is introduced in analogy tothe Burgers vector representing direction and mag-nitude of the shift. In an analytic definition, the Frankvector closes the rotation mismatch in a Nabarrocircuit [66], as the Burgers vector closes the shiftmismatch in a Burgers circuit. Wedge and twistdisclination characters are distinguished (Frank vec-tor parallel or perpendicular to the defect line alongthe cylinder axis, respectively) as edge and screwdislocation characters are distinguished (Burgersvector perpendicular or parallel to the defect line).

However, in contrast to a single dislocation, asingle disclination cannot exist in a crystal becauseof the requirement of coincidence of lattice sitesafter the Volterra process: after rotating or shiftingthe two shores of the cut with respect to each otherand after inserting the missing or withdrawing thesurplus matter, the lattice sites on the two shoresof the interfaces have to coincide. In a crystal, thisrequirement can only be fulfilled for angles of at least60 degrees which would obviously cause fracture. Itwas only in the early 1970s that scientists in theUSA [67,69,70] started to consider dipoles of par-tial disclinations and researchers in Russia proposednetworks [16,20] and other self-screeningconfigurations [71] of partial disclinations rather thansingle perfect disclinations. Such partial disclinationdipoles as new entities can fulfill the crystallographicrequirements because the two rotations compen-

1 For introductory texts on disclinations see, for instance,[23,66,67].

��0��,��1� �� 2�� 0+'.��

�� 3 ��,��1 ��1� �� 2��

� �� 1�� 4 ��,

�� 1 ��2�� 0+'.�

��

sate mutually at large distances from the dipole andleave only a net shift behind, comparable to a“superdis-location” [23]. Therefore, the dipole con-figuration allows the realization of partial disclinationswith low strenghts +ω and -ω in a crystal. Note thatit thus also allows the free variation1 of the magni-tude of the mutually compensating rotation anglesor disclination strengths +ω and -ω, and it introducesthe dipole width 2a as an additional variable. Whiledislocation modelling works with only one variable,namely the dislocation density, disclination model-ling has to deal with three, namely the disclinationdipole density and the average disclination strengthsand dipole widths.

A partial disclination with a low strength ω canthen be “translated” into the border line of a termi-nating low-angle grain boundary with misorientationϕ = ω in the sense that both representations exhibitthe same long-range stress field. More specific, apartial wedge disclination can be “translated” intothe border line of a terminating regular excess edgedislocation wall (see Fig. 5), and a partial twistdisclination can be “translated” into the border lineof a terminating regular screw dislocation net.

However, these “translations” are non-unique: apartial wedge disclination dipole can, for instance,be “translated” into a both-side terminating excessdislocation wall of only one sign as well as into twoparallel semi-infinite excess dislocation walls ofopposite signs (see Fig. 6). Both of theseconfigurations exhibit the same long-range stressfield.

In this way, in principle all partial disclinationconfigurations in a crystal can be “translated” into,possibly complex, dislocation configurations2 . Theadvantages of a disclination description are itsmathematical simplicity and its correspondence withcertain ideas of large-strain plastic deformation andwork-hardening: if fragmentation is considered asthe essential feature of large-strain plastic deforma-tion and is understood as a process of nucleationand growth of fragment boundary segments, i.e. asan activation of rotational modes of deformation

through cooperative motion of dislocations, thenthese ideas can be implanted in a plastic deforma-tion model by describing fragmentation in terms ofgeneration of partial disclination dipoles and propa-gation of partial disclinations through capturing ofmobile dislocations. If large-strain work-hardeningis supposed to be due to long-range stresses and ifthese long-range stresses are assumed to beRandspannungen (border stresses) around the bor-der lines of fragment boundary segments with a forthe rest ideal low-angle grain boundary structure,then the long-range stress fields can properly beexpressed as partial disclination stress fields. Infact, the situation is similar with dislocations: In prin-ciple, all dislocation configurations can be “trans-lated” into, possibly complex, atom configurations.The advantages of the dislocation description areagain its mathematical simplicity and, above all, itscorrespondence with the idea and experimental evi-dence that plastic deformation is carried out by pro-gressive slip on glide planes, i.e. by the activationof translational modes of deformation throughcollective motion of atoms. Therefore, it is improperto consider disclinations as a mere modelling toolintroduced only for mathematical convenience.Representing a complex configuration of many dis-locations, disclinations are probably just as muchphysical objects as dislocations, representing acomplex configuration of many atoms, are. The keyexperimental task is to confirm (or reject) that dis-locations are indeed arranged in configurationscorresponding to partial disclinations and that long-range stresses indeed exhibit the functional decaytypical for partial disclinations (see below).

The consideration of partial disclination dipoleswas the first keystone to the introduction of

1 If the partial disclination with strengths ω representsthe tip of a terminating low-angle grain boundary with

misorientation ϕ (see below), the allowed strength isdiscretized in multiples of ∆ϕ =b/hmin where hmin is theminimum dislocation spacing in a low-angle grain

boundary. This discretization is so fine that it can beneglected in the present considerations.2 This does, of course, not hold for non-crystalline ma-

terials where disclinations can directly act as carriersof plastic deformation.

��(*�� )��

��(��)�� ,� �%��

�� 5�� %

�� 1 ��

� �� 6� � �� %� � 1 ��

��2��

6�� %��

��

&� ��

��7�� 8��

��

disclinations into crystal plasticity because it be-came clear that (partial) disclinations could exist inreal crystals. The “translation” of partial dislocationdipoles into dislocation configurations was the sec-ond one because it opened the door to couplingdislocation and disclination dynamics and actuallyestablishing disclination dynamics in real crystals.

Motivated by TEM micrographs showing pairs ofparallel terminating fragment boundaries with oppo-site misorientations (see e.g. [10,18,72], cf. thescheme (A1) in Fig. 6), Romanov and Vladimirov[22,23] were the first to model the growth of amisorientation band in terms of the propagation ofa partial wedge disclination dipole at the top of theband. The partial disclination dipole was proposedto propagate by widening dislocation loops aheadof the dipole with the help of its long-range stressfield, capturing the edge components, attachingthem to the backward terminating boundaries,thereby shifting the border lines of the terminatingboundaries and propagating itself. Similarly, a both-side terminating excess dislocation wall, can as al-

ready discussed by Nabarro [73] and Li [74] in thecontext of polygonization, act as a “nucleus” of afragment boundary, expand or “grow” by capturingadditional mobile dislocations through the long-rangestress fields around its tips, and thus “polygonize”the matrix. A propagating partial disclination, incontrast to a perfect one, thus leaves a track be-hind, namely a fragment boundary, like a glidingpartial dislocation, in contrast to a perfect one, leavesa stacking fault behind.

If, finally, such a propagating partial disclinationgets stopped at an existing fragment boundary, arotation mismatch should be left behind around thenew triple junction (see Fig.7). Formally, thismismatch could be detected by carrying out aNabarro circuit [66] around the triple junction. Arotation mismatch around the triple junction wouldthen correspond to an immobile partial disclinationin the node line.

Experimental evidence for such a rotation mis-match around fragment boundary triple junctionsand thus for the presence of partial disclinations

��9�� :��

�� 2 �� *��

:��

��

was recently found by Klemm and coworkers [75,76]in TEM microdiffraction measurements at cold-rolledcopper mono- and polycrystals. The misorientationsacross the three joining fragment boundaries weredetermined and the product of the misorientationmatrices was found to deviate from the identity. Inthe linear approximation, the sum of themisorientation vectors was found to deviate fromzero. Furthermore, when shifting the electron beamalong the joining boundaries, the local orientationwas found to vary continuously.

Further methods were used to demonstrate thepresence of partial disclinations in the substructureof deformed metals: Koneva, Kozlov and coworkers[38-42] used TEM bending contour measurementsto analyse the magnitude and the functional decayof the long-range stress fields around terminatingsubboundaries and found them to match with theones of the partial disclination stress fields. Tyumen-tsev et al. [77] used TEM reflex width measurementsto determine lattice bend-twist tensor componentsalong grain boundaries in ultra-fine grained copperand nickel crystals produced through severe plas-tic deformation. The measured variation of the bend-twist tensor components along matched with theone to be expected for a pile-up of partial disclina-tions on the grain boundary. Recently, the presentauthor proposed to use scanning tunnel microscopy(STM) to study the surface etching around the pointswhere fragment boundary triple junctions leave tothe surface. Preferred etching might indicate thatthe nodes are non-compensated and include partialdisclinations [78].

2.3. Reaction kinetics

The evolution of a global average defect density withtime or strain is determined by the elementarymechanisms of generation and annihilation of thedefects and of reactions of defects in the populationunder consideration with defects in other popula-tions. These elementary mechanisms depend onthe material and process parameters. Therefore, arate equation which describes the time or strainevolution of a, say, global dislocation density ρ

i, has

the formal structure

d

dtF c c

S S m m p p

i

k i

m n o

ρρ ρ ρ= ( , , , , , , , ,

, , , , , , , , ),

1 1

1 1 1

� � �

� � �

�

(8)

where the c’s denote point defect concentrations,the ρ’s and S’s line and plane defect densities andthe m’s and p’s material and process parameters,

respectively. The earliest representant of this classof evolution equations was Gilman’s equation [79]for a dislocation density undergoing multiplicationand pairwise annihilation. Another commonly usedexample, proposed for the shear strain γ instead ofthe time as an evolution parameter, is Kocks’ semi-phenomenological equation [80]

d

d b

L

ba

ρ

γ βρ ρ= −

1 (9)

for a forest dislocation density ρ with storage anddynamic recovery. For single slip or symmetricmultiple slip, the strain increment can be expressedin terms of the shear strain increment, dε = m

Sn dγ,

giving Kocks’ and Mecking’s law [81]

d

dk k

ρ

ερ ρ= −

1 2 . (10)

However, at intermediate and large strains, dis-location patterning takes place, so that arepresentation of the dislocation structure in termsof one global dislocation density becomes improper.Two routes were followed to tackle this problem: Onone of them, the global defect densities werereplaced by local ones and the evolutions of thelocal defect densities were considered not only asa time-, but as a time- and space-dependent prob-lem. The evolution equations then have the struc-ture of continuity equations,

d

dtF c c

S S m m p p

i

k i

m n o

ρρ ρ ρ=

−

( , , , , , , , ,

, , , , , , , , )

1 1

1 1 1

� � �

� � �

�

∇ ji

(11)

Orlov [82] proposed already in 1965 to consider thedislocation density separately for the slip systems.According to Malygin [83], one obtains a system ofpartial differential rate equations with the structure

∂ρ

∂ρ ρ

( )( ) ( ) ( ) ( )

( )

( )

( ) .i

tv n v

v

l

i i i i

i

p

i

i

p

+ ∇ = + ∑� � (12)

While Malygin’s models are restricted on aconsistent description of the first three stages ofplastic deformation, including smooth transitionsbetween these three stages, the present paper aimsat an extension of the models to the later stages ofplastic deformation, that is to large strain plasticdeformation, by transferring the framework proposedby Malygin for dislocation kinetics to disclinationkinetics,

&� ��

∂θ

∂θ θ

( )

( ) ( ) ( ) ( )

( )

( )

( ) .w

w w w w

w

p

w

w

ptV n V

V

l+ ∇ = + ∑� � (13)

On the second route, the transport terms were ne-glected, and a substructure geometry was imposedinstead by implementing phenomenological scalinglaws like a principle of similitude (see below) intothe model. As far as patterning is concerned, thepresent paper follows this second route.

2.4. Plastic deformation and work-hardening

Following Kocks, Argon and Ashby [84], the localeffective shear stress1 is defined as2

� � �τ τ τij

eff

ij

ext

ij= − int , (14)

where the internal shear stress �τij

int is by definitiontaken positive in the direction opposing the appliedexternal shear stress �τ

ij

ext . The local effective shearstress �τ

ij

eff would then be opposed by the local shearresistance3 � *τ

ij so that the material would (plastically)shear, if

� � � � �* *τ τ τ τ τ

ij

ext

ij ij ij

eff

ij− − = − ≥int� � 0 (15)

would hold. In case of athermal deformation, theequality applies.

The local shear resistance of a crystalline metalis governed by glide obstacles as solute atomconcentrations, local dislocation densities, twin,grain or phase boundaries and/or, in the sense ofan organized shear resistance [58], by long-rangestress fields of static or dynamic origin, in as far asthese stress fields do not directly act as backstresses in the active slip systems. Since thepresent paper discusses only pure metals with in-termediate and high stacking fault energies, the fol-lowing considerations are restricted to local shearresistances due to local dislocation densities andlong-range stress fields. The local dislocation den-sity contributes to the local shear resistance throughthe elastic and contact interaction between inter-secting dislocations, as described, for instance, inthe Hirsch-Saada process [85,86]. The long-rangestress fields can hinder the dislocation motion intwo different ways: either directly by reducing the

local effective stress through the components act-ing as back stresses on the slip planes [87], orindirectly through an organized shear resistanceconnected to the locally stored elastic strain en-ergy due to the components not acting as backstresses [58].

Long-range stresses can be explained from twodifferent points of view: From the defect theory orstatic point of view, high-energy dislocation arrays,like pile-ups or terminating excess dislocation walls(with Randspannungen (border stresses) around theirborders, cf. [60]), cause long-range stresses. Suchdislocation configurations evolve from dislocationdynamics and reflect the deformation history on thedefect level. However, the evolution of dislocationconfigurations exhibiting long-range stresses is lim-ited by the micromechanical compatibility require-ments (see below). From the micromecha-nics ordynamic point of view, elastic strain mismatchescause long-range stresses. Such mismatches re-sult from compatible deformation of plastically het-erogeneous materials and reflect the deformationhistory on the stress and resistance level [48]. Forexample, at intermediate strains, dislocation pat-terning leads to spatially varying local shear resis-tances. Applying an external load to such a “com-pound” material (made up of a “soft phase” with lowdislocation density ρc and shear resistance τ

c

* anda “hard phase” with high dislocation density ρ

w and

shear resistance τw

* ) gives elastic strain mis-matches and thus elastic misfit stresses [48]. Thecorresponding plastic strain mismatches can, fol-lowing Nye and Kröner [88,89], be “translated” intodensities of misfit dislocations (also called “geo-metrically necessary dislocations” [90]). However,these densities do not provide any information onthe spatial distribution, on the configuration of themisfit dislocations and thus also not about possiblelong-range stress fields due to the configuration. Inan ideal case, a misfit dislocation configuration canbe found which exhibits a long-range stress fieldthat equals the elastic misfit stress field and dislo-cation dynamics can be used to study how thisconfiguration arose in the course of plastic defor-mation. However, in other cases it is impossible tofind such a configuration because the calculated“virtual” misfit dislocations are non-crystallographic[58].

In any case, local elastic misfit stresses ∆τc

mis

and ∆τw

mis due to different local shear resistances indifferent “phases” are self-equilibrated local internalstresses. In the global average over the specimen,they, weighted with the respective volume fractions(1-ξ) and ξ of the “phases”, have to sum up to zero,

1 The microscopic resolved shear stresses and shearstrains are denoted as τ and γ; the macroscopicstresses and strains are denoted as σ and ε.2Tensorial quantities are marked with �a.3The resistances are marked with an asterisk * in order

to distinguish them from the stresses.

��

( ) .1 0− + =ξ ξ∆τ∆τc

mis

w

mis (16).

Therefore, they can contribute to the local effec-tive stress and/or to the local shear resistance butdo not directly affect the global flow stress. If, how-ever, their contribution to the local shear resistance,in the sense of an organized shear resistance [58],is significant compared to the one of the local dislo-cation density, then the elastic misfit stresses doaffect the global flow stress.

If one considers, following Argon and Haasen[58] or Pantleon and Klimanek [61,91], a “compound”made up of a cell wall “phase” with high dislocationdensity and a cell interior “phase” approximately voidof dislocations, then the local shear resistance inthe cell interior “phase” is entirely controlled by thelong-range elastic misfit stresses. In the following,two simplified examples are discussed in more de-tail. During uniaxial deformation of copper singlecrystals with a high-symmetric orientation, multipleslip is activated and a cell structure with cell wallsparallel and perpendicular to the load axis is formed[55]. The elastic strain mismatches arising due tothe different local shear resistances cause internalmisfit stresses. The corresponding plastic strainmismatches can be “translated” into regular layersof misfit dislocations at the interfaces between cellwalls and cell interiors. As long as the local shearresistance in the cell walls increases faster thanthe one in the cell interiors, the plastic strainmismatch increases and misfit dislocations areaccumulated at the interfaces according to [61]

NGb m

mis

w c

res S

=− −1

2

ν τ τ* *

, (17)

where Nmis

and bres

denote the line density and theresultant Burgers vector of the misfit dislocationsand m

S is the Schmid factor. In this example, the

regular layers exhibit long-range stress fieldscorresponding to the elastic misfit stresses. Theselong-range stress fields assist the applied stress inthe cell walls and counteract it in the cell interiors.They have strong components which act as backstresses and reduce the local effective stress in thecell interiors and only weak components which donot act as back stresses but might produce anorganized shear resistance and raise the local shearresistance that the local effective stress stress hasto overcome in the cell walls. However, a detailedconsideration by Pantleon (appendix D in [61])showed that an estimation of the flow stress takingthe long-range stress fields into account differs onlyby 3% from an estimation ascribing the compound

deformation resistance fully to the local dislocationdensity in the cell walls or to the smeared-out glo-bal dislocation density. Therefore, the present pa-per neglects the long-range stresses due to misfitdislocation configurations.

One can conclude that there are two “damping”or “inhibiting” mechanisms limiting the growth ofelastic and plastic strain gradients or the accumu-lation of misfit dislocations: One of them is dynamicrecovery which counteracts and finally balances dis-location storage in the cell walls, and the other isthe generation of internal misfit stresses. In this way,the difference between the two local shearresistances is restricted. This is required for stableplastic deformation [92] because misfit stressesassociated with elastic strain gradients scale withthe (huge) shear modulus G according to

( ) ( ).σ σ α ε εw c w cG− = − − (18)

In the case of single slip with cell walls perpen-dicular to the slip direction, the situation is moredifficult. Again, elastic strain mismatches and misfitstresses arise, but if one would “translate” thecorresponding plastic strain mismatches into regularlayers of misfit dislocations at the interfaces, theselayers would not exhibit any long-range stress fieldsbecause they would represent ideal low-angle grainboundary (LAGB) configurations. This problem couldbe overcome by, for instance, considering the twosingle layers to the both sides of a (narrow) cell walltogether as a double layer which does not have long-range stresses outside the double layer (i.e. in thecell interiors in this case), but inside (i.e. in the cellwalls). Gil Sevillano and Torrealda [93,94] proposedthat the long-range stresses inside the layer wouldresult in the activation of additional slip systems toensure homogeneous deformation, so that the cellwalls would rotate with respect to the cell interiors.Straight long excess dislocations stopping at theinterface would then serve to accommodate latticerotation mismatches rather than strain mismatches.Another way to overcome the problem would be toconsider layers with irregular excess dislocationdistributions which do exhibit long-range stresses[95,96]. The latter approach would result in locallyvarying misorientations across the layers, i.e. be-tween cell walls and cell interiors, and, in general,also to finite locally varying net misorientationsacross the double layers, i.e. between neighbouringcells.

The present paper follows the latter approach ina more “schematic” way (for details, see below).The excess dislocations are assumed to be arranged

�" ��

in layer segments with a regular dislocation distri-bution, so that the misorientation varies only aroundthe border lines of the segments. These layer seg-ments are considered as a result of incidentalcollective stopping of neighbouring straight longexcess dislocations at the interfaces. If a layer seg-ment has an ideal low-angle grain boundary (LAGB)configuration, it corresponds to a dipole of partialdisclinations at the borders lines of the group.

In the early stages I and II of plastic deforma-tion, the spacing between the excess dislocationsin the layer segments would remain large, so thatthe misorientation across the layer segment andthe end stresses around its border lines wouldremain too small to capture additional mobile dislo-cations and thus too small to let the layer segmentexpand. Consequently, short excess dislocationlayer segments would cover the cell wall on bothsides in a more or less irregular way, with locallyvarying, but largely mutually balancing misorien-tations (see Fig. in subsection 3.2.3.). However, instage III of plastic deformation, the misorientationacross the layer segments and the end stressesaround its border lines would get sufficiently largeto capture additional mobile dislocations and toexpand along the cell wall (see Fig. in subsection3.2.4.). If such a layer segment is not balanced byits counterpart on the other side of the cell wall, itcauses local net misorientations across the cell walland spreads this net misorientation out over thewhole cell wall. In this way, it acts as a nucleus of afragment boundary layer which grows along the cellwall. If a layer segment has an ideal LAGB configu-ration, its growth corresponds to the propagation ofthe two partial disclinations along the cell wall.

In this picture, the build-up of strain mismatchesat the interfaces between cell walls and cell interi-ors is restricted by the transition from balanced tonon-balanced stopping of straight long excess dis-locations. The stopped excess dislocations thenserve to accommodate lattice rotation rather than

strain mismatches. The transition from translationalto rotational modes of deformation is thus requiredby micromechanics, namely to inhibit further build-up of strain mismatches, and is realized by dislo-cation dynamics through switching from balancedto non-balanced stopping, essentially due to areduction of the spacing between the stopped ex-cess dislocations as a result of bundling mobile dis-locations in slip bands (for details, see below). Ifone assumes an ideal LAGB configuration of thelayer segments, the transition from translational torotational modes of deformation corresponds to thetransition from dislocations gliding individually alongslip planes to disclinations propagating along cellwalls (which is, of course, realized by dislocationsgliding collectively along slip planes).

In the proposed picture, the more or less irregu-lar distribution of (strain) misfit dislocation layer seg-ments on both side of the obstacle in the early stagesI and II exhibits long-range stresses – not throughthe layers itself, but through their ends – which re-semble the elastic misfit stresses. These long-rangestress fields are required by micromechanics, forma self-equilibrated set and do not contribute to theglobal flow stress. The expanding (lattice rotation)misfit dislocation layers on one side of the cell wallin the late stages III and IV also exhibit long-rangestresses – again not through the layers themselves,but through their ends. However, these long-rangestress fields are a consequence of the nucleationand growth mechanism of fragment boundary layerformation, are connected to the propagating partialdisclinations as carriers of the rotational modes ofdeformation, do not form a self-equilibrated set anddo contribute to the global flow stress. The partialdisclination dipoles (PDDs) have the same pleas-ing feature as traditional pile-ups that their long-rangestress fields affect the local effective stress by act-ing as back stresses on mobile dislocations on theprimary slip plane.

Consequently, the global flow stress resultingfrom the substructure is calculated from twocontributions:· the local shear resistances τ

w

* and τc

* of the cellwalls and cell interiors, respectively,

· the long-range internal back stresses τ ω

int due tothe partial disclination dipoles (PDDs).

According to Eq.(15), the local effective stresshas to overcome the local shear resistances τ*,

τ τ τ τeff ext= − ≥int * . (19)

In the present paper, only the contributions ofthe local dislocation densities to the local shear

�� 0��,�� 1��

� ��

� �� :��

4 ��,�� 1�� 1

��

�

��

resistances in the cell walls and cell interiors aretaken into account. The contributions of long-rangestress fields due to misfit dislocation configurationsaccommodating plastic strain mismatches, in thesense of organized shear resistances, are neglected.Furthermore, the local dislocation density ρ

c in the

cell interiors is considered to be negligible comparedto the one ρ

w in the cell walls, so that the com-

pound shear resistance is entirely controlled by thelocal dislocation density in the cell walls. There-fore, one ends up with

τ α ρ τ α ρc c w wGb Gb* *= << = (20)

for the local shear resistances and with (ρw –

smeared-out global dislocation density)

τ ξτ ξ τ ξτ

ξαρ

ξξα ρ

* * * *( )= + − ≈ =

=

w c w

w

wGb Gb

1

(21)

for the compound shear resistance.Correspondingly, the local internal stresses are

proposed to be entirely controlled by the long-rangestress fields due to partial disclination dipoles whichcorrespond to both-side terminating excess dislo-cation layers accomodating plastic lattice rotationmismatches. Long-range stress fields due to misfitdislocation configurations accommodating plasticstrain mismatches, are again neglected. In the caseof single slip, irregular distributions of excess dislo-cation layer segments on both sides of the cell wallswould cause back stresses, but, according to Saada[95,96], those would remain small compared to thestress fields due to PDDs. In the case of multipleslip, regular distributions of misfit dislocations fromthe symmetrically active slip systems would gener-ate back stresses, but, according to Pantleon [61],taking into account these effects would only slightlychange an estimation of the flow stress. Since the“wavelength” of the fragment structure is largecompared to the one of the cell structure (or themean distance between the partial disclinations islarge compared to the one between the dislocationcell walls), and since, in the case of single slip, thedipole arm is perpendicular to the slip plane, a glo-bal average internal back stress τint is considered.The long-range stress field of such PDDs withdisclination strength ω and dipole arm 2a are thesame as the ones of pile ups with a “super Burgersvector” B = ω a, so that an internal back stress arises,

τ α θ α ω θ

α ω θ θ α ω θ

ω

int = = =

+ ≈

GB G a

G a G a

tot tot

p i i

2 2

2 2 , (22)

where θp denotes the density of PDDs with propa-

gating partials and θi and θ

tot denote the immobile

and total partial disclination density, respectively1.If the fragment structure arises from generation andexpansion of PDDs and immobilization of their par-tials and consists of fragment boundaries with idealLAGB configurations and non-compensated triplejunctions with partial disclinations (see Fig. 8), thenthe average fragment boundary spacing d

f

corresponds to the dipole width 2ai between the

immobilized partials of the PDDs.With

2a dK

i f

f

i

≈ =θ

, (23)

where Kf is a geometry factor depending on the as-

sumed mosaic geometry of the fragment structure(see below), one finds

τ α ωω

int ≈1

2GKf

. (24)

Assuming the disclination interaction coefficient βto be β α= K m

f S/ ,2 this matches with the flow

stress contribution

∆σω β ω= G (25)

proposed by Romanov and Vladimirov [23]. Distrib-uting the available excess dislocation density ρ

exc

on the total surface per unit volume of the fragmentboundary layers “drawn” by the propagating partialdisclinations gives the misorientation

ϕρ

θ≈ = =

b

hbN f

bexc f

geom exc

i

, (26)

where ffgeom is another geometry factor depending

on the assumed mosaic geometry of the fragmentstructure (see below). It is now assumed that theimmobile partial disclination strength ω, i.e. the meanorientation mismatch around a non-compensatedtriple junction, is equal to the current meanmisorientation ϕ. In a simplistic picture for a triplejunction formed by the two halfs of an “old” fragmentboundary and a “new” boundary, all with the same

1 Note that θp thus is a disclination dipole density, whileθi and θtot are single disclination densities.

ω

�/ ��

mean misorientation, this makes sense becausethe misorientations across the two halfs of the “old”boundary compensate each other, so that the netorientation mismatch is equal to the misorientationacross the “new” boundary. Finally, one ends upwith

τ α ϕ

αρ

θ

ω

int ≈ =1

2

1

2

GK

GbK f

f

f f

geom exc

i

. (27)

The flow stress to be applied for athermal plasticdeformation thus is

σ τ τ

ξα ρ α ϕ

ξα ρ αρ

θ

ωf

S

S

w f

S

w f f

geom exc

i

m

mGb GK

mGb GbK f

= + =

+ =

+

��

��

��

��

1

1 1

2

1 1

2

*

.

int� �

(28)

This corresponds with the phenomenological lawsproposed by Gil Sevillano, Van Houtte and Aernoudt[97],

∆σ ∝ ∝ϕ

ϕd d

f f

with ,1

(29)

and by Kozlov and Koneva [42],

∆σ ∝ ϕ . (30)

3. A WORK-HARDENING MODEL FORSTABLE ORIENTATIONS

3.1. Schematization of the substructure

The work-hardening model to be discussed in thissection is designed for a single crystal with a stablesymmetric orientation, i.e. for a constant Schmidfactor m

S and for multiple slip right from the begin-

ning of plastic deformation. A cell structure is thenformed soon. Therefore, the model starts from a well-developed cell structure consisting of a cell wall“phase” with a high local dislocation density ρ

w and

a high shear resistance τW

* and a cell interior “phase”with a low local dislocation density ρ

c and a low

shear resistance τc

* [48,53].

As mentioned in subsection 2.4, most of thework-hardening models for intermediate and largestrains are based on such a two-“phase” compoundrepresentation of a cell structure and ascribe theshear resistance of the cell interior “phase” to a localdislocation density. Argon and Haasen [58] andPantleon and Klimanek [61], on the contrary, referredto abundant TEM evidence – on the unloaded aswell as on the neutron-pinned loaded state – for anonly negligibly small dislocation density in the cellinterior and ascribed the shear resistance of the cellinterior “phase” to long-range stresses. The presentwork follows the latter reasoning and considers thelocal dislocation density in the cell interiors to benegligible compared to the one in the cell walls,

ρ ρρ

ξc w

tot<< = , (31)

so that the total (global average) dislocation den-sity ρ

tot is obtained by “smearing out” the wall dislo-

cation density from the wall volume fraction ξ to thewhole volume.

It is assumed that the mean cell size dc and the

total (“smeared-out”) dislocation density ρtot

areconnected by a Holt-type scaling law [98,99],

dK K

c

tot w

= =ρ ξρ

, [32]

where K denotes a material constant (cf. e.g. thereview paper by Raj and Pharr [100] ). Furthermore,it is assumed that a principle of similitude holdsthroughout the whole course of plastic deformation,i.e. that the volume fraction ξ of the cell wall “phase”remains constant,

ξ = fw

dgeom

c

, (33)

where fgeom

= 2 is a geometrical factor reflecting theelongated shape of the cells and w denotes the cellwall width.

From the elementary processes of fragmentboundary nucleation and growth as discussed be-low in subsections 3.2.3 and 3.2.4, it follows thatfragment boundaries evolve as excess dislocationlayers along existing cell walls. These excess dis-location layers accomodate the lattice rotationmismatch and in this sense resemble Ashby’s “geo-metrically necessary dislocations” or Mughrabi’s“interface dislocations” accomodating either a strainor a lattice rotation mismatch. They are arranged inlow-angle grain boundary (LAGB) configurations, so

��

that the misorientation ϕ can be calculated fromthe dislocation spacing h in the wall via the Read-Shockley law which in the case of a single-component LAGB (i.e. an LAGB made up of ex-cess dislocations from only one slip system) reads

ϕ ≈b

h. (34)

The dislocation spacing h or the excess dislo-cation line density per area on the fragment bound-aries can be derived by distributing the (smeared-out) excess dislocation volume density ρ

exc on the

available total fragment boundary surface S. In orderto calculate the latter, one has to make an assump-tion on the geometry of the fragment structure. Whileduring the early stages of fragmentation, a Pois-son-Voronoi mosaic geometry seems to be appro-priate (i.e. a random distribution of nucleation pointsis generated and each place in space is then as-signed to the closest of these nucleation points, cf.e.g. [101,102], – a principle which is known fromthe Wigner-Seitz cells in Solid State Physics), inthe developed or later stages a simpleparallelepipedal mosaic with “brick-shaped” frag-ments seems to reflect reality better [58,103]. Inthis paper, the first will be applied.

In a Poisson-Voronoi mosaic, the mean nodeline length per unit volume corresponds to the meantriple junction line length per unit volume in a frag-ment boundary mosaic, i.e. to the immobile partialdisclination density θ

i. The mean chord length of

the Poisson-Voronoi cells corresponds to the meanchord length of the fragments which is a goodmeasure for the mean fragment size d

f [102]. The

mean surface area of the Poisson-Voronoi cells, fi-nally, corresponds to the mean fragment boundaryarea S

fb per unit volume in a fragment boundary

mosaic. All three quantities can be expressedthrough the density of nucleation points of the Pois-son-Voronoi mosaic (for details, see [101,102]), sothat all three quantities can also be expressedthrough each other,

df

i

≈166.

,θ (35)

Sfb i

≈ 121. .θ (36)

Distributing the available (smeared-out) excess dis-location density ρ

exc on the total fragment boundary

surface area Sfb per unit volume then gives the mean

excess dislocation spacing in the fragment bound-ary or, using the Read-Shockley formula Eq. (34),

the mean misorientation across the fragment bound-aries,

ϕρ

θ≈ b exc

i121.

. (37)

The dislocation substructure is thus charac-terizedin terms of the following dislocation densities:– the (smeared-out) redundant cell wall disloca-

tion density ρw,

– the (smeared-out) non-redundant excess dislo-cation density ρ

exc in the fragment boundary lay-

ers,– the density of dipoles of propagating partial

disclinations θp at the border lines of the excess

dislocation layers– the density of immobile partial disclinations θ

i in

the fragment boundary triple junctions.

3.2. Elementary processes

3.2.1. Dislocation storage: trapping of mobiledislocations into cell walls. Mobile dislocationsget stored in the material by trapping into cell walls.Slip line measurements by Ambrosi and Schwink[104] indicated that the mean free path of mobiledislocations is about three times larger than themean cell size d

c and thus revealed that cell walls

are semi-transparent obstacles. A fraction of Pc ≈

1/3 of the arriving mobile dislocations gets trappedin the cell wall, while the remaining fraction of about2/3 penetrates through the wall. (The point of viewthat all arriving mobile dislocations get trapped atthe one side of the wall and in 2/3 of all trappingevents another dislocation gets remobilized andemitted at the other side, is formally equivalent tothe reasoning suggested above.) For single slip, thisresults in a storage rate for the local cell wall dislo-cation density of [61]

d

dt

P

wv

P

w bw c

m

cρ

ργ�

��

+

= =�

, (38)

or, for multiple slip on n symmetrically activated slipsystems,

d

dt

P

wnv

P

wn

bw c

m

cρ

ργ�

��

+

= =�

, (39)

where v is the average dislocation velocity, w thecell wall width, and γ denotes the shear strain rate.Using the principle of similitude (33) and the Holt-type scaling law (32), this can be transformed intothe well-known Kocks storage term (cf. eq. (9)), nowfor a smeared-out cell wall dislocation density,

.

��

d

dt

d

dt

P f

dn

b

P f

Kn

b

w m c geom

c

c geom

w

ρ ρ γ

ργ

��

��

��

+ −

= − = − =

−

�

�

. (40)

3.2.2. Dislocation dynamic recovery: annihila-tion of mobile with cell wall dislocations. Storedcell wall dislocations can, on the other hand, anni-hilate with mobile dislocations of opposite sign pass-ing by. In the case of edge dislocations, suchannihilation events can take place through dipoledisintegration, as proposed by Essmann andMughrabi [105], or after vacancy-assisted climb. Inthe case of screw dislocations, annihilation eventstake place after cross-slip. As discussed in sub-section 2.1, room temperature deformation of copperis characterized by comparable annihilation lengthsy

s of screws after cross-slip and y

e of edges after

dipole disintegration. Annihilation of edges after va-cancy-assisted climb can be neglected becausebulk diffusion of plastic vacancies is too slow at lowhomologous temperatures, while core or pipe diffu-sion would require a significant dislocation densityin the cell interior (where the plastic vacancies wouldbe produced) to let the vacancies diffuse from thereto the cell walls (where they are needed to assistclimb and subsequent annihilation of cell wall withmobile dislocations). Therefore, the two dislocationcharacters can be treated in the same commonframework in the present context. The commonrecovery term scales with the same annihilationlength y

e=6b proposed by Essmann and Mughrabi

[105],

d

dty

by

nn

bw

e w e

wρ

ργ ρ γ�

��

−

= − = −2 2� �

, (41)

for single slip as well as for multiple slip on n sym-metrically activated slip systems.

3.2.3. Generation of partial disclination dipoles.The keystones of any dynamic model of fragmenta-tion are the mechanisms of nucleation and growthof fragment boundary segments. In a disclination-based model, these mechanisms correspond to theelementary mechanisms of generation of partialdisclination dipoles (PDD) and of propagation ofpartial disclinations (PD). Following subsection 2.2,the generation of a wedge PDD corresponds to theformation of a both-side terminating array of parallelexcess edge dislocations of the same sign, or tothe formation of two parallel semi-infinite walls of

excess edge dislocations of opposite signs. Thegeneration of a twist PDD corresponds to the for-mation of a both-side terminating net of two sets ofparallel excess screw dislocations of the same sign,or of two parallel semi-infinite nets of parallel ex-cess screw dislocations of opposite signs. A vari-ety of different mechanisms was proposed to ex-plain this nucleation of fragment boundary segmentsthrough collective behaviour of a couple of disloca-tions of the same sign [8,87,106-108]. The modelssuggested until 1960 [87,106] aimed at an under-standing of nucleation of kink bands on the grainscale, the ones suggested later [107,108] wereprompted by TEM observations of misorientationbands on the mesoscopic scale arising in stage IIIof plastic deformation [9,10,17].

From the beginning, two different routes werefollowed to explain kink banding on the grain scaleand misorientation banding on the mesoscopic scale.On the one route, an “imposed” macroscopic back-ground of kink band nucleation or a grain-scale back-ground of misorientation band nucleation was sug-gested, e.g. constraints imposed by the holders [5]or stress concentrations at grain boundary ledges[108]. On the other route, an “intrinsic” microscopicbackground of kink band as well as misorientationband nucleation was proposed on the level of dislo-cation dynamics, namely collective behaviour or self-organization of dislocations.

Frank and Stroh [106] defined a kink band as athin plate of sheared material in a non-sheared ma-trix, transverse to a slip direction and bounded byopposite tilt walls of dislocations (while in most ofthe later papers it was considered as a non-shearedplate in a sheared matrix). In such a configuration,the applied shear stress would drive the two oppo-site tilt walls not towards, but away from each other.Nevertheless, in case of finite tilt walls, the tips ofthe walls would still attract each other. Frank andStroh determined the equilibrium between therepulsive force due to the external shear stress andthe attracting force between the edges andcalculated equilibrium band widths and lengths forparallel walls and for an elliptical band. Furthermore,an elliptical band with a sufficient angle of kinkingcould grow by generating new dislocation loops orpairs at its tips. The authors noted that the stressfields of kink bands were similar to those of slipbands, so that they “might act as sources for eachother”.

As already mentioned in subsection 1.1, Maderand Seeger [8] distinguished two different types ofkink bands, one on the macroscopic scale arisingalready at the end of stage I and mainly in stage II

��

(kink bands belonging to the deformation band struc-ture in the terminology used in this paper) and oneon the mesoscopic scale arising only in stage III(misorientation bands belonging to the fragmentstructure in the terminology used here). The onsetof stage III is characterized by the resolved shearstress reaching a critical value to trigger off extensivecross-slip. Furthermore, the intensity of kink band-ing of the second type turns out to scale with thestacking fault energy. Therefore, cross-slip is mostlikely to be involved in misorientation banding.

Mader and Seeger suggested that cross-slip ofscrew dislocation segments piled-up at Lomer-Cottrell barriers would result in pairwise annihilationof screw segments from neighbouring glide zonesleaving edge dislocations in the primary slip planeas well as edge segments in the cross slip planesbehind (see Fig. 9). The glide motion of the primaryedge dislocations could be hindered by Lomer-Cottrell barriers formed through reactions amongsecondary dislocations, and by the edge segmentsleft behind in the cross-slip planes. On the one hand,the primary edges could then arrange in anenergetically favourable wall configuration perpen-dicular to their slip plane. This both-side terminat-ing wall might act as a nucleus of a kink band, andits stress fields might join the Lomer-Cottrell barri-ers in limiting the free path of the mobile disloca-tions. On the other hand, the edge segments in thecross-slip planes would always act as forest ob-stacles to the primary edges, but would, in general,not represent a low-angle grain boundary configura-

tion. However, with the help of Ergänzungsgleitung(completion glide) [64,65], they could be completedto become a both-side terminating low-angle grainboundary. This could then limit the free path of themobile dislocations in the slip direction and act asa nucleus of a second kink band.

Vladimirov [107] proposed a similar nucleationmechanism based on the redistribution of piled-upedge dislocations into an energetically favourablewall configuration with the help of vacancy-assistedclimb. For such a mechanism, the intensity ofmisorientation banding should scale with theactivation enthalpy of vacancy diffusion.

However, any mechanism based on impenetrableobstacles, e.g. Lomer-Cottrell barriers, can onlyexplain the formation of symmetric kink bands ordouble walls with mutually compensatingmisorientations because the impenetrable obstaclesalso stop dislocations which travel in the oppositedirection in the glide zone on the other side of theobstacle. In order to initiate the formation of an asym-metric kink band or double wall or a single wall witha non-vanishing total misorientation, either globallyor locally different slip rates on the two sides of theobstacle are required. Since a negligible disloca-tion content of the cell interiors and an initially ho-mogeneous orientation throughout the crystallite areassumed, there is no way to a global slip imbal-ance between the two sides of a cell wall (whichwould result in the formation of an excess disloca-tion layer on the cell wall in a straight-forward man-ner).

��3�� *�� 1��

� �� 0��1;�� *��

� �� 2��*�� 1��

��1�� <��

�� <�� 1��

��

��

A local slip imbalance, however, can arise dueto incidental collective behaviour of neighbouringmobile dislocations at a semi-transparent obstaclewith stochastic trapping. If several neighbouringmobile dislocations approaching the cell wallincidentally all together get trapped, while theircounterparts of opposite sign approaching the wallfrom the other side incidentally all together do notget trapped, then a both-side terminating excessdislocation wall representing a nucleus of a frag-ment boundary arises. Whether and on whichconditions such a nucleus grows or gets “dissolved”again, will be discussed in the next subsection. Asalready pointed out by Pantleon [109], the semi-transparency of the obstacle and the stochasticcharacter of the trapping process1 thus are of crucialimportance for misorientations to be formed. If theobstacle was an impenetrable one, then only a doubleexcess dislocation wall with no net misorientationcould be formed. If the trapping process would nothave a stochastic character, incidental collectivebehaviour would not be possible and again only adouble excess dislocation wall with no netmisorientation would result.

As mentioned in subsection 1.1, Koneva, Kozlovand coworkes [38-40] proposed the coexistence oftwo fragment boundary nucleation mechanisms. Inthe first case, straight misorientation bands wouldbe nucleated at grain boundary ledges and growinto the grain interior, in the second case disclinationloops would be generated in the grain interior in the

course of dislocation dynamics. The nucleation partof the first mechanism was recently modelled byGutkin et al. [108] as a splitting event of partialdisclinations in faults of grain boundaries triggeredby the applied shear stress. The growth part of thefirst mechanism was already modelled by Romanovand Vladimirov [22] as propagation of partialdisclination dipoles at the tips of the growingmisorientation bands consisting of two parallel ter-minating excess dislocation walls with oppositesigns. The present paper, on the contrary, focusseson modelling the second case, the nucleation andgrowth of loop-like fragment boundary layers on thegrounds of dislocation dynamics.

As discussed in sections 2.1 and 3.1, the modelpresented in this paper starts from a well-developedcell structure, i.e. from a substructure which is typi-cal for stage III of plastic deformation. Detailed lightand electron microscopy studies of slip line pat-terns on the surface of deformed copper andaluminium single crystals [87,110] revealed alreadyin the 1950s that during stage III the gliding disloca-tions get increasingly organized in bundles or slipbands and plastic deformation gets increasinglyconcentrated in the slip bands. This bundling or slipbanding was attributed to extensive double crossslip [87,110-113]. Therefore, the model presentedhere assumes that the mobile dislocations areformed through double cross slip in the cell interi-ors and thus also supposes that they are groupedinto bundles or slip bands (see Fig. 10).

In f.c.c. single crystals with intermediate and highstacking fault energy, there is a striking coincidence of· the onset of massive cross-slip,· the bundling of mobile dislocations in slip bands,

1 Of course, this stochastic character of the trappingprocess is not a fundamental one, but just due to fluc-tuations in the dislocation structure of the cell wall and

poor knowledge of its details.

��3�� 8��

�� *��1��

�� *��

�� 1��,��

��

��

· the formation of a dislocation cell structure,· and the onset of misorientation banding at the be-

ginning of stage III.If one takes into account that the mobile dislo-

cations get bundled in slip bands in the course ofstage III of plastic deformation (see the nextsubsubsection), one can use data from slip linemeasurements to estimate typical parameters of apartial disclination dipole representing a fragmentboundary nucleus. Mader [114] studied low-sym-metry oriented copper single crystals after tensionat room temperature and found slip bands with 3-10slip lines per band and lamella widths of about 300Å. If one then assumes incidental collective trap-ping of the mobile dislocations in such a slip bandat a cell wall, one gets an excess dislocation wallsegment made up of N≈ 5 dislocations with an aver-age spacing h of about 300 Å. This gives an initialPDD width of 2a

0≈ 0.15 µm and, according to Eq.

(34), a misorientation of ϕ ≈0.085. Müller andLeibfried [115] investigated low-symmetry orientedaluminium single crystals after tension at liquid airand at room temperature for a series of different strainrates. They found slip bands with 2-4 slip lines forε > 10-2 s-1 at liquid air temperature and with 10-30slip lines for ε <10-2 s-1 and about 5 slip lines forε >10-2 s-1 at room temperature. The lamella widthswere found around 700 Å at liquid air temperatureand around 400 Å at room temperature, without cleardependence on the strain rate. This gives 2a

0 ≈ 0.21

µm and ϕ ≈ 0.0041 for fast deformation at liquid airtemperature, 2a0 ≈ 0.80 µm and ϕ ≈ 0.0072 for slow

deformation at room temperature, and 2a0 ≈ 0.20

µm and ϕ ≈ 0.0072 for fast deformation at room tem-perature.

It has to be emphasized that slip line measure-ments do not provide any information about the dy-namics of plastic deformation. Since the strain in-tervals used to generate the slip lines are ratherlarge, one slip line can correspond to several mo-bile dislocations stepping out to the surface at dif-ferent times. Consequently, the spacing betweenthe slip lines does not equal the spacing betweenmobile dislocations arriving at an obstacle (perpen-dicular to the slip plane) at the same time. The spac-ings between collectively trapped dislocations de-rived above from the spacings between slip linesthus have to be considered as a lower limit or as arough aproximation.

In order to obtain a net misorientation acrossthe arising fragment boundary, it is of crucialimportance that the excess dislocation layer on theone side of the cell wall does not get compensatedby its counterpart with opposite sign on the otherside of the cell wall (compare Figs. 11 and 12 be-low) - which is only possible for a semi-transparentobstacle with stochastic trapping. For a quantita-tive description of PDD generation through incidentalcollective trapping of mobile dislocations at cell walls,the two extreme cases of single slip (with cell wallsroughly perpendicular to the slip plane, see Figs.11 and 12) and of symmetric multiple slip (with cellwalls bisecting the equivalent slip planes, see Fig.13) are considered.

�� 1�� ,�'�� 7==�� 1�� %��

�� %�� %��

��1�� %��1� ��

��1�� %�� %� �� >��?��0��

�� 1��2��

� ��

..

.

��

The loss rate of mobile dislocations due to PDDgenerating trapping can be set up taking patternfrom the loss rate due to ordinary trapping. The prob-ability P

c of individual trapping in Eq. (40) has to be

replaced by the probability (Pc(1-P

c))N of collective

and non-compensated trapping of N dislocations.The probability product reflects the requirement thatthe trapped mobile dislocations must not be com-

�� 1�� ,� '��

7==�� 1�� %��

��

� �� (��)� � �� %

� ��2��

��1�� %��

�%� �� (��)��0��

��

��1��2��

��3�� 1�� ,�'�� 7==�� %��

�� (��)��

��1�� %��3�� %��

�� 2�� %� ��

��2�� %�� %� �� (��)��0��

�� 1��2��

pensated by their counterparts of the opposite signat the other side of the cell wall, the exponent re-flects the collective behaviour of the N neighbouringmobile dislocations. In the case of single slip, onefinds a generation rate for the (smeared-out) excessdislocation density of

d

dt

d

dt

P P f

Nd b

P P f

NK b

exc m

c c

N

geom

c

c c

N

geom

w

ρ ρ

γ

ργ

��

��

��

��

+ −

= − =

−=

−

( ( )) �

( ( )) �

.

1

1

(42)

The same collective trapping process occurs at frag-ment boundaries. The generation rate can be esti-mated taking patting from the one for the cell walls,but replacing the trapping probability P

c by the one

Pf and replacing Holt’s scaling law (32) for cells by

the one (35) for fragments,

d

dt

d

dt

P P

d b

P P

b

exc m

f f

N

f

f f

N

i

ρ ρ

γ

θγ

��

��

��

��

+ −

= − =

−=

−

( ( )) �

( ( ))

.

�

.

1

1

166

(43)

��

Since N collectively trapped mobile dislocationsconstitute one new partial disclination dipole, thecorresponding PDD generation rate at cell wallsreads

d

dt

P P f

Nd b

P P f

NK b

m c c

N

geom

c

c c

N

geom

w

θ γ

ργ

��

��

+

=−

=

−

( ( )) �

( ( )) �

.

1

1 (44)

The major difference with the case of symmetricmultiple slip is that the cell walls are not perpen-dicular then to the slip plane, but roughly bisectingthe equivalent active slip planes (see Fig. 13).Consequently, neighbouring mobile dislocations fromone slip system getting trapped collectively at a cellwall would not form a both-side terminating low-anglegrain boundary configuration, so thatErgänzungsgleitung (completion glide) is required,or, more specific, mobile dislocations from theconjugate slip system are required to get trappedcollectively as well and to react formally with thealready trapped ones in order to form a low-anglegrain boundary segment. As a result, collectivebehaviour of twice as much dislocations is requiredin the case of symmetric multiple slip which makesthe probability of a PDD generating event droppingmuch below the probability in case of single slip.An example: If the individual trapping probability isP

c = 1/3, then the probability (P

c(1-P

c))N=5 of collec-

tive and non-compensated trapping under single slipis about 5.10-4, the probability (P

c(1-P

c))N=10 of col-

lective and non-compensated trapping under mul-

��=�� *��

�� ,�� *��

�� ,��

(�� )��

tiple slip is about 3.10-7. While the first probabilityis already small, but still large enough to generatea sufficient density of partial disclinationsfragmentating the crystallite, the second one turnsout to be insufficient for fragmentation.

Since the long-range stress field around such anew PDD would drive the dislocation loop segmentswith the opposite sign at the other end of the loopsaway from the PDD and towards the neighbouringcell wall, a local slip imbalance would also be in-duced at this neighbouring cell wall. As a result, aloop-like fragment boundary layer as sketched inFig.14 would form.

3.2.4.Propagation of partial disclinations. Theconditions of growth of a finite excess edge dislo-cation wall through capturing and attaching addi-tional excess dislocations were first discussed inthe context of polygonization during recovery[66,73,74]. The total stress field of the finite dislo-cation wall was calculated as the sum of the stressfields of the individual dislocations. For the long-range stress field at large distances from the wall,the sum can be replaced by an integral. This givesa shear stress perpendicular to the wall of

τπ νxy

Gbx

h

y a

x y a

y a

x y a

=−

×

+

+ +−

−

+ −��

��

2 1

2 2 2 2

( )

( ) ( ), (45)

where h is the dislocation spacing in the wall, a =Nh/2 denotes the half height of the wall, and thereference system is chosen as indicated in Fig.15.

@" ��

If one expresses the dislocation spacing throughthe misorientation according to Eq. (34), one findsthe long-range stress field of the finite wall with dis-location spacing a and height 2a to be equivalent tothe one of a partial disclination dipole withdisclination strength ω = ϕ = b/h and dipole width2a,

τω

π νxy

G x

h

y a

x y a

y a

x y a

=−

×

+

+ +−

−

+ −��

��

2 1

2 2 2 2

( )

( ) ( ). (46)

As Nabarro showed already in 1952 [73], it followsfrom Eq. (45) that the finite wall or the PDD exertsan attracting Peach-Koehler force F

x = b

x τ

xy on an

edge dislocation with matching sign inside thecapture hyperbola (see Fig.15) given by

y x ac c

2 2 2− = . (47)

However, the applied shear stress due to the finitewall or the PDD has to overcome the Peierls-Na-barro stress and the local effective shear stress,i.e. the external shear stress reduced by the meaninternal shear stresses due to e.g. other disloca-tions or due to plastic strain or lattice rotation mis-fits,

��(;�� )��

��

τ τ τ τ τ τyx PN eff PN ext

> − = − +int

. (48)

Or, phrased in another way, the applied shearstresses due to the externally applied load and thePDD have to overcome the local shear resistancecomposed of the Peierls-Nabarro resistance and theshear resistance due to the current defect struc-ture,

τ τ τ τyx ext PN

+ > +* * . (49)

Determining the local internal and thus the local ef-fective stress is a formidable task, so that in gen-eral strongly simplified representations of the de-fect structure are used to estimate the internal stressat least by order of magnitude. In the following, tworough estimations are carried out in order to find aminimum required partial disclination strength ω

p

which allows PD propagation, i.e. PDD expansionor fragment boundary growth, and to find a capturelength y

c, i.e. a distance ahead of a PD within which

the PD is still able to capture a mobile dislocationof matching sign passing by.

From a graphical representation of the shearstress field τxy of a finite excess dislocation wall asshown by Li [74], one can read that the loop-shapedcurves of attracting shear stresses of magnitude 0.3,0.2 and 0.1 times a stress unit of Gb/2π(1-ν)h =Gϕ/2π(1-ν), end up at distances of about 1.5, 2.5and 5 times the wall half length a in front of theexcess dislocation wall tip. In the case of, forinstance, copper at room temperature this means

| ( )| .( )

.( )

MPa.

τπ ν

ϕ

π νϕ

xy y aGb

h

G

= <−

=

−≈

5 0 12 1

0 12 1

1000 (50)

A bundle of N = 5 dislocations with a slip planedistance of h =100b ≈ 26nm corresponding to amisorientation ϕ = 0.01 would then at y = 5a, i.e. at4a ≈ 0.26 µm ahead of the wall tip, exert a shearstress of not more than 10 MPa (and of no morethan 30 MPa at 0.5a=1.25h ahead of the wall tip). Agroup of N = 5 dislocations with h = 1000b ≈ 260nm corresponding to ϕ =0.001, however, would at4a ≈ 2.6 µm ahead of the wall tip exert a shearstress of only less than 1 MPa (and of only lessthan 3 MPa at 0.5a=1.25h ahead of the wall tip).From this simple example, one can already guessthat a partial disclination of strength ϕ = ω = 0.01would have a chance to grow by capturing additionalmobile dislocations, whereas a partial disclination

��

of strength ϕ = ω =0.001 would not because the formerwould have a chance to overcome the local effectivestress or the externally applied load plus the localshear resistance, whereas the latter would not.

For a quantitative evaluation of Eq. (49), thePeierls-Nabarro resistance is neglected for f.c.c.materials and the coordinates are rescaled in unitsof a to give ~x = x/a and ~y = y/a, giving

τω

π νxy

G x

y

x y

y

x y

=−

×

+

+ +−

−

+ −��

��

~

( )~

~ (~ )

~

~ (~ ).

2 1

1

1

1

12 2 2 2

(51)

The limiting curve where the applied shear stressdue to the PDD equals the local effective stress isthen defined by

~~

( )~ (~ )~

(~ )~

( )~ (~ ) .

yG x

y x y

G x xy x

eff

eff

4 3 2 2

2

2 2

12 1

1

11 0

−−

+ − −

−

−+ + =

ω

π ν τ

ω

π ν τ

(52)

A simple special case is x = a or ~x =1 giving

~( )

~yG

yeff

4 3

14 0−

−+ =

ω

π ν τ. (53)

To estimate the local effective stress, firstconsider an idealized framework of only the finitedislocation wall and a random distribution of mobiledislocations being present. Then the scaling law

τ α ρeff mGb= (54)

proposed by Kocks, Argon and Ashby [84] for thelocal effective stress would apply. The mobile dislo-cation density ρ

m can be estimated from Orowan’s

law

� .ε ρ= m bvS m

(55)

To estimate an average dislocation velocity in thiscontext, one should better refer to the TEM in-situmeasurements rather than to the pulse load andetching techniques because the continuous loadingused in the in-situ tests is similar to the conditionsof macroscopic straining [116]. In copper as well asin aluminium plastic deformation is carried out byfew but fast mobile dislocations – Appel andMesserschmidt [117] measured an average dislo-cation velocity of v = 10-6 ms-1 in a TEM in-situ teston aluminium at room temperature. For a strain rate

of ε =10-2 s-1, this gives a mobile dislocation densityof about ρ

m ≈ 1014 m-2 which, according to Eq. (53),

corresponds to an effective stress of about τeff

≈ 20MPa. Inserting this into Eq. (52) gives two real rootsof y

c,1 ≈ 0.75a and y

c,2 ≈ 10.2a for ω = 0.01 and only

complex roots for ω = 0.001. An estimation of thecapturing length carried out by the author [118] forthe case of a PDD propagating at the top of a grow-ing misorientation band also gave a capturing lengthof about y

c ≈ 10a.

It can be concluded that the Peach-Koehler forceexerted by a partial disclination with a strength be-yond a required minimum strength of about ω

p =

0.01 is sufficient to capture mobile dislocations pass-ing by within a capturing length of about y

c ≈ 10a, to

attach them to the terminating excess dislocationwall and to let the partial disclination dipole expand.As Li [74] could show, such a process is alsoenergetically feasible. It has to be emphasized thatit is only the bundling of slip in slip bands whichmakes possible the nucleation of fragment bound-ary segments that are able to grow: this bundling inslip bands reduces the mean dislocation slip planedistance h in the slip bands of stage III compared tothe one in the homogeneous fine glide regime ofstage II, so that the resulting partial disclinationstrength exceeds the critical value of about ω

p =

0.01. Thereby, slip banding and fragmentation, twodifferent manifestations of collective behaviour in thedislocation ensemble, are closely related to eachother.

The coupling of fragmentation to slip bandingdoes not only lead to a minimum required disclinationstrength. Since the width of the lamellae betweenthe slip lines in a slip band is only slightly changingin the further course of plastic deformation, it alsooffers an average strength of a propagating partialdisclination, i.e. a step-width of the increase inmisorientation. In this way, one can eliminate oneof the parameters unknown in disclination in con-trast to dislocation dynamics.

While Romanov and Vladimirov [22] “fed” thepropagating partial disclination dipole at the top of agrowing misorientation band by widening of dislo-cation loops from a static dislocation distributionbetween the two boundary planes ahead of the bandtip, the present work follows a dynamic approachas proposed by Essmann and Mughrabi [105] inthe context of dynamic recovery. The reasoning fora wedge PDD capturing a passing edge dislocationof the matching sign is exactly the same as for acell wall edge dislocation spontaneously disintegrat-ing with a passing edge dislocation of opposite sign(see subsection 3.2.2 and Fig. 16).

.

@/ ��

Formally, the density θp of dipoles of propagat-

ing partial disclinations replaces the density ρw of

cell wall dislocations as sink density, and thecapture length y

c replaces the annihilation length

ye,

d

dt

d

dty

b

ab

exc m

c p

p p

ρ ρθ

γ

θγ

��

��

��

��

+

= − = − =

−

2

20

�

�

, (56)

where ap denotes the average half width of the ex-

panding dipoles of propagating partial disclinations.The same mechanism, of course, also controls

the mean propagation speed V of the partialdisclinations: To calculate this velocity, one can es-timate the time t

c which is needed by the mobile

dislocation current γ/b to provide the Nc disloca-

tions of the matching sign and character requiredby the PD to propagate by its own capturing lengthy

c,

tN

y

b

c

c

c

p= =

4

4

� �,

γ

ω

γ (57)

where Eq. (34) was used. This results in an averagespeed of

��%�= � ��

�� %�;��

�� 2�� %��%�;��

��

�� %�

.

y

t

ac

c

p

p

= ≈10

4ωγ� . (58)

3.2.5. Disclination storage: immobilization ofpropagating partial disclinations at fragmentboundaries. For simplicity, it is assumed that eachpropagating partial disclination is definitely stoppedat the next fragment boundary, i.e. in the next meshof the fragment boundary mosaic (which is sug-gested by the analytical formula describing the in-teraction between a wide sessile and a narrow propa-gating PDD [23,119]). Then, the mean free path of apropagating PD equals the mean fragment size d

f,

giving an immobilization term of

d

dt

d

dt d

V

V

i p

f

p

i

p

θ θ θ

θθ

��

��

+ −

= − = =1

2

0 6022

. , (59)

where Eq. (35) was used. The factor of 1/2 is due tothe transition from the PD dipole density θ

p to the

single PD density θi. Note that the disclination im-

mobilization term has the same structure as thedislocation storage term proposed by Kocks (cf. Eq.(9). Using expression (58) for the mean disclinationspeed finally gives

��

d

dt

d

dt

a b

bi p p

p

i p

θ θ

ωθ θ

γ��

��

+ −

= − = 0 60210

8.

�

. (60)

3.2.6. An open question: disclination recovery.In order to study the substructural background ofstage V of plastic deformation, one probably has tosearch for a mechanism of dynamic recovery ofpartial disclinations. A promising hint for such amechanism might be the coupling between work-hardening in stage V and the hydrostatic pressure[120] because the wedge disclination density is alsocoupled to the hydrostatic stress [23]. Another pos-sible mechanism might be the activation of addi-tional slip systems around the PD triggered by theirlong-range stress fields. This would result in an ac-cumulation of misfit-relieving dislocations around thePD, as proposed by Argon and Haasen for othersources of long-range stresses [58].

3.3. Substructure development

Collecting all the reaction terms from Eqs. (40), (41),(42), (43), (44), (56) and (60), transforming the evo-lution equations with the help of Orowan’s law (m

S –

Schmid factor)

�ε ρ= m nbvS m

(61)

from the evolution parameter time to the evolutionparameter strain and abbreviating all the reactioncoefficients gives the following set of evolution equa-tions:

d

dI Rw

w w

ρ

ερ ρ= − , (62)

d

dS S Eexc

c w f i p

ρ

ερ θ θ= + + , (63)

d

d

S

N

S

NJp c

w

f

i i p

θ

ερ θ θ θ= + − , (64)

d

dJi

i p

θ

εθ θ= . (65)

If one neglects collective trapping at cell walls com-pared to collective trapping at fragment boundaries,this system of ordinary differential equations hasthe advantage that the evolution equations (62) and(65) for the redundant dislocation density and theimmobile partial disclination density which describethe substructure development and control the work-hardening can be solved analytically. Equation (62))for the redundant dislocation density has the struc-

ture suggested by Mecking and Kocks [81] for thetotal dislocation density (cf. Eq. (10) in subsection2.3) and can directly be integrated by separatingthe variables,

ρ εε ε

w

I C e

R

y

( ) .( )/

=− − −�

��

1

2 2

(66)

The integration constant C1 can be determined from

the starting condition ρw(ε = ε

y) = ρ

w. The stationary

values of the cell wall dislocation density and of thedensity of dipoles of propagating partial disclinationscan directly be calculated and classified as stablestationary values:

ρ

θ

w

s

p

s f

I

R

S

NJ

=

=

��

2

and

. (67)

Inserting these stationary values into (64), one findsa simple law for the evolution of the slowly varyingstructure variable θ

i in the quasi-stationary state,

d

d

S

Ni

qs

f

i

θ

εθ�

�� = , (68)

or integrated (C2 – integration constant)

θ ε εi

qs fS

NC( ) ,= +�

��2

2

2

(69)

and through the relationship (35) also for the meanfragment size df:

d d

d

S

N

S

Ndf

qs

f

i

f

fε θ�� = − = −0 83

10 30 2. . , (70)

or integrated (C3 – integration constant):

dS N C

f

qs

f

( ). ( / )

.εε

=+

1

0 303

(71)

Such a “hyperbolic” decrease matches with experi-mental observations [26].

The reaction coefficients R for dynamic recov-ery, E for capture of mobile dislocations throughpropagating partial disclination dipoles, and J forimmobilization of propagating partial disclinationswere calculated according to the expressions in (41),(56) and (60) with an annihilation length of y

e = 6b, a

capturing length of yc = 10a, an average width of

0

@� ��

expanding partial disclination dipoles of 2ap =0.1

µm, and a strength of the propagating partialdisclinations of ω

p = 0.01, see Table 2. The reaction

coefficients I for “individual” trapping of mobile dis-locations in cell walls and S

f for “collective” trapping

of mobile dislocations at fragment boundaries wereused as fitting parameters. Their values were deter-mined through fitting the modelled to an experimen-tal flow curve for uniaxial compresssion of a [001]-oriented copper single crystal at room temperature,see Table 2.

The obtained values were then evaluated accord-ing to the expressions in (40), (42) and (43) withtrapping probabilities of P

c = 1/3 at cell walls and P

f

= 1/2 at fragment boundaries, a Holt constant of K= 16 and a geometric cell shape factor of f

geom = 2.

The fitted coefficients then corresponded to a numbern = 3 of locally active slip systems and to a numberof N = 5 excess dislocations constituting a newpartial disclination dipole. The first number corre-

Table 2. Values of the reaction coefficients used in the model. Schmid factor ms = 0.408.

coefficient value reference

I 1.24.109 m-1 fitting parameter for ξ = 0.20 [121], Eq. (40)I 8.29.108 m-1 fitting parameter for ξ = 0.45 [55], Eq. (40)R 29.4 Eq. (41)E 2.39.104 Eq. (56)S

f3.83.108 m–1 fitting parameter, Eq. (43)

J 9.22.10-4 m-1 Eq. (60)

sponds to the experimental observation (cf., for in-stance, [58]) that, for symmetric orientations, usu-ally much less than all the slip systems expectedfrom a Schmid factor analysis (in this case eight)are locally activated. The second number corre-sponds well with the slip line measurements at thebeginning of stage III as discussed in subsections3.2.3 and 3.2.4.

The resulting fragment size evolution accordingto Eq. (35)) and the resulting fragment misorientationevolution according to Eq. (37) are displayed in Figs.17 and 18. The results are in good agreement withthe (rare) experimental data, see, for instance,[32,122].

3.4. Work-hardening

Fig. 19 displays the resulting flow stress contribu-tions due to the cell structure or the redundant cellwall dislocations and due to the fragment structure

��<�� %��

�� 1� �� 3��

��ε�A�B"1/��1B�.

��

�� <��

�� %�� 1� �� 3��

��ε�A�B"1/��1B�

��9��

�� 1� ��

� �� *��

�� A�"��"��

��ε�A�B"1/��1B��9��

� ��

��13��

.

.

@� ��

or the immobile partial disclinations in the fragmentboundary triple junctions as well as the total flowstress according to Eq. (28). For comparison, anexperimental flow curve from a uniaxial compres-sion test at a [001] oriented copper single crystal atroom temperature is also shown. In Fig. 20, thecorresponding Kocks-Mecking plot is presented.

The stages III and IV of work-hardening can clearlybe distinguished. Stage III can be attributed to thesaturating local shear resistance of the cell struc-ture due to the redundant dislocations in the cellwalls. Stage IV, on the contrary, can be connectedto the monotonously growing long-range internalstresses of the fragment structure due to the im-mobile partial disclinations in the fragment bound-ary triple junctions. This matches with the work-hardening concept presented by Kozlov, Koneva andcoworkers [39-42].

ACKNOWLEDGEMENTS

The author wishes to thank P. Klimanek, Freiberg,and A.E. Romanov, St. Petersburg, for the intro-duction into the concept of disclinations and forcontinuous encouragement, support and help. W.Pantleon, Roskilde, V. Klemm, Freiberg, P. VanHoutte, Leuven, and E. Aernoudt, Leuven,contributed in many fruitful discussions to a criticaland better understanding.

Financial support was given by the DeutscheForschungsgemeinschaft (Graduate School “Physi-cal Modelling in Materials Science”) and the Fondsvoor Wetenschappelijk Onderzoek – Vlaanderen(contract number G.0228.98).

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��

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