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

DISCLINATIONS IN LARGE-STRAIN PLASTIC DEFORMATION 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 temperatures results in the subdivision of monocrystals or polycrystal grains into mesoscopic fragments and deformation bands. Stage IV of single crystal work-hardening and the substructural contribution to the mechanical anisotropy emerge at about the same equivalent strain at which the fragment structure becomes the dominant substructural feature. Therefore, the latter is likely to be the reason for the new features in the macroscopic mechanical response. The present paper reviews some experimental and theoretical work on deformation banding and fragmentation as well as a recent model which tackles the fragment structure development as well as its impact on the macroscopic mechanical response with the help of disclinations. Incidental or stress-induced formation of disclination dipoles and non-conservative propagation of disclinations are considered as “nucleation and growth” mechanisms for fragment boundaries. Propagating disclinations get immobilized in fragment boundaries to form new triple junctions with orientation mismatches and thus immobile disclinations with long-range stress fields. The substructure development is described in terms of dislocation and disclination density evolution equations; the immobile defect densities are coupled to flow or critical resolved shear stress contributions.

1. INTRODUCTION in the 1960s the attention of metal physicists was drawn to individual dislocation behaviour in the early The activation of rotational modes of deformation, stages of plastic deformation. However, deformation i.e. the spatially and temporally progressing re- banding was rediscovered in texture research where orientation of parts of monocrystals or polycrystal the disregard of grain subdivision was understood grains in the course of plastic deformation, and its to be one reason for the deficiencies in texture pre- impact on the macroscopic mechanical response diction, especially in local texture prediction. were intensively studied in Physical Metallurgy since Already at the beginning of the 1960s the 1930s. The phenomenon was discovered through monocrystal subdivision on a smaller scale than the X-ray diffraction at deformed metals exhibiting Laue one of the deformation bands was discovered in TEM: asterism, i.e. a subdivision of the ideal Bragg reflexes misorientation bands were found, a band substruc- into sets of subreflexes, corresponding to the sub- ture arising during stage III of work-hardening. Ex- division of grains into sets of subgrains. tensive TEM studies in the 1970s and 1980s Monocrystal or grain subdivision was studied in revealed deformation banding on the II. mesoscopic a variety of different contexts: In the early years of scale and fragmentation on the I. mesoscopic scale Physical Metallurgy, deformation banding, especially to be the characteristic features of the substructure kink banding, was studied in order to understand development at intermediate and large strains, in kinking as an additional mechanism of plastic de- the developed stage of plastic deformation, as Rybin formation next to dislocation gliding and deforma- phrased it. At the same time, the stages IV and V tion twinning [1,2] (and next to grain boundary glid- of work-hardening as well as the substructural ing and deformation-induced phase transformations contribution to the mechanical anisotropy were dis- which were discovered later). With the new possi- covered. However, only in Russia it was tried from bilities of transmission electron microscopy (TEM) the beginning to connect these new features in the Corresponding author: M.Seefeldt, e-mail: [email protected]

© 2001 Advanced Study Center Co. Ltd. Disclinations in large-strain plastic deformation and work-hardening 45 macroscopic mechanical response to the new fea- oriented crystal lamellae, and b) bands of second- tures in the substructure evolution, namely to frag- ary slip are regions nearly free of primary slip traces mentation. Still today, the connection between stage and almost parallel to the primary slip planes in the IV and fragmentation is not yet generally accepted, early stages of plastic deformation. The influence of neither on experimental nor on theoretical grounds. the crystal orientation, the temperature, the defor- This development was also driven by technologi- mation speed and the crystal purity was studied. cal interest: All the industrially relevant forming pro- The essential feature of the kink bands is the bend- cesses, as, for example, flat rolling, deep drawing ing which gives rise to a diffuse, streak-like Laue or wire drawing, involve large strains, high strain asterism. The bands of secondary slip act as pref- rates and multiaxial deformation modes. Recently, erential sites of activation of unexpected slip sys- the production of ultra-fine grained and bulk tems and thus also exhibit disorientations, but they nanostructured materials by means of severe plas- are essentially free of curvature. tic deformation – which is essentially based on ex- Kink bands were almost absent in deformed tensive fragmentation – found increasing interest also crystals with highly symmetric initial orientations in industry. (deforming through conjugate slip on two or more This section gives a brief overview on deforma- slip systems from the beginning) and were most tion banding and fragmentation and large strain work- pronounced in crystals with initial orientations close hardening – the framework within which the present to the centre of the standard triangle of the stereo- work is situated. The second section explains the graphic projection (deforming mainly through primary concepts of reaction kinetics and disclinations used slip). Bands of secondary slip were observed for all in the proposed model – the approach or point of initial orientations, but were most pronounced and view taken by the present work. The third section even of macroscopic size in multiple-slip oriented presents a semi-phenomenological application to crystals (deforming through conjugate slip e.g. on the cell and fragment structure development and their four active systems). Kink bands were observed in impact on the flow stress for single crystals with a wide range of temperatures. However, the spac- symmetric orientation. The model is capable of ing between the bands turned out to be about 2-3 reproducing stages III and IV of the room tempera- times larger at high temperatures and too small for ture work-hardening curve for copper. observation with optical microscopy at liquid nitro- gen temperature. At temperatures above 450 °C, 1.1. Deformation banding and frag- polygonization of the bent regions was observed, mentation resulting into a sharp, small spot-like Laue aster- ism. The influence of the deformation speed on both Since the 1930s and extensively in the 1950s light types of banding was found to be small. and electron microscopy studies of slip line pat- From recovery experiments [6], it was concluded terns and X-ray studies of Laue asterism were used that the Laue asterism was due to deformation-in- to investigate deformation banding during plastic de- duced banding. Honeycombe [5] attributed the for- formation. In most cases, aluminium single crystals mation of kink bands to the restraints imposed by after tensile deformation at room temperature were the grips on the crystals. In case of multiple slip, studied. Barrett and Levenson [3,4] used macro- the stresses due to these restraints could be scopic etching methods to reveal orientation differ- reduced by slip on alternative slip systems. The for- ences and found band-shaped surface structures. mation of bands of secondary slip was ascribed to They defined a deformation band as a region in unpredicted cross-slip at the beginning of plastic which the orientation progressively rotates away from deformation which could hinder slip on primary slip that in the neighbouring parts in the crystal and systems. So as early as 1951 two different types of suggested that “deformation bands form as a result deformation banding were distinguished and traced of the operation of different sets of slip systems in back to two different reasons, namely to an imposed different band-shaped regions”. one on the macroscopic scale – the restraints ow- Honeycombe [5,6] distinguished, based on op- ing to the deformation mode – and to an intrinsic tical microscopy surface studies and X-ray aster- one on the microscopic scale – dislocation dynam- ism measurements at aluminium single crystals, ics leading to unexpected cross-slip. two types of bands: a) kink bands are narrow bent Staubwasser [7] used surface slip line and X- regions (typically with double curvature) normal to ray asterism studies on aluminium single crystals the active slip direction which separate slightly dis- after tensile deformation at room and liquid air tem- 46 M. Seefeldt

perature in order to investigate the orientation de- monocrystals after tensile deformation at room tem- pendence of excess dislocation boundary and kink perature. At the beginning of stage III, they found band formation and work-hardening. Low-symmet- flat, disk-shaped areas parallel to the primary slip ric single-slip orientations in the lower central part planes which were limited by secondary disloca- of the standard triangle showed strong asterism right tions and included long bowed-out primary disloca- from the beginning of plastic deformation, tions, but only few forest dislocations. According to corresponding to the formation of excess disloca- Essmann, the disks were misoriented by 0.2-1.0° tion tilt boundaries perpendicular to the glide direc- with respect to each other and the boundaries had tion. With increasing deformation, strong kink-band- a mixed tilt-twist character. He drew a remarkable ing was found, also perpendicular to the glide direc- conclusion from a comparison of Steeds’ and his tion and with net misorientation across the band. micrographs: while both of them agreed on the disk High-symmetric multiple-slip orientations, on the structure parallel to the primary slip planes, their contrary, exhibited only weak asterism, statements on the height of the disks varied: Steeds corresponding to the formation of dislocation walls found 1-2 µm, Essmann 0.4 µm. Following perpendicular to all of the active slip planes. No kink- Essmann, Steeds’ result matched with the period banding at all was observed. Furthermore, of the large misorientations which he also observed, Staubwasser suggested a mechanism for kink band but found to be subdivided on a finer scale. In spite formation. Groups of primary dislocations could get of these early TEM findings of a misorientation band stopped at (unspecified) obstacles, and “such on substructure, the attention of metal physicists was neighbouring planes could be held fast by the former then mainly drawn to the dislocation structures in ones”. The arising excess dislocation boundaries the stages I and II of work-hardening, especially to were still penetrable to mobile dislocations. How- individual dislocations, Lomer-Cottrell barriers, pile- ever, at the end of stage I, the stress fields around ups, dislocation bundles and cell walls (for an over- the dislocation boundaries would activate a second- view see e.g. [11]). ary slip system, so that immobile barriers could be However, in the mid 1960s, deformation banding formed through dislocation reactions. Mobile dislo- and fragmentation were rediscovered in another cations would then pile up at the now intransparent context, namely in texture research. A long-stand- boundaries and induce the characteristic S-shaped ing problem in this field was the mechanism for the lattice curvature of the kink bands. Fully developed formation of a <111> and <100> double fibre texture kink bands would then be impenetrable obstacles in wire-drawing of f.c.c. metals. Ahlborn [12,13] stud- and have a serious impact on work-hardening. ied the orientation development of f.c.c. single Mader and Seeger [8] distinguished, based on crystals (Ag, Au, Cu, Al, brass) during wire-drawing replica studies of slip lines patterns on copper single using the X-ray rotation crystal method. He found crystals deformed at room temperature, two differ- the single crystals to split up into deformation bands ent types of kink bands: the kink bands of the first with volume fractions of the two components strongly type arise already at the end of stage I and mainly varying among the metals. He connected this varia- in stage II, are of macroscopic size (length between tion to the variation in stacking fault energy. Later several 100 µm and the specimen size, width be- on, Ahlborn and Sauer [14] used TEM to study de- tween 10 and 100 µm) and are very straight. The formation banding on the (second) mesoscopic scale kink bands of the second type arise at the begin- as well as fragment and cell structure development ning of stage III, are of mesoscopic size (only about on the (first) mesoscopic and microscopic scales 100-200 µm long and very narrow) and are less at copper single crystals during wire-drawing. The straight, a bit wavy and connected to each other extensive TEM study by Malin and Hatherly [15] through branches. They correspond to the also aimed at a better understanding of the coupling misorientation bands on the mesoscopic scale stud- between fragmentation, deformation banding and ied later on in TEM. The present paper focusses on texture evolution. these misorientation bands on the (first) mesoscopic In line with Essmann’s remark, Likhachev, Rybin scale rather than on the kink and deformation bands and co-workers were the first to distinguish between on the grain-size or (second) mesoscopic or even the cell structure development on the microscopic macroscopic scale. scale and the fragment structure development on In the early TEM era at the beginning of the the (first) mesoscopic scale [16-18]. The cell struc- 1960s, Steeds [9] and Essmann [10] investigated ture development, as reflected in the average cell the dislocation arrays in single-slip oriented copper size, cell wall width or misorientation across cell Disclinations in large-strain plastic deformation and work-hardening 47 walls, was found to be very slow at large strains, so TEM results [36]. According to the Risø school, frag- that Rybin [18] called the cell structure “frozen” 1. mentation is due to the activation of different sets of Therefore, large-strain plastic deformation and locally less than five slip systems (giving lower den- work-hardening were ascribed to the fragment struc- sities of intersecting dislocations and thus of jogs) ture evolution. Fragmentation was recognised as [25] or due to different slip rates within the same an almost universal phenomenon which takes place set of active slip systems [36]. in f.c.c., b.c.c., and h.p. mono- as well as polycrys- Recently, the orientation dependence of grain talline metals and alloys. Cell sizes, fragment sizes subdivision was investigated. In a TEM study on as well as orientations and misorientations of frag- cold-rolled aluminium polycrystals, Liu et al. [28] ment boundaries were measured mainly at molyb- classified the subdivision patterns into three types: denum (as a representative for the b.c.c. materials) Type AI grains (mainly near the Goss and S texture mono- and polycrystals but also at nickel (for the components) split up into a quite uniform cell block f.c.c. materials) and α-titanium (for the h.p. materi- structure with crystallographic boundaries parallel als) [17,18]. The St. Petersburg school of plasticity to the active {111} slip planes. Type AII grains (scat- interpreted the arising fragment boundary mosaic tered) split up into a non-uniform structure with crys- as a network of partial disclinations in the mosaic’s tallographic boundaries. Type B grains (roughly triple junctions [16,20,21] and ascribed fragmenta- around the cube type texture components) split up tion to the elementary processes of generation, with non-crystallographic boundaries and on two propagation [18,22,23] and immobilization of par- different scale levels, on the (first) mesoscopic one tial disclinations (see below). and on the grain-size or (second) mesoscopic one. Hansen, Juul Jensen, Hughes and co-workers The latter probably corresponds to deformation band- studied grain subdivision mainly in pure aluminium ing in cube-oriented single crystals. The subdivision [24-28] and nickel [25,26,29-31] but also in Ni-Co patterns of channel-die compressed and cold-rolled, and Al-Mn alloys [25,29] as well as in copper [32]. respectively, aluminium single crystals with the Cold-rolling [24-28,30,32] as well as torsion brass, the copper and the cube orientation were dis- [25,29,30] were applied. Recently, single crystal frag- cussed in [33-35]. mentation was investigated in pure aluminium after To the author’s knowledge, Koneva, Kozlov and channel-die compression [33,34] and cold-rolling co-workers [38-42] were the first [35]. The cell and cell block sizes as well as the · to distinguish two nucleation mechanisms for wall and boundary orientations and misorientations fragmentation, namely an intrinsic one and a were characterized extensively. Whereas the cell grain boundary-induced one and size and the misorientation across cell walls (CW) · to distinguish the discrete misorientations across remain almost constant at large strains, the cell the fragment boundaries and continuous block size continues to decrease, and the misorientations bending and twisting the frag- misorientation across the cell block boundaries ment interiors (for a scheme, cf. Fig. 1). (CBB) continues to increase. The cells are roughly Their conclusions were based on TEM studies equiaxed, while the cell blocks are elongated and of compressed f.c.c. Cu-, Ni- and Fe-based alloys show a strong «directionality». The CW were inter- in the long-range and short-range ordered solid so- preted as «incidental dislocation boundaries» (IDB), lution states. the CBB as «geometrically necessary boundaries» According to Koneva, Kozlov and co-workers, (GNB). Both of them are considered as low energy the beginning of stage III is characterized by the dislocation structures (LEDS). Especially, the in- emergence of terminating as well as continuous dis- terpretation of the CBB as Frank boundaries free of location boundaries with discrete misorientations. long-range stresses turned out to correspond with These boundaries appear in band-like configurations made up of parallel boundaries which rotate the lattice inside the band with respect to the matrix 1 This steady state of the cell structure is a dynamic (and grow in band direction) as well as in loop-like one: Dislocations would constantly get trapped into configurations made up of a closed boundary ring and emitted from the cell walls. However, there is ex- (see and in Subsection 3.2.3. below) which rotates perimental evidence [19] that at least in copper cell the loop interior with respect to the matrix (and grows refinement is realised by cell subdivision rather than perpendicular to the loop). The different configura- by cell wall motion. Strictly speaking, the geometry of tions are the result of different nucleation mecha- the cell structure is frozen in the steady state, but not nisms: the cell structure itself. 48 M. Seefeldt

Fig. 1. Schematic representation of substructure development under plastic deformation at low homologous temperatures or without signifcant influence of diffusion. Simplified and slightly modified after Kozlov, Koneva and coworkers ([40], p. 96).

· grow of band systems from polycrystal grain deformations, distortions of a mixed type predomi- boundaries, especially from steps in the grain nated. boundaries and Kozlov, Koneva and co-workers used the follow- · nucleation and development of closed subboun- ing quantities to characterize the substructure: the dary rings bordered by partial disclination loops. redundant1 (or “incidentally stored” or “scalar”) dis- ρ Which of the mechanisms operates, depends, location density r and the non-redundant or excess for instance, on the previous substructure and on (or “geometrically necessary” or “tensorial”) dislo- ρ the presence of grain boundaries. Both mechanisms cation density exc, the subboundary spacing df , the lead to a misorientation band substructure with first radial and azimuthal subboundary misorientations ω ω one and later two or more intersecting band sets rad and az, the lattice curvature or lattice bend twist κ ρ which gradually cover the whole volume of the speci- . The non-redundant dislocation density exc was men. At the beginning of stage III, the sets were calculated as the density of dislocations accom- oriented parallel to non- or only slightly active octa- modating lattice rotation mismatches through divid- hedral slip planes and were either of a pure tilt or ing the lattice rotation difference by the bending pure twist character. With increasing deformation, extinction contour width l the sets deviated more and more from the octahe- κ ∆ϕ dral slip planes and were of a mixed tilt-twist ρ == exc . (1) character. In stage IV, the misorientation band sub- b l structure is present in the whole specimen volume. While the growth rate of the redundant dislocation Beyond that, continuous misorientations with small density was maximum in stage II, the one of the but increasing orientation gradients develop inside non-redundant dislocation density was maximum the misorientation bands due to local bending and twisting. At the beginning of stage IV, mostly either 1 pure bending or pure twisting was found. At larger The distinction between redundant and non-redun- dant dislocations was introduced by Weertman [37]. Disclinations in large-strain plastic deformation and work-hardening 49 in stage III. It was observed that extensive genera- varying lattice rotation rates but not in varying strain tion of non-redundant dislocations starts at a critical rates, i.e. they have to be different solutions of the redundant dislocation density of about 1.5 - 3.1014 same Taylor problem. The Risø school proposed m-2. Plotting the mean misorientation ω and the av- the selection of different sets of less than five active erage lattice curvature κ over the redundant disloca- slip systems in different cell blocks because with ρ tion density r clearly revealed the transition from fewer active slip systems less dislocation jogs have stage III to stage IV. The results can be summa- to be produced and less energy is dissipated [25]. rized in the scheme presented in Fig. 1 taking pat- The selection of different sets of active slip systems tern from Kozlov et al. [40]. results in different lattice rotation rates and thus Continuous and discrete misorientations were requires the formation of geometrically necessary also distinguished and analysed quantitatively in a boundaries to accommodate the lattice rotation recent OIM study at cold-rolled aluminium AA1050 mismatch and to separate the evolving cell blocks by Delannay et al. [43]. The subdivision of each grain from each other. Dislocation dynamics has to pro- was characterized by measuring the orientation vide the dislocations to construct these boundaries. spread within the grains. Two parameters were de- In this approach the choice of different sets of fined: the average misorientation of all grid points active slip systems is thus prior to the formation of within one grain with respect to the mean orientation the geometrically necessary boundaries. This would of that grain (reflecting a large orientation spread require spontaneous collective behaviour of dislo- across the grain), and the average misorientation cations in a mesoscopic volume which is not likely between neighbouring grid points in the orientation to occur. Supposing that local dislocation dynam- map (reflecting the subdivision into highly misoriented ics is controlled only by the locally present stress fragments). Both parameters were calculated sepa- fields. rately for each grain. It was concluded that grains In the author’s opinion, fragmentation should be with orientations close to cube have higher triggered off by local rather than global slip imbal- orientation spreads than grains along the β-fibre. ances across an obstacle. This results in the gen- Furthermore, among the grains with a strong lattice eration of a both-side terminating excess disloca- distortion, only those with orientations close to tion wall which can be completed to form a both- copper and TD-rotated cube (cube rotated by 22.5 ° side terminating small-angle grain boundary as a around the transverse direction) have high nucleus of a misorientation boundary which would neighbouring-points misorientations. then, with the help of its long-range stress fields, The models which were proposed to describe impose different slip rates. In this approach, the and explain fragmentation, can roughly be grouped nucleation and growth of a misorientation boundary into two families. The first family is based on tex- would thus be prior to the choice of different sets of ture-type approaches. These models start from ei- slip rates. In the framework of the model to be dis- ther different sets of active slip systems or different cussed in this paper, the nucleation and growth of slip rates within the same set of active systems in fragment boundaries is only possible in the different crystal parts (see, for instance, [44-47]). combination of bundling of mobile dislocations in However, in general they lack an explanation for the slip bands and of stochastic trapping in a semi-trans- initiation of the fluctuations in the slip regime. The parent obstacle. second family is based on nucleation and growth approaches. These models discuss the generation 1.2. Large strain work-hardening of short segments of fragment boundaries and the conditions for their growth. The arising fragment The coupling between the macroscopic mechani- boundaries then impose the variations in the slip cal properties of a material and its micro- and regime. However, the models in this family usually mesoscopic structure is the key to the development fail in describing the transition from a homogeneous of new materials with high strength and good form- to an inhomogeneous slip regime quantitatively, i.e. ability. Since all the industrially relevant forming pro- they fail in treating the fragments as separate tex- cesses as, for instance, flat rolling, deep drawing or ture components starting from a certian critical wire drawing, involve intermediate and large strains, misorientation. a better understanding of the coupling between work- Considering any inhomogeneous slip regime, it hardening and substructure development is strongly has to be emphasized that compatible deformation desirable especially for intermediate and large of all crystal parts has to be ensured, so that spa- strains. tially varying sets of slip rates may only result in 50 M. Seefeldt

The substructural background of the transition from stage III to stage IV of work-hardening is still a matter of controversial discussion. What all stage IV models have in common is the use of (at least) two substructural variables, mostly dislocation den- sities, as suggested by Mughrabi [48] for any model considering substructure development and work- hardening at intermediate and large strains. The early stage IV models [49-52] started from a dislocation cell structure, treated it as a compound made up of the cell interiors as a «soft phase» and of the cell Fig. 2. Generation of “virtual misfit dislocations” walls as a «hard phase» [48,53], used the cell inte- by mobile dislocations travelling across rior and cell wall dislocation densities as substruc- misorientation boundaries. To the left, two mobile tural parameters and ascribed the shear resistances dislocations of opposite sign which do not fully of the two phases to the respective local disloca- compensate each other any more and thus tion densities. The ideas on the underlying disloca- leave a “virtual misfit dislocation” behind. To the tion kinetics varied. It was noticed that the low right, a mobile dislocation penetrating through hardening rate of stage IV was rather close to the the boundary leaving a “virtual misfit dislocation” behind. one of stage I of single crystal plasticity and sug- gested that the underlying elementary processes should also be related to each other (see e.g. [54]). In line with this assumption, Prinz and Argon [49] ing dislocation kinetics was concluded to cause the proposed the formation of hardly mobile dislocation transition to stage IV, while recent X-ray studies in- dipoles in the cell interiors and their drift to the cell dicate that such a change in the dominating dislo- walls to be the rate-controlling mechanism of stage cation character takes place already in stage III [57]. IV. Haasen [51] and Zehetbauer [52], on the contrary, With the increasing amount of experimental were prompted by different features of stage IV in results on grain subdivision it became clear that a low and high temperature deformation to focus on substructure-based model of stage IV work- the different storage and recovery mechanisms of hardening should take fragmentation into account. screw dislocations in the cell interiors and of edge Haasen [51] had already assumed that different sets dislocations in the cell walls. The model by of slip systems were active in different cells and Zehetbauer [52] took the concentration of plastic causing «geometrically necessary» boundaries be- vacancies as an additional substructural variable into tween the cells, but Argon and Haasen were the account. With four fitting parameters (which could first to model the coupling between increasing quantitatively be interpreted on physical grounds), misorientation and work-hardening [58]. They as- it was able to reproduce work-hardening curves for cribed the stage IV work-hardening rate to the fact a variety of different temperatures in excellent agree- that dislocations of opposite signs, approaching a ment with experiment. misorientation boundary from the two adjacent frag- However, the model was based on questionable ments, do not fully compensate each other any assumptions concerning the elementary processes more. Consequently, elastic misfit stresses accu- and the substructural framework: interactions be- mulate, or, in defect language, “virtual misfit dislo- tween screw and edge dislocations were neglected, cations” with a “Burgers vector” equaling the differ- and experimental evidence of fragmentation (e.g. ence of those of the two dislocations from the two misorientation bands, cell blocks) as well as of long- sides (see Fig. 2). These elastic stresses or “virtual range stresses (bow-out of dislocations in the cell misfit dislocations” were translated into an organized interiors in TEM [11,55], asymmetric XRD line broad- shear resistance of the cell interiors. The ening [56]) was disregarded. Furthermore, the model misorientation was first assumed to accumulate with predicted substructural features which were in the square root of the shear strain, ϕ = B γ . Nabarro conflict with experimental results: considerable dis- [59] proposed the coefficient B to scale with the location densities in the cell interiors were predicted, inverse square root of the fragment size, whereas TEM studies in the unloaded as well as in the neutron-pinned load-applied state revealed only b very small densities [11,55], and a transition from ϕγ= . (2) the screw to the edge dislocation density dominat- df Disclinations in large-strain plastic deformation and work-hardening 51

Fig. 3. Above: Argon and Haasen: alternating tensile and compressive misfit stresses due to “virtual misfit dislocations” resulting from mobile dislocations travelling across misorientation boundaries. Below: This work: the misorientation boundary mosaic represented in terms of partial disclinations generating the same pattern of alternating tensile and compressive misfit stresses.

Based on this scaling law, a work-hardening rate of mismatch stresses). An important conclusion from this reasoning was that the decomposition and sub- τ 2 structural interpretation of X-ray profiles is not d c 8 BG = γ = unique. For example, a similar misfit stress distri- dγν331()− bution as proposed by Argon and Haasen on the 8 Gb γ 8 G (3) = ϕ2 basis of “virtual misfit dislocations” can be suggested − ν − ν on the basis of a regular array of partial disclinations 31()df 31 () describing the mosaic of fragment boundaries (see for the shear resistance of the cell interiors was Fig.3). derived. However, the mechanism proposed by Argon and Argon and Haasen demonstrated that the misfit Haasen does not distinguish between the cell and stresses obtained from their model were compatible the fragment structure. If one applies it to the cell with the X-ray residual stress measurements by structure, problems arise in case if the cell wall width Mughrabi and coworkers [56] (as one might already is of the same order of magnitude as the cell size, guess from the arrangement of “virtual misfit dislo- as it is typical for substructures evolving during de- cations” in the case of single slip, see Fig. 3, which formation at low homologous temperatures, i.e. resembles the arrangement of “interface dislocations” without significant influence of diffusion. Mobile dis- in the case of multiple slip according to Mughrabi locations of opposite signs from the two adjacent [56]). While Mughrabi and coworkers decomposed cells would hardly get into contact with each other, the asymmetric X-ray peak profiles into two sym- and penetration would more and more replaced by metric components and ascribed them to the two trapping of one dislocation plus emission of another. “phases” of cell walls and cell interiors (with the peak The consequences on the accumulation of elastic widths reflecting the respective local dislocation misfit stresses would be unclear and would strongly densities and the peak shifts reflecting the respec- depend on the cell wall morphology. If one applies it tive local residual stresses), Argon and Haasen to the fragment structure, the scaling law for the decomposed them into three symmetric compo- misorientation, Eq. (2) might have to be modified nents and ascribed one to the “phase” of cell walls and the coupling with the cell stucture evolution (also with the width reflecting the local dislocation model would get lost. density and the shift reflecting the local residual To the author’s knowledge, Koneva, Kozlov and stress) and the remaining pair of peaks to the coworkers [38-42] were the first to propose the ex- “phase” of cell interiors (with the two widths reflect- cess dislocation density as the second substruc- ing the alternating tensile and compressive elastic tural variable required to describe substructure de- 52 M. Seefeldt

velopment at intermediate and large strains and to number to explain the stage II work-hardening rate couple the increasing excess dislocation density [11]. Later on, they were often mentioned as a pos- and the increasing misorientation to an additional sible source of long-range stresses (see, for instance, flow-stress contribution. They suggested [29,48]), but, to the author’s knowledge, not incorpo- · to connect the transition to stage IV of work- rated into a work-hardening theory so far. hardening to misorientation bands and fragments becoming the predominant substructural features 2. MODELLING FRAMEWORK and ρ In the theory of crystal plasticity, plastic deforma- · to consider the excess dislocation density exc as the substructural variable which controls stage tion and work-hardening are understood as a result IV work-hardening. of the generation, motion, immobilization and annihilation of point, line and plane crystal defects. In contrast to Argon and Haasen, they did not These elementary mechanisms are controlled by ascribe this connection to elastic misfit stresses the external applied stress field as well as by mate- due to the presence of the misorientation bound- rial and process parameters as, for example, stack- aries themselves but to the boundaries acting as ing fault energy, vacancy formation enthalpy, tem- glide barriers, to Randspannungen (end stresses) perature and deformation speed. A major task of around the edges of terminating boundaries and to plasticity modelling is to make a choice of defect the lattice curvature due to the continuous miso- populations which are relevant under given material rientations in the fragment interiors. and process conditions. This choice should be based The kinetics of band substructure development on a physical consideration of the dependence of and of the growth of its volume fraction was observed the elementary mechanisms on the material and to be connected to the growth of subboundaries process parameters (see subsection 2.1). In order through the material, in the course of which the ratio to couple the defect populations to the macroscopic of the number of terminating subboundaries and of behaviour and properties of the material, the volume the total number of subboundaries remained densities of the point, line and the defects popula- constant. The terminating subboundaries were iden- tions are represented by plane defects. A dynamic tified as defects of disclination type on grounds of consideration of plastic deformation and work-hard- an analysis of the magnitude and functional decay ening then requires a description of the evolution of of the long-range stress fields around them. In the these densities with time or strain (see subsection same measure as the non-redundant dislocation 2.2). In a third step, the impact of the stored de- density, the long-range stress fields grew. Starting fects on the local shear resistance and on the long- from the middle of stage III, when the volume frac- range internal stresses has to be discussed and tion of the band substructure became significant, the defect densities have to be coupled to critical the macroscopic flow stress proved to be propor- resolved shear stress or flow stress contributions tional to the average lattice curvature. The origin of (see subsection 2.3). this flow stress contribution was proposed to be closely related to the long-range stress fields. The 2.1. Materials and processes macroscopic flow stress would then be controlled not only by the local shear resistance due to elas- In the present paper, the quasistatic deformation of tic and contact interaction between the dislocations, f.c.c. metals with intermediate and high stacking but also by the barrier resistance due to the new fault energies at room temperature is discussed. In boundaries and by long-range stresses due to the the model presented in section 3, uniaxial new excess dislocation configurations. The flow compression of a multiple-slip oriented pure copper stress was observed to be proportional to monocrystal is considered. In the following, some · excess dislocation density, conclusions are drawn from the material parameters · subboundary density, and on the relevance of certain defect populations in the · misorientation. deformation process. Especially, the respective Randspannungen (end stresses) were first sug- contributions to dynamic recovery through gested by Seeger and Wilkens [60] around the edges annihilation of screw dislocations after cross-slip and of terminating boundaries as an additional source through annihilation of edge dislocations after climb of long-range stresses in stage II of plastic are estimated. On this basis, it can be decided, defomation, after it had turned out that classical pile- whether the two dislocation characters have to be ups were well observed in TEM, but not in sufficient treated separately, and whether vacancies have to Disclinations in large-strain plastic deformation and work-hardening 53

Table 1. Effective screw dislocation annihilation length for copper and aluminium at room temperature for different local cell wall dislocation densities. Cu Al γ -2 . -3 . -2 SF/J m 55 10 200 10

RT/Tm/1 0.2 0.3 ρ 14 -2 µ ys,eff ( w = 10 m or dc = 2.4 m)/b 6 584 ρ 15 -2 µ ys,eff ( w = 10 m or dc = 0.75 m)/b 6 831 ρ 16 -2 µ ys,eff ( w = 10 m or dc = 0.24 m)/b 36 1591

be taken into account as a relevant defect popula- The quantity χ in Eq. (4) expresses a total probabil- tion. ity of cross-slip events during penetration of a mo- Materials with intermediate and high stacking bile dislocation through a cell wall, but is irrelevant fault energies are chosen in order to avoid deforma- in the present context because the term in large tion twinning. Pure metals were used because they brackets in Eq. (4) approximately equals 1 for do form cells. The annihilation rate of immobile screw quasistatic deformation of copper as well as dislocations cancelling out after cross slip with aluminium at room temperature. For more details, mobile screw dislocations of the opposite sign pass- the reader is referred to [61]. Table 1 displays the ing by scales with an effective annihilation length effective annihilation lengths for copper and, as a ys,eff. If the mobile dislocation passes by within this point of reference, also for aluminium at room tem- length, an annihilation event takes place (see also perature together with their stacking fault energies subsubsection 3.2.2). Pantleon [61] used the cross- [63] and homologous temperatures for three differ- slip model by Escaig [62] – in a version slightly ent local cell wall dislocation densities corresponding modified for large strains – to calculate the effective to three different cell sizes (see below). annihilation length according to One can conclude that – within the framework of the cross-slip model by Pantleon – for cold deforma- l F 1 I =+−+$ SF G bg− χ J tion of copper up to intermediate strains with cell sizes yys,, eff s Pecs1 1 y s 0 . (4) b H χ K of more than 0.1 µm, the screw dislocation annihila- tion length for cross-slip is very similar to the edge where the stacking fault width l is connected to SF dislocation annihilation length for dipole disintegra- the stacking fault energy γ through SF tion, so that the both characters do not need to be distinguished. For aluminium, on the contrary, the Gb 2 l = . annihilation length of screws is much larger than the SF πγ (5) 16 SF one of edges from the beginning of cross-slip, so that one can assume, as a first approximation, that the The cross-slip probability P , in addition, scales cs cell structure consists mainly of edge dislocations. with the cross-slip energy Annihilation of edge dislocations after vacancy-as- sisted climb can be neglected because low homolo- Gb 2 l = l SF gous temperatures are considered: bulk diffusion of ECS SF ln 23 (6) 37. 3 b plastic vacancies is too slow due to the low mobility of the vacancies, and core or pipe diffusion would and, especially, with the local shear resistance due require significant dislocation densities in the cell in- to the local dislocation density in the cell walls ac- teriors (where plastic vacancies are generated by mov- cording to ing jogs in the mobile dislocations) to transport the vacancies to the cell walls (where they would be −1 F E F τb I I needed to assist climb and subsequent annihilation =−CS G +J = PCS expG 13 J of cell wall with mobile dislocations). H H γ K K kTBSF (7) 2.2. Disclinations F E −1 I expG −+CS di148παl ρ J . H kT wSF w K If the formation of fragment boundaries is consid- B ered as a nucleation and growth process, then it is natural to model fragmentation in terms of disclina- 54 M. Seefeldt

Fig. 4. Left: both-side terminating excess dislocation wall with an ideal LAGB segment structure at an obstacle perpendicular to the primary slip planes. Middle: non-ideal both-side terminating excess dislocation wall with long-range stresses at an obstacle inclined to the primary slip planes. Right: completion slip to complete the non-ideal excess dislocation wall to an ideal LAGB.

tion dynamics. In the present paper, it is assumed early as in 1907 [68]. In a Volterra process, that the nucleus of a fragment boundary is a both- disclinations are generated by introducing a cut into side terminating excess dislocation wall. Such a wall an elastically isotropic hollow cylinder, rotating the can be formed by collective stopping of mobile dis- two shores with respect to each other, and inserting locations of the same sign on neighbouring parallel the missing or withdrawing the surplus matter. Dis- slip planes at an obstacle. If the excess disloca- locations, on the contrary, are produced by shifting tions in the wall are arranged at equal and minimum the two shores with respect to each other. In a syn- possible distances, i.e. if they form an ideal low- thetic definition, a Frank vector representing axis angle grain boundary (LAGB) segment, then the bor- and angle of the rotation is introduced in analogy to der lines of the both-side terminating wall are partial the Burgers vector representing direction and mag- disclinations. This is, for instance, the case for single nitude of the shift. In an analytic definition, the Frank slip and an obstacle perpendicular to the slip planes vector closes the rotation mismatch in a Nabarro (see Fig. 4, left). If it is not the case, e.g. for single circuit [66], as the Burgers vector closes the shift slip and an obstacle inclined to the slip planes (see mismatch in a Burgers circuit. Wedge and twist Fig. 4, middle), the long-range stress fields around disclination characters are distinguished (Frank vec- the non-ideal dislocation configuration would trigger tor parallel or perpendicular to the defect line along Ergänzungsgleitung [64,65] (completion slip) on the cylinder axis, respectively) as edge and screw additional slip systems to complete the non-ideal dislocation characters are distinguished (Burgers both-side terminating excess dislocation wall to an vector perpendicular or parallel to the defect line). ideal low-angle grain boundary segment (see Fig. However, in contrast to a single dislocation, a 4, right). single disclination cannot exist in a crystal because Disclinations are the rotational “twin-sisters” of of the requirement of coincidence of lattice sites the translational dislocations1: disclinations are lines after the Volterra process: after rotating or shifting of discontinuity of misorientation, they separate the two shores of the cut with respect to each other regions where crystal parts are (already) misoriented and after inserting the missing or withdrawing the with respect to each other from regions where this surplus matter, the lattice sites on the two shores is not (yet) the case, they accommodate lattice of the interfaces have to coincide. In a crystal, this rotation mismatches, while dislocations are lines requirement can only be fulfilled for angles of at least of discontinuity of shear, they separate regions 60 degrees which would obviously cause fracture. It where crystal parts are (already) sheared with respect was only in the early 1970s that scientists in the to each other from regions where this is not (yet) USA [67,69,70] started to consider dipoles of par- the case, they accomodate strain mismatches. tial disclinations and researchers in Russia proposed Disclinations were, together with dislocations, in- networks [16,20] and other self-screening troduced into the theory of elasticity by Volterra as configurations [71] of partial disclinations rather than single perfect disclinations. Such partial disclination dipoles as new entities can fulfill the crystallographic 1 For introductory texts on disclinations see, for instance, requirements because the two rotations compen- [23,66,67]. Disclinations in large-strain plastic deformation and work-hardening 55 sate mutually at large distances from the dipole and leave only a net shift behind, comparable to a “superdis-location” [23]. Therefore, the dipole con- figuration allows the realization of partial disclinations with low strenghts +ω and -ω in a crystal. Note that it thus also allows the free variation1 of the magni- tude of the mutually compensating rotation angles ω ω or disclination strengths + and - , and it introduces Fig. 5. “Translation” of a partial wedge disclination the dipole width 2a as an additional variable. While into the “language” of dislocations: (a) wedge of dislocation modelling works with only one variable, matter to be inserted in the Volterra process, (b) namely the dislocation density, disclination model- package of semi-infinite crystallographic planes ling has to deal with three, namely the disclination with equidistant border lines, (c) semi-infinite dipole density and the average disclination strengths wall of parallel excess edge dislocations with and dipole widths. equal and minimum possible distances, (d) partial A partial disclination with a low strength ω can wedge disclination. then be “translated” into the border line of a termi- nating low-angle grain boundary with misorientation ϕ ω = in the sense that both representations exhibit through cooperative motion of dislocations, then the same long-range stress field. More specific, a these ideas can be implanted in a plastic deforma- partial wedge disclination can be “translated” into tion model by describing fragmentation in terms of the border line of a terminating regular excess edge generation of partial disclination dipoles and propa- dislocation wall (see Fig. 5), and a partial twist gation of partial disclinations through capturing of disclination can be “translated” into the border line mobile dislocations. If large-strain work-hardening of a terminating regular screw dislocation net. is supposed to be due to long-range stresses and if However, these “translations” are non-unique: a these long-range stresses are assumed to be partial wedge disclination dipole can, for instance, Randspannungen (border stresses) around the bor- be “translated” into a both-side terminating excess der lines of fragment boundary segments with a for dislocation wall of only one sign as well as into two the rest ideal low-angle grain boundary structure, parallel semi-infinite excess dislocation walls of then the long-range stress fields can properly be opposite signs (see Fig. 6). Both of these expressed as partial disclination stress fields. In configurations exhibit the same long-range stress fact, the situation is similar with dislocations: In prin- field. ciple, all dislocation configurations can be “trans- In this way, in principle all partial disclination lated” into, possibly complex, atom configurations. configurations in a crystal can be “translated” into, The advantages of the dislocation description are 2 possibly complex, dislocation configurations . The again its mathematical simplicity and, above all, its advantages of a disclination description are its correspondence with the idea and experimental evi- mathematical simplicity and its correspondence with dence that plastic deformation is carried out by pro- certain ideas of large-strain plastic deformation and gressive slip on glide planes, i.e. by the activation work-hardening: if fragmentation is considered as of translational modes of deformation through the essential feature of large-strain plastic deforma- collective motion of atoms. Therefore, it is improper tion and is understood as a process of nucleation to consider disclinations as a mere modelling tool and growth of fragment boundary segments, i.e. as introduced only for mathematical convenience. an activation of rotational modes of deformation Representing a complex configuration of many dis- locations, disclinations are probably just as much 1 ω If the partial disclination with strengths represents physical objects as dislocations, representing a the tip of a terminating low-angle grain boundary with complex configuration of many atoms, are. The key misorientation ϕ (see below), the allowed strength is experimental task is to confirm (or reject) that dis- discretized in multiples of ∆ϕ =b/hmin where hmin is the locations are indeed arranged in configurations minimum dislocation spacing in a low-angle grain corresponding to partial disclinations and that long- boundary. This discretization is so fine that it can be range stresses indeed exhibit the functional decay neglected in the present considerations. typical for partial disclinations (see below). 2 This does, of course, not hold for non-crystalline ma- The consideration of partial disclination dipoles terials where disclinations can directly act as carriers was the first keystone to the introduction of of plastic deformation. 56 M. Seefeldt

Fig. 6. Partial disclination dipoles of the four configurations considered above. On the left in the language of dislocations, on the right in the language of disclinations.

disclinations into crystal plasticity because it be- ready discussed by Nabarro [73] and Li [74] in the came clear that (partial) disclinations could exist in context of polygonization, act as a “nucleus” of a real crystals. The “translation” of partial dislocation fragment boundary, expand or “grow” by capturing dipoles into dislocation configurations was the sec- additional mobile dislocations through the long-range ond one because it opened the door to coupling stress fields around its tips, and thus “polygonize” dislocation and disclination dynamics and actually the matrix. A propagating partial disclination, in establishing disclination dynamics in real crystals. contrast to a perfect one, thus leaves a track be- Motivated by TEM micrographs showing pairs of hind, namely a fragment boundary, like a gliding parallel terminating fragment boundaries with oppo- partial dislocation, in contrast to a perfect one, leaves site misorientations (see e.g. [10,18,72], cf. the a stacking fault behind. scheme (A1) in Fig. 6), Romanov and Vladimirov If, finally, such a propagating partial disclination [22,23] were the first to model the growth of a gets stopped at an existing fragment boundary, a misorientation band in terms of the propagation of rotation mismatch should be left behind around the a partial wedge disclination dipole at the top of the new triple junction (see Fig.7). Formally, this band. The partial disclination dipole was proposed mismatch could be detected by carrying out a to propagate by widening dislocation loops ahead Nabarro circuit [66] around the triple junction. A of the dipole with the help of its long-range stress rotation mismatch around the triple junction would field, capturing the edge components, attaching then correspond to an immobile partial disclination them to the backward terminating boundaries, in the node line. thereby shifting the border lines of the terminating Experimental evidence for such a rotation mis- boundaries and propagating itself. Similarly, a both- match around fragment boundary triple junctions side terminating excess dislocation wall, can as al- and thus for the presence of partial disclinations

Fig. 7. Formation of a new fragment boundary triple junction through a partial disclination propagating at the top of an arising fragment boundary and getting locked in an existing one. The new triple junction carries an orientational mismatch and is thus a noncompensated node. Disclinations in large-strain plastic deformation and work-hardening 57 was recently found by Klemm and coworkers [75,76] respectively. The earliest representant of this class in TEM microdiffraction measurements at cold-rolled of evolution equations was Gilman’s equation [79] copper mono- and polycrystals. The misorientations for a dislocation density undergoing multiplication across the three joining fragment boundaries were and pairwise annihilation. Another commonly used determined and the product of the misorientation example, proposed for the shear strain γ instead of matrices was found to deviate from the identity. In the time as an evolution parameter, is Kocks’ semi- the linear approximation, the sum of the phenomenological equation [80] misorientation vectors was found to deviate from ρ zero. Furthermore, when shifting the electron beam d 1 La =−ρρ (9) along the joining boundaries, the local orientation dbγβ b was found to vary continuously. Further methods were used to demonstrate the for a forest dislocation density ρ with storage and presence of partial disclinations in the substructure dynamic recovery. For single slip or symmetric of deformed metals: Koneva, Kozlov and coworkers multiple slip, the strain increment can be expressed ε γ [38-42] used TEM bending contour measurements in terms of the shear strain increment, d = mSn d , to analyse the magnitude and the functional decay giving Kocks’ and Mecking’s law [81] of the long-range stress fields around terminating ρ subboundaries and found them to match with the d =−kkρρ. (10) ones of the partial disclination stress fields. Tyumen- dε 12 tsev et al. [77] used TEM reflex width measurements However, at intermediate and large strains, dis- to determine lattice bend-twist tensor components location patterning takes place, so that a along grain boundaries in ultra-fine grained copper representation of the dislocation structure in terms and nickel crystals produced through severe plas- of one global dislocation density becomes improper. tic deformation. The measured variation of the bend- Two routes were followed to tackle this problem: On twist tensor components along matched with the one of them, the global defect densities were one to be expected for a pile-up of partial disclina- replaced by local ones and the evolutions of the tions on the grain boundary. Recently, the present local defect densities were considered not only as author proposed to use scanning tunnel microscopy a time-, but as a time- and space-dependent prob- (STM) to study the surface etching around the points lem. The evolution equations then have the struc- where fragment boundary triple junctions leave to ture of continuity equations, the surface. Preferred etching might indicate that the nodes are non-compensated and include partial dρ disclinations [78]. i = KKKρρρ Fc(,,,,,,,,11 ckil dt (11) KKK− ∇ 2.3. Reaction kinetics SSmmpp111,,mno , ,, ,,,) ji

The evolution of a global average defect density with Orlov [82] proposed already in 1965 to consider the time or strain is determined by the elementary dislocation density separately for the slip systems. mechanisms of generation and annihilation of the According to Malygin [83], one obtains a system of defects and of reactions of defects in the population partial differential rate equations with the structure under consideration with defects in other popula- tions. These elementary mechanisms depend on ∂ρ()i v ()i +∇bgvnv()iiρρ () = () ii () +∑ ()i . the material and process parameters. Therefore, a ()i (12) ∂t l rate equation which describes the time or strain p p evolution of a, say, global dislocation density ρ, has i While Malygin’s models are restricted on a the formal structure consistent description of the first three stages of plastic deformation, including smooth transitions dρ i = KKKρρρ between these three stages, the present paper aims Fc(,,,,,,,,11 ckil dt (8) at an extension of the models to the later stages of KKK plastic deformation, that is to large strain plastic SSmmpp111,,mn , ,, ,,,o ), deformation, by transferring the framework proposed where the c’s denote point defect concentrations, by Malygin for dislocation kinetics to disclination the ρ’s and S’s line and plane defect densities and kinetics, the m’s and p’s material and process parameters, 58 M. Seefeldt

local effective stress through the components act- ∂θ()w V ()w +∇bgVnV()wwθθ () = () w () w +∑ ()w . ing as back stresses on the slip planes [87], or ∂ ()w (13) t p l p indirectly through an organized shear resistance connected to the locally stored elastic strain en- On the second route, the transport terms were ne- ergy due to the components not acting as back glected, and a substructure geometry was imposed stresses [58]. instead by implementing phenomenological scaling Long-range stresses can be explained from two laws like a principle of similitude (see below) into different points of view: From the defect theory or the model. As far as patterning is concerned, the static point of view, high-energy dislocation arrays, present paper follows this second route. like pile-ups or terminating excess dislocation walls (with Randspannungen (border stresses) around their 2.4. Plastic deformation and borders, cf. [60]), cause long-range stresses. Such work-hardening dislocation configurations evolve from dislocation dynamics and reflect the deformation history on the Following Kocks, Argon and Ashby [84], the local defect level. However, the evolution of dislocation effective shear stress1 is defined as2 configurations exhibiting long-range stresses is lim- τττ$$$eff =−ext int , (14) ited by the micromechanical compatibility require- ij ij ij ments (see below). From the micromecha-nics or where the internal shear stress τ$ int is by definition ij dynamic point of view, elastic strain mismatches taken positive in the direction opposing the applied cause long-range stresses. Such mismatches re- external shear stress τ$ ext . The local effective shear ij sult from compatible deformation of plastically het- stress τ$ eff would then be opposed by the local shear ij erogeneous materials and reflect the deformation 3 τ$ * resistance ij so that the material would (plastically) history on the stress and resistance level [48]. For shear, if example, at intermediate strains, dislocation pat-

ext int **eff terning leads to spatially varying local shear resis- bττ$$−−=−≥g τττ$$$0 (15) ij ij ij ij ij tances. Applying an external load to such a “com- would hold. In case of athermal deformation, the pound” material (made up of a “soft phase” with low equality applies. dislocation density ρ and shear resistance τ * and c c The local shear resistance of a crystalline metal ρ a “hard phase” with high dislocation density w and is governed by glide obstacles as solute atom shear resistance τ* ) gives elastic strain mis- w concentrations, local dislocation densities, twin, matches and thus elastic misfit stresses [48]. The grain or phase boundaries and/or, in the sense of corresponding plastic strain mismatches can, fol- an organized shear resistance [58], by long-range lowing Nye and Kröner [88,89], be “translated” into stress fields of static or dynamic origin, in as far as densities of misfit dislocations (also called “geo- these stress fields do not directly act as back metrically necessary dislocations” [90]). However, stresses in the active slip systems. Since the these densities do not provide any information on present paper discusses only pure metals with in- the spatial distribution, on the configuration of the termediate and high stacking fault energies, the fol- misfit dislocations and thus also not about possible lowing considerations are restricted to local shear long-range stress fields due to the configuration. In resistances due to local dislocation densities and an ideal case, a misfit dislocation configuration can long-range stress fields. The local dislocation den- be found which exhibits a long-range stress field sity contributes to the local shear resistance through that equals the elastic misfit stress field and dislo- the elastic and contact interaction between inter- cation dynamics can be used to study how this secting dislocations, as described, for instance, in configuration arose in the course of plastic defor- the Hirsch-Saada process [85,86]. The long-range mation. However, in other cases it is impossible to stress fields can hinder the dislocation motion in find such a configuration because the calculated two different ways: either directly by reducing the “virtual” misfit dislocations are non-crystallographic [58]. 1 The microscopic resolved shear stresses and shear ∆τ mis In any case, local elastic misfit stresses c strains are denoted as τ and γ; the macroscopic ∆τ mis and w due to different local shear resistances in stresses and strains are denoted as σ and ε. different “phases” are self-equilibrated local internal 2Tensorial quantities are marked with a$. stresses. In the global average over the specimen, 3The resistances are marked with an asterisk * in order they, weighted with the respective volume fractions to distinguish them from the stresses. (1-ξ) and ξ of the “phases”, have to sum up to zero, Disclinations in large-strain plastic deformation and work-hardening 59

mis mis deformation resistance fully to the local dislocation ()10−+=ξ∆τ ξ∆τ . (16). c w density in the cell walls or to the smeared-out glo- Therefore, they can contribute to the local effec- bal dislocation density. Therefore, the present pa- tive stress and/or to the local shear resistance but per neglects the long-range stresses due to misfit do not directly affect the global flow stress. If, how- dislocation configurations. ever, their contribution to the local shear resistance, One can conclude that there are two “damping” in the sense of an organized shear resistance [58], or “inhibiting” mechanisms limiting the growth of is significant compared to the one of the local dislo- elastic and plastic strain gradients or the accumu- cation density, then the elastic misfit stresses do lation of misfit dislocations: One of them is dynamic affect the global flow stress. recovery which counteracts and finally balances dis- If one considers, following Argon and Haasen location storage in the cell walls, and the other is [58] or Pantleon and Klimanek [61,91], a “compound” the generation of internal misfit stresses. In this way, made up of a cell wall “phase” with high dislocation the difference between the two local shear density and a cell interior “phase” approximately void resistances is restricted. This is required for stable of dislocations, then the local shear resistance in plastic deformation [92] because misfit stresses the cell interior “phase” is entirely controlled by the associated with elastic strain gradients scale with long-range elastic misfit stresses. In the following, the (huge) shear modulus G according to two simplified examples are discussed in more de- σσ−=−− αεε (18) tail. During uniaxial deformation of copper single ()().wcG wc crystals with a high-symmetric orientation, multiple In the case of single slip with cell walls perpen- slip is activated and a cell structure with cell walls dicular to the slip direction, the situation is more parallel and perpendicular to the load axis is formed difficult. Again, elastic strain mismatches and misfit [55]. The elastic strain mismatches arising due to stresses arise, but if one would “translate” the the different local shear resistances cause internal corresponding plastic strain mismatches into regular misfit stresses. The corresponding plastic strain layers of misfit dislocations at the interfaces, these mismatches can be “translated” into regular layers layers would not exhibit any long-range stress fields of misfit dislocations at the interfaces between cell because they would represent ideal low-angle grain walls and cell interiors. As long as the local shear boundary (LAGB) configurations. This problem could resistance in the cell walls increases faster than be overcome by, for instance, considering the two the one in the cell interiors, the plastic strain single layers to the both sides of a (narrow) cell wall mismatch increases and misfit dislocations are together as a double layer which does not have long- accumulated at the interfaces according to [61] range stresses outside the double layer (i.e. in the cell interiors in this case), but inside (i.e. in the cell 1−−ντ** τ = wc walls). Gil Sevillano and Torrealda [93,94] proposed Nmis , (17) 2 Gbres m S that the long-range stresses inside the layer would result in the activation of additional slip systems to where N and b denote the line density and the mis res ensure homogeneous deformation, so that the cell resultant Burgers vector of the misfit dislocations walls would rotate with respect to the cell interiors. and m is the Schmid factor. In this example, the S Straight long excess dislocations stopping at the regular layers exhibit long-range stress fields interface would then serve to accommodate lattice corresponding to the elastic misfit stresses. These rotation mismatches rather than strain mismatches. long-range stress fields assist the applied stress in Another way to overcome the problem would be to the cell walls and counteract it in the cell interiors. consider layers with irregular excess dislocation They have strong components which act as back distributions which do exhibit long-range stresses stresses and reduce the local effective stress in the [95,96]. The latter approach would result in locally cell interiors and only weak components which do varying misorientations across the layers, i.e. be- not act as back stresses but might produce an tween cell walls and cell interiors, and, in general, organized shear resistance and raise the local shear also to finite locally varying net misorientations resistance that the local effective stress stress has across the double layers, i.e. between neighbouring to overcome in the cell walls. However, a detailed cells. consideration by Pantleon (appendix D in [61]) The present paper follows the latter approach in showed that an estimation of the flow stress taking a more “schematic” way (for details, see below). the long-range stress fields into account differs only The excess dislocations are assumed to be arranged by 3% from an estimation ascribing the compound 60 M. Seefeldt

strain mismatches. The transition from translational to rotational modes of deformation is thus required by micromechanics, namely to inhibit further build- up of strain mismatches, and is realized by dislo- cation dynamics through switching from balanced to non-balanced stopping, essentially due to a Fig. 8. Left: a brick-shaped fragment structure reduction of the spacing between the stopped ex- with the immobilized partials of partial disclination cess dislocations as a result of bundling mobile dis- dipoles in the fragment boundary triple junctions. locations in slip bands (for details, see below). If Right: a pile-up configuration with the same long- one assumes an ideal LAGB configuration of the range stress field. . layer segments, the transition from translational to rotational modes of deformation corresponds to the transition from dislocations gliding individually along in layer segments with a regular dislocation distri- slip planes to disclinations propagating along cell bution, so that the misorientation varies only around walls (which is, of course, realized by dislocations the border lines of the segments. These layer seg- gliding collectively along slip planes). ments are considered as a result of incidental In the proposed picture, the more or less irregu- collective stopping of neighbouring straight long lar distribution of (strain) misfit dislocation layer seg- excess dislocations at the interfaces. If a layer seg- ments on both side of the obstacle in the early stages ment has an ideal low-angle grain boundary (LAGB) I and II exhibits long-range stresses – not through configuration, it corresponds to a dipole of partial the layers itself, but through their ends – which re- disclinations at the borders lines of the group. semble the elastic misfit stresses. These long-range In the early stages I and II of plastic deforma- stress fields are required by micromechanics, form tion, the spacing between the excess dislocations a self-equilibrated set and do not contribute to the in the layer segments would remain large, so that global flow stress. The expanding (lattice rotation) the misorientation across the layer segment and misfit dislocation layers on one side of the cell wall the end stresses around its border lines would in the late stages III and IV also exhibit long-range remain too small to capture additional mobile dislo- stresses – again not through the layers themselves, cations and thus too small to let the layer segment but through their ends. However, these long-range expand. Consequently, short excess dislocation stress fields are a consequence of the nucleation layer segments would cover the cell wall on both and growth mechanism of fragment boundary layer sides in a more or less irregular way, with locally formation, are connected to the propagating partial varying, but largely mutually balancing misorien- disclinations as carriers of the rotational modes of tations (see Fig. in subsection 3.2.3.). However, in deformation, do not form a self-equilibrated set and stage III of plastic deformation, the misorientation do contribute to the global flow stress. The partial across the layer segments and the end stresses disclination dipoles (PDDs) have the same pleas- around its border lines would get sufficiently large ing feature as traditional pile-ups that their long-range to capture additional mobile dislocations and to stress fields affect the local effective stress by act- expand along the cell wall (see Fig. in subsection ing as back stresses on mobile dislocations on the 3.2.4.). If such a layer segment is not balanced by primary slip plane. its counterpart on the other side of the cell wall, it Consequently, the global flow stress resulting causes local net misorientations across the cell wall from the substructure is calculated from two and spreads this net misorientation out over the contributions: whole cell wall. In this way, it acts as a nucleus of a · the local shear resistances τ* and τ * of the cell w c fragment boundary layer which grows along the cell walls and cell interiors, respectively, wall. If a layer segment has an ideal LAGB configu- τ int · the long-range internal back stresses ω due to ration, its growth corresponds to the propagation of the partial disclination dipoles (PDDs). the two partial disclinations along the cell wall. According to Eq.(15), the local effective stress In this picture, the build-up of strain mismatches has to overcome the local shear resistances τ*, at the interfaces between cell walls and cell interi- ors is restricted by the transition from balanced to ττττeff=−≥ ext int * . (19) non-balanced stopping of straight long excess dis- In the present paper, only the contributions of locations. The stopped excess dislocations then the local dislocation densities to the local shear serve to accommodate lattice rotation rather than Disclinations in large-strain plastic deformation and work-hardening 61 resistances in the cell walls and cell interiors are ταint ==GB22 θαωθ G a = taken into account. The contributions of long-range ω tot tot stress fields due to misfit dislocation configurations (22) αωGa θ+≈22, θ αω Ga θ accommodating plastic strain mismatches, in the pi i sense of organized shear resistances, are neglected. where θ denotes the density of PDDs with propa- ρ p Furthermore, the local dislocation density c in the θ θ gating partials and i and tot denote the immobile cell interiors is considered to be negligible compared and total partial disclination density, respectively1. to the one ρ in the cell walls, so that the com- w If the fragment structure arises from generation and pound shear resistance is entirely controlled by the expansion of PDDs and immobilization of their par- local dislocation density in the cell walls. There- tials and consists of fragment boundaries with ideal fore, one ends up with LAGB configurations and non-compensated triple junctions with partial disclinations (see Fig. 8), then τα**=<<= ρ τα ρ (20) ccwwGb Gb the average fragment boundary spacing df

ρ corresponds to the dipole width 2ai between the for the local shear resistances and with ( w – smeared-out global dislocation density) immobilized partials of the PDDs. With τξτ**=+−≈=()1 ξτξτ ** wcw K 2ad≈=f , ρ if (23) w (21) θ ξαGb= ξα Gb ρ i ξ w

where Kf is a geometry factor depending on the as- for the compound shear resistance. sumed mosaic geometry of the fragment structure Correspondingly, the local internal stresses are (see below), one finds proposed to be entirely controlled by the long-range 1 stress fields due to partial disclination dipoles which ταωint ≈ ω GKf . (24) correspond to both-side terminating excess dislo- 2 cation layers accomodating plastic lattice rotation β mismatches. Long-range stress fields due to misfit Assuming the disclination interaction coefficient to be βα= this matches with the flow dislocation configurations accommodating plastic KmfS/,2 strain mismatches, are again neglected. In the case stress contribution of single slip, irregular distributions of excess dislo- ∆σ = βωG (25) cation layer segments on both sides of the cell walls ω would cause back stresses, but, according to Saada proposed by Romanov and Vladimirov [23]. Distrib- [95,96], those would remain small compared to the ρ uting the available excess dislocation density exc stress fields due to PDDs. In the case of multiple on the total surface per unit volume of the fragment slip, regular distributions of misfit dislocations from boundary layers “drawn” by the propagating partial the symmetrically active slip systems would gener- disclinations gives the misorientation ate back stresses, but, according to Pantleon [61], taking into account these effects would only slightly b bρ ϕ ≈=bN = f geom exc , (26) change an estimation of the flow stress. Since the h exc f θ “wavelength” of the fragment structure is large i compared to the one of the cell structure (or the geom where ff is another geometry factor depending mean distance between the partial disclinations is on the assumed mosaic geometry of the fragment large compared to the one between the dislocation structure (see below). It is now assumed that the cell walls), and since, in the case of single slip, the immobile partial disclination strength ω, i.e. the mean dipole arm is perpendicular to the slip plane, a glo- orientation mismatch around a non-compensated τint bal average internal back stress ω is considered. triple junction, is equal to the current mean The long-range stress field of such PDDs with misorientation ϕ. In a simplistic picture for a triple disclination strength ω and dipole arm 2a are the junction formed by the two halfs of an “old” fragment same as the ones of pile ups with a “super Burgers boundary and a “new” boundary, all with the same vector” B = ω a, so that an internal back stress arises,

1 θ Note that p thus is a disclination dipole density, while θ θ i and tot are single disclination densities. 62 M. Seefeldt

mean misorientation, this makes sense because As mentioned in subsection 2.4, most of the the misorientations across the two halfs of the “old” work-hardening models for intermediate and large boundary compensate each other, so that the net strains are based on such a two-“phase” compound orientation mismatch is equal to the misorientation representation of a cell structure and ascribe the across the “new” boundary. Finally, one ends up shear resistance of the cell interior “phase” to a local with dislocation density. Argon and Haasen [58] and Pantleon and Klimanek [61], on the contrary, referred 1 to abundant TEM evidence – on the unloaded as ταϕint ≈=GK ω f well as on the neutron-pinned loaded state – for an 2 only negligibly small dislocation density in the cell 1 ρ (27) αGbK f geom exc . interior and ascribed the shear resistance of the cell ff θ interior “phase” to long-range stresses. The present 2 i work follows the latter reasoning and considers the The flow stress to be applied for athermal plastic local dislocation density in the cell interiors to be deformation thus is negligible compared to the one in the cell walls,

ρ ρρ<< = tot 1 cw , (31) σττ=+=b * int g ξ f ω mS so that the total (global average) dislocation den- 11F I G ξαGb ρ+= α GK ϕJ sity ρ is obtained by “smearing out” the wall dislo- H wfK tot mS 2 cation density from the wall volume fraction ξ to the (28) whole volume. 11F ρ I ξα ρ+ α geom exc G Gb GbK f J. It is assumed that the mean cell size dc and the H wffθ K mS 2 ρ i total (“smeared-out”) dislocation density tot are connected by a Holt-type scaling law [98,99], This corresponds with the phenomenological laws KK proposed by Gil Sevillano, Van Houtte and Aernoudt d ==, c [32] [97], ρξρ tot w

ϕ 1 where K denotes a material constant (cf. e.g. the ∆σ ∝ϕ ∝ with , (29) review paper by Raj and Pharr [100] ). Furthermore, d d f f it is assumed that a principle of similitude holds and by Kozlov and Koneva [42], throughout the whole course of plastic deformation, i.e. that the volume fraction ξ of the cell wall “phase” ∆σ ∝ ϕ. (30) remains constant,

w ξ= 3. A WORK-HARDENING MODEL FOR fgeom , (33) d STABLE ORIENTATIONS c where f = 2 is a geometrical factor reflecting the 3.1. Schematization of the geom substructure elongated shape of the cells and w denotes the cell wall width. The work-hardening model to be discussed in this From the elementary processes of fragment section is designed for a single crystal with a stable boundary nucleation and growth as discussed be- symmetric orientation, i.e. for a constant Schmid low in subsections 3.2.3 and 3.2.4, it follows that

factor mS and for multiple slip right from the begin- fragment boundaries evolve as excess dislocation ning of plastic deformation. A cell structure is then layers along existing cell walls. These excess dis- formed soon. Therefore, the model starts from a well- location layers accomodate the lattice rotation developed cell structure consisting of a cell wall mismatch and in this sense resemble Ashby’s “geo- ρ “phase” with a high local dislocation density w and metrically necessary dislocations” or Mughrabi’s * a high shear resistance τ and a cell interior “phase” “interface dislocations” accomodating either a strain W ρ with a low local dislocation density c and a low or a lattice rotation mismatch. They are arranged in τ * shear resistance c [48,53]. low-angle grain boundary (LAGB) configurations, so Disclinations in large-strain plastic deformation and work-hardening 63 that the misorientation ϕ can be calculated from the mean misorientation across the fragment bound- the dislocation spacing h in the wall via the Read- aries, Shockley law which in the case of a single- ρ component LAGB (i.e. an LAGB made up of ex- ϕ ≈ b exc . cess dislocations from only one slip system) reads θ (37) 121. i

b The dislocation substructure is thus charac-terized ϕ≈ . (34) in terms of the following dislocation densities: h – the (smeared-out) redundant cell wall disloca- The dislocation spacing h or the excess dislo- ρ tion density w, cation line density per area on the fragment bound- – the (smeared-out) non-redundant excess dislo- aries can be derived by distributing the (smeared- ρ cation density exc in the fragment boundary lay- ρ out) excess dislocation volume density exc on the ers, available total fragment boundary surface S. In order – the density of dipoles of propagating partial to calculate the latter, one has to make an assump- θ disclinations p at the border lines of the excess tion on the geometry of the fragment structure. While dislocation layers during the early stages of fragmentation, a Pois- θ – the density of immobile partial disclinations i in son-Voronoi mosaic geometry seems to be appro- the fragment boundary triple junctions. priate (i.e. a random distribution of nucleation points is generated and each place in space is then as- 3.2. Elementary processes signed to the closest of these nucleation points, cf. e.g. [101,102], – a principle which is known from 3.2.1. Dislocation storage: trapping of mobile the Wigner-Seitz cells in Solid State Physics), in dislocations into cell walls. Mobile dislocations the developed or later stages a simple get stored in the material by trapping into cell walls. parallelepipedal mosaic with “brick-shaped” frag- Slip line measurements by Ambrosi and Schwink ments seems to reflect reality better [58,103]. In [104] indicated that the mean free path of mobile this paper, the first will be applied. dislocations is about three times larger than the mean cell size d and thus revealed that cell walls In a Poisson-Voronoi mosaic, the mean node c are semi-transparent obstacles. A fraction of P ≈ line length per unit volume corresponds to the mean c triple junction line length per unit volume in a frag- 1/3 of the arriving mobile dislocations gets trapped ment boundary mosaic, i.e. to the immobile partial in the cell wall, while the remaining fraction of about θ 2/3 penetrates through the wall. (The point of view disclination density i. The mean chord length of the Poisson-Voronoi cells corresponds to the mean that all arriving mobile dislocations get trapped at chord length of the fragments which is a good the one side of the wall and in 2/3 of all trapping events another dislocation gets remobilized and measure for the mean fragment size df [102]. The mean surface area of the Poisson-Voronoi cells, fi- emitted at the other side, is formally equivalent to nally, corresponds to the mean fragment boundary the reasoning suggested above.) For single slip, this results in a storage rate for the local cell wall dislo- area Sfb per unit volume in a fragment boundary mosaic. All three quantities can be expressed cation density of [61] through the density of nucleation points of the Pois- + F dρ I P P γ& son-Voronoi mosaic (for details, see [101,102]), so wc==vρ c , H K m (38) that all three quantities can also be expressed dt w wb through each other, or, for multiple slip on n symmetrically activated slip systems, 166. d ≈ , + f (35) F ρ I γ& θ d wcP Pc i ==nvρ n , (39) H dt K w m w b S ≈ 121..θ (36) where v is the average dislocation velocity, w the fb i . cell wall width, and γ denotes the shear strain rate. Distributing the available (smeared-out) excess dis- Using the principle of similitude (33) and the Holt- location density ρ on the total fragment boundary exc type scaling law (32), this can be transformed into surface area S per unit volume then gives the mean fb the well-known Kocks storage term (cf. eq. (9)), now excess dislocation spacing in the fragment bound- for a smeared-out cell wall dislocation density, ary or, using the Read-Shockley formula Eq. (34), 64 M. Seefeldt

excess edge dislocations of opposite signs. The +− F ρρI F I Pf γ& d d c geom generation of a twist PDD corresponds to the for- wm=− =−n = H K H K mation of a both-side terminating net of two sets of dt dt dc b parallel excess screw dislocations of the same sign, Pf γ& (40) − c geom ρ n . or of two parallel semi-infinite nets of parallel ex- K w b cess screw dislocations of opposite signs. A vari- ety of different mechanisms was proposed to ex- 3.2.2. Dislocation dynamic recovery: annihila- plain this nucleation of fragment boundary segments tion of mobile with cell wall dislocations. Stored through collective behaviour of a couple of disloca- cell wall dislocations can, on the other hand, anni- tions of the same sign [8,87,106-108]. The models hilate with mobile dislocations of opposite sign pass- suggested until 1960 [87,106] aimed at an under- ing by. In the case of edge dislocations, such standing of nucleation of kink bands on the grain annihilation events can take place through dipole scale, the ones suggested later [107,108] were disintegration, as proposed by Essmann and prompted by TEM observations of misorientation Mughrabi [105], or after vacancy-assisted climb. In bands on the mesoscopic scale arising in stage III the case of screw dislocations, annihilation events of plastic deformation [9,10,17]. take place after cross-slip. As discussed in sub- From the beginning, two different routes were section 2.1, room temperature deformation of copper followed to explain kink banding on the grain scale is characterized by comparable annihilation lengths and misorientation banding on the mesoscopic scale. y of screws after cross-slip and y of edges after s e On the one route, an “imposed” macroscopic back- dipole disintegration. Annihilation of edges after va- ground of kink band nucleation or a grain-scale back- cancy-assisted climb can be neglected because ground of misorientation band nucleation was sug- bulk diffusion of plastic vacancies is too slow at low gested, e.g. constraints imposed by the holders [5] homologous temperatures, while core or pipe diffu- or stress concentrations at grain boundary ledges sion would require a significant dislocation density [108]. On the other route, an “intrinsic” microscopic in the cell interior (where the plastic vacancies would background of kink band as well as misorientation be produced) to let the vacancies diffuse from there band nucleation was proposed on the level of dislo- to the cell walls (where they are needed to assist cation dynamics, namely collective behaviour or self- climb and subsequent annihilation of cell wall with organization of dislocations. mobile dislocations). Therefore, the two dislocation Frank and Stroh [106] defined a kink band as a characters can be treated in the same common thin plate of sheared material in a non-sheared ma- framework in the present context. The common trix, transverse to a slip direction and bounded by recovery term scales with the same annihilation opposite tilt walls of dislocations (while in most of length y =6b proposed by Essmann and Mughrabi e the later papers it was considered as a non-sheared [105], plate in a sheared matrix). In such a configuration, the applied shear stress would drive the two oppo- F ρ I − γργ&& d w w site tilt walls not towards, but away from each other. =−22y ρ =− y n , (41) H dt K ewb en b Nevertheless, in case of finite tilt walls, the tips of the walls would still attract each other. Frank and for single slip as well as for multiple slip on n sym- Stroh determined the equilibrium between the metrically activated slip systems. repulsive force due to the external shear stress and 3.2.3. Generation of partial disclination dipoles. the attracting force between the edges and The keystones of any dynamic model of fragmenta- calculated equilibrium band widths and lengths for tion are the mechanisms of nucleation and growth parallel walls and for an elliptical band. Furthermore, of fragment boundary segments. In a disclination- an elliptical band with a sufficient angle of kinking based model, these mechanisms correspond to the could grow by generating new dislocation loops or elementary mechanisms of generation of partial pairs at its tips. The authors noted that the stress disclination dipoles (PDD) and of propagation of fields of kink bands were similar to those of slip partial disclinations (PD). Following subsection 2.2, bands, so that they “might act as sources for each the generation of a wedge PDD corresponds to the other”. formation of a both-side terminating array of parallel As already mentioned in subsection 1.1, Mader excess edge dislocations of the same sign, or to and Seeger [8] distinguished two different types of the formation of two parallel semi-infinite walls of kink bands, one on the macroscopic scale arising already at the end of stage I and mainly in stage II Disclinations in large-strain plastic deformation and work-hardening 65

Fig. 9. Mechanism of kink band nucleation according to Seeger. To the left, piling-up of mobile dislocations at Lomer-Cottrell barriers limiting a glide zone. The glide zone is, for simplicity, sketched with a rectangular rather than with a hexagonal shape. To the right, the piled-up dislocations after cross-slip. Edge dislocations remaining in the primary slip planes can form nuclei of kink bands perpendicular to the primary slip plane. Edge dislocation segments remaining in parallel cross-slip planes can form nuclei of kink bands perpendicular to the cross slip plane.

(kink bands belonging to the deformation band struc- tion. However, with the help of Ergänzungsgleitung ture in the terminology used in this paper) and one (completion glide) [64,65], they could be completed on the mesoscopic scale arising only in stage III to become a both-side terminating low-angle grain (misorientation bands belonging to the fragment boundary. This could then limit the free path of the structure in the terminology used here). The onset mobile dislocations in the slip direction and act as of stage III is characterized by the resolved shear a nucleus of a second kink band. stress reaching a critical value to trigger off extensive Vladimirov [107] proposed a similar nucleation cross-slip. Furthermore, the intensity of kink band- mechanism based on the redistribution of piled-up ing of the second type turns out to scale with the edge dislocations into an energetically favourable stacking fault energy. Therefore, cross-slip is most wall configuration with the help of vacancy-assisted likely to be involved in misorientation banding. climb. For such a mechanism, the intensity of Mader and Seeger suggested that cross-slip of misorientation banding should scale with the screw dislocation segments piled-up at Lomer- activation enthalpy of vacancy diffusion. Cottrell barriers would result in pairwise annihilation However, any mechanism based on impenetrable of screw segments from neighbouring glide zones obstacles, e.g. Lomer-Cottrell barriers, can only leaving edge dislocations in the primary slip plane explain the formation of symmetric kink bands or as well as edge segments in the cross slip planes double walls with mutually compensating behind (see Fig. 9). The glide motion of the primary misorientations because the impenetrable obstacles edge dislocations could be hindered by Lomer- also stop dislocations which travel in the opposite Cottrell barriers formed through reactions among direction in the glide zone on the other side of the secondary dislocations, and by the edge segments obstacle. In order to initiate the formation of an asym- left behind in the cross-slip planes. On the one hand, metric kink band or double wall or a single wall with the primary edges could then arrange in an a non-vanishing total misorientation, either globally energetically favourable wall configuration perpen- or locally different slip rates on the two sides of the dicular to their slip plane. This both-side terminat- obstacle are required. Since a negligible disloca- ing wall might act as a nucleus of a kink band, and tion content of the cell interiors and an initially ho- its stress fields might join the Lomer-Cottrell barri- mogeneous orientation throughout the crystallite are ers in limiting the free path of the mobile disloca- assumed, there is no way to a global slip imbal- tions. On the other hand, the edge segments in the ance between the two sides of a cell wall (which cross-slip planes would always act as forest ob- would result in the formation of an excess disloca- stacles to the primary edges, but would, in general, tion layer on the cell wall in a straight-forward man- not represent a low-angle grain boundary configura- ner). 66 M. Seefeldt

Fig. 10. Mechanism of fragment boundary nucleation used in the present model. On the left, double cross slip events in the cell interior resulting into bundles of mobile dislocations. The cross-slip planes are, for simplicity, sketched as perpendicular to the primary slip plane. To the right, the effect of a semi-transparent obstacle: some of the edge dislocations penetrate through the cell wall, others get stuck.

A local slip imbalance, however, can arise due course of dislocation dynamics. The nucleation part to incidental collective behaviour of neighbouring of the first mechanism was recently modelled by mobile dislocations at a semi-transparent obstacle Gutkin et al. [108] as a splitting event of partial with stochastic trapping. If several neighbouring disclinations in faults of grain boundaries triggered mobile dislocations approaching the cell wall by the applied shear stress. The growth part of the incidentally all together get trapped, while their first mechanism was already modelled by Romanov counterparts of opposite sign approaching the wall and Vladimirov [22] as propagation of partial from the other side incidentally all together do not disclination dipoles at the tips of the growing get trapped, then a both-side terminating excess misorientation bands consisting of two parallel ter- dislocation wall representing a nucleus of a frag- minating excess dislocation walls with opposite ment boundary arises. Whether and on which signs. The present paper, on the contrary, focusses conditions such a nucleus grows or gets “dissolved” on modelling the second case, the nucleation and again, will be discussed in the next subsection. As growth of loop-like fragment boundary layers on the already pointed out by Pantleon [109], the semi- grounds of dislocation dynamics. transparency of the obstacle and the stochastic As discussed in sections 2.1 and 3.1, the model character of the trapping process1 thus are of crucial presented in this paper starts from a well-developed importance for misorientations to be formed. If the cell structure, i.e. from a substructure which is typi- obstacle was an impenetrable one, then only a double cal for stage III of plastic deformation. Detailed light excess dislocation wall with no net misorientation and electron microscopy studies of slip line pat- could be formed. If the trapping process would not terns on the surface of deformed copper and have a stochastic character, incidental collective aluminium single crystals [87,110] revealed already behaviour would not be possible and again only a in the 1950s that during stage III the gliding disloca- double excess dislocation wall with no net tions get increasingly organized in bundles or slip misorientation would result. bands and plastic deformation gets increasingly As mentioned in subsection 1.1, Koneva, Kozlov concentrated in the slip bands. This bundling or slip and coworkes [38-40] proposed the coexistence of banding was attributed to extensive double cross two fragment boundary nucleation mechanisms. In slip [87,110-113]. Therefore, the model presented the first case, straight misorientation bands would here assumes that the mobile dislocations are be nucleated at grain boundary ledges and grow formed through double cross slip in the cell interi- into the grain interior, in the second case disclination ors and thus also supposes that they are grouped loops would be generated in the grain interior in the into bundles or slip bands (see Fig. 10). In f.c.c. single crystals with intermediate and high 1 Of course, this stochastic character of the trapping stacking fault energy, there is a striking coincidence of process is not a fundamental one, but just due to fluc- · the onset of massive cross-slip, tuations in the dislocation structure of the cell wall and · the bundling of mobile dislocations in slip bands, poor knowledge of its details. Disclinations in large-strain plastic deformation and work-hardening 67

Fig. 11. Single-slip conditions: Generation of PDDs at semi-transparent obstacles. (a) balanced stopping of mobile dislocations, (b) double wall formed as a result of balanced stopping, (c) spatially non-balanced stopping of mobile dislocations, (d) both-side terminating double wall segment formed as a result of non-balanced stopping, (e) translation of (d) into disclination «language». Large edge symbols denote mobile, small ones redundant cell wall, bold ones non-redundant excess edge dislocations.

≈ · the formation of a dislocation cell structure, deformation at room temperature, and 2a0 0.20 · and the onset of misorientation banding at the be- µm and ϕ ≈ 0.0072 for fast deformation at room tem- ginning of stage III. perature. If one takes into account that the mobile dislo- It has to be emphasized that slip line measure- cations get bundled in slip bands in the course of ments do not provide any information about the dy- stage III of plastic deformation (see the next namics of plastic deformation. Since the strain in- subsubsection), one can use data from slip line tervals used to generate the slip lines are rather measurements to estimate typical parameters of a large, one slip line can correspond to several mo- partial disclination dipole representing a fragment bile dislocations stepping out to the surface at dif- boundary nucleus. Mader [114] studied low-sym- ferent times. Consequently, the spacing between metry oriented copper single crystals after tension the slip lines does not equal the spacing between at room temperature and found slip bands with 3-10 mobile dislocations arriving at an obstacle (perpen- slip lines per band and lamella widths of about 300 dicular to the slip plane) at the same time. The spac- Å. If one then assumes incidental collective trap- ings between collectively trapped dislocations de- ping of the mobile dislocations in such a slip band rived above from the spacings between slip lines at a cell wall, one gets an excess dislocation wall thus have to be considered as a lower limit or as a segment made up of N≈ 5 dislocations with an aver- rough aproximation. age spacing h of about 300 Å. This gives an initial In order to obtain a net misorientation across ≈ µ PDD width of 2a0 0.15 m and, according to Eq. the arising fragment boundary, it is of crucial (34), a misorientation of ϕ≈0.085. Müller and importance that the excess dislocation layer on the Leibfried [115] investigated low-symmetry oriented one side of the cell wall does not get compensated aluminium single crystals after tension at liquid air by its counterpart with opposite sign on the other and at room temperature for a series of different strain side of the cell wall (compare Figs. 11 and 12 be- rates. They found slip bands with 2-4 slip lines for low) - which is only possible for a semi-transparent . ε > 10-2 s-1 at liquid air temperature and with 10-30 obstacle with stochastic trapping. For a quantita- . slip lines for ε <10-2 s-1 and about 5 slip lines for tive description of PDD generation through incidental . ε >10-2 s-1 at room temperature. The lamella widths collective trapping of mobile dislocations at cell walls, were found around 700 Å at liquid air temperature the two extreme cases of single slip (with cell walls and around 400 Å at room temperature, without clear roughly perpendicular to the slip plane, see Figs. ≈ dependence on the strain rate. This gives 2a0 0.21 11 and 12) and of symmetric multiple slip (with cell µm and ϕ ≈ 0.0041 for fast deformation at liquid air walls bisecting the equivalent slip planes, see Fig. ≈ µ ϕ ≈ temperature, 2a0 0.80 m and 0.0072 for slow 13) are considered. 68 M. Seefeldt

pensated by their counterparts of the opposite sign at the other side of the cell wall, the exponent re- flects the collective behaviour of the N neighbouring mobile dislocations. In the case of single slip, one finds a generation rate for the (smeared-out) excess dislocation density of

+− F dρρI F d I exc=− m = H dt K H dt K ((PPf1− ))N γ& ccgeom = Nd b c (42) Fig. 12. Single-slip conditions: Generation of ((PPf1− ))N γ& PDDs at semi-transparent obstacles. (a) spatially ccgeom ρ . and directionally unbalanced stopping of mobile NKw b dislocations at a “hard” kinetic obstacle, (b) disordered excess dislocation wall formed as a The same collective trapping process occurs at frag- result of non-balanced stopping, (c) translation of ment boundaries. The generation rate can be esti- (b) into disclination “language”. Large edge symbols mated taking patting from the one for the cell walls, denote mobile, small ones redundant cell wall, but replacing the trapping probability P by the one bold ones non-redundant excess edges. c Pf and replacing Holt’s scaling law (32) for cells by the one (35) for fragments,

+− F dρρI F d I The loss rate of mobile dislocations due to PDD exc=− m = H dt K H dt K generating trapping can be set up taking pattern from the loss rate due to ordinary trapping. The prob- ((PP1− ))N γ& ff = ability P of individual trapping in Eq. (40) has to be c db replaced by the probability (P (1-P ))N of collective f (43) c c − N γ& and non-compensated trapping of N dislocations. ((PPff1 )) θ . The probability product reflects the requirement that 166. i b the trapped mobile dislocations must not be com-

Fig. 13. Multiple-slip conditions: Generation of PDDs at cell walls. (a) formation of a cell wall through mobile dislocations either being stopped at a “hard” kinetic obstacle or interacting with each other to form non-glissile combined dislocations, (b) cell wall of the Mughrabi type, (c) cell wall as a glide obstacle triggering the storage of excess dislocations, (d) incidental irregularities in the dipolar character of the excess dislocation walls, (e) translation of (d) into disclination “language”. Large edge symbols denote mobile, small ones redundant cell wall, bold ones non-redundant excess edges. Disclinations in large-strain plastic deformation and work-hardening 69

Fig. 14. Defect configuration resulting from fragment boundary nucleation. To the left, in the language of dislocations: a set of parallel dislocation loops along the cell wall obstacles. To the right, in the language of disclinations: a pair of parallel disclination loops at the border lines of the dislocation loop “cylinder” along the cell wall obstacles.

Since N collectively trapped mobile dislocations tiple slip is about 3.10-7. While the first probability constitute one new partial disclination dipole, the is already small, but still large enough to generate corresponding PDD generation rate at cell walls a sufficient density of partial disclinations reads fragmentating the crystallite, the second one turns out to be insufficient for fragmentation. + F dθγI ((PPf1− ))N & Since the long-range stress field around such a m = ccgeom = new PDD would drive the dislocation loop segments H dt K Nd b c with the opposite sign at the other end of the loops ((PPf1− ))N γ& (44) away from the PDD and towards the neighbouring ccgeom ρ w . cell wall, a local slip imbalance would also be in- NK b duced at this neighbouring cell wall. As a result, a The major difference with the case of symmetric loop-like fragment boundary layer as sketched in multiple slip is that the cell walls are not perpen- Fig.14 would form. dicular then to the slip plane, but roughly bisecting 3.2.4.Propagation of partial disclinations. The the equivalent active slip planes (see Fig. 13). conditions of growth of a finite excess edge dislo- Consequently, neighbouring mobile dislocations from cation wall through capturing and attaching addi- one slip system getting trapped collectively at a cell tional excess dislocations were first discussed in wall would not form a both-side terminating low-angle the context of polygonization during recovery grain boundary configuration, so that [66,73,74]. The total stress field of the finite dislo- Ergänzungsgleitung (completion glide) is required, cation wall was calculated as the sum of the stress or, more specific, mobile dislocations from the fields of the individual dislocations. For the long- conjugate slip system are required to get trapped range stress field at large distances from the wall, collectively as well and to react formally with the the sum can be replaced by an integral. This gives already trapped ones in order to form a low-angle a shear stress perpendicular to the wall of grain boundary segment. As a result, collective behaviour of twice as much dislocations is required Gbx τ = × in the case of symmetric multiple slip which makes xy 21πν()− h the probability of a PDD generating event dropping much below the probability in case of single slip. F ya+ ya− I (45) − , An example: If the individual trapping probability is H xya2222++()xya+− ()K N=5 Pc = 1/3, then the probability (Pc(1-Pc)) of collec- tive and non-compensated trapping under single slip where h is the dislocation spacing in the wall, a = . -4 N=10 is about 5 10 , the probability (Pc(1-Pc)) of col- Nh/2 denotes the half height of the wall, and the lective and non-compensated trapping under mul- reference system is chosen as indicated in Fig.15. 70 M. Seefeldt

ττττττ>−=−+ yx PN eff PN ext int . (48) Or, phrased in another way, the applied shear stresses due to the externally applied load and the PDD have to overcome the local shear resistance composed of the Peierls-Nabarro resistance and the shear resistance due to the current defect struc- ture,

ττ+>+ τ** τ yx 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 stress at least by order of magnitude. In the following, two rough estimations are carried out in order to find a ω minimum required partial disclination strength p which allows PD propagation, i.e. PDD expansion or fragment boundary growth, and to find a capture

length yc, i.e. a distance ahead of a PD within which Fig. 15. “Capturing hyperbolae” above for single the PD is still able to capture a mobile dislocation slip and below for multiply slip. of matching sign passing by. From a graphical representation of the shear τ stress field xy of a finite excess dislocation wall as shown by Li [74], one can read that the loop-shaped If one expresses the dislocation spacing through curves of attracting shear stresses of magnitude 0.3, the misorientation according to Eq. (34), one finds 0.2 and 0.1 times a stress unit of Gb/2π(1-ν)h = the long-range stress field of the finite wall with dis- Gϕ/2π(1-ν), end up at distances of about 1.5, 2.5 location spacing a and height 2a to be equivalent to and 5 times the wall half length a in front of the the one of a partial disclination dipole with excess dislocation wall tip. In the case of, for disclination strength ω = ϕ = b/h and dipole width instance, copper at room temperature this means 2a,

Gb Gxω |(τ ya=<501 )|. = τ = × xy xy 21πν()− h 21πν()− h Gϕ (50) F ya+ ya− I (46) 01. ≈ 1000ϕMPa. − . 2222 21πν()− H xya++()xya+− ()K A bundle of N = 5 dislocations with a slip plane As Nabarro showed already in 1952 [73], it follows distance of h =100b ≈ 26nm corresponding to a from Eq. (45) that the finite wall or the PDD exerts misorientation ϕ = 0.01 would then at y = 5a, i.e. at an attracting Peach-Koehler force F = b τ on an x x xy 4a ≈ 0.26 µm ahead of the wall tip, exert a shear edge dislocation with matching sign inside the stress of not more than 10 MPa (and of no more capture hyperbola (see Fig.15) given by than 30 MPa at 0.5a=1.25h ahead of the wall tip). A 22−= 2 (47) group of N = 5 dislocations with h = 1000b ≈ 260 yxacc . nm corresponding to ϕ =0.001, however, would at However, the applied shear stress due to the finite 4a ≈ 2.6 µm ahead of the wall tip exert a shear wall or the PDD has to overcome the Peierls-Na- stress of only less than 1 MPa (and of only less barro stress and the local effective shear stress, than 3 MPa at 0.5a=1.25h ahead of the wall tip). i.e. the external shear stress reduced by the mean From this simple example, one can already guess internal shear stresses due to e.g. other disloca- that a partial disclination of strength ϕ = ω = 0.01 tions or due to plastic strain or lattice rotation mis- would have a chance to grow by capturing additional fits, mobile dislocations, whereas a partial disclination Disclinations in large-strain plastic deformation and work-hardening 71 . of strength ϕ = ω =0.001 would not because the former of ε =10-2 s-1, this gives a mobile dislocation density ρ ≈ 14 -2 would have a chance to overcome the local effective of about m 10 m which, according to Eq. (53), τ ≈ stress or the externally applied load plus the local corresponds to an effective stress of about eff 20 shear resistance, whereas the latter would not. MPa. Inserting this into Eq. (52) gives two real roots ≈ ≈ ω For a quantitative evaluation of Eq. (49), the of yc,1 0.75a and yc,2 10.2a for = 0.01 and only Peierls-Nabarro resistance is neglected for f.c.c. complex roots for ω = 0.001. An estimation of the materials and the coordinates are rescaled in units capturing length carried out by the author [118] for ~ of a to give x = x/a and y~ = y/a, giving the case of a PDD propagating at the top of a grow- ing misorientation band also gave a capturing length Gxω~ of about y ≈ 10a. τ = × c xy It can be concluded that the Peach-Koehler force 21πν()− exerted by a partial disclination with a strength be- ~ + ~ − F y 1 y 1 I (51) yond a required minimum strength of about ω = − . p H xy~2222++(~ 1) xy~ +−( ~ 1) K 0.01 is sufficient to capture mobile dislocations pass- ≈ ing by within a capturing length of about yc 10a, to The limiting curve where the applied shear stress attach them to the terminating excess dislocation due to the PDD equals the local effective stress is wall and to let the partial disclination dipole expand. then defined by As Li [74] could show, such a process is also energetically feasible. It has to be emphasized that Gxω~ it is only the bundling of slip in slip bands which ~4322− ~ +−−~ ~ y yxy21( ) makes possible the nucleation of fragment bound- πντ()1− eff ary segments that are able to grow: this bundling in Gxω(~2 − 1) x~ (52) slip bands reduces the mean dislocation slip plane yx~ ++=(~2210). πντ− distance h in the slip bands of stage III compared to ()1 eff the one in the homogeneous fine glide regime of A simple special case is x = a or x~=1 giving stage II, so that the resulting partial disclination ω strength exceeds the critical value of about p = Gω 0.01. Thereby, slip banding and fragmentation, two ~43− ~ += y y 40. (53) different manifestations of collective behaviour in the πντ()1− eff dislocation ensemble, are closely related to each To estimate the local effective stress, first other. consider an idealized framework of only the finite The coupling of fragmentation to slip banding dislocation wall and a random distribution of mobile does not only lead to a minimum required disclination dislocations being present. Then the scaling law strength. Since the width of the lamellae between the slip lines in a slip band is only slightly changing τα= ρ (54) effGb m in the further course of plastic deformation, it also offers an average strength of a propagating partial proposed by Kocks, Argon and Ashby [84] for the disclination, i.e. a step-width of the increase in local effective stress would apply. The mobile dislo- misorientation. In this way, one can eliminate one cation density ρ can be estimated from Orowan’s m of the parameters unknown in disclination in con- law trast to dislocation dynamics. ερ& = mbv . (55) While Romanov and Vladimirov [22] “fed” the Sm propagating partial disclination dipole at the top of a To estimate an average dislocation velocity in this growing misorientation band by widening of dislo- context, one should better refer to the TEM in-situ cation loops from a static dislocation distribution measurements rather than to the pulse load and between the two boundary planes ahead of the band etching techniques because the continuous loading tip, the present work follows a dynamic approach used in the in-situ tests is similar to the conditions as proposed by Essmann and Mughrabi [105] in of macroscopic straining [116]. In copper as well as the context of dynamic recovery. The reasoning for in aluminium plastic deformation is carried out by a wedge PDD capturing a passing edge dislocation few but fast mobile dislocations – Appel and of the matching sign is exactly the same as for a Messerschmidt [117] measured an average dislo- cell wall edge dislocation spontaneously disintegrat- cation velocity of v = 10-6 ms-1 in a TEM in-situ test ing with a passing edge dislocation of opposite sign on aluminium at room temperature. For a strain rate (see subsection 3.2.2 and Fig. 16). 72 M. Seefeldt

Fig. 16. a) Disintegration of an immobile edge dislocation with a mobile one of opposite sign passing by within the annihilation length y . b) Capture of a mobile dislocation passing by within the capturing e length y ahead of the tip of a terminating excess dislocation wall (dislocation language). c) Capture c of a mobile dislocation passing by within the capturing length y ahead of a partial disclination (the c same in disclination language).

Formally, the density θ of dipoles of propagat- p y 10a ing partial disclinations replaces the density ρ of =≈c p γ& w . (58) t 4ω cell wall dislocations as sink density, and the c p capture length y replaces the annihilation length c 3.2.5. Disclination storage: immobilization of y , e propagating partial disclinations at fragment

+ boundaries. For simplicity, it is assumed that each F dρρI F d I γ& exc=− m =−2y θ = propagating partial disclination is definitely stopped H dt K H dt K cpb at the next fragment boundary, i.e. in the next mesh γ& (56) of the fragment boundary mosaic (which is sug- − θ 20app , gested by the analytical formula describing the in- b teraction between a wide sessile and a narrow propa- gating PDD [23,119]). Then, the mean free path of a where a denotes the average half width of the ex- p propagating PD equals the mean fragment size d , panding dipoles of propagating partial disclinations. f giving an immobilization term of The same mechanism, of course, also controls the mean propagation speed V of the partial + − disclinations: To calculate this velocity, one can es- F dθ I F dθθI 1 V i =−G p J =p = timate the time t which is needed by the mobile H K c dt H dtK d 2 γ. f dislocation current /b to provide the Nc disloca- Vθ (59) tions of the matching sign and character required θ p 0., 602 i by the PD to propagate by its own capturing length 2

yc, where Eq. (35) was used. The factor of 1/2 is due to ω θ N 4 the transition from the PD dipole density p to the t ==c p , c single PD density θ. Note that the disclination im- y γ& γ& i c (57) mobilization term has the same structure as the 4 b dislocation storage term proposed by Kocks (cf. Eq. where Eq. (34) was used. This results in an average (9). Using expression (58) for the mean disclination speed of speed finally gives Disclinations in large-strain plastic deformation and work-hardening 73

+ − ture suggested by Mecking and Kocks [81] for the F dθ I F dθ I 10ab γ& i =−G ppJ =0. 602 θθ . total dislocation density (cf. Eq. (10) in subsection H K H K ω ip (60) dt dt 8 p b 2.3) and can directly be integrated by separating the variables, 3.2.6. An open question: disclination recovery.

−−()/εε 2 2 In order to study the substructural background of y F ICe− I stage V of plastic deformation, one probably has to ρε()= G 1 J . w H K (66) search for a mechanism of dynamic recovery of R partial disclinations. A promising hint for such a The integration constant C can be determined from mechanism might be the coupling between work- 1 the starting condition ρ (ε = ε ) = ρ0. The stationary hardening in stage V and the hydrostatic pressure w y w values of the cell wall dislocation density and of the [120] because the wedge disclination density is also density of dipoles of propagating partial disclinations coupled to the hydrostatic stress [23]. Another pos- can directly be calculated and classified as stable sible mechanism might be the activation of addi- stationary values: tional slip systems around the PD triggered by their long-range stress fields. This would result in an ac- F I 2 cumulation of misfit-relieving dislocations around the s I ρ = and PD, as proposed by Argon and Haasen for other w H R K sources of long-range stresses [58]. S (67) θ s = f p . 3.3. Substructure development NJ Collecting all the reaction terms from Eqs. (40), (41), Inserting these stationary values into (64), one finds (42), (43), (44), (56) and (60), transforming the evo- a simple law for the evolution of the slowly varying structure variable θ in the quasi-stationary state, lution equations with the help of Orowan’s law (mS – i Schmid factor) qs F dθ I S & i = f θ ερ= m nbv (61) i , (68) Sm H dε K N from the evolution parameter time to the evolution or integrated (C – integration constant) parameter strain and abbreviating all the reaction 2 coefficients gives the following set of evolution equa- 2 F S I tions: θεqs =+f ε ()C2 , (69) i H 2N K dρ w =−ρρ IRww, (62) and through the relationship (35) also for the mean dε fragment size df: dρ qs exc =++ρθθ F I SSE, (63) ddf Sf 1 Sf 2 ε cw f i p =−083.., =−030 d d H εθK f (70) d N i N dθ p Sc Sf =+−ρθθθ or integrated (C3 – integration constant): w iipJ , (64) dε N N 1 qs ε = θ df () . (71) d i ε + = θθ 030.(SNf /) C3 J ip. (65) dε Such a “hyperbolic” decrease matches with experi- If one neglects collective trapping at cell walls com- mental observations [26]. pared to collective trapping at fragment boundaries, The reaction coefficients R for dynamic recov- this system of ordinary differential equations has ery, E for capture of mobile dislocations through the advantage that the evolution equations (62) and propagating partial disclination dipoles, and J for (65) for the redundant dislocation density and the immobilization of propagating partial disclinations immobile partial disclination density which describe were calculated according to the expressions in (41), the substructure development and control the work- (56) and (60) with an annihilation length of ye = 6b, a hardening can be solved analytically. Equation (62)) capturing length of y = 10a, an average width of for the redundant dislocation density has the struc- c 74 M. Seefeldt

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) . 8 –1 Sf 3.83 10 m fitting parameter, Eq. (43) J 9.22.10-4 m-1 Eq. (60)

expanding partial disclination dipoles of 2ap =0.1 sponds to the experimental observation (cf., for in- µm, and a strength of the propagating partial stance, [58]) that, for symmetric orientations, usu- ω disclinations of p = 0.01, see Table 2. The reaction ally much less than all the slip systems expected coefficients I for “individual” trapping of mobile dis- from a Schmid factor analysis (in this case eight)

locations in cell walls and Sf for “collective” trapping are locally activated. The second number corre- of mobile dislocations at fragment boundaries were sponds well with the slip line measurements at the used as fitting parameters. Their values were deter- beginning of stage III as discussed in subsections mined through fitting the modelled to an experimen- 3.2.3 and 3.2.4. tal flow curve for uniaxial compresssion of a [001]- The resulting fragment size evolution according oriented copper single crystal at room temperature, to Eq. (35)) and the resulting fragment misorientation see Table 2. evolution according to Eq. (37) are displayed in Figs. The obtained values were then evaluated accord- 17 and 18. The results are in good agreement with ing to the expressions in (40), (42) and (43) with the (rare) experimental data, see, for instance,

trapping probabilities of Pc = 1/3 at cell walls and Pf [32,122]. = 1/2 at fragment boundaries, a Holt constant of K

= 16 and a geometric cell shape factor of fgeom = 2. 3.4. Work-hardening The fitted coefficients then corresponded to a number n = 3 of locally active slip systems and to a number Fig. 19 displays the resulting flow stress contribu- of N = 5 excess dislocations constituting a new tions due to the cell structure or the redundant cell partial disclination dipole. The first number corre- wall dislocations and due to the fragment structure

Fig. 17. Evolution of the mean fragment or cell block size with the (logarithmic) strain according to the simplified dislocation-disclination reaction kinetics model. Material parameters for copper, process parameters for room temperature and ε. = 10-2 s-1. Disclinations in large-strain plastic deformation and work-hardening 75

Fig. 18. Evolution of the mean misorientation across fragment or cell block boundaries with the (logarithmic) strain according to the simplified dislocation-disclination reaction kinetics model. Material parameters for copper, process parameters for room temperature and ε. = 10-2 s-1.

Fig. 19. Flow stress contributions of the redundant cell wall dislocation density and of the immobile partial disclination density and their sum according to the simplified dislocation-disclination reaction kinetics model. Trapping and cooperative trapping reaction coefficients I and S used as fitting f parameters, Schmid factor m = 0.408, material parameters for copper, process parameters for room S temperature and .ε = 10-2 s-1. For comparison, the compression flow curve of a single crystal with highly symmetric orientation is shown.

Fig.20. Kocks-Mecking plot corresponding to the modelled flow curve. 76 M. Seefeldt

or the immobile partial disclinations in the fragment [9] I.W. Steeds // Proc. Roy. Soc. A 292 (1966) boundary triple junctions as well as the total flow 343. stress according to Eq. (28). For comparison, an [10] U. Essmann // Phys. stat. sol. 3 (1963) 932, experimental flow curve from a uniaxial compres- in German. sion test at a [001] oriented copper single crystal at [11] H. Mughrabi, In: Constitutive Equations in room temperature is also shown. In Fig. 20, the Plasticity, ed. by A.S. Argon (M.I.T. Press, corresponding Kocks-Mecking plot is presented. Cambridge, MA, 1975), p. 199. The stages III and IV of work-hardening can clearly [12] H. Ahlborn // Z. Metallkde. 56 (1965) 205 and be distinguished. Stage III can be attributed to the 411, in German. saturating local shear resistance of the cell struc- [13] H. Ahlborn // Z. Metallkde. 57 (1966) 887, in ture due to the redundant dislocations in the cell German. walls. Stage IV, on the contrary, can be connected [14] H. Ahlborn and D. Sauer // Z. Metallkde. 59 to the monotonously growing long-range internal (1968) 658, in German. stresses of the fragment structure due to the im- [15] A.S. Malin and M. Hatherly // Met. Sci. 13 mobile partial disclinations in the fragment bound- (1979) 463. ary triple junctions. This matches with the work- [16] V.A. Likhachev and V.V. Rybin // Izv. AN hardening concept presented by Kozlov, Koneva and SSSR, ser. fiz. 37 (1973) 2433, in Russian. coworkers [39-42]. [17] A.S. Rubtsov and V.V. Rybin // Fiz. met. metalloved. 44 (1977) 611, in Russian. ACKNOWLEDGEMENTS [18] V.V. Rybin, Bolshie plasticheskie deformatsii i razrushenie metallov (Large Plastic Defor- The author wishes to thank P. Klimanek, Freiberg, mations and Fracture of Metals), and A.E. Romanov, St. Petersburg, for the intro- (Metallurgiya, Moskva, 1986), in Russian. duction into the concept of disclinations and for [19] F. Prinz and A.S. Argon // Phys. stat. sol. (a) continuous encouragement, support and help. W. 57 (1980) 741. Pantleon, Roskilde, V. Klemm, Freiberg, P. Van [20] V.I. Vladimirov and I.M. Zhukovskiy // Fiz. Houtte, Leuven, and E. Aernoudt, Leuven, tverd. tela 16 (1974) 346, in Russian. contributed in many fruitful discussions to a critical [21] V.A. Likhachev and V.V. Rybin // Fiz. tverd. and better understanding. tela 18 (1976) 163, in Russian. Financial support was given by the Deutsche [22] V.I. Vladimirov and A.E. Romanov // Fiz. Forschungsgemeinschaft (Graduate School “Physi- tverd. tela 20 (1978) 3114, in Russian. cal Modelling in Materials Science”) and the Fonds [23] A.E. Romanov and V.I. Vladimirov, In: Dislo- voor Wetenschappelijk Onderzoek – Vlaanderen cations in Solids, vol. 9, ed. by F.R.N. (contract number G.0228.98). Nabarro (North Holland, Amsterdam, 1992), p. 191. REFERENCES [24] N. Hansen and D. Kuhlmann-Wilsdorf // [1] E. Orowan // Nature 149 (1947) 643. Mater. Sci. Engng. 81 (1986) 141. [2] A.S. Argon, In: Physical Metallurgy, 4th [25] B. Bay, N. Hansen, D. Hughes and D. edition, ed. by R.W. Cahn and P. Haasen Kuhlmann-Wilsdorf // Acta metall. mater. 40 (North Holland, Amsterdam, 1996), vol. 3, p. (1992) 205. 1877. [26] N. Hansen and D.A. Hughes // Phys. stat. [3] C.S. Barrett and L.H. Levenson // Trans. sol. (b) 149 (1995) 155. Amer. Inst. Min. Met. Eng. 135 (1939) 327. [27] Q. Liu and N. Hansen // Phys. stat. sol. (a) [4] C.S. Barrett and L.H. Levenson // Trans. 149 (1995) 187. Amer. Inst. Min. Met. Eng. 137 (1940) 112. [28] Q. Liu, D. Juul Jensen and N. Hansen // Acta [5] R.W.K. Honeycombe // J. Inst. Met. 80 (1951- mater. 46 (1998) 5819. 52) 49. [29] D.A. Hughes and W.D. Nix // Mater. Sci. [6] R.W.K. Honeycombe // J. Inst. Met. 80 (1951- Engng. A 122 (1989) 153. 52) 45. [30] D.A. Hughes and N. Hansen // Mater. Sci. [7] W. Staubwasser // Acta metall. 7 (1959) 43. Technol. 7 (1991) 544. [8] S. Mader and A. Seeger // Acta metall. 8 [31] D.A. Hughes and N. Hansen // Acta mater. (1960) 513, in German. 48 (2000) 2985. Disclinations in large-strain plastic deformation and work-hardening 77

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