eﬀect of deformation mode and grain orientation on...

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Acta Materialia 55 (2007) 4935–4946

Effect of deformation mode and grain orientationon misorientation development in a body-centered cubic steel

J.-Y. Kang a,b, B. Bacroix a,*, H. Regle a,c, K.H. Oh b, H.-C. Lee b

a LPMTM-CNRS, Universite Paris 13, Institut Galilee, 93430 Villetaneuse, Franceb School of Materials Science and Engineering, Seoul National University, Sillim 9-dong, Gwanak-gu, Seoul 151-744, Republic of Korea

c Arcelor Research SA, Voie Romaine, 57283 Maizieres-les Metz, France

Received 18 September 2006; received in revised form 1 April 2007; accepted 2 May 2007Available online 10 July 2007

Abstract

Strain-induced misorientation development was studied in an IF steel as a function of strain for two deformation modes, plane straincompression and simple shear. Using electron back-scattered diffraction, orientation maps of ‘‘large’’ areas were obtained, from whichseveral individual grains associated with the principal texture components could be extracted so that only intragranular misorientationscould be estimated for these orientations. It was observed that the increase of the misorientation angle was more prominent in simpleshear than in plane strain compression and that the orientation influence was different for each mode. Considering texture evolutionas a possible source of misorientation development, the lattice spin tensor was estimated with the Taylor model for the two deformationmodes; both reorientation axis and angle were compared with misorientation angle and axis. The striking concordance of both quantitiesallows us to conclude that there is a direct contribution of texture evolution to misorientation accumulation with strain.� 2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Misorientation; Orientation; Deformation mode; Texture; IF steel

1. Introduction

During plastic deformation of crystalline materials,some new boundary structures composed of collecting dis-locations are usually formed [1–6]. These boundaries subdi-vide the original grains into several volume elements whichhave different orientations from one another. Then, asstrain increases, the evolution of these boundaries may leadto a clear subdivision of the initial grains and an increasingmisorientation between adjacent cells is often reported.

It is evident that these sub-structures have a significantinfluence on the material’s properties and behaviors and,consequently, have important implications for their applica-tions. First of all, it is obvious that the creation of dislocationboundaries as well as the subdivision of the initial grains into

1359-6454/$30.00 � 2007 Acta Materialia Inc. Published by Elsevier Ltd. All

doi:10.1016/j.actamat.2007.05.014

* Corresponding author.E-mail address: [email protected] (B. Bacroix).

smaller elements are two hardening sources. Also, a spe-cific morphology of these sub-volumes can induce someanisotropy of the as-deformed materials in addition to thatdeveloped by the deformation texture evolution [7,8]. Fur-thermore, intragranular misorientation is also very oftenused to estimate the stored energy of deformation – whichis considered to be the driving force for primary recrystalli-zation – through the well-known Read–Shockley equation[9–12]. Finally, it is also well known that dislocation bound-aries may act as potential nucleation sites for phase transfor-mation just like grain boundaries, resulting in intragranularnucleation which in turn can induce grain refinement [13].

With the development of severe plastic deformationmodes to create ultrafine microstructures [14,15], it appearsclear that the deformation-induced boundaries may havetwo different origins which are the simple accumulationand organization of dislocations into walls and/or the sub-sequent fragmentation of grains due to different activatedslip systems on each side of a pre-existing sub-boundary

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Fig. 1. Initial texture of the material (ODF /2 = 45� section); the ideal a-fiber is located at (/1 = 0�, U = 0–54.74�) and the c-fiber at (/1 = 0–180�,U = 54.74�).

4936 J.-Y. Kang et al. / Acta Materialia 55 (2007) 4935–4946

[16]. These two types of boundaries may have differentinfluences on the hardening behaviour of the grains,whether they can act as barriers to further dislocationmovement or not and some hardening laws have beendeveloped in order to distinguish between these two contri-butions [14,17]. For boundaries which act as barriers tofurther dislocation movement, the hardening contributionarises from grain refinement (Hall–Petch relationship),whereas when the misorientation accross the boundary issmall enough to allow crossing by new dislocations, it issimply translated into dislocation density through the Readand Shockley equation and this density in turn is intro-duced into the hardening law. However, depending onthe deformation mode (monotonic rolling/tension or severeplastic deformation through sequences of shear deforma-tion as in equal channel angular extrusion (ECAE)) andon the technique used to measure misorientations (trans-mission electron microscopy (TEM) or electron back-scat-tered diffraction (EBSD)), there is some discrepancy in theidentification of these two types of boundaries. In the caseof monotonic rolling or tension, usually investigated byTEM, mostly intragranular boundaries are studied andthese are usually designated as incidental dislocationboundaries (IDB) and geometrically necessary boundaries(GNB) [16]; the first type are generally classified as lowangle boundaries (LAB) and assumed to affect hardeningonly through a dislocation effect, whereas the second typeare often referred to as high angle boundaries (HAB),which are claimed to act as barriers to further dislocationmovement and which thus affect hardening through a sizeeffect. In the case of severe plastic deformation through acombination of shear steps (as found in ECAE for exam-ple), the refinement of the structure may be so importantthat the distinction between intra- and intergranularboundaries is somewhat lost and the distinction is simplymade between HAB and LAB, again affecting hardeningthrough grain size or dislocation effects [14].

For a given deformation mode, the active slip systemswithin a given grain depend on its orientation, whichmeans that the deforming behavior and the subsequentmicrostructure of each grain of a polycrystalline materialis a function of its crystallographic orientation as well asof the imposed deformation mode (of course, as the defor-mation mode imposed on each grain may differ from themacroscopic one, this may also have an influence on thedirect neighborhood of the grain considered). Concerningthe morphological aspect of these microstructures, therehave already been many previous studies which relate typ-ical deformation microstructures with grain orientationsfor various deformation modes [5,18–25]. In these works,which mainly deal with monotonic rolling and tension orshear strain to moderate strain levels, the alignment of dis-location boundaries with planes which are close to theactive slip planes is often reported. Some complementarystudies using EBSD relate somewhat similar observations,but the correlation between grain orientation and misorien-tation is usually less clear in those cases [26]. In all these

cases, the evolution of misorientation (both in terms ofangle and axis) across these boundaries as a function ofstrain and crystallographic orientation is never reported.However, in the case of larger strains, the orientation ofthe deformation-induced boundaries as well as the misori-entation matrix (angle and axis) will depend on the specificorigin of this boundary (dislocation accumulation or frag-mentation) and will thus also be a function of grain orien-tation [26] as well as deformation mode.

Therefore, the aim of the present study is first to evalu-ate completely the evolution of deformation-induced intra-granular misorientation as a function of grain orientationfor two different deformation modes, which are rollingand simple shear. The comparison of these two deforma-tion modes is interesting in the sense that it may corre-spond to an intermediate case between the previousstudies performed on monotonic rolling and the investiga-tion of severe plastic deformation which implies simpleshear. Also, it is expected that even without change ofstrain path, simple shear allows the formation of both typesof boundaries, because of the large rotation rates which areinvolved. Thus, experimental data obtained by EBSD arereported in some detail. Then, the use of a crystal plasticitymodel enables some physical interpretation to be providedfor the different observed behaviors.

2. Experimental material and procedures

An extra-low-carbon IF steel sheet was studied in thepresent investigation. The hot rolled specimen of initialthickness 4 mm was cold rolled to 2 mm, then annealed at800 �C for 1 h. The resulting texture, measured by X-raydiffraction, is composed of the so-called a = {hk l}Æ110æand c = {11 1}Æuvwæ fibers that are typical for cold rollingand subsequent recrystallization, as shown in Fig. 1; theoverall intensity of the texture is, however, quite low (i.e.,the orientation spread around the two fibers is relativelylarge) and the resulting grain size is �40 lm on average.For the sake of comparison between rolling and simpleshear, the Euler angles (/1, U, /2) vary from 0� to 180�,90� and 90�, respectively, as imposed by the symmetry ofthe shear process [24].

Then, two types of cold deformation were imposed stepby step. For plane strain deformation, rolling was carried

Table 1Fraction of indexed points in the orientation maps as a function of strainand strain mode

eVM 0 0.17 0.23 0.29 0.35 0.40 0.78

Indexingrate (%)

Rolling 99.85 99.86 99.79 99.78 99.64 99.58 97.64Simple shear 99.75 99.74 99.07 99.32 99.41 94.63

J.-Y. Kang et al. / Acta Materialia 55 (2007) 4935–4946 4937

out on one part of the initial sheet, conserving the rollingdirection and plane of the previous cold rolling step priorto annealing. Some other part of the initial sheet was usedfor simple shear tests. The dimensions and geometry of thesimple shear samples are illustrated in Fig. 2. In order tocompare the resulting microstructures and texturesobtained in these two deformation modes at the samestrain level, the von Mises equivalent strain was used:

eVM ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

3eij : eij

r¼ 2ffiffiffi

3p e ¼ 1ffiffiffi

3p c ð1Þ

where e is the rolling reduction in true strain and c theshear strain in simple shear. The amount of equivalentstrain was then varied between 0.17 and 0.78.

Global texture evolution with strain was then measuredby X-ray diffraction on the mid-thickness region of thesamples for each deformation mode and orientation distri-bution functions were then calculated with the harmonicmethod in the Euler space. The misorientation evolutionwas obtained from orientation maps measured by EBSDon a Cambridge S360 (W-GUN) scanning electron micro-scope (SEM) equipped with automatic OIM (orientationimaging map) software from TSL. Channel 5 softwarefrom HKL and some in-house software were also used toanalyze the misorientation distribution in more detail.Although the EBSD technique generally fails to revealthe direct relation between individual dislocation bound-aries and their misorientations because of its limited spatialand angular resolution, it allows us to obtain very large ori-entation data sets which can therefore have a statistical sig-nificance due to the fast and fully automated measurementprocedure. For each deformed state, four maps covering a250 · 250 lm area were measured and concatenated; thestep size was chosen to be 1 lm (although our experimental

Fig. 2. Simple shear sample geometry for a positive or clockwise shear:SD is the shear direction and SPN the shear plane normal.

equipment can go down to 0.2 lm [27]), in order to realizea good compromise between acquisition time and preci-sion. As a consequence, the total examined area in eachsample was then equal to 0.25 mm2 and contained 252004 data points. Of course, this does not allow the studyof misorientations between small adjacent cells, whichwould be better taken into account by TEM, but as thisprocedure allows the gathering of enormous data setswhich can therefore have a statistical significance, it isexpected that the data obtained may nevertheless contrib-ute to the understanding of misorientation developmentwith strain. The sample section examined was the longitu-dinal transverse section (or ND-RD section) for the coldrolled samples and the sheet normal plane (or SD-SPN sec-tion) for the simple sheared samples. Post-processing of theraw data enabled removal of the so-called wild spikes, i.e.,the single isolated points which were misoriented >5� fromthe adjacent points, but no further treatment was applied inorder to fill out these ‘‘empty’’ points. Table 1 presents thevarious fractions of indexed points for each strain level anddeformation mode.

3. Results

3.1. Identification of the grains in the EBSD maps

Some typical EBSD maps obtained after a von Misesstrain of 0.4 in rolling and simple shear are shown inFig. 3. It is first clear that the morphology of the deformedgrains depends on the deformation mode. On these maps,using a widespread practice [11], the threshold value ofthe misorientation angle for grain boundary definitionwas taken equal to 15� in most cases, but where this defini-tion led to evident conglomerations of several grains, grainboundaries were drawn manually, following boundaries ofsmaller misorientation angle. The latter procedure couldalso be aided by the morphology revealed in the imagequality maps. Such a semi-manual technique is often used[26]; it is believed that it allows the researcher to avoid los-ing track of too many original grain boundaries, especiallyin the case of simple shear, for which it is suspected that thehigh involved reorientation rates (see below) can createlarge intragranular misorientations.

Concerning the orientation dependence of deformationmicrostructure development, it is first necessary to considera paradox which is inevitably encountered. Because whatactually governs the difference in deformation behaviorwithin each grain is the full reorientation history, i.e., theorientation path of the grain rather than simply its final

Fig. 3. Typical EBSD maps obtained after rolling (a) and simple shear (b)for a von Mises strain of 0.4 (lines representing misorientations: gray formisorientations >2.5�, black >5�, bold black >15�).

Fig. 4. Representation of the selected orientation groups in the /2 = 45�section.


orientation, the initial orientation of each grain could havea clearer influence on misorientation development than thefinal one [26,28,29]. Nevertheless, there is no experimen-tally reliable way to retrace the initial orientation of a grainfrom its final orientation inside the volume of a polycrystal-line sample, except in some specific cases, where thepresence of a strong initial texture enables an almost one-to-one correspondence to be made between initial and finalorientations [30]. Consequently, there is no other way ofinvestigating the orientation dependence of deformationmicrostructure but to extract average parameters over sev-eral grains of similar final orientations. Additionally, withregard to the final orientation, because of the significantorientation gradient developed inside the grains, a discretedefinition of grain orientation in a highly deformed micro-structure is rather elusive.

Therefore, in order to take these problems into account,some orientation groups were defined with quite a wide tol-

erance angle from ideal orientations, then several grainswere collected into each group and some parameters describ-ing the microstructure were averaged over all the grains ineach group. When the most part of one grain (i.e., >80%of its area) belongs to a given orientation group, the grainis classified into that group. In this way, the average param-eters describing intragranular misorientations could be cor-related with some representative grain orientations.

Four groups of grains were selected in this way andextracted from the original orientation maps, thus exclud-ing completely the grain boundary misorientations. Thecentral ideal orientations of the four investigated groupswere chosen from the representative orientations of theinitial texture in Fig. 1. Two of them belong to the c-fiber– namely, the F = {111}Æ11 2æ and D = {111}Æ110æorientations – and the other two to the a-fiber: one is theRC = {001}Æ110æ orientation and the fourth group iscomposed of all remaining orientations along the a-fiberapart from D and RC. All are also major components ofthe conventional cold rolling texture in low-carbon ferriticsteels. Although three of them decline in simple shear, theycould still be detected in the simple shear texture up to anequivalent strain of 0.40, whereas the F = {111}Æ112æ ori-entation becomes the major component of the shear texturefor all strains. The selected orientation groups are shownschematically in Fig. 4. The three former groups are repre-sented as spherical domains with a maximum spread angleof 12.5� from the central orientations, while the last one is atube domain also with a 12.5� radius from the ideal fiber.The rather large tolerance of 12.5� was chosen in orderto have as many grains as possible within each group(but without overlapping of the groups) and to take intoaccount the difficulties mentioned above.

3.2. Texture evolution

Fig. 5 shows the texture evolution with strain for bothdeformation modes. Throughout this analysis, the Euler

Fig. 5. Evolution of texture with strain in rolling for eVM = 0.17 (a), 0.23 (b) and 0.40 (c) and for simple shear for eVM = 0.17 (d), 0.23 (e) and 0.40 (f).

Fig. 6. The absolute number of misorientations detected on the concat-enated orientation maps (for all misorientation angles h > 2�) as a functionof strain and deformation mode; the map size is always constant(0.25 mm2) and contains 252 004 data points.


angles are always defined from the reference sample sys-tems (RD, TD, ND) for rolling and (SD, SPN, ND) forsimple shear. As the simple shear direction always coin-cides with the rolling direction in the present case, thesetwo reference sample systems indeed coincide. Also, themetallurgical notation of the orientations employed in roll-ing {hk l}Æuv wæ is maintained for the two deformationmodes, which means that {hk l} refers to the Miller indicesof the original rolling plane, while Æuvwæ refers to theMiller indices of the rolling (or shear) direction.

In cold rolling, the texture is always composed of thetwo initial a and c fibers, whose overall intensity firstslightly decreases and then increases with strain (this is trueup to e = 0.78, not shown here for the sake of brevity). Insimple shear, much more substantial reorientation takesplace towards the F = {111}Æ112æ orientation, which ishighly stable in simple shear [21,22,24] and, as strainincreases, the exact position of the maximum in the sheartexture approaches – more and more slowly – the idealposition of the F orientation, as expected [24].

3.3. Intragranular misorientation development

Fig. 6 shows the overall misorientation angle histogramsobtained for three strain levels and for the two deformationmodes with the OIM software. It should be noted thatthese histograms contain both intergranular and intragran-ular misorientations – calculated between all adjacent

points within the complete scanned area for each state –but the latter are far more numerous than the former. Asstrain increases, the strain-induced low-angle peaks becomemuch higher and also broaden toward higher misorienta-tion angles, while the absolute numbers of boundaries asso-ciated with large angles (i.e., >40�) do not vary significantlywith strain; in this part of the diagram, grain boundariesdominate and, as shown in Fig. 7 where the large angle part

Fig. 7. Absolute number of high angle misorientations (h > 20�) extracted from Fig. 6 for: (a) cold rolling and (b) simple shear.


of Fig. 6 is redrawn, it is clear that with increasing strainthe symmetry of the deformation texture induces a peakat large angles (i.e., >50�), which is not observed in the ran-dom texture where the maximum is �40�.

This texture-driven misorientation distribution couldindeed be reproduced with the aid of the viscoplastic Tay-lor model, which is widely employed in crystal mechanics[24,31,32]. The results of these calculations (which take intoaccount two possible slip system families, namely,{11 0}Æ111æ and {112}Æ11 1æ), are presented in Fig. 8 forboth deformation paths. In order to obtain these calculatedmisorientation distributions, an initial set of 1800 grain ori-entations was extracted from the EBSD maps measured onthe deformation-free sample. Then, both plane strain andsimple shear were simulated with the Taylor model andmisorientation angles were calculated for all orientationpairs taken in the final predicted orientation set. The agree-ment between the experimental and predicted profilesclearly indicates a texture effect in the building up of the60� peak; this specific misorientation is indeed foundbetween some different maxima of the ODF, such as, forexample, the ð111Þ½12�1� and ð111Þ½�1�12� orientations whichare simultaneously present in both rolling and shear tex-

Fig. 8. Predicted misorientation angle distribution of grain

tures because of the so-called sample symmetry (orthotro-pic in one case and centro-symmetric in the other).

It is also clear from Fig. 7 that the strain-induced misori-entation development is always stronger in simple shearthan in rolling at the same equivalent strain, consideringboth the peak height and the spread toward high angles.For example, at eVM = 0.23, the angle range of inducedmisorientations reaches 10� in rolling and 13� in simpleshear, whereas at eVM = 0.40, these values become 20� incold rolling and 25� in simple shear when compared tothe angle histogram of the initial deformation-free sample.

In order to study the development of intragranular mis-orientation as a function of grain orientation, some averagemisorientation angles were calculated for each grain. Suchangles have been calculated in two different ways. The firstis the average of the so-called ‘‘correlated’’ misorientationangle (or more simply ‘‘grain average misorientation’’),i.e., the misorientation angle between adjacent pointswithin the grain being studied (as in the case of the histo-gram in Fig. 6). The second way consists in calculatingthe average of the misorientations between a representativegrain orientation (which is calculated as the arithmeticaverage of Rodrigues–Frank orientation vectors) and the

boundaries after cold rolling (a) and simple shear (b).


orientations of all the measured points within a grain.Obviously, the first definition depends on the spatial distri-bution of orientations within the grain, whereas the seconddoes not; for this reason, it is called orientation spread. Thetwo parameters were averaged within each of the fourselected orientation groups to assess the orientation depen-dency. As can be seen in Fig. 9, the former is by definitiondependent on the measurement step size, whereas the latteris not, if the number of points within each grain is suffi-cient. Therefore, for an absolute quantitative estimate,the latter definition could be more appropriate.

Figs. 10 and 11 show the average intragranular misori-entation and orientation spread calculated at variousstrains for the two deformation modes and the four orien-tation groups. It is clear that both values are generally wellproportional to strain, but the orientation spread (Fig. 11)does not show a clear dependence on the grain orientation,unlike the grain average misorientation (Fig. 10). Thus,although the uncorrelated definition works effectively in

Fig. 9. Step size dependence of: (a) grain average misorientation and (b) orientFor each calculated average value, the bar indicates the value of the associate

Fig. 10. Grain average misorientation for the four selected orientation grouorientation group, the dashed line corresponds to the calculation performed forperformed for the orientation associated with the maximum value of the ODF

some cases, such as for discriminating between deformedand recrystallized grains in partially recrystallized steels[33], it does not seem adequate for a detailed analysis ofdeformed microstructure. This is why the average valuewhich takes into account the spatial distribution of theintragranular boundaries is preferentially considered inthe present work.

In Figs. 9–11, the standard deviations have also beencalculated and plotted for each point; the values associatedwith Figs. 10 and 11 are given in Tables 2 and 3. Like theaverage misorientation angles, these standard deviationsincrease with strain and can reach quite large values forthe higher strain values (and especially for the cases inwhich the number of considered grains is quite small).

It is clear from Fig. 10a that, in cold rolling, the twogroups F and D that belong to the c-fiber are associatedwith larger misorientation values than the A and RCgroups associated with the a-fiber; however, the differencebetween F and D or A and RC is quite small. Therefore,

ation spread for the four selected orientation group in rolling (eVM = 0.40).d standard deviation.

ps as a function of strain in (a) rolling and (b) simple shear. For the Fthe exact orientation, whereas the green line corresponds to the calculation.

Fig. 11. Grain orientation spread for the four selected orientation groups as a function of strain in (a) rolling and (b) simple shear (note that for simpleshear at eVM = 0.78, grains belonging to the D group cannot be identified because of a strong texture development and severe fragmentation). For the Forientation group, the dashed line corresponds to the calculation performed for the exact orientation, whereas the green line corresponds to the calculationperformed for the orientation associated with the maximum value of the ODF.

Table 2Standard deviation values of the average misorientation (Fig. 10) calcu-lated for the four orientation groups, the two deformation modes and thesix strain values

Orientation, eVM 0 0.17 0.23 0.29 0.35 0.40 0.78

(a) Cold rolling

A: a-fiber 0.045 0.169 0.203 0.320 0.306 0.355 0.343RC: {001}Æ110æ 0.052 0.127 0.184 0.228 0.287 0.207 0.370D: {111}Æ110æ 0.045 0.159 0.280 0.440 0.600 0.651 0.876F: {111}Æ112æ 0.041 0.202 0.256 0.435 0.626 0.586 0.652

(b) Simple shear

A: a-fiber 0.045 0.272 0.453 0.435 0.548 0.417 1.560RC: {001}Æ110æ 0.052 0.132 0.335 0.306 0.349 0.551D: {111}Æ110æ 0.045 0.342 0.402 0.599 0.414 0.924F: {111}Æ112æ 0.041 0.255 0.201 0.934 0.420 0.457 0.514Maximum ODF 0.678

Table 3Standard deviation values of the orientation spread (Fig. 11) calculated forthe four orientation groups, the two deformation modes and the six strainvalues

Orientation, eVM 0 0.17 0.23 0.29 0.35 0.40 0.78

(a) Cold rolling

A: a-fiber 0.232 0.705 0.736 0.978 1.005 2.113 2.014RC: {001}Æ110æ 0.212 0.461 1.026 1.141 1.183 1.477 1.603D: {111}Æ110æ 0.206 0.660 0.810 1.054 1.106 1.610 1.253F: {111}Æ112æ 0.261 0.732 0.977 1.010 1.427 1.344 2.589

(b) Simple shear

A: a-fiber 0.232 0.986 1.841 1.396 2.936 1.326 2.217RC: {001}Æ110æ 0.212 0.631 2.590 1.247 0.968 1.715D: {111}Æ110æ 0.206 2.362 2.213 2.211 0.844 2.010F: {111}Æ112æ 0.261 0.552 0.975 1.224 1.497 1.427 1.742Maximum ODF 2.744


for cold rolling, the grains can simply be classified into twogroups, namely, the a and c fibers associated respectivelywith small and large intragranular misorientations. Thisis completely consistent with the analysis of Regle [34],

which distinguishes between the elongated a grains andthe fragmented c grains.

In simple shear, the situation is quite different from coldrolling: the F group, which is one of the two that have largemisorientation values in rolling, is now associated with thesmallest value, while the D group is associated with thelargest value. The two a components are in between thosetwo. Also, we can see in Fig. 10 that the average misorien-tation of each group is always larger in simple shear than inrolling except at eVM = 0.78; for this deformation strain,many grains display higher levels of fragmentation thanthe selected tolerance value of 15� and can no longer beassociated with one of the few selected orientation groups(the numbers of grains and area fractions associated witheach level of strain and each orientation group in Figs.10 and 11 are given in Table 4). As a result, from theseselected orientation groups, there remains only the stableF orientation, the other orientations almost disappearingcompletely in the shear texture. But we can still concludethat for this strain level too, the average misorientationwithin grains is larger in simple shear than in rolling.

It should also be noted that, for the F orientation group,two different calculations are presented for eVM = 0.78: thedotted lines in Figs. 10b and 11b are associated with thestrict definition of the F orientation. But, in this case, only7% of the sample is considered in the analysis (see Table 4)because of the wide spread present in the grains and theselected procedure. So, in order to improve the statisticsslightly, the strict definition of the F orientation (associatedwith the Euler angles 30, 55, 45) was replaced by the orien-tation of the ODF maximum (slightly shifted from the Forientation) and the calculations of grain average misorien-tation and orientation spread were repeated for this modi-fied F group. The corresponding results are the greencontinuous lines in Figs. 10b and 11b. As shown in Table1, the considered percentage of the material rises to 25%and the average misorientation slightly increases in thisgroup (from 2.44 to 2.92 – see Fig. 10).

Table 4Number of grains (area fraction in %) taken into account in each orientation group for the investigated six strain levels and two deformation modes

Orientation, eVM 0.17 0.23 0.29 0.35 0.40 0.78

(a) Cold rolling

A: a-fiber 75 (15.45) 44 (11.97) 62 (14.80) 55 (14.18) 37 (11.91) 32 (12.24)RC: {001}Æ110æ 16 (3.63) 25 (4.20) 17 (2.37) 14 (2.85) 20 (5.11) 14 (3.44)D: {111}Æ110æ 47 (11.72) 38 (17.44) 42 (11.98) 39 (10.35) 23 (11.38) 26 (12.03)F: {111}Æ112æ 40 (12.95) 42 (15.54) 53 (12.83) 46 (14.49) 26 (10.26) 20 (8.07)Total 178 (43.75) 149 (49.15) 174 (41.98) 106 (38.66) 92 (35.78)

(b) Simple shear

A: a-fiber 18 (5.65) 19 (4.41) 13 (2.41) 9 (2.01) 9 (0.73) 3 (0.49)RC: {001}Æ110æ 4 (2.03) 5 (1.44) 12 (1.73) 3 (2.85) 7 (0.91)D: {111}Æ110æ 10 (2.42) 14 (4.21) 7 (1.30) 5 (1.55) 10 (2.97)F: {111}Æ112æ 27 (16.39) 26 (14.49) 38 (8.24) 24 (10.09) 37 (11.07) 15 (6.93)Maximum ODF 74 (24.39)Total (with F) 59 (26.49) 63 (15.68) 19 (7.87)Total (with max. ODF) 78 (25.33)

For simple shear at eVM = 0.78, the definition of the F group was changed: the F orientation was replaced by the orientation corresponding to themaximum of the ODF.


Fig. 12 now shows the distribution of the intragranu-lar misorientation axes expressed in the sample frame foreach deformation mode and each orientation group ateVM = 0.40. It is noteworthy that the rotation axes of mis-orientations are mostly concentrated near TD in rollingand ND in simple shear, irrespective of the grain orienta-tions and also the strain level. In simple shear, the generalconcentration at ND is stronger than the concentration atTD in rolling. The degree of concentration and the shapesof distributions are somewhat dependent on the grain ori-entations and especially characteristic in the D group. Inrolling, the most dispersed nature of distribution is visiblein the D group, while it shows the strongest concentratingbehavior in simple shear.

Fig. 12. Misorientation (h > 2�) axes distributions in (a) rolling and

4. Discussion

It is first worth noting that the present data are consis-tent with previous works dealing with rolling or simpleshear. For example, Nave and Barnett [26] observed thesame trend for the difference in average misorientationbetween a and c grains, i.e., more fragmentation in the lat-ter during rolling. They also mention a strong tendency forthe misorientation axis to be aligned along TD. Similarly, arecent TEM investigation of dislocation microstructures inan IF steel deformed in simple shear showed quite clearlythat a strong degree of fragmentation, associated withincreasing misorientations, was present within the mostunstable orientations for intermediate levels of strain [21].

(b) simple shear for eVM = 0.40 expressed in the sample frame.


4.1. Degree of misorientation development associated with

orientation instability

Various explanations have been provided to explain thelink between the features of the dislocation microstruc-tures, the degree of intragranular fragmentation and theorientation of the grain. They usually mention the degreeof slip resistance, or the sum of shears required to attaina given deformation, i.e., the Taylor factor [25,35–37]. Ahigher Taylor factor would mean a higher dislocation den-sity and thus larger intragranular misorientations whenconsidering only the statistical accumulation of disloca-tions. But concerning the misorientation due to dislocationwall organization, the Taylor factor may not explain theirdevelopment satisfactorily because it may also include theamount of homogeneous slips, or plastic dissipation ratherthan dislocation accumulation due to wall organization. Itis well known that the Taylor factor is higher for c than fora orientations in rolling and usually smaller in simple shearthan in rolling, due to a smaller number of activated sys-tems. If this explanation can thus more or less ‘‘satisfacto-rily’’ account for the higher degree of fragmentation in thec grains in rolling, it should lead to a lesser degree of frag-mentation in simple shear, due to the reduced number ofactive slip systems. This could explain the data obtainedin the stable orientations at very large shear strains, butnot the situation observed at intermediate strain levels inwhich the average misorientation is quite high in the unsta-ble grains (in fact, the Taylor factor calculated in simpleshear takes exactly the same value for both F and D orien-tations). In addition, Dillamore et al. [38] expected that theorientation metastability and the rotation of orientationsapart from a metastable one would affect the developmentof transition bands which are a prominent source of largeintragranular misorientation. Here, the instability of thetexture must also be explored as a possible source of mis-orientation increase.

As already shown in Fig. 5, the texture evolution is morepronounced for the given initial texture in simple shearthan in rolling. The higher degree of lattice reorientation,therefore, can be regarded as a reason for more pro-nounced misorientation development in simple shear, asshown in Figs. 6 and 10. It can thus be of interest to esti-mate the rate of lattice reorientation in relation to the

Fig. 13. Lattice spin norm map (in the /2 = 45� section) in: (a) rolling (contou2.0–2.5); the lowest-value region is shaded.

observed misorientation development. In order to do so,the viscoplastic Taylor model has again been used, by tak-ing into account both {110}Æ1 11æ and {112}Æ111æ slip sys-tems with the critical resolved shear stress ratio of the twosystems set to 1. With this model, it is possible to calculatefor any orientation a lattice spin tensor, defined as the dif-ference between the macroscopic rigid body spin and theplastic spin due to slip; this spin tensor is an anti-symmet-rical tensor which is thus composed of three terms usuallydefined as x1, x2 and x3 which represent the lattice rota-tion rates around the 1, 2 and 3 axes of the reference frame,respectively [20]. From this tensor, we then calculate thenorm of the lattice spin tensor |X(g,D)|, which dependson the orientation g of the considered grain and theimposed plastic strain rate tensor D, and which is definedas

jXðg;DÞj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

1 þ x22 þ x2

3

qð2Þ

This parameter is minimum for a stable orientation andmaximum for an unstable one [21,35].

Fig. 13 shows the variation of this parameter as a func-tion of orientation for rolling and simple shear. If thesemaps are superimposed on the initial ODF of the material(Fig. 1), it is clear that this initial texture falls mostly in thelow value region for rolling, whereas it corresponds to bothhigh and low value regions for simple shear. Also, it is clearthat the lattice spin is larger in simple shear than in rollingbecause of the macroscopic rigid body rotation involved insimple shear. Therefore, it can be said that the generalprevalence of misorientation development in simple shearwhich is shown in Fig. 6 may have a significant correlationwith texture evolution rate. Fig. 14a shows the averagenorm of the spin tensor for each of the selected orientationgroups organized in ascending order, whereas Fig. 14b pre-sents the corresponding grain average misorientation val-ues calculated for eVM = 0.4 (already presented in Fig. 6).It can thus be seen that there is a strong correlationbetween the two parameters. It can also be seen that, inrolling, only a and c grains can be distinguished from eachother, whereas in simple shear the four groups are associ-ated with four distinct values of both misorientation andlattice spin. We can thus conclude that, at least duringthe process of reorientation, the grains which correspond

r levels 0.15–0.5–1.0–1.5) and (b) simple shear (contour levels 0.5–1.0–1.5–

Fig. 14. (a) Average lattice spin norm and (b) grain average misorientation (at eVM = 0.40) for the four orientation groups in rolling and simple shear.


to a large lattice spin experience a progressive reorientationtowards stable orientations, and a stronger orientation gra-dient is consequently developed within these grains.

4.2. Distribution of misorientation axis associated with

reorientation axis

If the misorientation development is significantlyaffected by the lattice rotation towards stable orientations,it can be further expected that the distributions of misori-entation axes also have some coincidence with those of lat-tice reorientation axes. In order to verify this hypothesis,the reorientation axis distribution was derived again withthe help of the Taylor model. For orientations regularlyspaced within each orientation group, one strain increment(in rolling or simple shear) was applied, then, from the mis-orientations between these initial and rotated orientationpairs, the distribution of reorientation axes could be plot-

Fig. 15. Predicted reorientation axes distribution in (a) cold ro

ted in the sample reference system. The concordancebetween Figs. 13 and 15 is striking, although the case ofthe RC orientation is not completely reproduced. In partic-ular, for this orientation, reorientation takes place aroundthe TD or the ND axis for rolling, whereas only the TDmisorientation axis is seen in Fig. 13. This exception isnot surprising since this orientation is highly symmetrical,resulting in a fragmentation process, both in rolling andsimple shear [21], which cannot be reproduced by a simplemodel such as the Taylor model, which assumes the homo-geneity of the deformation.

It is also worth noting that the stronger concentration ofthe reorientation rotation rates in simple shear can beexplained by the smaller number of active slip systems.Therefore, considering both the angle and the axis aspect,it can be concluded that within the strain range of the cur-rent study up to eVM = 0.4, the characteristic misorienta-tion development actually reflects the evolution of the

lling and (b) simple shear for the four orientation groups.


texture towards the stable configuration. However, itshould be noted that this behavior may no longer hold atlarge strain levels where the overall texture evolution satu-rates; in this case, some dispersion of the misorientationaxes is usually observed.

5. Conclusions

Intragranular misorientation characteristics were ana-lyzed for specific orientations (representative of the macro-scopic texture of the material) in an IF steel, after rollingand simple shear; the principal results of the present inves-tigation are the following.

(1) The average intragranular misorientation angle cal-culated for the selected orientations was found tobe higher in simple shear than in rolling, except forvery large strains, for which the highly dispersedgrains were no longer included in the analysis.

(2) These average misorientation angles have been foundto be proportional to the lattice rotation rate due totexture evolution for all investigated orientationgroups and for both deformation modes.

(3) The distribution of misorientation axes was very simi-lar to the distribution of reorientation axes due totexture evolution for a wide strain range; the intragran-ular misorientation development thus reflects the tex-ture evolution, at least up to a strain level of 0.4.

Therefore, at least when there is significant texture evo-lution, it can be said that this texture development plays adominant role in misorientation accumulation of disloca-tion boundaries. This investigation should now be com-pleted by examination of the highly fragmented grainsfor both deformation paths; this is, however, a complextask since in this case many EBSD patterns fail to beindexed.

Acknowledgements

The authors acknowledge J.-P. Fondere and J.-L.Dournaux for the rolling and the simple shear tests, andT. Chauveau for X-ray pole figure measurements.

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