Stacking fault energy and microstructure effects on...

Stacking fault energy and microstructure ereg ectson torsion texture evolution

By Darcy A Hughes1 R icardo A Lebensohn2Han s Rudolf W e n k3 a n d Ashish K uma r4

1Center for Materials and Applied Mechanics Sandia National LaboratoriesLivermore CA 94550 USA

2Instituto de Fisica Rosario (CONICET-UNR) 2000 Rosario Argentina3Department of Geology University of California Berkeley CA 94720 USA

4Division of Engineering Brown University Providence RI 02912 USA

Received 28 January 1999 accepted 28 April 1999

A series of experiments and simulations that vary the texture and microstructuresimultaneously are used to establish the role of the microstructure in texture forma-tion in FCC metals The stacking fault energy (SFE) of the metal which is knownto have a strong impact on texture and microstructure is the vital parameter usedto make these variations It was determined that the wide variety of textures andmicrostructures observed as a function of SFE and temperature was developed byslip processes alone twinning was not necessary as previously thought The dinoterenttextures are caused by (i) variations in local slip patterns within a single grain asrevealed by grain subdivision into dinoterently deforming cell blocks and (ii) moresubtly by the cell-block shape The local selection of slip systems creating the lat-tice rotations within a cell block is altered by the planarity of slip Slip planarityis controlled by the SFE and temperature It is hypothesized that the new texturecomponents that are distinct from the generally accepted ideal components arecreated by the dinoterent slip processes occurring as a result of low SFE and low tem-perature A more subtle enotect of grain subdivision is related to the cell-block shapesthat develop as a function of SFE and temperature and correspond to the dinoterenttextures observed The shape of the cell block is related to the level of constraintrequired by the deformation The slip pattern changes and cell-block shapes correlatewith the presence or absence of certain ideal texture components whose evolutionis not simulated Materials and conditions with similar deformation microstructuresdeveloped similar textures in the experiments

Keywords texture microstructure twinning dislocationsstacking fault energy torsion

1 Introduction

The formation of a preferred crystallographic texture during deformation has beenstudied extensively in the past through both experimental measurement and com-puter simulation (see for example Williams 1962 van Houtte amp Aernoudt 1976 GilSevillano et al 1980 Sekine et al 1981 Hecker amp Stout 1982 Canova et al 1984Montheillet et al 1984 Stout et al 1988 Harren et al 1989 Stout amp OrsquoRourke1989 Toth et al 1989) Signishy cant advances in texture simulation have occurred

Proc R Soc Lond A (2000) 456 921953

921

creg 2000 The Royal Society

922 D A Hughes and others

in the last 20 years (Kocks et al 1998) including variations of the Taylor model(Taylor 1938) eg the full-constraints (FC) model (van Houtte amp Aernoudt 1976Canova et al 1984 Harren et al 1989 Toth et al 1989) the relaxed-constraints(RC) model (Honnenot amp Mecking 1981 Canova et al 1984) the self-consistent model(Molinari et al 1987 Tome amp Canova 1998 Lebensohn amp Tome 1993) and shy nite-element based schemes (eg Becker 1991 Bronkhorst et al 1992 Beaudoin et al 1995) However discrepancies still exist between the measured texture evolution andthe simulation results (Stout et al 1988) Notable among these dinoterences are thepresence or absence of certain ideal components dinoterent ratios of ideal texture com-ponents as a function of strain a much slower texture evolution in experiment thanmodel and the spread of orientations around the ideal components that is relatedto the intensity of a preferred orientation These dinoterences have remained over theyears despite the introduction of dinoterent types of models In these models withthe exception of some shy nite-element (eg Beaudoin et al 1996 Becker 1995) andn-site self-consistent approaches (see for example Canova et al 1992) the slip pat-tern is modelled across a whole grain and the microstructural changes relate onlyto grain shape and possibly to twinning Consequently a radical new approachmay be required to resolve these issues One such approach may be to consider theevolution of the deformation-induced dislocation microstructures which takes placesimultaneously with the texture evolution This approach is introduced because thedislocation structure both reregects and modishy es the slip pattern thereby altering thetexture development

Before the microstructure can be included in a texture simulation however therole of the microstructure must be established by a simultaneous and careful consid-eration of the measured texture texture simulations and quantitative observations ofthe dislocation structure It is the aim of this paper to make this connection througha series of experiments and simulations that varies the texture and microstructuresimultaneously The vital parameter of choice to make these variations is the stack-ing fault energy (SFE) of the metal which is known to have a strong impact ontexture and microstructure For the dinoterent SFEs the range of possible texture andmicrostructure combinations is further increased by changing the temperature

To further clarify the impact of the SFE on texture and microstructures torsiondeformation is examined In torsion with its dyadic symmetry (only one twofoldsymmetry axis) the enotect of low SFE on the texture development is especially evidentdue to the formation of one special texture component (111)[1middot12] that is generallyeither lacking or very weak in the texture developed by the high-SFE metals (vanHoutte amp Aernoudt 1976 Gil Sevillano et al 1980) Torsion has other advantagesover rolling in trying to ascertain the various roles of the slip pattern and twinningStress-induced deformation twinning is postponed to larger strains in torsion sinceregow stresses are lower in torsion thereby allowing the texture to form by slip priorto twinning Shear banding is very minimal in torsion but would add complicationsto a similar analysis of texture during rolling

The outline of the paper is as follows First the experimental textures are pre-sented as a function of SFE and increasing strain for pure nickel and nickelcobaltsolid solutions Next the simulated textures are described including the assump-tions and parameters used to vary the texture in the simulations Both Taylor andself-consistent models are used A step-by-step comparison is made with experimentto see what can and cannot be explained currently by the simulations Finally the

Proc R Soc Lond A (2000)

Stacking fault energy and microstructure ereg ects 923

Table 1 Materials and deformation conditions

stacking initial von Misesfault recrystallized shear equivalent

energya grain sizeb starting strain strain temperaturematerial (mJ m iexcl 2 ) ( m) texture reg z sup3 vM (K)

nickel (9999) 240 80 random 20 12 296

36 21

69 40

37 21 573

Ni + 30 wt Co 150 53 random 19 11 296

38 22

83 48

Ni + 60 wt Co 20 44 random 20 12 296

32 18

58 36

38 22 573

a Beeston et al (1968)b Heyn intercept distance

microstructures are described and discussed with respect to their inreguence on tex-ture formation Brief summaries and explanations are used in the text to aid thereader and maintain a connection between these diverse areas

2 Experimental procedures

Three metals with a wide range of SFEs were studied high-purity nickel (9999)Ni + 30 wt Co and Ni + 60 wt Co (see table 1) Cobalt additions to nickel sys-tematically lower the SFE (Beeston et al 1968) and consequently change thedeformation behaviour with increasing cobalt concentration Other contributions tobehaviour changes such as solid solution hardening are minimized because nickeland cobalt have very similar atomic sizes elastic moduli and melting temperaturesConstant strain rate torsion tests reg zsup3 = 10iexcl3 siexcl1 were performed at room tempera-ture and 573 K using short thin-walled tube samples the deformation conditions areoutlined in detail in Hughes amp Nix (1989) All samples were recrystallized prior totesting It was ascertained from pole shy gure measurements that the starting texturewas random The starting material parameters and deformation conditions are listedin table 1 for the three materials

Texture and microscopy samples were prepared from the starting materials andfrom the torsion samples following deformation Optical metallography showed thatthese materials initially had equiaxed medium-large sized grains (table 1) Addition-ally recrystallized grains in both Ni + 30 wt Co and Ni + 60 wt Co containedannealing twins whereas pure nickel had very few annealing twins consistent withtheir SFEs (Hughes amp Nix 1988)

Texture samples were made from the ring-shaped gauge section of torsion sam-ples which were cut into segments unrolled assembled to increase the surface areamechanically polished and electropolished The surface of the texture samples cor-responds to the zsup3 -plane of the torsion sample Throughout the text z refers to the



Table 2 Ideal texture components for FCC torsion (see also macrgures 1h and 2h)

crystal orientation

label fhklghuvwia

Acurren1 (111)[2sup11sup11]

Acurren2 (sup1111)[211]

A f111gh1sup110iB f112gh1sup110iC f001gh1sup110i

f111g macrbre f111ghuvwih110i macrbre fhklgh110i

a The designation fhklghuvwi refers to the shear plane and shear direction respectively fortorsion

shear plane normal sup3 to the maximum shear direction and r to the radial direc-tion of the torsion sample Conventional X-ray techniques in reregection geometrywere used to measure incomplete (111) (200) and (220) pole shy gures Fe K not radia-tion was used to avoid reguorescence from cobalt The intensity data were correctedfor background and defocusing The orientation distribution functions (ODFs) werecalculated from incomplete pole shy gures using WIMV (WilliamsImhofMatthiesVinel) implemented in Beartex (Wenk et al 1998) Crystal ODFs are represented inRodrigues space (Frank 1988) Volume fractions of the various ideal texture compo-nents (shown in table 2) were calculated from the ODFs using a three-dimensionalangular spread of 15macr about an ideal component These ideal texture componentsare labelled according to the nomenclature of Montheillet et al (1984)

Transmission electron microscopy (TEM) analysis was performed on samples madein the zsup3 -plane which contains the maximum shear strain Orientations of individ-ual crystallites were obtained in the TEM from convergent-beam Kikuchi patternsThe Kikuchi patterns were analysed using a computer method based on Young etal (1973) and Liu (1994) to obtain orientation matrices for individual crystallitesThe minimum angle misorientation relationship (disorientation) between adjacentcrystallites separated by dislocation boundaries was calculated by considering all24 symmetry operations for the orientation matrices in a standard manner Theangleaxis pairs for the disorientations were also calculated A negative or positivedisorientation angle was assigned by considering whether the disorientation axis isin a left-hand or right-hand triangle respectively

3 Measured texture results

In the following section the experimental textures are quantishy ed by using ODFsThe ODF relates the orientation of the crystallographic axes to the macroscopicsample axes and is plotted in orientation space (ie Euler space) For symmetriccrystals such as the current cubic case the whole of orientation space containsmuch redundant information As a result crystal symmetry is frequently used toreduce the orientation space to a fundamental region in which these redundancies areremoved (or nearly removed) Herein instead of the Euler angle spaces conventionallyemployed in texture analysis Rodrigues space is used The advantages and use ofRodrigues space have been discussed by various authors (see for example Frank



1988 Becker amp Panchanadeeswaran 1989 Kumar amp Dawson 1995) For exampletexture comparisons are particularly convenient because the inherent symmetry ofthe space relates the axes of the space directly to the sample axes Also most idealcomponents of the torsion texture lie either on or close to the surface (or boundary)of the cubic fundamental region As a result the structures of the textures developedcan be inferred from single plots of the ODF on the surface of the fundamental regionBecause the ideal components and shy bre textures created by the deformation lie onor near the surface these presentations of the ODF can be readily interpreted likepole shy gures In contrast to pole shy gures the three-dimensional information is moredistinctly retained Note however that some minor components and recrystallizationcomponents may lie inside the fundamental region The absence or presence of thesecomponents should be checked by cross-sectioning the space Cross-sectioning in thepresent case did not reveal any minor components lying inside the space Thus thesurface plots of the cubic fundamental region are representative

Figure 1 displays complete f111g pole shy gures recalculated from the ODF for allsamples The maximum pole density observed is 326 multiples of a random distribu-tion whereas the minimum is 023 (The pole shy gure for nickel at the smallest strainshows several spurious maxima They are attributed to regions of rather coarse grainsize and small sample dimensions) All textures display a statistical monoclinic sym-metry with a dyad parallel to the radial direction in accordance with the torsiondeformation geometry The pole shy gures illustrate considerable variation in texturesconsistent with changes in SFE (cobalt content) temperature and strain Thesechanges will be described with respect to the ideal orientations

The ideal orientations for torsion texture development listed in table 2 have beenplotted in both a pole (shy gure 1) and in Rodrigues space (shy gure 2) to aid the readerDinoterent ideal torsion textures include the development of orientations with theslip plane parallel to the shear plane along the f111ghuvwi shy bre that includes thecomponents A f111gh1middot10i A curren

1 (111)[2middot1middot1] and A curren2 (middot111)[211] This f111g shy bre may

be present as only a partial shy bre in which orientations near either Acurren1 or A curren

2 aremissing thus creating the monoclinic dyadic symmetry of shy gure 1a d e il A secondcommon shy bre forms in which the slip direction h110i is parallel to the shear directionOrientations in this shy bre include the B f112gh1middot10i C f001gh1middot10i and A f111gh1middot10icomponents Note that A is common to both f111g and h110i shy bres Additionally theC and A curren

1 and A curren2 components are linked together through the macroscopic spin in the

radial h110i direction Rodrigues space is particularly useful in distinctly separatingthese various components many of which blur together on a pole shy gure

The ODFs for the three materials (SFEs) with increasing values of strain andtemperature are plotted in shy gure 2 Figure 2 shows that well-formed torsion textureshave already developed even at the lowest strain for all of the materials at bothtemperatures The relative volume fractions Vf (shy gure 3) of orientations associatedwith the ideal components depends on the material (SFE) and temperature andevolves with strain

For the high-to-medium SFE tests conducted at 296 K there is a close similarity inthe texture development for the nickel and Ni + 30 wt Co (table 3 and shy gure 2ag)The only dinoterence is the consistently higher intensities and volume fractions ofthe A and B components in nickel compared with Ni + 30 wt Co At all of thestrain levels measured the volume fractions of orientations are fairly well dividedbetween the A B and C components in nickel and Ni + 30 wt Co (shy gure 3) (The



e = 21(l)

e = 36(k)e = 48(g)e = 40(c)

e = 18( j )e = 22( f )e = 21(b)

e = 12(i)e = 11(e)e = 12Ni + 60 wt CoNi + 30 wt Conickel296 K

537 K

(a)

e = 21(d )

(h)A A1

A1

A2

A2

A2

A1

A1

A1A2

Z

q

A2

A1 A2

A

C

C

CB

B

B

B

B

B

A

A AA

AA B

CC

C

(111)

Figure 1 Experimental f111g pole macrgures in equal area projections showing the texturetransition with increasing strain decreasing SFE and increasing temperature (a) Nickel296 K vM = 12 (b) nickel 296 K vM = 21 (c) nickel 296 K vM = 40 (d) nickel573 K vM = 21 (e) Ni + 30 wt Co 296 K vM = 11 (f) Ni + 30 wt Co 296 KvM = 22 (g) Ni + 30 wt Co 296 K vM = 48 (h) The locations of the ideal texturecomponents are labelled according to the designations in table 2 (i) Ni + 60 wt Co 296 KvM = 12 (j) Ni + 60 wt Co 296 K vM = 18 (k) Ni + 60 wt Co 296 K vM = 36(l) Ni + 60 wt Co 573 K vM = 21



(a)

296 K

nickel Ni + 30 wt Co Ni + 60 wt Co

537 K

e = 12 e = 11 e = 12

e = 21 e = 22 e = 18

e = 40 e = 48 e = 36

e = 21 e = 21

70

z

q

r

55

40

25

10

(b)

(c)

(d )

(e)

( f )

(g)

(h)A1

A2

A2

B2

B1

C

(i)

( j)

(k)

(l)

A1

Figure 2 Some experimental ODFs plotted in Rodrigues space showing the texture transi-tion with increasing strain decreasing SFE and increasing temperature (a) Nickel 296 KvM = 12 (b) nickel 296 K vM = 21 (c) nickel 296 K vM = 40 (d) nickel 573 KvM = 21 (e) Ni + 30 wt Co 296 K vM = 11 (f) Ni + 30 wt Co 296 K vM = 22(g) Ni + 30 wt Co 296 K vM = 48 (h) The locations of the ideal texture components in thefundamental region are labelled according to the designations in table 2 Note that the angulardistance of the dashed line from corner to corner is 90macr (i) Ni + 60 wt Co 296 K vM = 12(j) Ni + 60 wt Co 296 K v M = 18 Note that the intensities that peak around the bottomcorner of the left face are part of the macrbre running between Acurren

2 and B (k) Ni + 60 wt Co296 K vM = 36 (l) Ni + 60 wt Co 573 K vM = 21



(b)

(a)

0

0

004

008

012

Vf

randomlevel

nickelC component

Ni + 30 wt CoNi + 60 wt Co

016

012

008

004

Vf

random level

nickelA component B component


nickelNi + 30 wt CoNi + 60 wt Co

vMe

Figure 3 Volume fractions of the texture components with increasing strain for the threematerials (a) A and B components (b) C component

special A curren components will be discussed at the end of this section) While the A andB components are the stronger components from vM = 1222 in terms of bothvolume fractions (shy gure 3) and peak intensities (shy gure 2) the C component becomesthe strongest at vM = 4 Both the B and C components increase with increasingstrain (shy gure 3a b) albeit this increase is much stronger for the C component Asshown in the ODFs orientations are strongly spread in a short shy bre h110i from Ctowards B at all strain levels

Overall except for the A and B components the texture development for the low-SFE Ni + 60 wt Co at 296 K is very dinoterent A very weak C component formsand disappears below random intensity with increasing strain Uniquely the low-SFE Ni + 60 wt Co at 296 K develops a strong partial shy bre between the Acurren

1 B andA components that increases with increasing strain as shown by the peak intensi-



004

008

012

(c)

5

vMe

Vf

0 4321

randomlevel

nickelA2ndash

A1ndash



Figure 3 (Cont) (c) Acurren1 and Acurren

2 components

Table 3 Texture types observed in experiment as a function of materialSFEand temperature at intermediate-to-large strain

intermediate strain large strain

material and major ideal macrbre major ideal macrbretemperature components textures components textures

573 K nickel andNi + 60 wt Co

A B C Acurren1 Acurren

2 completef111ghuvwifhklgh110i




A B C Acurren1 partial

f111ghuvwifhklgh110i

C partialfhklgh110i

296 KNi + 60 wt Co

A B Acurren1 partial

A2 B2 Acurren1 A1

macrbres


A2 B2 Acurren1 A1

macrbres

ties (shy gure 2ik) and volume fractions (shy gure 3) Note importantly that the peakintensities along this shy bre are located 1520macr away from the exact location of theideal components This low-SFE texture may be better described by a new idealcomponent fmiddot545gh652i

At the higher temperature of 573 K all of the ODFs are similar and independent ofeither high or low SFE eg nickel (shy gure 2d) and Ni + 60 wt Co (shy gure 2l) TheseODFs are characterized by long and wide shy bre texture of moderate intensity thatcan be seen to wrap around the cubic fundamental region This long shy bre textureencompasses a uniform distribution of orientations connecting all of the A B and Ccomponents along the h110i shy bre to the A curren

1 Acurren2 and A along the f111g shy bre This

full shy bre texture is in contrast to the dominance of a partial h110i shy bre strongly



centred at C for the lower temperature 296 K and large strain ODFs of nickel andNi + 30 wt Co (shy gure 2b f)

While both A curren1 and A curren

2 components are observed at 573 K regardless of SFE thepresence of either A curren

1 or Acurren2 depends on SFE at the lower temperature of 296 K

For high SFE only a moderately weak A curren2 component is observed at low strain

which disappears at strains above e = 12 (shy gures 1a e and 2a e) For low SFE andtemperature no A curren

2 is observed (shy gure 2ik) instead a moderate A curren1 is present at

all strainsTable 3 summarizes the dinoterent texture results as functions of material SFE and

temperature Note the very large dinoterence in texture components formed in thelow-SFE metal at low temperature compared with medium-to-high SFE

4 Polycrystal simulation

Deformation-induced texture development has been frequently explored by usingmodels for polycrystal behaviour The important factors of SFE and temperaturegenerally enter into these models only as factors that activate twinning and changethe hardening response Two of these models will be considered herein as examples tohelp explain the experimental observations as well as the dinoterences between simula-tions and experiment an FC Taylor model and the self-consistent model (Lebensohnamp Tome 1993) These two models like all models of polycrystal behaviour are basedon deriving polycrystal response from the collective response of a representative dis-crete aggregate of single crystals Two components comprise a typical polycrystalmodel a model for the mechanical response of individual crystals and a homogeniz-ing hypothesis to link the response of individual crystals to that of the polycrystalThe single-crystal model determines the way in which the applied deformation ispartitioned among the participating accommodation mechanisms such as elasticitydislocation glide (slip) and twinning This model then determines the stress devel-oped in the single crystal as a consequence of this partitioning as well as its latticerotations That lattice rotation leads to the development of a preferred orientationof the crystal with increasing deformation The evolving orientations of all the crys-tals provide the sample texture development The homogenizing hypothesis on theother hand relates the stress and strain associated with individual crystals to thatof the polycrystal by appropriate averaging Hypotheses range from the Taylor FChypothesis which assumes that individual crystals experience the polycrystal defor-mation identically to the Sachs hypothesis which assumes that individual crystalstresses are equal to the polycrystal stress The self-consistent hypothesis performsthe homogenizing by embedding crystals in a homogeneous enotective medium (HEM)representing the average properties of the polycrystal

(a) Simulation results

In this section we present and discuss predictions of torsion textures in FCCmaterials using Lebensohn amp Tomersquos (1993) implementation of the viscoplastic self-consistent (VPSC) model as well as some Taylor FC results Details of the simulationmethod are given in Appendix A All the simulations shown here were performedusing an initial random texture of 1000 grains The active deformation mechanismsconsidered were the f111gh110i slip and the f111gh112i twinning (the latter has



150

118

85

52

20

= 40 z

qe= 40e= 40e

= 10

TaylorFC

n = 19

self-consistentfull grain shape

evolution n = 47


evolution n = 19e= 10e= 10e

r

(h)(e)(b)

= 40e= 40e= 40e

(i)( f )(c)

(g)(d)(a)

Figure 4 Simulated ODFs plotted in Rodrigues space and f111g pole macrgures in equal areaprojections for the case of deformation by slip only (a) VPSC model with grain-shape updatinginverse rate sensitivity n = 19 vM = 1 (b) same as (a) with vM = 4 (c) same as (b) (d) VPSCmodel with grain-shape updating n = 47 vM = 1 (e) same as (d) with vM = 4 (f) same as(d) (g) FC Taylor model n = 19 vM = 1 (h) same as (g) with vM = 4 (i) same as (h)

associated with it a characteristic twin shear of 0707) For those simulations thatincluded twinning the initial critical stresses of both slip and twinning were arbi-trarily assumed to be equal (10 arbitrary units) The rules governing the evolutionof the critical stresses with deformation are given in Appendix A A total strainof vM = 40 was imposed in incremental steps of vM = 001 Both the shy nal andcertain interesting intermediate textures are presented and discussed Some of thesetextures are shown as (111) pole shy gures but for most of them we rely on the moretransparent Rodrigues representation

Dinoterent aspects of texture formation were investigated using the Taylor FC andthe VPSC models namely

(a) the presence or absence of twinning

(b) the enotect of rate sensitivity



150

= 40

118

85

52

20

(h)

(i)

(e)

( f )

(b)

(g)(d )(a)

(c)

z

qe= 40e= 40e

= 25e= 25e= 25e

= 25e= 25e= 25

TaylorFC

n = 19

self-consistentlimited grain shapeevolution n = 19


evolution n = 19e

r

Figure 5 Simulated ODFs plotted in Rodrigues space and f111g pole macrgures in equal area pro-jections for the case of deformation by slip and twinning In all cases the deformation consistedof an early stage (up to vM = 1) of only slip and a late stage of slip plus twinning (a) VPSCmodel with grain-shape updating inverse rate sensitivity n = 19 htw = 01 vM = 25 (b) sameas (a) (c) same as (a) for vM = 4 (d) VPSC model without grain-shape updating n = 19htw = 01 vM = 25 (e) same as (d) (f ) same as (d) for vM = 4 (g) FC Taylor model n = 19htw = 01 vM = 25 (h) same as (e) (i) same as (e) vM = 4

(c) the enotect of grain rotations by both slip and twinning

(d) the inreguence of shy nal amount of twinned volume fraction

(e) the grain-shape enotect as predicted by the VPSC model

We use twinning to implicitly model one possible enotect of temperature and SFEIn order to discuss these aspects we present intermediate and shy nal textures corre-sponding to the following cases

1 Only slip cases (macrgure 4) Taylor FC and VPSC simulations for moderate ratesensitivity (n = 1=m = 19) and VPSC with low rate sensitivity (n = 1=m = 47)



z

q

r

180

135

90

45

00

00

ndash45

ndash90

ndash135

ndash180

(a) (b)

Figure 6 The ODFs showing the orientations of regions that twin in the simulations and theirorientations after twinning (a) Orientations near Acurren

2 preferentially twin (b) All regions reori-ented to Acurren

1 by twinning

2 Slip plus twinning cases with and without grain-shape updating (macrgure 5) TaylorFC and VPSC simulations consisting of an early stage (up to vM = 10) of onlyslip and a late stage of slip plus twinning (up to the shy nal strain of vM = 40)assuming htw = 01 regular grain-shape updating and moderate rate sensi-tivity (n = 1=m = 19) and same VPSC simulation but preventing grain-shapeupdating during the late stage

In general for the case of slip alone both the Taylor FC and the VPSC models witha moderate rate sensitivity n = 19 predict the formation of the A curren

2 A B and Ccomponents following moderate-to-large strains (shy gure 4ac gi) The B componentis stronger relatively in the VPSC simulation compared with the Taylor simulationespecially if the grain shape is not evolved in the VPSC model For the case of Taylora strong partial f111g shy bre plus a weak h110i shy bre connects the sharp peaks of thesecomponents (shy gure 4g h) whereas a complete f111g shy bre plus a stronger h110ishy bre connects these peaks in the VPSC case (shy gure 4a b) This complete f111g shy brecontains both the Acurren

1 and Acurren2 components in the VPSC simulation Consequently the

VPSC model produces a texture that approaches orthotropic symmetry in contrastto the clear dyadic symmetry of the Taylor results

The volume fraction of all the components increases with increasing strain frommoderate to large strains A transition in this trend occurs at large strains (vM = 3n = 19 VPSC) in which the C component increases rapidly with strain at the expenseof the decreasing A B and A curren

2 components The strain level at which this reversaloccurs depends on the value of the rate sensitivity and on the grain shape Decreasingthe rate sensitivity signishy cantly retards the formation of the C component to verylarge strains above 4 as shown by the strong B component and lack of C componentin shy gure 4e compared with shy gure 4b Decreasing the rate sensitivity also increasesthe texture peak intensities weakens the connecting shy bre textures and changes therelative ratios of the intensity peaks of A curren

1=C=A curren2 with strain Grain shape retards or

enhances the formation of the C component If the grain shape is kept spherical inthe simulations then very little C component forms following vM 40

If twinning was included in the simulations twinning was suppressed below astrain of 1 which corresponds with the experimental observations As the strain isincreased above 1 the start of twinning enhances the formation of the opposite A curren

1



component and its adjacent partial f111g shy bre at the expense of the Acurren2 for both

the VPSC and Taylor models with slip plus twinning (shy gure 5) In the simulationgrains that twinned had orientations near one of three types of orientation all ofwhich twinned to orientations near A curren

1 and its adjacent partial f111g shy bre Thoseorientations associated with twinning shown in shy gure 6 were Acurren

2 regions on eitherside of the C component near the h110i shy bre and near f56middot4gh7middot8middot3i Thus the increasein the Acurren

1 component with twinning is due primarily to the twinning of A curren2 and

to a lesser extent to twinning of regions near C and f56middot4gh7middot8middot3i The density oforientations near both A curren

2 and C diminishes for the case of twinning compared withjust slip (compare shy gures 4b and 5b) With increasing strain the amount of twinningtapers onot as the critical stress of twinning relative to slip increases For the presentset of parameters the shy nal stable twinned volume fraction reached in the VPSCcase a value of 289 at around 24 strain Once the twin activity drops to a verylow value the orientations near the A curren

1 component are not stable with respect tocontinued slip Thus they rotate back to near the A curren

2 and C orientations and theresulting ODF becomes similar to the case for only slip (shy gure 5c f i)

As for only slip the amount of C component in the VPSC simulations plus twinningcan be varied by playing with the grain-shape updating (compare shy gure 5b c with5e f) If the grain shape is not updated a signishy cant amount of Acurren

1 is retained at25 strain At 40 strain the A curren

1 are less intense but still practically no C is formedOn the contrary if the grain shape is continuously updated a high-SFE-like shy naltexture with a strong C component is obtained once more Finally the FC model(completely insensitive to grain shape) gives an intermediate result (shy gure 5i) someA curren

1 is retained but some C starts to reappear at the 40 strain

5 Comparison of measured and simulated texture

(a) Medium-to-high SFE andor high temperature

For high temperature or medium-to-high SFE the major types of texture compo-nents that are observed in the experiments are those that are predicted in the simu-lations However the evolution of a preferred component or the relative proportionsof components with increasing deformation are not simulated well One exception isthe better match between the VPSC model and the complete shy bres observed in thehigher temperature results (compare shy gure 2d l with shy gure 4a b) Generally it isdimacr cult to simulate the critical strain ranges for the appearance and disappearanceof the C A curren

1 or Acurren2 components as well as their relative intensities especially in the

high-to-medium SFE materials at room temperatureThe simulations predict that the C A curren

1 or Acurren2 components should rise and fall

as observed experimentally and best illustrated in the analytical model in Gilorminiet al (1990) While simulations predict a cyclic development they do not predictthe experimentally observed timing and the proportion of components in the cycleConsider that the Acurren

1 A curren2 and C components lie along a special shy bre deshy ned by

h110i along the radial direction Analyses show that orientations can rotate alongthis shy bre between the A curren

2 C and A curren1 components in a cyclical manner due to the

magnitude of rigid-body spin about the radial direction (Toth et al 1989 Gilorminiet al 1990) During this very long cycle densities of crystallites build up and decayin a sequence of orientations from A curren

1 to A curren2 to C to A curren

1 and back In the analyticalcalculations the A curren

2 component peaks in the shy rst quarter of the cycle and then decays



to random by the middle of the cycle at which point the C component has peakedThe length of this cycle period depends on the rate sensitivity m (ie n = 1=m) Theperiod extends to inshy nity for rate-insensitive material and decreases to vM = 20 form = 005 (ie n = 1=m = 20) (Harren et al 1989 Neale et al 1990) Consequentlyas shown in either the rate-sensitive Taylor FC or VPSC simulations in shy gure 4 astrong A curren

2 still remains at vM = 4 In contrast with the simulations a rather shortcycle is observed in the experiments at 296 K with only a moderate build up oforientation densities around A curren

2 occurring before a strain of vM = 1 and decayingquickly to a level below random with increasing strain thereafter (shy gure 2a b e f)

Similar shortened A curren2 C cycles are observed experimentally in other FCC metals

deformed in shear or by torsion including aluminium (Montheillet et al 1984 Rollett1988) copper (Montheillet et al 1984 Sekine et al 1981 Stout et al 1988 Stoutamp OrsquoRourke 1989 Williams 1962) and dilute copperzinc alloys (Sekine et al 1981)The initial texture can alter this result For example if the A curren

2 component is fed witha favourable starting texture as in a sample with common rolling texture with theprior rolling plane normal parallel to the shear direction then the A curren

2 intensity peaksearlier and at a higher intensity before decaying as shown in the data of Williams(1962) Since the A curren

2 peak in that case occurred at very low strains vM = 02where textures are not usually measured it could be easy to miss Generally A curren

2 hasbeen observed only as a very weak component in experiments because the textureis measured at larger strains vM gt 15 (see for example Montheillet et al 1984Sekine et al 1981 Stout et al 1988 Stout amp OrsquoRourke 1989) Rather than a strongAcurren

2 component as suggested by the simulations the C component is the strongest inboth the current experiments for nickel and nickelcobalt and in previous ones formoderate-to-high-SFE metals

The C component in the experiments begins to dominate very early in the defor-mation as A curren

2 disappears vM = 2 and strongly increases in intensity with increasingstrain In comparison for a rate-sensitive material in the Taylor FC simulations adominant C component and the associated disappearance of Acurren

2 is only observed fol-lowing extremely large strains vM gt 5 If relaxed constraints are used in the Taylormodel then the development of the A curren

2 component is somewhat suppressed while theC is increased as shown shy rst by Canova et al (1984) Similarly in the VPSC modelif the grain shape is evolved to match the geometry of the deformation then A curren

2 isdecreased and the amount of C is enhanced at lower strains Thus the proportion ofAcurren

2 C and to a lesser extent Acurren1 can be varied in the simulations by changing the

rate sensitivity the strain level and level of constraint based on the evolution of thegrain shape

However even with these modishy cations and by using a higher rate sensitivity thanobserved experimentally Acurren

2 is much too strong while the relative proportion of C istoo weak in simulations compared with experiment This much shorter experimentalcycle compared with prediction is related to the very regat shape of the dislocationmicrostructure and to grain subdivision This microstructural contribution to thetexture evolution will be discussed in the next sections

The A and B components are also steady contributors to the experimental texturesthat are observed However their signishy cance is lost in the Taylor simulations thatpredict the overriding dominance of A curren

2 The VPSC model does a better job ofpredicting the A and B components although the location of the `Brsquo texture peak inthe VPSC model is displaced 15macr from the ideal B along the h110i shy bre



(b) Low SFE and twinning

The low-SFE Ni + 60 wt Co deformed at 296 K developed a signishy cantly dif-ferent texture than the high-SFE samples The principal dinoterences include thenew texture components fmiddot545gh652i near the Acurren

1 component and the lack of aC component Simulations including this one have been inspired by the early sug-gestions of Wasserman (1963) and Haessner amp Kiel (1967) to introduce reorienta-tion by twinning to enotect the observed change in texture (Chin et al 1969 vanHoutte 1978) Later experimental studies in rolling have also lent support to thisapproach (Hu amp Cline 1988 Donadille et al 1989 Ray 1995) based on microstruc-tural observations

When twinning is added to the simulations then reorientation of the texture occursby both slip and twinning Twinning in the simulations principally had one majorchange in the predicted texture taking the strong A curren

2 component formed by slipand reorienting it to the twin related A curren

1 component Thus twinning occurred pre-dominantly in one region of orientation space and along one twin system that wasoriented to have the largest resolved shear stress with respect to the macroscopicapplied stress This change produces a texture similar to that observed in the exper-iments in which a medium-strong partial shy bre near Acurren

1 is observed (compare shy g-ures 2j and 5e) Note however that the experimental texture is characterized bynew components 20macr from Acurren

1Reorientation by twinning also removed a small amount of the orientation density

near the C component in the simulations thereby lowering the amount of C that wasformed While some C is decreased in the simulations it was dimacr cult to eliminatethe C in the simulations to a degree comparable with the experiment In the VPSCmodel this decrease in C was aided by not evolving the grain shape

While several early investigators (eg Dillamore amp Roberts 1965) have suggestedthat slip was more important to the texture change than twinning only a coupleof later investigators have been proponents of the contribution of slip For exampleLenoters amp Bilde-Sfrac12rensen (1990) have suggested that the physical constraint of thetwins on further slip in the matrix has a bigger enotect on the texture developmentthan the contribution from the actual crystal orientation of the twins Duggan et al (1978) also postulated that slip was more important than the small twin volumebased on their microstructural observations following rolling The two more recentsets of investigators note that the rolling case is complicated by the early onset oftwinning and the microstructure and texture changes due to shear banding followingtwinning Quantitative observations of the associated microstructure must be usedto determine whether or not twinning is important and to what extent slip plays arole

(c) Strength and intensity of the preferred texture

The simulations generally overpredict the degree of texture formation The volumefractions and peak intensities in the experiments are on average 23 times less thanthose in the simulations In the worst case at the largest strain the Taylor FCmodel predicts a sharp peak for the A curren

2 component that is 20 times that observedin the experiments Generally the VPSC model has lower peak intensities than theTaylor model and is thus a better match with experiment in this regard Ratherthan having a sharp peak of intensity surrounding an ideal component as in the



simulations the experiments show a very broad distribution of moderate intensityabout a preferred orientation (compare shy gure 2 with shy gure 4) The wide spread inorientations about an ideal component in the experiments described by the half-width of the texture peak at half its maximum height is twice that predicted bythe VPSC and thrice that for the Taylor model Note that much of the texturespread observed in the experiment lies outside of the regions counted in the volumefraction summation for the ideal components A larger texture spread in experimentscompared with the simulations is a common observation for many other deformationmodes including rolling (Hansen et al 1993) One factor in this dinoterence may begrain statistics in the simulations As Matthies amp Wagner (1996) have shown thetexture strength decreases for increasing numbers of individual orientations Otherexplanations should be sought such as grain subdivision by dislocation boundariessince the dinoterence is so large

6 Microstructure

The microstructural aspects considered by the simulations were only the grain shapeand deformation-induced twins However deformation creates a microstructure thatincludes volume elements surrounded by dislocation boundaries at a shy ner scale thana grain These structures and the local orientation environment are considered nextwith respect to their impact on the observed macroscopic textures The structuraland orientation dinoterences that occur as a function of temperature and SFE arehighlighted

The dislocation structures observed for the three materials and dinoterent defor-mation temperatures shy t within the framework of grain subdivision by deformation-induced dislocation boundaries on two size-scales (Bay et al 1992 Hansen amp Hughes1995) The smallest size-scale comprises equiaxed cells while the larger scale com-prises long regat geometrically necessary boundaries (GNBs) (Kuhlmann-Wilsdorfamp Hansen 1991) that surround groups of cells arranged in long cell blocks (CBs)These GNBs include double-walled microbands (MBs) single dense dislocation walls(DDWs) and lamellar boundaries (LBs) which have been deshy ned and described else-where (Bay et al 1992) Microbands and DDWs are characteristic of the low-strainstructures vM 08 whereas the LBs are characteristic of the large-strain struc-tures above vM = 19 All types of GNBs may be observed at intermediate strainsThe nearly parallel groups of GNBs that subdivide a grain are aligned macroscop-ically with respect to the torsion axis At small strain either one family or twointersecting families of MBsDDWs are observed with one family nearly parallelto the macroscopic shear plane (shy gure 7a) and the other at 6090macr to that plane(shy gure 7b) The number of subdividing GNBs within a grain increases from a few atvery small strains vM 02 to many at vM 06

The aspect ratio of the cell blocks deshy ned by the GNBs is not the same as theaspect ratio deshy ned by the evolving grain shape Optical metallography showed thatthe initially equiaxed grains elongated with increasing strain as expected from geo-metrical considerations As a result the aspect ratio for grains changes very slightlyfrom 1 at vM = 0 to 101 at vM = 04 In contrast with the whole grain the aspectratio of cell blocks deshy ned by the GNBs ranges from 1 for grains with two families ofintersecting GNBs to roughly 3 in grains with one family of nearly parallel GNBseven at a medium strain (compare parts (a) and (c) of shy gure 7) At large strains



(a)

(b)

(c)

1 m m

1 m m

5 m m

Z

q

Figure 7 For description see opposite



where vM 1 the structure is made up of closely spaced LBs nearly parallel tothe shear plane (shy gures 7c and 8ac) These regat LBs sandwich single layers of cellsforming cell blocks This LB structure has a very high aspect ratio ranging approx-imately from 3 to 5 Another common feature within the large-strain microstructureis groups of lenticular to nearly equiaxed subgrains that alternate with groups ofLBs (shy gure 8b) The proportion of subgrains compared with LBs is much higherat the higher temperature of 573 K than at 296 K (compare parts (a) and (b) inshy gure 8)

While the framework of grain subdivision describes the general features of themicrostructure for all of the materials SFEs temperatures and strain levels thesedinoterent conditions systematically inreguence the slip pattern and thereby the tex-ture For example the spacing between all types of dislocation boundaries decreaseswith increasing strain decreasing SFE and decreasing temperature The misorienta-tion angle across dislocation boundaries also increases with increasing strain Thisincrease occurs at a higher rate for the GNBs compared with the cell boundariesresulting in the formation of high angle boundaries and a wide spread of dinoterentorientations within a single original grain This result is shown in shy gure 9a in whichthe minimum misorientation angle (disorientation) across dislocation boundaries inadjacent crystallites is plotted as a function of distance within a single grain Manyof the measured misorientation angles across the deformation-induced dislocationboundaries are very large and have angles characteristic of regular grain boundaries(see for example Hughes amp Hansen 1997)

Note that the dislocation boundaries separate dinoterent texture components andthat the grain has split up into many local orientations spread out widely alongthe various texture shy bres This result is superposed on the misorientation plot ofshy gure 9a in which the crystal orientation of adjacent crystallites is indicated byshading An orientation type is assigned based on the criterion that the crystalliteorientation is within 15macr of an ideal texture component from table 2 Strikinglynearly all of the ideal texture components are observed over a very short distancein one original grain Similar behaviour is observed in Ni + 60 wt Co (shy gure 9b)in which many orientations are observed over a short distance in two locations Thedisorientations were measured mainly across adjacent dislocation boundaries Only10 of the boundaries measured were twin boundaries as determined by their angleaxis pair near 60macr h111i Note that these twin boundaries did not separate the A curren

1

and A curren2 components although the A curren

1 was the predominant ideal orientation in bothof these small regions The A curren

1 appeared to be part of the matrix material ratherthan the deformation twin

Deformation twinning occurred as a deformation mechanism in conjunction withgrain subdivision by dislocation boundaries depending on the SFE strain level andtemperature Twinning was only observed in Ni + 60 wt Co ie the material with

Figure 7 TEM micrographs of the small-to-medium strain deformation microstructures showing that longdislocation boundaries form very early in the deformation The shear directions are marked by doublearrows (a) Nickel deformed at 296 K vM = 035 This region is in a random orientation near fmiddot12middot1gh41middot2ibut within the texture spread about the B orientation A cell block is marked CB (b) Ni + 60 wt Codeformed at 296 K vM = 035 showing two intersecting families of microbands (MBs) and DDWs thatare nearly parallel to the f111g slip planes marked by four white lines in the bottom right-hand corner(c) Ni + 30 wt Co deformed at 296 K vM = 12 The MBs delineate lamellar regions along the shearplane



(a)

(b)

05 m

1 m

05 m(c)




the lowest SFE These observations based on TEM were corroborated with opticalmetallography At medium strains vM = 035 and 296 K there were a few isolatedvery thin and small faulted regions observed in the TEM which may have beenincipient deformation twins The deformation microstructure at vM = 035 howeverwas dominated by the dislocation structure of intersecting MBs and DDWs FollowingvM = 1 at 296 K a few very thin deformation twins were observed in isolated grainscomprising less than a 1

2vol based on area measurements As the strain increased

to vM = 2 many more thin deformation twins were observed on intersecting twinsystems within the dislocation microstructure (shy gure 10a) but the actual volumefraction of these twins ca 5 was still small

These thin twins exhibited a wide variety of orientations and sizes Frequentlythe twins were oriented to cut across the lamellar cell blocks enotectively changingthe aspect ratios of the lamellar cell blocks (shy gure 10b) These intersecting twinsare macroscopically oriented at a high angle to the shear plane and are in contrastwith other twins that lie in bundles roughly along the shear plane and parallel to thelamellar cell blocks Twins were found in grains having a wide variety of crystal orien-tations as well as in grains oriented near either A curren

1 or A curren2 The dominating dislocation

microstructure had many short and thin eg 10 nm wide lamellae that could be mis-taken morphologically for twins However measurements showed that the boundariesof these thin lamellae had only low-to-moderate misorientations quite far from thehigh 60macr=[111] misorientation angleaxis characteristic of twin boundaries Overallthe GNBs Ni + 60 wt Co deshy ned more fragmented and square-shaped cell blockscompared with the lamellar cell blocks observed at similar strains in the materialswith the higher SFE (compare parts (a) and (c) of shy gure 7 and also shy gure 8a withshy gure 10a) As another dinoterence the cell boundaries were ill-deshy ned and someuniform distributions of dislocations in Taylor lattices were observed

In contrast with the observations at 296 K deformation twins still comprised lessthan 1

2vol following deformation at 573 K and the largest strain of vM = 22

(shy gure 8c) The dislocation structures including cells were more reshy ned than at thelower temperature While many square-shaped cell blocks were still observed theproportion of lamellar cell blocks increased signishy cantly Thus the overall microstruc-ture was intermediate between the low-SFE low-temperature and the high-SFE low-temperature microstructures

7 Relation between texture and microstructure

A variety of both textures and microstructures has been shown as functions of SFEmaterial temperature and strain in the preceding sections When these textures andmicrostructures are plotted together as in shy gure 11 then it is observed that condi-tions leading to similar textures also lead to similar microstructures This remark-able commonality will be discussed below using some instructive insight from thesimulations What is important to note is that dinoterences in the dislocation slip pat-tern alone without crystal reorientation by twinning cause dramatic changes in the

Figure 8 TEM micrographs of the large strain dislocation structure composed of lamellar dislocationboundaries sandwiching thin layers of cells (C) (a) Nickel deformed at 296 K vM = 21 (b) nickeldeformed at 573 K vM = 21 (c) Ni + 60 wt Co deformed at 573 K vM = 22 showing a regionwith local orientations along the h110i shy bre between the A and B ideal components



60

40

20

0

ndash 40

ndash 60

ndash 20

16distance along z axis ( m)

diso

rien

tatio

n (d

eg)

0 2 4 6

A2 A1 A1 A2 B2 B1 C 110 fibre random

8 10 12 14

111 fibre

60(a)

(b)

40

20

0

ndash 40

ndash 60

ndash 20


diso

rien

tatio

n (d

eg)

0 2 4 6 8




(a)

(b)

05 m m

1 m m

Figure 10 TEM micrographs of deformation microstructures that develop in the low-SFENi + 60 wt Co at 296K (a) Dislocation structure composed of LBs and equiaxed subgrains(ESs) there are no deformation twins in this region vM = 12 (b) TEM dark macreld micrographshowing whitersquo deformation twins (TWs) cutting across lamellar bands vM = 21

observed texture The dinoterent orientations that develop within a grain demonstratethe inreguence of the local slip pattern (shy gure 9) Because the slip pattern is importantto the texture evolution the temperature and SFE which anotect slip are thereby alsovery important

Figure 9 The disorientation angles measured across adjacent dislocation boundaries in the axial direc-tion z for torsion show an alternating character with distance These boundaries separate shy nely dis-tributed texture components as shown by the shading (a) Nickel deformed at 296 K vM = 40(b) Ni + 60 wt Co deformed at 296K vM = 12 Two separate regions in the sample were measuredas marked by the arrows



296

temperature (K)

C A B C A1 A2[ 111] ltuvwgt[ hkl] lt110gt

A B C A1 A2[ 111] ltuvwgt[ hkl] lt110gt

A B A2

ES

LB

LB

TW

LB

LB

LB

ES

ES

ES

stac

king

fau

lt e

nerg

yhigh

low

nick

el

Ni + 30 w

t C

o

Ni + 60 w

t C

o

573

Figure 11 Schematic showing the major characteristics of the three types of textures andmicrostructures that develop at large strain as a function of material SFE and temperatureOnly the GNBs are shown in the schematic LB indicates lamellar dislocation boundaries ESindicates equiaxed subgrains TW indicates short thin deformation twins that either cut acrossthe lamellar bands or lie along the lamellar boundaries

(a) Slip or twinning

The microstructure and texture observations show that the distinctive low-SFEand low-temperature texture (shy gure 2i) was formed before the onset of twinning Inthe absence of twinning the microstructure and textures are controlled by disloca-tion slip and the ease with which dislocations cross-slip High SFE and temperatureencourage cross-slip Thus although twinning in the simulation appeared to give thecorrect texture it is not the principal mechanism that is active in the experimentsas revealed by the microstructural observations Secondly the twinning was onlyobserved following large strain in the experiment These twins were not just alongone or two systems in special grain orientations but occurred in a wide variety ofgrain orientations and along 23 twin systems per grain Most twin systems werenot the twin system with the highest resolved shear stress based on the macroscopicapplied stress state The variety of systems observed is explained by the fact thatdeformation twins have a threshold stress for initiation of a twin Therefore twinformation is linked to the very local stress state at the level of the dislocation bound-ary Thus twin systems can be activated by local stresses that are dinoterent from theones predicted by the applied macroscopic or even grain-level stresses



Consequently only in a few very special cases do the observed twins reorientmaterial to a main texture component Thus most of the twins reorient the twinnedmaterial to locations in orientation space that are away from the main texture com-ponents thereby adding to the random level Of course some twins of A curren

1 orientationdo form in the experiments in both A curren

2 oriented grains and in grains near C as pre-dicted but the twin volume fraction is far too small to make up the strong intensityof orientations near A curren

1 that are observed in the experimental ODFsGrains near the A curren

1 component arise by slip mechanisms as well as by twinningThis possibility is demonstrated by both the experiment and the VPSC simula-tion without twinning The A curren

1 component was observed for experimental texturesdeveloped at 573 K without twinning (shy gure 2d nickel) or with very little twinning(shy gure 2l Ni + 60 wt Co) Several A curren

1 oriented crystallites that were not deforma-tion twins|as determined by their size shape and boundary character|were alsoobserved in the TEM for Ni + 60 wt Co deformed at 296 K (eg shy gure 9b) TheVPSC simulation for only slip also predicts a moderate A curren

1 component at low-to-medium strains Thus the Acurren

1 component observed in the low-SFE texture at 296 Karises mostly from slip with twinning making up a smaller fraction at the very largeststrains

Neither the new texture components nor the low value for the A curren2 in the low-SFE

low-temperature experiment is explained by the microstructural results discussedthus far It may be that texture development is more retarded in the low-SFE metalso that the texture never develops much beyond the formation of the initial A curren

1

component suggested by the analysis of Gilormini et al (1990) The more completeanswer must however be found in the development of the slip pattern within grainsas a function of SFE and temperature

(b) Slip patterns revealed by grain subdivision

Geometrically necessary dislocation boundaries subdivide a grain into dinoterentlydeforming regions that slip with fewer slip systems than the number required by aTaylor analysis (Kuhlmann-Wilsdorf amp Hansen 1991 Bay et al 1992) These regionstherefore act collectively to accommodate the deformation on average Neighbouringregions deform with either dinoterent selections of slip systems dinoterent partitions ofslip on the same slip systems andor dinoterent amounts of strain These GNBs accom-modate the lattice mismatch that develops as a consequence of the slip selection

This selection and partitioning of slip systems within cell blocks will be inreguencedby the SFE and the temperature Low SFE and low temperature restrict dislocationmobility and thus promote planar glide As the SFE andor temperature isareincreased thermally activated cross-slip of dislocations is easier thus reducing thetendency for planar glide More slip systems would also be active Thus the choiceand number of slip systems in cell blocks for a low-SFE metal will be biased by thematerialrsquos tendency for planar slip This bias would dinoter from that of high-SFEmaterials

Dinoterences in local slip system selections in the cell blocks will result in dinoterencesin local lattice rotations as a function of SFE Thus it is postulated that some grainsin a low-SFE metal favouring planar glide like Ni + 60 wt Co deformed at 296 Kwill subdivide along dinoterent shy bre textures than a high-SFE material such as purenickel This view is supported by the local orientation data in shy gure 9 which show



that one grain in Ni + 60 wt Co has broken into the distinctive shy bre texture fromB2 to A2 As temperature is increased the bias towards planar slip is reduced andthe microstructure and texture evolution approach that of the high-SFE metal

Note that this postulate regarding planar slip is new because it considers planarslip within the context of grain subdivision into dinoterently deforming regions Theassociation of grain subdivision into dinoterent regions of planar slip is distinct fromthe lower bounds or Sachs-type hypothesis applied to a grain as a whole

The local lattice rotations will reregect not only the slip system activity caused by theapplied deformation but also slip activity caused by the presence of the dislocationboundaries (and deformation twins) These dinoterent sources of slip-system activitywill further modify the texture development

(c) Cell-block shape as a result of grain subdivision and twinning

The simulations showed that subtle changes to the texture can be made by incorpo-rating dinotering deformation constraints due to the crystal shape Both grain subdivi-sion and deformation twinning create new volumes in cell blocks within a grain Thecell-block shape as a function of SFE and temperature correlates with the observedchanges in the texture The evolving cell-block shapes dinoter from the evolving grainshape (see shy gures 7 and 8) Crystal or grain shape inreguences the progression of ori-entations along the A curren

1 A curren2 to C radial shy bre as shown by the simulations While an

evolving grain shape was applied in the simulations at the level of the original grainsfor illustration in the future these constraints need to be applied with respect tothe shape evolution of the deformation microstructure

For the case of medium-to-high-SFE materials at 296 K the early developmentof lamellar-shaped cell blocks (shy gure 7a) in the shear direction correlates with astrongly enhanced evolution of the C component relative to the other texture com-ponents along the h110i and f111g shy bres At the same time lamellar cell blocks dis-favour the weakly observed Acurren

2 component This strong enhancement of C decreaseswith increasing temperature as relatively larger proportions of more equiaxed sub-grains are formed in combination with the lamellar substructure Consequently moreuniform and complete f111g and h110i shy bre textures are observed at the higher tem-perature of 573 K The low-SFE Ni + 60 wt Co has a mixed equiaxed and lamel-lar microstructure like nickel at this higher temperature as well as a similar shy bretexture

For the low-SFE material at 296 K the dislocation structure generally tendstowards more equal sided blocks and is not as well deshy ned as that for the high-SFE material over the entire strain range The deformation constraints required bythe more equal sided structure suppress the formation of the C component whichwould be favoured for regat shapes along the shear direction At larger strains regionswith cell blocks formed by lamellar boundaries also contain a small volume fractionof twins This twinning occurs on several twin systems that cut across the lamellarcell blocks creating square rather than lamellar-shaped volumes (see schematic inshy gure 11) Thus the texture is anotected by how the twins change the shape of thecell block rather than by the crystal orientation of the twin Consequently in thesimulations a better match with the experiment is found if the grain shape is arti-shy cially limited in its evolution when twinning starts (compare parts (e) and (b) ofshy gure 5) thereby suppressing the C component as in the low-SFE Ni + 60 wt Coat 296 K



8 Summary and conclusions

The dinoterent slip patterns formed within a single grain are at odds with the centralTaylor assumption How well the Taylor assumption is fulshy lled on average depends onthe SFE and deformation temperature as is clearly shown by the development of newtexture components The wide variety of textures and microstructures observed as afunction of SFE and temperature was developed by slip processes alone It is hypoth-esized that the low-SFE slip pattern is a cause of the observed new texture compo-nents which are distinct from the generally accepted ideal components obtained bymodelling These discrepancies are related to the microstructure enotects determinedby the slip pattern Materials and conditions with similar deformation microstruc-tures developed similar textures in the experiments regardless of the SFE Somedeformation twins formed in the low-SFE material but only at large strains after thetexture was formed by slip Upon further evolution at large strain this twinning didnot alter the distinctive character of the low-SFE texture formed originally by slip

The most important mechanism that creates the new texture components is thelocal slip pattern reregected in the grain subdivision into cell blocks The slip-systemselection and partitioning creating the dinoterent lattice rotations within a cell blockis altered by the planarity of slip Slip planarity is controlled by the SFE and tem-perature Thus some grains in low-SFE metals deformed at low temperature willsubdivide along dinoterent shy bre textures than those in high-SFE metals as shown bydetailed measurements of the microstructure Additionally new slip systems are acti-vated as a result of the grain subdivision that are not included in the texture modelsIn the future detailed analyses of the dislocations that compose the GNBs surround-ing cell blocks will provide additional keys to these slip-pattern dinoterences Futurereshy nements of polycrystal plasticity simulations will need to include the character-istics of the microstructure Clearly a grain cannot be considered as a homogeneousunit in realistic models of the deformation process

More subtle enotects of microstructure on the texture arise from the dinoterent defor-mation constraints allowed by dinoterent shaped cell blocks Grain subdivision bydislocation boundaries in these materials created dinoterent cell-block shapes that cor-related with the dinoterent textures observed as a function of SFE and temperatureThe evolution of the cell-block shapes within the grains dinoters from the grain-shapeevolution These dinoterent shapes correlated with the presence or absence of certainideal texture components whose evolution could not be simulated

DAH acknowledges the support of the Oplusmn ce of Basic Energy Sciences US DOE under con-tract no DE-AC04-94AL85000 HRW was supported through NSF grant EAR 94-17580 RLacknowledges a Fulbright Scholarship which provided a visit to Berkeley Our discussions withU F Kocks N Hansen A D Rollett and C Tomparae are gratefully acknowledged We are appre-ciative of the assistance of I Houver with the texture measurements

Appendix A

A brief overview of the simulation methods including a description of how twinning isimplemented in the model is given in this appendix Detailed accounts of the VPSCsimulation methods used herein have been published before For readers wantingmore detail the development of the VPSC model can be traced in terms of theEshelby (1957) inclusion formalism by Hill (1965) and Hutchinson (1970 1976) and



macro rotation

W

plastic rotation local rotation(a) (b) (c)

w locwndash pl

Figure 12 Schematic of the three rotation terms in equation (A 1)

the general n-site approach for large-strain viscoplasticity of Molinari et al (1987)leading to the 1-site approximation by Lebensohn amp Tome (1993) For both Taylorand VPSC the viscoplastic approach is used at the single-crystal level Thus if astrain-rate is applied to a single crystal (grain) and the geometry and critical stressesof the active slip and twinning systems are known the stress in the grain can becalculated by solving a system of nonlinear equations In the FC case whereas thelocal strain rates in all grains are assumed to be equal to the macroscopic appliedstrain rate the local stresses can be readily calculated and the macro stress obtainedas an average of those local stresses

Unlike the FC approach within the VPSC formulation the stress and strain ratein the grains can be dinoterent from the corresponding macroscopic magnitudes Thedeviations of the local magnitudes with respect to the macroscopic ones depend onthe directional plastic properties of the grains and the whole polycrystal Basicallythe VPSC model regards each grain of the polycrystal as a viscoplastic inclusiondeforming in a viscoplastic homogeneous enotective medium (HEM) having the aver-age properties of the polycrystal The VPSC formulation is based on the determi-nation of the interaction tensor that relates the local deviations in stress and strainrate with respect to the macroscopic state The interaction tensor depends on theviscoplastic Eshelby tensor (which in turn depends on the grain shape) and onthe macroscopic modulus The latter is not known a priori but can be determinedself-consistently by imposing the conditions that the averages of the local stressesand strain rates over all of the grains should match with the corresponding appliedmacroscopic magnitudes Consequently the inreguence of grain morphology entersmore gradually in the VPSC approach (Tiem et al 1986) compared with the com-plete relaxation of some components of the strain as a consequence of the grainmorphology assumed by the Taylor RC theory (Honnenot amp Mecking 1981 Canova etal 1984)

The total deformation is applied to the polycrystal as a sequence of incremen-tal deformation steps After each step the grain shape the texture and the criticalstresses are updated due to stretching grain rotation and strain hardening respec-tively

(a) Grain rotations by slip and twinning

In order to predict texture development it is necessary to determine the rotationsof the grains When slip is the only operative mechanism of plastic deformation the



total rotation rate of a grain is given by

_ = _laquo iexcl _ p l + _loc (A 1)

where the terms on the right-hand side are the macro plastic and local rotationrates respectively The shy rst term (macro) is related to the rotation of the sampleduring a torsion test The second one (plastic) can be readily derived from the shearrates The third term (local) depends on the magnitude of the local deviations instrain and on the assumed grain shape and it is zero in the FC case On the otherhand in the VPSC case the local rotation rate can obtained as

_loc = ~_ = brvbar Siexcl1 ~_ (A 2)

where S and brvbar are the symmetric and skewsymmetric (Lebensohn amp Tome 1993)viscoplastic Eshelby tensors respectively and ~_ = _ iexcl _E and _E are the macroscopicand local strain rates respectively Particularly the brvbar tensor vanishes for sphericalgrains while its norm increases|and so does the weight of _loc in expression (A 2)|as the grain shape becomes more severely elongated

Figure 12 clarishy es the meaning of the three rotation terms in equation (A 1) Whena polycrystal undergoes a rigid-body (macro) rotation (shy gure 12a) each grain in thepolycrystal rotates by the same amount If the polycrystal deforms plastically and sodoes a grain inside the slip activity involves a rotation of the crystal lattice due tothe constraints that the polycrystal as a whole exerts on the grain Figure 12b showsa simple example of a polycrystal undergoing uniaxial tension with an embeddedgrain whose lattice rotates while the grain deforms plastically by slip Finally if anon-equiaxed grain deforms dinoterently from the polycrystal the local rotation tendsto align the principal directions of the grain and the strain path applied to thepolycrystal Figure 12c shows the extreme case of a highly elongated non-deforminggrain embedded in a polycrystal deforming under uniaxial tension In this case thelong direction of the grain tends to align with the tensile direction

An additional source of rotations is related to twinning activity When a twin isformed inside a given crystal some volume fraction of it adopts a completely dinoterentbut crystallographically related orientation Several schemes have been proposed(Tome et al 1991 Lebensohn amp Tome 1993 van Houtte 1978) to deal with theenotect of twinning reorientation in texture development Particularly for moderatetwinning activities (ie when slip participates more than twinning in plastic strainaccommodation as in FCC materials) the so-called predominant twin reorientation(PTR) scheme (Tomacutee et al 1991) has been shown to be good enough to capture themain role of twinning reorientation in texture development Brieregy the PTR schemeconsists of the following steps

(1) Keep track of three magnitudes associated with twinning namely (a) F gs thevolume fraction of each twin system (s) inside each grain (g) given by theaccumulated shear associated with that twin divided by the characteristic twinshear (b) F tot the sum of the individual F g s over all systems and grainsand (c) F enot the enotective twinned fraction reoriented according to the randomselection described below

(2) If F enot is less than F tot make a random choice of a grain (g) and perform acomplete reorientation of that grain to the orientation associated with the twin



system (s) if

F g s gt 025 + 025 pound F enot

F tot (A 3)

In this way reorientation by twinning takes place in those twin systems that exhibitthe highest activity in the grain throughout the deformation process Furthermoreas the actual twinned fraction F tot grows larger than the enotective twinned frac-tion F enot the second term of the sum in the right-hand side decreases and furtherreorientation by twinning is favoured until F enot `catches uprsquo Such a procedure isself-correcting and both fractions tend to remain approximately equal throughoutdeformation When both fractions are equal condition (A 3) will be fulshy lled if thevolume of the twin represents more than half of the total volume of the grain If thisis the case the whole grain will adopt the orientation of that `predominantrsquo twinThe two values 025 in equation (A 3) are rather arbitrary but the choice of any otherpair of positive numbers that sum to 05 does not signishy cantly change the results

(b) Hardening law

Texture development predictions of single-phase materials do not depend in prac-tice on the actual value of the macroscopic stress applied to the polycrystal or onthe absolute values of the critical stresses but only on their relative values Thusthe known changes in hardening as a function of temperature would not changethe texture results in these simulations Since we are only interested in texture thetemperature enotect on hardening was neglected Deformation temperature was onlyconsidered implicitly as a factor in determining when twinning was active in thesimulation Furthermore when both slip and twinning were assumed to be activewe adopted a simple linear hardening law given by

cent frac12 s lc = h s lcent iexcl cent frac12 tw

c = htwcent iexcl (A 4)

where cent frac12 s lc and cent frac12 tw

c are the increments in the critical stresses of every slip andtwinning system respectively cent iexcl is the increment of the total shear in the grain(ie the sum of shears in every slip and twinning systems) and h s l and htw are twoindependent hardening parameters for slip and twinning respectively In the slip plustwinning simulation that follows we adopted h s l equal to zero and htw equal to agiven positive value In this way we prevented an unbounded increase of the twinnedvolume fraction keeping it in a good agreement with the experimental evidence Onthe other hand when only slip was considered as the active deformation mechanismthe critical stress of slip was assumed to be constant along the calculation

References

Bay B Hansen N Hughes D A amp Kuhlmann-Wilsdorf D 1992 Evolution of FCC deforma-tion structures in polyslip Acta Metall Mater 40 205219

Beaudoin A J Dawson P R Mathur K K amp Kocks U F 1995 A hybrid macrnite-element for-mulation for polycrystal plasticity with consideration of macrostructural and microstructurallinking Int J Plast 11 501521

Beaudoin A J Mecking H amp Kocks U F 1996 Development of local orientation gradientsin FCC polycrystals Phil Mag 73 15031517



Becker R 1991 Analysis of texture evolution in channel die compression|ereg ects of grain inter-action Acta Metall 39 12111230

Becker R 1995 A material size scale through crystal boundary energy Mod Sim Mater SciEngng 3 417435

Becker R amp Panchanadeeswaran S 1989 Crystal rotations represented as Rodrigues vectorsText Microstruct 10 167194

Beeston B E P Dillamore I L amp Smallman R E 1968 The stacking fault energy of somenickel cobalt alloys Metal Sci J 2 1214

Bronkhorst C A Kalidindi S R amp Anand L 1992 Polycrystalline plasticity and the evolutionof crystallographic texture in FCC metals Phil Trans R Soc Lond A 341 443477

Butler G C Graham S McDowell D L Stock S R amp Ferney V C 1998 Application ofthe Taylor polycrystal plasticity model to complex deformation problems J Engng MaterTech Trans ASME 120 197205

Canova G R Kocks U F amp Jonas J J 1984 Theory of torsion texture development ActaMetall 32 211226

Canova G R Wenk H R amp Molinari A 1992 Deformation modeling of multiphase polycrys-tals case of a quartz mica aggregate Acta Metall Mater 40 15191530

Chin G Y Hosford W F amp Mendorf D R 1969 Accommodation of constrained deformationin fcc metals by slip and twinning Proc R Soc Lond A 309 433456

Dillamore I L amp Roberts W T 1965 Preferred orientation in wrought and annealed metalsMetall Rev 10 271380

Donadille C Valle R Dervin P amp Penelle R 1989 Development of texture and microstructureduring cold-rolling and annealing of FCC alloys example of an austenitic stainless steel ActaMetall 37 15471571

Duggan B J Hatherly M Hutchinson W B amp Wakemacreld P T 1978 Deformation structuresand textures in cold-rolled 7030 brass Metal Sci 12 343351

Eshelby J D 1957 The determination of the elastic macreld of an ellipsoidal inclusion and relatedproblems Proc R Soc Lond A 451 376396

Frank F C 1988 Orientation mapping In Proc 8th Int Conf Textures of Materials (ed J SKallend amp G Gottstein) pp 313 Warrendale PA The Metallurgical Society

Gil Sevillano J van Houtte P amp Aernoudt E 1980 Large strain work hardening and texturesProg Mater Sci 25 69412

Gilormini P Toth L S amp Jonas J J 1990 An analytical method for the prediction of ODFswith application to the shear of FCC polycrystals Proc R Soc Lond A 430 489507

Haessner F amp Kiel D 1967 Orientierungsverteilung der kristallite in gewalztem Ms 70 bes-timmt mit Hilfte elektronenmikroskopischer feinbereichsbeugung Z Metallk 58 220227

Hansen N amp Hughes D A 1995 Analysis of large dislocation populations in deformed metalsPhysica Status Solidi (b) 149 155172

Hansen N Juul Jensen D amp Hughes D A 1993 Textural and microstructural evolution duringcold-rolling of pure nickel In Proc 10th Int Conf Textures of Materials (ICOTOM 10) (edH J Bunge) pp 693700 Clausthal Trans Tech

Harren S Lowe T Asaro R amp Needleman A (eds) 1989 Analysis of large-strain shear inrate-dependent face-centred cubic polycrystals correlation of micro- and macromechanicsPhil Trans R Soc Lond A 328 443500

Hecker S S amp Stout M G 1982 Strain hardening of heavily cold worked metals In Deformationprocessing and structure (ed G Krauss) pp 146 Metals Park OH ASM

Hill R 1965 Continuum micromechanics of elasto-plastic polycrystals J Mech Phys Solids13 89101

Honnereg H amp Mecking H 1981 In Proc 6th Int Conf Textures of Materials (ICOTOM 6) (edS Nakashima) pp 347355 Tokyo The Iron and Steel Institute of Japan



Hu H amp Cline R S 1988 On the transition mechanism of texture transition in face centeredcubic metals Text Microstruct 89 191206

Hughes D A amp Hansen N 1997 High angle boundaries formed by grain subdivision mecha-nisms Acta Mater 45 38713886

Hughes D A amp Nix W D 1988 The absence of steady-state deg ow during large strain defor-mation of some FCC metals at low and intermediate temperatures Metall Trans A 1930133024

Hughes D A amp Nix W D 1989 Strain hardening and substructural evolution in NiCo solidsolutions at large strains Mater Sci Engng A 122 153172

Hutchinson J W 1970 Elastic-plastic behavior of polycrystalline metals and composites ProcR Soc Lond A 319 247272

Hutchinson J W 1976 Bounds and self-consistent estimates of creep of polycrystalline materi-als Proc R Soc Lond A 348 101127

Kocks U F Tomparae C N amp Wenk H R 1998 Texture anisotropy preferred orientations inpolycrystals and their ereg ects on material properties Cambridge University Press

Kuhlmann-Wilsdorf D amp Hansen N 1991 Geometrically necessary incidental and subgrainboundaries Scripta Metall Mater 25 15571562

Kumar A amp Dawson P R 1995 Polycrystal plasticity modeling of bulk forming with macrniteelements over orientation space Comp Mech 17 1025

Lebensohn R A amp Tomparae C N 1993 A self-consistent anisotropic approach for the simulationof plastic deformation and texture development of polycrystals|application to zirconiumalloys Acta Metall Mater 41 26112624

Lereg ers T amp Bilde-Siquest rensen J B 1990 Intra- and intergranular heterogeneities in the plasticdeformation of brass during rolling Acta Metall Mater 38 19171926

Liu Q 1994 A simple method for determining orientation and misorientation of the cubic crystalspecimen J Appl Cryst 27 755761

Matthies S amp Wagner F 1996 On a 1=n law in texture related single orientation analysisPhysica Status Solidi (b) 196 K11K15

Molinari A Canova G R amp Ahzi S 1987 A self consistent approach of the large deformationpolycrystal viscoplasticity Acta Metall 35 29832994

Montheillet F Cohen M amp Jonas J J 1984 Axial stresses and texture development duringthe torsion testing of Al Cu and not -Fe Acta Metall 32 20772089

Neale K W Toth L S amp Jonas J J 1990 Large strain shear and torsion of rate-sensitiveFCC polycrystals Int J Plast 6 4561

Ray R K 1995 Rolling textures of pure nickel nickeliron and nickelcobalt alloys Acta MetallMater 43 38613872

Rollett A D 1988 Strain hardening at large strains in aluminum alloys PhD thesis DrexelUniversity

Sekine K van Houtte P Gil Sevillano J amp Aernoudt E 1981 The transition of torsionaldeformation textures in FCC metals In Proc 6th Int Conf Textures and Materials (ICO-TOM 6) (ed S Nagashima) pp 396407 Tokyo The Iron and Steel Institute of Japan

Stout M G amp Orsquo Rourke J A 1989 Experimental deformation textures of OFE copper and7030 brass from wire drawing compression and torsion Metall Trans A 20 125131

Stout M G Kallend J S Kocks U F Przystupa M A amp Rollett A D 1988 Materialdependence of deformation texture development in various deformation modes In Proc 8thInt Conf Textures of Materials (ed J S Kallend amp G Gottstein) pp 479484 WarrendalePA The Metallurgical Society

Taylor G I 1938 Plastic strain in metals J Inst Met 62 307324

Tiem S Berveiller M amp Canova G R 1986 Grain shape ereg ects on the slip system activityand on the lattice rotations Acta Metall 34 21392149



Tomparae C N amp Canova G R 1998 Self-consistent modeling of heterogeneous plasticity InTexture and anisotropy preferred orientations in polycrystals and their ereg ects on materialproperties (ed U F Kocks C N Tomparae amp H R Wenk) pp 467509 Cambridge UniversityPress

Tomparae C N Lebensohn R A amp Kocks U F 1991 Acta Metall Mater 39 26672680

Toth L S Neale K W amp Jonas J J 1989 Stress response and persistence characteristics ofthe ideal orientations of shear textures Acta Metall 37 21972210

van Houtte P 1978 Simulation of the rolling and shear texture of brass by the Taylor theoryadapted for mechanical twinning Acta Metall 26 591604

van Houtte P amp Aernoudt E 1976 Considerations on the crystal and the strain symmetry inthe calculation of deformation textures with the Taylor theory Mater Sci Engng 23 1122

Wasserman G 1963 Der Eindeg ufrac14 mechanischer Zwillingsbildung auf die Entstehung der Walz-texturen Kubisch Flachenzentrierter Metalle Z Metallk 54 6165

Wenk H R Matthies S Donovan J amp Chateigner D 1998 BEARTEX a windows-basedprogram system for quantitative texture analysis J Appl Cryst 31 262269

Williams R O 1962 Trans Metall Soc AIME 224 129

Young C T Steele J H amp Lytton J L 1973 Characterization of bicrystals using Kikuchipatterns Metall Trans 4 20812088



in the last 20 years (Kocks et al 1998) including variations of the Taylor model(Taylor 1938) eg the full-constraints (FC) model (van Houtte amp Aernoudt 1976Canova et al 1984 Harren et al 1989 Toth et al 1989) the relaxed-constraints(RC) model (Honnenot amp Mecking 1981 Canova et al 1984) the self-consistent model(Molinari et al 1987 Tome amp Canova 1998 Lebensohn amp Tome 1993) and shy nite-element based schemes (eg Becker 1991 Bronkhorst et al 1992 Beaudoin et al 1995) However discrepancies still exist between the measured texture evolution andthe simulation results (Stout et al 1988) Notable among these dinoterences are thepresence or absence of certain ideal components dinoterent ratios of ideal texture com-ponents as a function of strain a much slower texture evolution in experiment thanmodel and the spread of orientations around the ideal components that is relatedto the intensity of a preferred orientation These dinoterences have remained over theyears despite the introduction of dinoterent types of models In these models withthe exception of some shy nite-element (eg Beaudoin et al 1996 Becker 1995) andn-site self-consistent approaches (see for example Canova et al 1992) the slip pat-tern is modelled across a whole grain and the microstructural changes relate onlyto grain shape and possibly to twinning Consequently a radical new approachmay be required to resolve these issues One such approach may be to consider theevolution of the deformation-induced dislocation microstructures which takes placesimultaneously with the texture evolution This approach is introduced because thedislocation structure both reregects and modishy es the slip pattern thereby altering thetexture development

Before the microstructure can be included in a texture simulation however therole of the microstructure must be established by a simultaneous and careful consid-eration of the measured texture texture simulations and quantitative observations ofthe dislocation structure It is the aim of this paper to make this connection througha series of experiments and simulations that varies the texture and microstructuresimultaneously The vital parameter of choice to make these variations is the stack-ing fault energy (SFE) of the metal which is known to have a strong impact ontexture and microstructure For the dinoterent SFEs the range of possible texture andmicrostructure combinations is further increased by changing the temperature

To further clarify the impact of the SFE on texture and microstructures torsiondeformation is examined In torsion with its dyadic symmetry (only one twofoldsymmetry axis) the enotect of low SFE on the texture development is especially evidentdue to the formation of one special texture component (111)[1middot12] that is generallyeither lacking or very weak in the texture developed by the high-SFE metals (vanHoutte amp Aernoudt 1976 Gil Sevillano et al 1980) Torsion has other advantagesover rolling in trying to ascertain the various roles of the slip pattern and twinningStress-induced deformation twinning is postponed to larger strains in torsion sinceregow stresses are lower in torsion thereby allowing the texture to form by slip priorto twinning Shear banding is very minimal in torsion but would add complicationsto a similar analysis of texture during rolling

The outline of the paper is as follows First the experimental textures are pre-sented as a function of SFE and increasing strain for pure nickel and nickelcobaltsolid solutions Next the simulated textures are described including the assump-tions and parameters used to vary the texture in the simulations Both Taylor andself-consistent models are used A step-by-step comparison is made with experimentto see what can and cannot be explained currently by the simulations Finally the






nickel (9999) 240 80 random 20 12 296

36 21

69 40

37 21 573

Ni + 30 wt Co 150 53 random 19 11 296

38 22

83 48

Ni + 60 wt Co 20 44 random 20 12 296

32 18

58 36

38 22 573










crystal orientation

label fhklghuvwia
























e = 21(l)

e = 36(k)e = 48(g)e = 40(c)

e = 18( j )e = 22( f )e = 21(b)


537 K

(a)

e = 21(d )

(h)A A1

A1

A2

A2

A2

A1

A1

A1A2

Z

q

A2

A1 A2

A

C

C

CB

B

B

B

B

B

A

A AA

AA B

CC

C

(111)




(a)

296 K


537 K

e = 12 e = 11 e = 12

e = 21 e = 22 e = 18

e = 40 e = 48 e = 36

e = 21 e = 21

70

z

q

r

55

40

25

10

(b)

(c)

(d )

(e)

( f )

(g)

(h)A1

A2

A2

B2

B1

C

(i)

( j)

(k)

(l)

A1





(b)

(a)

0

0

004

008

012

Vf

randomlevel

nickelC component


016

012

008

004

Vf

random level




vMe







004

008

012

(c)

5

vMe

Vf

0 4321

randomlevel

nickelA2ndash

A1ndash




2 components












C partialfhklgh110i

296 KNi + 60 wt Co


A2 B2 Acurren1 A1

macrbres


A2 B2 Acurren1 A1

macrbres






















150

118

85

52

20

= 40 z

qe= 40e= 40e

= 10

TaylorFC

n = 19


evolution n = 47



r

(h)(e)(b)

= 40e= 40e= 40e

(i)( f )(c)

(g)(d)(a)








150

= 40

118

85

52

20

(h)

(i)

(e)

( f )

(b)

(g)(d )(a)

(c)

z

qe= 40e= 40e

= 25e= 25e= 25e

= 25e= 25e= 25

TaylorFC

n = 19



evolution n = 19e

r









z

q

r

180

135

90

45

00

00

ndash45

ndash90

ndash135

ndash180

(a) (b)



1 by twinning











1







































































6 Microstructure






(a)

(b)

(c)

1 m m

1 m m

5 m m

Z

q







1








(a)

(b)

05 m

1 m

05 m(c)


















60

40

20

0

ndash 40

ndash 60

ndash 20


diso

rien

tatio

n (d

eg)

0 2 4 6


8 10 12 14

111 fibre

60(a)

(b)

40

20

0

ndash 40

ndash 60

ndash 20


diso

rien

tatio

n (d

eg)

0 2 4 6 8




(a)

(b)

05 m m

1 m m






296

temperature (K)



A B A2

ES

LB

LB

TW

LB

LB

LB

ES

ES

ES

stac

king

fau

lt e

nerg

yhigh

low

nick

el

Ni + 30 w

t C

o

Ni + 60 w

t C

o

573

















1

























Appendix A




macro rotation

W


w locwndash pl




















system (s) if


F tot (A 3)


(b) Hardening law






References






































































nickel (9999) 240 80 random 20 12 296

36 21

69 40

37 21 573

Ni + 30 wt Co 150 53 random 19 11 296

38 22

83 48

Ni + 60 wt Co 20 44 random 20 12 296

32 18

58 36

38 22 573










crystal orientation

label fhklghuvwia
























e = 21(l)

e = 36(k)e = 48(g)e = 40(c)

e = 18( j )e = 22( f )e = 21(b)


537 K

(a)

e = 21(d )

(h)A A1

A1

A2

A2

A2

A1

A1

A1A2

Z

q

A2

A1 A2

A

C

C

CB

B

B

B

B

B

A

A AA

AA B

CC

C

(111)




(a)

296 K


537 K

e = 12 e = 11 e = 12

e = 21 e = 22 e = 18

e = 40 e = 48 e = 36

e = 21 e = 21

70

z

q

r

55

40

25

10

(b)

(c)

(d )

(e)

( f )

(g)

(h)A1

A2

A2

B2

B1

C

(i)

( j)

(k)

(l)

A1





(b)

(a)

0

0

004

008

012

Vf

randomlevel

nickelC component


016

012

008

004

Vf

random level




vMe







004

008

012

(c)

5

vMe

Vf

0 4321

randomlevel

nickelA2ndash

A1ndash




2 components












C partialfhklgh110i

296 KNi + 60 wt Co


A2 B2 Acurren1 A1

macrbres


A2 B2 Acurren1 A1

macrbres






















150

118

85

52

20

= 40 z

qe= 40e= 40e

= 10

TaylorFC

n = 19


evolution n = 47



r

(h)(e)(b)

= 40e= 40e= 40e

(i)( f )(c)

(g)(d)(a)








150

= 40

118

85

52

20

(h)

(i)

(e)

( f )

(b)

(g)(d )(a)

(c)

z

qe= 40e= 40e

= 25e= 25e= 25e

= 25e= 25e= 25

TaylorFC

n = 19



evolution n = 19e

r









z

q

r

180

135

90

45

00

00

ndash45

ndash90

ndash135

ndash180

(a) (b)



1 by twinning











1







































































6 Microstructure






(a)

(b)

(c)

1 m m

1 m m

5 m m

Z

q







1








(a)

(b)

05 m

1 m

05 m(c)


















60

40

20

0

ndash 40

ndash 60

ndash 20


diso

rien

tatio

n (d

eg)

0 2 4 6


8 10 12 14

111 fibre

60(a)

(b)

40

20

0

ndash 40

ndash 60

ndash 20


diso

rien

tatio

n (d

eg)

0 2 4 6 8




(a)

(b)

05 m m

1 m m






296

temperature (K)



A B A2

ES

LB

LB

TW

LB

LB

LB

ES

ES

ES

stac

king

fau

lt e

nerg

yhigh

low

nick

el

Ni + 30 w

t C

o

Ni + 60 w

t C

o

573

















1

























Appendix A




macro rotation

W


w locwndash pl




















system (s) if


F tot (A 3)


(b) Hardening law






References




































































crystal orientation

label fhklghuvwia
























e = 21(l)

e = 36(k)e = 48(g)e = 40(c)

e = 18( j )e = 22( f )e = 21(b)


537 K

(a)

e = 21(d )

(h)A A1

A1

A2

A2

A2

A1

A1

A1A2

Z

q

A2

A1 A2

A

C

C

CB

B

B

B

B

B

A

A AA

AA B

CC

C

(111)




(a)

296 K


537 K

e = 12 e = 11 e = 12

e = 21 e = 22 e = 18

e = 40 e = 48 e = 36

e = 21 e = 21

70

z

q

r

55

40

25

10

(b)

(c)

(d )

(e)

( f )

(g)

(h)A1

A2

A2

B2

B1

C

(i)

( j)

(k)

(l)

A1





(b)

(a)

0

0

004

008

012

Vf

randomlevel

nickelC component


016

012

008

004

Vf

random level




vMe







004

008

012

(c)

5

vMe

Vf

0 4321

randomlevel

nickelA2ndash

A1ndash




2 components












C partialfhklgh110i

296 KNi + 60 wt Co


A2 B2 Acurren1 A1

macrbres


A2 B2 Acurren1 A1

macrbres






















150

118

85

52

20

= 40 z

qe= 40e= 40e

= 10

TaylorFC

n = 19


evolution n = 47



r

(h)(e)(b)

= 40e= 40e= 40e

(i)( f )(c)

(g)(d)(a)








150

= 40

118

85

52

20

(h)

(i)

(e)

( f )

(b)

(g)(d )(a)

(c)

z

qe= 40e= 40e

= 25e= 25e= 25e

= 25e= 25e= 25

TaylorFC

n = 19



evolution n = 19e

r









z

q

r

180

135

90

45

00

00

ndash45

ndash90

ndash135

ndash180

(a) (b)



1 by twinning











1







































































6 Microstructure






(a)

(b)

(c)

1 m m

1 m m

5 m m

Z

q







1








(a)

(b)

05 m

1 m

05 m(c)


















60

40

20

0

ndash 40

ndash 60

ndash 20


diso

rien

tatio

n (d

eg)

0 2 4 6


8 10 12 14

111 fibre

60(a)

(b)

40

20

0

ndash 40

ndash 60

ndash 20


diso

rien

tatio

n (d

eg)

0 2 4 6 8




(a)

(b)

05 m m

1 m m






296

temperature (K)



A B A2

ES

LB

LB

TW

LB

LB

LB

ES

ES

ES

stac

king

fau

lt e

nerg

yhigh

low

nick

el

Ni + 30 w

t C

o

Ni + 60 w

t C

o

573

















1

























Appendix A




macro rotation

W


w locwndash pl




















system (s) if


F tot (A 3)


(b) Hardening law






References















































































e = 21(l)

e = 36(k)e = 48(g)e = 40(c)

e = 18( j )e = 22( f )e = 21(b)


537 K

(a)

e = 21(d )

(h)A A1

A1

A2

A2

A2

A1

A1

A1A2

Z

q

A2

A1 A2

A

C

C

CB

B

B

B

B

B

A

A AA

AA B

CC

C

(111)




(a)

296 K


537 K

e = 12 e = 11 e = 12

e = 21 e = 22 e = 18

e = 40 e = 48 e = 36

e = 21 e = 21

70

z

q

r

55

40

25

10

(b)

(c)

(d )

(e)

( f )

(g)

(h)A1

A2

A2

B2

B1

C

(i)

( j)

(k)

(l)

A1





(b)

(a)

0

0

004

008

012

Vf

randomlevel

nickelC component


016

012

008

004

Vf

random level




vMe







004

008

012

(c)

5

vMe

Vf

0 4321

randomlevel

nickelA2ndash

A1ndash




2 components












C partialfhklgh110i

296 KNi + 60 wt Co


A2 B2 Acurren1 A1

macrbres


A2 B2 Acurren1 A1

macrbres






















150

118

85

52

20

= 40 z

qe= 40e= 40e

= 10

TaylorFC

n = 19


evolution n = 47



r

(h)(e)(b)

= 40e= 40e= 40e

(i)( f )(c)

(g)(d)(a)








150

= 40

118

85

52

20

(h)

(i)

(e)

( f )

(b)

(g)(d )(a)

(c)

z

qe= 40e= 40e

= 25e= 25e= 25e

= 25e= 25e= 25

TaylorFC

n = 19



evolution n = 19e

r









z

q

r

180

135

90

45

00

00

ndash45

ndash90

ndash135

ndash180

(a) (b)



1 by twinning











1







































































6 Microstructure






(a)

(b)

(c)

1 m m

1 m m

5 m m

Z

q







1








(a)

(b)

05 m

1 m

05 m(c)


















60

40

20

0

ndash 40

ndash 60

ndash 20


diso

rien

tatio

n (d

eg)

0 2 4 6


8 10 12 14

111 fibre

60(a)

(b)

40

20

0

ndash 40

ndash 60

ndash 20


diso

rien

tatio

n (d

eg)

0 2 4 6 8




(a)

(b)

05 m m

1 m m






296

temperature (K)



A B A2

ES

LB

LB

TW

LB

LB

LB

ES

ES

ES

stac

king

fau

lt e

nerg

yhigh

low

nick

el

Ni + 30 w

t C

o

Ni + 60 w

t C

o

573

















1

























Appendix A




macro rotation

W


w locwndash pl




















system (s) if


F tot (A 3)


(b) Hardening law






References



































































(a)

296 K


537 K

e = 12 e = 11 e = 12

e = 21 e = 22 e = 18

e = 40 e = 48 e = 36

e = 21 e = 21

70

z

q

r

55

40

25

10

(b)

(c)

(d )

(e)

( f )

(g)

(h)A1

A2

A2

B2

B1

C

(i)

( j)

(k)

(l)

A1





(b)

(a)

0

0

004

008

012

Vf

randomlevel

nickelC component


016

012

008

004

Vf

random level




vMe







004

008

012

(c)

5

vMe

Vf

0 4321

randomlevel

nickelA2ndash

A1ndash




2 components












C partialfhklgh110i

296 KNi + 60 wt Co


A2 B2 Acurren1 A1

macrbres


A2 B2 Acurren1 A1

macrbres






















150

118

85

52

20

= 40 z

qe= 40e= 40e

= 10

TaylorFC

n = 19


evolution n = 47



r

(h)(e)(b)

= 40e= 40e= 40e

(i)( f )(c)

(g)(d)(a)








150

= 40

118

85

52

20

(h)

(i)

(e)

( f )

(b)

(g)(d )(a)

(c)

z

qe= 40e= 40e

= 25e= 25e= 25e

= 25e= 25e= 25

TaylorFC

n = 19



evolution n = 19e

r









z

q

r

180

135

90

45

00

00

ndash45

ndash90

ndash135

ndash180

(a) (b)



1 by twinning











1







































































6 Microstructure






(a)

(b)

(c)

1 m m

1 m m

5 m m

Z

q







1








(a)

(b)

05 m

1 m

05 m(c)


















60

40

20

0

ndash 40

ndash 60

ndash 20


diso

rien

tatio

n (d

eg)

0 2 4 6


8 10 12 14

111 fibre

60(a)

(b)

40

20

0

ndash 40

ndash 60

ndash 20


diso

rien

tatio

n (d

eg)

0 2 4 6 8




(a)

(b)

05 m m

1 m m






296

temperature (K)



A B A2

ES

LB

LB

TW

LB

LB

LB

ES

ES

ES

stac

king

fau

lt e

nerg

yhigh

low

nick

el

Ni + 30 w

t C

o

Ni + 60 w

t C

o

573

















1

























Appendix A




macro rotation

W


w locwndash pl




















system (s) if


F tot (A 3)


(b) Hardening law






References



































































(b)

(a)

0

0

004

008

012

Vf

randomlevel

nickelC component


016

012

008

004

Vf

random level




vMe







004

008

012

(c)

5

vMe

Vf

0 4321

randomlevel

nickelA2ndash

A1ndash




2 components












C partialfhklgh110i

296 KNi + 60 wt Co


A2 B2 Acurren1 A1

macrbres


A2 B2 Acurren1 A1

macrbres






















150

118

85

52

20

= 40 z

qe= 40e= 40e

= 10

TaylorFC

n = 19


evolution n = 47



r

(h)(e)(b)

= 40e= 40e= 40e

(i)( f )(c)

(g)(d)(a)








150

= 40

118

85

52

20

(h)

(i)

(e)

( f )

(b)

(g)(d )(a)

(c)

z

qe= 40e= 40e

= 25e= 25e= 25e

= 25e= 25e= 25

TaylorFC

n = 19



evolution n = 19e

r









z

q

r

180

135

90

45

00

00

ndash45

ndash90

ndash135

ndash180

(a) (b)



1 by twinning











1







































































6 Microstructure






(a)

(b)

(c)

1 m m

1 m m

5 m m

Z

q







1








(a)

(b)

05 m

1 m

05 m(c)


















60

40

20

0

ndash 40

ndash 60

ndash 20


diso

rien

tatio

n (d

eg)

0 2 4 6


8 10 12 14

111 fibre

60(a)

(b)

40

20

0

ndash 40

ndash 60

ndash 20


diso

rien

tatio

n (d

eg)

0 2 4 6 8




(a)

(b)

05 m m

1 m m






296

temperature (K)



A B A2

ES

LB

LB

TW

LB

LB

LB

ES

ES

ES

stac

king

fau

lt e

nerg

yhigh

low

nick

el

Ni + 30 w

t C

o

Ni + 60 w

t C

o

573

















1

























Appendix A




macro rotation

W


w locwndash pl




















system (s) if


F tot (A 3)


(b) Hardening law






References



































































004

008

012

(c)

5

vMe

Vf

0 4321

randomlevel

nickelA2ndash

A1ndash




2 components












C partialfhklgh110i

296 KNi + 60 wt Co


A2 B2 Acurren1 A1

macrbres


A2 B2 Acurren1 A1

macrbres






















150

118

85

52

20

= 40 z

qe= 40e= 40e

= 10

TaylorFC

n = 19


evolution n = 47



r

(h)(e)(b)

= 40e= 40e= 40e

(i)( f )(c)

(g)(d)(a)








150

= 40

118

85

52

20

(h)

(i)

(e)

( f )

(b)

(g)(d )(a)

(c)

z

qe= 40e= 40e

= 25e= 25e= 25e

= 25e= 25e= 25

TaylorFC

n = 19



evolution n = 19e

r









z

q

r

180

135

90

45

00

00

ndash45

ndash90

ndash135

ndash180

(a) (b)



1 by twinning











1







































































6 Microstructure






(a)

(b)

(c)

1 m m

1 m m

5 m m

Z

q







1








(a)

(b)

05 m

1 m

05 m(c)


















60

40

20

0

ndash 40

ndash 60

ndash 20


diso

rien

tatio

n (d

eg)

0 2 4 6


8 10 12 14

111 fibre

60(a)

(b)

40

20

0

ndash 40

ndash 60

ndash 20


diso

rien

tatio

n (d

eg)

0 2 4 6 8




(a)

(b)

05 m m

1 m m






296

temperature (K)



A B A2

ES

LB

LB

TW

LB

LB

LB

ES

ES

ES

stac

king

fau

lt e

nerg

yhigh

low

nick

el

Ni + 30 w

t C

o

Ni + 60 w

t C

o

573

















1

























Appendix A




macro rotation

W


w locwndash pl




















system (s) if


F tot (A 3)


(b) Hardening law






References


















































































150

118

85

52

20

= 40 z

qe= 40e= 40e

= 10

TaylorFC

n = 19


evolution n = 47



r

(h)(e)(b)

= 40e= 40e= 40e

(i)( f )(c)

(g)(d)(a)








150

= 40

118

85

52

20

(h)

(i)

(e)

( f )

(b)

(g)(d )(a)

(c)

z

qe= 40e= 40e

= 25e= 25e= 25e

= 25e= 25e= 25

TaylorFC

n = 19



evolution n = 19e

r









z

q

r

180

135

90

45

00

00

ndash45

ndash90

ndash135

ndash180

(a) (b)



1 by twinning











1







































































6 Microstructure






(a)

(b)

(c)

1 m m

1 m m

5 m m

Z

q







1








(a)

(b)

05 m

1 m

05 m(c)


















60

40

20

0

ndash 40

ndash 60

ndash 20


diso

rien

tatio

n (d

eg)

0 2 4 6


8 10 12 14

111 fibre

60(a)

(b)

40

20

0

ndash 40

ndash 60

ndash 20


diso

rien

tatio

n (d

eg)

0 2 4 6 8




(a)

(b)

05 m m

1 m m






296

temperature (K)



A B A2

ES

LB

LB

TW

LB

LB

LB

ES

ES

ES

stac

king

fau

lt e

nerg

yhigh

low

nick

el

Ni + 30 w

t C

o

Ni + 60 w

t C

o

573

















1

























Appendix A




macro rotation

W


w locwndash pl




















system (s) if


F tot (A 3)


(b) Hardening law






References



































































150

= 40

118

85

52

20

(h)

(i)

(e)

( f )

(b)

(g)(d )(a)

(c)

z

qe= 40e= 40e

= 25e= 25e= 25e

= 25e= 25e= 25

TaylorFC

n = 19



evolution n = 19e

r









z

q

r

180

135

90

45

00

00

ndash45

ndash90

ndash135

ndash180

(a) (b)



1 by twinning











1







































































6 Microstructure






(a)

(b)

(c)

1 m m

1 m m

5 m m

Z

q







1








(a)

(b)

05 m

1 m

05 m(c)


















60

40

20

0

ndash 40

ndash 60

ndash 20


diso

rien

tatio

n (d

eg)

0 2 4 6


8 10 12 14

111 fibre

60(a)

(b)

40

20

0

ndash 40

ndash 60

ndash 20


diso

rien

tatio

n (d

eg)

0 2 4 6 8




(a)

(b)

05 m m

1 m m






296

temperature (K)



A B A2

ES

LB

LB

TW

LB

LB

LB

ES

ES

ES

stac

king

fau

lt e

nerg

yhigh

low

nick

el

Ni + 30 w

t C

o

Ni + 60 w

t C

o

573

















1

























Appendix A




macro rotation

W


w locwndash pl




















system (s) if


F tot (A 3)


(b) Hardening law






References



































































z

q

r

180

135

90

45

00

00

ndash45

ndash90

ndash135

ndash180

(a) (b)



1 by twinning











1







































































6 Microstructure






(a)

(b)

(c)

1 m m

1 m m

5 m m

Z

q







1








(a)

(b)

05 m

1 m

05 m(c)


















60

40

20

0

ndash 40

ndash 60

ndash 20


diso

rien

tatio

n (d

eg)

0 2 4 6


8 10 12 14

111 fibre

60(a)

(b)

40

20

0

ndash 40

ndash 60

ndash 20


diso

rien

tatio

n (d

eg)

0 2 4 6 8




(a)

(b)

05 m m

1 m m






296

temperature (K)



A B A2

ES

LB

LB

TW

LB

LB

LB

ES

ES

ES

stac

king

fau

lt e

nerg

yhigh

low

nick

el

Ni + 30 w

t C

o

Ni + 60 w

t C

o

573

















1

























Appendix A




macro rotation

W


w locwndash pl




















system (s) if


F tot (A 3)


(b) Hardening law






References







































































































































6 Microstructure






(a)

(b)

(c)

1 m m

1 m m

5 m m

Z

q







1








(a)

(b)

05 m

1 m

05 m(c)


















60

40

20

0

ndash 40

ndash 60

ndash 20


diso

rien

tatio

n (d

eg)

0 2 4 6


8 10 12 14

111 fibre

60(a)

(b)

40

20

0

ndash 40

ndash 60

ndash 20


diso

rien

tatio

n (d

eg)

0 2 4 6 8




(a)

(b)

05 m m

1 m m






296

temperature (K)



A B A2

ES

LB

LB

TW

LB

LB

LB

ES

ES

ES

stac

king

fau

lt e

nerg

yhigh

low

nick

el

Ni + 30 w

t C

o

Ni + 60 w

t C

o

573

















1

























Appendix A




macro rotation

W


w locwndash pl




















system (s) if


F tot (A 3)


(b) Hardening law






References



































































(a)

(b)

(c)

1 m m

1 m m

5 m m

Z

q







1








(a)

(b)

05 m

1 m

05 m(c)


















60

40

20

0

ndash 40

ndash 60

ndash 20


diso

rien

tatio

n (d

eg)

0 2 4 6


8 10 12 14

111 fibre

60(a)

(b)

40

20

0

ndash 40

ndash 60

ndash 20


diso

rien

tatio

n (d

eg)

0 2 4 6 8




(a)

(b)

05 m m

1 m m






296

temperature (K)



A B A2

ES

LB

LB

TW

LB

LB

LB

ES

ES

ES

stac

king

fau

lt e

nerg

yhigh

low

nick

el

Ni + 30 w

t C

o

Ni + 60 w

t C

o

573

















1

























Appendix A




macro rotation

W


w locwndash pl




















system (s) if


F tot (A 3)


(b) Hardening law






References






































































1








(a)

(b)

05 m

1 m

05 m(c)


















60

40

20

0

ndash 40

ndash 60

ndash 20


diso

rien

tatio

n (d

eg)

0 2 4 6


8 10 12 14

111 fibre

60(a)

(b)

40

20

0

ndash 40

ndash 60

ndash 20


diso

rien

tatio

n (d

eg)

0 2 4 6 8




(a)

(b)

05 m m

1 m m






296

temperature (K)



A B A2

ES

LB

LB

TW

LB

LB

LB

ES

ES

ES

stac

king

fau

lt e

nerg

yhigh

low

nick

el

Ni + 30 w

t C

o

Ni + 60 w

t C

o

573

















1

























Appendix A




macro rotation

W


w locwndash pl




















system (s) if


F tot (A 3)


(b) Hardening law






References



































































(a)

(b)

05 m

1 m

05 m(c)


















60

40

20

0

ndash 40

ndash 60

ndash 20


diso

rien

tatio

n (d

eg)

0 2 4 6


8 10 12 14

111 fibre

60(a)

(b)

40

20

0

ndash 40

ndash 60

ndash 20


diso

rien

tatio

n (d

eg)

0 2 4 6 8




(a)

(b)

05 m m

1 m m






296

temperature (K)



A B A2

ES

LB

LB

TW

LB

LB

LB

ES

ES

ES

stac

king

fau

lt e

nerg

yhigh

low

nick

el

Ni + 30 w

t C

o

Ni + 60 w

t C

o

573

















1

























Appendix A




macro rotation

W


w locwndash pl




















system (s) if


F tot (A 3)


(b) Hardening law






References

















































































60

40

20

0

ndash 40

ndash 60

ndash 20


diso

rien

tatio

n (d

eg)

0 2 4 6


8 10 12 14

111 fibre

60(a)

(b)

40

20

0

ndash 40

ndash 60

ndash 20


diso

rien

tatio

n (d

eg)

0 2 4 6 8




(a)

(b)

05 m m

1 m m






296

temperature (K)



A B A2

ES

LB

LB

TW

LB

LB

LB

ES

ES

ES

stac

king

fau

lt e

nerg

yhigh

low

nick

el

Ni + 30 w

t C

o

Ni + 60 w

t C

o

573

















1

























Appendix A




macro rotation

W


w locwndash pl




















system (s) if


F tot (A 3)


(b) Hardening law






References



































































(a)

(b)

05 m m

1 m m






296

temperature (K)



A B A2

ES

LB

LB

TW

LB

LB

LB

ES

ES

ES

stac

king

fau

lt e

nerg

yhigh

low

nick

el

Ni + 30 w

t C

o

Ni + 60 w

t C

o

573

















1

























Appendix A




macro rotation

W


w locwndash pl




















system (s) if


F tot (A 3)


(b) Hardening law






References



































































296

temperature (K)



A B A2

ES

LB

LB

TW

LB

LB

LB

ES

ES

ES

stac

king

fau

lt e

nerg

yhigh

low

nick

el

Ni + 30 w

t C

o

Ni + 60 w

t C

o

573

















1

























Appendix A




macro rotation

W


w locwndash pl




















system (s) if


F tot (A 3)


(b) Hardening law






References














































































1

























Appendix A




macro rotation

W


w locwndash pl




















system (s) if


F tot (A 3)


(b) Hardening law






References




















































































Appendix A




macro rotation

W


w locwndash pl




















system (s) if


F tot (A 3)


(b) Hardening law






References



































































macro rotation

W


w locwndash pl




















system (s) if


F tot (A 3)


(b) Hardening law






References














































































system (s) if


F tot (A 3)


(b) Hardening law






References


































































Stacking fault energy and microstructure effects on...

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Transcript of Stacking fault energy and microstructure effects on...