A study of through-thickness texture gradients in rolled sheets

A Study of Through-Thickness T exture Gradients in Rolled Sheets
O.ENGLER,M.-Y .HUH,and C.N.TOME
´A method to simulate shear effects and through-thickness texture gradients in rolled sheet materials
is introduced.The strain history during a rolling pass is idealized by superimposing a sine-shaped evolution of the ␧˙13shear component to a plane-strain state.These generic strain histories are enforced in a visco-plastic self-consistent (VPSC)polycrystal deformation model to simulate texture evolution as a function of through-thickness position.The VPSC scheme is deemed superior to a full constraints (FC)or relaxed constraints (RC)approach,because it allows one to fully prescribe diagonal and shear-strain-rate components while still accounting for grain-shape effects.The idealized strain states are validated by comparison with deformation histories obtained through finite-element method (FEM)calculations.The through-thickness texture gradients are accounted for by introducing a relative variation of the sine-shaped ␧˙13shear with respect to the plane-strain component.The simulation results are validated,in turn,by comparison with typical examples of through-thickness texture gradients observed experimentally in rolled plates and in sheets of fcc and bcc materials.
I.INTRODUCTION
strongly affect the plastic properties of the rolled products (e.g.,References 5through 7).Severe through-thickness I N the literature,there are many attempts to model the
texture gradients have been reported in cases where the ratio evolution of the crystallographic texture accompanying roll-of the contact length between the roll and sample (l C )to the ing deformation with the help of Taylor-type deformation sheet thickness (t )is smaller than 1(with l C ϷΊR ⌬t ,where models;extensive reviews have been given,e.g.,by Hirsch R is the roll radius and ⌬t is the thickness reduction per and Lu ¨cke [1]and Aernoudt et al.[2]In such models,the pass).[8–13]However,also,in cases characterized by l C /t Ͼindividual crystallites are assumed to deform by slip on a 1,shear textures have been observed,which have been attrib-number of crystallographic slip systems,so as to accommo-uted to the friction between the rolls and sheet.[14–17]Such date the prescribed macroscopic strain rate (␧˙ij ).[3,4]The term conditions,viz.,large rolling draughts combined with high ␧˙ij is the symmetric part of the velocity gradient e ˙ij ϵu ˙i ,j .The friction,are typically encountered in practical materials pro-antisymmetric part,denoted ␻˙ij ,represents the plastic spin.cessing during breakdown rolling and multistand hot roll-In most simulations of rolling textures,the deformation ing operations.
in every grain is either approximated by a plane-strain state As a flat plate or sheet enters the rolling gap,the rolls (␧˙11ϭϪ␧˙33),and all other components (␧˙22,␧˙12,␧˙13,and induce a compressive stress in the normal direction (ND).␧˙23)are imposed to be zero (full constraints (FC)),or else,The sheet is thinned along the ND and is free to expand in the grain-stress components ␴13and ␴23are made zero based the rolling direction (RD),whereas lateral expansion in the on grain shape and stress continuity considerations (relaxed transverse direction (TD)is effectively restrained by fric-constraints (RC)).*However,both approximations represent
tion.[18]For the center layer of the workpiece,the net effect *Throughout this article,the rolling direction (RD),transverse direction is a condition of plane strain,whereas for the outer layers,(TD),and normal direction (ND)of a sheet are identified as directions 1,strong deviations from the plane-strain state prevail.As illus-2,and 3,respectively.
trated in Figure 1,the geometrical changes within the rolling a strong simplification,in that away from the center plane,gap lead to a nonzero component e ˙31ϭu
˙3,1of the displace-shear deformation takes place as a consequence of the rolling ment-rate gradient.It is clear that the value of e ˙31changes
boundary conditions.Factors like roll-gap geometry,includ-magnitude as well as sign during a rolling pass.[12]
Another,ing the geometrical changes during a rolling pass,friction even more severe,deviation from the plane-strain condition between the roll and the sheet contact surface,and tempera-may result from the friction between the roll surface and ture gradients upon hot rolling can cause severe deviations the sheet surface,which can be understood in terms of the from the plane-strain condition.Furthermore,these parame-friction-hill effect.[18]At the beginning of the deformation ters depend on the distance from the surface of the rolled within the plastic zone,i.e.,left from the neutral point,the sheet,giving rise to nonhomogeneous strain states and,as velocity of the material (V 0)is lower than that of the rolls a consequence,to different rolling textures at different (V R ),and the friction between the metal and the rolls tends through-thickness layers of the sheet,which,in turn,may
to draw metal into the roll gap (Figure 1).To the right from the neutral point,the metal velocity (V e )is higher than the roll velocity,so that the friction direction is reversed.The O.ENGLER and C.N.TOME
´are with the Materials Science and Technol-resulting friction hill imposes a component e ˙31ϭu ˙3,1,which ogy Division (MST-8),Los Alamos National Laboratory,Los Alamos,NM is positive at the entry of the rolling mill,zero at the neutral 87545U.S.A..M.-Y .HUH,Professor,is with the Division of Materials point,and negative at the exit of the rolling mill (Figure 1).Science and Engineering,Korea University,Seoul 136-701,Korea.Manuscript submitted October 26,1999.
In the present article,we introduce a novel method to
FE mesh,and texture,hardening,and anisotropy are updated
as deformation proceeds.[21–24]However,for the case of
rolling,the overall constraints are such that local deformation
is not likely to be sensitive to details in the local hardening
and anisotropy evolution(constitutive law).In addition,
FEM computations have the major disadvantage of being
extremely time consuming,especially if they are coupled to
a polycrystal constitutive law.
As an alternative,the evolution of strain can be deduced
using certain simplifying assumptions.The resulting strain
states are eventually fed into a deformation model so as to
derive the texture variations as a function of the strain state
and,thus,the through-thickness texture gradient.Lee and
Duggan[12]described the strain field in the rolling gap by Fig.1—Schematic representation showing the formation of the shear com-means of a highly simplified analytical model.This model
ponents e13and e31in a roll gap.considers both the shear component e
31,introduced by the
geometrical changes during a rolling pass,and the compo-
nent e13,caused by the friction between the rolls and sheet
surface.Based on these analytical expressions,Huh et
al.[17,19]used a Taylor approach to estimate shear textures simulate shear textures and through-thickness texture gradi-
by an idealized strain history during a rolling pass.Fedosseev ents in rolled sheet materials.Based on the ideas put forward
et al.derived the strain distribution for various rolling by Huh et al.,[19]the strain history during a rolling pass is
idealized by a plane strain superimposed on a simple sine-sequences based on the concepts of fluid dynamics(method shaped profile of the e˙13and e˙31shear-rate components.The of superposition of harmonic currents)to model through-resulting strain history is rationalized by comparing it with thickness texture gradients.[25,26]
results obtained by the finite-element method(FEM).To In all these approaches,the strain distribution derived with simulate the texture evolution and,in particular,the through-the various assumptions was eventually fed into a Taylor thickness texture gradients,the strain history at different FC deformation model to simulate the resulting textures. layers is input in a visco-plastic self consistent(VPSC)However,it is nowadays acknowledged that the Taylor FC deformation model.The simulation results obtained for fcc model is not well suited for simulating the texture changes and bcc structures are compared with typical examples of accompanying rolling deformation.With increasing reduc-through-thickness texture gradients observed experimentally tion,the grain thickness decreases and the prescription of
in rolled plates and sheets of various aluminum alloys and the␧˙
13and␧˙23shear components in the Taylor FC model
steels.becomes increasingly less meaningful;for infinitely flat It has erroneously been stated in the literature that the grains,only the remaining components are defined.This led textures observed at the sheet surface are caused by an to the development of the so-called RC models,[27,28]which accumulated shear induced by the friction between the roll since have proven to be superior in simulating the evolution and sheet surface.However,the required strong shear would
of rolling textures,particularly at high strains(e.g.,Refer-lead to an unrealistic shape change of the sheets.Further-
ences1and2).In the present application,however,the␧˙13 more,this view cannot account for the shear textures in the
shear has to be prescribed such as to account for the shear surface of reversibly rolled sheets.In this article,we will
superimposed on the plane-strain state.This means that the show that,even for a final strain distribution where the
␧˙13shear cannot be relaxed,preventing the application of accumulated shear strain is completely reversed,pronounced
an RC model.
through-thickness variations in texture may take place in
Alternatively,deformation textures can be modeled by the sheet.
means of a VPSC scheme.[29,30]Each orientation or“grain”
of a polycrystalline aggregate is regarded as an inclusion that II.SIMULATION OF SHEAR TEXTURES AND is embedded in and interacts with a homogeneous equivalent THROUGH-THICKNESS TEXTURE medium(HEM)with the average properties of the aggregate.
GRADIENTS The properties of the matrix(the HEM)are not known
a priori,however,but are adjusted“self-consistently”to The conventional way to determine the local strain evolu-
coincide with the average of all inclusions forming the aggre-tion in inhomogeneously deforming specimens is to employ
gate.In contrast to the Taylor-type models,in the VPSC the FEM.In what concerns rolling operations,the strain
model each grain deforms differently,depending on its rela-states in the sheet layers follow from enforcing an appro-
tive stiffness with respect to the HEM.In addition,a relax-priate set of boundary conditions.The information about
ation of the shear components takes place in individual grains the resulting strain history can then be input in polycrystal
as a consequence of evolving grain shape,while all compo-plasticity codes to model the corresponding crystallographic
nents of the overall strain rate may still be prescribed.Thus, texture and,hence,to simulate texture gradients.[15,20]This
a fully prescribed strain-rate tensor can be used,which favors method can be further refined by a coupling of the FE code
this approach for the simulation of through-thickness tex-with a polycrystal constitutive law.Within this approach,a
polycrystal texture is associated with each element of the ture gradients.
Table ler Indices and Euler Angles of the Orientations Characteristic of fcc and bcc Plane Strain and Shear Textures Miller Indices Euler Angles
{hkl}͗uvw͘(“Designation”)␸1⌽␸2Remarks {112}͗111͘(“C”)90deg35deg45deg fcc plane strain/bcc shear {123}͗634͘(“S”)59deg34deg65deg fcc plane strain
{011}͗211͘(“B”)35deg45deg0deg/90deg fcc plane strain/bcc shear {011}͗100͘(“Goss”)0deg45deg0deg/90deg(fcc plane strain)/bcc shear {111}͗112͘30deg/90deg54.7deg45deg bcc plane strain/fcc shear {111}͗110͘0deg/60deg54.7deg45deg bcc plane strain/fcc shear {112}͗110͘0deg35deg45deg bcc plane strain/fcc shear {001}͗110͘(“rotated cube”)0deg0deg45deg/45deg0deg0deg/90deg(bcc plane strain)/fcc shear III.THROUGH-THICKNESS TEXTURE symmetry is not generally justified in the case of pronounced GRADIENTS IN ROLLED SHEETS
shear deformation.
A.Analysis and Representation of Crystallographic
Textures B.Examples of Through-Thickness Texture Gradients in
fcc Metals
The experimental results of texture gradients reproduced
in this section were determined by conventional X-ray mac-Gradients of strain and strain rate manifest themselves in
the pronounced through-thickness texture gradients that have rotexture analysis(e.g.,Reference31).Pole figures were
measured from the sheets in back-reflection using a standard been described for various inhomogeneously deformed fcc X-ray texture goniometer.In order to analyze through-thick-
metals and alloys(e.g.,References9through11,15,33, ness texture gradients,various layers of the sheets have to and34).The example shown here to illustrate the texture be prepared sequentially by careful grinding,polishing,or
gradients in fcc materials pertains to a laboratory cold-rolled etching,or by an appropriate combination thereof.In the sample of a direct chill–cast commercial-purity aluminum,
AA1145.A specimen was machined from13mm hot gage following text,the layer within the sheet is indicated by the
parameter s,with sϭϩ1and sϭϪ1denoting the upper to6mm,so as to achieve an initially uniform through-and lower surface of the sheet,respectively,such that sϭ
thickness structure.Then,the specimen was cold rolled 0identifies the center layer.Note that,with an average reversibly to a0.6mm final thickness,corresponding to penetration depth or X-rays of the order of100␮m,X-ray
a total thickness reduction of90pct.In order to enforce diffraction is well suited for analysis of texture gradients in inhomogeneous deformation,the rolling was performed dry, sheets with a thickness in excess of,for example,1mm.
i.e.,without using a lubricant,which resulted in pronounced Comparison of X-ray results with data obtained by means of through-thickness texture gradients.[35]
the more tedious and time-consuming local-texture analysis
The texture in the center layer of the hot band mainly done by electron back-scattering diffraction showed very comprised orientations that are typical of plane-strain defor-good agreement between both techniques(e.g.,References
mation of fcc metals and alloys.In such textures,most 13and16).orientations are assembled along the so-called␤fiber,which After correction of the pole-figure data for background
runs through the Euler-angle space from the C orientation irradiation and defocusing error,complete orientation distri-{112}͗111͘through the S orientation{123}͗634͘to the B bution functions(ODFs),f(g),were computed according to
orientation{011}͗211͘(Figure2(a)and Table I).Close to the method of series expansion with spherical harmonic the surface,in contrast,typical shear textures were found functions.[32]In texture analysis,crystal orientations are
(Figure2(b)).The maximum texture intensity was obtained commonly denoted by the Miller indices{hkl}͗uvw͘,where in the45deg ND-rotated cube orientation{001}͗110͘.Fur-the first set of Miller indices indicates the direction of the
thermore,a pronounced scatter of{001}͗110͘toward crystal that is parallel to the ND and the second refers to{112}͗110͘and minor intensities of{111}͗uvw͘orientations
were also observed.
the crystal direction parallel to the RD.For quantitative
texture analysis,the orientations(g)are expressed as a set Figure3shows another example of an fcc shear texture, of three Euler angles:␸1,⌽,and␸2.Table I lists the Miller
obtained in a hot-rolled specimen of the aluminum alloy Al-indices and Euler angles of the most commonly observed 5.8pct Cu-0.4pct Zr.This material,with a composition orientations of rolled fcc and bcc metals.The textures are
equivalent to the superplastic alloy SUPRAL100but with represented by plotting isodensity lines in sections of con-a coarser grain size,of the order of20␮m,was laboratory stant␸1or␸2through the three-dimensional Euler space.
hot rolled at310ЊC to a thickness of6mm.The texture All experimental pole figures discussed in this article,irre-obtained for the layer sϭ0.67may be regarded as a transi-spective of the sheet layer analyzed,revealed orthotropic
tion between the two cases shown in Figure2.The typical sample symmetry within experimental accuracy.Therefore,plane-strain-texture␤fiber is still present,although weak, the ODF representation was confined to the familiar sub-
but it shows strong scatter toward the rotated cube orientation space of Euler space,with0degՅ(␸1,⌽,␸2)Յ90deg.It{001}͗110͘.This is obvious from the smear of the individual is noted,however,that the assumption of orthotropic sample
texture maxima in the various␸2sections“upward,”i.e.,
(b )
(a )
Fig.2—Hot band texture of commercial purity aluminum serving as an example of the through-thickness texture variations in fcc materials:(a )center layer,s ϭ0.0;and (b )surface layer,s ϭ1.0(by courtesy of S.Benum).
toward ⌽ϭ0deg.The rotated cube orientation again shows with ␣and ␥fiber orientations prevail,the surfaces show shear textures consisting of the Goss orientation {011}͗100͘scatter toward the {112}͗110͘shear component,whereas {111}͗uvw ͘orientations were not observed.Similar textures and orientations close to {112}͗111͘.[8,17,40]In contrast,in hot-rolled low-carbon steels,usually weak,rather uniform,have been observed in a variety of nonrecrystallizing high-strength Al alloys.
textures were observed,which can be attributed to the ran-domizing effect of the ␥-␣phase transformation during sub-It should be noted that,in addition to the friction and geometry effects,the evolution of through-thickness texture sequent cooling.[41]Lowering the finishing temperature has the effect that more deformation takes place in the ferrite gradients also depends on the material investigated.[10,11,36]While most Al alloys develop pronounced through-thickness range,which strengthens the hot-band texture and,in particu-lar,enhances through-thickness texture gradients.[42]
texture gradients,other fcc materials,in particular,materials with a low stacking-fault energy like brass,silver,and aus-The typical texture gradients of bcc sheet materials are illustrated in Figure 4for an interstitial-free (IF)steel sheet.tenitic steels,tend to show more-uniform textures throughout the sheet thickness.In high-purity copper,even under very A specimen of a Ti-alloyed IF steel with 0.0026pct C was warm rolled at 650ЊC (i.e.,in the ferritic range)from 5to inhomogeneous deformation conditions,no through-thick-ness texture gradients,but rather,strongly enhanced shear 3mm in one pass.[43]Similarly as described for the commer-cial-purity aluminum (Figure 2),the rolling was performed band formation,were observed.[37]This material dependence is not the subject of this article,however,and will not,without lubricant in order to enforce nonuniform deforma-tion throughout the sheet thickness.
therefore,be discussed here any further.
The center layer of the specimen (s ϭ0.0)shows a texture that is characteristic of a plane-strain deformation state (Fig-C.Examples of Through-Thickness Texture Gradients in ure 4(a)):under plane-strain conditions,bcc metals and bcc Metals
alloys tend to form fiber textures where most orientations are assembled along two characteristic fibers (marked in Very pronounced through-thickness texture gradients form in alloyed steel grades that undergo none or only partial Figure 4(a)),as follows.(1)The (mostly incomplete)␣fiber comprises orientations with a common ͗110͘direction phase transformation during hot rolling.[14,38,39]Whereas in the center layers of such inhomogeneously rolled sheets,parallel to the RD,i.e.,the orientations {hkl }͗110͘,including the orientations {001}͗110͘,{112}͗110͘,and {111}͗110͘.
typically the well-known plane-strain deformation textures
Fig.3—Hot band texture of Al-5.8pct Cu-0.4pct Zr(regular grain sized SUPRAL100)at the layer sϭ0.7showing a characteristic mixture of plane strain and shear texture components.(a)Complete ODF in␸2sections,(b)representation in the␸2ϭ45deg section that contains the most important rolling and shear texture components,and(c)schematic representation of the orientations in the␸2ϭ45deg section.
(2)The␥fiber comprises orientations with{111}parallel{112}͗111͘appear.The maximum intensity of these shear-
texture components is obtained at sϭ0.8,whereas in the to the ND,i.e.,the orientations{111}͗uvw͘,including
{111}͗110͘and{111}͗112͘.immediate surface texture,the sharpness decreases
slightly.[43]Raabe and Lu¨cke[39]reported very similar results The surface texture of the specimen mainly consisted of
a strong{011}͗100͘Goss orientation plus minor intensities in the hot bands of Cr-containing ferritic stainless steels.
Microstructural investigations of the microband arrange-close to{011}͗211͘and{112}͗111͘(Figure4(b)).From
Figure4and Table I,it may be seen that most of the relevant ment through the thickness of a rolled steel sheet with pro-
nounced through-thickness texture gradients have shown bcc orientations and fibers can be found in the␸2ϭ45deg
section of the Euler space.Therefore,the representation of that,in the center layers,where plane-strain conditions pre-
vail,the microbands formed at angles of approximately35 ODFs can be highly condensed by focusing merely on this
section rather than showing the entire ODF.Figure5shows deg to the RD,which is indicative of the arrangement of
the activated slip planes.[17]Close to the surface,where well-the␸2ϭ45deg sections of the textures in the IF steel as
obtained at the various layers.Again,at the center layer,the defined shear textures were observed,the microbands were typical plane-strain texture with the characteristic␣and␥
arranged approximately parallel to the RD.This highly fibers prevails.With increasing the parameter s,the rolling unusual arrangement of slip planes can only be achieved if
the local-stress tensor is strongly rotated by the superposition texture degrades;at sϭ0.4,a minimum in texture sharpness
is observed.This minimum is due to the transition from the of friction on the plane-stress state imposed by the rolls.
Note that fcc plane-strain and bcc shear textures(and, plane strain to the shear texture that dominates closer to the
sheet surface.Accordingly,with further increasing s,the vice versa,the bcc plane-strain and fcc shear textures)show
some characteristic similarities(Figures2through5and
shear components{011}͗100͘(Goss),{011}͗211͘,and
(b )
(a )
Fig.4—Hot band texture of an IF steel serving as an example of the through-thickness texture variations in bcc materials:(a )center layer,s ϭ0.0;and (b )surface layer,s ϭ1.0(by courtesy of B.Beckers).
Fig.5—(a )through (h )␸2ϭ45deg sections of the ODFs determined at various through-thickness layers s of the IF hot band steel shown in Fig.4(by courtesy of B.Beckers).
the sine-shaped profiles of e ˙13and e ˙31and the resulting
components ␧˙ij ,of the symmetric,and ␻˙ij ,of the antisymmet-ric,part of the velocity gradient (Eq.[2])are characterized by a single parameter,viz.,the value of its first maximum (step 4in Figure 6).For example,the curve of e ˙13in Figure
6will be denoted e ˙max
13ϭ3and the one of e ˙31as e ˙max 31ϭϪ1.
In the case of plane-strain deformation,e ˙31and e ˙13are expected to be so small that the resulting shear component ␧˙13becomes negligible.Conversely,for pronounced shear deformation to take place,i.e.,a large ␧˙13component,either the friction-induced shear component e ˙13or the geometry-induced shear component e ˙31must be large.However,because of the sine-shaped shear profiles,after a complete rolling pass,the integrated values e 13and e 31are zero,
inde-pendent of the respective amounts of e ˙13and e ˙31.As a conse-quence,a volume element would experience zero net shear Fig.6—Idealized evolution of the geometry-induced component e ˙31,the (␧13)as well as zero net rotation (␻13),such that the shape friction-induced component e ˙13,and the resulting shear ␧˙13ϭ␧˙31during a rolling pass.This sine-shaped strain rate history is subdivided into 13of an originally orthogonal element would still be orthogonal steps and used as a generic input for the present rolling texture simulations.
after deformation.Thus,the deformation of the overall roll-ing bite is plane strain,although the instantaneous local values may strongly deviate from the plane-strain condition.We will show here that,although the accumulated shear Table I).This resemblance is addressed in detail else-adds up to zero,the final texture depends strongly on the where.[44,45]
deformation history.IV .MODELING OF THROUGH-THICKNESS TEXTURE GRADIENTS IN ROLLED SHEETS B.Texture Simulations
A.Determination of the Strain Distribution
To simulate the rolling textures,the strain history was simulated by means of a VPSC deformation model briefly As already mentioned in the introduction,the strain state described in Section II.The calculations were performed during practical rolling operations may exhibit severe devia-with a fully prescribed velocity gradient e ˙ij of the form given tions from the idealized plane-strain condition that is defined by Eq.[1]and adopting various values of the shear strains as e ˙ij ϭ0for i j .Considering rolling as a two-dimensional e ˙13and e
˙31.problem,i.e.,e ˙22ϭe ˙12ϭe ˙21ϭe ˙23ϭe ˙32ϭ0and e ˙33ϭϪe ˙11,these deviations from the plane-strain state manifest themselves as nonzero contributions of the geometry-e ˙ij ϭ΂
1
e ˙130
00e ˙31
0Ϫ1
΃
[3]
induced shear component e ˙31and the friction-induced shear component e ˙13(Figure 1).Thus,the displacement gradient tensor e ˙ij becomes
To simulate the texture evolution,an aggregate composed of 500initially random orientations was deformed in 78steps (6rolling passes of 13steps each).In each step,a e ˙ij ϭ΂
e ˙11
0e ˙130
00e ˙31
0Ϫe ˙11
΃
[1]
different displacement-rate tensor is imposed,and the incre-mental deformation is controlled by enforcing ⌬e 11ϭ0.0175up to a total accumulated strain of e 11ϭ1.365,which The components ␧˙13,␧
˙31,␻˙13,and ␻˙31of the symmetric and approximately corresponds to a 75pct thickness reduction.antisymmetric parts of the velocity gradient e ˙ij are Simulations of fcc textures were performed with the usual twelve {111}͗110͘slip systems,i.e.,four {111}slip planes ␧˙13ϭ␧
˙31ϭ12(e ˙13ϩe ˙31)and ␻˙13ϭϪ␻˙31ϭ1
2
(e ˙13Ϫe ˙31)each containing three ͗110͘slip directions.In the simulations of bcc textures,two slip-system families,{110}͗111͘and [2]
{112}͗111͘,were considered (with equal critical resolved shear stress (␶crss )),which is commonly assumed to give a It has already been pointed out that the friction-induced
shear component e ˙13is positive at the entry of the rolling reasonable description of the slip characteristics in many bcc structures (e.g.,Reference 2).Since,for this highly mill,zero at the neutral plane,and negative at the exit of the rolling mill;for the geometry-induced shear component constrained forming problem,local deformation is mostly controlled by the boundary conditions,all simulations were e ˙31,the opposite behavior is anticipated (Figure 1).Accord-ingly,in the present model,the evolution of e ˙31and e ˙13and performed without hardening.
As outlined previously,the shear rates e ˙13and e ˙31were of the resulting shear rates (␧˙13ϭ␧˙31)and rotations (␻˙13ϭϪ␻˙31)is assumed to follow a simple sine-shaped profile varied according to a sine-shaped profile,so as to account for the changes in strain evolution from the entry to the during a rolling pass,as indicated in Figure 6.The numerical values of e ˙ij ,␧˙ij ,and ␻˙ij are referred to the principal strain exit of the rolling mill during one rolling pass.For the deformation texture simulations,the sine profiles were sub-rate e ˙11ϭ␧˙11,which is assumed to remain constant during a rolling pass and to be the same for all layers.Hereinafter,
divided into 13steps (Figure 6),which are regarded as。