TEXTURE EVOLUTION DURING THE BIAXIAL STRETCHING OF FCC SHEET METALS

Using a rate-sensitive crystal plasticity model together with the full constraint Taylor theory, the 
formation of textures during biaxial stretching of FCC sheet metals is investigated in detail. Three-dimensional 
lattice rotation fields, orientation evolution and polycrystalline texture development are 
simulated for the entire range of biaxial strain ratio. The investigation discloses the paths of orientation 
development and respective stable end orientations, as well as the relation between the evolution paths 
and the biaxial strain ratio. Our results show that the formation of textures depends mainly on the 
behaviour of the α- and βρ fibres in biaxial stretching. The strain ratio affects the composition of the βρ-fibre, as well as the flow direction and velocity of orientations towards and along α and βρ, and thus results in different biaxial-stretching textures. The predictions of FCC biaxial-stretching textures are compared with experimental observations reported in literature. Finally, we discuss the influence of complex strain paths on texture formation.


INTRODUCTION
The regular arrangement of atoms which exists in a single crystal leads to anisotropy in the single crystal.For a polycrystal comprised of grains with the same crystal structure, its macroscopic properties are anisotropic if there exists a non-uniform distribution of grain orientations.This type of preferred orientation distribution is termed "texture".
Accordingly, a textured polycrystal merely reflects the well known anisotropy of single crystals.Such macroscopic anisotropy of metals is a critical factor for subsequent fabrication processes, and influences as well the mechanical, thermal and electrical properties of products.To study the behaviour of such anisotropy, it is necessary to investigate the formation of crystallographic textures during manufacturing processes.
Almost all manufacturing and metallurgical processes can result in textures.For instance, the cube and Goss textures in FCC metals result mainly from recrystallization in hot rolling or annealing processes.Cold rolling usually produces the copper-type textures (for high stacking-fault energy FCC metals) or the brass-type textures (for low stacking-fault energy FCC metals).Numerous investigations have been carded out for numerical simulations of texture formation.Because the simulations must consider the orientation change of every grain in a polycrystal, they involve very lengthy calculations.Such simulations have become feasible only quite recently, because of the development copper and Cu-30%Zn.The initial textures in the sheets were rolling textures.For all the metals considered, Kohara (1981) observed that the main component for equibiaxial stretching was represented as < O11 >//ND, which is significantly different from rolling textures.He also observed that even for a relative small equivalent stretching strain (0.2) the sheet textures were quite different from the initial textures.These features were also observed by Starczan et al. (1981) for an aluminum-manganese alloy.Using the Taylor theory, Bunge (1970) investigated the dependence of the rolling texture on relative broadening..In that case, the strain state is equivalent to the biaxial stretching.He found that rolling textures were strongly influenced by the relative broadening.The above investigations suggest that the effects of texture development are important during biaxial stretching of sheet metals and that such effects should not be neglected.
In this study, the formation of FCC textures during biaxial stretching of sheet metals is investigated.Three-dimensional lattice rotation fields in Euler space (1, and 2, Bunge's notation (Bunge, 1969) are investigated numerically, using a rate-sensitive crystal plasticity model together with the full-constraint Taylor theory (T6th et al., 1989;   Neale et al., 1990; Zhou, 1992; Zhou et al., 1992).By examining the characteristics of the three-dimensional fields, the development paths of crystallographic orientations during stretching are obtained in Euler space, and the corresponding stable end orientations are determined for different strain ratios.The evolution of individual orientations during deformation are traced to demonstrate the obtained formation paths of FCC biaxial-stretching textures.Texture evolution is simulated for various f'txed strain ratios, and results are compared with the experimental observations of Kohara (198i) and Starczan et al., (1981).Finally, resultant textures under biaxial stretching with complex strain paths are simulated and compared with the experiments reported by Starczan et al., (1981).The change of the resultant textures with strain paths and initial textures are discussed.

RATE-SENSITIVE CRYSTAL PLASTICITY MODEL AND CONSTRAINT CONDITIONS
The crystal plasticity model adopted in the present work is based on the rate-sensitivity of slip in the slip systems of a crystal.Such sensitivity is expressed by a power-law relationship between the shear rate )'s" and the resolved shear stress xs (Hutchinson, 1976  , x o sgn (,)IT'I X0--' where rn is the strain-rate sensitivity index, Td is the reference shear rate, assumed to be constant and equal for all slip systems in this analysis and x 0 is the reference shear stress, which depends on the amount of strain hardening.All geometrically possible slip systems are activated with this rate-sensitive model.In the present work, the 12 111 (110 ) slip systems of a FCC crystal are considered.
For a specified strain rate D, the deviatoric stress can be derived uniquely from the following: In this relation, rn'=b'xn' is the Schmid tensor for the slip system s, b' and n* being the unit slip direction and unit slip plane normal of the system in the deformed configuration, respectively.The relation between g and x, on system s is: (3) From ( 1) and (3), shear rates T" on slip systems can be determined.
The orientation change of a given grain during deformation can be described by the corresponding rotation velocity field I ( 2 ( 32 sin 1-hi3 cos l)/Sin where is the lattice spin defined with respect to the sample axes, which is related to the difference between the imposed velocity gradient L and the velocity gradient produced by plastic slip (Hutchinson, 1976 The re-orientation of the grain is thus determined from the shear rates ,, (5) and (4).
In the present work, the stretching axes X and X of a thin FCC sheet metal are assumed to remain fixed with respect to the laboratory reference system during biaxial stretching.Here the X and X axes lie in the plane of the sheet and the X axis is aligned with the sheet normal direction.Using the full constraint Taylor theory, the strain-rate tensor D of each grain in the sheet, which is equal to the velocity gradient L, can be specified in the sample system by where Dll> 0 and the strain ratio r D22/ D. We only consider the cases of 0 < p < 1.This implies that Dll and D22 are the major and minor strain-rates, respectively.
Using the rate-sensitive crystal plasticity model and the boundary conditions described above, the three-dimensional lattice rotation fields in Euler space and paths of orientation development during deformation, as well as biaxial-stretching textures, are investigated for FCC sheet metals.The m-value is rn 0.005 in this investigation to represent cold working.The calculation procedure is as follows.First, for a crystal with a specific orientation, the strain increment state is set to correspond to the appropriate boundary conditions.Then (2) is solved using an iterative Newton-Raphson scheme, and the deviatoric stress state and the shear slip increment distribution are determined.Finally, the corresponding lattice rotation increment of the crystal and its re-orientation are obtained from (5) and (4), respectively.The final orientations of grains in a sheet describe the respective polycrystalline texture.

LATTICE ROTATION FIELDS IN EULER SPACE
Three-dimensional rotation fields in Euler space are now investigated under biaxial stretching for various strain ratios p.For a specific orientation in Euler space, the lattice rotations of orientations in the range 20 x 20 x 20 around it are calculated with respect to a von Mises equivalent-strain increment A, -A l/1 + p + p2 0.02, using the rate-sensitive crystal plasticity model (m 0.005).Some sections of these fields are shown in Figures 1 6 (p 0, 0.1, 0.17, 0.25, 0.5, 1).In these figures, the directions of the arrows specify the orientation change, and their lengths indicate the corresponding magnitude (multiplied by the magnification MA 3) for the equivalent-strain increment A The central orientations of these rotation fields are Goss (0 , 45 , 90), brass (35.26 , 45 , 90), S (56.79 , 29.21 , 63.43) and Taylor (90 , 27.37 , 45), respectively, which are the so-called "ideal orientations" of FCC rolling textures.The behaviour of these ideal orientations also plays an important role under biaxial stretching (Zhou et al., 1993).It is seen that, when 0 _< p < 0.16 (Figures 1 and 2), the corresponding rotation fields are similar to that for p 0 (see Zhou et al., 1992).In such cases, the rotation fields converge to Goss in the directions of and 2, but diverge away from Goss in the direction of .Therefore, the preferred flow of the rotation fields is along the direction of , away from Goss towards positions near brass.At the Goss orientation, 0. Orientations first converge to positions (where 0) at or near brass with a pattern similar to an eddy, and then flow away towards positions (where -0) at or near Taylor, passing positions near S. In Figures 1-3d (left), the rotation field around Taylor appears to diverge for the section 2 45" By examining the corresponding three-dimensional rotation fields, however, we find that orientations finally stabilize at positions near Taylor, i.e. three-dimensional convergency occurs in these cases.The main disparity among the described rotation fields is that, by increasing the p-value from 0, the positions of their ideal orientations change gradually from brass or Taylor to those near brass or Taylor.In addition, the larger the p-value in this range, the more slowly orientations rotate.
The rotation fields for 0.21 _< p _< 1 (Figures 4-6) are totally different from those corresponding to 0 _< p < 0.16.Instead of reaching positions near Taylor, the preferred Figure 1 Lattice rotation fields around Goss, brass, S and Taylor for the case p O. ttttttttttt Figure 2 Lattice rotation fields around Goss, brass, S and Taylor for the case p 0.1.TEXTURE DURG BIAXIAL STRETCHING 93 Figure 3 Lattice rotation fields around Goss, brass, S and Taylor for the case p 0.17   flow of these rotation fields is towards positions between Goss and brass for 0.21 < p < 0.5, towards Goss for 0.5 < p < 1, or towards a fibre including Goss and brass under equibiaxial stretching (p 1).In this p-value range, the larger the p-value, the more quickly orientations rotate if they are in the vicinity of Taylor and S, and the slower are the rotations for orientations in the vicinity of Goss and brass.For 0.16 < p < 0.21, the corresponding rotation fields are more complex.Some orientations rotate towards positions near Taylor, which is similar to the cases of p < 0.16.The others rotate towards positions between Goss and brass, which is similar to the cases where p > 0.21.We shall refer to these as "transition fields".
Examining these rotation fields shows that there are three important types of fibres in the fields.During biaxial stretching, orientations will rotate first to the fibres and them move along the fibres.These fibres are the o, lip and /fibres.Here the o-fibre is such that the < 011 > crystallographic direction is parallel to the sheet normal direction, with Euler angles (0 < 1 < 90, 45, 90), including the Goss and brass orientations.
The Euler angles of the ),-fibre are comprised of (90 , 0 < < 45 , 45), including the Taylor and copper (90 , 35.26 , 45).The Euler angles of the l]o-type fibres will change with the applied strain ratio p, but they include a position Bo between Goss and brass in the o-fibre and a position T o between Taylor and 19.47 in the ),-fibre, respectively, by passing through a position S near the S orientation.The positions of the three fibre types are shown schematically in Euler space in Figure 7. cube G Figure 7 The schematic positions of the ct-, Ipand T-fibres in Euler space.
To better understand the predicted rotation fields, we first consider certain positions of the I0-type fibres and then investigate the behaviour of the three fibres under biaxial s.tretchin.g.Figures 8, 9 and 10 illustrate the distributions of the relative lattice spin f-+IKMD along the three fibres for various strain ratios2 where D is the von Mises equivalent strain-rate.The position and negative values of f in Figures 8, 9 or 10 signify that the respective orientations rotate along the fibres towards and away from the G, O or B 0 positions, respectively From these figures, we observe the end (B and T P 0 and intermediate (So) points of the 13-type fibres, which are shown in Table 1 for p 0, 0.1, 0.17, 0.25, 0.5 and 1.The -values are the same as those obtained by Bunge (1970).For 0.5 < p < 1, the B position is the same as Goss.Under plane strain stretching (p 0), B and T are identical to brass and Taylor, respectively.In fact, a -fibre does not exist under equibiaxial stretching (p 1) since, for this case, orientations move directly to the o-fibre and no rotation occurs along the entire o-fibre.Consequently we may consider the entire tx-fibre as B1.0.The curve for p= 1 in Figure 10 corresponds the development path of an initial orientation near T1.0.This curve is only a reference used to compare the rotation velocities with those for the other strain ratios.
Figure 8 shows that orientations in the tx-fibre rotate along the fibre towards the corresponding B 0 positions during biaxial stretching for 0 < p < 0.5.They move very slowly towards the Goss position for the cases of 0.5 < p < 1, and do not rotate under equibiaxial stretching (p 1).It is seen in Figure 9 that, for all the strain ratios (0 < p < 1) considered, orientations in the ),-fibre ow along this fibre towards the respective T o positions during deformation.The larger the strain ratio p, the slower the rotation is along the ct-and T-fibres.Figu.re 10 Distribution of the relative lattice spin along the 13p-fibre.Positive and negative values of f signify that the respective orientations rotate along the fibre towards and away from the Bp position, respectively.Figure 10 describes the rotation behaviour of the Ip-fibres (B_p-S-To) During biaxial p stretching, orientations rotate along these fibres towards the T position for the cases where 0 < p < 0.16, but towards the B p position for 0.21 < p < 1.For the cases 0.16 < p < 0.21, some orientations rotate towards B p while others move towards Tp.
However, the rotation is extremely slow for such p-values (e.g., p 0.17) and can be neglected in these cases.The closer the p-value is to this range, the slower the corresponding rotation is along the [p-fibre.The rotation velocity is relatively large for p > 0.5, but it is small for p < 0.25, particularly for orientations between S_ and Tp.Among the orientations in the I-fibre, the orientations near a position corresponding to 2 78 rotate with the greatest rate for the p-values (0 < p < 1) considered in the present work.
The above behaviour of the three fibre-types, particularly the 0-and p-fibres, plays a determinant role in the formation of FCC biaxial-stretching textures, resulting in different types of textures.This will be discussed later.Following the investigation of the behaviour of the three fibre types, we now examine the stability of orientations during biaxial stretching, using the criterion of stability for orientations discussed by Zhou et al. (1992).In the criterion, an orientation g=(, , 2) remains stable during deformation only if the following conditions are satisfied: (7) Using the criterion (7) together with the obtained lattice rotation fields, the stable end orientations under biaxial stretching have been determined for the various p-values.
For the cases of 0 < p < 0.16, the stable end orientations are the respective T positions When the p-value is in the range of 0.21 < p < 0.5, by contrast, the stable orientations are the B p positions.Both Tp and B p are the stable end orientations for 0.16 < p < 0.21.There is the same stable end orientation for 0.5 < p < 1, which is the Goss orientation.Under equibiaxial stretching, all orientations rotate towards the a-fibre and finally stabilize in this fibre if stretching is sufficiently large.Table 2 lists the stable end orientations and their rotation rates (dh d d2), gradients (, -, ), divergences div and relative ODF intensity changes (f/f)g for p 0, 0.1 0.17, 0.2, 0.5 and 1.Here the (/f)g values are calculated from the following continuity equation   The values of at T o are larger than zero.These do not satisfy the condition (7) for stability.However, as mentioned earlier, examining the corresponding three- dimensional rotation fields shows that orientations around T_ move indirectly to T o and P finally stabilize at positions near T 0. Therefore, we can still consider T o as a stable end orientation.

TEXTURES
It is seen from the above investigations that different p-values result in different lattice rotations fields, which in turn lead to different paths of orientation development during biaxial stretching.In order to further explore the paths of orientation development during biaxial stretching, the orientation evolution of individual grains are now investigated in Euler space using the rate-sensitive model (m 0.005).As examples, Figure 11 illustrates the rotation development of three typical grains in Euler space during biaxial stretching with p 0.1, 0.25, 0.5 and 1 (for p 0, see Figure 7a in Zhou et al., 1992).The original orientations of these grains are (1 , 1 , 89), (20 , 20 , 89) and (89 , 36 , 46), respectively.Their rotation paths are representative.
From this investigation and the previously obtained rotation fields, we see that an orientation rotates towards and along either the tx-or I -fibres during biaxial stretching p Figure 12 illustrates the schematic models of orientation development during biaxial stretching.For the cases 0 < p < 0.16 (Figure 12a), orientations develop during deformation either directly into the 13p-fibre, or first into the 0t-fibre, then move slowly along tx to I, and finally rotate very slowly along [p towards their stable end positions near T o.By ontrast, for 0.21 < p < i (Figure 12c), orientations rotate first either into the ix-fibre or the 3p_-fibre, then move slowly along tz or 13, and f'mally stabilize at their stable end positions near B. With an increase of the p-value in this range, more orientations rotate directly towards the ix-fibre than the 13_-fibre.Although Bo Goss for 0.5 < p < 1, orientations rotating along 1 first arrive at positions close to the ix-fibre and then move very slowly towards Goss.For 0.16 < p < 0.21 (Figure 12b), the rotation features of orientations close to the T-fibre are similar to the cases of 0 < p < 0.16.However, those close to the tz-fibre are similar to the cases of Figure 12 Schematic models of orientation development during biaxial stretching: (a) 0 < p < 0.16, (b) 0.16 < p < 0.21, (c) 0.21 < p < 1, (d) p=l.0.21 < p < 0.5.Under equibiaxial stretching (Figure 12d), all orientations rotate towards the t-fibre directly and finally stabilize at their end positions in this fibre, for instance (Figures 1 la, c), the orientations in the vicinity of the cube and copper positions rotate towards and stabilize at positions near Goss and brass at large equibiaxial stretching, respectively.
The rotating velocity towards the o-and Ip-fibres is much greater than that along the fibres.As indicated in the last section, changing the p-value affects both the rotation velocities towards and along the fibres.The larger the strain ratio p, the slower is the rotation of orientations towards and along the t-fibre.With an increase of the p-value, the rotation velocity of orientations rotating towards and along the Ip-fibre decreases for the range 0 < p < 0.16, but increases for the range 0.21 < p < 1. Orientations rotate very slowly along the -fibre for 0.16 < p < 0.21.Under equibiaxial stretching, orientations far removed frorn the t-fibre rotate quickly towards the fibre, but when they arrive at positions in the vicinity of the fibre, they rotate towards the fibre with small velocities.
The formation of a deformation texture obviously depends on the rotation paths of orientations and the rotation velocities.The above results show that changing the pvalue not only changes the rotation paths, e.g., the compositions of the I fibre and the flow direction along the fibre, but also changes the rotation velocities towards and along the t-and -fibres.Therefore, different p-values result in different biaxial- stretching textures.Since an orientation rotates towards and along either the t-or Ifibres during biaxial stretching for 0 < p < 1, the formation of the corresponding textures depends on the behaviour of the two fibres and initial textures.Therefore, the possible components of biaxial-stretching textures can be predicted from the characteristics of the t-and it_-fibres as well as the obtained paths of orientation development, by using a criterion of stability for a texture componen.tdiscussed by Zhou et al. (1992).In the criterion, a texture component at position g is stable during deformation as long as (, 2) 0, ((g)ff(g))g > 0.
(9) Here (g) is the ODF intensity at g, which can be approximated by a finite number of discrete orientations, or can be deduced from the continuity equation (8) of texture development (Clement et al., 1979; Gilormini et al., 1990).
A strong I-fibre, mainly the part between Sand T, is expected to occur in the biaxial-stretclaing textures for 0 < p < 0.16.This is because, in these cases, most orientations rotate into the p-fibre and then move very slowly along the fibre towards T_, particularly after they pass the S position.Therefore, most of them will still be destdbuted between S and T at large deformation.If stretching is not very large, a weak t-fibre would also a component in the textures.For the case of 0.21 < p < 0.5, both ct-(from Goss to B o) and l-fibres would consist of the corresponding texture components, since odenthtions rotate first either into the t-fibre or the Ip-fibre, then move slowly along t or i towards the B_ position.By increasing the p-value in this range, the t-and -fibres become relatively strong and weak, respectively, as the rotation velocity along I increases with increasing p-value.Since orientations rotate very slowly along theand I-fibres, both fibres also form the textures for 0.16 < p < 0.21.When 0.5 < p < 1, the corresponding textures would only be comprised of the entire 0-fibre with a density peak of orientation distribution at the Goss position.This is because the rotation of orientations is very slow along the p-fibre but is relatively quick along the -fibre, and more orientations rotate directly towards the m-fibre than The final texture of equibiaxial stretching (p 1) would be the entire the Ip-fibre.
tx-fibre, with a random orientation-distribution if grains are randomly oriented before stretching.
In order to corroborate the above results regarding the formation of textures in biaxial stretching, deformation textures of FCC polycrystals have been simulated using the rate- sensitive crystal plasticity model (m 0.005).The stretching increment was specified as A( 0.0125.At the beginning of deformation, a thin sheet comprised of 800 grains randomly oriented was considered.Isotropic slip-hardening is employed here.Figure 13 illustrates the ODF plots of the simulated textures at a total von Mises equivalent strain A( 0.75 for p 0, 0.1, 0.17, 0.25, 0.5 and 1. These simulated textures demonstrate the formation of textures in biaxial stretching described above.It is shown that most grains have reached the positions in the vicinity of the ct-and [-fibres at ee 0.75.A strong 1-fibre dominates the texture with a weak ix-fibre forp 0 and 0.1 (Figure 13a, b).As the p-value increases, the -fibre deteriorates gradually and meanwhile the weight of the ix-fibre increases (Figure 13c, d).When p > 0.5, the I.-fibre disappears and the t-fibre gains dominance over the corresponding textures (Figure 13e, f).Such biaxial-stretching textures were predicted by Bunge (1970).Figure 13f demonstrates that the distribution of grain orientations along the or-fibre is uniform for the case of equibiaxial stretching.Figure 14 gives the simulated ODF plots at e= 0.25 for p 0.1 and 1.It is shows that, at this relatively small level of strain, many orientations have arrived at positions in the vicinity of theand [-fibres for p 0.1, and at positions in the vicinity of the t-fibre under equibiaxial stretching (p 1).This implies that biaxial-stretching textures form even at small levels of strain.Consequently, their effects on the behaviour of sheet metals during biaxial stretching cannot be ignored.Uniaxially compressed FCC metals have been reported to have a fibre texture with < 011 > normal to the compressive plane (i.e., the it-fibre) (Dillamore et al., 1965).Because the strain for equibiaxial stretching is identical to that of uniaxial compression, the texture of equibiaxial stretching sho.uld be similar to the compression texture.Kohara (1981) measured texture development in equibiaxial stretching for aluminum, copper and Cu-30%Zn.The observed initial textures-at the beginning of deformation are comprised of the S plus cube components for aluminum, the cube component for copper and the brass plus 113 < 211 > components for Cu-30%Zn, respectively.For all the metals considered, Kohara observed that the main component of equibiaxial stretching was represented as < 011 > // ND (X3).Our predicted texture of FCC polycrystals in equibiaxial stretching is identical to this experimental result.
In his experimental measurements, Kohara also observed that there exist different minor components in the equibiaxial-stretching textures.The ideal orientations of these minor components also belong to the t-fibre, and are 110} < 112 > (brass) for aluminum, {110} <111> for copper and |110} < 110> plus {110} <001 > (Goss) for Cu 30% Zn, respectively.The presence of these minor components implies that the distribution of grain orientations along the t-fibre is not uniform.Kohara did not discover experimentally how the minor components evolved from the initial textures, but he attributed the observed differences in the minor components to the different stacking-fault energies of these metals.Such minor components of equibiaxial-stretching textures are not present in the predicted textures obtained from the simulation with a initial random distribution of orientations in this section (Figure 13f).It is worth mentioning that our predictions of FCC biaxial-stretching textures are only based on crystallographic slip in the 12{ 111 < 110 > slip systems.Other deformation micromechanisms (e.g., twinning) are not considered.Therefore, the corresponding predictions only really correspond to the behaviour of thigh SFE metals such as aluminum and copper.In reference (Zhou et al., 1992), we have shown that the minor brass component in the equibiaxial-stretching texture for aluminum results from the initial texture which contains the S component, since the stable end position of the S orientation under equibiaxial stretching is near the brass orientation.This demonstrates how important a role an initial texture plays in the formation of biaxial-stretching texture.Unfortunately, the paths of orientation development predicted in the last section cannot explain the 110 < 111 > component present in the final texture of stretching copper sheets.Figure 1 la shows that orientations in the vicinity of the cube orientation rotate towards the Goss orientation and finally stabilize in the vicinity of the Goss position.
Starczan et al. (1981) measured texture development in biaxial stretching for an aluminum-manganese alloy.At the beginning of deformation, orientations of grains in the metal sheets distribute between ideal orientations {110} <322> and {113} < 111 > (a copper-type rolling texture).As predicted in the present work, they found that equibiaxial stretching reinforces the weight of 110} < 322 >, which is located in the tz-fibre, and weakens all others.For uniaxial tension, they observed that the deformation almost equally reinforces all initial preferred orientations.Their measured textures with respect to plane strain stretching show that the initial I]-fibre is reinforced with a peak formed around the S position.Our prediction for the formation of plane- strain stretching textures is consistent with their measurements.Actually, the strain state of plane strain stretching is identical to that of plane strain compression (ideal flat rolling).It is well known that the ]3-fibre texture with a S peak is a typical rolling texture for FCC pure metals.
The formation of textures during biaxial stretching has not received much attention in the past.It was thought perhaps that forming limit strains for sheet metals are not large enough to allow a strong texture to evolve, so that the effects of textures formed during stretching could be neglected.However, in their measurements Kohara (1981) and Starzcan et a/.(1981) observed that textures in sheets at a relatively small equivalent strain (0.2) were quite different from the respective initial textures.Our predicted textures (Figure 14) at the strain = 0.25 for p 0.1 and p=l exhibit the same features as their mesurements.Therefore, both their measurements and our predictions show that the effects of textures developing during biaxial stretching cannot be ignored for annealed FCC sheet (pure or close to pure) metals.

Resultant Textures Under Complex Strain Paths
In addition to direct strain paths, Starzcan et al. (1981) also measured texture evolution along complex strain paths.In their measurements there were two samples.One experienced larger equibiaxial stretching (co 0.32) followed by uniaxial tension (ee 0.06), while the other was subjected to a smaller uniaxial tension (e e= 0.085) followed by equibiaxial stretching (e = 0.32).Thus, both samples had approximately the same final strain.It was observed "curiously" that both samples had practically the same final texture.Starzcan et al. wondered if the resultant texture was only dependent on the final values of strain, indepentantly of the the successive strain paths, or whether their observation was merely a coincidence.
By examining the formation of biaxial-stretching textures and lattice rotation rates predicted in the present investigation, we can explain this "curious" phenomenon.As shown in Figure 10, orientations rotate relatively quickly towards the t-flbre under equibiaxial stretching, if the initial texture is comprised of the I-fibre.When they arrive at positions in the vicinity of the t-fibre, their rotation becomes slow.Consider first the complex uniaxial tension-equibiaxial stretching path.Since uniaxial tension reinforces the initial I-fibre texture, as observed by Starzcan et al., the deformation of e 0.085 would not change the components of the initial texture.Instead, orientations would converge further to the -fibre.Therefore, during the following equibiaxial stretching, orientations would rotate from the -fibre towards the t-fibre.As a result, the resultant texture at the equivalent strain 0.32 would be dominated by the t-fibre component such that the < 011 > crystallographic direction is parallel to the sheet normal.On the other hand, for the complex equibiaxial-uniaxial stretching path, orientations first rotate towards the t-fibre during equibiaxial stretching and at e = 0.32 the main component of the corresponding texture would be the t-fibre.The following uniaxial tension o 0.06 would produce little change in such a texture.Consequently, the resultant textures for the two samples would be almost identical to each other.
To simulate Starzcan's experiments for the complex strain paths, two thin sheets comprised of 800 grains were analyzed numerically using the rate-sensitive crystal plasticity model (m 0.005).At the beginning of deformation, their initial texture was assumed to be a copper-type rolling texture comprised of orientations distributed around the I]0-fibre (see Figure 7a in Zhou et al., 1992), which is similar to the initial texture used by Starzcan et al. (1981).One sheet was subjected to equibiaxial stretching (e 0.32) followed by uniaxial tension (e 0.06), and the other subjected to uniaxial tension (e 0.085) followed by equibiaxial stretching (ee 0.32).Figure 15 shows the simulated resultant textures corresponding to the two strain paths.As observed by Starzcan et al. in their experiments, the two textures are indeed identical to each other.
It is worth pointing out that, for such a coincident result, the large equibiaxial stretching and initial copper-type rolling texture play an important role.For instance, there are two strain paths: equibiaxial stretching-plane strain stretching and plane strain stretching-equibiaxial stretching.The equivalent strain is the same for both plane strain stretching and equibiaxial stretching in these paths.Suppose that the initial texture were a R-type rolling texture having orientations distributed around the cube and S positions.
Then, according to the orientation evolution results described in last section, a I0-fibre with a minor cube component would comprise the resultant textures corresponding to both paths.However, the texture with respect to the former path would have more orientations distributed in the 0-fibre between S and Taylor than that for the latter path, while the latter path would result in more orientations distributed in the I-fibre between S and brass than the former path.This is because in the former path orientations first rotate towards the t-fibre during equibiaxial stretching and then towards the 130fibre during the following plane strain stretching.By contrast, in the latter path, orientations rotate into both t-and 130-fibres during plane strain stretching and then move towards the tz-fibre during the following equibiaxial stretching.Therefore, the two strain paths would not result in the same final texture.The above features of the final textures are illustrated in Figure 16, which shows two simulated textures at e--0.(a) (b) Figure 16 ODF intensities of pricted result texes under biiM setcng wi e complex s pas for e R-initiM texture: (a) equibiiM setchMg ( 0.20) ple sMn setchMg (6 0.20) d (b) ple sain setching (6  0.20) equibiiM setching (6 0.20).
for the above complex strain paths.Here the initial R-type rolling texture is the same as Figure 7b in reference (Zhou et al., 1992), which is similar to the initial texture used by Kohara (1981) in his experimental investigation of texture development under equibiaxial stretching for aluminum.
By examining the paths of orientation development and lattice rotation rates during the biaxial stretching of FCC sheet metals, we see that virtually identical final textures can occur at the same final equivalent strain for certain cases of two different complex stretching paths applied in reverse order.To explain this, we first discuss the case of an initially random texture.According to our predictions (Figure 13), three types of texture would occur under biaxial stretching.,,Theyare,,the 13p-fibre (0 < p < 0.16, "range 1"), the p-plus o-fibres (0.16 < p < 0.5, range 2 ) and the t-fibre (0.5 < p < 1, "range 3").This implies that, during deformation, a change of strain path within the above three ranges of p-value has little influence on the final texture.However, a change of strain path from range 1 to 3 or from range 3 to 1 would lead to differences in the f'mal textures.For example, suppose again that the strain paths are equibiaxial stretching (te 0.2) and plane strain stretching (ee 0.2) and plane strain stretching ( 0.2) equibiaxial stretching (e 0.2).The final textm'es, as predicted in Figure 17, show somewhat more orientations distributed in the 0-fibre in the former path than the latter.Such a feature also exists if the initial texture is the copper-type texture, as shown in Figure 18 where the initial texture is the same as that used for Figure 15. Figure 17 ODF intensities of predicted resultant textures under biaxial stretching with the complex strain paths for an initially random texture: (a) equibiaxial stretching (e 0.20) plane strain stretching ( --0.20) and (b) plane strain stretching (6 0.20) equibiaxial stretching (6 0.20).

CONCLUSIONS
Using the rate-sensitive crystal plasticity model together with the full constraint Taylor theory, the formation of textures for the biaxial stretching of FCC sheet metals has been investigated in the present work.Three-dimensional lattice rotation fields, the orientation evolution during stretching and polycrystalline textures were simulated for the entire range of strain ratio p.In these investigated, the paths of orientation development and respective stable end orientations have been obtained, and the relation between the evolution path and applied strain ratio determined.It has been observed that, during biaxial stretching, an orientation rotates towards and along either the or-or p-fibres, and finally stabilizes at its stable end positions.
The strain ratio affects the composition of the -fibres as well as the flow direction and velocity of orientations towards and along the fibres, thus results in different biaxial- stretching textures.If an initially random texture is considered, three types of texture will occur under biaxial stretching: the lp-fibre (0 < p < 0.16), range 1), the l-plus p oc-fibres (0.16 < p < 0.5, range 2), and the oc-fibre (0.5 < p < 1, range 3).Therefore, the effects of textures developing during biaxial stretching cannot be ignored.
In the present work, we have also investigated the influence of complex strain paths on the formation of textures.It has been found that a change of strain path within the previously defined ranges "1", "2" or "3" of the p-value has little influence on the final textures.However, a change of strain path from range "1" to range "3" or from range "3" to range "1" would lead to observable differences in the textures.IX: 16.8   (a) (b) Figure 18 ODF intensities of predicted resultant textures under biaxial stretching with the complex strain paths for the copper-type initial texture: (a) equibiaxial stretching (Ee 0.20) plane strain stretching (Ee 0.20) and (b) plane strain stretching f 0.20) equibiaxial stretching ( 0.20) Figure 3  Lattice rotation fields around Goss, brass, S and Taylor for the case p 0.17.

Figure 4 Figure 5
Figure 4  Lattice rotation fields around Goss, brass, S and Taylor for the case p 0.25.

Figure 6
Figure 6 Lattice rotation fields around Goss, brass, S and Taylor for the case p 1.
fibre F.igure $ Distribution of the relative lattice spin along the tz-fibre.Positive and negative values of signify that the respective Orientations rotate along the fibre towards and away from the G position, 25 30 35 4-4.5   p olong the 7 fibre F.igure 9 Distribution of the relative lattice spin along the 7-fibre.Positive and negative values of flsignify that the respective orientations rotate along the fibre towards and away from the O position, respectively.

Figure 14 ODF
Figure 14 ODF intensities of predicted FCC biaxial-stretching textures at

Figure
Figure l$ ODF intensities of predicted resultant textures under stretching with the complex strain paths for the copper-type initial texture: (a) equibiaxial stretching E 0.32) uniaxial tension E

Table 2
Rates of change (),.gradients ()dl/O), divergences (div ) and relative rates of change of the ODF intensity (f/f) for the stable end orientations