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Materials Science and Engineering A 380 (2004) 365–377

Texture development in the cold rolling of IF steel P.S. Bate∗ , J. Quinta da Fonseca Manchester Materials Science Centre, The University of Manchester, Grosvenor Street, Manchester M1 7HS, UK Received 1 December 2003; received in revised form 31 March 2004

Abstract The development of deformation texture in ferrite has been measured in cold rolled IF steel. This has been compared, in a quantitative way, to the predictions of Taylor models—including those with relaxed constraints—and a finite element model with crystal plasticity constitutive laws. The finite element model gave much better prediction of the overall levels of orientation density but failed to predict the relatively high level of {0 0 1}00021 1 00003 texture which occurred at strains greater than about unity. That feature was predicted by relaxed constraint Taylor models. It is argued that that prediction is a coincidence, and either the finite element model cannot readily deal with the intragranular inhomogeneity of deformation in an adequate way, or that factors such as high-angle boundary migration may be important in the development of deformation texture. © 2004 Elsevier B.V. All rights reserved. Keywords: Texture; Plasticity; Modelling; Finite element analysis; Iron

1. Introduction The study of deformation textures has formed an important part of research into the plasticity of metals. Even though it does not reach the levels of complexity and difficulty involved in the prediction of recrystallisation textures—and is also, perhaps, less practically useful—the quantitative prediction of deformation textures remains a challenge. This quantitative prediction depended on the availability of adequate electronic computers, but the use of models based on Taylor’s [1] assumptions of uniform deformation and local equality of slip stresses on equivalent systems began to give quite reasonable approximations to deformation textures in cubic metals by the 1970s [2–5]. This coincided with the application of texture analysis techniques which removed some of the possible ambiguity associated with individual pole figures [see [6]], and allowed adequate comparison of experimental with predicted textures. Despite the initial success, it was clear that the Taylor model, at least in its original form, gave imperfect predictions. Perhaps the most famous example of this is the ‘texture transition’ occurring in FCC metals, where the texture developed depends on the stacking fault energy. Low stacking fault energy metals, such as silver, give a single com-

∗

Corresponding author. Tel.: +44-161-200-8842. E-mail address: [email protected] (P.S. Bate).

0921-5093/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2004.04.007

ponent {0 1 1}00022 1 10003 texture following flat rolling, whereas the Taylor model, using the usual octahedral slip systems, predicts a fibre of orientations from near {2 2 5}00025 5 40003 to {0 1 1}00022 1 10003, with the highest density at the first of those orientations. Various proposals have been made to account for the FCC ‘texture transition’, but it is fair to say that no wholly satisfactory solution has been found. Even where the stacking fault energy is high, as in aluminium, the original Taylor formulation only gives a first approximation to real texture evolution. The predicted textures are nearly always too well developed, i.e. too ‘sharp’, and the precise locations and relative orientation densities of the components deviate from those measured. Various attempts have been made to modify the Taylor model with a view to improving texture predictions. The self-consistent model [7], where the behaviour of sample orientations embedded in an ‘equivalent medium’ are modelled, has been used. A significant modification was the implementation of ‘relaxed constraints’, whereby specific components of the imposed deformation were excluded from the deformation imposed on the crystal volumes involved in the simulations [8,9]. This relaxation was justified in terms of grain shape: in a grain sufficiently elongated by deformation, a simple shear along the long axis could occur without incurring a large penalty associated with the inhomogeneous deformation needed to maintain material continuity. The success of this approach in FCC metals is questionable, but in BCC there have been indications that introducing relaxed

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constraints could be of benefit [10–12]. There are potential problems with such relaxed constraint models, however, one of which is that they, in common with the original ‘full constraint’ models, predict too sharp a texture. The use of finite element models with constitutive laws based on crystal plasticity have had a major influence on deformation texture predictions. Using such models allows the uniformity of deformation assumed in the Taylor models to be abandoned in a much less arbitrary way than in relaxed constraint Taylor models. The magnitude of the inhomogeneous deformation between grains, and the effect on texture development, predicted using such models is significant [13–18]. The predictions also indicate that simple rules for deviations from full constraint of particular grain orientations are unlikely to be appropriate, and the spatial disposition of neighbourhood orientations will have a significant effect. Most of this work has been done on FCC systems, particularly aluminium, and finite element predictions are generally better than those of Taylor models. Work on BCC metals has been more limited, but a comparison by Van Houtte et al. [19] showed that here, as well, finite elements predictions were generally better than Taylor model simulations. However, as mentioned above, there are indications that certain relaxed constraints can give quite good Taylor model predictions and this issue is worthy of further investigation. Another feature of most finite element simulations is that the number of elements used to represent a grain is very limited, often to one. This limits modelling the influence of intragranular inhomogeneity of deformation. In the work presented below, a series of experimental measurements in BCC iron is compared with both Taylor and finite element models to assess the effect of the intergranular and intragranular inhomogeneity of deformation—in so far as they can be readily modelled by finite elements—on texture development.

2. Experimental work 2.1. Material The material used was interstitial-free [IF] steel, with the composition given in Table 1, supplied by SSAB Tunnplåt, Sweden, and provided as hot rolled plate 28 mm thick. Sections of this plate, 50 mm wide, were cold rolled to 10 mm thickness, maintaining the original rolling direction [RD] throughout, using a rolling mill with 250 mm diameter rolls operating with a roll surface speed of 150 mm s−1 and with light oil lubrication. This material was then annealed at 750 ◦ C for 30 min in air. This gave full recrystallisation with

Table 1 The composition, in weight percent, of the IF steel C

N

Mn

Ti

Al

Fe

0.003

0.003

0.15

0.08

0.05

Remainder

Fig. 1. Section through the ODF at Euler angle ϕ2 = 45◦ for the recrystallised plate. The contour levels shown are multiples of random density.

a mean linear intercept grain size of 30 m. The texture was determined from EBSD measurements on the middle third of the thickness of a transverse cross section, with about 35 000 sample points covering an area of 17.6 mm2 . The continuous orientation distribution function [ODF] was determined using—as all textures in this work—harmonic series fitting [6] with a truncation at lmax = 22, and a representative section of this is shown in Fig. 1. The texture is quite typical of recrystallised ferritic steel plate, showing a tendency for 00021 1 10003 to align near the rolling plane normal [ND]. That recrystallisation texture is close to stable deformation texture components, and as such does not provide an ideal start for a study of deformation texture development. A texture that is not so stable was generated by cutting and machining the plate into five rectangular section strips, rotating these about the RD by 90◦ and joining them by autogenous TIG welds at the surface. The heat affected zones of this welding were assessed on as-rolled material, and only affected material within a radius of 2 mm. This compounded plate was then rolled, with 50 mm long sections taken from the leading edge to give material with a series of reductions. These are detailed in Table 2. Samples for texture determination were cut from the back end of the central component,

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Table 2 The dimensions, and resulting strains in the thickness [h] and width [w], of the rolled samples Specimen

Thickness [mm]

Width [mm]

−εh

A0 A1 A2 A3 A4 A5

9.52 7.44 5.73 4.48 3.46 2.18

38.02 38.88 39.27 39.57 40.00 Not recorded

0 0.247 0.508 0.754 1.012 1.474

which were then ground and polished to half thickness. The sampling ensured that the material had undergone a very close approximation to plane strain deformation. EBSD measurements were made on the middle third of width, which corresponds to the middle third of thickness in the original plate orientation, over a length in the RD of about 10 mm. Despite the cold rolled state, about half of the sample points gave valid orientation measurements in all the specimens. The number of orientations used for each sample texture was about 20 000, which satisfies any reasonable statistical criterion. The step size of this sampling was of the order of the grain size, the aim being to sample texture rather than provide local and spatially correlated orientation data. Spatial correlation of orientation is, of course, not included in the conventional definition of texture. There was concern about whether the sampling was influenced by orientation: it is known that X-ray line broadening and substructural density [20,21] vary with orientation in deformed

εw ± ± ± ± ±

0.004 0.004 0.005 0.005 0.008

0 0.024 0.034 0.041 0.051 –

−εw /εh ± ± ± ±

0.001 0.001 0.001 0.001

– 0.09 0.06 0.05 0.05 –

ferrite. This effect was assessed by using x-ray diffraction, to effectively give inverse pole figure densities in the ND, and by comparing textures derived using different levels of EBSD pattern quality. Some results from this exercise are shown in Fig. 2. No systematic effect was found when comparing EBSD and XRD data, and there was no significant effect of pattern quality cut-off until this was set artificially high. Because of these factors, there is a high degree of confidence that the measured textures are truly representative, with errors less than those likely to arise from the measurement of partial pole figures by X-ray diffraction and the subsequent manipulation of that data to give ODFs. 2.2. Texture development Sections of the ODFs of the rolled material are shown in Fig. 3. There is a very rapid change from the original texture—which is simply that shown in Fig. 1 rotated by

Fig. 2. Some results of trials used to check the validity of EBSD texture data. On the left are shown the ratios of XRD integrated peak intensities, for diffraction vectors parallel to the ND, to the relevant inverse pole figure densities predicted from the EBSD textures, at different strains. The unstrained specimen had consistently lower XRD intensities, but the ratios for the two reflections are in proportion. On the right, the effect of changing the pattern quality cut-off, and so reducing the number of data included, on the derived orientation densities for two representative orientations. That result is for the specimen rolled to a thickness strain of 0.75.

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Fig. 3. Sections through ODFs at Euler angle ϕ2 = 45◦ for the rolled specimens. The contour levels shown are multiples of random density, and the thickness strains for the relevant section are indicated.

90◦ about the rolling direction—to that with components characteristic of rolled ferrite. These lie on two fibres, one with 00021 1 00003//RD extending from {1 0 0}//ND to {1 1 1}//ND, known as the a fibre, and the other with 00021 1 10003//ND known as the g fibre. This latter fibre tends to occur, in modified form, after recrystallisation as shown in Fig. 1. The first significant texture component to form is near {1 1 2}00021 1 00003, followed by the gradual development of orientations comprising the α and γ fibres, with a near uniform orientation density along the α fibre after a strain of about 1.5. The highest orientation density on the γ fibre remains near 00021 1 00003//RD throughout. The maximum orientation density increases very little after a strain of 0.75. The general trend observed is similar to that observed previously, e.g. by Schläffer and Bunge [22].

3. Modelling The two basic techniques used to model the deformation texture of the IF steel—Taylor modelling and crystal plasticity finite element modelling [CPFEM]—have a common background in standard geometrical crystal plasticity, which will be detailed before describing the models and the results of their application.

3.1. Crystal plasticity If attention is restricted in scope to slip [via dislocation glide] as the plastic deformation mechanism and, for the time being, ignoring the elastic component of deformation, then slip on a system α with direction bα and slip plane normal nα at a rate of simple shear γ˙ α will give a contribution to the local rate of plastic deformation [velocity gradient] of: α α α LPα ij = bi nj γ˙

(1)

The basic kinematic problem in geometric crystal plasticity is to find the set of slip system activities which satisfy: 0001 1 mαij γ˙ α (2) DijP = (LPij + LPji ) = 2 α where DP is the symmetric plastic deformation [strain] rate and m is the Schmid tensor: 1 mPij = (biα nαj + bjα nαi ) (3) 2 There is no requirement for slip to occur to satisfy the antisymmetric (spin) component of LP . The mismatch between the antisymmetric components of LP and of the sum of mαij γ˙ α will lead to a change in crystal orientation with respect to the spatial coordinates. This is fundamental to the

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formation of deformation texture. It is also very easy to calculate once the slip activities are known. The problem is determining those activities, and the simplest way is to use the method proposed by Taylor [1]. 3.2. Taylor modelling The Taylor hypothesis—that the local deformation rate is equal to the overall deformation rate—has been used for the vast majority of deformation texture prediction. Even when a local DP is specified, the equations [2] are overdetermined. There will be five independent components of DP in the general case [six minus one because of volume constancy], and more than five independent slip systems. Elasticity is ignored, and subsets of five of the available slip systems which give non-singular solutions to equations [2] are found. From these, the ones that give minimum values of the plastic power: 0001 ˙ = W τ α γ˙ α (4) α

are selected as solution sets. The slip stresses, τ, are usually taken to be equal on all systems, though this is not necessary. There are generally multiple solutions for a particular deformation rate which give the same minimal plastic power. This leads to what has become known as “Taylor indeterminacy”. There are several ways of dealing with that indeterminacy. The individual solutions can all be accepted, though that leads to a dramatic growth in the population of orientations being used when predicting texture evolution. A single solution can be selected at random [5], or one can derive a continuous rotation field and use a type of fluid mechanical approach in orientation space [23]. It is also possible to invoke slip rate sensitivity, as in the finite element method. In this work, an average of the Taylor solutions was used which is essentially the same as the rate-insensitive limit of such viscoplastic solutions. The model was applied to sets of discrete orientations, with small steps of deformation L dt ≈ 0.01 and a simple forward Euler integration scheme. The full set of about 30 000 orientations from the EBSD measurement of the annealed material was used. The condition that the local deformation is equal to the overall imposed rate can be relaxed. The original, full constraint (FC) model is modified in relaxed constraint [RC] models by allowing specified components of LP to be free. This gives freedom of corresponding DP and the overall stress associated with those components is set to zero. The free components then adopt values which minimise plastic work with reduced subsets of slip systems in the kinematic condition, equation [2]. FC Taylor modelling was carried out using 00021 1 10003 slip directions, with either {1 1 0}, {1 1 0} + {1 1 2} or {1 1 0} + {1 1 2} + {1 2 3} slip planes. All slip systems were assigned equal slip stresses. The last of those slip system sets gives results which are very close to those of ‘pencil glide’, where any plane containing the slip direction is considered

369

[24]. The intermediate case, {1 1 0} + {1 1 2}, was used in RC modelling. With axis 1 being the RD, 2 being the transverse direction (TD) and 3 being the ND, the simple shears 13, 23, 12 and 13 + 23 combined were relaxed. The textures predicted using Taylor models are much sharper than those measured, and for comparison with experimental results have been convoluted with a Gaussian spread function [6] to allow better comparison of the forms of the predicted textures with the experimental ones. These are shown, for a plane strain of 0.75, in Fig. 4. The locations of high orientation density predicted using FC modelling are somewhat different to those observed experimentally. The addition of more slip planes shifts the dominant component from near{1 1 2}00024 6 10003 to {1 1 2}00021 1 00003, but even with the {1 1 0} + {1 1 2} + {1 2 3} set, the γ fibre and the associated spreading of the α fibre observed in the experimental textures are not predicted. The RC models gave slightly lower orientation densities—though still much greater than the experimental ones unless artificially spread. Relaxing the 13 or 12 simple shears gave no improvement in prediction, but relaxing 23 and especially a combination of 13 + 23 improved the situation significantly. The fit of the simulations to the experimental results was quantified using a correlation index ξ given by ξ =1−

J0017 2(0002J0003 − 1)

(5)

where J denotes texture index, corresponding to the integral of the square of the orientation distribution function [6]. JD is the texture index of the difference of the two textures and 0002J0003 is the mean of their texture indices. This correlation value ranges from unity for perfect correlation [the textures are the same] to zero with no correlation. Numerical trials showed that this function gave good agreement with different proportions of common orientations in point sets, including cases with different total numbers of points in the two sets, and serves as a useful single measure of the difference between model and experimental textures. The correlation index as a function of strain for the different Taylor models is shown in Fig. 5, and the beneficial effect of introducing 23, and especially 13 + 23, relaxed constraints at levels of strain greater than about 0.75 is clear. In contrast, relaxation of the 13 shear has a rather deleterious effect until the strains are greater than about 1. The Taylor model predicts textures that are too sharp, and this contributes to the generally poor correlation shown in Fig. 5. It is possible to compensate for this effect by introducing an artificial Gaussian spread to the predictions, as carried out to give the results shown in Fig. 4. Fig. 6 shows the best fit correlation index, and the degree of spread giving that best correlation, as functions of strain. This reinforces the observation that a 23 relaxed constraint, especially in conjunction with 13 freedom at the higher strains, leads to significantly better predictions of deformation texture. There are two deficiencies with this, however. The most obvious is the rather arbitrary spread used to give a good prediction.

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Fig. 4. Sections through ODFs at Euler angle ϕ2 = 45◦ of textures calculated using Taylor modelling at a strain of 0.75, which have been convoluted with a Gaussian of 8◦ spread. The contour levels shown are multiples of random density, note that these are significantly greater than used for the experimental results. The top row shows the effect of slip planes on FC prediction, the lower four show the effect of relaxed constraints.

The second is the lack of a simple physical basis for using the 23 relaxed constraint. The conventional argument for introducing relaxed constraints is related to the aspect ratio of deformed grains. It is difficult on that basis to see why a 23 relaxation should occur at relatively small strains and certainly not before the operation of 13 relaxation. One further factor that can be usefully assessed using Taylor modelling is the effect of deviations from ideal plane strain. Rolling does give a small TD extension, which leads to an approximately uniform velocity gradient L22 when steady state rolling deformation occurs. Measurements, given in Table 1, indicate that an upper bound for this would be about −0.1L33 . Another deviation from ideal plane strain is the possibility of an L13 simple shear due to macroscopic inhomogeneous deformation in rolling. This would not be large near the mid-plane of the plate and would partially reverse. A notional upper bound estimate of this was provided by using: Fig. 5. The correlation between Taylor model predictions and experimental textures as functions of strain.

L13 (t) = A{t(t − η)(t − 1)} L11 (t)

(6)

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371

than are given by the Taylor models. There have also been some indications that inhomogeneous deformation within grains can be linked to forms of relaxed constraint [25,26]. The relative effects of both the intergranular and intragranular inhomogeneity of deformation on deformation texture can best be investigated using finite element modelling. 3.3. CPFEM

Fig. 6. The correlation between Taylor model predictions and experimental textures as functions of strain, with the Taylor predictions convoluted with Gaussians having angular widths which gave the optimum correlation. Those angular widths are shown in the lower graph.

where t is the time as a fraction of the total deformation interval, the multiplier A was set to 1.27 and η was set to 0.667. This gave a maximum shear velocity gradient 0.1 of the plane strain components. The step size of the Taylor model integration was adjusted to give a reasonable approximation to this shear occurring in the correct way with the rolling pass schedule used. The effects of those two deviations from ideal plane strain on texture development are shown in Fig. 7. The shear has virtually no effect, and although there is some apparent improvement in the prediction of the {1 1 2}00021 1 00003 component by including lateral spread, this is offset by the introduction of a uniform 00021 0 00003//ND component and a generally poorer prediction of the a fibre. The overall improvement is then marginal, and the level of lateral spread used was significantly greater than that measured, except at small strains. In view of these results, deviations of the macroscopic deformation from ideal plane strain were discounted as significant factors in the texture evolution. Previous work using finite element modelling has shown that the mechanical interaction between grains can lead to much more realistic levels of deformation texture intensity

The formulation used has been described previously [25] for two-dimensional implementation. It uses an incremental-iterative scheme, with equilibrium in the current configuration being solved at the end of each small deformation step using nested iteration to solve for the elasto-plastic state and overall stress equilibrium. The three dimensional variant used here follows that scheme, and simply uses different element types and boundary conditions. For the majority of simulations, 20 node isoparametric brick elements with 8 integration points were used. In one case, arrangements of seventeen 10 node isoparametric tetrahedral elements with 4 integration points to form rhombic dodecahedra were employed; those groups fill space in a close packed manner. The initial domain was cubic in all cases, and deformational boundary conditions were imposed which maintained planar boundaries but allowed nodes to move within those boundary planes. Relative displacement steps of about 0.001 were used, with the boundary displacements ensuring constant domain volume. With such small steps, convergence to a fractional displacement norm <10−3 typically occurred in two iterations. 3.3.1. Mechanical model The material model in this, as in almost all crystal plasticity finite element models, was elasto-viscoplastic. The linear elastic compliance modulii used were: S1111 = 7.67 × 10−6 (MPa)−1 , S1122 = −2.83 × 10−6 (MPa)−1 , S1212 = 8.57 × 10−6 (MPa)−1 Plasticity was assumed to occur by the operation of rate-sensitive slip, so that on slip system α, the slip rate, γ˙ α , was given by: 0002 α 00031/m γ˙ α τ = (7) γ˙ 0 τ0α where τ α is the resolved shear stress, τ0α is the current slip resistance on that system, γ˙ 0 is a nominal reference slip rate and m is the systems slip rate sensitivity exponent. That exponent was set at a small value of 0.02, so that the formulation approached a rate insensitive one. The evolution of slip resistance is characterised by the plastic hardening of system α as a result of slip on system β: H αβ =

∂τ0α ∂γ β

(8)

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Fig. 7. The effect of deviations from ideal plane strain predicted by the FC Taylor model with {1 1 0} + {1 1 2} slip planes. ODF sections at Euler angle ϕ2 = 45◦ are show at the top for a strain of 0.75. The graph at the bottom shows the correlation of the predictions, using optimal Gaussian spreading, with experimental results as functions of strain. The effect of the additional shear is negligible.

In the basic implementation of crystal plasticity used, the slip resistance on all systems involved was taken to be equal at a given location and time, and to evolve with the total [absolute] slip activity occurring on all systems. In that case, all H are equal to a single hardening modulus, Θ, and this can be termed ‘isotropic’ hardening. A modified Voce law was used, which in terms of plastic hardening rate is 0005 00040005 00030006 0002 0005 τ 0005n τ Θ = ΘIV + Θ0 000500051 − 00050005 sgn 1 − (9) τ τ s

s

where ΘIV is the hardening rate in stage IV, the linear regime of hardening typically occurring at high strain. The hardening rate at τ = 0 would be Θ0 + QIV ≈ Θ0 . This formula is implemented for finite [small step] deformation via fourth-order Runge–Kutta integration. The ‘saturation’ slip resistance, τ s , can be made rate sensitive by a power law, but it was rate insensitive in this work. Uniaxial compression testing was carried out on annealed material, with the stress in the initial plate normal direction. This was carried out at a strain rate of about 10−3 s−1 ,

using PTFE film and oil lubrication. This lubrication gives low friction, and even at a strain of 0.8, there was little “barrelling” of the compression specimens; the maximum difference in lateral dimensions was 6% of the mean after that strain. The experimental stress-strain data could be converted to slip system resolved values using a constant Taylor factor: 0007 σ γ ˙ M= = ≈ 3.0 (10) τ ε˙ appropriate for 00021 1 10003 pencil glide with the essentially stable σ//00021 1 10003 texture. A good fit was given by: ΘIV = 21 MPa,

Θ0 = 1350 MPa,

τs = 250 MPa,

n = 5.8 These parameters were used to generate the material model lines shown in Fig. 8, together with experimental results. An initial value of slip resistance of 12 MPa was used in the modelling.

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Fig. 8. Experimental and fitted stress–strain curves (left) and work hardening vs. stress curves (right) for the IF steel. There are two experimental stress–strain curves, one interrupted to renew the PTFE lubrication and one continuous; this demonstrates the low significance of friction. The approach of the work hardening to a finite steady value at higher stress—stage IV hardening—is shown in the right hand plot.

3.3.2. Results The basic finite element prediction was a plane strain simulation using an 18 × 18 × 18 array of 20 node elements, each of which had equal initial integration point orientations taken at random from the EBSD measured set of the recrystallised steel. Representative sections of the ODFs from simulations using different slip plane sets are shown in Fig. 9 for a strain of 0.75, together with the texture correlation index as a function of strain. The immediate improvement over the Taylor models is that the sharpness of the predicted textures is now much closer to the experimental ones, and no artificial spreading is required. There is clearly a benefit in including {1 1 2} slip planes in addition to {1 1 0}, but the improvement that results from including {1 2 3} as well is marginal, and less pronounced than in the Taylor FC simulations. The CPFEM gives a significantly better prediction of both the γ fibre location and the orientation density distribution along the α fibre than the FC Taylor, even with additional spreading. However, these CPFEM predictions still deviate from the experimental measurements. The major improvement to the Taylor predictions when the 23 relaxed constraint was involved, especially when combined with 13 relaxation, can be interpreted as an indication that shears within grains might be occurring. The simulations above only have one element per grain and so the opportunity for such intragranular effects is limited. This can be demonstrated by modelling the development of a single orientation on the α fibre, (2 2 3)[1 1¯ 0]. The effect of increasing the number of elements on the development of that orientation in plane strain is shown in Fig. 10. It is clear that a single element, even a quadratic one as used here, does not have sufficient freedom whereas a group of 3 × 3 × 3 elements approaches the result as a much larger group. The contour plot, also shown in Fig. 10, reveals the tendency

for this orientation to give inhomogeneous 23 deformation associated with the spread in orientation. To assess the effect of intragranular inhomogeneity on overall texture development, simulations were carried out in which grains were represented by groups of 27 (3 × 3 × 3) elements. Each simulation then had 216 distinct start orientations, which is a rather small number. To improve the statistical significance, five of these simulations was carried out and the resulting texture data combined. Simply taking orientations at random from the measured set often gave poor start texture correlation in this case. Instead, several small orientation sets were generated from the texture of the complete experimental set. This was done by generating random orientations and testing the orientation density of these, divided by the maximum orientation density. If that scaled orientation density was greater than a random number in the range 0–1, the orientation was accepted, otherwise it was not. From those sets, five were selected which gave the best correlation with the full start texture. Even with five sets, the statistics are still relatively sparse, and so five simulations using the same initial orientations but with one element per grain, i.e. 6×6×6 array domains, were also carried out. The average predicted textures of those simulations is shown in Fig. 11, which also shows the result from a simulation using 10 node tetrahedral elements arranged in rhombic dodecahedral groups to represent grains: that type of arrangement was previously used by Mika and Dawson [17]. The use of 27 elements to represent each grain gives a marginal improvement in correlation with the experimental results. The main improvement is a slightly better prediction in the spread of the ␣ fibre towards {1 1 1}00021 1 00003. The use of the dodecahedral groups gives no benefit, and it is likely that these do not give sufficient internal freedom to allow the inhomogeneous deformation possible in the 27 hexahedral element groups. The use of multiple elements to represent

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Fig. 9. Results from CPFEM modelling using one element per grain with different slip plane families involved. ODF sections at Euler angle ϕ2 = 45◦ are show at the top for a strain of 0.75. The graph at the bottom shows the correlation of the predictions—without any additional spreading—with experimental results as functions of strain.

grains fails to give an adequate prediction of the spreading out of orientation density in the α fibre which occurs at high strains, and it seems unlikely that this would be predicted using more elements per grain. This feature can best be shown by plots of orientation density as a function of & with ϕ1 = 0◦ and ϕ1 = 45◦ . These are shown for strains of 0.75 and 1.47 in Fig. 12. Those plots also include the Taylor 13 + 23 RC results, with optimal spread. Although the RC Taylor predictions give too much spread towards {0 0 1}00021 1 00003 at low strain, the prediction at the higher strain is rather better than the CPFEM one. The conventional view of relaxed constraints depending on grain shape would appear capable of resolving that because the relaxation would occur progressively with straining. However, it is difficult to see why the finite element model would not include any relaxation which would actually occur. There is, in fact, good visual correlation between the deformed grain configurations predicted by the CPFEM with multiple elements per grain and those observed by optical metallography: a feature of which is grain ‘curling’ in sections normal to the RD. The RC Taylor

model takes no account of the mechanical behaviour of the material, and work hardening tends to reduce inhomogeneity of deformation. However, reducing the ΘIV hardening contribution in the material model, Eq. (9), to zero fails to improve the ␣ fibre prediction of the CPFEM at high strains.

4. Discussion The main point arising from the work described above is that, although the CPFEM gives absolutely better predictions of deformation texture than Taylor models at strains up to about 0.75, it fails to predict the increase in ␣ fibre spread towards {0 0 1}00021 1 00003 which occurs at higher strains. Although that spread is predicted by Taylor modelling with 13 + 23 relaxed constraints, the CPFEM predictions are generally superior to those of Taylor models. A major advantage of the CPFEM is that it predicts textures with the correct order of orientation densities. These are much too high in Taylor model predictions, even with

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Fig. 10. The effect of the number of elements used to represent the single orientation (2 2 3)[1 1 0] on the spread of orientation following a plane strain of 0.5. The orientation spread is shown as 0 0 1 pole figures [x = RD vertical] in the top four images. The contour plot of the 10 × 10 × 10 domain, shown at the bottom, is coded by the Euler angle, Φ, and shows the association of that with the inhomogeneous deformation.

relaxed constraints, and this demonstrates in a clear way the importance of inhomogeneity of deformation on texture development. This inhomogeneity is sufficient to ensure that, after quite moderate strains, there are typically no regions of orientation space that are unpopulated. The finite element model only takes into account mechanical differences due to orientation, and there may be others, such as grain size. Fine grained materials typically have higher flow stress at low temperatures, and a region of relatively small grains

would locally require a higher stress. This argument may extend to individual grains, in which case there should be even more inhomogeneity of deformation than predicted by the CPFEM and so greater spread of texture. That this does not seem to be necessary might be due to the grain size becoming irrelevant as orientation differences develop within grains. It is possible to use data for the inhomogeneity of deformation predicted by CPFEM in a very general way to perturb

Fig. 11. Results from CPFEM modelling showing the effect of using multiple elements to represent ‘grains’, compared with a result using one element per grain with the same set of start orientations. For the result at the right, there were 512 dodecahedral groups and partial groups of quadratic tetrahedra.

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Fig. 12. Experimental and predicted orientation densities along the α fibre. The results for the Taylor RC model have been subject to the additional Gaussian spread which gives best correlation with the relevant experimental texture.

local deformation states in a Taylor model [27], and this hybrid approach gives better prediction of orientation density. However, it ignores the connections between inhomogeneity and the actual local configurations of orientations which are responsible for that inhomogeneity [16]. Although such local configurations are dealt with to some extent in recent modifications to the Taylor model [28,29], they can only be dealt with in anything approaching a realistic manner by methods such as CPFEM. The prediction of an inhomogeneity by a simpler model which is not predicted by CPFEM is likely to be an artefact of that simpler model. In other words, because CPFEM must be a more realistic model of the intragranular mechanics than any Taylor model, it is suggested that the prediction of the α fibre spread by the RC model is a coincidence. The occurrence of 23 shear, and rotation, in α fibre orientations is largely responsible for the distribution of orientations in that fibre. Using multiple elements to represent

grains allows a greater degree of intragranular inhomogeneity, but this still does not give adequate prediction of the α fibre evolution. It may be that an even greater subdivision of grains would give better prediction, but this would mean very large models—and very great computer time—and the indications from the single orientation modelling presented above are that the improvement may only be marginal. There are fundamental limits to the extent to which microstructural inhomogeneity can be modelled by continuum methods [26], and it is unlikely that some of the features which are observed in reality—for example, the in-grain shear bands occurring in γ fibre orientations [30] which are potentially important in determining the orientations of recrystallisation nuclei [31]—could be predicted using CPFEM. Further insight into the effects of deformation inhomogeneity could potentially be obtained from investigations involving different initial grain sizes and also deforming under conditions giving different mechanical response. It is not clear whether some deficiency in the ability of the CPFEM to deal with inhomogeneity of deformation is ultimately responsible for the discrepancy between the experimental and predicted textures. Other factors could influence the development of deformation texture. It may be possible to devise a material model which gives better predictions, although limited investigations using latent hardening in the present model did not give any benefit. An example is shown in Fig. 13, where slip on a given system was taken to contribute an additional factor of 0.2 to hardening on all other slip systems, i.e. a cross hardening of 1.2. This has had a slight negative effect on the correlation of prediction and experiment. There are, of course, many other possible ways in which the mechanical behaviour could be changed and it is possible that some could lead to better

Fig. 13. The correlation of CPFEM predictions with experimental textures, using 18 × 18 × 18 meshes and one element per grain, using {1 1 1} + {1 1 2} slip planes with isotropic and with latent hardening. The latent hardening in this example used a cross-hardening factor of 1.2, i.e. slip on a system hardened all other systems 20% more than the self-hardening on that system.

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deformation texture prediction, but physical justification would naturally be desirable. Another possibility, unlikely though it may seem, is that high angle boundary migration is taking place and is affecting the texture. A increase in {0 0 1}00021 1 00003 has been observed during annealing of ferrite and this can be associated with ‘broad front’ boundary migration driven by the lower substructural energy of that orientation [32]. Although the temperature involved here was much lower than that required for static annealing, there is clear evidence that high angle boundary migration occurs in the cold deformation of aluminium and this is associated with dynamic recovery [33]. This process effectively limits the high angle boundary spacing which can be achieved with cold deformation. Although there was no clear evidence of this phenomenon in the present material, only 5% of the material volume would need to be involved to satisfy the discrepancy between the experimental result and the CPFEM prediction. This must remain a highly speculative hypothesis at this stage. It must be remembered that a major disadvantage of the CPFEM, in comparison to Taylor models, is the amount of computation involved. Using a 2 GHz processor, the typical CPU times for the CPFEM simulations reported here were about 30 h, compared to about 2 min for Taylor models. This factor is a clear driver for the development and application of models such as those of Van Houtte et al. [28] and Crumbach et al. [29]. 5. Conclusions The deformation texture of ferrite measured here agrees with that found elsewhere. A well-defined start texture, significantly different from the stable deformation texture, has been used and the development of that texture has been compared with both Taylor and CPFEM predictions. In general, the CPFEM gave better predictions, but at high strain did not adequately predict the distribution of orientation density in the α fibre. That distribution, if not the overall level, was predicted quite well using a Taylor model with relaxed simple shears having shear planes lying in the sheet plane. Such shearing can also be associated with intragranular inhomogeneity, but the level of this predicted by the CPFEM, even with multiple elements used to represent the initial grains, was not sufficient to give good prediction of the α fibre texture. The prediction of the relaxed constraint Taylor model is taken to be a coincidence, and the disparity between the finite element predictions and observed textures at high strain is due to subtle deficiencies in the modelling of inhomogeneity or the involvement of a factor such as high angle boundary migration during deformation.

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Acknowledgements The authors thank W.B. Hutchinson for his encouragement and constructive criticism. Thanks also to Mark Harris and his colleagues in the MMSC workshop for their skilful manipulation of samples. This work was supported by the EPSRC via grant GR/R69341.

References [1] G.I. Taylor, J. Inst. Metals 62 (1938) 307. [2] J.S. Kallend, G.J. Davies, in: J. Karp, S. Gorczyca, H.-J. Bunge, J. Pospiech, W. Dabrowski (Eds.), Quantitative Analysis of Textures, Society of Polish Metallurgical Engineers, Kracow, 1971, p. 297. [3] H.-J. Bunge, Kristall. Tech. 5 (1970) 145. [4] H.-J. Bunge, T. Leffers, Scripta Metall. 5 (1971) 143. [5] I.L. Dillamore, H. Katoh, Met. Sci. J. 8 (1974) 21. [6] H.-J. Bunge, Texture Analysis in Materials Science, Butterworths, London, 1982. [7] A. Molinari, G.R. Canova, S. Ahzi, Acta Metall. 35 (1987) 2983. [8] H. Honneff, H. Mecking, in: G. Gottstein, K. Lücke (Eds.), Textures of Materials, Proc. ICOTOM 5, Springer, Berlin, 1978, p. 265. [9] U.F. Kocks, H. Chandra, Acta Metall. 30 (1982) 695. [10] U. von Schlippenbach, F. Emren, K. Lücke, Acta Metall. 34 (1986) 1289. [11] P. Van Houtte, in: C.M. Brakman, P. Jongenburger, E.J. Mittemeijer (Eds.), Textures of Materials, Proc. ICOTOM7, Netherlands Society for Materials Science, Zwijndrecht, 1984, p. 7. [12] D. Raabe, K. Lücke, Mater. Sci. Forum 157–162 (1994) 597. [13] S.V. Harren, R.J. Asaro, J. Mech. Phys. Solids. 37 (1989) 191. [14] R. Becker, Acta Metall. Mater. 39 (1991) 1211. [15] A.J. Beaudoin, P.R. Dawson, K.K. Mathur, U.F. Kocks, Int. J. Plasticity 11 (1995) 501. [16] G.B. Sarma, P.R. Dawson, Acta Mater. 44 (1996) 1937. [17] D.P. Mika, P.R. Dawson, Mat. Sci. Eng. A257 (1998) 62. [18] V. Bachu, S.R. Kalidindi, Mat. Sci. Eng. A257 (1998) 108. [19] P. Van Houtte, L. Delannay, S.R. Kalidindi, Int. J. Plasticity 18 (2002) 359. [20] I.L. Dillamore, C.J.E. Smith, T.W. Watson, Metal Sci. J. 1 (1967) 49. [21] R.L. Every, M. Hatherly, Texture 4 (1974) 183. [22] D. Schläffer, H.-J. Bunge, Texture 1 (1974) 157. [23] H. Klein, H.-J. Bunge, Mater. Sci. Forum 157–162 (1994) 1791. [24] J.M. Rosenerg, H.R. Piehler, Metall. Trans. 2 (1971) 257. [25] J. Gil Seviliano, P. Van Houtte, E. Aernoudt, Prog. Mat. Sci. 25 (1980) 69. [26] P. Bate, Phil. Trans. R. Soc. Lond. A 357 (1999) 1589. [27] O. Engler, Adv. Eng. Mater. 4 (2002) 181. [28] P. Van Houtte, L. Delanay, I. Samajdar, Text. Microstruct. 31 (1999) 109. [29] M. Crumbach, G. Pomana, P. Wagner, G. Gottstein, in: G. Gottstein, D.A. Molodov (Eds.), Recrystallization and Grain Growth: Proceddings of the First International Conference, Springer, Berlin, 2001, p. 1053. [30] B. Hutchinson, Phil. Trans. R. Soc. Lond. A. 357 (1999) 1471. [31] B. Hutchinson, P. Bate, in: H. Takechi (Ed.), IF Steels 2003, The Iron and Steel Institute of Japan, Tokyo, 2003, p. 337. [32] B. Hutchinson, H. Magnusson, J.-M. Feppon, in: T. Sakai, H.G. Suzuki (Eds.), Proceedings of the 4th International Conference on Recrystallisation. The Japan Institute of Metals, Tsukuba, 1999, p. 49. [33] H. Jazaeri, F.J. Humphreys, Mater. Sci. Forum 396–403 (2002) 551.

Texture development in the cold rolling of IF steel P.S. Bate∗ , J. Quinta da Fonseca Manchester Materials Science Centre, The University of Manchester, Grosvenor Street, Manchester M1 7HS, UK Received 1 December 2003; received in revised form 31 March 2004

Abstract The development of deformation texture in ferrite has been measured in cold rolled IF steel. This has been compared, in a quantitative way, to the predictions of Taylor models—including those with relaxed constraints—and a finite element model with crystal plasticity constitutive laws. The finite element model gave much better prediction of the overall levels of orientation density but failed to predict the relatively high level of {0 0 1}00021 1 00003 texture which occurred at strains greater than about unity. That feature was predicted by relaxed constraint Taylor models. It is argued that that prediction is a coincidence, and either the finite element model cannot readily deal with the intragranular inhomogeneity of deformation in an adequate way, or that factors such as high-angle boundary migration may be important in the development of deformation texture. © 2004 Elsevier B.V. All rights reserved. Keywords: Texture; Plasticity; Modelling; Finite element analysis; Iron

1. Introduction The study of deformation textures has formed an important part of research into the plasticity of metals. Even though it does not reach the levels of complexity and difficulty involved in the prediction of recrystallisation textures—and is also, perhaps, less practically useful—the quantitative prediction of deformation textures remains a challenge. This quantitative prediction depended on the availability of adequate electronic computers, but the use of models based on Taylor’s [1] assumptions of uniform deformation and local equality of slip stresses on equivalent systems began to give quite reasonable approximations to deformation textures in cubic metals by the 1970s [2–5]. This coincided with the application of texture analysis techniques which removed some of the possible ambiguity associated with individual pole figures [see [6]], and allowed adequate comparison of experimental with predicted textures. Despite the initial success, it was clear that the Taylor model, at least in its original form, gave imperfect predictions. Perhaps the most famous example of this is the ‘texture transition’ occurring in FCC metals, where the texture developed depends on the stacking fault energy. Low stacking fault energy metals, such as silver, give a single com-

∗

Corresponding author. Tel.: +44-161-200-8842. E-mail address: [email protected] (P.S. Bate).

0921-5093/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2004.04.007

ponent {0 1 1}00022 1 10003 texture following flat rolling, whereas the Taylor model, using the usual octahedral slip systems, predicts a fibre of orientations from near {2 2 5}00025 5 40003 to {0 1 1}00022 1 10003, with the highest density at the first of those orientations. Various proposals have been made to account for the FCC ‘texture transition’, but it is fair to say that no wholly satisfactory solution has been found. Even where the stacking fault energy is high, as in aluminium, the original Taylor formulation only gives a first approximation to real texture evolution. The predicted textures are nearly always too well developed, i.e. too ‘sharp’, and the precise locations and relative orientation densities of the components deviate from those measured. Various attempts have been made to modify the Taylor model with a view to improving texture predictions. The self-consistent model [7], where the behaviour of sample orientations embedded in an ‘equivalent medium’ are modelled, has been used. A significant modification was the implementation of ‘relaxed constraints’, whereby specific components of the imposed deformation were excluded from the deformation imposed on the crystal volumes involved in the simulations [8,9]. This relaxation was justified in terms of grain shape: in a grain sufficiently elongated by deformation, a simple shear along the long axis could occur without incurring a large penalty associated with the inhomogeneous deformation needed to maintain material continuity. The success of this approach in FCC metals is questionable, but in BCC there have been indications that introducing relaxed

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constraints could be of benefit [10–12]. There are potential problems with such relaxed constraint models, however, one of which is that they, in common with the original ‘full constraint’ models, predict too sharp a texture. The use of finite element models with constitutive laws based on crystal plasticity have had a major influence on deformation texture predictions. Using such models allows the uniformity of deformation assumed in the Taylor models to be abandoned in a much less arbitrary way than in relaxed constraint Taylor models. The magnitude of the inhomogeneous deformation between grains, and the effect on texture development, predicted using such models is significant [13–18]. The predictions also indicate that simple rules for deviations from full constraint of particular grain orientations are unlikely to be appropriate, and the spatial disposition of neighbourhood orientations will have a significant effect. Most of this work has been done on FCC systems, particularly aluminium, and finite element predictions are generally better than those of Taylor models. Work on BCC metals has been more limited, but a comparison by Van Houtte et al. [19] showed that here, as well, finite elements predictions were generally better than Taylor model simulations. However, as mentioned above, there are indications that certain relaxed constraints can give quite good Taylor model predictions and this issue is worthy of further investigation. Another feature of most finite element simulations is that the number of elements used to represent a grain is very limited, often to one. This limits modelling the influence of intragranular inhomogeneity of deformation. In the work presented below, a series of experimental measurements in BCC iron is compared with both Taylor and finite element models to assess the effect of the intergranular and intragranular inhomogeneity of deformation—in so far as they can be readily modelled by finite elements—on texture development.

2. Experimental work 2.1. Material The material used was interstitial-free [IF] steel, with the composition given in Table 1, supplied by SSAB Tunnplåt, Sweden, and provided as hot rolled plate 28 mm thick. Sections of this plate, 50 mm wide, were cold rolled to 10 mm thickness, maintaining the original rolling direction [RD] throughout, using a rolling mill with 250 mm diameter rolls operating with a roll surface speed of 150 mm s−1 and with light oil lubrication. This material was then annealed at 750 ◦ C for 30 min in air. This gave full recrystallisation with

Table 1 The composition, in weight percent, of the IF steel C

N

Mn

Ti

Al

Fe

0.003

0.003

0.15

0.08

0.05

Remainder

Fig. 1. Section through the ODF at Euler angle ϕ2 = 45◦ for the recrystallised plate. The contour levels shown are multiples of random density.

a mean linear intercept grain size of 30 m. The texture was determined from EBSD measurements on the middle third of the thickness of a transverse cross section, with about 35 000 sample points covering an area of 17.6 mm2 . The continuous orientation distribution function [ODF] was determined using—as all textures in this work—harmonic series fitting [6] with a truncation at lmax = 22, and a representative section of this is shown in Fig. 1. The texture is quite typical of recrystallised ferritic steel plate, showing a tendency for 00021 1 10003 to align near the rolling plane normal [ND]. That recrystallisation texture is close to stable deformation texture components, and as such does not provide an ideal start for a study of deformation texture development. A texture that is not so stable was generated by cutting and machining the plate into five rectangular section strips, rotating these about the RD by 90◦ and joining them by autogenous TIG welds at the surface. The heat affected zones of this welding were assessed on as-rolled material, and only affected material within a radius of 2 mm. This compounded plate was then rolled, with 50 mm long sections taken from the leading edge to give material with a series of reductions. These are detailed in Table 2. Samples for texture determination were cut from the back end of the central component,

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Table 2 The dimensions, and resulting strains in the thickness [h] and width [w], of the rolled samples Specimen

Thickness [mm]

Width [mm]

−εh

A0 A1 A2 A3 A4 A5

9.52 7.44 5.73 4.48 3.46 2.18

38.02 38.88 39.27 39.57 40.00 Not recorded

0 0.247 0.508 0.754 1.012 1.474

which were then ground and polished to half thickness. The sampling ensured that the material had undergone a very close approximation to plane strain deformation. EBSD measurements were made on the middle third of width, which corresponds to the middle third of thickness in the original plate orientation, over a length in the RD of about 10 mm. Despite the cold rolled state, about half of the sample points gave valid orientation measurements in all the specimens. The number of orientations used for each sample texture was about 20 000, which satisfies any reasonable statistical criterion. The step size of this sampling was of the order of the grain size, the aim being to sample texture rather than provide local and spatially correlated orientation data. Spatial correlation of orientation is, of course, not included in the conventional definition of texture. There was concern about whether the sampling was influenced by orientation: it is known that X-ray line broadening and substructural density [20,21] vary with orientation in deformed

εw ± ± ± ± ±

0.004 0.004 0.005 0.005 0.008

0 0.024 0.034 0.041 0.051 –

−εw /εh ± ± ± ±

0.001 0.001 0.001 0.001

– 0.09 0.06 0.05 0.05 –

ferrite. This effect was assessed by using x-ray diffraction, to effectively give inverse pole figure densities in the ND, and by comparing textures derived using different levels of EBSD pattern quality. Some results from this exercise are shown in Fig. 2. No systematic effect was found when comparing EBSD and XRD data, and there was no significant effect of pattern quality cut-off until this was set artificially high. Because of these factors, there is a high degree of confidence that the measured textures are truly representative, with errors less than those likely to arise from the measurement of partial pole figures by X-ray diffraction and the subsequent manipulation of that data to give ODFs. 2.2. Texture development Sections of the ODFs of the rolled material are shown in Fig. 3. There is a very rapid change from the original texture—which is simply that shown in Fig. 1 rotated by

Fig. 2. Some results of trials used to check the validity of EBSD texture data. On the left are shown the ratios of XRD integrated peak intensities, for diffraction vectors parallel to the ND, to the relevant inverse pole figure densities predicted from the EBSD textures, at different strains. The unstrained specimen had consistently lower XRD intensities, but the ratios for the two reflections are in proportion. On the right, the effect of changing the pattern quality cut-off, and so reducing the number of data included, on the derived orientation densities for two representative orientations. That result is for the specimen rolled to a thickness strain of 0.75.

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Fig. 3. Sections through ODFs at Euler angle ϕ2 = 45◦ for the rolled specimens. The contour levels shown are multiples of random density, and the thickness strains for the relevant section are indicated.

90◦ about the rolling direction—to that with components characteristic of rolled ferrite. These lie on two fibres, one with 00021 1 00003//RD extending from {1 0 0}//ND to {1 1 1}//ND, known as the a fibre, and the other with 00021 1 10003//ND known as the g fibre. This latter fibre tends to occur, in modified form, after recrystallisation as shown in Fig. 1. The first significant texture component to form is near {1 1 2}00021 1 00003, followed by the gradual development of orientations comprising the α and γ fibres, with a near uniform orientation density along the α fibre after a strain of about 1.5. The highest orientation density on the γ fibre remains near 00021 1 00003//RD throughout. The maximum orientation density increases very little after a strain of 0.75. The general trend observed is similar to that observed previously, e.g. by Schläffer and Bunge [22].

3. Modelling The two basic techniques used to model the deformation texture of the IF steel—Taylor modelling and crystal plasticity finite element modelling [CPFEM]—have a common background in standard geometrical crystal plasticity, which will be detailed before describing the models and the results of their application.

3.1. Crystal plasticity If attention is restricted in scope to slip [via dislocation glide] as the plastic deformation mechanism and, for the time being, ignoring the elastic component of deformation, then slip on a system α with direction bα and slip plane normal nα at a rate of simple shear γ˙ α will give a contribution to the local rate of plastic deformation [velocity gradient] of: α α α LPα ij = bi nj γ˙

(1)

The basic kinematic problem in geometric crystal plasticity is to find the set of slip system activities which satisfy: 0001 1 mαij γ˙ α (2) DijP = (LPij + LPji ) = 2 α where DP is the symmetric plastic deformation [strain] rate and m is the Schmid tensor: 1 mPij = (biα nαj + bjα nαi ) (3) 2 There is no requirement for slip to occur to satisfy the antisymmetric (spin) component of LP . The mismatch between the antisymmetric components of LP and of the sum of mαij γ˙ α will lead to a change in crystal orientation with respect to the spatial coordinates. This is fundamental to the

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formation of deformation texture. It is also very easy to calculate once the slip activities are known. The problem is determining those activities, and the simplest way is to use the method proposed by Taylor [1]. 3.2. Taylor modelling The Taylor hypothesis—that the local deformation rate is equal to the overall deformation rate—has been used for the vast majority of deformation texture prediction. Even when a local DP is specified, the equations [2] are overdetermined. There will be five independent components of DP in the general case [six minus one because of volume constancy], and more than five independent slip systems. Elasticity is ignored, and subsets of five of the available slip systems which give non-singular solutions to equations [2] are found. From these, the ones that give minimum values of the plastic power: 0001 ˙ = W τ α γ˙ α (4) α

are selected as solution sets. The slip stresses, τ, are usually taken to be equal on all systems, though this is not necessary. There are generally multiple solutions for a particular deformation rate which give the same minimal plastic power. This leads to what has become known as “Taylor indeterminacy”. There are several ways of dealing with that indeterminacy. The individual solutions can all be accepted, though that leads to a dramatic growth in the population of orientations being used when predicting texture evolution. A single solution can be selected at random [5], or one can derive a continuous rotation field and use a type of fluid mechanical approach in orientation space [23]. It is also possible to invoke slip rate sensitivity, as in the finite element method. In this work, an average of the Taylor solutions was used which is essentially the same as the rate-insensitive limit of such viscoplastic solutions. The model was applied to sets of discrete orientations, with small steps of deformation L dt ≈ 0.01 and a simple forward Euler integration scheme. The full set of about 30 000 orientations from the EBSD measurement of the annealed material was used. The condition that the local deformation is equal to the overall imposed rate can be relaxed. The original, full constraint (FC) model is modified in relaxed constraint [RC] models by allowing specified components of LP to be free. This gives freedom of corresponding DP and the overall stress associated with those components is set to zero. The free components then adopt values which minimise plastic work with reduced subsets of slip systems in the kinematic condition, equation [2]. FC Taylor modelling was carried out using 00021 1 10003 slip directions, with either {1 1 0}, {1 1 0} + {1 1 2} or {1 1 0} + {1 1 2} + {1 2 3} slip planes. All slip systems were assigned equal slip stresses. The last of those slip system sets gives results which are very close to those of ‘pencil glide’, where any plane containing the slip direction is considered

369

[24]. The intermediate case, {1 1 0} + {1 1 2}, was used in RC modelling. With axis 1 being the RD, 2 being the transverse direction (TD) and 3 being the ND, the simple shears 13, 23, 12 and 13 + 23 combined were relaxed. The textures predicted using Taylor models are much sharper than those measured, and for comparison with experimental results have been convoluted with a Gaussian spread function [6] to allow better comparison of the forms of the predicted textures with the experimental ones. These are shown, for a plane strain of 0.75, in Fig. 4. The locations of high orientation density predicted using FC modelling are somewhat different to those observed experimentally. The addition of more slip planes shifts the dominant component from near{1 1 2}00024 6 10003 to {1 1 2}00021 1 00003, but even with the {1 1 0} + {1 1 2} + {1 2 3} set, the γ fibre and the associated spreading of the α fibre observed in the experimental textures are not predicted. The RC models gave slightly lower orientation densities—though still much greater than the experimental ones unless artificially spread. Relaxing the 13 or 12 simple shears gave no improvement in prediction, but relaxing 23 and especially a combination of 13 + 23 improved the situation significantly. The fit of the simulations to the experimental results was quantified using a correlation index ξ given by ξ =1−

J0017 2(0002J0003 − 1)

(5)

where J denotes texture index, corresponding to the integral of the square of the orientation distribution function [6]. JD is the texture index of the difference of the two textures and 0002J0003 is the mean of their texture indices. This correlation value ranges from unity for perfect correlation [the textures are the same] to zero with no correlation. Numerical trials showed that this function gave good agreement with different proportions of common orientations in point sets, including cases with different total numbers of points in the two sets, and serves as a useful single measure of the difference between model and experimental textures. The correlation index as a function of strain for the different Taylor models is shown in Fig. 5, and the beneficial effect of introducing 23, and especially 13 + 23, relaxed constraints at levels of strain greater than about 0.75 is clear. In contrast, relaxation of the 13 shear has a rather deleterious effect until the strains are greater than about 1. The Taylor model predicts textures that are too sharp, and this contributes to the generally poor correlation shown in Fig. 5. It is possible to compensate for this effect by introducing an artificial Gaussian spread to the predictions, as carried out to give the results shown in Fig. 4. Fig. 6 shows the best fit correlation index, and the degree of spread giving that best correlation, as functions of strain. This reinforces the observation that a 23 relaxed constraint, especially in conjunction with 13 freedom at the higher strains, leads to significantly better predictions of deformation texture. There are two deficiencies with this, however. The most obvious is the rather arbitrary spread used to give a good prediction.

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Fig. 4. Sections through ODFs at Euler angle ϕ2 = 45◦ of textures calculated using Taylor modelling at a strain of 0.75, which have been convoluted with a Gaussian of 8◦ spread. The contour levels shown are multiples of random density, note that these are significantly greater than used for the experimental results. The top row shows the effect of slip planes on FC prediction, the lower four show the effect of relaxed constraints.

The second is the lack of a simple physical basis for using the 23 relaxed constraint. The conventional argument for introducing relaxed constraints is related to the aspect ratio of deformed grains. It is difficult on that basis to see why a 23 relaxation should occur at relatively small strains and certainly not before the operation of 13 relaxation. One further factor that can be usefully assessed using Taylor modelling is the effect of deviations from ideal plane strain. Rolling does give a small TD extension, which leads to an approximately uniform velocity gradient L22 when steady state rolling deformation occurs. Measurements, given in Table 1, indicate that an upper bound for this would be about −0.1L33 . Another deviation from ideal plane strain is the possibility of an L13 simple shear due to macroscopic inhomogeneous deformation in rolling. This would not be large near the mid-plane of the plate and would partially reverse. A notional upper bound estimate of this was provided by using: Fig. 5. The correlation between Taylor model predictions and experimental textures as functions of strain.

L13 (t) = A{t(t − η)(t − 1)} L11 (t)

(6)

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371

than are given by the Taylor models. There have also been some indications that inhomogeneous deformation within grains can be linked to forms of relaxed constraint [25,26]. The relative effects of both the intergranular and intragranular inhomogeneity of deformation on deformation texture can best be investigated using finite element modelling. 3.3. CPFEM

Fig. 6. The correlation between Taylor model predictions and experimental textures as functions of strain, with the Taylor predictions convoluted with Gaussians having angular widths which gave the optimum correlation. Those angular widths are shown in the lower graph.

where t is the time as a fraction of the total deformation interval, the multiplier A was set to 1.27 and η was set to 0.667. This gave a maximum shear velocity gradient 0.1 of the plane strain components. The step size of the Taylor model integration was adjusted to give a reasonable approximation to this shear occurring in the correct way with the rolling pass schedule used. The effects of those two deviations from ideal plane strain on texture development are shown in Fig. 7. The shear has virtually no effect, and although there is some apparent improvement in the prediction of the {1 1 2}00021 1 00003 component by including lateral spread, this is offset by the introduction of a uniform 00021 0 00003//ND component and a generally poorer prediction of the a fibre. The overall improvement is then marginal, and the level of lateral spread used was significantly greater than that measured, except at small strains. In view of these results, deviations of the macroscopic deformation from ideal plane strain were discounted as significant factors in the texture evolution. Previous work using finite element modelling has shown that the mechanical interaction between grains can lead to much more realistic levels of deformation texture intensity

The formulation used has been described previously [25] for two-dimensional implementation. It uses an incremental-iterative scheme, with equilibrium in the current configuration being solved at the end of each small deformation step using nested iteration to solve for the elasto-plastic state and overall stress equilibrium. The three dimensional variant used here follows that scheme, and simply uses different element types and boundary conditions. For the majority of simulations, 20 node isoparametric brick elements with 8 integration points were used. In one case, arrangements of seventeen 10 node isoparametric tetrahedral elements with 4 integration points to form rhombic dodecahedra were employed; those groups fill space in a close packed manner. The initial domain was cubic in all cases, and deformational boundary conditions were imposed which maintained planar boundaries but allowed nodes to move within those boundary planes. Relative displacement steps of about 0.001 were used, with the boundary displacements ensuring constant domain volume. With such small steps, convergence to a fractional displacement norm <10−3 typically occurred in two iterations. 3.3.1. Mechanical model The material model in this, as in almost all crystal plasticity finite element models, was elasto-viscoplastic. The linear elastic compliance modulii used were: S1111 = 7.67 × 10−6 (MPa)−1 , S1122 = −2.83 × 10−6 (MPa)−1 , S1212 = 8.57 × 10−6 (MPa)−1 Plasticity was assumed to occur by the operation of rate-sensitive slip, so that on slip system α, the slip rate, γ˙ α , was given by: 0002 α 00031/m γ˙ α τ = (7) γ˙ 0 τ0α where τ α is the resolved shear stress, τ0α is the current slip resistance on that system, γ˙ 0 is a nominal reference slip rate and m is the systems slip rate sensitivity exponent. That exponent was set at a small value of 0.02, so that the formulation approached a rate insensitive one. The evolution of slip resistance is characterised by the plastic hardening of system α as a result of slip on system β: H αβ =

∂τ0α ∂γ β

(8)

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Fig. 7. The effect of deviations from ideal plane strain predicted by the FC Taylor model with {1 1 0} + {1 1 2} slip planes. ODF sections at Euler angle ϕ2 = 45◦ are show at the top for a strain of 0.75. The graph at the bottom shows the correlation of the predictions, using optimal Gaussian spreading, with experimental results as functions of strain. The effect of the additional shear is negligible.

In the basic implementation of crystal plasticity used, the slip resistance on all systems involved was taken to be equal at a given location and time, and to evolve with the total [absolute] slip activity occurring on all systems. In that case, all H are equal to a single hardening modulus, Θ, and this can be termed ‘isotropic’ hardening. A modified Voce law was used, which in terms of plastic hardening rate is 0005 00040005 00030006 0002 0005 τ 0005n τ Θ = ΘIV + Θ0 000500051 − 00050005 sgn 1 − (9) τ τ s

s

where ΘIV is the hardening rate in stage IV, the linear regime of hardening typically occurring at high strain. The hardening rate at τ = 0 would be Θ0 + QIV ≈ Θ0 . This formula is implemented for finite [small step] deformation via fourth-order Runge–Kutta integration. The ‘saturation’ slip resistance, τ s , can be made rate sensitive by a power law, but it was rate insensitive in this work. Uniaxial compression testing was carried out on annealed material, with the stress in the initial plate normal direction. This was carried out at a strain rate of about 10−3 s−1 ,

using PTFE film and oil lubrication. This lubrication gives low friction, and even at a strain of 0.8, there was little “barrelling” of the compression specimens; the maximum difference in lateral dimensions was 6% of the mean after that strain. The experimental stress-strain data could be converted to slip system resolved values using a constant Taylor factor: 0007 σ γ ˙ M= = ≈ 3.0 (10) τ ε˙ appropriate for 00021 1 10003 pencil glide with the essentially stable σ//00021 1 10003 texture. A good fit was given by: ΘIV = 21 MPa,

Θ0 = 1350 MPa,

τs = 250 MPa,

n = 5.8 These parameters were used to generate the material model lines shown in Fig. 8, together with experimental results. An initial value of slip resistance of 12 MPa was used in the modelling.

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Fig. 8. Experimental and fitted stress–strain curves (left) and work hardening vs. stress curves (right) for the IF steel. There are two experimental stress–strain curves, one interrupted to renew the PTFE lubrication and one continuous; this demonstrates the low significance of friction. The approach of the work hardening to a finite steady value at higher stress—stage IV hardening—is shown in the right hand plot.

3.3.2. Results The basic finite element prediction was a plane strain simulation using an 18 × 18 × 18 array of 20 node elements, each of which had equal initial integration point orientations taken at random from the EBSD measured set of the recrystallised steel. Representative sections of the ODFs from simulations using different slip plane sets are shown in Fig. 9 for a strain of 0.75, together with the texture correlation index as a function of strain. The immediate improvement over the Taylor models is that the sharpness of the predicted textures is now much closer to the experimental ones, and no artificial spreading is required. There is clearly a benefit in including {1 1 2} slip planes in addition to {1 1 0}, but the improvement that results from including {1 2 3} as well is marginal, and less pronounced than in the Taylor FC simulations. The CPFEM gives a significantly better prediction of both the γ fibre location and the orientation density distribution along the α fibre than the FC Taylor, even with additional spreading. However, these CPFEM predictions still deviate from the experimental measurements. The major improvement to the Taylor predictions when the 23 relaxed constraint was involved, especially when combined with 13 relaxation, can be interpreted as an indication that shears within grains might be occurring. The simulations above only have one element per grain and so the opportunity for such intragranular effects is limited. This can be demonstrated by modelling the development of a single orientation on the α fibre, (2 2 3)[1 1¯ 0]. The effect of increasing the number of elements on the development of that orientation in plane strain is shown in Fig. 10. It is clear that a single element, even a quadratic one as used here, does not have sufficient freedom whereas a group of 3 × 3 × 3 elements approaches the result as a much larger group. The contour plot, also shown in Fig. 10, reveals the tendency

for this orientation to give inhomogeneous 23 deformation associated with the spread in orientation. To assess the effect of intragranular inhomogeneity on overall texture development, simulations were carried out in which grains were represented by groups of 27 (3 × 3 × 3) elements. Each simulation then had 216 distinct start orientations, which is a rather small number. To improve the statistical significance, five of these simulations was carried out and the resulting texture data combined. Simply taking orientations at random from the measured set often gave poor start texture correlation in this case. Instead, several small orientation sets were generated from the texture of the complete experimental set. This was done by generating random orientations and testing the orientation density of these, divided by the maximum orientation density. If that scaled orientation density was greater than a random number in the range 0–1, the orientation was accepted, otherwise it was not. From those sets, five were selected which gave the best correlation with the full start texture. Even with five sets, the statistics are still relatively sparse, and so five simulations using the same initial orientations but with one element per grain, i.e. 6×6×6 array domains, were also carried out. The average predicted textures of those simulations is shown in Fig. 11, which also shows the result from a simulation using 10 node tetrahedral elements arranged in rhombic dodecahedral groups to represent grains: that type of arrangement was previously used by Mika and Dawson [17]. The use of 27 elements to represent each grain gives a marginal improvement in correlation with the experimental results. The main improvement is a slightly better prediction in the spread of the ␣ fibre towards {1 1 1}00021 1 00003. The use of the dodecahedral groups gives no benefit, and it is likely that these do not give sufficient internal freedom to allow the inhomogeneous deformation possible in the 27 hexahedral element groups. The use of multiple elements to represent

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Fig. 9. Results from CPFEM modelling using one element per grain with different slip plane families involved. ODF sections at Euler angle ϕ2 = 45◦ are show at the top for a strain of 0.75. The graph at the bottom shows the correlation of the predictions—without any additional spreading—with experimental results as functions of strain.

grains fails to give an adequate prediction of the spreading out of orientation density in the α fibre which occurs at high strains, and it seems unlikely that this would be predicted using more elements per grain. This feature can best be shown by plots of orientation density as a function of & with ϕ1 = 0◦ and ϕ1 = 45◦ . These are shown for strains of 0.75 and 1.47 in Fig. 12. Those plots also include the Taylor 13 + 23 RC results, with optimal spread. Although the RC Taylor predictions give too much spread towards {0 0 1}00021 1 00003 at low strain, the prediction at the higher strain is rather better than the CPFEM one. The conventional view of relaxed constraints depending on grain shape would appear capable of resolving that because the relaxation would occur progressively with straining. However, it is difficult to see why the finite element model would not include any relaxation which would actually occur. There is, in fact, good visual correlation between the deformed grain configurations predicted by the CPFEM with multiple elements per grain and those observed by optical metallography: a feature of which is grain ‘curling’ in sections normal to the RD. The RC Taylor

model takes no account of the mechanical behaviour of the material, and work hardening tends to reduce inhomogeneity of deformation. However, reducing the ΘIV hardening contribution in the material model, Eq. (9), to zero fails to improve the ␣ fibre prediction of the CPFEM at high strains.

4. Discussion The main point arising from the work described above is that, although the CPFEM gives absolutely better predictions of deformation texture than Taylor models at strains up to about 0.75, it fails to predict the increase in ␣ fibre spread towards {0 0 1}00021 1 00003 which occurs at higher strains. Although that spread is predicted by Taylor modelling with 13 + 23 relaxed constraints, the CPFEM predictions are generally superior to those of Taylor models. A major advantage of the CPFEM is that it predicts textures with the correct order of orientation densities. These are much too high in Taylor model predictions, even with

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Fig. 10. The effect of the number of elements used to represent the single orientation (2 2 3)[1 1 0] on the spread of orientation following a plane strain of 0.5. The orientation spread is shown as 0 0 1 pole figures [x = RD vertical] in the top four images. The contour plot of the 10 × 10 × 10 domain, shown at the bottom, is coded by the Euler angle, Φ, and shows the association of that with the inhomogeneous deformation.

relaxed constraints, and this demonstrates in a clear way the importance of inhomogeneity of deformation on texture development. This inhomogeneity is sufficient to ensure that, after quite moderate strains, there are typically no regions of orientation space that are unpopulated. The finite element model only takes into account mechanical differences due to orientation, and there may be others, such as grain size. Fine grained materials typically have higher flow stress at low temperatures, and a region of relatively small grains

would locally require a higher stress. This argument may extend to individual grains, in which case there should be even more inhomogeneity of deformation than predicted by the CPFEM and so greater spread of texture. That this does not seem to be necessary might be due to the grain size becoming irrelevant as orientation differences develop within grains. It is possible to use data for the inhomogeneity of deformation predicted by CPFEM in a very general way to perturb

Fig. 11. Results from CPFEM modelling showing the effect of using multiple elements to represent ‘grains’, compared with a result using one element per grain with the same set of start orientations. For the result at the right, there were 512 dodecahedral groups and partial groups of quadratic tetrahedra.

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Fig. 12. Experimental and predicted orientation densities along the α fibre. The results for the Taylor RC model have been subject to the additional Gaussian spread which gives best correlation with the relevant experimental texture.

local deformation states in a Taylor model [27], and this hybrid approach gives better prediction of orientation density. However, it ignores the connections between inhomogeneity and the actual local configurations of orientations which are responsible for that inhomogeneity [16]. Although such local configurations are dealt with to some extent in recent modifications to the Taylor model [28,29], they can only be dealt with in anything approaching a realistic manner by methods such as CPFEM. The prediction of an inhomogeneity by a simpler model which is not predicted by CPFEM is likely to be an artefact of that simpler model. In other words, because CPFEM must be a more realistic model of the intragranular mechanics than any Taylor model, it is suggested that the prediction of the α fibre spread by the RC model is a coincidence. The occurrence of 23 shear, and rotation, in α fibre orientations is largely responsible for the distribution of orientations in that fibre. Using multiple elements to represent

grains allows a greater degree of intragranular inhomogeneity, but this still does not give adequate prediction of the α fibre evolution. It may be that an even greater subdivision of grains would give better prediction, but this would mean very large models—and very great computer time—and the indications from the single orientation modelling presented above are that the improvement may only be marginal. There are fundamental limits to the extent to which microstructural inhomogeneity can be modelled by continuum methods [26], and it is unlikely that some of the features which are observed in reality—for example, the in-grain shear bands occurring in γ fibre orientations [30] which are potentially important in determining the orientations of recrystallisation nuclei [31]—could be predicted using CPFEM. Further insight into the effects of deformation inhomogeneity could potentially be obtained from investigations involving different initial grain sizes and also deforming under conditions giving different mechanical response. It is not clear whether some deficiency in the ability of the CPFEM to deal with inhomogeneity of deformation is ultimately responsible for the discrepancy between the experimental and predicted textures. Other factors could influence the development of deformation texture. It may be possible to devise a material model which gives better predictions, although limited investigations using latent hardening in the present model did not give any benefit. An example is shown in Fig. 13, where slip on a given system was taken to contribute an additional factor of 0.2 to hardening on all other slip systems, i.e. a cross hardening of 1.2. This has had a slight negative effect on the correlation of prediction and experiment. There are, of course, many other possible ways in which the mechanical behaviour could be changed and it is possible that some could lead to better

Fig. 13. The correlation of CPFEM predictions with experimental textures, using 18 × 18 × 18 meshes and one element per grain, using {1 1 1} + {1 1 2} slip planes with isotropic and with latent hardening. The latent hardening in this example used a cross-hardening factor of 1.2, i.e. slip on a system hardened all other systems 20% more than the self-hardening on that system.

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deformation texture prediction, but physical justification would naturally be desirable. Another possibility, unlikely though it may seem, is that high angle boundary migration is taking place and is affecting the texture. A increase in {0 0 1}00021 1 00003 has been observed during annealing of ferrite and this can be associated with ‘broad front’ boundary migration driven by the lower substructural energy of that orientation [32]. Although the temperature involved here was much lower than that required for static annealing, there is clear evidence that high angle boundary migration occurs in the cold deformation of aluminium and this is associated with dynamic recovery [33]. This process effectively limits the high angle boundary spacing which can be achieved with cold deformation. Although there was no clear evidence of this phenomenon in the present material, only 5% of the material volume would need to be involved to satisfy the discrepancy between the experimental result and the CPFEM prediction. This must remain a highly speculative hypothesis at this stage. It must be remembered that a major disadvantage of the CPFEM, in comparison to Taylor models, is the amount of computation involved. Using a 2 GHz processor, the typical CPU times for the CPFEM simulations reported here were about 30 h, compared to about 2 min for Taylor models. This factor is a clear driver for the development and application of models such as those of Van Houtte et al. [28] and Crumbach et al. [29]. 5. Conclusions The deformation texture of ferrite measured here agrees with that found elsewhere. A well-defined start texture, significantly different from the stable deformation texture, has been used and the development of that texture has been compared with both Taylor and CPFEM predictions. In general, the CPFEM gave better predictions, but at high strain did not adequately predict the distribution of orientation density in the α fibre. That distribution, if not the overall level, was predicted quite well using a Taylor model with relaxed simple shears having shear planes lying in the sheet plane. Such shearing can also be associated with intragranular inhomogeneity, but the level of this predicted by the CPFEM, even with multiple elements used to represent the initial grains, was not sufficient to give good prediction of the α fibre texture. The prediction of the relaxed constraint Taylor model is taken to be a coincidence, and the disparity between the finite element predictions and observed textures at high strain is due to subtle deficiencies in the modelling of inhomogeneity or the involvement of a factor such as high angle boundary migration during deformation.

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Acknowledgements The authors thank W.B. Hutchinson for his encouragement and constructive criticism. Thanks also to Mark Harris and his colleagues in the MMSC workshop for their skilful manipulation of samples. This work was supported by the EPSRC via grant GR/R69341.

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