MODELLING HIGH TEMPERATURE ROLLING TEXTURES OF FCC METALS

A partially relaxed constraints grain deformation model is proposed to explain the influence of temperature on the rolling textures of fcc metals. The effects of the grain plastic shear in the TD/RD plane and the role of the rate sensitivity of crystallographic slip on the evolution of the texture have been investigated by numerical simulations for a random initial texture. The rate sensitivity and the TD/RD shear are assumed to increase with temperature. The progression from the Copper {112}(111) component towards Brass {110}(112) and S {123}(634) type textures is predicted at higher values of the rate sensitivity and the TD/RD shear. These model predictions compare well with published hot rolling textures of aluminium alloys. The concept of grain shear partial relaxation has been validated by room and high temperature channel die tests on {110}(112) oriented AI crystals constrained between aluminium polycrystals.


INTRODUCTION
The typical evolution of deformation textures during the cold rolling of medium to high stacking fault energy fcc metals can be adequately described by classical Taylor-type models, i.e. the Full Constraints (FC) or Relaxed Constraints (RC) models, see for example the reviews by Van Houtte (1986) and Hirsch and Liicke (1988). These models can be extended to low stacking fault energy alloys, or equivalently to low temperature deformation, by allowing for twinning in addition However, it is well known that high temperature rolling in the homologous temperature range 0.4-0.8Tm, (and when dynamic recrystallization does not occur) also leads to the development of texture components which can be significantly different from those developed by cold rolling. In particular, hot rolling often develops a strong Bs component which, at these temperatures, cannot be ascribed to twinning. A common example is the strong Bs component of aluminium alloys rolled in the temperature range 300-500C. The origin of this high temperature texture component has been discussed in detail by Bacroix and Jonas, (1988a). The most frequent explanations are (i) non-octohedral slip 211 (Bacroix and Jonas, 1988b) and (ii) a Sachs type RC model in which the grain TD/RD shear is completely relaxed (Hirsch and Liicke, 1988). Neither explanation is totally free from criticism. Non-octahedral slip has been observed at low strain rates, <10 -3 S -1, but not, as yet, at the high strain rates, >10 s-1, typical of industrial hot rolling schedules. Secondly, it is generally agreed that the Sachs type grain relaxation leads to unreasonably high strain rate incompatibilities between grains.
These physical problems have hindered texture modelling for hot rolling, one of the most technologically important metallurgical processes. The absence of adequate high temperature deformation texture predictions also limits the usefulness of texture modelling during subsequent treatments, such as cold rolling or recrystallization, which depend upon the metallurgical state of the hot deformed material.
It is the purpose of this paper to present a simple variation of the RC model that can be used to quantitatively predict the development of hot rolling textures. The model is applied to the calculation of ODFs (orientation distribution functions) of hot rolled fcc metals as a function of strain. The results are compared with the textures of hot rolled Al alloys taken from the literature.

THE PARTIALLY RELAXED CONSTRAINTS MODEL Grain Shear Strains
As mentioned above, a strong Bs rolling texture component is predicted by a Taylor type model in which the TD/RD shear is completely relaxed (this is the 12 shear strain rate in a reference frame for which X1--RD, X2 TD and X3 ND). However, during rolling the grain shape tends towards a "surf board" shape which means that a totally free 12 shear rate would involve large intergranular strain rate incompatibilities, extending over relatively large volumes of the grains. The high reaction stresses developed in the surrounding grains would therefore strongly reduce the /12 shear rate. In fact a totally free 12 shear strain rate is associated with a zero shear stress o'12 as, for example, in channel die compression of a single crystal. Under these conditions 12 takes a value, denoted 2 which is, of course, orientation dependent, being, for example, very high for the Bs orientation (0.35/0 and zero for the Goss orientation {011) (100).
It is physically more realistic to allow the grains to undergo small 12 strain rates, significantly less than /12. There is some experimental evidence for this from channel die experiments on Al polycrystals, (Fortunier (1987) and Mizera (1988)). Recent finite element calculations of grain deformations in a polycrystal undergoing plane strain compression, (Becker and Lalli (1991)), also reveal el2 grain shears, particularly for grains near to the Bs orientation. Experimental results described below on partially constrained crystals illustrate this effect quite clearly.
It is therefore proposed that during high temperature rolling individual grains undergo a shear strain rate 12 reduced by a factor compared with the RC value 2, where 0 < < 1. For most conditions, is of the order of 0.2 so that the shears are relatively small. At the same time, in conformity with the usual RC lath model, the e13 shear (eRD/ND) is taken at the value e3 corresponding to cr13 0.
The stress and strain boundary conditions imposed on the grains in the present model are therefore 0 0"23 0"33/ 3 0

The Rate Sensitivity Model
With increasing temperatures the flow stresses of metallic materials become increasingly sensitive to the deformation rate. High temperature deformation and texture evolution can then be reasonably treated by a rate sensitive model.
The model used here is that developed by T6th, Gilormini and Jonas (1988) which has been applied to the cases of torsion texture development (T6th et al. (1988)) and single crystal channel die deformation (T6th, Jonas and Neale (1990)). Following Pierce, Asaro and Needleman (1983), the rate sensitivity of slip is modelled by a power law relation between the shear rate and the resolved shear stress r o (1) r r sgn () ,-0 m is the slip system rate sensitivity (rn -> 0), : and 2o are reference quantities for the shear stresses and the slip rates. For the case of isotropic hardening 0 considered here : to. Equation (1) implies the existence of a stress potential which is rounded, strictly convex and tends to the Bishop and Hill yield surface when ro is set equal to rc the critical resolved shear stress for rate insensitive slip, and m tends to zero.
The stress response of a grain to imposed strain rates is obtained by solving the following system of non-linear equations o (m + m)mtok, [mnOmn[ (1 (2) kij where the quantities m are defined by the slip plane normal n and the slip direction b by mq b in.
The number of unknowns in Eq.
(2) (the components of ) usually equals the number of imposed independent strain rate components, i.e. 5 for the fully constrained deformation mode.
In the partially relaxed constrained mode described above, the 3 fully constrained strain rate components k, 33 and 3 are first used to calculate an initial stress state with 3 unknowns, 0, 33 and o3 (01, 03 0). The relaxed constraints values of the non-prescribed strain rate components and 3 are then obtained directly from Eq.
The stress state corresponding to the boundary conditions for the partially relaxed grain deformation mode (= ) is then recalculated from Eq. (2) with 4 unknown stress components (11, 33, 23 and o). This value of is then used to evaluate the resolved shear stress via the usual relation The slip rates are then obtained directly from Eq. (1). It is evident from this equation and the existence of a rounded yield potential that rate sensitive slip always leads to a unique slip rate distribution as well as to a unique stress state. Finally the lattice rotation rate is obtained from where d# is the displacement gradient velocity tensor of the material with respect to the laboratgry. For the grain deformation modes considered here, the components of d that take zero values can be used to define as follows: 12 ml EXPERIMENTAL Channel die plane strain compression tests on partially constrained A1 single crystals have been carried out to measure the amount of/12 shears at different temperatures.
High purity aluminium single crystals with a (110)(112)  inscribed with a I mm spacing on a compression and ,a transverse face to measure the shear strains and the degree of deformation homogeneity. These "constrained" crystals were deformed to compressive true strains of about 0.6 at temperatures of 22C and 350C. A standard teflon lubricated punch and die apparatus was used at room temperature (Orlans-Joliet B. et al. (1990)).
High temperature tests were carried out in a suitably modified channel die using Teflon and graphite lubrication at a strain rate of 10 -2 s-x. Previous tests have shown that homogeneous uniform deformations up to strains of 1.5 can be achieved, up to 400C, with this equipment (Maurice (1992)). Figures 2(a) and (b) show the grid displacements on the compression face due to the e12 shears after true strains of 0.59 at 22C (a) and 0.53 at 350C (b). The important feature illustrated by these figures is the shear strain that occurs in the central {110}(112) crystal despite the presence of the polycrystalline ends. The latter partially constrain the crystal since the average measured shear strains are roughly half the theoretical value for unconstrained shear (2 (V'2/4)/). The measured shears are in fact 0.14 at room temperature (i.e. 49% of the theoretical finite shear corresponding to a compressive true strain of 0.59) and 0.11 at 350C (i.e. 42% of the theoretical value obtained for a compressive true strain of 0.53).
These values indicate apparently similar behaviour at room and high temperature. However, close inspection of the reference grids on the polycrystals indicate significant differences: at room temperature, the polycrystals are sheared by the central crystal for distances over half their length, whereas at high temperature the e12 shear of the polycrystals is negligible, implying large shear strain rate gradients at the poly/monocrystal interface. This suggests that at high temperature shear strain gradients between neighbouring grains can be accommodated relatively easily. Under the more typical rolling conditions of a small grain constrained by similar sized grains, one can therefore expect easier accommodation of the grain shear strains at high temperature, i.e. the individual grains can shear more easily leading to higher values.

NUMERICAL SIMULATION PROCEDURE
Most calculations were performed with a set of 936 initial grain orientations distributed in Euler space so that the value of the orientation distribution function f(g) 1 + 0.03 (Fortunier (1987)). The numerical simulations using these orientations then describe the theoretical texture evolution from a set of initial "random" orientations.
For each grain orientation, the lattice rotations and the new orientations were calculated by the method described in Section 2, using strain increments of 0.025 for finite strains up to 3. At fixed strains, the file of discrete orientations was converted into ODFs using the Gaussian function method with a spread of 5. ODFs were printed out as such, or to facilitate quantitative comparisons, the orientation densities f(g) were evaluated along standard skeleton lines such as the c and fl fibres.
Two parameters have been used in the simulations to characterize the influence of temperature; the strain rate sensitivity m and the 12 shear rate parameter .I t is well known that m increases with temperature. Tegart (1966) quotes values for aluminium from 0.02 (20C) to 0.07 (300C) and 0.15 (500C). More recent data indicate somewhat higher values; 0.23 for a 1100 Al alloy measured by Brown, Kim and Anand (1989) in the range 300-500C and 0.25 to 0.29 for AI-2% Mg (Chen and Kocks (1991)) at 300-400C. We have used a range of m values from 0.05 to 0.25 as being typical of the behaviour of most Al alloys in the temperature range 20C-500C.
The variation of with temperature during rolling is scarcely known at all. Our experiments, for the particular case of a partially constrained crystal in channel die compression, indicate values of about 0.5. However, we strongly suspect that for rolling of small grains, such large values can only be achieved at high temperatures. At low temperatures the intergranular strain rate incompatibilities should substantially reduce .V alues used here vary systematically from 0 to 0.5 which can be considered to represent extremes of material behaviour.
The entire set of values of the parameters used in the simulations is given below m:0.05, 0.1, 0.15, 0.2, 0.25 :0, 0.1, 0.25, 0.5 e:0,5, 1, 1.5, 2, 2.5, To simplify the interpretation, results will be described in detail for 3 conditions which can be considered to qualitatively represent 3 temperature regimes.
(i) low temperature (ii) medium temperature (iii) high temperature Influence of m and Figure 5 illustrates the influence of the parameter on the fl fibre texture components at a constant strain rate sensitivity of rn 0.05. At 0 a strong Cu component is developed as expected from the classical relaxed constraints model with a free e13 shear. At higher values, however, it is the S component that increases most and dominates for ->0.25. To the authors' knowledge, the role of a partial relaxation of 12 in strengthening the S component has not been previously considered. For example, Hirsch and Liicke (1988) have shown that total relaxation of 2 ( 1) virtually eliminates the S component at the expense of Bs. They argue that the 23 shear is necessary to stabilise the S component. The present results show that a partially relaxed 12 shear can also stabilise S. Figure 5(c) also shows that at 0.5 the Bs component can become stronger than Cu. However if < 1 very strong Bs components can only be achieved with a higher m value. This is demonstrated in Figure 6 by the fl fibre plots at constant =0.25 and increasing m. As the rate sensitivity increases, the S component decreases in intensity, while Bs increases. Finally if 0, corresponding to the more conventional Taylor models, then increasing m results in a slightly weaker Cu and a somewhat stronger Bs components. This effect has also been pointed out by Chen and Kooks (1991). Note also that increasing m tends to decrease the fmax values; this is a well-known effect of high rate sensitivity values which tend to distribute the slip rate more uniformily, (T6th et al. (1988)) so that the overall lattice rotation rates decrease leading to slightly weaker textures.
Another interesting feature of higher m and values is the influence on the stability of the Cube and Goss orientations, i.e. along the RD rotated cube fibre (0< q < 45). This is illustrated in Figure 7 by normalized f(g)/fmax plots for simulated textures using the 3 "temperature regimes." According to the model the stability of both the cube and Goss components should generally increase with increasing temperature.

DISCUSSION
A simple model has been presented to describe the influence of elevated temperatures on rolling texture formation of fcc metals. The model assumes that the rate sensitivity rn increases with temperature which is supported by most experimental data. However, as noted by Chen and Kocks (1991), the influence of the strain rate sensitivity by itself is insufficient to explain the large variations of rolling textures of most fcc alloys with increasing temperature. Our model further assumes that the amount of the e12 shear increases with temperature (although always significantly less that the value corresponding to the Sachs type RC model). This hypothesis is the most difficult to justify theoretically at the present time. There are good grounds for arguing that/h2 takes values between 0 and the Sachs type RC value 2, as clearly shown by the channel die tests of partially constrained crystals. Other experimental measurements of grain shears in compressed AI polycrystals, even at room temperature, indicate significant e12 shears. Furthermore, as pointed out by Becker (1991), the stress states required to enforce 12 0 for the near-Bs orientations are physically unreasonable; they require large th2 values of the order of the compressive stress. Some relaxation of 12 is therefore inevitable. The problem is to quantify the degree of relaxation of grains in a rolled polycrystal as a function of temperature.
It is clear nevertheless that, using reasonable values of m and , the model gives qualitatively good agreement with the high temperature rolling textures of aluminium. In particular, the model predicts an evolution of the fl fibre components from Cu to Bs + S at high temperatures.  (1990)).
Quantitative comparison of the model simulations with experimental hot rolling textures of fcc metals is hampered by the absence of good quantitative texture data. Most available data does not specify in sufficient detail the important parameters such as the initial texture, the rolling strain, strain rates or even temperature (since the latter vary during rolling). Qualitative comparisons do however show the trends predicted by the model. For example, most hot rolled high strength aluminium alloys (7000 series, A1-Li) develop, in the absence of recrystallization, very strong Bs textures (Hirsch (1990), Liicke et al. (1991)).
Furthermore inspection of these data reveals that the S component increases during hot rolling at the expense of the Cu component--as predicted by the present model. This is also confirmed by the ODFs of hot rolled Al-1% Mn-1.2% Mg (Bate and Oscarsson (1990)), shown in Figure 8. At a strain of 2, Cu, S and Bs have roughly similar densities (with a maximum near Cu). At a strain of 5 both S and Bs have doubled in intensity and Cu has decreased slightly.
There is another useful feature of the present model predictions; both cube and Goss components should strengthen at higher rolling temperatures. Very recent work by Daaland et al. (1992) on a hot rolled 3104 aluminium alloy confirms this general trend.