Introduction

The main purpose of texture simulation is the reliable understanding and prediction of textures. A good understanding of the main underlying mechanisms for the presence of different components has been achieved but the models have not been so successful in the prediction of texture sharpness or component intensities. It is quite well understood that slight differences in strain rates, grain size and/or alloying compositions, through stacking fault energy changes, can influence the rate of development and the final sharpness of textures. However a more general disagreement between simulations and experiments is the evident lack of sharpness matching by, sometimes, an order of magnitude at high deformations. While component texture prediction and sharpness matching is quite good at medium strains the simulations do not do so well at very low and high deformations. The main argued reason for that disagreement is a well accepted one: Each grain does not follow exactly the equal deformation Taylor assumption neither follows some of the Self-Consistent model calculations. The technical procedure has long been an appropriate Gaussian smoothing, capable to reduce the intensities to levels compatible with experiments. However minor components are usually not attainable by such procedure.
Briefing the sources of heterogeneity, in the same way as the residual strains are usually classified, we can identify three main types: a) 1st kind are those heterogeneities comprising many grains in a macroscopic arrangement reaching even the level of the sample. FEM models are mandatory for dealing with this level of heterogeneity. b) 2nd kind are those heterogeneous strains spanning over a few neighboring grains. They are usually approached by SC models and/or FEM working on the basis of "one grain-one element". Variations from grain-to-grain acquire special importance by the presence of two phases. Heterogeneities are larger and the models should not be stressed beyond their limits. c) 3rd kind heterogeneities are developed internally in each grain by dislocation arrangements built up and consequent fragmentation. They are approached by the FEM scheme "one grain-many elements". In the understanding of the current authors the fragmentation process is well simulated by using a spin sharing approach, technique that will be applied in the current paper [1-2]. 
Macroscopic heterogeneities (1st kind), those stemming from the test constrains, are among the mechanisms responsible for texture smoothing processes. Different regions of a sample are subject to different strain paths due to interaction with the machine tools and geometrical constraints imposed by the desired final shape. In rolled materials the sample surface undergoes through a process of shearing and counter-shearing due to the roll action that is absent in the middle part of the sample. The superposition of texture patterns characteristic of each deformation path confers to the composed texture a lower severity or sharpness, comparing with homogeneously deforming materials. The main reasons for heterogeneous deformation are usually combined: geometrical as well as friction effects, which are not easily separable. In extruded or wire drawn materials the characteristic of the die (friction coefficient), geometry (die angle) and speed are combined to give lower sharpness textures. Die angle is the mayor modifier of the uni-axial velocity gradient. The most appropriate technique to evaluate strain heterogeneity at that level is FEM [3-4].
Regarding the ability of SC models to simulate grain-to-grain heterogeneity it is worthy to emphasize the many neighbors surrounding a single crystal: 12 in an ideal compact regular arrangement and from 5 to 7 in a real polycrystal. Those surrounding grains should actually be representative of a local cluster reproducing in average the behavior of the matrix in which SC models assume the main grain is embedded. 2nd kind heterogeneity is quite likely well simulated by having each crystal embedded in a matrix with visco-plastic properties averaged over the rest of the crystals. The main variations from grain to grain are due to the grain orientation and the statistical homogeneity of closest neighbor orientations.
On the opposite end of the scale, the individual grains actually deform very heterogeneously, achieving strain compatibility and equilibrium by developing different internal dislocation structures (3rd kind). Those dislocation structures are grain orientation dependent and interrelated, at different deformations, in a not completely well known way. Notwithstanding they bear the common feature of being able to accumulate dislocations of the kind that are activated by the general Taylor rules. Minimum energy principles for the accumulated dislocations are the master rules to decide which dislocations are indeed active. The experiments conducted in this field are quite profuse. It is by now pretty clear that which were in principle assumed to be the main regions of a polycrystal acting as dislocation sink areas, the grain boundaries, are actually not. The dislocations are accumulated in regions internal to the grains and, there, they develop quite complex interactions and consequently carry the local deformations and crystal lattice spin (rotation rate). The experiments also show quite large misorientations among the many fragments developed in a grain. Some dislocation arrangements, known as Geometrically Necessary Boundaries (GNB), introduce larger misorientations than others, called Incidental Dislocation Boundaries (IDB) or statistically accumulated dislocations [5-7]. Those intra-grain misorientations, and consequent texture development, are simulated in the current paper by using a co-spin scheme between two close neighbor grains in a VPSC model. Details of the model have been published elsewhere [1] and will be just briefed in the third section: Micromechanical model. A detailed justification from the microscopic viewpoint and an application to a single phase material can be found in Bolmaro et al. [2].
The relative importance of macroscopic vs. grain-to-grain and intra-grain heterogeneities will be assessed by a combination of an n-Sites Visco-plastic Self-Consistent (nS-VPSC used alternatively in its 1 and 2-sites versions) micromechanical model and Finite Element Methods (FEM). The resultant textures will be compared with a micromechanical model applied to a seemingly wire drawing test sample undergoing homogeneous deformation. The micromechanical model will alternatively reckon texture development on the Taylor assumption or Self Consistent approach, which takes care of the 2nd kind heterogeneity. All together will shed light on experimental data obtained by neutron diffraction of Cu-Fe wire-drawn samples. 

Experimental data

The experimental data consists in texture measurements performed in extrusion-wire drawing of Cu-Fe powder composites. The starting samples are 25%Cu-75%Fe hot extruded samples prepared by powder metallurgy (800 oC). The samples were afterwards wire drawn to a Von Mises strain of 3.0 at room temperature. The starting texture and the texture every 0.25 Von Mises strain for both phases were measured by neutron diffraction using the TEX-2 diffractometer [8] at the Geesthacht Neutron Facility (GeNF), Germany. Experimental information was processed by using software packages supported in popLA [9] and Beartex [10]. The same packages are later on used for processing the simulation results ensuring a similar transfer function. That is to say, the Euler angles orientation-weight files used for simulations and the orientation-intensity files used for ODF plotting are smoothed only by the numerical procedure used to process the experimental data. A 5º x 5º grid is used for both experiments and simulations. Details about sample fabrication and measurement procedure have been provided elsewhere [11]. Fig. 3 g) shows the inverse pole figures for an equivalent strain of 3.0. Three main components are noticeable at almost the same level of strength for Cu: Preferential <100>, <111> and <112> orientations along the cylindrical axes of the sample with a low sharpness. Fe shows the characteristic <110> orientation, along the same sample direction, with an incomplete fiber or spread to the <113> direction with a minor intermediate maximum.

 
Micromechanical model

The crystal behavior follows a potential law relating the applied stress  in each plane and the Critical Resolved Shear Stress (CRSS) of that system .  The total strain rate in each grain is obtained adding the shear strain rates  of each active slip system and/or twining:

₍₁₎

where  is the crystalline microscopic strain rate,  is the microscopic deviatoric stress,  is the Schmid tensor describing the geometry of the slip system s in the single crystal, n^s is the slip plane normal, b^s is the slip direction,  is a reference strain rate velocity, n is the inverse of the strain rate sensitivity and  is the secant viscoplastic compliance crystal modulus.

The material behavior under a viscoplastic regime is described through the following expression:

                                                                                (2)

where  is the macroscopic stress,  is the macroscopic deviatoric strain rate tensor and  is the secant material modulus. Currently  is calculated by means of a Self-Consistent scheme leading to the interaction equation that relates stress and strain rate deviations at crystal level [12-13].

                                                                  (3a)

                                                               (3b)

where  is the accommodation tensor, a is a tuning factor and S is the Eshelby tensor. The  is a scalar parameter that tunes the interaction as follows:

                                                               (4a)

                                                                  (4b)

where

                                                 (5)

where  is the tangent viscoplastic crystal modulus and  the macroscopic one. Simultaneously solving equations 4 a) and 4 b) leads to add a supplementary iteration level to the classic SC formulation. When different kind of slip systems are active or different phases are interacting a general secant-tangent relation can be written as:

                                                                          (6)

where  is now a 5x5 matrix. Particularly, when all matrix components n_ij are equal,  reduces to a diagonal matrix with all components equal to a scalar parameter  as in eq. 3 b). Otherwise Hutchinson’s relation between tangent and secant viscoplastic compliances is no longer valid [14]. Turner et al. [15] solved the mixed creep problem that we currently extend to a viscoplastic regime. The physical meaning and implies of this extension will be discussed later.

The current model assumes that the dislocation structures formed inside each grain are responsible for both strain rate and spin. In such a way two companion grains will be allowed to freely deform following the rules coming from Taylor or SC models but spinning together like a unique entity. Statistically, each grain will represent only half of a grain, which other part is assigned as a companion to other randomly chosen grain. The two parts forming the same grains are initially oriented in the same direction but will spin and deform following the closest influence of the first neighbor. From the statistical viewpoint it is not even necessary to have both fractions present. The many randomly chosen grains, representative of the starting texture, are capable of representing the textures in a statistical sense [16].

The volume fraction can be controlled by the relative size of both ellipsoids, with both close companions always belonging to different phases, or by the number of grains assigned to each phase with mixed-phases and one-phase pairs statistically proportional to the volume fraction. No large differences were found by using both approaches and the first one, being cheaper in terms of computational time, was chosen. 
Without a careful evaluation of other microscopic features of the model, its predictions could be considered just a matter of chance. The main feature to be compared between experiments and simulations, in an average sense, would be the misorientation between first neighbors.Higher orders of correlation are, for the time being, not available in the literature. Hughes et al. [17] measured the average misorientation vs. deformation for Al. We have calculated such misorientations by the simple way of enforcing co-spin between closest neighbors and compare the results with experiments and Mika and Dawson FEM results [18]. They have been already discussed in [2] but Fig. 1 shows a reproduction to help fixing the ideas behind the model. Fig. 1 shows the power law relationship between the average misorientation angles and the equivalent strain. For large deformations the slope is 0.68, which is fairly close to the experimental value of 2/3. The values simulated by Mika and Dawson [18] are also shown and they seem to be underestimated in comparison with the experimental values. In our simulation the bias is probably coming from three effects not considered in the model: a) The original grain boundaries do evolve in real materials to allow the grains to reorient b) IDBs are also misorientation boundaries permitting texture development. c) The model allows a single GNB to be formed inside each grain, which is representative of small size grain materials. From 5 to 27 GNBs (for rolling and torsion respectively) have been observed in deformed materials [19]. The agreement between the current model and the experiments is clearly better at low deformations when the average number of GNBs per grain is close to two. At larger strains the GNBs allowed by the model develop misorientations larger than the experimental ones.
Both grains in the simulation were alternatively subject to the independent spin and co-spin schemes. Both schemes are representative of no-fragmentation and fragmentation processes.

FEM model and coupling strategy

The velocity field and pressure distribution, solution of the boundary problem over the sample, is provided by the finite element method. We assumed that inertial effects and body forces in the global balance and the elastic components are negligible. The basic equations are solved using a finite element discretization over the workpiece volume. In standard form we found:

                                          (7)

The evolution of the deformation was calculated using an Eulerian scheme. The resolution method is based on a velocity-pressure formulation: v (velocity field), P (pressure field).  is the deviatoric part of the symmetric component of the velocity gradient and T represents the constrains over the workpiece surfaces. b

Fig. 2: FEM mesh used to perform 12 successive wire drawing processes of 0.25 equivalent Von Mises strain each 

The discretization equation leads to the classic system of linear algebraic equations to get the velocity and pressure fields. The discretization is performed using 8 nodes “brick” elements. The interpolation of the velocity field is linear, and the pressure field is assumed constant inside each element (elements P1-P0). The pressure term is integrated using only one Gauss point localized in the center of the element while other volumetric terms are integrated using usual 8 Gauss points. 
The mesh is iteratively adjusted to prevent mesh degenerations. Internal nodes are over the streamlines, which are recalculated in the iteration. Therefore, the mesh structure in the flow direction is obtained adding homomorphic sections between them.

Once the convergence is achieved, the streamline concept appears naturally and it can be followed from the elements localized in equivalent positions in the successive homomorphic transversal sections. The strain path calculation is obtained immediately from the streamline and constitutes the input for the texture calculation. c
Using the constitutive equation given by the micromechanical model (eq. 2),  can be eliminated in eq. 7. In function of the calculated velocity gradient in each element, the corresponding compliance tensor  is updated through the SC polycrystalline model for a subsequence calculation [20]. In the current paper the velocity gradients were calculated along “stream lines” for each set of elements representative of a fixed proportional radial distance. Results came out very much alike for both cases, coupled and un-coupled, and the last approach was chosen because of computing time economy.
Fig. 2 shows the mesh used to perform 12 successive wire drawing processes of 0.25 equivalent Von Mises deformation each to accumulate a total deformation of 3.00 (one quarter of a real sample).
Friction at the wire-die wall interface was considered negligible and shearing strains are assumed related to geometrical effects. Just a thin surface layer would receive influence from friction interaction but the volume fraction compromised by such process is considered very small at the current strain rate.

Results and discussion

Due to the high combined crystal and sample symmetry the textures will be shown as inverse pole figures.

Fig. 3 a) shows the texture simulated under the assumption of homogeneous deformation at both macroscopic and microscopic levels (Case 1). The sharpness is 50 and 10 times larger than the experimental values, respectively for Cu and Fe, showing the known result of excessively intense simulated textures by comparison with experiments.

Fig 3 b) shows the texture simulated under the assumption of a unique source of heterogeneity provided by the grain fragmentation (Case 2). This is the less likely behavior but it is included, by the sake of completeness, to compare the relative influence of each mechanism. The dramatic influence of co-spin over texture sharpness is considered credible, judging by the current results, because of other two new features. Cu texture slightly develops the (112) component present in the experiments and Fe texture develops an incomplete fiber going from (011) component to (113) direction. An unwanted result is the division of the (101) component creating a lower (203) component that has no similar in experiments.

The influence of macroscopic change of velocity gradient over texture, without any other source of heterogeneity, is shown in Fig. 3 c). The macroscopic heterogeneity is able to generate Fe fiber and slightly reduce the strength of both textures. The Cu (112) component is barely growing but most likely due to a simple smoothing process. g

By allowing pairs of grain to co-spin the intra-grain heterogeneity is induced and the results are shown in Fig. 3 d). Most components are shown but they lack the right strengths and locations.

Fig. 3 e) shows the influence of the usual grain-to-grain variations calculated by SC models (Case 5). It is worthy to say some few words regarding stress exponents and interaction between phases. The strain rate sensitivity obtained from an experiment, at the same temperature from a wire drawing experiment in single phase materials, could not be representative of the current behavior because the interaction between both phases could impose quite different behaviors to them [21,22]. However it is known that high symmetry metals show very low strain rate sensitivity regardless of texture. Thus, poly-crystal stress exponents are assumed representative of single crystal behavior. Stress exponents of 100 and 50 for Cu and Fe were used as coming from previous experiments and were found to be  reasonable values to achieve a fair match with experimental textures.
However interaction matrix (or coefficient of eq. 6) between both phases obeys to microscopic behaviors different from the ones acting at atomic level in a single-phase material. It is not a property that can be known in advance and it is currently inferred from the simulations and comparison with the experiments. An almost Sachs [23-24] behavior has been enforced by using a large interaction coefficient of approximately 63. It comes out as a natural selection by the mechanism proposed by Turner et al. [15] (a=1). a can be changed at will in order to simulate different polycrystalline or poly-phase behaviors.
A low number of active slip systems is expectable because of at least three reasons: a) The deformation reaches high levels where relaxation of some strain components due to grain shape certainly happen. b) It is proved that inside each sub-grain, or grain fragment, a low number of slip systems is active (usually 1 or 2). c) The Fe is prone to deform in plane strain and that will enforce ribbon shapes both in Cu and Fe. The Cu (112) component has clearly developed, this time due to the Sachs behavior of the SC models for large n exponents. However the sharpness is still 10 times larger than the experiments for both phases and showing almost no variation for the Fe texture with respect to the Taylor model values.
Fig. 3 f) shows the results considering microscopic heterogeneity stemming from grain fragmentation and grain-to-grain variation by using the SC model under the assumption of spin sharing between grains (Case 6). Intensities are approaching experimental values but the (112) component has moved slightly down to (113) direction.
Fig. 3 g) shows the textures obtained by superposition of the different velocity gradients in the final layer of the FEM mesh without spin sharing (Case 7). Texture in the central element showed the usual pattern and intensities of a homogeneous extrusion texture, with no shearing strains, at the same equivalent deformation. Texture sharpness is higher than the experimental ones, which is a common characteristic of almost every simulation. Redundant deformation, in the outermost layer elements, generated by the shearing strain represents approximately 30% of the total deformation and is negligible in the central elements. The outer layer elements undergo through a weakening process due to the combination of texture patterns characteristic of pure radial compression and shearing induced by a purely geometrical effect because of die shape. Friction at the die walls is considered to be able to modify a thin boundary layer at the surface of the sample but the main shearing effect is independent of friction coefficient election. After the combination of all elements, most components are almost in place but sharpness are still to high.
Finally, Fig. 3 h) shows the results obtained by considering a combined action of microscopic (grain-to-grain heterogeneity simulated by SC model plus fragmentation simulated by spin sharing) and macroscopic (different velocity gradients obtained by FEM) heterogeneities (Case 8). All simulated textures show the main characteristics of the experimental results being the main differences related with the intensities and slightly misplaced minor components.

Summing up the results:

a)          Main components are qualitatively well simulated by all models.

b)          Minor components are shown by different combinations of three effects:

b1) Sachs behavior and co-spin tend to explain, in some degree, the emergence of the Cu (112) component.

b2) Co-spin and macroscopic heterogeneity tend to explain (101)-(113) fiber in Fe.

c)          Minor and main sharpness of all components are quantitatively well simulated just by concurrence of all three levels of heterogeneity.

Some questions arise from the following criticism: improbability of having a casual combination of two or three different causes contributing to the same result. Analyzing the nature of the main and minor components can shed light on the general behavior and elucidate the underlying phenomena in all mechanisms. Main components, (001) and (111) for Cu and (101) for Fe, are called stable components. The grains tend to spin reaching those orientations and permanently settling there, provided that the strain path is not changed. Stable components are usually developed when the velocity gradient has no unti-symmetrical components included (no spins). When spins are included, whether they come from macroscopic set-up  (case of pure shear vs. simple shear, Bolmaro et al.[25]) or from redundant local velocity gradients (microscopic heterogenities) the trend is to develop unstable components. Those unstable components are permanently fed with new grains, which are lately drawn out of those orientations due to the spin component. Either by enforcing macroscopic spin (FEM) at different radii of the sample or by the SC spin calculated for each grain (SC models) or spin sharing between grains we make the orientation of the grains become unstable and create intermediate components. In fact we found that the orientation of the minor components is very sensitive to the die angle of extrusion (more or less determining the macroscopic spin), stress exponent  (influencing the SC microscopic spin) or spin sharing magnitude (microscopic intra-grain spin). In the limit of Sachs assumption the shear component is, in average, approximately 10% of the total strain. That deviation is accompanied by a larger SC spin component.

Regarding sharpness, the simulations appear much more intense than the experimental ones for Case 1 and with decreasing sharpness by adding different combinations of heterogeneity levels. Fig. 3 h) (Case 8) shows an almost perfect agreement with experimental results, exception made of a lack of well balanced distribution of (001) and (111) components in the Cu phase and because the (112) component, present in the experimental data, has slightly moved to larger indexes (around (225)).
The unstable nature of that component has been studied elsewhere [11]. It is also apparently built upon the continuous transference of grains from component (100) along the whole section <100> a <111> of the IPF to direction <111>. Fig 4 a) shows the experimental intensities for that IPF section for different deformation stages. Fig 4 b) shows a similar behavior obtained by applying SC simulation plus spin sharing assuming a macroscopic homogeneous situation (Case 6). The results were achieved by an slightly different approach: allowing the stress exponents to evolve along the deformation process in order to most accurately match textures for both phases at every 0.25 deformation step. The fact that the same effect is not very noticeable in the current simulations is due to a different pace of texture development not yet matched once FEM modeling is included. Current results actually emphasize the other aspect of the origin and development of the <112> component; its direct connection with plane deformation due to interaction with Fe which tend to deform in plane strain by itself.
The agreement is actually surprisingly good for the kind of very complicated deformation path plus phase interaction and very large imposed deformation. The large number of adjusting parameters along the whole simulation process, either in the micromecanical model or the FEM simulation, can certainly contribute to such good matching. However the selection of those parameters has always been made over physically grounded base: They can be rationalized in terms of volume fraction, CRSS ratio, phase interaction, average number of slip systems, etc.. Moreover, Fe texture is slightly softer than the experimental results showing than the imposed co-spin constraint might be stronger than the one coming from the grain-to-grain interaction and consequent fragmentation.

(a)

Fig. 6. Fe (110) Pole Figure with respect to ellipsoidal axes showing the preferential orientation of (100) direction along the shortest axes. Ellipsoid axes are not on scale.

The well-known “curling effect” is also partially tractable with the current model [26]. Each Eshelby tensor was updated independently for each grain together with orientation and axes for each hole.  We obtained, at Von Mises equivalent deformation =3.0, ribbon like grains with relative axes lengths of a₁= 0.12,  a₂= 0.71 and a₃= 11.93 leading to an aspect ratio a₂/a₁= 7. That aspect ratio is in close agreement with experimental results obtained in a few two-phase materials [27-28]. It is clear that real curling effect can not be obtained in the sense that Cu and Fe develop a bent entanglement only likely to be achievable by a finer scale FEM simulation.

Current simulations show that the interaction between Fe and Cu induce a plane strain deformation also for Cu phase. The aspect ratio of the shortest dimensions of grains is however much smaller, reaching a value of 3 at the largest deformation (Table I). Fig. 5 a) shows schematically a mixture of Cu and Fe crystals deformed by wire drawing. In this case the interaction coefficient is representative of the constraints exerted by the Fe presence. Cu by itself is not prone to develop plane strain under uniaxial deformation but the Fe grains are acting as internal tools over Cu phase. Besides deforming in plane strain, the Fe crystals enforce plane strain over the Cu phase. The crystal bending around the longitudinal axes does not contribute much to texture development because of the high symmetry of the test. Fortunately for the model the situation depicted in Fig. 5 b) is close enough to situation 5 a) in terms of texture developments. Fig. 6 shows results obtained for the orientation of Fe crystals with respect to the ellipsoids representative of the matrix holes where the crystals are located. It is clear that the main orientation along the longest ellipsoid axes is one of the <011> while the shortest is a <100> and the intermediate one is another <011>. Preferential orientations, with respect to the ellipsoid axes, is not so well developed for the Cu phase except for a preferential orientation along the longest axes in coincidence with the <112> crystalline orientation. This is a new result coming out from the simulations that should be confirmed by local measurements through EBSD techniques.

It is not expected in any case that the co-spin model hold for larger deformations. For heavily deformed Cu-Nb wire drawn samples a few authors have observed Nb crystals almost free of dislocations [27-29]. The deformation mechanism has been substituted at that deformation (Von Mises equivalent deformation = 9 ) by a dislocation shuttling between both sides of each grain. The grain boundaries are apparently acting as dislocation sources and sinks. In such situation the textures should develop further and misorientation between grains would not be kept constant, even not approximately.

Finally, we have to keep in mind that the texture intensities are obtained without any Gaussian smoothing and comprise, for the FEM-micromechanics simulations, 30000 grains distributed in 5^o x 5^o x 5^º bins in the Eulerian orientation space. It means that, to get the current numerical values, numerical massage has been kept at the minimum possible.  

Conclusions

The simulated textures show the expected symmetries and patterns in each case. Weakening process, in the average texture, is due to the superposition of the different element patterns, consequence of radial variation of velocity gradient (1^st kind or macroscopic), grain-to-grain heterogeneity (2^nd kind) and intra-grain heterogeneity (3^rd kind or microscopic). Clearly all levels of heterogeneity must be included to achieve texture sharpness results close enough to experimental values. Otherwise the inference of hardening parameters, stress exponents, etc. by means of simulation approaches can be misleading.

The three levels of heterogeneity have been shown contributing to the build up of minor texture components, and partial depletion of mayor ones, in different degrees. When the contributions of more of one level are shown the mechanisms are clearly complementary and due to the same underlying phenomena.

The simulated texture intensities are lower than the experimental values in Fe, may be due to an overestimation of the misorientation effect obtained through co-spin between companion grains. Notwithstanding the very simple scheme of having co-spinning grains, even in absence of macroscopic heterogeneity, takes the texture sharpness to values very close to experimental ones.

A discussion is due to the large values of stress exponents usually coming from experiments and simulations for metals and alloys. Other than the fact that we are enforcing plane strain, they can also be rationalized understanding the presence of just a few slip systems in a sole grain (which takes the exponent n to higher values) as a consequence of fragmenting grains. The rate independence law for metallic materials would naturally induce fragmentation. Different parts of a grain are subject to slightly different velocity gradient fields and compatibility is achieved by averaging over a few of those different fractions. In the current simulations the number of slip systems is kept around 2.5 in average for Cu and Fe along the whole deformation process.

A search and checking, either from experiments or simulations, for the best parameters (n exponents, hardening laws, interaction coefficients, etc.) was performed exhaustively for the macroscopically homogeneous cases. The search could not be done as exhaustively as in that case for the FEM based simulations due to the larger computing time. However, the paper shows the capability of the model to integrate all three major levels of heterogeneity in a still computationally tractable way.

References 

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10.- See web page at http://www.seismo.berkeley.edu/~wenk/beartex.htm 

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19.- D.A. Hughes and N. Hansen. Acta mater., 45, 3871, 1997. 

20.- J.W. Signorelli, PhD Thesis. Rosario National University. Argentina , 1999. 

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23.- G.Z. Sachs, Der Verein dutsher Ingenieur, 72, 734, 1928.   

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28.- E. Snoeck, F. Lecouturier, L. Thilly, M.J. Casanove, H. Rakoto, G. Coffe, S. Askénazy, J.P. Peyrade, C. Roucau, V. Pantsyrny, A. Shikov and A. Nikulin. Scripta mater. , 38, 1643, 1998. 

29.- S.I. Hong and M.A. Hill, Mat. Sc. Eng. A281, 189, 2000.

Acknowledgements: 
           This project was partially funded by the German-Argentine Intergovernmental Agreement # ARG 6 L1A 1A/2 and CONICET-Argentina.

Table I

Main mechanical parameters. Inputs and outputs of the simulations

Figure captions

Fig. 1. Power law relationships between average misorientation angle and equivalent deformation. Hughes et al. experiments [17]. Linear fit: 0.656. Mika and Dawson [18]. Linear fit: 0.95. Current simulations: linear fit for small strains: 0.83; idem for large strains: 0.69

Fig. 2: FEM mesh used to perform 12 successive wire drawing processes of 0.25 equivalent Von Mises strain each (total deformation accumulated = 3.00).

Fig. 3. Simulations performed by: a) Case 1.Taylor assumption of equal deformation for each grain. b) Case 2. Taylor model but allowing grains to fragment by using the co-spin scheme. c) Taylor model plus Heterogeneous macroscopic deformation. d) Taylor model, heterogeneous macroscopic strain and co-spin. e) Case 3. Grain to grain heterogeneity calculated by SC models. f) Case 4. Considering microscopic heterogeneity stemming from grain fragmentation by using the SC model under the assumption of spin sharing between grains. g) Case 5. Superposition of the different velocity gradients in the final layer of the FEM mesh without spin sharing. h) Case 6. Considering a combined action of microscopic (fragmentation simulated by spin sharing and grain to grain heterogeneity by SC models) and macroscopic (different velocity gradients obtained by FEM) heterogeneities. 
Experimental texture. i) Inverse pole figures for 25%Cu-75%Fe powder composite. Wire drawn until Von Mises equivalent deformation of 3.0. All results in log scale.

Fig 4 a). Experimental intensities for section <100> a <111> of the IPF for different deformation stages. b) behavior obtained by applying SC simulation plus spin sharing assuming a macroscopic homogeneous situation (Case 3).

Fig 5. a) Curling effect induced by the Fe grains over both phases. b) Geometry and orientation of grains provided by the SC model for large stress exponents.

Fig. 6. Fe (110) Pole Figure with respect to ellipsoidal axes showing the preferential orientation of (100) direction along the shortest axes. Ellipsoid axis are not on scale.

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Fig. 3. Simulations performed by: a) Case 1.Taylor assumption of equal deformation for each grain. b) Case 2. Taylor model but allowing grains to fragment by using the co-spin scheme. c) Taylor model plus Heterogeneous macroscopic deformation. d) Taylor model, heterogeneous macroscopic strain and co-spin. e) Case 3. Grain to grain heterogeneity calculated by SC models. f) Case 4. Considering microscopic heterogeneity stemming from grain fragmentation by using the SC model under the assumption of spin sharing between grains. g) Case 5. Superposition of the different velocity gradients in the final layer of the FEM mesh without spin sharing. h) Case 6. Considering a combined action of microscopic (fragmentation simulated by spin sharing and grain to grain heterogeneity by SC models) and macroscopic (different velocity gradients obtained by FEM) heterogeneities. Experimental texture. i) Inverse pole figures for 25%Cu-75%Fe powder composite. Wire drawn until Von Mises equivalent deformation of 3.0. All results in log scale.