Keywords: plasticité cristalline, texture cristallographique, visco-plasticité, modè le intermé diaire, maclage, interaction inter-granulaire, modè le- Polycrystalline materials, crystal plasticity, texture evolution, visco-plasticity, intermediate model, mechanical twinning, grain interaction strength,  -model

The computation of the macroscopic response of polycrystalline aggregates from the properties of their constituent single-crystal grains is considered as one of the main problems in materials mechanics. During the mechanical deformation processing, all the grains in the polycrystalline material sample are reoriented. A crystallographic texture may thus be developed which is responsible for the material anisotropy. Therefore, the modeling of the texture evolution in polycrystals is important to predict the anisotropy effects that are present in many industrial processes.

The formulation of metallic polycrystals plasticity has been the subject of many studies and different approaches have been proposed. These homogenization techniques span from the classical bounds (Taylor and Sachs) to the more sophisticated self-consistent approaches. Ahzi and M"Guil developed a novel viscoplastic  -model. This model takes into account the In the study of BCC metal, we compare our predicted results with those predicted by the viscoplastic self-consistent (VPSC) model. We study the slip activities, texture evolutions and the evolution of yield loci. We also present a comparison with experimental textures from literatures for several BCC metals under cold rolling tests. Possible links between the parameter  and the microstructure characteristics such as grain size are briefly mentioned.

The model has also been extended to HCP metals. We predict the deformation behaviors of the magnesium alloy (AZ31) for different interaction strengths. We also compare our predicted results with experimental data from literatures. Tests are carried out in plane strain compression, tensile and compression on the rolled AZ31 sheets. We show that the results predicted by the  -model are in good agreement with the experimental ones.

Ré sumé

Les maté riaux polycristallins sont constitué s d'un grand nombre de monocristaux (appelé s grains). Chaque grain montre un comportement anisotrope fort. Les comportements mé caniques macroscopiques des polycristaux dé pendent fortement de la distribution des orientations des cristaux et du comportement du monocristal. Si les grains ont une distribution d'orientation alé atoire alors les proprié té s mé caniques macroscopiques sont les mê mes dans toutes les directions et donc le maté riau est isotrope. En revanche, si les grains sont de pré fé rence orienté s dans des directions cristallographiques donné es, le maté riau est anisotrope.

La distribution d'orientation cristallographique d'un é chantillon polycristallin est dé finie comme la texture cristallographique.

Le calcul de la ré ponse macroscopique des agré gats polycristallins à partir des proprié té s de leurs constituants (grains monocristallins) est considé ré comme l'un des principaux problè mes de mé canique des maté riaux. Lors de la dé formation plastique, tous les grains de l'é chantillon du maté riau polycristallin sont ré orienté s. Une texture cristallographique peut alors se dé velopper. Cette derniè re est responsable de l'anisotropie du maté riau, voir Fig. R-1 qui pré sente le sché ma de l'é volution de l'orientation des grains lors des essais de laminage. Par consé quent, la modé lisation de l'é volution de la texture dans les polycristaux est importante afin de pré voir les effets d'anisotropie pré sents dans de nombreux procé dé s industriels. (Taylor et Sachs) à des approches auto-cohé rente plus sophistiqué es. L'insuffisance principale de la plupart des modè les existants de plasticité cristalline, c'est qu'ils sont incapables de pré dire les ré sultats d'une interaction rigide à une interaction plus souple entre les grains. Cette insuffisance conduit à un manque de capacité de pré vision des résultats à grande échelle. Par exemple, lors d"un essai de laminage, les mé taux CFC dé veloppent gé né ralement 2 types de texture: cuivre et laiton.

Du point de vue de la simulation, les modè les de type Sachs ne peuvent pas pré dire la texture de type cuivre. Les modè les de Taylor-type et le modè le visco-plastique auto-cohé rent (VPSC) ne peuvent pas pré dire la texture de type laiton à moins que le mé canisme de maclage soit pris en compte. Toutefois, certains maté riaux, ayant des microstructures spé cifiques ou pour des conditions particuliè res de chargement, peuvent dé velopper une texture de type cuivre mê me si le maclage n"ait pas été détecté. Ce manque de capacité des modèles existants à capturer correctement les transitions de texture pousse au dé veloppement de nouveaux modè les de plasticité cristalline.

Ahzi et M'Guil ont dé veloppé un nouveau modè le viscoplastique. Ce modè le- prend en compte les effets d'interaction entre les grains sans utiliser le problè me de l'inclusion Eshelby.

Ce modè le est formulé par la minimisation d'une fonction spé cifique combinant les champs de dé viations locaux (vitesse de dé formation et de contrainte) aux champs macroscopiques. Cette fonction dé pend é galement d'un paramè tre de ré glage, entre 0 et 1 qui permet à la force interaction du grain de varier d'une interaction rigide (petites valeurs de  ) à une interaction plus souple (valeurs é levé es de  ). La loi d'interaction proposé e permet de couvrir les ré sultats des modè les de la borne supé rieure à la borne infé rieure en variant un seul paramè tre de force d'interaction  , voir Fig. R-2. Ce modè le fournit la nouveauté sur la formulation et montre une bonne performance sur la pré diction des comportements de dé formation plastique et des é volutions des textures cristallographiques des mé taux au cours des procé dé s de mise en forme. Bien que l'avantage de ce modè le par rapport aux modè les existants soit significatif, il peut encore ê tre modifié et complé té . Par consé quent, l'objectif principal de ce travail est d'améliorer, d'étendre et d"appliquer ce modèle aux prévisions d'évolution de texture dans diffé rents mé taux au cours de la dé formation plastique. Un tel travail ré pond au besoin d'une meilleure compré hension du comportement mé canique lors de la mise en forme des mé taux par dé formation plastique ; ce qui permettra d'optimiser les procé dé s industriels. L'effet couplé du maclage mé canique et la force d'interaction entre les grains (contrôlé e par le paramè tre  ) sur l'é volution de la texture a é té é tudié pour les mé taux CFC dans des conditions de chargement diffé rents. La possibilité de la transition de texture de laminage du type cuivre au type laiton a é té discuté e. Le maclage mé canique permet é galement d'appliquer le modè le aux mé taux HC. En outre, le modè le a é té é tendu aux mé taux CC.

Le dé tail de chaque chapitre est pré senté comme suit:

Chapitre I

Le chapitre I se compose d'une bibliographie sur la modé lisation de la plasticité cristalline.

Nous pré sentons briè vement la modé lisation microscopique y compris le cadre constitutive des lois d"écrouissage des modèles. Ensuite, le concept de la texture dans les polycristaux et les principaux modè les d'homogé né isation existants sont ré sumé s.

Chapitre II

Le nouveau modè le viscoplastique  est pré senté en dé tail dans ce chapitre. Aprè s la pré sentation de la formulation de ce modè le, nous avons ré sumé les aspects numé riques de la modé lisation. En plus, nous proposons une é tude des dé viations de contrainte et vitesse de dé formation normalisé es. À la fin de ce chapitre, l'é volution du tenseur d'interaction est étudiée. Nous avons prédit l"évolution du tenseur d'interaction du modèle  et comparé ces ré sultats avec ceux issus du modè le VPSC. Nous pouvons voir que les tenseurs d'interaction entre les deux modè les montrent diffé rentes tendances d'é volution. Par ailleurs, nous avons analysé l'information contenue dans les tenseurs d'interaction de ces deux modè les. Celui du modèle VPSC peut prendre en compte l'effet de la forme des grains via le tenseur d"Eshelby.

Toutefois, le tenseur d'interaction du modè le  contient les informations de l'orientation cristallographique ce qui est absent dans le modè le VPSC

Chapitre III

Dans le chapitre III, la transition de la texture CFC de laminage du type-cuivre au type-laiton a é té é tudié e en utilisant le modè le- . Les exemples de textures type-cuivre et type-laiton sont pré senté s dans la Fig. R-3. La pré diction de l'é volution de la texture de laminage des mé taux CFC est contrôlé e par les lois d'interaction, les mé canismes de la dé formation plastique ainsi que de la dé finition de la rotation du ré seau. L'effet couplé de ces trois facteurs sur l'é volution de la texture de laminage est analysé . Dans ce travail, deux dé finitions de la rotation du ré seau, l'analyse mathé matique (MA) et l'analyse en dé formation plane (PSA), sont prises en compte dans le modè le. L'influence des dé finitions de MA et de PSA sur l'é volution de la CFC texture de laminage est analysé e en dé tail en conjonction avec le maclage et la force d'interaction entre les grains. Une texture typique de type-laiton a é té pré dite, ce qui est gé né ralement observé dans les mé taux avec une é nergie de dé faut d'empilement (EDE) faible.

Chapitre IV

Dans ce chapitre, nous montrons que le modè le- peut-ê tre utilisé pour calculer simplement la transition de texture cristallographique lors de l"essai de cisaillement des mé taux CFC en grandes dé formations. Les ré sultats pré dits sont comparé s aux textures expé rimentales de cisaillement pour les mé taux avec EDE haute/moyenne (i.e. cuivre) et EDE faible (i.e. argent).

Nous montrons que le modè le est capable de pré dire une transition de texture claire ce qui caracté rise une gamme de mé taux CFC ayant une EDE é levé e/moyenne à faible. Le mé canisme de maclage est pris en compte afin d'amé liorer les textures de cisaillement pré dites pour les mé taux avec une EDE faible. L'effet du maclage sur les composantes idé ales de texture de cisaillement est repré senté et est comparé avec les ré sultats expé rimentaux de la litté rature. Nous avons montré que le mé canisme de maclage joue un rôle important dans la pré diction de la texture de cisaillement des mé taux avec une EDE faible.

Chapitre V

Dans ce chapitre, les textures de laminage des mét aux CC et les surfaces de charge correspondantes sont simulé es en utilisant le modè le- . Nous comparons nos ré sultats à ceux pré dits par le modè le viscoplastique auto-cohé rent (VPSC). Les ré sultats sont comparé s en termes d"activité de glissement, de l"évolution de la texture et de surface de charge. Nous pré sentons é galement une comparaison qualitative avec les textures expé rimentales de laminage à froid issues de la litté rature pour plusieurs mé taux CC . Les liens possibles entre le paramè tre de force d'interaction entre les grains et les caracté ristiques microstructurales telles que la taille des grains sont briè vement mentionné s.

Chapitre VI

Le modè le  a é té é tendu aux mé taux HC. Dans la premiè re partie de ce chapitre, le modè le- est utilisé pour pré dire l'é volution de la texture de laminage dans les mé taux HC, voir Dans la deuxiè me partie, nous avons simulé le comportement mé canique du AZ31 qui est un alliage de magné sium. Nous avons considé ré diffé rentes forces d'interaction intergranulaire et comparé les ré sultats pré dits avec les ré sultats expé rimentaux tiré s de la litté rature. Les tests sont effectué s pour le laminage (dé formation en compression plane), ainsi que pour les tests de traction et de compression sur des feuilles laminé es. Nous montrons que le modè le- avec de grandes valeurs de  pré dit des ré sultats en bon accord avec les textures expé rimentales. grains interaction effects without involving the Eshelby inclusion problems. The proposed interaction law allows spanning the results from the upper to the lower bound models by varying a single interaction strength parameter  . This model provides the novelty on the formulation and shows a good performance on the prediction of plastic deformation behavior and texture evolution in metals during forming processes.

In this thesis, the  -model was applied to different crystallographic structures and under different loading conditions. The mechanical twinning has been taken into account in the model. The FCC rolling texture transition from copper-type to brass-type texture is studied. In this part of the work, two definitions for the crystal lattice spin are considered: mathematical analysis (MA) and plane strain analysis (PSA). The influence of these two definitions, MA and PSA, on the rolling textures evolution in FCC metals is analyzed. In addition, the influence of twinning and the interaction strength between the grains (controlled by the parameter  ) is also analyzed.

The shear tests in FCC metals are also studied. The predicted results are compared with experimental shear textures for a range of metals having a high/medium stacking fault energy (SFE) to low SFE. We have shown that the  -model is able to predict a transition from shear texture characterizing a range of FCC metals with high/medium SFE to low SFE. The twinning mechanism is included in the  -model to improve the shear texture predictions for low SFE metals.

Introduction

The polycrystalline materials consist of a large number of single crystals (called grains). Each grain shows a strong anisotropic behavior. The macroscopic mechanical behaviors of the polycrystals strongly depend on the distribution of the grain orientations and the single crystal behavior of the polycrystalline material. If the grains have a random orientation distribution, the macroscopic mechanical properties are the same in all directions and therefore the material is isotropic. In contrast, if the grains are preferably oriented in some given crystallographic directions, the material is therefore anisotropic. The distribution of crystallographic orientations of a polycrystalline sample is defined as the crystallographic texture.

For polycrystalline metals, the crystallographic texture is the main source of plastic anisotropy.

It strongly influences the mechanical properties of the metals such as formability, which is important for the sheet metal forming in industrial applications. Therefore, the texture evolution should be taken into account for the modeling of anisotropy effects in forming processes.

The formulation of metallic polycrystals plasticity has been the subject of many studies and different approaches have been proposed. These homogenization techniques span from the classical bounds Taylor (i.e. Asaro and Needleman, 1985;Parks and Ahzi, 1990;Taylor, 1938) and Sachs (i.e. Ahzi et al., 2002;Leffers and Ray, 2009;Sachs, 1928) models to the more sophisticated self-consistent approaches (Abdul-Latif, 2004;Abdul-Latif and Radi, 2010;Lebensohn and Tomé , 1993;Lebensohn et al., 2007;Mercier and Molinari, 2009;Molinari et al., 1987). The main insufficiency of most existing crystal plasticity models is that they are unable to predict the results from a stiff interaction to a more compliant interaction. This insufficiency leads to a lacking ability of prediction of large scale results. For example, in rolling tests, FCC metals usually develop 2 types of texture: copper-and brass-type (Leffers and Ray, 2009). From the simulation point of view, the Sachs-type models cannot predict the copper-type texture. The Taylor-type model and the well-known visco-plastic self-consistent (VPSC) model cannot predict the brass-type texture unless the twinning mechanism is considered (Leffers and Ray, 2009;M"Guil et al., 2009M"Guil et al., , 2011)). However, certain materials with specific microstructures or under specific loading conditions can develop a brass-type texture even if no twinning has been detected [START_REF] Engler | Deformation and texture of copper-manganese alloys[END_REF][START_REF] Engler | Rolling texture development in Cu-Mn alloys[END_REF]. This lacking ability of correctly capturing texture transitions calls for new crystal plasticity models. Ahzi and M"Guil (2008) developed a novel viscoplastic  -model which includes grain interactions effects in a new and original way. The interaction law of the  -model is expressed similarly to the self-consistent one but the interaction tensor is independent of the Eshelby tensor. This model is formulated by the minimization of a specific function combining the local fields" deviations (strain rate and stress) from the macroscopic ones. This function depends also on a tuning parameter,  between 0 and 1 which allows the grain strength interaction to vary from a stiff interaction (small values of  ) to a more compliant interaction (high values of  ). This model provides the novelty on the formulation and shows a good performance on prediction of plastic deformation behavior and texture evolution in metals during forming processes. Although the advantage of this model is significant, it can still be modified and complemented. Therefore, the main objective of this work is to improve and extend the  -model and to apply this model on the texture evolution predictions in various metals during the plastic deformation. Such a work will fulfill the need for a better understanding of plastic deformation behavior and forming of metals which allows optimizing industrial processing.

In this work, the mechanical twinning, which is one of the basic plastic deformation mechanisms in polycrystals materials, has been implemented into the  -model. The coupled effect of mechanical twinning and the interaction strength (controlled by the  parameter) on the texture evolution has been studied in FCC metals under different loading conditions.

The possibility of the FCC rolling texture transition from copper-to brass-type textures has been discussed. The consideration of mechanical twinning also allows applying the  -model to HCP metals. In addition, the  -model has been extended to BCC metals. The BCC rolling textures are predicted and compared to experimental results.

The details of each chapter are presented as follow:

Chapter I

The chapter I consists of a literature survey on the crystal plasticity modeling. We briefly present the microscopic modeling including the constitutive framework, hardening laws and twinning models. Then, the concept of texture in polycrystals and the main existing homogenization models are summarized.

Chapter II

The new visco-plastic  -model is introduced in detail in this chapter. After the presentation of the  -model formulation, we shortly summarize the numerical aspects of the  -model in literatures (Ahzi and M"Guil, 2008;M"Guil et al., 2009M"Guil et al., , 2011)). At the end of this chapter, we propose a study of the  -model on the normalized stress and strain rate deviations and the evolution of the interaction tensor.

Chapter III

In the chapter III, the FCC rolling texture transition from copper-to brass-type texture has been studied using the visco-plastic  -model. The prediction of the rolling texture evolution in FCC metals is controlled by interaction laws, deformation mechanisms and definition of the lattice spin. The coupled effect of these three factors on the FCC rolling texture evolution is hereby analyzed. In this work, two definitions of the lattice spin, the mathematical analysis (MA) and the plane-strain analysis (PSA), are considered in the  -model. The influence of the MA and PSA definitions on the FCC rolling texture evolution is deeply analyzed in conjunction with twinning and grain interaction strength, from a stiff interaction to a more compliant interaction. A typical brass-type texture has been achieved which is usually observed in metals with low Stacking Fault Energy (SFE).

Chapter IV

In this chapter, we show that the polycrystalline  -model can be used to compute simple shear crystallographic texture transition for FCC at large strains. Predicted results are compared to experimental shear textures for high/medium SFE metals (i.e. copper) and low SFE metals (i.e. silver). We show that the  -model is able to predict a clear shear texture transition characterizing a range of FCC metals having high/medium to low SFE. The twinning mechanism is included to improve the predicted shear textures for low SFE metals.

The effect of twinning on the ideal shear texture components is shown and is consistent with experimental results from literature.

Chapter V

The BCC rolling textures and the corresponding yield surfaces are simulated using the  -model in this chapter. We compare our results to those predicted by the visco-plastic self-consistent (VPSC) model. The results are compared in terms of predicted slip activity, texture evolution and yield loci. We also present a qualitative comparison with experimental cold rolling textures taken from the literature for several BCC metals.

Chapter VI

The  -model has been extended to HCP metals. We simulated the deformation behavior of AZ31 magnesium alloy with different intergranular interaction strengths and compared the predicted results with experimental ones taken from the literatures. Tests are performed for rolling (plane strain compression) as well as for tensile and compressive tests on rolled sheets.

We show that the  -model predicts results in good agreement with the experimental ones.

Chapter I. Background: crystal plasticity modeling

I.1 Introduction of polycrystals

Metallic materials belong to the group of polycrystals, and it is one of the most common used materials in industrial application. This kind of material can usually be largely deformed during forming process due to its relative high ductility. The works of this thesis mainly focus on the micro-macro mechanic modeling of metallic materials under large plastic deformation. It is therefore important to understand the microstructure of polycrystals as well as their micro mechanisms of plastic deformation.

I.1.1 Microstructure in single crystal

The single crystal (called grain) is the basic unit in polycrystalline materials such as

I.1.2 Plastic deformation mechanisms in polycrystals

Usually, if an elastic deformation occurs in a material, its microstructure will not show permanent changes. The material will return to its original shape and dimension when the stress is removed. The plastic deformation usually involves the microstructure changes. In the forming process of metals, large strain is usually required. In this case, the plastic deformation is important whereas the elastic deformation can be neglected. In single crystals, the plastic deformation can be activated by several mechanisms. Two basic plastic deformation mechanisms: crystallographic slip and twinning, will be introduced in the next section.

I.1.2.1 Deformation by slip

The crystallographic slip is produced by the dislocation motion of the atoms in the single crystals. As shown in Fig. 1-2, if the shear stress suffered by a single crystal exceeds a threshold value, some parts of the crystal lattice will glide along each other.

The atoms dislocate a whole space between atoms. The slip occurs only on certain planes and directions which depend on the crystal systems. Usually, the slips occur on the plans and directions in which the atoms are most closely packed due to energy reasons.

Fig. 1-2. Schematic representation of crystallographic slip.

I.1.2.2 Deformation by twinning

Mechanical twinning can be found in metals with certain properties or under certain loading conditions. The twinning mechanism is proved to be influenced by Stacking Fault Energy (SFE), temperature, strain rate etc. (Christian and Mahajan, 1995). The mechanical twinning is activated by a strong shear stress. If the twinning is activated, part of the lattice structure will be rearranged and the twinned region (child region)

will become mirrored to the original region (parent region). The schematic description of twinning is presented in Fig. 123.

We note that the mechanical twinning can be activated only on one direction whereas the slip can be activated on both positive and negative directions. The volume fraction of the twinned region has been reported by some experimental investigations (e.g. Leffers and Ray, 2009). For example, under rolling tests, the experimental volume fraction of deformation twins is less than 25% for brass with 30% zinc content at high reduction. In HCP lattice structure, the crystallographic directions and plans are identified using a reference frame with four coordinate axes instead of three. As shown in Fig. 12345,

1 A O  , 2 A O  , 3 A O  and C O 
are the four coordinate axis. Thus, the planes and direction in HCP structures should be identified by four integers: [u h v h m h w h ] for directions and (h h k h n h l h ) for planes. Those integers can be obtained in the same way as in cubic systems.

  110 ,   101 ,   011 ,   10 1 ,   01 1 ,   1 1 0
. It is good to keep in mind that the mechanisms occurred on directions and planes from the same families are normally equivalent.

I.3 Constitutive framework for single crystal

In order to describe the deformation behavior of single crystal, both elastic and inelastic parts of deformation need to be considered. The inelastic deformation of single crystal is mainly accommodated by the plastic deformation mechanisms such as slip and twinning. The elastic part should include the elastic deformation and the lattice spin. Thus, it makes sense to make a polar decomposition of deformation gradient, Asaro and Needleman (1985) proposes the decomposition of the deformation gradient as follows:

p * F F F  (1-1)
Where P F is the plastic deformation gradient and * F represents the rotation of the crystal lattice in which the elastic deformation is included, see Fig. 123456. . Fig. 123456. Decomposition of deformation gradient, kinematic scheme proposed by Asaro and Needleman (1985), presented by Kalidindi (1998) The Eq. 1-2 shows the expression of the microscopic velocity gradient L which can be divided into the elastic part and a plastic part:

p * 1 L L F F L      (1-2) P
L and * L can also be divided as:

    T P P T P P P P 1 P P P 2 1 2 1 L L L L Ω D F F L          (1-3) * e 1 e e * Ω D F F L     
(1-4)

Here, P D and e D are the plastic and elastic strain rate tensors, respectively. P Ω is the plastic spin, and Thus, the plastic velocity gradient P L can be expressed as function of the shear rate

   :     p p n n p Ω D n s A P L           with     D (elasticity is neglected)
and the deviatoric Cauchy stress tensor S of the single crystal:

    S S M S P P S P D D                            
Here, M is the microscopic fourth order visco-plastic compliance tensor.

 Polycrystal constitutive law

In order to numerically describe the macroscopic behavior of polycrystals, we need to firstly find the relation between the macroscopic strain and stress. This relation can be described by a macroscopic constitutive law which is proposed in a similar way of the microscopic one (Eq. 1-14):

S M D   or D L S   (1-15) Where 1   M L .

D , S and

M represent the macroscopic plastic strain rate, the macroscopic deviatoric Cauchy stress and the macroscopic fourth order visco-plastic compliance tensor, respectively. The consistency conditions need to be fulfilled which are given by the averaging conditions:

D D  and S S 
.

(1-16)

Here,  denotes the volume average over the polycrystals.

I.5

Hardening laws

In polycrystals, hardening corresponds to the phenomena that, during the plastic deformation, the plastic mechanisms become harder to be activated due to the modification of microstructure. As results, the strength characteristics (hardness, yield strength, etc.) of materials are increased whereas the ductility is decreased.

For the modeling point of view, the hardening is usually described by the modification of the critical resolved shear stress  g in the nonlinear constitutive law (Eq. 1-14). This parameter is the reference value of shear stress which describes the resistance of the activation of plastic mechanisms such as slip and twinning. The hardening laws of Latent and Voce are commonly used to describe the microscopic strain hardening in polycrystals.

I.5.1 Latent hardening law

In the latent hardening law, the increment of CRSS in system  can be calculated as follow:

         H g (1-17)
Here, the  H represents the n n  hardening matrix. [START_REF] Kalidindi | Crystallographic texture evolution in bulk deformation processing of FCC metals[END_REF] proposed the expression of  H as follow:

a sat 0 g g 1 h q           

H

(1-18)

Here, 0 h represents the initial hardening slope. sat g is the stress level for the hardening saturation. The matrix  q described the Latent hardening behavior of the single crystal (Franciosi et al., 1980;[START_REF] Kocks | The relation between polycrystalline deformation and single crystal deformation[END_REF]Tomé et al., 1984). This n n  matrix (n = the total number of considered slip and twinning systems) is populated by 1 (for coplanar systems) and q (for non-coplanar systems). The q is the latent hardening ratio usually taken between 1 and 1.4 (Franciosi et al., 1980;[START_REF] Kocks | The relation between polycrystalline deformation and single crystal deformation[END_REF]Tomé et al., 1984).

The Eq. 1-19 shows the form of matrix  q when the 12 FCC slip systems are considered.

              A A A A A A A A A A A A A A A A q q q q q q q q q q q q q with            1 1 1 1 1 1 1 1 1 A (1-19)

I.5.2 Voce hardening law

In this thesis, we use the voce hardening law (Voce, 1948) to describe the hardening effect in polycrystals. In this hardening law, the evolution of the CRSS of slip or twinning system  for a single crystal is calculated as follows:

                                  1 0 1 1 0 exp 1 g  (1-20)
In Eq. 1-20,  is the shear rate accumulated in this grain. The parameters 0  , 1  , 0



and 1  can be determined from the experimental strain-stress curves. The parameters 0  and 0  describe the initial flow and the initial rate of hardening in the grain, respectively. The parameters 1  , 1  describe the asymptotic characteristics of strain hardening. The condition

1 0    , 0 1  
corresponds to an increase in yield stress and a decrease in the hardening rate to a linear saturation. The linear hardening is the limiting case of this law where the four hardening parameters are set to be

1 0   , 0 1   , 1 0   and 1 1   .
A "self" and "latent" hardening coupling parameter  h is introduced to describe the barrier effect between the activated slip or twinning modes (Tomé et al., 1984):

           h d dg g
(1-21)

We denote that  h is a n n  matrix where n is the total number of activated slip or twinning systems. The component of  h should be set as 1 if the  and  systems belong to the same slip or twinning mode. We denote that the voce hardening law can be used for both crystallographic slip systems and the slip systems associated with mechanical twinning.

I.6 Modeling of mechanical twinning

The mechanical twinning may affect the plastic deformation in two ways: via the orientations of twinning and via the slip associated with the formation of twinning.

Several models have been proposed to describe the effect of twinning on the plastic deformation. In this section, we propose an introduction of the Van Houtte model (Van Houtte, 1978), the Anand-Kalidindi model (Kalidindi, 1998;[START_REF] Staroselsky | Inelastic deformation of polycrystalline face centered cubic materials by slip and twinning[END_REF] and the Predominant Twin Reorientation (PTR) scheme (Tomé et al., 1991). The PTR scheme is the one used in this thesis.

I.6.1 Van Houtte twinning model

In the model of Van Houtte, each grain is divided into several sub-grains. The sub-grains have the same orientation and are calculated independently. During the simulation, those grains are randomly picked with a fixed probability at each deformation step and the picked grains are orientated along a given twinning system.

We note that the probability should be set the same as the twinning volume fraction in this step.

The Van Houtte model can successfully describe the effect of twinning in the simulations (Van Houtte, 1978). However, this model can only rotate the grains along the given twinning system which may not be the most active twinning system (Tomé et al., 1991). This disadvantage may reduce the accuracy of simulations. This model may also largely increase the required computational time because large numbers of sub-grains are needed to ensure the accuracy of the twinning volume fraction (Tomé et al., 1991).

I.6.2 Anand-Kalidindi twinning model

The Anand-Kalidindi twinning model (Kalidindi, 1998;[START_REF] Staroselsky | Inelastic deformation of polycrystalline face centered cubic materials by slip and twinning[END_REF] proposes another way to describe the twinning orientation. In this model, the deformation gradient is decomposed in the same way as Asaro and Needleman (1985), see section I.3. However, the deformed matrix is divided into twined region and untwined region (see Fig. 1234567). Therefore, the mechanical response needs to be expressed by the combination of those regions.

Fig. 1234567. Decomposition of deformation gradient, kinematic scheme proposed by Kalidindi (1998) and [START_REF] Staroselsky | Inelastic deformation of polycrystalline face centered cubic materials by slip and twinning[END_REF] Based on this idea, Kalidindi (1998) and [START_REF] Staroselsky | Inelastic deformation of polycrystalline face centered cubic materials by slip and twinning[END_REF] split the Schmid tensor fraction in the matrix:

                         Nt w unt w tw tw unt w s Ns tw s t Ns p f f 1 f m m m L    (1-22)
Here, f is the twinning volume fraction. Ns-t, Ns and Ntw represent the number of slip systems in twined region, number of slip systems in untwined region, and number of twinning systems in untwined region, respectively.  f is the twinning volume fraction accumulated by twinning system  . tw  refers to the shear strain of twinning systems which is suggest to be constant (Kalidindi, 1998[START_REF] Kalidindi | Modeling anisotropic strain hardening and deformation textures in low stacking fault energy fcc metals[END_REF]. In the last term,

tw f   
represents the equivalent shear rate in twinning system  . We note that in this model, twinning is not associated with slip.

The deviatoric Cauchy stress tensor can also be divided into untwined part u n tw S and the twined part tw S . We note that, when we calculated   (see Eq. 1-15) for a slip (or twinning) system in twined (or untwined) region, the corresponding Cauchy stress tensor and Schmid tensor should be used. The Cauchy stress tensor in the single crystal can be calculated by a linear combination of the twined and untwined parts:

  tw u n tw S S S      f f 1 (1-23)
At each deformation step, certain amount of volume fraction may be transferred from the untwined region to the twined regions. We should notice that each twinning system may create a twined region. For a twinning system  , the transferred volume fraction can be calculated as:

t g f n tw 0                   (1-24)
We note that the  f should be set to 0 if the calculated   is negative as the twinning can be only activated on the positive direction. The orientation of the twined region is obtained according to the direction of corresponding twinning system.

I.6.3 The Predominant Twin Reorientation (PTR) scheme

In the single crystal plasticity framework for single crystals (see section I.3), the twinning is associated with slip and should be calculated as normal slip but only activated on the positive direction. Meanwhile, the twinning orientation (twinning volume fraction, direction of orientation, etc.) needs to be described.

In this thesis, we used the Predominant Twin Reorientation (PTR) scheme proposed by Tomé et al. (1991) which was used by several authors [START_REF] Abdolvand | Internal strain and texture development during twinning: Comparing neutron diffraction measurements with crystal plasticity finite-element approaches[END_REF]Beyerlein et al. 2011;[START_REF] Prakash | Twinning models in self-consistent texture simulations of TWIP steels[END_REF], Schmid et al., 2007). In this model, the associated volume fraction is defined in each twinning system of one grain, as:

t g , t g , t S V  
. Here, g , t  represents the shear strain contributed by the twin systems and t S is the characteristic twin shear. The twinning system in this grain with the highest associated volume fraction is identified as the predominant twin system (PTS).

The sum of the associated volume fraction over all twinning systems in a given twinning mode is defined as the accumulated twin fraction:

   t t g , t g , acc S V
. At each deformation incremental step, a grain is picked randomly, and the grain will be fully reoriented if the accumulated twin fraction exceeds a threshold value which is defined as:

acc eff 2 th 1 th th V V A A V   (1-25)
In Eq. 1-25,

  g t g , t acc V V
. Once the condition is fulfilled, the grain will be completely reoriented and only the PTS is considered in the reorientation. Then, both eff V and acc V will be updated. This process will be repeated until either all grains are randomly picked or the eff V exceeds the accumulated twin volume.

The PTR scheme can well define the direction of the twinning as only the PTS is considered in the twinning reorientation. The main disadvantage of this model is the reorientation of the entire grain. In polycrystals, only a part of the grain should be reoriented due to the twinning. Therefore, the PTR scheme could overestimate the twinning volume fraction (Beyerlein et al. 2011;[START_REF] Prakash | Twinning models in self-consistent texture simulations of TWIP steels[END_REF], Schmid et al., 2007).

I.7 Crystallographic texture and its evolution

The crystallographic texture is an important source of anisotropy in polycrystals. The macroscopic mechanical behavior cannot be well predicted if the texture is not taken into account. In each single crystal, the mechanisms of the plastic deformation usually occur on certain planes and directions. Therefore, its mechanical properties are strongly influenced by the orientation of the grain and therefore anisotropic.

Furthermore, the microscopic anisotropy can influence the macroscopic behavior of the polycrystals. If the grains have a random orientation distribution, the macroscopic mechanical properties are the same in all directions and therefore isotropic. In contrast, if the grains are preferably oriented in some given crystallographic directions, the material is therefore anisotropic.

The crystallographic texture is defined as the distribution of crystallographic orientations of a polycrystalline sample. The textures are considered as the main source of the anisotropy in the polycrystals. In modeling of polycrystals, the evolution of the texture needs to be followed in the entire forming process as the grains eager to rotate to some favored orientations during the plastic deformation. These rotations depend on the mode of deformation imposed on the material and the activated mechanisms in each grain. The crystallographic texture can be predicted by the models applied on polycrystals such as Taylor-type models, Sachs-type models, self-consistent models (see section I.8).

I.7.1 Euler angles

The three Euler angles, 1  ,  and 2  , are usually used to identify the grain orientations (Bunge definition). The Euler angles describe the transition from the reference frame of sample (XYZ) into the crystallographic reference frame of each individual grain (xyz) of the polycrystals. To find the orientation g, as shown in Fig.

1-8, the system xyz needs to be rotated by 1  about Z axis, then rotated by  about x axis, finally rotated by 2  about z axis.

Update of grain orientations

During the simulations, the 3 Euler angles representing the grain orientation are transformed into an orientation matrix a:

                                                1 0 0 0 cos sin 0 sin cos cos sin 0 sin cos 0 0 0 1 1 0 0 0 cos sin 0 sin cos 1 1 1 1 2 2 2 2 a (1-26)                                               
1 2 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 a (1-27)
The lattice spin * Ω is used to update the grain orientation at each calculation step. In the calculation, we should firstly obtain the rotation angle  from the * Ω (Lee et al., 2002):

  2 tr 2    with t ˆ*   Ω
(1-28)

The incremental rotation matrix rot A is defined to calculate the update of the orientation matrix a. The expression of rot A is presented as follows:

  2 2 rot ĉos 1 ŝin I Ω Ω A          
(1-29)

Then, the grain orientation can be updated as follows:

t rot t t a A a     (1-30)
The matrix a of each grain should be calculated in each incremental steps. After the simulation, in order to output the final texture, the a matrix of each grain needs to be transformed back to the three Euler angles. To make the results more visual, the output textures are represented in three different ways: the pole figures, inverse pole figures and the orientation distribution function. The methods to translate the three Euler angles into those visual figures are introduced in the Appendix.

I.8

Main existing models for polycrystalline plasticity I.8.1

Taylor-type model

The Taylor model (or called the upper bound approximation) is proposed in 1938 (Taylor, 1938). This model is based on the assumption of uniform strain, which means that each grain undergoes the same plastic deformation as the macroscopic. This model is usually formulated by the condition that the velocity gradient at the local level is equal to the macroscopic one:

L L  (1-31)
Based on the idea of Taylor (1938), several extensions have been proposed. Lin (Lin, 1957) assumes that the elasticity is isotropic and has proposed an interaction law to solve the distribution between the elastic and plastic deformation which are not imposed. Asaro and Needleman (1985) have extend the Taylor theory into elasto-viscoplastic based on the assumption of uniform deformation gradient:

F F  .
We should note that the global condition (see section I-4, Eq. 1-16) should also be fulfilled in Taylor-type model. Since the Taylor model is based on the iso-strain assumption, the average strain condition (Incompressibility condition) is automatically fulfilled. The macroscopic stress can be calculated from the average stress condition (equilibrium condition).

I.8.2 Sachs-type model

The Sachs model is proposed in 1928 (Sachs, 1928). This model is established by assuming that the stress tensor in each grain is proportional to the macroscopic one.

This assumption results in a proportional relationship between the local and global stress:

σ σ g   (1-32)
In the above equation, g  depends on the orientation of the grain. The static model (or called lower bound approximation) is a limited case of Sachs model and is based on the assumption of the uniform stress, that is to say that the stress in each grain is equal to the macroscopic stress:

σ σ  (1-33)
We should keep in mind that, in the visco-plastic cases, the strain and stress are usually represented by plastic strain rate tensor D and deviatoric Cauchy stress Since the Static model is based on the iso-stress assumption, the equilibrium condition (see Eq. 1-16) will be achieved automatically. The macroscopic strain can be calculated from the Incompressibility condition.

I.8.3 Relaxed-constraints model

The classical Taylor model is too strict and shows a limit in the texture predictions especially when the grain is largely flatted. Thus, Honeff and Mecking (1981) have proposed a relaxed-constraints model. In this model, the local strain is equal to the global one as the Taylor theory and the shear deformations of relaxation are added into local strain. For the rolling cases, the relaxed-constraints models include the lath model (with one shear deformation of relaxation along rolling direction) and the pancake model (with two shear deformations of relaxation along rolling and transverse directions). [START_REF] Van Houtte | Deformation texture prediction: from the Taylor model to the advanced Lamel model[END_REF] have summarized those models by proposing a general expression of relaxation velocity gradient:

    R 1 r rlx r r K  rlx L (1-34)
In Eq. 1-34, R is total number of relaxation modes and r K denotes the velocity gradient of each mode. rlx r   is a reference value. The rlx L is a part of the local velocity gradient, therefore, the local strain rate, which is the asymmetric part of local velocity gradient (see Eq. 1-11 and Eq. 1-12), can be expressed as:

          R 1 r rlx r r n P B   P D (1-35)
Here, r B refers to the asymmetric part of r K . The Eq. 1-35 can be used to calculate rlx r   via a linear programming (Van Houtte, 1988;[START_REF] Van Houtte | Deformation texture prediction: from the Taylor model to the advanced Lamel model[END_REF].

For each grain, the shear rates of slip systems can be determined by the minimization of the plastic work [START_REF] Van Houtte | Deformation texture prediction: from the Taylor model to the advanced Lamel model[END_REF] which is expressed as:

            r r rlx r rlx g P  
(1-36)

In Eq. 1-36, the first term is the Taylor deformation energy (Taylor, 1938). r rlx  represents the friction coefficient.

I.8.4 Multigrain model

The multigrain models refer to the models taking into account the local interaction between neighboring grains [START_REF] Van Houtte | Deformation texture prediction: from the Taylor model to the advanced Lamel model[END_REF]. The Lamel model (Kanjarla et al., 2010;Lee et al., 2002;Van Houtte et al., 2002, 2005, 2006) is a simplified multigrain model which is designed for the rolling tests. This model divides the material into a large amount of clusters. Each cluster contains two grains and the interface between those two grains remains parallel to the rolling plane during the deformation. The velocity gradient of each cluster is equal to the macroscopic one as the Taylor theory, whereas the relaxation shear deformation is allowed on the interface between the two grains. The relaxation, in this model, is calculated as the relaxed-constraints model (section I.8.3), but with the r K of a grain is opposite to the one of its neighboring grain. The sum of plastic work of the two grains should be used in the minimization process to calculate the slip shear rates. Lee et al., (2002) have proposed a generalized Lamel model which also considers two grains in each cluster. This model is formulated in a similar way than the Lamel model but the bicrystal interfaces are not necessarily parallel to certain plane during the deformation. The directions of the interfaces are taken into account in the formulation. Lee et al., (2002) The grain interaction (GIA) model (Crumbach et al., 2001(Crumbach et al., , 2006;;Engler et al., 2005;Leffers and Ray, 2009;[START_REF] Van Houtte | Deformation texture prediction: from the Taylor model to the advanced Lamel model[END_REF] is also a multigrain model but with eight grains in one cluster. The grain boundaries and the interface between the cluster and the matrix are assumed to be orthogonal arranged. The energies of dislocations on these grain boundaries and the cluster interfaces are represented by the energies of geometrically necessary dislocations (GND) which is shown as follows:

2 b G P 2 GND      (1-37)
Here,  is one of the GNDs. GND  is the density of GNDs which depends on the shear strain, GND directions and the grain size. b is the Burgers vector of the GNDs.

G is the shear modulus. Therefore, the total plastic work of the cluster can be presented as:

            P g P nor cluster  (1-38)
Here, nor represents all the eight grains in this cluster. Then, by the minimization of cluster P , the relaxation deformation and the shear rate of each grain can be determined (Crumbach et al., 2006).

I.8.5 Self-consistent modeling

The self-consistent theory plays an important role in the micro-macro mechanical modeling. This type of model is widely used to predict the mechanical properties for heterogeneous materials, particularly polycrystals (e.g. Abdul-Latif, 2004;Abdul-Latif and Radi, 2010;Berveiller and Zaoui, 1979;Hill, 1965;Kröner, 1961;Mercier et al., 2005;Mercier and Molinari, 2009;Molinari et al., 1987). Unlike the classic Taylor-type and Sachs-type models, the self-consistent models can take into account the grain shape via the Eshelby theory (1957) and then describe the interaction between the grains and the polycrystals. In the self-consistent theory applied in polycrystals, each grain or grain group are treated as an inhomogeneous inclusion embedded in the "homogeneous effective medium" (HEM) which has the average properties of the aggregate. The macroscopic response of the polycrystals results from the contribution of each grain. An interaction law should be proposed for each self-consistent model in order to solve the problem of interaction between a grain and the aggregate.

A solution of the inclusion problem is proposed by Eshelby (1957). For the case that the elasticity is isotropic and homogeneous, the relation between the total deformation T ε and the plastic deformation p ε of an ellipsoidal inclusion is expressed as:

p kl ijkl T ij S    (1-39)
Here, S is the Eshelby tensor which depends on the grain shape and the elastic properties of the matrix. For the spherical inclusions, the expression of the Eshelby tensor is shown as follow:

      jk il kl ik kl ij ijkl δ δ δ δ ν 1 15 ν 5 4 δ δ ν 1 15 1 ν 5 S        (1-40)
Then, for the case that 0 p ii   , the relation in Eq. 1-40 becomes:

p ij T ij    with       1 5 4 15 2
(1-41)

In Eq. 1-41,  denotes the Poisson ratio. The internal stress of the inclusion can be obtained as:

                         ij e ij e ij ij 2 1 2 (1-42)
Here,  represents the elastic shear modulus. e ij  is the elastic deformation and can be calculated as:

  P ij P ij T ij e ij 1          (1-43)
Therefore, we can have:

  p ij ij 1 2       
(1-44)

I.8.5.1 Kröner model

Over the past several decades, the modeling studies based on the self-consistent theory is an active field of research. Kröner (1961) proposed an elasto-plasitc self-consistent approach for the polycrystals under small deformation. The inclusions are assumed spherical. The interaction law of Kröner model is expressed as:

   p p 1 2 ε ε σ σ       (1-45) Here,       1 5 4 15 2
as spherical inclusions are assumed to be spherical. P ε and P ε are the local and global plastic deformations, respectively. σ and σ are the local and global internal stresses. We note that the  value is quite large and thus the plastic deformation deviation is suppressed (

0 p p   ε ε
). Therefore, the predicted results of Kröner model are close to the ones of Taylor-type models (see section I.8.1).

I.8.5.2 Hill model

Hill (1965) has proposed a more complex model, which successfully take into account elasto-plastic interaction between the grains and the aggregate. The interaction law of Hill model is represented as follow:

  p p * ε ε L σ σ     (1-46)
The interaction tensor * L is expressed as:

    * 1 L L L L L L        (1-47)
Here, L and L represent the local and global tangent modulus tensors, respectively.

I.8.5.3 Berveiller and Zaoui model

Berveiller and Zaoui (1979) proposed a model based on the interaction law of Kröner (1961). The spherical grains are taken into consideration and the elastic propriety is assumed to be isotropic in this model. As mentioned above (see section I.8.5.1), the predicted results of Kröner model are close to the ones of Taylor-type model due to the high value of  . In order to solve this problem, Berveiller and Zaoui (1979) added a plastic accommodation factor  into the interaction law of Kröner. When the Poisson ratio is about 3 1 (

5 . 0  
), the interaction law of Berveiller and Zaoui (1979) can be expressed as:

  p p ε ε σ σ    
(1-48)

where:

h 1 1    
(1-49)

The parameter h depends on the state of plastification of the matrix. This parameter can be determined by strain-stress response of the tensile tests:

p ε ζ 3 2 h  (1-50) Therefore:       p ε ζ 3 2 1 1
(1-51)

I.8.5.4 Visco-plastic self-consistent model

The visco-plastic self-consistent (VPSC) model, which is formulated by Molinari et al. (1987), is a widely used model in polycrystal plasticity at large strain. The first version of the interaction law of the VPSC model is shown as follow (Molinari et al., 1987;Ahzi, 1987):

) ( ) ( t 1 D D A Γ S S       (1-52)
Here, Γ is a fourth order interaction tensor. t A is also a fourth order which represents tangent modulus of the polycrystals. [START_REF] Tóth | Tuning a self consistent viscoplastic model by finite element results-II. Application to torsion textures[END_REF] and [START_REF] Tóth | Tuning a self consistent viscoplastic model by finite element results-II. Application to torsion textures[END_REF] have improved the proposed interaction law by introducing a tuning parameter  . With the consideration of the tuning parameter, the new interaction law of VPSC model is presented as:

) ( ) ( s 1 s D D A Γ S S        (1-53)
Here, s Γ is a fourth order interaction tensor with

1 m   s -1 Γ Γ .
s A is the fourth order tensor of secant modulus where

s t A A m  .
Here, m is the strain-rate sensitivity

( 1 n m  
). We can see that if the  parameter is set to be 0, the local stress will be equal to the global one. Therefore the interaction law is equivalent to the lower bound Static approximation. In the same way, the upper bound Taylor approximation can be obtained when

  
. The Secant interaction can be found when 1  

. If the  parameter is equal to m, the new interaction law (Eq. 1-53) will be the same as the first version of VPSC interaction law (Eq. 1-52). The interaction between grains and aggregate is therefore Tangent. Lebensohn and Tomé (1993) have proposed the following interaction law which is given by:

  S S K D D -1     (1-54) or   D D K S S     (1-55) with   E I E M K -1 -1      1 eff n (1-56)
Here, K is the interaction tensor.

I represents the fourth-order identity tensor. E is the Eshelby tensor. We denote that the interaction law in Eq. 1-55 is in the similar form of the one in Eq. 1-53. The parameter eff n , as the inverse of parameter , may be adopted to obtain different interactions: 0 n eff  for a Taylor case,   eff n for a Static case, 1 n eff  for a Secant case and n n eff  for a Tangent case (Lebensohn and Tomé , 1993; Lebensohn et al., 2007).

The work of this thesis mainly focuses on the visco-plastic  -model (Ahzi and M"Guil, 2008) which also belongs to the group of self-consistent model. The  -model will be introduced in chapter II. Christian, J.W., Mahajant, S., 1995. Deformation twinning. Prog. Mater. Sci. 39, 1-157. Crumbach, M., Pomana, G., Wagner, P., Gottstein, G., 2001. Recrystallisation andgrain growth. In: Gottstein, G., Molodov, D. (Eds.) Mercier, S., Jacques, N., Molinari, A., 2005. Validation of an interaction law for the Eshelby inclusion problem in elasto-viscoplasticity. Int. J. Sol. Struct. 42, 1923-1941. Mercier, S., Molinari, A., 2009. Homogenization of elastic-viscoplastic heterogeneous materials: Self-consistent and Mori-Tanaka schemes. Int. J. Plasticity 25, 1024-1048. Molinari, A., Canova, G.R., Ahzi, S., 1987. A self-consistent approach of the large deformation polycrystal viscoplasticity. Acta Metall. 35, 2983-2994[START_REF] Tóth | Tuning a self consistent viscoplastic model by finite element results-II. Application to torsion textures[END_REF]. Tuning a self consistent viscoplastic model by finite elements reultats-I. Modeling, Acta Metall. 42, 2453-2458[START_REF] Prakash | Twinning models in self-consistent texture simulations of TWIP steels[END_REF] Twinning models in self-consistent texture simulations of TWIP steels. Steel. Res. Int. 79, 645-652. Sachs, G., 1928. Plasticity problems in metals. Z. Ver. Deutsch. Ing.72, 734-736. Schmid, C., Kawalla, R., Walde, T., Riedel, H., Prakash, A., Poizat, C., 2007. Experimental and numerical investigation of texture development during hot rolling of magnesium alloy AZ31. Mater. Sci. Forum., 539-543, 3448-3453. Staroselsky, A., Anand, L., 1998. Inelastic deformation of polycrystalline face centered cubic materials by slip and twinning. J. Mech. Phys. Solids. 46, 671-696. Taylor, G.I., 1938. Plastic strain in metals. J. Inst. Metals. 62, 307-324. Tomé , C.N., Lebensohn, R.A., Kocks, U.F., 1991. A model for texture development dominated by deformation twinning: Application to zirconium alloys. Acta Metall.

Mater. 39, 2667-2680. Tomé , C., Canova, G.R., Kocks, U.F., Christodoulou, N., Jonas, J.J., 1984. The relation between macroscopic and microscopic strain hardening in fcc polycrystals.

Acta Metall. 32, 1637-1653[START_REF] Tóth | Tuning a self consistent viscoplastic model by finite element results-II. Application to torsion textures[END_REF] Van Houtte, P., Kanjarla, A.K., Van Bael, A., Seefeldt, M., Delannay, L., 2006. Multiscale modeling of the plastic anisotropy and deformation texture of polycrystalline materials. European Journal of Mechanics A/Solids 25, 634-648.

Voce, E., 1948. The relationship between stress and strain for homogeneous deformation. J. Inst. Metals. 74, 537-562.

Chapter II. Visco-plastic -model

II.1 Introduction

The work of this thesis mainly focus on the application, extension and evaluation of the visco-plastic  -model. The  -model is proposed by Ahzi and M"Guil (2008).

This model is based on the micromechanical modeling of large visco-plastic deformation of polycrystalline aggregates using the physical mechanisms of the plastic deformation. The novelty and originality of this approach is the way of obtaining the interaction laws of the polycrystals. The considered polycrystalline model makes use of an interaction law which is based on a new intermediate approach that allows spanning from the upper to the lower bound models using a single interaction strength parameter  . This parameter is naturally introduced in a given potential function that is minimized to obtain the interaction law (Ahzi and M"Guil, 2008). With this approach a wide range of results can be obtained by varying  between 0 and 1. In addition, unlike other self-consistent approaches, this intermediate model is not based on the Eshelby-type interaction laws which are difficult to numerically implement. The simplicity of the  -model against the self-consistent models is a real advantage for the numerical simulations. This new model will give a new scale to the domain of crystal plasticity mainly dominated by the self-consistent model (e.g. Molinari et al., 1987;Ahzi, 1987) and other approaches such as the upper bound (Taylor, 1938) or the lower bound (Sachs, 1928) models.

II.2 Development of the visco-plastic -model  Review of the single crystal constitutive law used in the -model

As mentioned in section I.3, the visco-plastic power law is used to describe the mechanism of crystallographic slip and twinning. This law is used to express the relationship between the shear-rate,    , of a given slip system  and the corresponding resolved shear stress,   . Thus, the rigid visco-plastic single crystal behavior can be described by a rate-sensitive constitutive power law relating the plastic strain rate tensor D (elasticity is neglected) and the deviatoric Cauchy stress tensor S of the single crystal (see also section I.4):

    S S M S P P S P D 0                        1 n g g  (2-1)
Here, 0   is a reference shear rate, n is the inverse rate-sensitivity coefficient,  P represents the symmetric part of the Schmid tensor and  g is the critical resolved shear stress (CRSS) of the slip (or twinning) system  which evolves according to the Voce hardening law (see section I.5.2) in this thesis.

M is the macroscopic fourth order visco-plastic compliance tensor. The macroscopic behavior can be described by a similar relationship to the microscopic constitutive law (Eq. 2-1):

S M D   (2-2)
Here D , S and M are the macroscopic plastic strain rate, the macroscopic deviatoric Cauchy stress and the macroscopic fourth-order visco-plastic compliance tensor, respectively.

The global conditions need to be fulfilled which is given by the averaging conditions:

D D  and S S  . (2-2)
Here,  denotes the volume average over the polycrystals.

 Formulation of the -model

In order to guarantee that the global conditions (Eq. 2-2) can be fulfilled, the macroscopic strain rate tensor, D ~ and the corresponding macroscopic deviatoric stress tensor, S ~ have been introduced by Ahzi and M" Guil (2008). The authors assume that:

D K D   D ~ and S K S   S ~ (2-3)
Here, D K and S K are fourth-order mapping tensors. The introduced macroscopic strain rate and stress are supposed to be related similarly to Eq. 2-2.

S M D   or D L S   with 1 ~  M L (2-4)
Here, M ~ is an average compliance tensor. The  -model is formulated by the minimization of a specific function E combining the local fields" deviations (strain rate and stress) from the macroscopic ones:

  S D E E E      1 (2-5) Where 2 0 2    D - D E D and 2 0 2   S - S E S (2-6)
Here, 0  is the initial resolved shear stress. 0   represents the reference shear rate.

The function E in Eq. 2-5 depends on the parameter  between 0 and 1 which allows spanning the entire solution domain between the upper and lower bounds. This tuning parameter  allows the control of the grains interactions strength in the polycrystals. The mathematical development of the  -model can be found in the work of Ahzi and M"Guil (2008). By minimization of the function

E ( 0    D E ),
the dual interaction laws are derived as:

D A A D 1       and S B B S 1       (2-7) with                                           M M M M A 1 n 1 n 1 1 1 (2-8)                                           M M I M M I B 1 n 1 n 1 (2-9)
In Eq. 2-8 and Eq. 2-9,

1 0    ,   2 0 0      is a normalizing coefficient, n is
the inverse strain rate coefficient, A and B are fourth-order strain-rate and stress interaction tensors, respectively. The inverse of the ensemble average compliance tensor M ~ is given by:

         A M B M 1 1 1 ~ (2-10)
The interaction tensors A and B depend on the local and global compliance tensors, on the inverse rate sensitivity coefficient n and on the parameter  . The computation of these interaction tensors is much easier to handle than those involved in a self-consistent scheme. We note that in our interaction law, the Taylor and the static models results can be retrieved by setting

0   or 1   , respectively.
We notice that original stress and deformation tensors (e.g. strain rate tensor D and stress tensor S) are 3x3 matrixes. However, in the numeric calculations of  -model, these tensors are vectorized into 5x1 matrixes which can be expressed as (Ahzi, 1987):

  T 12 13 23 33 11 22 5 4 3 2 1 D 2 , D 2 , D 2 , D 2 3 , 2 D D D , D , D , D , D          D (2-11) And   T 12 13 23 33 11 22 5 4 3 2 1 S 2 , S 2 , S 2 , S 2 3 , 2 S S S , S , S , S , S          S (2-12)
Other tensors should be vectorized in the same way.

The general algorithm of the  -model is presented in Fig. 2-1. After the calculation of local strain rate D , the Newton-Raphson method is used to solve the non-linear constitutive law and calculate the local stress tensor S . We note that the first guess of S is made based on the Static model in which the local stress is assumed to be equal to the macroscopic one.

II.3 General numerical aspects

In previous works (Ahzi and M"Guil, 2008;M"Guil et al., 2009M"Guil et al., , 2011)), this model has been applied to FCC metals under tensile, compression and rolling tests. By varying the  value from 0 to 1, the  -model is able to predict large scale results from a stiff interaction (small values of  ) to a more compliant interaction (high values of  ). The result transitions of the  -model can also be found in the predicted textures. We present the textures of FCC metals under rolling and tensile tests in Fig. 2-3 and Fig.

2-4 as examples.

Compared with the VPSC model, the  -model shows a better performance in the texture transition (M" Guil et al., 2009).

Finally, we underline that the computational time required for the  -model simulations is of the same order as that for the classical Taylor model. This low computing time makes the  -model suitable for large scale computations and simulations of metal forming processes (M" Guil et al., 2011). ) at % 60 eq   for  -model. ) at % 60 eq   for  -model.

According to these results, we can conclude that the  -model with small  values (< 0.1) predicts standard deviations (and thus results) that are far from the ones predicted by the Taylor upper bound model. The  -model, with small  (<0.1) values, should not be considered as an almost iso-strain approach. In the same way, the  -model predictions with high  values (> 0.8) predicts standard deviations (and thus results)

that are quite different from those predicted with the lower Static bound model. The  -model, with large  (>0.8) values, should not be seen as an almost iso-stress model.

II.5 Evolution of the interaction tensor

The interaction tensor is one of the indicators of the interaction behavior in self-consistent models. The evolution of the interaction tensor in self-consistent models has been studied by several works (e.g. Ahzi, 1987). To define the interaction tensor, we need to express the interaction law in the following form:

  D D K S S     (2-14) or   S S K D D     1 (2-15)
Here, K is the interaction tensor. We notice that, for the  -model and the VPSC model, the strain rate tensor and the stress tensor are vectorized in the calculation (see Eq. 2-11 and Eq. 2-12). Therefore, the tensor K is a 5x5 matrix.

 Interaction tensor of the -model

For the visco-plastic  -model, the interaction law is presented in Eqs. 2-7 to 2-9. In order to calculate the interaction tensor K, we need to combine the Eq. 2-14 with the interaction law of  -model (Eq. 2-7). Thus, we can obtain:

  D A A D K S B B S            1 1 (2-16)
And then:

    D A A I K S B B I            1 1 (2-17)
The relation between D and S is described in Eq. 2-1. So we can get:

    M A A I K B B I           1 1 (2-18)
Finally, the interaction tensor K can be expressed as:

    1 1 1            A A I M B B I K 1 - (2-19)     1 1 1 1             B B I M A A I K (2-20)
To study the evolution of the interaction tensor, we perform simulations of tensile tests in FCC metals with only the {111}<110> slip considered. The simulations start with a random texture and the linear hardening is used. The boundary condition (macroscopic velocity gradient) is set to be:

              1 0 0 0 5 . 0 0 0 0 5 . 0 0  L (2-21)
We should note that the interaction tensor is different for each grain. We randomly pick one grain and its interaction tensor evolutions are calculated. We choose the  value of 0.01, 0.1, 0. From those results, we can see that, for all n values, a lower  value will lead to a stronger interaction tensor K (larger absolute value of K ij ). It is reasonable because for a stiff interaction (low  values), the difference between D and D will be limited whereas the difference between S and S will be enlarged. When  is close to 1, the values of K ij are very close to 0.

We note that the values of K ij evolve with the strain increase. This phenomenon can be explained by the texture evolution. The results for n=9 show the waviness at 99 . 0   . This may be explained by the fast transition of the grain orientation. That is to say, when a compliant interaction (high  values) is used, some grains may rotate very fast due to the kinematic freedom.

 Interaction tensor of the VPSC model

As shown in section I.8.5.4, the interaction tensor of the VPSC model, which is obtained directly from its interaction law, can be expressed as:

  E I E M K -1 -1 VPSC      1 eff n (2-22)
Then, we carried out a similar simulation with the same input parameters than the one using the  -model. The grain shape is calculated individually in each grain. The simulation is performed using the Tangent and Secant interactions and we present the evolution of the interaction tensors on the same grain as the previous simulation using 2-22), they both contain the n and M . The VPSC K can take into account the grain shape effect via the Eshelby tensor which cannot be found in  K . However, we believe that the grain shape effect may also be considered in

 K via the  value.
That is to say, the evolution of the grain shape may correspond to the evolution of the  value. The work on this point will be carried out in future. Furthermore, we should also notice that the forth order compliance tensor M can be found in the expression of  K (Eq. 2-19), which contains the information of the crystallographic orientation of the grains. Such information is absent from the interaction tensor of the VPSC model.

Chapter III. Application of -model in FCC rolling tests: coupled effects of the lattice rotation definition, twinning and interaction strength III.1 Introduction

The development of texture during rolling is one of the most important factors that influence the properties of the rolled sheets and their behavior in subsequent forming processes. The FCC rolling textures have been analyzed in many works (e.g. [START_REF] Carstensen | Texture development in Ni-Co alloys rolled to moderate reductions[END_REF][START_REF] El-Danaf | Role of stacking fault energy on the deformation characteristics of copper alloys processed by plane strain compression[END_REF][START_REF] Leffers | On the development of the brass-type texture[END_REF][START_REF] Leffers | The brass-type texture once again[END_REF]Leffers, , 2001aLeffers, ,b, 2012;;[START_REF] Leffers | Intra-and intergranular heterogeneities in the plastic deformation of brass during rolling[END_REF][START_REF] Leffers | The activation energy for the fcc rolling-texture transition as related to the activation energy for cross slip[END_REF][START_REF] Leffers | The early stages of the development of rolling texture in copper and brass[END_REF]Leffers and Ray, 2009;[START_REF] Leffers | Calculated and experimental orientation distributions of twin lamellae in rolled brass[END_REF][START_REF] Staroselsky | Inelastic deformation of polycrystalline face centered cubic materials by slip and twinning[END_REF]Van Houtte, 1978;Wen et al., 2012a). Two different types of FCC rolling texture, the copper-type and brass-type textures, can be observed in FCC metals with different properties and loading conditions (Leffers and Ray, 2009).

Usually, the copper-type textures were observed in FCC metals with high stacking fault energy (SFE) whereas the brass-type texture can be obtained for low SFE.

Sources influencing the development of copper-or brass-type texture are : stacking fault energy (SFE), twinning, micro-shear banding, twin barrier effect, hardening etc. (Beyerlein et al., 2011;[START_REF] El-Danaf | Deformation texture transition in brass: critical role of micro-scale shear bands[END_REF][START_REF] Kalidindi | Modeling anisotropic strain hardening and deformation textures in low stacking fault energy fcc metals[END_REF], 2011;Lebensohn et al., 2007;Leffers and Ray, 2009;[START_REF] Miraglia | On the influence of mechanical environment on the emergence of brass textures in FCC metals[END_REF][START_REF] Paul | The role of shear banding on deformation texture in low stacking fault energy metals as characterized on model Ag crystals[END_REF], 2009).

From a simulation point of view, the prediction of FCC rolling texture transition can be controlled by the interaction law (Leffers and Ray, 2009;[START_REF] Leffers | The brass-type texture-How close are we to understand it?[END_REF] and the consideration of {111}<112> mechanical twinning (Leffers andRay, 2009, Van Houtte, 1978). Another factor that influences the predicted rolling texture is the definition of the lattice spin [START_REF] Hosford | On orientation changes accompanying slip and twinning[END_REF][START_REF] Lebensohn | The rules for the lattice rotation accompanying slip as derived from a self-consistent model[END_REF][START_REF] Wierzbanowski | Problem of lattice rotation due to plastic deformation. example of rolling of f.c.c materials[END_REF].

The mathematical analysis (MA) and the plane-strain analysis (PSA) are usually used to calculate the lattice spin in solid mechanics [START_REF] Hosford | On orientation changes accompanying slip and twinning[END_REF][START_REF] Lebensohn | The rules for the lattice rotation accompanying slip as derived from a self-consistent model[END_REF][START_REF] Wierzbanowski | Problem of lattice rotation due to plastic deformation. example of rolling of f.c.c materials[END_REF]. The MA derives naturally from kinematic analysis in continuum mechanics whereas the PSA is derived from geometrical condition under rolling. The PSA calculation is suitable to simulate rolling process at large strain when the grains are elongated and flattened [START_REF] Hosford | On orientation changes accompanying slip and twinning[END_REF][START_REF] Lebensohn | The rules for the lattice rotation accompanying slip as derived from a self-consistent model[END_REF][START_REF] Wierzbanowski | Problem of lattice rotation due to plastic deformation. example of rolling of f.c.c materials[END_REF].

The effects of each of these three factors (interaction laws, consideration of twinning and definition of lattice spin) on FCC rolling texture transition, from copper-to brass-type, have been addressed by many authors (Beyerlein et al., 2011;[START_REF] Lebensohn | The rules for the lattice rotation accompanying slip as derived from a self-consistent model[END_REF]Leffers, 1968[START_REF] Leffers | On the development of the brass-type texture[END_REF][START_REF] Leffers | The brass-type texture once again[END_REF][START_REF] Leffers | The brass-type texture-How close are we to understand it?[END_REF]Leffers and Ray, 2009;[START_REF] Leffers | Calculated and experimental orientation distributions of twin lamellae in rolled brass[END_REF]Van Houtte, 1978;[START_REF] Wierzbanowski | Problem of lattice rotation due to plastic deformation. example of rolling of f.c.c materials[END_REF]. The plastic deformations in FCC metals are mainly accumulated by the activation of {111}<110> slip and {111}<112> mechanical twinning. The copper-type texture is usually achieved when only {111}<110> slip mode is considered using Taylor model, the upper bound approximation (e.g. Taylor, 1938[START_REF] Van Houtte | Deformation texture prediction: from the Taylor model to the advanced Lamel model[END_REF]. In this case, the effect of the lattice spin definition can be neglected as the MA and PSA calculations are theoretically equal when the Taylor model is used. [START_REF] Leffers | The brass-type texture once again[END_REF] have obtained a brass-type texture using PSA and a modified Sachs approach (e.g. [START_REF] Ahzi | Application of crystal plasticity theory for mechanically processed BSCCO superconductors[END_REF], 2002, Sachs, 1928) where only {111}<110> slip mode is considered. However, the {111}<112> mechanical twinning was suggested to be the explanation of the formation of the brass-type texture (Leffers and Ray, 2009). Van Houtte (1978) obtained a typical brass-type texture by using a Taylor model with the consideration of twinning.

In

III.2 Calculation of the lattice spin: MA and PSA definitions

The lattice spin * Ω was defined in two ways, *(MA) Ω in MA and

*(PSA) Ω in PSA.
The single crystal velocity gradient L can be decomposed into the strain rate D and the spin Ω as follows:

Ω D L   (3-1) where   T L L D   2 1 and   T L L Ω   2 1
. In the MA definition, Ω is described as the sum of the plastic spin P Ω and MA lattice spin *(MA) Ω of a grain:

p *(MA) Ω Ω Ω   (3-2)
Here, the grain shape is assumed to be spherical. Therefore, the local spin Ω is equal to the macroscopic one:

Ω Ω  .
We note that the macroscopic spin is null for the case of rolling. Hence, the MA definition for rolling can be expressed as:

    T p p p (MA) * L - L Ω Ω 2 1 - -  (3-3)
The PSA definition is based on the assumption that the shape of each grain should follow that of the sample [START_REF] Hosford | On orientation changes accompanying slip and twinning[END_REF][START_REF] Lebensohn | The rules for the lattice rotation accompanying slip as derived from a self-consistent model[END_REF][START_REF] Wierzbanowski | Problem of lattice rotation due to plastic deformation. example of rolling of f.c.c materials[END_REF]. Therefore, one can impose the following constraint: the shear components of local and global velocity gradient should remain the same:

ij ij L L  ( j i  ).
In the rolling case, we have 0

L ij  for j i  , hence: 0 L L ij ij   for j i  . Moreover, * ij p ij ij L L   
. Therefore, the definition of PSA can be expressed as:

p ij ) PSA *( ij L    (3-4)
It can be seen that the difference in the shear components between the MA and PSA rotation defined by Eqs. 3-3 and 3-4 can be present as [START_REF] Wierzbanowski | Problem of lattice rotation due to plastic deformation. example of rolling of f.c.c materials[END_REF]:

        ij p ij T p ij p ij p ij T p ij p ij ) PSA ( * ij ) MA ( * ij D D L L 2 1 L L L 2 1            (3-5)
In the upper-bound approximation, the local strain is assumed to be equal to the macroscopic one: D D  . Meanwhile, the shear components of D are equal to 0 for the rolling process, so: ij ij D D  ( j i  ). Thus, for rolling, the MA and PSA definition are theoretically equal when the Taylor model is used [START_REF] Wierzbanowski | Problem of lattice rotation due to plastic deformation. example of rolling of f.c.c materials[END_REF]. The evolution of these FCC rolling components can affect the metal properties during the forming applications. For instance, the material formability decreases with the rise of the G component whereas stronger C, B and S components lead to higher formability (M" Guil et al., 2010Guil et al., , 2011)). For the two types of FCC rolling texture (copper-and brass-type), their specific components evolution have been described in many works (e.g. [START_REF] Carstensen | Texture development in Ni-Co alloys rolled to moderate reductions[END_REF][START_REF] Leffers | The early stages of the development of rolling texture in copper and brass[END_REF]Leffers and Ray, 2009). These works usually focus on the evolution of the Cu and the B components. [START_REF] Leffers | The early stages of the development of rolling texture in copper and brass[END_REF]

III.3 Simulations under plane strain compression test

III.3.2 Input parameters

The simulation of FCC rolling texture was approximated by the plane strain compression test which neglects the effect of shear involved in the rolling process.

The initial texture is represented by 500 random orientations (grains). The simulations are conducted with a linear strain hardening using Voce law. The inverse strain rate sensitivity coefficient is chosen as 11 n  . In order to explain the crystallographic reorientation by twinning, we used the PTR scheme of Tomé et al. (1991). When twinning is considered, the parameters of the PTR scheme are set to be the same as the ones used in the work of Beyerlein et al. (2011):

25 . 0 A 1 th  , 1 . 0 A 2 th  and 707 . 0 S t 
. The ratio between the critical resolved shear stress (CRSS) in twinning and in slip (called  value by Van Houtte (1978)) is chosen to be 0.8 as in the work of Van Houtte (1978). In order to study the effect of the interaction strength, we choose several values of  : 0.1, 0.3, 0.5, 0.7, as well as extreme values of  to be close to Taylor case with

0   ( 6 10   
) and close to the lower bound (static model) with

1   ( 999 . 0  
). We note that the low values of  , such as 0.1, allow for an important deviation from the upper bound Taylor model. Similarly, high  values, such as 0.9, also allow for an important deviation from the static model (M" Guil et al., 2009).

The plane strain compression test is a crude approximation of a symmetric rolling process. In fact, we neglect the effect of shear involved in the rolling process. Thus, we describe only the neutral (central) zone of the rolled sheet where shear effects vanish. This simplification allows for the use of homogeneous boundary conditions.

In the following, deformation is simulated imposing successive deformation increments. At each deformation step, the boundary conditions are imposed in terms of the overall velocity gradient, L , corresponding to a constant strain rate   0   . In this study, the macroscopic velocity gradient is assumed constant and fully prescribed.

Here, x  represents the flow direction and z  the normal direction: and without twinning (Fig. 3-2), we predict a texture with a relatively strong Cu component. In the corresponding ODF sections, for the case without twinning (Fig. 34), the Cu and B components do not show an evident intensity difference for different  values. However, the B component presents an evident shift towards lower 1  values. Meanwhile, the effect of twinning on the intensity of these two components is much stronger than the effect of the interaction strength. With twinning (Fig. 345), the intensity of the Cu component is strongly decreased and the B component is strongly increased. The reduction of Cu component is known to be caused by twinning and responds to the formation of brass-type texture [START_REF] Vercammen | Cold rolling behavior of an austenitic Fe-30Mn-3Al-3Si TWIP-steel: the importance of deformation twinning[END_REF].

            1 0 0 0 0 0 0 0 1 0  L ( 3

 PSA definition

The predicted textures with the PSA definition are presented in Figs. 3-6 to 3-9. Since the MA and PSA definitions are theoretically equal when Taylor model is used (see section III.2), the pole figures predicted by these two definitions are similar when 0   . For the case without twinning, the predicted pole figures (Fig. 3456) show a complete texture transition from copper-to brass-type with the decreasing of the interaction strength (increasing of  ). When twinning is considered, the predicted pole figures (Fig. 34567) tend to yield a texture close to brass-type texture for all  values. In their corresponding ODF sections, for the case without twinning (Fig. 345678), the intensity of the B component increases while the intensity of the Cu component decreases gradually as the  value increases. The shift phenomenon of component B that was reported in the MA case (Fig. 34) with high  values is not found with the PSA case. When twinning is considered (Fig. 3456789), the effect of the parameter  on the component B is not significant. The Cu component shifts from its ideal position ), the volume fractions of Cu and S components first increase and then decrease during the rolling process. Hence, the textures are dominated by the B component for higher strain. Moreover, the G component seems to be independent of  value and does not show a significant change with increasing strain.

III.3.4.2 Results with twinning

 MA definition

When twinning is considered, with the MA definition (Fig. 3-12), the results for low  values show a strong B component followed by G and S components. The superposition of B and G components leads to a brass-type texture (M" Guil et al., 2010Guil et al., , 2011)). We note that the strongest B component can be found at

PSA definition

The results for PSA with twinning are shown in Fig. 3

III.4 Effect of relative slip/twinning activities on texture evolutions

The activation of deformation twinning can affect the texture evolution. This influence can be more evident if the relative activity of twinning to slip is stronger.

The relative activities of crystallographic slip and twinning can be controlled by the

 value ( slip 0 tw 0    
) defined by Van Houtte (1978). Previously, we choose 8 . 0   as in the work of Van Houtte (1978). In this section, we will compare the slip/twinning activities and predicted textures for  values of 0.8, 1.0 and 1. . In these results, we can see that the effect of  value is explicit in some cases. For example, a higher  value leads to a stronger Cu component for both MA and PSA definitions (

III.5 Volume effects of deformation twinning

The volume effect of twinning is one of the ways that the twinning affects the texture evolution. The twinning volume fraction represents the volume fraction of the twinned region in grains during the deformation. Several experimental works have been carried out to determine the twin volume fraction in FCC metals with low SFE (e.g. Leffers and Ray, 2009;Leffers and Kayworthl, 1973;Hutchinson et al. 1979).

According to these works, the twinning volume fraction is less than 25% for about 75% rolling reduction ( % 140 eq  

) and can be neglected under about 40% rolling reduction ( % 51 eq  

).

In order to correctly predict the texture evolutions, the predicted twin volume fraction should meet an agreement with the experimental ones. However, in the works above, the predicted volume fraction cannot be related to the experiments as it is strongly overestimated with the PTR scheme (Beyerlein et al. 2011;[START_REF] Prakash | Twinning models in self-consistent texture simulations of TWIP steels[END_REF], Schmid et al., 2007).

In this section, we carried out a simulation study of the volume effect of twinning. In order to relate our results with experiments, we used the voce hardening parameters of silver that are presented in the work of Beyerlein et al. (2011), see Tab. 3-1. In the PTR scheme, the characteristic twin shear parameter t S can be linked with the twin volume fraction. It is suggested to be 0.707 for cubic material (e.g. Beyerlein et al., 2011;[START_REF] Prakash | Twinning models in self-consistent texture simulations of TWIP steels[END_REF]. However, this value will lead to an overestimation of the twin volume fraction.

Here, we tried to control the twin volume fraction by increasing the value of t S . In Fig. 3-24, the twin volume fractions for 707 . 0 S t  are around 20% at 50% equivalent strain (40% rolling reduction) and are around 40% to 80% for larger strain up to 140% equivalent strain (75% rolling reduction). The volume fractions are therefore overestimated by this value of t S . By increasing t S up to 2.5, the twin volume fraction can be reduced to a reasonable range for both MA and PSA lattice spin definitions. We therefore analyzed texture evolution for three values of

III.6 Conclusion

In this work, two definitions for the calculation of the lattice rotation, MA and PSA, have been compared under plane strain compression by using the visco-plastic  -model. Firstly, the coupled effect of twinning, interaction strength (inversely proportional to  ) and the used definition for the lattice spin on the texture evolution has been deeply analyzed. In the second part, we perform parametric studies by varying the parameters controlling the slip/twinning activities and the twin volume fractions. The effects of these parameters on the texture evolutions are studied.

In the first part, we showed that the MA and PSA definitions are equal when the ) or using both MA or PSA definitions with twinning. However, these two methods of the brass-type texture prediction show an evident difference in the G component. The latter method presents a stronger G component which is suggested to be caused by mechanical twinning [START_REF] Vercammen | Cold rolling behavior of an austenitic Fe-30Mn-3Al-3Si TWIP-steel: the importance of deformation twinning[END_REF].

The predict results, in terms of pole figures and ODF sections, show a tendency towards brass-type texture when  is small with twinning and when  is high with twinning. However, when we analyze texture components, in a more detailed way, the brass-type texture is well predicted only by high  value with twinning (around

7 . 0  
). This is proven by the comparison of our results with the experimental results of 70/30 brass [START_REF] El-Danaf | Deformation texture transition in brass: critical role of micro-scale shear bands[END_REF] reported in Fig. 3 The Cu components are relatively weak when twinning is considered. We believe that this can be explained by the strong activation of the twinning systems and the overestimation of twinning volume fraction by the PTR scheme. This point is supported by the following parameter studies in which we showed that the Cu component can be elevated by high activities of twinning and high twin volume fraction.

The mechanical twinning can be detected in most rolled materials developing brass-type texture such as brass and silver [START_REF] El-Danaf | Deformation texture transition in brass: critical role of micro-scale shear bands[END_REF][START_REF] Leffers | The early stages of the development of rolling texture in copper and brass[END_REF]Leffers and Ray, 2009;[START_REF] Sekine | Characteristic features of rolling texture development in FCC alloys having very low stacking fault energies[END_REF]. Hence, the consideration of twinning for simulating texture evolution in these materials is logical. However, certain materials with specific microstructures or under specific loading conditions can develop a rolling texture that deviates from copper-type such as Cu-Mn alloy [START_REF] Engler | Deformation and texture of copper-manganese alloys[END_REF][START_REF] Engler | Rolling texture development in Cu-Mn alloys[END_REF]. This alloy develops a brass-type texture even if no twinning has been detected in this metal [START_REF] Engler | Deformation and texture of copper-manganese alloys[END_REF][START_REF] Engler | Rolling texture development in Cu-Mn alloys[END_REF]. In this case, the rolling texture evolution of the Cu-Mn alloy may be predicted by the  -model with high  values without twinning and using the PSA definition.

To precisely predict the rolling texture transition, other mechanisms such as shear banding and twin barrier effect were suggested. However, the modeling studies of these mechanisms are still in progress. The shear banding is suggested to be responsible for the formation of the brass-type texture and have been studied in many works (e.g. [START_REF] Duggan | Recrystallization studies in 70:30 brass using a high voltage electron microscope[END_REF][START_REF] El-Danaf | Deformation texture transition in brass: critical role of micro-scale shear bands[END_REF], 2011;[START_REF] Gil Sevillano | The contribution of macroscopic shear bands to the rolling texture of FCC metals[END_REF][START_REF] Kalidindi | Modeling anisotropic strain hardening and deformation textures in low stacking fault energy fcc metals[END_REF]Leffers and Ray, 2009;[START_REF] Paul | The role of shear banding on deformation texture in low stacking fault energy metals as characterized on model Ag crystals[END_REF]2009). However, shear banding is an extremely complex process and therefore the modeling work of this mechanism is complicated. Quantitative modeling of shear banding effects on texture has not been reported except several highly simplified models [START_REF] Gil Sevillano | The contribution of macroscopic shear bands to the rolling texture of FCC metals[END_REF][START_REF] Kalidindi | Modeling anisotropic strain hardening and deformation textures in low stacking fault energy fcc metals[END_REF]. The twin barrier effect occurs due to the boundaries of the fine twins. When the twinning region appears in a grain, non-coplanar slip with the twinning plane becomes harder to be activated. Lebensohn et al. (2007) and Beyerlein et al. (2011) have taken into account this effect via the latent hardening. However, the simulated textures show some deviations from the typical brass-type texture.

Wen, W., M"Guil, S., Ahzi., 2012a. Rolling texture transition in FCC metals using the viscoplastic  -model and considering mechanical twinning. Mater. Sci. Forum.

702-703, 241-244

Wen, W., M"Guil, S., Ahzi, Gracio, J.J., 2012b. Coupled effects of the lattice rotation definition, twinning and interaction strength on the FCC rolling texture evolution using the viscoplastic  -model. Int. J. Plasticity. http://dx.doi.org/10.1016/j.ijplas.2012.09.009

Chapter IV. Application to FCC metals: analysis of shear deformation by slip and twinning in low and high stacking fault energy metals

IV.1 Introduction

In order to validate new plasticity modeling theories, simulations until very large plastic deformation must be performed. In such a way, severe plastic deformation methods such as torsion test [START_REF] Canova | Theory of torsion texture development[END_REF][START_REF] Lin | A comparative study of hardening theories in torsion using the Taylor polycrystal model[END_REF][START_REF] Tóth | Texture development and length changes in copper bars subjected to free end torsion[END_REF] or Equal Channel Angular Extrusion (ECAE) method [START_REF] Beyerlein | Role of twinning on texture evolution of silver during equal channel angular extrusion[END_REF][START_REF] Beyerlein | Texture evolution in equal-channel angular extrusion[END_REF][START_REF] Gazder | Microstructure and texture evolution of bcc and fcc metals subjected to equal channel angular extrusion[END_REF][START_REF] Li | Orientation stability in equal channel angular extrusion. Part I: Face-centered cubic and body-centered cubic materials[END_REF]Li et al., 2005aLi et al., ,b, 2006a,b;,b;[START_REF] Segal | Materials processing by simple shear[END_REF][START_REF] Suwas | Evolution of crystallographic texture during equal channel angular extrusion of silver[END_REF][START_REF] Suwas | Evolution of texture during equal channel angular extrusion of commercially pure aluminum: Experiments and simulations[END_REF][START_REF] Suwas | Evolution of crystallographic texture during equal channel angular extrusion of silver[END_REF] can be used. These tests involving shear stresses allow material to be deformed to very high plastic strain that cannot be obtained with more conventional processes, such as rolling. These tests usually lead to a very fine microstructure with extraordinary mechanical properties, such as simultaneous ultrahigh strength and high ductility. An extended review on texture evolution during ECAE can be found in the work of [START_REF] Beyerlein | Texture evolution in equal-channel angular extrusion[END_REF]. Except subtle differences in terms of location of the respective components (see Fig. 4-1), the textures that develop during ECAE are similar to those seen in torsion, in which deformation occurs by simple shear [START_REF] Beyerlein | Role of twinning on texture evolution of silver during equal channel angular extrusion[END_REF][START_REF] Gazder | Microstructure and texture evolution of bcc and fcc metals subjected to equal channel angular extrusion[END_REF][START_REF] Li | Orientation stability in equal channel angular extrusion. Part I: Face-centered cubic and body-centered cubic materials[END_REF]Li et al., 2005aLi et al., ,b, 2006a;;[START_REF] Suwas | Evolution of crystallographic texture during equal channel angular extrusion of silver[END_REF][START_REF] Suwas | Evolution of crystallographic texture during equal channel angular extrusion of silver[END_REF][START_REF] Tóth | Tuning a self consistent viscoplastic model by finite element results-II. Application to torsion textures[END_REF]. Thus, it is commonly accepted that the deformation process during ECAE can be well approximated by simple shear [START_REF] Segal | Materials processing by simple shear[END_REF].

From the modeling point of view, many of crystal plasticity models have been applied to the study of shear texture development in polycrystalline metals [START_REF] Beyerlein | Role of twinning on texture evolution of silver during equal channel angular extrusion[END_REF]Li and Havner, 1996;Li et al., 2005aLi et al., ,b, 2006aLi et al., ,b, 2008;;[START_REF] Suwas | Evolution of texture during equal channel angular extrusion of commercially pure aluminum: Experiments and simulations[END_REF][START_REF] Tóth | Tuning a self consistent viscoplastic model by finite element results-II. Application to torsion textures[END_REF]. In this chapter, we focus on the simulation of large plastic deformation and texture evolution under simple shear by using the  -model. Our results show that the visco-plastic  -model can be used to correctly predict shear texture evolution in FCC metals and to predict texture transition that other models such as the VPSC cannot predict. Although some similarities in the shear texture components are obtained, a wide range of shear texture predictions in both low stacking fault energy (SFE) metals (i.e. silver) and medium SFE metals (i.e. copper) is only obtained with the  -model. We note that in this chapter, we do not consider dynamic recrystallization, we do not take into account the Swift effect [START_REF] Tóth | Texture development and length changes in copper bars subjected to free end torsion[END_REF]Swift, 1947) and we neglect the effect of substructure evolution [START_REF] Beyerlein | Texture evolution in equal-channel angular extrusion[END_REF]. The results presented in this chapter can be found in the work of M"Guil et

 0°9 0°* 1 A * 2 A C  1     0 2  0°9 0°* 1 A * 2 A C  1     0 2  0°9 0° 1     0 2 * 1 A   C * 2 A   0°9 0° 1     0 2 * 1 A   C * 2 A  * 2 A  A B A B B C B * 1 A  0°9 0°* 2 A C  1     45 2 A B A B B C B * 1 A  0°9 0°* 2 A C  1     45 2  0°9 0° 1     45 2  A  B  B  B * 1 A  * 2 A   C  0°9 0° 1     45 2  A  B  B  B * 1 A  * 2 A   C

IV.2 Experimental shear textures for FCC metals

IV.2.1 Preferred orientations for simple shear textures

The main ideal simple shear texture components of FCC metals are usually presented  , 2  ) and Miller indices of these components are listed in Tab. 4-1 (Li et al., 2005b). Simple shear textures are characterized by two partial fibres: the " A " partial fiber indexed as {111}<uvw> and the " B " partial fiber indexed as {hkl}<110>. There exist also the " C " ideal orientation indexed as {001}<110>. However, the C position is just a special position on the B fiber (Canova et al., 1984). As shown in

IV.2.2 Description of experimental shear texture for a medium SFE metal

In this section, we describe the shear texture for a medium SFE metals such as copper.

This description is based on the work of [START_REF] Tóth | Tuning a self consistent viscoplastic model by finite element results-II. Application to torsion textures[END_REF] In this chapter, we will compare our predicted shear textures of silver to those of [START_REF] Suwas | Evolution of crystallographic texture during equal channel angular extrusion of silver[END_REF] which where obtained for an ECAE test. However, we note again that the texture development during ECAE is close to that obtained under a simple shear test [START_REF] Beyerlein | Role of twinning on texture evolution of silver during equal channel angular extrusion[END_REF][START_REF] Gazder | Microstructure and texture evolution of bcc and fcc metals subjected to equal channel angular extrusion[END_REF][START_REF] Li | Orientation stability in equal channel angular extrusion. Part I: Face-centered cubic and body-centered cubic materials[END_REF]Li et al., 2005Li et al., a,b, 2006a;;[START_REF] Suwas | Evolution of crystallographic texture during equal channel angular extrusion of silver[END_REF][START_REF] Suwas | Evolution of crystallographic texture during equal channel angular extrusion of silver[END_REF][START_REF] Tóth | Tuning a self consistent viscoplastic model by finite element results-II. Application to torsion textures[END_REF]. [START_REF] Tóth | Tuning a self consistent viscoplastic model by finite element results-II. Application to torsion textures[END_REF] In this section, we present the main concluding remarks reported by [START_REF] Suwas | Evolution of crystallographic texture during equal channel angular extrusion of silver[END_REF] relative to the description of shear texture evolution. During ECAE of silver, strong texture develops and it continues to strengthen with the number of passes.

 0°9 0° 1     0 2 * 1 A * 2 A C  0°9 0° 1     0 2 * 1 A * 2 A C  0°9 0° 1     0 2 * 1 A * 2 A C  0°9 0° 1     45 2 A B A B B C B * 1 A * 2 A C  0°9 0° 1     45 2 A B A B B C B * 1 A * 2 A C  0°9 0° 1     45
These authors observed, that for the first pass, the main ideal components are: a strong component near to the ideal B and B , a relatively strong component * 1 A , a weak * 2 A and a very weak C components. These components, however, are significantly rotated from their respective ideal positions : 20° for the * 1 A , 16° for C and 9° for the * 2 A components. All these rotations are in the decreasing 1  direction (Suwas et al., 2003).

 0°9 0° 1     0 2 * 1 A * 2 A C  0°9 0° 1     0 2 * 1 A * 2 A C  0°9 0° 1     0 2 * 1 A * 2 A C  0°9 0° 1     0 2 * 1 A * 2 A C  0°9 0° 1     0 2 * 1 A * 2 A C  0°9 0° 1     0 2 * 1 A * 2 A C  0°9 0° 1     45 2 A B A B B C B * 1 A * 2 A C  0°9 0° 1     45 2 A B A B B C B * 1 A * 2 A C  0°9 0° 1     45 2 A B A B B C B * 1 A * 2 A C  0°9 0° 1     45 2 A B A B B C B * 1 A * 2 A C  0°9 0° 1     45 2 A B A B B C B * 1 A * 2 A C  0°9 0° 1     45 2 A B A B B C B * 1 A * 2 A C
These tendencies for silver are different from those of high/medium SFE FCC metals like aluminium, copper and nickel. For high/medium SFE metals, the C , * 1 A and B / B components are the strongest whereas the * 2 A component is the weakest. In fact, for high SFE FCC metals, simple shear textures are typically found to consist of a partial <110> fibre and a less strong partial {111} fibre. On the other hand, the simple shear texture of silver has been reported to be {112}< 0 1 1 >, which are the B / B components (Suwas et al., 2003).

In the extended work of [START_REF] Beyerlein | Texture evolution in equal-channel angular extrusion[END_REF], the authors enumerate the reasons in texture differences between copper and silver: differences in SFE, initial texture effects (initially silver had a random texture and copper had a strong initial fiber texture) and deformation twinning. [START_REF] Beyerlein | Role of twinning on texture evolution of silver during equal channel angular extrusion[END_REF], using experimental observations and simulation tool, have attributed these significant differences in the texture evolution between silver and copper to deformation twinning in silver [START_REF] Beyerlein | Role of twinning on texture evolution of silver during equal channel angular extrusion[END_REF]. For instance, experimental microscopy investigations confirmed the presence of twins in each pass of ECAE as well as significant grain refinement for silver [START_REF] Beyerlein | Role of twinning on texture evolution of silver during equal channel angular extrusion[END_REF][START_REF] Suwas | Evolution of crystallographic texture during equal channel angular extrusion of silver[END_REF]. [START_REF] Beyerlein | Role of twinning on texture evolution of silver during equal channel angular extrusion[END_REF] performed polycrystalline simulations of the ECAE process with the PTR scheme for twinning (Lebensohn et al., 1991) which confirmed that twinning caused the texture differences between a low and medium/high SFE. [START_REF] Beyerlein | Texture evolution in equal-channel angular extrusion[END_REF] also note that the strain path changes in ECAE process is important and may influence the twinning in each pass. However, in our study, we do not perform ECAE test but simple shear test and thus, we totally neglect the effect of strain path. To conclude this paragraph, under the same processing conditions, the shear textures for silver (a low SFE) are clearly distinct from those of copper (a medium SFE) [START_REF] Beyerlein | Role of twinning on texture evolution of silver during equal channel angular extrusion[END_REF].

IV.3 Shear texture transition as function of SFE IV.3.1 SFE effect on plastic deformation mechanisms

The SFE is a microstructural material parameter which varies in pure metals and can be lowered by alloying. Examples of high to medium SFE FCC materials are aluminum (167.5 mJ/m 2 ), nickel (128 mJ/m 2 ) and copper (78 mJ/m 2 ). Low SFE FCC materials include gold (45 mJ/m 2 ) and silver (19-22 mJ/m 2 ) [START_REF] Beyerlein | Texture evolution in equal-channel angular extrusion[END_REF]. The SFE has a strong impact on the plastic deformation mechanisms and on the dislocations movements. The mechanisms of deformation in FCC metals are strongly influenced by the SFE which controls the crystallographic slip planarity and the dislocations motion. Materials with high SFE permit dislocations to cross slip easily around obstacles. In materials with low SFE, cross slip is difficult and dislocations are constrained to move in a more planar fashion and the dislocation mobility decreases [START_REF] Beyerlein | Role of twinning on texture evolution of silver during equal channel angular extrusion[END_REF]. Low SFE metals may present mechanical twinning in addition to the crystallographic slip during plastic deformation [START_REF] Beyerlein | Role of twinning on texture evolution of silver during equal channel angular extrusion[END_REF]. Thus, metals with medium to high SFE values deform by slip only, while those with low SFE values deform by both slip and twinning. When the SFE of metals is lowered, the twinning mechanism is increased and some noticeable texture transitions take place [START_REF] Beyerlein | Texture evolution in equal-channel angular extrusion[END_REF][START_REF] Suwas | Evolution of crystallographic texture during equal channel angular extrusion of silver[END_REF]. [START_REF] Hu | Texture transition in high-purity silver and its correlation with stacking fault[END_REF] have studied the texture transition in High-Purity silver and correlated this transition with the SFE. Leffers (1968) suggested that the Taylor upper bound assumption is too strict and is only applicable to FCC metals having high SFE where extensive cross-slip lead to a reasonably homogeneous deformation. We note that Van Houtte (1978) have used the Taylor upper bound model in order to simulate and interpret the rolling and torsion textures in FCC metals with a wide variety of SFE . [START_REF] Stout | Proceedings of the ICOTOM[END_REF] and Hughes and Wenk (1988) studied the variation in torsion texture development after relatively large deformation with the variation of the SFE.

IV.3.2 Torsion texture development as function of SFE

Below, we summarize some of their observations during torsion tests for low and high SFE metals. These authors noted that for shear texture the B / B fiber is present to various degrees in all materials. Futhermore, [START_REF] Hughes | Stacking fault energy and microstructure effects on torsion texture evolution[END_REF] found that the B / B component was strong after large strains for a wide range of SFE. [START_REF] Hughes | Stacking fault energy and microstructure effects on torsion texture evolution[END_REF] also found that the C component strengthens for the medium to high SFE metals while it weakens in a low SFE metals.

For low SFE metals, Hughes and Wenk (1988) showed that the C component is weaker in silver than in aluminum and copper (results reported in [START_REF] Beyerlein | Role of twinning on texture evolution of silver during equal channel angular extrusion[END_REF]. The C component weakens with increasing amount of shear and vanishes after a very large shear strain ( 8 . 5  

). These authors have identified the major components for low SFE metals as A , B / B and * 1 A .

During the first pass of ECAE deformation of silver, the A  and  C components with increasing pass number [START_REF] Beyerlein | Role of twinning on texture evolution of silver during equal channel angular extrusion[END_REF].

IV.4 Shear texture predictions without twinning mechanism IV.4.1 Input parameters

In this section, we performed various simulations in order to power the  -model against experimental and predicted (using other widely used models) shear textures for FCC polycrystals. In this way, for the visco-plastic  -model, we choose several values for the parameter,  , between 0 and 1. This parameter is microstructure and strain path dependent (M" Guil et al., 2009Guil et al., , 2010Guil et al., , 2011a,b),b), but, in this chapter, no specific way exists for deriving this parameter and thus, it can simply be regarded as a tuning parameter. In the case of simple shear tests of FCC metals, we selected For the second value of 0.1, the results deviate drastically from those of the Taylor model and are surprisingly close to those obtained by the static lower bound predictions.

In addition to the comparison of the  -model and upper and lower bounds, we also compared these results to those obtained using the VPSC model. For this, we select two values of the tuning parameter eff n corresponding to different formulations of the macroscopic moduli: the secant formulation ( 1 n eff  ) and the tangent formulation ( n n eff  ). These simulations were conducted with strain hardening according to the Voce law.

The Voce hardening parameters are obtained by fitting the results of the upper bound Taylor model to experimental shear stress/shear strain curves given in the work of Tóth and Molinari (1994) for OFHC copper. The best fitting is obtained for:   0 16 MPa,   1 120 MPa,   0 190 MPa and   1 7 MPa. These parameters are kept the same for all simulations with different models. Small modifications of the hardening coefficients will lead to a better agreement between the measured and the predicted stress-strain curves but not in a significant change in the texture development. Tóth and Molinari (1994) have pointed out that important differences in the texture can only be seen when the model assumptions are changed, i.e. when the Taylor assumption, self-consistent approach, shape effects, or changes in the value of the strain rate sensitivity were considered. However, in this work, the hardening effect will be analyzed, through a comparison between predicted results with non-linear hardening, with a linear hardening and also without hardening. These simulations showed that the  -model and the lower bound approach are sensitive to hardening parameters. As in the work Tóth and Molinari (1994), the strain rate sensitivity parameter for the crystallographic slip is chosen to be 7 n  .

In order to compare our predictions to the results of Tóth and Molinari (1994), we used their initial texture, for OFHC copper, represented by 400 orientations and shown in Fig. 4-2. For silver, since we do not have an experimental starting texture, we used the same initial texture as the one shown in Fig. 4-2. Based on the results for these two FCC materials, we attempt to qualitatively correlate the experimental shear textures and SFE of FCC metals based on the  -model predictions.

In the simulations, deformation is simulated imposing successive deformation increments using the 12 {111}<110> FCC slip systems. At each deformation step, the boundary conditions corresponding to a constant equivalent strain rate, 0   , are imposed. In other words, we assume homogeneous boundary conditions where the imposed macroscopic velocity gradient L is constant. The prescribed velocity gradient L of simple shear is thus taken as:

            0 0 0 0 0 0 0 1 0 0  L (4-1)

IV.4.2 Results for shear texture evolutions

In order to compare our predictions to the experimental shear textures of Tóth and Molinari (1994), we simulated simple shear tests up to a total shear strain of 2   varying  between 0 to 1. This shows that this parameter allows for the prediction of the texture transition and therefore it can be linked to the microstructural features controlling this transition in FCC metals (M" Guil et al., 2009Guil et al., , 2010Guil et al., ,2011a)). We should note that the grain shape is not updated in the VPSC calculations and this effect is not included in the  -model. Since, we do not take into account the effect of substructure evolution which is due to the severe plastic deformation [START_REF] Beyerlein | Texture evolution in equal-channel angular extrusion[END_REF], we limit our analysis to a shear strain of 2   . Under larger strains and strain path changes, [START_REF] Beyerlein | Texture evolution in equal-channel angular extrusion[END_REF] have pointed out that some deficiencies in models (Taylor and VPSC) can affect texture predictions.

0°9 0° 1     0 2 * 1 A * 2 A C  0°9 0° 1     0 2 * 1 A * 2 A C  0°9 0° 1     0 2 * 1 A * 2 A C  0°9 0° 1     45 2 A B A B B C B * 1 A * 2 A C  0°9 0° 1     45 2 A B A B B C B * 1 A * 2 A C  0°9 0° 1     45 2 A B A B B C B * 1 A * 2 A C  0°9 0° 1     0 2 * 1 A * 2 A C  0°9 0° 1     0 2 * 1 A * 2 A C  0°9 0° 1     0 2 * 1 A * 2 A C  0°9 0° 1     45 2 A B A B B C B * 1 A * 2 A C  0°9 0° 1     45 2 A B A B B C B * 1 A * 2 A C  0°9 0° 1     45 2 A B A B B C B * 1 A * 2 A C  0°9 0° 1     0 2 * 1 A * 2 A C  0°9 0° 1     0 2 * 1 A * 2 A C  0°9 0° 1     0 2 * 1 A * 2 A C  0°9 0° 1     45 2 A B A B B C B * 1 A * 2 A C  0°9 0° 1     45 2 A B A B B C B * 1 A * 2 A C  0°9 0° 1     45 2 A B A B B C B * 1 A * 2 A C  0°9 0° 1     0 2 * 1 A * 2 A C  0°9 0° 1     0 2 * 1 A * 2 A C  0°9 0° 1     0 2 * 1 A * 2 A C  0°9 0° 1     45

IV.4.3 Effect of microscopic hardening on predicted shear texture

In this section, the hardening effect is analyzed through a comparison between predicted shear textures with non-linear hardening (   0 16 MPa,   1 120 MPa,

  0 190 MPa;   1 7 MPa), with a linear hardening (   0 1 MPa,   1 0 MPa,   0 1 MPa;   1 1 MPa) and without hardening (   0 1 MPa,   1 0 MPa,   0 0 MPa and   1 0 MPa).
The predictions show that the  -model is quite sensitive to variations of hardening However, we should note that the A , B , and C textures components appear in all of the hardening assumptions but with different intensities as was already reported by others (Li and Havner, 1996). On the other hand, for the VPSC model, the effect of hardening is negligible and the predicted texture without hardening is almost indistinguishable from that with hardening. We have not reported these results in this work since these observations for the VPSC model are in accord with other results in literature such as those of Li et al. (2006). Similar conclusion is applied to the Taylor model. Thus, the VPSC model and the Taylor model are negligibly sensitive to variations in hardening parameters. On the other hand, the high sensitivity of the  -model to hardening parameters makes it the most capable model to reproduce a large number of experimental observations by correctly adjusting the value of hardening parameter versus the value of  . We note that the static model (results not reported) is also found to be sensitive to variations in hardening parameters.

 0°9 0° 1     0 2 * 1 A * 2 A C  0°9 0° 1     0 2 * 1 A * 2 A C  0°9 0° 1     0 2 * 1 A * 2 A C  0°9 0° 1     45 2 A B A B B C B * 1 A * 2 A C  0°9 0° 1     45 2 A B A B B C B * 1 A * 2 A C  0°9 0° 1     45 2 A B A B B C B * 1 A * 2 A C  0°9 0° 1     0 2 * 1 A * 2 A C  0°9 0° 1     0 2 * 1 A * 2 A C  0°9 0° 1     0 2 * 1 A * 2 A C  0°9 0° 1     45 2 A B A B B C B * 1 A * 2 A C  0°9 0° 1     45 2 A B A B B C B * 1 A * 2 A C  0°9 0° 1     45

IV.5 Shear texture predictions with twinning mechanism

In this section, we take into account the twinning mechanism to improve predicted shear texture for silver which is an FCC metal that can twin during plastic deformation [START_REF] Beyerlein | Role of twinning on texture evolution of silver during equal channel angular extrusion[END_REF]. In fact, silver is a low SFE metal and thus presents twinning deformation mode in addition to crystallographic slip during plastic deformation. On the other hand, a high/medium SFE metal, such as copper, deforms mainly by crystallographic slip.

 0°9 0° 1     0 2 * 1 A * 2 A C  0°9 0° 1     0 2 * 1 A * 2 A C  0°9 0° 1     0 2 * 1 A * 2 A C  0°9 0° 1     45 2 A B A B B C B * 1 A * 2 A C  0°9 0° 1     45 2 A B A B B C B * 1 A * 2 A C  0°9 0° 1     45 2 A B A B B C B * 1 A * 2 A C  0°9 0° 1     0 2 * 1 A * 2 A C  0°9 0° 1     0 2 * 1 A * 2 A C  0°9 0° 1     0 2 * 1 A * 2 A C  0°9 0° 1     45 2 A B A B B C B * 1 A * 2 A C  0°9 0° 1     45 2 A B A B B C B * 1 A * 2 A C  0°9 0° 1     45 2 A B A B B C B * 1 A * 2 A C  0°9 0° 1     0 2 * 1 A * 2 A C  0°9 0° 1     0 2 * 1 A * 2 A C  0°9 0° 1     0 2 * 1 A * 2 A C  0°9 0° 1     45 2 A B A B B C B * 1 A * 2 A C  0°9 0° 1     45 2 A B A B B C B * 1 A * 2 A C  0°9 0° 1     45 2 A B A B B C B * 1 A * 2 A C  0°9 0° 1     0 2 * 1 A * 2 A C  0°9 0° 1     0 2 * 1 A * 2 A C  0°9 0° 1     0 2 * 1 A * 2 A C  0°9 0° 1     45 2 A B A B B C B * 1 A * 2 A C  0°9 0° 1     45 2 A B A B B C B * 1 A * 2 A C  0°9 0° 1     45 2 A B A B B C B * 1 A * 2 A C  0°9 0° 1     0 2 * 1 A * 2 A C  0°9 0° 1     0 2 * 1 A * 2 A C  0°9 0° 1     0 2 * 1 A * 2 A C  0°9 0° 1     45 2 A B A B B C B * 1 A * 2 A C  0°9 0° 1     45 2 A B A B B C B * 1 A * 2 A C  0°9 0° 1     45 2 A B A B B C B * 1 A * 2 A C  0°9 0° 1     0 2 * 1 A * 2 A C  0°9 0° 1     0 2 * 1 A * 2 A C  0°9 0° 1     0 2 * 1 A * 2 A C  0°9 0° 1     45 2 A B A B B C B * 1 A * 2 A C  0°9 0° 1     45 2 A B A B B C B * 1 A * 2 A C  0°9 0° 1     45 2 A B A B B C B * 1 A * 2 A C
In order to check the influence of twinning, we conducted simulations adding 12 twinning systems {111}<112> to the 12 slip systems {111}<110>. We consider that the critical resolved shear stress of slip and twinning are equal ( ) twin ( 0) slip ( 0

  

). The PTR scheme (Tomé et al., 1991) is used here to describe the twinning orientations.

For the parameters relative to twinning, we used the same values as [START_REF] Beyerlein | Role of twinning on texture evolution of silver during equal channel angular extrusion[END_REF]:

25 . 0 A 1 th  , 1 . 0 A 2 th  and 707 . 0 S t 
. For cubic FCC metals, the parameter t S is usually equated equal to 0.707 (Beyerlein et al., 2011). In these simulations, we consider a linear hardening for both twinning and slip modes.

In this section, we choose A , respectively, for simple shear test (see Fig. 4-1). In fact, the * 2 A  component is the least likely to twin, which explains why this component is the strongest [START_REF] Beyerlein | Role of twinning on texture evolution of silver during equal channel angular extrusion[END_REF]. These results are consistent with our simulations which show a decrease of the component C (see Fig. A , C , the shift is observed in a decreasing 1  direction while the * 2 A shifts in the increasing 1  direction. We note that a shift, for silver, was already reported by [START_REF] Suwas | Evolution of crystallographic texture during equal channel angular extrusion of silver[END_REF]. These authors have obtained the smallest shift for this component * We also note, that the evolution, as function of  , of the B / B components, see Fig. 4-13, is quite stable which is in accord with literature [START_REF] Suwas | Evolution of crystallographic texture during equal channel angular extrusion of silver[END_REF]. In fact, without (Fig. 4-13a) and with twinning (Fig. 4 if twinning is taken into account, see Fig. 4-14. To close this section, we conclude that in order to accurately predict the shear texture evolution in silver, twinning mechanism must be taken into account. This is in accord with the results from literature such as those of [START_REF] Beyerlein | Role of twinning on texture evolution of silver during equal channel angular extrusion[END_REF].

IV.6 Parametric studies

IV.6.1 Effect of relative slip/twinning activities

In this section, we study the influence of slip/twinning activities on the FCC shear textures. The relative slip/twinning activities are controlled by the  value

( slip 0 tw 0    
) as in chapter II. We note that, in the previous sections of this chapter, the CRSS of slip and twinning are set to be equal ( 1   ).

Here we carry out simulations with a linear hardening and the  values are chosen to be 0.8, 1.0 and 1.2. All other parameters, including the twinning parameters, are set to be the same than the previous simulations of this chapter. In Fig. 4

IV.6.2 Volume effects of deformation twinning

The volume effects of twinning on the FCC shear will be investigated in this section.

The voce hardening parameters of silver (see Tab. 2-1 in chapter II) are used here to relate the predicted results with experiment. The twin volume fraction will be controlled by the characteristic twin shear parameter t S of the PTR scheme (Tomé et al., 1991). As shown in chapter II, the twin volume fraction can be suppressed by increasing this parameter. The predicted twin volume fraction are presented in Fig . By increasing the t S value from 0.707 (suggested value for cubic materials, e.g. Beyerlein et al., 2011;[START_REF] Prakash | Twinning models in self-consistent texture simulations of TWIP steels[END_REF] to 2.5, the predicted twin volume fraction is reduced from over 35% to around 15%. Higher  value will leads to lower twin volume fraction. . We notice that, for both  values, the results for 5 . 2 S t  (lowest twin volume fraction) fit the best with the experimental results of silver [START_REF] Beyerlein | Role of twinning on texture evolution of silver during equal channel angular extrusion[END_REF]

IV.7 Conclusion

In this chapter, we have extended the validation of the visco-plastic  -model under shear test up to large strains by comparing predicted results for texture evolution to the experimental ones from the literature.

In the first part of the chapter, by considering only crystallographic slip and by varying  , we showed that the  -model predictions are consistent with experimental shear textures of high/medium to low SFE FCC metals. As in the case of plane strain compression (or rolling) test where a texture transition is obtained from the copper-type to the brass-type, a shear texture transition is also predicted by varying  between the values 0 and 1.

In the second part of the chapter, in order to improve the predicted results for low SFE metals, we added the twinning mechanism to the crystallographic slip. We also observed a shear texture transition by varying  and show that our predicted results In this work, we showed that in order to predict the shear texture evolution of low SFE metals such as silver, twinning was needed to accommodate plastic deformation.

By comparison of our results, as function of the parameter  , to the experimental shear textures of both medium and low SFE FCC metals, the predicted texture-transitions suggest that the parameter  should be correlated to the SFE of the polycrystalline material. Here again, we can roughly assume that a high value of  is adequate to low SFE and that a low value for  is adequate for to high/medium SFE.

We also proposed the parametric studies by varying the slip/twinning activities (controlled by  value) and twin volume fraction (controlled by t S value). The effects of those parameters on the shear texture evolution are analyzed. We have shown that when the real hardening parameters of low SFE metal are used, the best fitted results with experiment could be found when the high  value is used and the twin volume fraction is reduced to reasonable values.

Chapter V. Application to BCC metals: large plastic deformation behavior and anisotropy evolution in cold rolled steels V.1 Introduction

The simulation of large plastic deformation behavior of BCC metals have been studied using different approaches. The Taylor type models (including full constraint and "Relaxed constrained" models) have been mostly used for the simulation of the behavior of BCC metals such as low carbon steels (Hölsher et al., 1991;[START_REF] Liao | Texture development and plastic anisotropy of B.C.C. strain hardening sheet metals[END_REF]Raabe, 1995a,b;[START_REF] Raabe | Texture and microstructure of hot rolled steel[END_REF][START_REF] Raphanel | Simulation of the rolling textures of b.c.c. metals by means of the relaxed taylor theory[END_REF], ferritic stainless steels [START_REF] Bate | Texture development in the cold rolling of IF steel[END_REF]Hölsher et al., 1991Hölsher et al., , 1994;;[START_REF] Raabe | Textures of ferritic stainless steels[END_REF], Interstitial-Free (IF) steel [START_REF] Bate | Texture development in the cold rolling of IF steel[END_REF], iron aluminides [START_REF] Raabe | Taylor simulation and experimental investigation of rolling textures of polycrystalline iron aluminides with special regard to slip on {112} planes[END_REF] and tantalum [START_REF] Lee | Modeling the mechanical behavior of tantalum[END_REF]. In other few existing works, the Sachs type models have also been applied to BCC metals such as martensite [START_REF] Raabe | Texture and microstructure evolution during cold rolling of a strip cast and of a hot rolled austenitic stainless steel[END_REF]. Raabe (1995c) explained the appropriate choice of the Sachs model for the martensite because of the low stacking fault energy of this phase.

The self-consistent approaches have been also applied to BCC metals such as low carbon steels [START_REF] Paquin | Micromechanical modeling of the elastic-viscoplastic behavior of polycrystalline steels having different microstructures[END_REF], duplex stainless steel [START_REF] Jia | Micromechanical behavior and texture evolution of duplex stainless steel studied by neutron diffraction and self-consistent modeling[END_REF] and IF-steel [START_REF] Delannay | Influence of grain shape on the planar anisotropy of rolled steel sheets -evaluation of three models[END_REF]. To account for non-homogeneous boundary conditions, the Crystal Plasticity Finite Element Methods (CPFEM) have also been applied to BCC metals such as low carbon steels (Raabe et al., 2005a,b), ferritic steel [START_REF] Tikhovskiy | Simulation of the deformation texture of a 17%Cr ferritic stainless steel using the texture component crystal plasticity finite element method considering texture gradients[END_REF][START_REF] Tikhovskiy | Simulation of earing of a 17% Cr stainless steel considering texture gradients[END_REF] and IF-steel [START_REF] Bate | Texture development in the cold rolling of IF steel[END_REF][START_REF] Delannay | Influence of grain shape on the planar anisotropy of rolled steel sheets -evaluation of three models[END_REF]). An extended review on CPFEM models is given by Roters et al. [START_REF] Roters | Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finite-element modeling: Theory, experiments, applications[END_REF].

In this chapter, we consider either the restricted glide ({110}<111>) or the pencil glide ({110}<111>+{112}<111>+{123}<111>) as the slip modes [START_REF] Becker | Pencil glide formulation for polycrystal modeling[END_REF][START_REF] Liao | Texture development and plastic anisotropy of B.C.C. strain hardening sheet metals[END_REF]Raabe, 2005a,b;[START_REF] Raabe | Textures of ferritic stainless steels[END_REF]. We also show that the bi-plane slip mode ({110}<111>+{112}<111>) yields results similar to the pencil glide case. In this chapter, as in the work of Raabe (1995a), we do not consider the twinning mechanism. We should note that the assumption used here for pencil glide is more restrictive than the real {hkl}<111> pencil glide.

The evolution of texture in BCC metals under cold rolling boundary conditions is simulated using the VPSC model (Molinari et al., 1987) and the intermediate  -model (Ahzi and M"Guil, 2008). We also present the results of the predicted yield Moreover, we present a comparison of our predicted texture results with various experimental BCC cold rolling textures, taken from the literature, for the electrical steels FeCr/FeSi (e.g. Fe3%Si) [START_REF] Hölscher | Rolling and recrystallization textures of bcc steels[END_REF], ferritic steels (e.g. Fe16%Cr and Fe11%Cr) [START_REF] Raabe | Textures of ferritic stainless steels[END_REF], IF-steels [START_REF] Bate | Texture development in the cold rolling of IF steel[END_REF][START_REF] Nicaise | Coupled effects of grain size distributions and crystallographic textures on the plastic behaviour of IF steels[END_REF][START_REF] Wang | Texture comparison of an ordinary IF steel and a high-strength IF steel under ferritic rolling and high-temperature coiling[END_REF] and low carbon steel (Raabe, 2005b). The comparison study shows that the  -model predictions are in accord with the experimental results for BCC metals. In particular, the evolution of the  -fiber is predicted during the cold rolling process as function of the parameter  , which also means for different interaction strengths (from a stiff to a more compliant interaction).

We also analyzed the evolution of the  -fiber as function of the rolling reduction ratio and number of available slip systems (restricted slip systems versus pencil glide).

We show that the  -model could predict a very large and sensitive range of cold rolling texture transitions as well as the corresponding yield loci. Lastly, we show that the pencil glide assumption yields predicted results that are in a better agreement with experimental results in the literature for low carbon and IF steels. On the other hand, the restricted slip assumption is well suited for ferritic and electrical steels. The results

shown in this chapter can be found in the work of M" Guil et al. (2011b).

V.2

Application and results for plane strain compression (cold rolling)

V.2.1 Slip systems for BCC metals

For the BCC crystallographic structure, the slip mainly occurs on the <111> direction.

In this work, we considered the restricted {110}<111>, the bi-plane {110}<111>+{112}<111>, or the pencil glide {110}<111>+{112}<111>+{123}<111> slip modes [START_REF] Liao | Texture development and plastic anisotropy of B.C.C. strain hardening sheet metals[END_REF]Raabe, 1995a,b;[START_REF] Tikhovskiy | Simulation of earing of a 17% Cr stainless steel considering texture gradients[END_REF]. Since using two ({110}<111>+{112}<111>) or three slip modes (pencil glide) give very similar results, we do not present results obtained with two slip modes. Consequently, we only present results for the restricted slip and pencil glide. As in the work of other authors [START_REF] Bate | Texture development in the cold rolling of IF steel[END_REF]Raabe, 2005a), we assumed the same initial critical resolved shear stress for all of these slip families.

V.2.2 Ideal cold rolling texture for BCC metals

The cold rolling texture of BCC metals could be described in terms of fibers. Some important fibers and ideal rolling texture components for BCC metals are presented in Tab. 5-1. In general, for BCC metals under cold rolling, the typical crystallographic texture components include the fibers  and  . Fig. 5-1 and Tab. 5-2 show the corresponding location of these two fibers and other ideal components in the Euler space. In this study, we particularly focused on the  -fiber (crystallographic fiber axis <110> parallel to the rolling direction including the major components:

{001}<110>, {112}<110> and {111}<110>) and the  -fiber (crystallographic fiber axis <111> parallel to the normal direction including the major components {111}<110> and {111}<112>).

Fiber name Fiber axis Important texture components

 -fiber <110> parallel to RD {001}<110>, {112}<110>, {111}<110>  -fiber <111> parallel to ND {111}<110>, {111}<112>  -fiber <001> parallel to RD {001}<100>, {011}<100>  -fiber <011> parallel to ND {011}<100>, {011}<211> {011}<111>, {011}<011>  -fiber <011> parallel to TD {001}<110>, {112}<111>, {4411}<11118>, {111}<112>, {11118}<4411>, {011}<100>  -fiber <001> parallel to ND {001}<100>, {001}<110>  -skeleton line  <111> close to ND {111}<110>, {557}<583>, {111}<112>
Tab. 5-1. Some important fibers and texture components for crystallographic textures of BCC alloys (RD : rolling direction, ND : normal direction and TD : transverse direction) [START_REF] Engler | A study of through-thickness texture gradients in rolled sheets Metall[END_REF] Fig. 5-1. Description of  -fiber and  -fiber [START_REF] Engler | A study of through-thickness texture gradients in rolled sheets Metall[END_REF])

  45 2  1   0 90°9 0°f iber   fiber   I G E 1 F 1 E 2 F 2 H C H   45 2  1   0 90°9 0°f iber   fiber   I G E 1 F 1 E 2 F 2 H C H H C H Components {hkl}<uvw> 1   2  C {001}<100> 45° 0° 45° H {001}<110> 0° 0° 45° 1 E {111}<110> 0° 55° 45° 2 E {111}<110> 60° 55° 45° 1 F {111}< 2 11 > 30° 55° 45° 2 F {111}< 2 11 > 90° 55° 45° I {112}< 10 1 > 0° 35° 45° G {110}<001> 90° 90° 45°
Tab. 5-2. Ideal rolling component for BCC metals

V.2.3 Parameters of the simulations

We consider an initial random texture represented by 500 orientations. We simulated plane strain compression test up to a total plastic strain % 100   corresponding to 63% cold rolling reduction. These simulations were conducted without strain hardening. However, the global conclusions do not change when strain hardening is used since the results follow a similar trend with or without hardening. The inverse strain rate sensitivity coefficient for the crystallographic slip is taken as To quantitatively address the differences in predicted textures from these models, we plot the texture results in terms of orientation distribution function (ODF) sections (see Fig. 5-5).

11 n  .
For both restricted and pencil glide, Fig. 5-5 shows the ODF sections for ). On these figures, one can observe the development of the two fibers  and  which correspond, more or less, to the ideal case. One may also note that the number of available slip systems (restricted or pencil) influences the evolution of these fibers. We can also see that the 5-8d and 5-9d). The predictions from the upper and lower bound models are also presented in Fig. 5-6 as well as those from the tangent VPSC model (Fig. 567).

For all models, the results show that the  -fiber increases continuously with strain. This result is in accord with the results of Raabe (1995b). These figures also show differences in the predictions of the evolution of the  -fiber as function of the considered slip systems (restricted versus pencil) for all models. However, these differences are more pronounced for the  -model and the bounds. These differences are expressed in terms of intensity, number of "peeks" and location (i.e. shift along the  axis) of the  -fiber. We note that the particular orientation {001}<110> (H component) increases drastically under restricted slip for all models except the tangent VPSC. The relative insensitivity of the tangent VPSC model to the number of available slip systems is in accord with previous studies such as those of Raabe (1995b). Thus, as for FCC metals (Ahzi and M"Guil, 2008;M"Guil et al., 2009M"Guil et al., , 2011a)), the  -model is very sensitive and may be more appropriate to predict texture transition in BCC metals. 5678. Simulated evolution of BCC cold rolling texture (  -fiber) using restricted slip for  -model: 56789. Simulated evolution of BCC cold rolling texture (  -fiber) using pencil glide for  -model: For all models, the case of pencil glide increases the intensity of the  -fiber. For example, the intensity increases from about 5.3 (for restricted glide) to about 11.36 (for pencil glide) for 1 . 0   (see Figs. 5-8a and 5-9a) and from about 5.94 to about 12.51 for 7 . 0   (see . As expected, the intensity of the  -fiber, obtained with the tangent VPSC model, is less sensitive to the number of available slip systems: from about 10.91 to about 13.42 in intensity (see Fig. 567).

2 4 0°30°60°90°2 0% 30% 40% 50% 60% 70% 80% 100% f(g)     45 0 2 1    {001} <110> {112} {111} <110> <110> {110} <110> 0 2 4 0°30°60°90°2 0% 30% 40% 50% 60% 70% 80% 100% f(g)     45 0 2 1    {001} <110> {112} {111} <110> <110> {110} <110> 0 2 4 0°30°60°90°2 0% 30% 40% 50% 60% 70% 80% 100% 20% 30% 40% 50% 60% 70% 80% 100% f(g)     45 0 2 1    {001} <110> {112} {111} <110> <110> {110} <110> 0 3.5 7 0°30°60°90°2 0% 30% 40% 50% 60% 70% 80% 100% f(g)     45 0 2 1    {001} <110> {112} {111} <110> <110> {110} <110> 0 3.5 7 0°30°60°90°2 0% 30% 40% 50% 60% 70% 80% 100% 20% 30% 40% 50% 60% 70% 80% 100% f(g)     45 0 2 1    {001} <110> {112} {111} <110> <110> {110} <110> 0 4 8 0°30°60°90°2 0% 30% 40% 50% 60% 70% 80% 100% f(g)     45 0 2 1    {001} <110> {112} {111} <110> <110> {110} <110> 0 4 8 0°30°60°90°2 0% 30% 40% 50% 60% 70% 80% 100% f(g)     45 0 2 1    {001} <110> {112} {111} <110> <110> {110} <110> 0 4 8 0°30°60°90°2 0% 30% 40% 50% 60% 70% 80% 100% 20% 30% 40% 50% 60% 70% 80% 100% f(g)     45 0 2 1    {001} <110> {112} {111} <110> <110> {110} <110> 0 4 8 0°30°60°90°2 0% 30% 40% 50% 60% 70% 80% 100% f(g)     45 0 2 1    {001} <110> {112} {111} <110> <110> {110} <110> 0 4 8 0°30°60°90°2 0% 30% 40% 50% 60% 70% 80% 100% 20% 30% 40% 50% 60% 70% 80% 100% f(g)     45 0 2 1    {001} <110> {112} {111} <110> <110> {110} <110> a)
0°30°60°90°2 0% 30% 40% 50% 60% 70% 80% 100% f(g)     45 0 2 1    {001} <110> {112} {111} <110> <110> {110} <110> 0 4 8 0°30°60°90°2 0% 30% 40% 50% 60% 70% 80% 100% f(g)     45 0 2 1    {001} <110> {112} {111} <110> <110> {110} <110> 0 4 8 0°30°60°90°2 0% 30% 40% 50% 60% 70% 80% 100% 20% 30% 40% 50% 60% 70% 80% 100% f(g)     45 0 2 1    {001} <110> {112} {111} <110> <110> {110} <110> 0 5 10 0°30°60°90°2 0% 30% 40% 50% 60% 70% 80% 100% f(g)     45 0 2 1    {001} <110> {112} {111} <110> <110> {110} <110> 0 5 10 0°30°60°90°2 0% 30% 40% 50% 60% 70% 80% 100% 20% 30% 40% 50% 60% 70% 80% 100% f(g)     45 0 2 1    {001} <110> {112} {111} <110> <110> {110} <110> a) 1 . 0   b) 2 . 0   c) 5 . 0   d) 7 . 0   Fig.
1 . 0   (a), 2 . 0   (b), 5 . 0   (c) and 7 . 0   (d) 0 2 4 0°30°60°90°2 0% 30% 40% 50% 60% 70% 80% 100% f(g)     45 0 2 1    {001} <110> {112} {111} <110> <110> {110} <110> 0 2 4 0°30°60°90°2 0% 30% 40% 50% 60% 70% 80% 100% 20% 30% 40% 50% 60% 70% 80% 100% f(g)     45 0 2 1    {001} <110> {112} {111} <110> <110> {110} <110> 0 2 4 0°30°60°90°2 0% 30% 40% 50% 60% 70% 100% f(g)     45 0 2 1    {001} <110> {112} {111} <110> <110> {110} <110> 0 2 4 0°30°60°90°2 0% 30% 40% 50% 60% 70% 100% f(g)     45 0 2 1    {001} <110> {112} {111} <110> <110> {110} <110> 0 2 4 0°30°60°90°2 0% 30% 40% 50% 60% 70% 100% 20% 30% 40% 50% 60% 70% 100% f(g)     45 0 2 1    {001} <110> {112} {111} <110> <110> {110} <110> 0 2 4 0°30°60°90°2 0% 30% 40% 50% 60% 70% 80% 100% f(g)     45 0 2 1    {001} <110> {112} {111} <110> <110> {110} <110> 0 2 4 0°30°60°90°2 0% 30% 40% 50% 60% 70% 80% 100% f(g)     45 0 2 1    {001} <110> {112} {111} <110> <110> {110} <110> 0 2 4 0°30°60°90°2 0% 30% 40% 50% 60% 70% 80% 100% 20% 30% 40% 50% 60% 70% 80% 100% f(g)     45 0 2 1    {001} <110> {112} {111} <110> <110> {110} <110> 0 3 6 0°30°60°90°2 0% 30% 40% 50% 60% 70% 80% 100% f(g)     45 0 2 1    {001} <110> {112} {111} <110> <110> {110} <110> 0 3 6 0°30°60°90°2 0% 30% 40% 50% 60% 70% 80% 100% 20% 30% 40% 50% 60% 70% 80% 100% f(g)     45 0 2 1    {001} <110> {112} {111} <110> <110> {110} <110> a) 1 . 0   b) 2 . 0   c) 5 . 0   d) 7 . 0   Fig.
1 . 0   (a), 2 . 0   (b), 5 . 0   (c) and 7 . 0   (d) 0 4 8 0°30°60°90°f (g)     45 0 2 1    {001} <110> {112} {111} <110> <110> {110} <110> 20% 30% 40% 50% 60% 70% 80% 90% 100% 0 4 8 0°30°60°90°f (g)     45 0 2 1    {001} <110> {112} {111} <110> <110> {110} <110>
   {001} <110> {112} {111} <110> <110> {110} <110> {110}<111> {112}<111> {123}<111> 0 9 18 0°30°60°90°f (g)     45 0 2 1    {001} <110> {112} {111} <110> <110> {110} <110> {110}<111> {112}<111> {123}<111> 0 9 18 0°30°60°90°f (g)     45 0 2 1    {001} <110> {112} {111} <110> <110> {110} <110> 0 9 18 0°30°60°90°f (g)     45 0 2 1    {001} <110> {112} {111} <110> <110> {110} <110> {110}<111> {112}<111> {123}<111> {110}<111> {112}<111> {123}<111> 0 9 18 0°30°60°90°f (g)     45 0 2 1    {001} <110> {112} {111} <110> <110> {110} <110> {110}<111> {112}<111> {123}<111> 0 9 18 0°30°60°90°f (g)     45 0 2 1    {001} <110> {112} {111} <110> <110> {110} <110> {110}<111> {112}<111> {123}<111> 0 9 18 0°30°60°90°f (g)     45 0 2 1    {001} <110> {112} {111} <110> <110> {110} <110> 0 9 18 0°30°60°90°f (g)     45 0 2 1    {001} <110> {112} {111} <110> <110> {110} <110> {110}<111> {112}<111> {123}<111> {110}<111> {112}<111> {123}<111> 0 9 18 0°30°60°90°f (g)     45 0 2 1    {001} <110> {112} {111} <110> <110> {110} <110> {110}<111> {112}<111> {123}<111> 0 9 18 0°30°60°90°f (g)     45 0 2 1    {001} <110> {112} {111} <110> <110> {110} <110> {110}<111> {112}<111> {123}<111> 0 9 18 0°30°60°90°f (g)     45 0 2 1    {001} <110> {112} {111} <110> <110> {110} <110> 0 9 18 0°30°60°90°f (g)     45 0 2 1    {001} <110> {112} {111} <110> <110> {110} <110> {110}<111> {112}<111> {123}<111> {110}<111> {112}<111> {123}<111>
simulations after 100% equivalent strain, under plane strain compression, and probing the polycrystalline aggregate by different loading directions, accounting for the actual texture at such deformation. In the case of restricted slip, Fig. 5 . In these figures, the anisotropy decreases in the case of pencil glide. This is in agreement with other results in the literature [START_REF] Liao | Texture development and plastic anisotropy of B.C.C. strain hardening sheet metals[END_REF]. [START_REF] Liao | Texture development and plastic anisotropy of B.C.C. strain hardening sheet metals[END_REF]

V.2.4.3 Slip activity

In order to compare the slip activity, we have represented the variation of the number of active slip systems (based on a threshold shear rate) as a function of the equivalent plastic strain for both restricted slip and pencil glide (see . Similar slip activities are reported on these figures for  -model (Fig. 5-14a) and the VPSC (Fig. 5-14b). The slip activity predicted by the VPSC approach shows a decrease of the number of slip systems as eff n increases. For the  -model, the slip activity decreases as we span from small to large values of  . This leads to higher anisotropy (due to lower slip activity) which is observed on the corresponding yield loci as well (see Figs. ). The contribution of the {123}<111> is quite important compared to those of the {110}<111> and {112}<111> slip modes. When the three slip modes are simultaneously used, slip activity is dominated by the {123}<111> slip, since it has 24 systems, as opposed to 12 for each of the other two slip modes. The pencil glide offers an increase in kinematic freedom as was already reported by other authors [START_REF] Liao | Texture development and plastic anisotropy of B.C.C. strain hardening sheet metals[END_REF]. These previous studies showed that the <111> pencil glide increases the slip activity and thus reduces the anisotropy which can be seen on the yield loci shape and size evolution (see Fig.

V.3 Comparison to experimental results under cold rolling of BCC metals

In this section, we present a comparative study with experimental cold rolling textures (  -fiber) taken from the literature for a few BCC metals: electrical steels FeCr/FeSi (e.g. Fe3%Si) [START_REF] Hölscher | Rolling and recrystallization textures of bcc steels[END_REF], ferritic steels: Fe16%Cr and Fe11%Cr [START_REF] Raabe | Texture and microstructure of hot rolled steel[END_REF], IF-steels [START_REF] Bate | Texture development in the cold rolling of IF steel[END_REF][START_REF] Nicaise | Coupled effects of grain size distributions and crystallographic textures on the plastic behaviour of IF steels[END_REF][START_REF] Wang | Texture comparison of an ordinary IF steel and a high-strength IF steel under ferritic rolling and high-temperature coiling[END_REF] and low carbon steel (Raabe, 2005b). This literature study of the simplify our comparative analysis of texture development, we normalized the intensity of the  -fiber relative to the corresponding maximum intensity. The parameters of the simulations are the same as those described in the previous section.

V.3.1 Comparison under restricted slip assumption

In this section, we have digitalized the experimental  -fibers, taken from the literature, for ferritic steels (Fe16%Cr, Fe11%Cr) [START_REF] Raabe | Texture and microstructure of hot rolled steel[END_REF] and electrical steel FeCr/FeSi (e.g. Fe3%Si) [START_REF] Hölscher | Rolling and recrystallization textures of bcc steels[END_REF]. ) for the electrical steel (Fig. 5-16a), the first ferritic steel Fe16%Cr (Fig. 5-16b), and the second ferritic steel Fe11%Cr (Fig. 5-16c). On these figures, we can observe that the  -model predictions, under restricted slip assumption, are in a relative good agreement with the experimental  -fiber texture for these BCC metals.

In particular, the  -model predicts well the position of the maximal intensity of the  -fiber corresponding to the strongest texture component (i.e. {111}<110> orientation) in all of the three considered materials. We can also point out that the Taylor model is not able to correctly predict the exact position (on the  -axis) of this texture component of the  -fiber for these considered BCC metals (see Fig. We can conclude that for ferritic and electrical steels, the best correlation with experimental results for texture development is obtained under restricted slip assumption and for relatively small value of the parameter  (around 0.2). However, we should note that the effects of microstructure (grain size, stacking fault energy, hardening…) may require some adjustment of the value of  . For example, [START_REF] Raabe | Textures of ferritic stainless steels[END_REF] have observe that increasing the Cr content in ferritic stainless steel causes a shift of the {112}<110> orientation to higher  angles on the  -fiber.

The difference between the ferritic stainless steels is the content of Cr which also means different stacking fault energies.

V.3.2 Comparison under pencil glide assumption

In this part, we present results where the pencil glide assumption gives the best correlation with the experimental  -fiber. This is the case for IF-steels (IF-steel-1 0.5 0.5 [START_REF] Hölscher | Rolling and recrystallization textures of bcc steels[END_REF], IF-steel-2 [START_REF] Nicaise | Coupled effects of grain size distributions and crystallographic textures on the plastic behaviour of IF steels[END_REF]), IF-steel-3 (Wang et al., 2006)) and low carbon steels (Raabe, 1995b). This part is divided in two subsections: a first one where small values of  yield a good correlation with experimental results and a second one where higher values of  give a relatively good trend in comparison to experimental results.

V.3.2.1 Pencil glide assumption and small values of 

The experimental  -fibers have been taken from the literature and digitalized for IF-steel-1 at 75 . 0   [START_REF] Hölscher | Rolling and recrystallization textures of bcc steels[END_REF] and low carbon steel (Raabe, 1995b) This qualitative comparison to the experimental results, for low carbon steel and IF-steel, shows that the pencil glide case gives the best rolling texture predictions than the restricted slip mode. This is in accord with previous studies which concluded that the pencil glide-based results can fit the experimental low carbon steel rolling texture better than the restricted slip-based predictions (Raabe. 1995a,b;[START_REF] Tikhovskiy | Simulation of earing of a 17% Cr stainless steel considering texture gradients[END_REF]. The herein considered low carbon steel has small globular grains of about m 12  size (Raabe, 1995b). M" Guil et al. (2009) have hypothesized that small values of  may correspond to metals which have small grain size. The results from our current simulations, for low carbon steel, are thus in accord with this hypothesis.

To conclude, contrary to ferritic and electrical steels, the texture development in IF-steel-1 and low carbon steel is better predicted by the  -model under pencil glide 0.5 0.5 assumption. In this subsection, the best correlation is obtained with small value of  which is not the case of the next subsection.

V.3.2.2 Pencil glide assumption and large value of 

From the literature, we have found a few experimental cases for which the  -model predictions are in good accord with experimental  -fiber for higher values of  and under pencil glide assumption. Among these materials are an IF-steel-2 and IF-steel-3. Fig. 5-18 shows the comparison of our simulations to these experimental ones. We can see that the best correlation with the experimental results is obtained for higher values of  (between 0.5 and 0.7). The maximal intensity, around accord with this hypothesis. However, we must note that the differences between the IF-steels" texture may be also explained by the differences in the mechanical processing and recrystallization, especially for IF-steel-3 [START_REF] Wang | Texture comparison of an ordinary IF steel and a high-strength IF steel under ferritic rolling and high-temperature coiling[END_REF] versus IF-steel-1 [START_REF] Hölscher | Rolling and recrystallization textures of bcc steels[END_REF]. In addition, grain size effects may also contribute to these texture differences as pointed out in the work of [START_REF] Nicaise | Coupled effects of grain size distributions and crystallographic textures on the plastic behaviour of IF steels[END_REF]. To conclude this part, we can note that the  -model predictions under pencil glide assumption, for different values of  , is well suited for low carbon steel and IF-steels.

V.4 Conclusion

With the visco-plastic  -model, a large and sensitive range in the predictions of cold rolling textures of BCC metals can be obtained by simply varying the value of the parameter  . Independently of the number of the available slip systems, one can note important differences in the predictions of the  -fiber as function of the value of  .

These differences are shown in terms of the intensity of the  -fiber, number of peeks and their positions along the  -axis of this fiber.

For BCC metals, taking into account the pencil glide, one may obtain predicted results closer to the experimental ones as already reported in the literature, especially for low carbon steels and IF-steels. However, for ferritic and electrical steels, the restricted slip assumption is well suited and gives predicted results closer to the experimental textures. The  -model is thus able to predict texture transition in BCC metals as already shown for FCC metals by simply varying the value of the parameter  . Our predicted yield surfaces reflect the effect of these texture transitions in BCC metals.

Thus, the parameter  controls the shape and size of these yield loci. In fact, the parameter  scales the strength of the grain/aggregate interaction from a stiff (small values of  ) to a more compliant interaction (high values of  ).

In a future work, it is necessary to attempt to link the parameter  to the microstructure of the considered metal"s microstructure such as grain size and stacking fault energy. For instance in the particular case of BCC metals, one can try to directly link the shift of the  -fiber along the  -axis to the stacking fault energy.

One may also analyze the evolution of the intensity of various preferential orientation as function of different microstructural features (grain size, stacking fault energy, …)

for several BCC metals. This may help in developing a way for adjusting the parameter  as function of these microstructural parameters.

Chapter VI. Application to HCP metals: deformation behavior and texture evolution in magnesium alloy

VI.1 Introduction

As light weight structural materials, several hexagonal metals have been used in industries mainly for a significant reduction of weight. For example, magnesium and its alloys, with a low density, have been used as structural components in automotive, computer, communication and consumer electronic appliances (Wang and Huang, 2003). Another example is titanium and its alloys, applied in high performance engineering industries such as aerospace industry. Zirconium alloys, also hexagonal materials, are used as cladding materials in nuclear reactor fuels.

The main challenges of hexagonal metals as structural materials are the limited ductility and the poor room temperature formability, which are primarily due to the restricted number of slip systems [START_REF] Agnew | Plastic anisotropy and the role of non-basal slip in magnesium alloy AZ31[END_REF]Parks and Ahzi, 1990;Scheonfeld et al., 1995). At room temperature, hexagonal materials possess fewer easy glide systems than cubic metals, resulting in greater crystal anisotropy. However, at high temperatures and moderate strain rates, hexagonal metals can be ductile and readily formable due to lower flow resistance and the activation of addition slip systmes.

The main crystallographic slip families in hexagonal close-packed (HCP) structures are the basal a , prismatic a and pyramidal a slip systems. The first-and the second-order pyramidal a c  slip systems occur mainly at high temperature and were held responsible for the good elevated temperature ductility of HCP metals such as magnesium alloys. The a slip comprises only four independent slip systems and, thus, cannot accommodate plastic deformation in the crystallographic c direction of the single crystal. The basal a and the prismatic a planes are mutually orthogonal. However, because the 0 2 11 (or a ) slip directions are perpendicular to the c-axis and confined to the basal plane, there are only two independent slip systems of each type. Thus, basal a and prismatic a slips together possess only four independent slip systems. Therefore, HCP metals are often nearly inextensible along their c-axis. Pyramidal a c  systems, if activated, will provide the additional fifth independent slip system, necessary for the accommodation of an arbitrary plastic deformation. However, the slip resistance of pyramidal a c  slip systems are usually much higher than those of basal and prismatic systems. Even when five independent slip systems are available, the difference of the critical resolved shear stresses between different deformation mechanisms (basal, prismatic and a c  pyramidal) can be large enough to introduce an important plastic anisotropy at the single crystal level. Therefore, the mechanical behavior of HCP metals is controlled by the relative strengths and substantially different hardening responses of the various slip modes [START_REF] Francillette | Grain orientation effects in Zr702α polycrystalline samples deformed in channel die compression at room temperature[END_REF][START_REF] Fundenberger | Modelling and prediction of mechanical properties for materials with hexagonal symmetry (Zinc, Titanium and Zirconium alloys)[END_REF]. The schematic diagram of these slip and twinning systems are presented in Fig. 6-1. Fig. 6-1. Schematic diagram of slip and twinning systems in HCP metals [START_REF] Mayama | Influence of twinning deformation and lattice rotation on strength differential effect in polycrystalline pure magnesium with rolling texture[END_REF].

For many HCP crystals, the resistance to plastic shearing can vary by an order of magnitude (or even higher) from one mechanism to another [START_REF] Francillette | Grain orientation effects in Zr702α polycrystalline samples deformed in channel die compression at room temperature[END_REF][START_REF] Fundenberger | Modelling and prediction of mechanical properties for materials with hexagonal symmetry (Zinc, Titanium and Zirconium alloys)[END_REF]. A random aggregate of such crystals will deform only by their soft modes (e.g. a slip). As the deformation proceeds, the aggregate becomes highly textured (elastically stiff) and the activation of the hard modes is possible only after a high degree of alignment that leads to materials" locking, resulting into microcraking and materials failure.

It is well known that in addition to crystallographic slip from dislocation movement, HCP materials exhibit a greater tendency to mechanically twin than cubic materials.

In the absence of pyramidal a c  slip, twinning may supplement the a slip for full kinematic freedom. Twinning provides additional deformation which relaxes the requirements for five independents slip modes and may help a material to satisfy the Taylor criterion [START_REF] Agnew | Application of texture simulation to understanding mechanical behavior of Mg and solid solution alloys containing Li or Y[END_REF][START_REF] Brown | Internal strain and texture evolution during deformation twinning in magnesium[END_REF][START_REF] Kocks | The importance of twinning for the ductility of CPH polycrystals[END_REF].

However, as a polar mechanism [START_REF] Agnew | Plastic anisotropy and the role of non-basal slip in magnesium alloy AZ31[END_REF], twinning depends strongly on temperature, alloying content, stacking fault energy and crystal lattice structure. It makes the modeling of the interaction between slip and twinning complicated [START_REF] Prantil | Modeling deformation induced textures in titanium using analytical solutions for constrained single crystal response[END_REF]. At low temperature, twinning will compete with slip to accommodate the crystal motion. In warmer processing regimes, however, twinning may become less favorable deformation mechanism than slip.

Polycrystals plasticity and texture evolution of hexagonal materials are characterized by the diversity of possible deformation mechanisms such as basal slip, prismatic slip, pyramidal slip and several twinning modes. The low symmetry nature of these polycrystals may presents high anisotropy, kinematic deficiencies, and associated locking nature at high strains. The difficulty of modeling of the plastic behavior of HCP metals is mainly due to the fact that less than five independent soft modes are available owing to the hexagonal symmetry of the lattice cell. In other words, hard glide systems should be activated to accommodate any arbitrary deformation [START_REF] Kocks | The relation between polycrystalline deformation and single crystal deformation[END_REF].

Several crystal plasticity models have recently been used to simulate the mechanical deformation behavior and texture evolution of HCP metals such as the Taylor type model (e.g. Inal and Mishra, 2012), fast Fourier transforms model (e.g. Lebensohn et al., 2012) and self-consistent type model (e.g. Brown et al., 2012;Oppedal et al., 2012;Wang et al., 2012). We note that self-consistent approaches, such as the VPSC approach (Molinari et al., 1987[START_REF] Tomé | Simulation of texture development in hexagonal materials using a self-consistent approach[END_REF], are well-suited to HCP crystals since the hard systems may not be activated as long as the polycrystal is not highly textured. This model has been applied by several authors to a wide variety of hexagonal metals and alloys, such as zirconium (Lebensohn and Tomé , 1993;[START_REF] Plunkett | Anisotropic yield function of hexagonal materials taking into account texture development and anisotropic hardening[END_REF], titanium [START_REF] Lebensohn | A self-consistent approach for modelling texture development of two-phase polycrystals: Application to titanium alloys[END_REF][START_REF] Suwas | Texture evolution in commercially pure titanium after warm equal channel angular extrusion[END_REF], and magnesium [START_REF] Yi | Deformation and texture evolution in AZ31 magnesium alloy during uniaxial loading[END_REF]). In the work of M" [START_REF] Guil | Modelling of texture evolution for HCP materials[END_REF], the Sachs, the self-consistent and the Constrained-Hybrid models were utilized to predict texture evolution in a class of low symmetry crystals comprising less than five independent slip systems. In the previous work, the  -model and the VPSC model have been used

to simulate the deformation behavior of HCP metals deforming by both soft and hard slip modes (M" Guil et al., 2009). The  -model is able to reproduce similar results as the VPSC model, particularly the plastic locking of HCP metals at high strains (see Fig. 6-2). Guil et al., 2009) In the first part of this chapter, the visco-plastic  -model is used to predict the texture evolution of rolled Mg alloy sheet. [START_REF] Agnew | Application of texture simulation to understanding mechanical behavior of Mg and solid solution alloys containing Li or Y[END_REF] have proposed a simulation of rolled Mg alloy sheet using VPSC model with different relative activities of a c  pyramidal slip. We carry out similar simulation and compare our results with the ones of Agnew et al. (2011). In the second part, the  -model is used for the analysis of texture evolution in HCP polycrystals with specific application to AZ31 magnesium alloy. In this alloy, the a slip (basal and prismatic) as well as the a c  pyramidal slip and tensile twinning are considered as the mechanisms of plastic deformation [START_REF] Jain | Modeling the temperature dependent effect of twinning on the behavior of magnesium alloy AZ31B sheet[END_REF][START_REF] Proust | Modeling the effect of twinning and detwinning during strain-path changes of magnesium alloy AZ31[END_REF][START_REF] Styczynski | Cold rolling textures in AZ31 wrought magnesium alloy[END_REF][START_REF] Wang | Evaluation of self-consistent polycrystal plasticity models for magnesium alloy AZ31B sheet[END_REF]. We have applied the  -model to simulate rolling texture development using an initially isotropic polycrystal. Tensile and compressive deformation behaviors of a rolled sample are also simulated. Comparing with experimental results in the literature [START_REF] Jain | Modeling the temperature dependent effect of twinning on the behavior of magnesium alloy AZ31B sheet[END_REF][START_REF] Styczynski | Cold rolling textures in AZ31 wrought magnesium alloy[END_REF]Ulacia et al., 2010a, b;[START_REF] Wang | Evaluation of self-consistent polycrystal plasticity models for magnesium alloy AZ31B sheet[END_REF], simulation results using the -model are discussed in terms of the effect of interaction strength, via the  parameter, on texture evolution and activation of different deformation mechanisms. ), % 34 eq   .

 Simulations with the -model

We carry out a similar simulation using the  -model. The n value is set to be 11. The ).

VI.2.1.2 Simulations with prismatic slip

 Simulations with the VPSC model [START_REF] Agnew | Application of texture simulation to understanding mechanical behavior of Mg and solid solution alloys containing Li or Y[END_REF] have also proposed a similar simulation but take into account the prismatic slip. The relative CRSS of these systems are set to be

VI.2.2 Application of -model on magnesium alloy AZ31

VI.2.2.1 Input parameters

The material studied here is magnesium alloy AZ31, a hexagonal material with parameter ratio c/a=1.624. The intermediate  -model was applied to simulate the deformation behavior of AZ31 and validated with experimental results from the literature [START_REF] Jain | Modeling the temperature dependent effect of twinning on the behavior of magnesium alloy AZ31B sheet[END_REF][START_REF] Styczynski | Cold rolling textures in AZ31 wrought magnesium alloy[END_REF]Ulacia et al., 2010a, b;[START_REF] Wang | Evaluation of self-consistent polycrystal plasticity models for magnesium alloy AZ31B sheet[END_REF].

Two series of tests were carried out in this study. First, we simulated cold rolling of heat treated AZ31 sheet with initial random texture. Second, we conducted simulations of compression and tension tests, of rolled AZ31, in different directions.

The plastic deformation mechanisms in AZ31 is assumed to be composed of three kinds of slip systems : (0001)<11 2 0> basal slip, (1000)<11 2 0> prismatic slip, (11 2 2)<11 2 3> pyramidal slip and tensile twinning (10 1 2)<10 1 1>. In all the numerical simulation, the reference shear rate and rate sensitivity parameters (see Eq. ( 1)) are assumed to be the same for all the considered systems: 0   =0.001s -1 and n=19. The other voce hardening parameters are listed in Tab. 6-1. Here, the first-order a c  pyramidal slip system has a much higher critical resolved shear stress than that of basal slip and prismatic slip systems. agrees well with the experimental results. With the increase of interaction parameter  , stress strain curves drop down and deviate from experimental data. It is possible to fit the curves for all cases of interaction parameters by changing the hardening law parameters, just like it is possible to simulate the stress-strain response by different models by changing the parameters. However, we will not fit the experimental data under different interaction parameters. Instead, a parametric study will be performed to investigate the influence of the interaction parameters on deformation systems activities and texture evolution. The goal of this paper is to validate the intermediate model in simulating mechanical behavior and texture evolution of hexagonal magnesium alloys.

VI.2.2.2 Rolling tests

In the first test set, rolling of initially random AZ31 polycrystals was simulated.

Texture evolution results are listed in Fig. 6-15. The experimental (0001) and (11 2 0) pole figures of rolled sheet are shown in Fig. 6-15a [START_REF] Styczynski | Cold rolling textures in AZ31 wrought magnesium alloy[END_REF]. The simulated polycrystalline aggregate is composed of 100 single crystals with initial random texture as shown by pole figures in Fig. 6-15b. After rolling, experimental texture [START_REF] Styczynski | Cold rolling textures in AZ31 wrought magnesium alloy[END_REF] has a strong component with c-axis aligned around the normal direction (ND) (Fig. 6-15c). Texture evolution and mechanical behavior have been simulated up to 20% equivalent strain and using 0   , 0.1, 0.2, 0.5, 0.7, 0.9. The results for 0   is the same as that from Taylor model (not reported here).

Although we observed a large difference in the simulated stress-strain curves for different values of  , the corresponding textures simulated are close. There is only small movement of the split of the basal texture components as shown by the ( 0001

VI.2.2.3 Tension and compression tests on rolled Mg sheet  Tensile tests along RD

In the second set of test, tension and compression of rolled magnesium sheets along different directions were investigated. For the simulation of tensile deformation test up to 20% strain, we used the rolling texture in Fig. 6-15d as initial texture. Tensile tests were performed along RD. The predicted textures for different values of  are plotted in Fig. 6-17 in comparison with the experimental results (Ulacia et al., 2010a, b). Fig. 6-17a shows the experimental (0001) and (10 1 0) pole figures of rolled Mg sheet uniaxially stretched along RD at room temperature (Ulacia et al., 2010a, b). The deformation texture has a strong texture component with c-axis aligned along ND. (10 1 0) axis distributed along the plane of RD (also tension direction here) and transverse However, as deformation proceeds, the polycrystal becomes highly textured with the soft modes becoming less favorable for activation and therefore, hard mode start to be more activated. As shown in Fig. 6-16d, the best correlation with the experimental texture evolution is obtained for high value of  and low interaction strength.

The predicted activities of different deformation mechanism in the case of tension along RD are presented in Fig. 6-18. Similar to the rolling case, contribution from twinning is small. This is an accord with the results of [START_REF] Wang | Evaluation of self-consistent polycrystal plasticity models for magnesium alloy AZ31B sheet[END_REF], who showed that the activity of tensile twinning is relatively low and decreases with increasing strain. In addition, there is no much change in the activities during deformation for high interaction strength (low  , see Figs.

 Compression tests along TD

For the uniaxial compression of the rolled sheet, different loading directions were investigated. First simulated texture evolution during uniaxial compression along TD was compared with Jain and Agnew"s experimental results [START_REF] Jain | Modeling the temperature dependent effect of twinning on the behavior of magnesium alloy AZ31B sheet[END_REF]

 Compression tests along RD

Compression of rolled Mg sheet along RD is also simulated. Fig. 6-21 shows the predicted texture at 20% strain for various  values. These predictions are compared to the experimental results of [START_REF] Jain | Modeling the temperature dependent effect of twinning on the behavior of magnesium alloy AZ31B sheet[END_REF]. Experimental data show that the c-axis is aligned with the RD, which is also the loading direction in this case. Using

VI.3 Conclusion

In the first part of this chapter, we studied the effects of the interaction strength and the relative activities of slip and twinning systems on the texture evolution using the visco-plastic intermediate  -model. We compared our results with the predicted ones for Mg alloys under rolling tests from literature. The results show that the activation of basal slip can be promoted when the activity of pyramidal slip is depressed (high X value) or when the interaction strength is low (high  value). The highly activated basal slip can reduce the spread of basal poles towards RD and lead to the formation of the typical basal texture.

In the second part, the  -model was applied to simulate the deformation behavior and texture evolution in magnesium alloys AZ31. Different cases, including rolling of initially random textured magnesium, tension as well as compression of rolled magnesium sheets along different directions were investigated. Our predicted texture results were validated by comparing to experimental results taken from literature. The influence of different intergranular interaction strengths on the predicted results (texture evolution and deformation systems activity) was studied.

In simulating rolling of initially random textured Mg sheet, the effect of  on texture development up to 20% strain is not significant. However, for both uniaxial tension and compression of rolled sheet, the simulated texture results are highly dependent on the value of  . Predicted deformation mechanism activities provide good explanation for the texture development. For rolling, the twinning activity is very fairly low for all  values, which affects slightly the texture development as observed by the tilt of the basal texture. For tension and compression of rolled sheet, slip/twinning activity is highly depending on the value of  . As expected, tensile twinning contributes significantly to plasticity under compression and negligibly under tension. For the uniaxial compression, high values of  induce high activity of the hard modes (twinning or pyramidal). In this case, twinning is highly activated at the lower strains and decreases at large strains where it becomes compensated by a higher activity of the pyramidal slip. Based on our predicted textures, particularly for uniaxial tension and compression, it is concluded that the best correlation with the experimental results was found for low interaction strength (high value of  ).

The visco-plastic  -model is a novel model for simulating the texture evolution and other mechanical behaviors in polycrystals materials during large deformation process.

This model provides the novelty on the formulation and is able to predict large scale results from a stiff interaction to a more compliant interaction by varying the interaction strength parameter  .

In this work, the  -model has been applied on metals with different lattice structures and under different loading conditions. The mechanical twinning has been implemented into the  -model. The coupled effect of the mechanical twinning and the interaction strength (controlled by the interaction strength parameter  ) has been deeply studied. We also used this model to predict the mechanical behaviors including the texture evolution on BCC and HCP metals and compared the predicted results with the experiments. The results show that the  -model is well suitable to predict the large deformation response of metals with different properties. 1. The modeling of other deformation mechanisms in polycrystals, beside crystallographic slip and twinning, is an active topic of research. The -model may be adopted to take into account those mechanisms such as the twin barrier effect and shear banding. Such a work will optimize the simulation capability of -model 2. In this thesis, the possible physics background of the -model has been briefly mentioned. We suggest that the  parameter should be linked with the SFE and grain size. However, such a suggestion needs to be quantitatively validated. Some experimental work should be carried out on this point.

3. The -model may be extended and applied on the multi-phase metals such as the dual-phase steel and the silver-copper cast eutectic nanocomposite. We suggest carrying out some experimental works on multi-phase metals in order to investigate the effect of the high densities of the interfaces on the microstructure evolutions. The predicted results of the -model should be compared with the experimental ones. This work may lead to a better understanding of the physics background of the -model.

4. In the previous works, only the velocity gradient can be imposed in the -model as the boundary condition. We suggest that the stress imposed boundary condition should be implemented into the -model. This work will enable the -model to predict the Lankford value.

5. The -model can be embedded into a FEM code to simulate complex metal forming processes which naturally involve non-homogeneous boundary conditions. The texture evolution will be known at each deformation step which will help to optimize the mechanical processes.

Fig

  Fig. R-1. Sché ma de l'é volution de l'orientation des grains lors des essais de laminage

Fig

  Fig. R-2. Courbes contrainte-dé formation pré dites pour diffé rentes valeurs de  lors d"un

  Fig. R-3. Exemples de texture de type-cuivre et type-laiton sous forme de (111) pole figure.

Fig

  Fig. R-4. Textures pré dites de laminage pour les mé taux CC (fibre  ) et comparaison avec

Fig.R- 5 .

 5 Fig.R-5. Les ré sultats sont comparé s avec le modè le VPSC. L'effet du glissement pyramidal

  Fig. R-5. Effect du glissement pyramidal sur les textures de laminage pour les mé taux HC

  Fig. 1-1. Representation of space lattice and unit cell in grains

Fig

  Fig. 1-3. Schematic representation of crystallographic twinning.

Fig. 1

 1 Fig. 1-5. Coordinate system to identify the crystallographic planes and directions in

  determined by either single-crystal experiments or fitting to a known polycrystals response. The acc V is the sum of the associated volume fraction over all twinning systems and over all grains:

Fig

  Fig. 1-8. Representation of the three Euler angles

  tensor S . Therefore, the Taylor model (upper bound approximation) should be expressed by the uniform plastic strain rate ( D D  ), whereas the Static model (lower bound approximation) should be expressed by the uniform deviatoric Cauchy stress (

  used the Taylor model (the strain rate of each cluster is equal to the corresponding macroscopic ones) and Static model (the deviatoric stress of each cluster is equal to the macroscopic one) to calculate the local fields of the clusters. The strain rate, spin and the deviatoric stress of each grain can be calculated from the local fields of the clusters considering the interface continuous condition and the global equilibrium and compatibility condition (volume average of the local fields are equal to the corresponding macroscopic ones)

  Lebensohn, R.A., Tomé , C.N., 1993. A self-consistent anisotropic approach for the simulation of plastic deformation and texture development of polycrystals: application to zirconium alloys. ActaMetall. Mater. 41, 2611-2624. Lebensohn, R.A., Tomé , C.N., Castañeda, P.P., 2007. Self-consistent modelling of the mechanical behaviour of viscoplastic polycrystals incorporating intragranular field fluctuations. Philosophical Magazine. 87, 4287-4322. Lee, B.J., Ahzi, S., Parks, D.M., 2002. Bicrystal-based modeling of plasticity in FCC metals. J. Eng. Mater. Tech. 124, 27-40.Leffers, T., Ray, R.K., 2009. The brass-type texture and its deviation from the copper-type texture.Prog. Mater.Sci. 54, 351-396. Lin, T.H., 1957. Analysis of elastic and plastic strains of a face-centred cubic crystal. J.Mech. Phys. Solids. 5,[143][144][145][146][147][148][149]

  . Tuning a self consistent viscoplastic model by finite elements reultats-II. Application to torsion textures. Acta Metall. Mater. 42, 2459-2466. Van Houtte, P., 1978. Simulation of the rolling and shear texture of brass by the Taylor theory adapted for mechanical twinning. Acta Metall. 26, 591-604. Van Houtte, P., 1988. A comprehensive mathematical formulation of an extended Taylor-Bishop-Hill model featuring relaxed constraints, the Renouard-Wintenberger theory and a strain rate sensitivity model. Textures Microstruct. 8-9, 313-350. Van Houtte, P., Delannay, L., Kalidindi, S.R., 2002. Comparison of two grain interaction models for polycrystal plasticity and deformation texture prediction. Int. J. Plasticity 18, 359-377. Van Houtte, P., Li, S., Seefeldt, M., Delannay, L., 2005. Deformation texture prediction: from the Taylor model to the advanced Lamel model. Int. J. Plasticity 21, 589-624.

Fig. 2

 2 Fig. 2-1. General calculation algorithm of the  -model.

  Fig. 2-2. Predicted strain-stress curves for different  values under tensile test in FCC metals(Ahzi and M"Guil, 2008). The curves from higher to lower correspond to  values from 0 to 1.

Fig. 2

 2 Fig. 2-3. FCC rolling texture transition predicted by  -model representing in

Fig. 2

 2 Fig. 2-5. Deviation of the norm of the local strain-rate under plane strain compression

Fig. 2

 2 Fig. 2-6. Deviation of the norm of the local stress under plane strain compression test

  7 and 0.99. The evolutions of the five components on the trace of the 5x5 matrixes are presented in Fig.2-7 to Fig.2-9for the n value of 3, 5 and 9. For the results of n=9, we present the inverse of K tensor

Fig. 2

 2 Fig. 2-7. Evolution of the interaction tensor of  -model for n=3.

  the  -model (Fig.2-10 to Fig.2-12).

Fig. 2 -

 2 Fig. 2-10. Evolution of the interaction tensor of the VPSC model for n=3.

Fig. 2 -

 2 Fig. 2-12. Evolution of the interaction tensor of the VPSC model for n=9.

  this work, both MA and PSA definitions have been considered into the visco-plastic  -model. The textures calculated by the PSA definition are compared with the ones of MA under plane strain compression tests (approximation of the rolling). The results are presented in terms of pole figures, Orientation Distribution Function (ODF)sections. The ideal rolling texture components (cube (C), brass (B), goss (G), copper (Cu) and S components) are analyzed as function of increasing strain. The coupled effect of the plastic deformation mechanisms (with or without twinning), the grain strength interaction (parameter  from 0 to 1) and the lattice spin definition (MA or PSA) on the FCC rolling texture transition is deeply studied. We note that in this work, no attempt to link the parameter  to microstructural features of the material is proposed. This parameter is merely used as a measure of interaction strength (from stiff to more compliant interaction). The results presented in this chapter can be found in the work ofWen et al. (2012b).

  Fig. 3-1. Schematic representation of the main orientations in FCC materials, ODF sections at    45 2

  have studied the rolling texture for copper and brass. Their results showed that the brass-type texture has a lower intensity of the Cu component and a higher intensity of the B component. As the Ni-Co alloys can give a texture transition from the copper-to brass-type texture by increasing the content of cobalt, Carstensen et al. (2002) have compared the evolution of textures measured from rolled Ni-30%Co (high SFE) to Ni-60%Co (low SFE). In the Ni-30%Co which develops a texture close to copper-type, the intensity of the Cu component increases gradually whereas the intensity of the B component does not show a significant change with increasing strain. Moreover, in the Ni-60%Co, the intensity of the Cu component remains at a constant level and the intensity of the B component increases linearly with increasing strain.

  figures

  Fig. 3-2. Effect of  on predicted textures in terms of (111) pole figures at

Fig. 3

 3 Fig. 3-10. Effect of  on the evolution of rolling components for MA definition

Fig. 3 -

 3 Fig. 3-11. Effect of  on the evolution of rolling components for PSA definition

  fraction of the B component begins to decrease with increasing  . Compared to the case of MA without twinning (Fig. 3-10), the volume fraction of the Cu component is strongly decreased whereas the one of B component and G component are strongly increased when twinning is accounted for. This is in agreement with the literature such as Vercammen et al. (2004) who suggested that the change in Cu and G components is due to the mechanical twinning by which the Cu component is transformed to a position close to the G component.

  Fig. 3-12. Effect of  on the evolution of rolling components for MA definition with

  Fig. 3-14. Effect of  value on slip/twinning activities for 0   (a), 1 . 0   (b),

  Fig. 3-15. Effect of  value on slip/twinning activities for 0   (a), 1 . 0   (b),

  Voce hardening parameters used to evolve the critical resolved shear stresses of the slip and twinning systems for sliver(Beyerlein et al., 2011) The effect of t S on FCC rolling components for different  are presented in Fig.3-25 toFig. 3-32. Generally, lower twin volume fraction (higher t S value) leads to stronger Cu and S components and weaker B and G components for all  values and for both MA and PSA definitions. The tendencies of texture components evolution are in agreement with Vercammen et al. (2004) who quoted that the twinning eagers to rotate the grains around Cu component to a position near the G component. For MA definition, the brass-type texture (B component stronger than Cu component) can be only obtained for high twin volume fraction. On the other hand, the brass-type texture can be obtained with PSA for high  values when twin volume fraction is low.

Fig. 3 -

 3 Fig. 3-24. Effect of S t on the twinning volume fraction for different  values and different lattice spin definitions (MA and PSA).

Fig. 3 -

 3 Fig. 3-25. Effect of S t on the FCC rolling components (B, Cu, S and G) for 0   ,

Fig. 3 -

 3 Fig. 3-27. Effect of S t on the FCC rolling components (B, Cu, S and G) for 5 . 0   ,

Fig. 3 -

 3 Fig. 3-29. Effect of S t on the FCC rolling components (B, Cu, S and G) for 0   ,

Fig. 3 -

 3 Fig. 3-31. Effect of S t on the FCC rolling components (B, Cu, S and G) for 5 . 0   ,

  show a significant difference between the MA and PSA definitions. The deviation between the predict texture by MA and PSA definition increases with increasing  values. At high interaction strength (low  values, definitions can be used to predict a copper-type texture if only the {111}<110> slip is considered. The brass-type texture can be obtained by either using PSA definition without twinning at low interaction strength (high 

  Fig. 3-33. Evolution of rolling components B, Cu and S for 70/30 brass and copper

  Fig. 4-1. Ideal orientations for simple shear (a) and ECAE (b) texture components observed in the    0 2

  ), see Fig.4-1a(Li et al., 2005b). This ODF representation allows for showing each ideal shear components separately. The corresponding Euler angles (1 ,

Fig. 4 -

 4 Fig. 4-1a and in Tab. 4-1, the partial " A " fiber contains the A / A , * 1 A and * 2 A

  Fig. 4-3 represents the measured textures at shear strains of 2   and 5 . 5   in

  Fig. 4-3. Experimental shear textures of OFHC copper at shear deformations of 2   (a) and 5 . 5   (b)[START_REF] Tóth | Tuning a self consistent viscoplastic model by finite element results-II. Application to torsion textures[END_REF]

  least likely to twin, and thus are the relatively more stable components and consequently remain the strongest components during shear deformation. On the other hand,[START_REF] Beyerlein | Role of twinning on texture evolution of silver during equal channel angular extrusion[END_REF] showed that the  C and * 1A  components are induced primarily by slip and weaken due to twinning activation.In contrast, in copper under ECAE deformation, the [START_REF] Beyerlein | Role of twinning on texture evolution of silver during equal channel angular extrusion[END_REF].As a consequence, the distinctive features of texture in silver (from those of copper)

.

  The first value of 0.005 yields results that significantly deviate from the Taylor upper bound results even though one may think the opposite.

  Fig. 4-4. Simulated shear textures obtained by the  -model: 005 . 0   (a) and



  Fig. 4-6a) are close to the secant VPSC model (see Fig. 4-5a) as well as  -model

  Fig. 4-6. Simulated shear textures obtained by the Taylor (a) and static (b) models

  difference in texture predictions (intensities and position of ideal shear components) is observed with different hardening assumptions; see Fig. 4-4 (with hardening) and Fig. 4-7a (without hardening) and Fig. 4-7b (with a linear hardening).

  Fig. 4-7. Simulated shear textures obtained by the  -model without hardening (a)

Fig

  Fig. 4-8. Simulated shear textures without (a-b) and with (b-d) twinning

  Fig. 4-10, the shift of the components * 1 A , C and * 2 A from the ideal position is

  Fig. 4-13. Evolution of the shear texture components: B / B , in function of the parameter  , without (a) and with twinning (b)

  -13b), we do not obtain any shift from ideal orientations for these B / B components. For the tangent VPSC model, the simulations, without hardening, show that both the components C and

  Fig. 4-14. Effect of  value on slip/twinning activities for 2 . 0   .

Fig

  Fig. 4-18. Effect of S t on the volume fraction for 2 . 0   and 8 . 0  

Fig

  Fig. 4-19. Effect of t S value on the shear texture * 1 A , C and * 2 A components for

(

  with twinning) are consistent with related results from literature and particularly for low SFE metals such as silver. The effect of twinning on the ideal shear texture components is clearly shown for silver. The tendencies of the evolution of the ideal shear texture component ( or without twinning, are in accord with experimental results in the literature.

  loci of rolled BCC metals. The visco-plastic Taylor model and the visco-plastic Static model are also used to define the upper and lower bound estimates of the results. The results of the VPSC model are also presented for comparison with those obtained with the intermediate  -model. The results are compared and discussed in terms of predicted textures, slip activity and yield loci.

  For

  the visco-plastic  -model, we choose several values for  :  (tangent formulation). The results obtained by the Taylor and Static models are also shown for comparison.

  tangent VPSC model and by the visco-plastic intermediate 

  Fig. 5-5. ODF"s sections for    45 2

  Fig. 5-6. Simulated evolution of BCC cold rolling texture (  -fiber) using restricted

  Fig. 5-7. Simulated evolution of BCC cold rolling texture (  -fiber) using restricted slip (a) and pencil glide (b) for tangent VPSC model

Fig. 5 -

 5 Fig. 5-10 shows the  -fiber and  -fiber at % 100   for the Taylor, Static, 

  Fig. 5-10. Simulated evolution of BCC cold rolling texture (  -fiber and  -fiber) for  -model ( 1 . 0   , 7 . 0  ) and tangent VPSC model using restricted slip (a) and

  Fig. 5-12. Yield loci for restricted slip at % 100   for  -model (a) and

  Fig. 5-13. Yield loci for the  -model at % 100   , 1 . 0   (a) and 7 . 0   (b)

  Fig. 5-14. Slip activity for the  -model (a) and the VPSC model (b)

  Fig. 5-15. Slip activity for the  -model ( 1 . 0   )

  experimental cold rolling textures of BCC metals allowed us to divide the comparison with our predicted results into two parts. The first part presents results where a good correlation with the  -model predictions are obtained under restricted slip assumption. The second part presents results where a good correlation with the experimental results is obtained under pencil glide assumption. Since we do not have the exact input parameters such as the initial texture, test temperature, exact hardening law, our analysis may be regarded as a qualitative comparison. Moreover, we only analyze the cold rolling texture in the center of the sheets and neglect the through-thickness texture gradient of the rolled BCC steels. In this section, we focus on the analysis of the  -fiber at %

  Fig. 5-16 presents the experimental and predicted  -fibers by the upper bound Taylor model and the 

  Fig. 5-16. Comparison to experimental  -fiber for FeCr/FeSi (Hölscher et al., 1991) (a), Fe16%Cr (Raabe and Lücke 1993) (b) and Fe11%Cr (Raabe and Lücke 1993) (c)

  Fig. 5-17. Comparison to experimental  -fiber for IF-steel-1 (Bate and Quinta da Fonseca, 2004) (a) and low carbon steel (Raabe, 1995b) (b) (under pencil glide

  Fig. 5-18. Comparison to experimental  -fiber for IF-steel-2 (Nicaise et al., 2011) (a) and IF-steel-3 (Wang et al., 2006) (b) (under pencil glide assumption)

Fig. 6

 6 Fig. 6-2. Stress-Strain curves for (a) visco-plastic  -model and (b) VPSC model (M"Guil et

Fig

  Fig. 6-2. Experimental   0002 and   0 1 10 textures for pure Mg, Mg-1Y and

  predicted results for various  values are presented in Figs. 6-4 to 6-6. We can see that lower  value leads to a stronger spread of basal poles towards RD in the (0001) pole figures. The spread of basal poles can be reduced by increasing the X value. The results for high  values are close to the one predicted by[START_REF] Agnew | Application of texture simulation to understanding mechanical behavior of Mg and solid solution alloys containing Li or Y[END_REF].

  Fig. 6-4. Predicted rolling texture for X=12 ( 2 : X : 1 : : tw a c basal       

.

  Fig.6-10. By reducing X from 12 to 4, the basal poles can still be spread but not as

Fig. 6 -

 6 Fig. 6-10. Predicted rolling texture by Agnew et al. (2001).   0002 and   0 1 10

  Fig. 6-11. Predicted rolling texture for X=12 ( 2 : X : 3 : 1 : : : tw a c prism basal        

Fig. 6 -

 6 Fig. 6-13b. Effect of  value on the slip and twinning activities for X=4

  Model parameters describing critical resolved shear stress and hardening parameters.The hardening parameters used in Tab. 6-1 are obtained by backfitting the experimental mechanical behavior for the case of high interaction with the 14 shows the stress strain curve of rolled magnesium sheet under uniaxial tension along rolling direction (RD). The initial texture for these simulations is that of rolled sheet. Experimental data are obtained fromJain and Agnew"s work (2007).

Fig. 6

 6 Fig. 6-14. Simulated and experimental stress strain curves of rolled AZ31 sheet under

  ) poles figures. For these reasons, we are showing only the simulated results for

  Fig. 6-15. (0001) pole figures of squeeze cast AZ31 sheet with random texture (a),

  Fig. 6-17. Experimental (a) and simulated   0001 and   0 1 10 pole figures of Mg sheet uniaxially stretched along RD using the  -model with different interaction parameter  (b, c, d) (experimental data after Ulacia et al., 2010a, b)

  , the activity of basal slip decreases while that of prismatic slip increases during tension. Higher  value leads to lower activity of pyramidal slip.

  Fig. 6-18. Predicted activities of different deformation mechanisms in Mg sheet during uniaxial tension along RD using the intermediate  -model, 01 . 0   (a),

  Fig. 6-19. Experimental and simulated (0001) and (10 1 0) pole figures of Mg sheet uniaxially compressed along TD using the intermediate  -model with different  .

  Fig. 6-20. Predicted activities of different deformation mechanisms in Mg sheet during uniaxial compression along TD using the intermediate  -model, 01 . 0   (a),

  Fig. 6-21. Experimental and simulated (0001) and (10 1 0) pole figures of Mg sheet uniaxially compressed along RD using the intermediate  -model with different 

  Fig. 6-23. Effect of parameter  on the predicted textures under uniaxial

  The possible links between the interaction strength parameter  and the microstructural features such as the grain size and the stacking fault energy (SFE) of the polycrystals are briefly mentioned. However, additional work is needed to clarify and quantify this link.With the advantage of the high computation speed, the intermediate -model will play an important role in integrated computational materials engineering (ICME) to provide guidance on processing optimization.The future work should focus on the development and application of the -model. It might include the following:

III.3.1 Ideal FCC rolling component

	 37 ,  63 ) for S component; and (  45 ,  0 ,  45 ) for C component. The positions of these
	orientations in the ODF section at	  45 
	The {001}<100> cube (C), {110}<112> brass (B), {110}<001> goss (G), {112}<111>
	copper (Cu) and the {123}<634> S components are the main orientations for FCC
	metals. The location of these texture components in the Euler space can be
	represented by the following Euler angles : ( 1  ,  , 2  ): ( o 90 , o 35 , o 45 ) for Cu
	component; (  55 ,  90 ,  45 ) for B component; (  90 ,  90 ,  45 ) for G component; (  59 ,

Conclusions & Perspectives

V.2.4 Results

V.2.4.1 Predicted cold rolling textures

For the restricted slip and pencil glide, Figs. 5-2 to 5-4 show the predicted rolling textures by the (100), ( 110) and ( 111) pole figures at % 100  

for the tangent VPSC model (Fig. 5-2) and the  -model ( 1 . 0   in Fig. 5-3 and 7 . 0   in Fig. 5-4). In these figures, one may note significant differences between the restricted slip results and pencil glide results for all models. In addition, one may also observe the development of the two fibers  and  with some differences between the models.

 Orientation distribution function (ODF)

The full 3-Dimension description of crystallographic texture can be given by the orientation distribution function (ODF) which is defined as the volume fraction of grains in a certain orientation g.

If we consider the three Euler angles as the three axes of Cartesian coordinates, a specific area of space can be obtained which is called the Euler orientation, see Fig.