F Delaire

J L Raphanel{{

C Rey}

PLASTIC HETEROGENEITIES OF A COPPER MULTICRYSTAL DEFORMED IN UNIAXIAL TENSION: EXPERIMENTAL STUDY AND FINITE ELEMENT SIMULATIONS

Keywords: Mechanical properties, Dislocations, Computer simulation, SEM, Copper

A copper sample made of a single layer of grains is plastically deformed by uniaxial tension at room temperature and low strain rate. The deformation field is measured by means of grids deposited on the polished surface of the undeformed specimen and local orientations are recorded using electron back scattering diagrams in a scanning electron microscope. These measures are compared with simulations made by a finite element code using a physically based model for the deformation and hardening of face centered cubic crystals. A good agreement is found between measured and computed values. The simulations give access to much more detail about the history of glide in each grain and help establish which systems are active at a local level. They also provide the evolution of internal variables such as dislocation densities. A new insight into intergranular accommodation as well as intragranular heterogeneities is provided.

Résumé: Un échantillon de cuivre constitué d'une seule couche de grains est déformé dans le domaine plastique par traction uniaxiale à température ambiante et à faible vitesse de déformation. Le champ de déformation est mesuré à l'aide de grilles déposées sur la surface polie de l'échantillon non déformé et les orientations locales sont enregistrées par traitement des diagrammes d'électrons rétrodiffusés dans un microscope électronique à balayage. Ces mesures sont comparées à des simulations effectuées avec un code aux éléments finis qui utilise un modèle physique pour la déformation et l'écrouissage des cristaux de structrure cubique à faces centrées. Un bon accord est trouvé entre valeurs mesurées et calculées. Les simulations donnent accès à plus de détails de l'histoire de la déformation par glissement dans chaque grain et aident à établir quels systèmes sont actifs à un niveau local. Elles fournissent aussi l'évolution de variables internes, comme les densités de dislocations. Une nouvelle approche peut ainsi être faite de l'accommodation intergranulaire et de l'hétérogénéité intragranulaire.

INTRODUCTION

The plastic response of crystalline materials is in general anisotropic and heterogeneous, which leads to the development of deformation textures and intragranular microstructures. These may be responsible for localization or loss of ductility of the material. It is thus important to gain a better understanding of how an imposed macroscopic stress or strain is accommodated at the local level of a grain interacting with its immediate neighbors. The many models of crystalline plasticity do not speci®cally address this matter: the interaction between grains is at best taken in an average sense. For instance, the relaxed Taylor models [START_REF] Kocks | 2nd RISé Symposium: Deformation, Mechanisms, Microstructure[END_REF][START_REF] Van Houtte | [END_REF] use arguments based on grain shapes and the self-consistent schemes [3±5] place the grain in interaction with a continuum, the equivalent homogeneous medium. This approach is justi®ed since the purpose of these models is the prediction of macroscopic properties, such as deformation textures and overall mechanical response. This class of models includes at their core an averaging scheme in order to allow the transition from the microscale (i.e. the grain scale) to the macroscale.

The aim of this paper is to present an attempt to calculate by ®nite element techniques the complete strain and stress ®elds in aggregates with a small number of grains and to apply the numerical simulation to the case of a copper multicrystal which has been experimentally tested. The use of ®nite elements in crystalline plasticity has been rapidly growing with the growth of the computing power of the average workstation. Finite element techniques were ®rst used to evaluate the state of stress and strain of bi-and tricrystals [6,7]. Monotonic, but also cyclic loadings have been investigated [8]. In these papers, however, the overall deformation remained small and no attempt at re-orienting the crystal was performed. Some other ®nite element codes using crystalline plasticity have aimed directly at describing the polycrystalline behavior, focusing on the development of anisotropy [9], or of deformation textures [10] including, in some instances, temperature and strain rate eects [11]. In most cases, an ideal polycrystal is imagined and relatively few attempts have been made to simulate the behavior of an actual sample [12,13], one of the reasons being that for a sample with more than one layer of grains, if surface observations provide a lot of detail about the strains and crystal orientation in the surface layer, the boundary conditions imposed by the deeper layers of material cannot be assessed. The problem of number of through thickness grains is addressed in the work by Buchheit et al. [14], but on an ideal polycrystal. Most of these simulations use a hardening rule derived from Asaro and Needleman [15], where quantities similar to critical shear rates are linearly related to glide velocities through a constant hardening matrix which in its simplest form has only two dierent kinds of term, the diagonal term for self-hardening and the o-diagonal for latent hardening.

The originality of the approach presented here, is to perform the simulation on an actual aggregate made of a single layer of grains and conduct the experiment on the same sample. Moreover, the modeling uses the framework of large deformations and accounts for step by step re-orientation of the crystalline material with a physically motivated hardening rule based on evolution of dislocation densities, so that the hardening matrix is no longer constant but is recalculated and updated at each step.

The special aggregate that we consider shall be called a multicrystal and the material is OFHC copper. It has been possible to grow by the Bridgman technique a multicrystal of about ten grains of millimetric sizes which provided a sample for uniaxial tension made of a single layer of grains with grain boundaries almost orthogonal to the free surfaces. The geometry of the sample and the experimental techniques are described in the next section. The scale of observation and simulation is not that of individual dislocations but rather that of the collective behavior of dislocations or in experimental terms, the scale of the scanning electron microscope (SEM), rather than the transmission electron microscope (TEM). The model and its implementation in the ®nite element code are detailed in Section 3. One assumes an elasto-viscoplastic behavior of the aggregate, with plastic deformation taking place solely by intragranular dislocation glide on crystallographic slip systems. One attempts to compare quantitatively results of experiments and simulations. The quantities obtained by both methods are the strain tensor (at least some of its components) and the local crystal orientations at predetermined levels of the overall deformation. The simulations are the only means to evaluate the activity of slip systems, the stress ®eld and the dislocation densities. The experimental quantities are computed over a rectangular grid of 1 Â 1 mm 2 squares in the undeformed state and the simulated ones at each integration point of the ®nite element mesh. One shall not only consider the various ®elds over the sample but also focus at certain selected points in order to illustrate how, within the same grain, dierent material points undergo dierent histories of stress and strain and the consequences of these for the heterogeneity of the deformed crystal.

SAMPLE AND EXPERIMENTAL PROCEDURE

The multicrystal has been taken from an ingot obtained by the Bridgman technique. The sample is made of a thin slice of the ingot, machined so that each surface has almost the same topology and a single layer of grains is present, with grain boundaries almost orthogonal to the free surfaces. The shape of the specimen, with an active length of 29 mm, a width of 16 mm and a thickness of 3 mm is chosen so that its ends, which are held by the grips of a tensile machine, do not deform noticeably during the uniaxial tensile test. The specimen is made of a total of 11 grains, with a central grain entirely surrounded by six neighbors (see Fig. 1).

The initial orientations of the grains are determined by the EBSD (electron back scattered diagram) technique performed in a scanning electron microscope (SEM). A microgrid of 1 Â 1 mm 2 squares is deposited on one side of the sample. The uniaxial tensile test takes place at constant crosshead velocity of 4 Â 10 À4 =s:The test is stopped at some intervals and the sample observed in the SEM in order to measure local orientations by EBSD and to record the displacements of the nodes of the deformed grid. The slip plane traces are also observed and recorded.

Measurements of displacements

The measurements are performed in a SEM. The in-plane displacements are obtained by the displacement of the stage along two orthogonal axes. The third component is given by a focusing length read on the SEM. The intersections of the grid with the grain boundaries are initially recorded from photographs of the undeformed sample. The position of these points at any later stage, are then estimated by a cubic spline interpolation using the position of the neighboring grid points. A direct measure becomes impossible owing to the contrast values near a grain boundary in the deformed state. The overall precision is 1 mm for in-plane displacements and about 50 mm for out-of-plane measurements.

Strain ®eld computations

Only six components of the transformation gradient F are determined by the measured displacements since one associates to a point in the reference con-®guration its position in the deformed one, namely Mt 0 X 1 , X 2 4 Mtx 1 t, x 2 t, x 3 t so that only the ®rst two columns of F may be computed, namely F ia @ x i @ X a , for i 1, 2, 3 and a 1, 2 allowing the computation of three of the Green Lagrange strain tensor components, E 11 , E 22 , E 33 , through the well-known relation:

E 1 2 F T F À 1:
In order to have an estimate of these strain components at any point of the surface, one may consider the grid of measured points as constituting a mesh in the ®nite element sense (see Fig. 2), and then use mapping and interpolation functions to compute the values of the components at any spatial location on the sample.

Crystalline orientations

The EBSD technique allows the measurement of local crystalline orientations in the SEM. To any material point in the deformed con®guration, one may thus associate a rotation matrix, which allows the passage from the macroscopic reference axes (namely, tensile axis, transverse axis and normal to the plane of the undeformed sample) to the cubic directions in the crystal. In this case also, one may estimate the local orientation at any point by interpolation from the measured ones. For this purpose, the use of the Rodrigues vector [16] is particularly appropriate, since it is de®ned in a metric space and can provide a measure of the ``distance'' between two orientations.

MODELING AND SIMULATIONS

The mechanical modeling follows the framework developed in Teodosiu and co-workers [START_REF] Raphanel | Finite element simulation of the large elastoplastic deformation of multicrystals[END_REF]. The main characteristics of the model and its special form that is implemented in the computer code are recalled and outlined in the following subsections.

Kinematics and constitutive relations

The kinematics of large transformations are used which are based on the multiplicative decomposition of the deformation gradient into elastic and plastic parts [START_REF] Lee | [END_REF], the velocity gradient L being subsequently expressed as the sum of an elastic part L e and a plastic part L p [19,[START_REF] Mandel | Plasticite classique et viscoplasticiteÂ,i n Lecture Notes CISM[END_REF]. This requires the choice of intermediate con®gurations. For crystalline plasticity, the most convenient choice is that of isoclinic, relaxed local con®gurations [START_REF] Teodosiu | [END_REF]. In the case of cubic crystals, one shall simply assume that the cubic axes are aligned in the local con®gurations, with the frame of a Cartesian system of coordinates. For each grain, plasticity is due to glide on crystallographic slip systems. A system (s ) is de®ned by a glide direction Å g s and a glide plane normal Å n s in the isoclinic local con®gurations. In face centered cubic (f.c.c.) materials such as copper, these planes and directions are well known as the {111} planes and h110i directions. There are 12 such slip systems which are referred to by a letter and a number according to the Schmid and Boas notation (see Table 1).

One natural way of coupling the elastic behavior with the ¯ow theory of plasticity is to derive from a linear elastic law an hypoelastic law by time dierentiation, so that one may express the time derivative of the Kirchho stress t in terms of total strain rates, total rotation rates and plastic rate quantities

_t t t K e :D t t tW À Wt t t À X s _ g s R s 1
where R s K e :D s W s t t t À t t tW s with D (s ) and W (s ) being, respectively, the symmetric and antisymmetric parts of g s n s and K e being simply related to the fourth order elasticity tensor (see for instance Ref. [22]).

The rate of slip on an activated slip system is related to the resolved shear by the viscoplastic power law

_ g s _ g 0 t s t s C n sgn t s 2
where _ g 0 is a reference slip rate and the exponent n is of the order of 100 for copper at room temperature. The large value of n implies an almost purely plastic behavior, where _ g s remains negligible unless jt s j is close to t s C which plays the role of a critical shear stress, as in purely plastic models with use of the Schmid law.

Hardening law

One of the advantages of this approach is to use a physically based argument in order to express evolution laws for the critical shears. The more common and more phenomenological approach has been to relate linearly slip rates and critical shear through a hardening matrix with constant coecients, in general limited to a diagonal one characterizing self-hardening and equal o-diagonal terms associated with latent hardening (see for instance Refs [23±25]). Here, the linear structure of the relation is kept, but one considers that the coecients of the hardening matrix are not constant and depend on internal variables that are related to the physics of plasticity by dislocation glide.

The model is based on the evolution of dislocation densities [26,[START_REF] Kroè Ner | Problems of Plasticity[END_REF]. One introduces total dislocation densities on a system (s ), r (s ) , whose evolutions are governed by a production term, following Orowan's law and thus linked to a mean free path L (s ) and a dynamic recovery term controlled by a critical annihilation length y C . One writes

_ r s 1 b 1 L s À 2y C r s _ g s 3
where the mean free path is related to the density of point obstacles encountered by the gliding dislocations:

L s K X uT s r u ! À1=2 :
The critical shear stress on a system (s ) can be related to the dislocation densities by the hardening relation:

t s C mb X u a su r u ! 1=2 4
where m is taken as the shear modulus of an isotropic aggregate made of the same material and a (su ) is a matrix taking into account various types of dislocation interactions [START_REF] Franciosi | [END_REF]. The dierentiation of this relation with respect to time and some computation lead to the expression of the coecients of the hardening matrix in terms of the total dislocation densities.

It has been shown by Tabourot et al. [29] that this model can describe the ®rst three stages of the hardening curve of copper single crystals initially oriented for single slip and can also correlate satisfactorily multislip data.

Choice of parameters for the model

The model requires few materials parameters in order to be implemented. Some are well-known values for a material such as copper, others require an identi®cation from curves found in the literature. They fall into the following categories: elastic anisotropy the moduli c 11 , c 12 , c 44 for copper single crystals; viscoplasticity a reference glide velocity _ g 0 , the exponent n of the viscoplastic power law, an initial critical shear t 0 C , which is the same on all the glide systems and shall also be used in order to determine an initial dislocation density r 0 ; hardening the annihilation distance (for edge dislocations) y C , a material parameter K (needed for the evaluation of the mean free path), the modulus of the Burgers vector b, the isotropic shear modulus m, and the terms of the interaction matrix, a su , which we shall reduce to two values, one for the diagonal terms (self-hardening) and one for the o-diagonal terms (latent hardening).

The values used for our computations are given in Table 2.

FINITE ELEMENT COMPUTATION

The evolution of the deformation process is described by an updated Lagrangian scheme, that is to say that the con®guration at time t is taken as reference con®guration for the time interval t, t Dt: At the end of this interval, the con®guration and the internal variables are updated and the new con®guration is taken as reference for the next time increment. Let V be the region occupied by the multicrystal at time t and assume that its boundary S can be composed into two parts S t and S v , such that the traction vector t à is prescribed on S t and the velocity v à is prescribed on S v at any time t.I n the particular case of uniaxial tension simulations, these conditions become:

. no traction on the free surfaces of the sample;

. imposed velocities at the end surfaces, modeling the grips of the tension machine: one ®xed end with zero velocity and the other with a constant axial speed V 0 .

The position vector x, the velocity ®elds v and dv are approximated by the same shape functions. The discretization of the kinematical ®elds leads to the classic discretization of the principle of virtual power that in turn leads to the system of linear algebraic equations:

KfDugfDfg: 5

After solving the system, the con®guration and the state variables are updated at each Gauss point by a simple Euler explicit scheme, except for the glide increments that are calculated by the forward gradient approximation [24]. This method presents the drawback to impose very small time increments, but has the advantage of allowing a progressive intragranular lattice reorientaion at each Gauss point so that the incipience of glide on a new slip system at any point of the sample is well accounted

for. Attempts at implicit schemes, worked well for single crystals but were less ecient for multicrystals.

The meshing of the structure had the constraint to follow the grain boundaries and also to provide a ®ner mesh in regions where stronger gradients are expected. The chosen element is a six-node pentahedron, with two integration points. Two layers are used so that there are four integration points through the thickness of the sample (see Fig. 3).

The number of elements has been kept relatively small (469) in order to make the computations in a reasonable time on a powerful workstation. One must compute and update a very large number of variables at each integration point, since in addition to the local orientation (three angles, but in fact a 3 Â 3 matrix), one updates the dislocation densities on each system (12 quantities) and the stresses. With intermediate quantities used in the computation, it is a total of 134 values per Gauss point that are retained and updated. Another mesh with about the same number of quadratic hexahedral elements has also been tested with very similar numerical results. In order to have a better estimate of quantities in the high gradient regions, one would have to go to the next order of magnitude for the discretization of the structure and consider thousands rather than hundreds of elements. The choice retained here, while partly dictated by the available hardware, was also consistent with the scale of the experimental observations which dealt with millimetric grids deposited on the surface of the sample and manual recording of local orientations.

MAIN FEATURES OF A COMPARISON BETWEEN EXPERIMENT AND SIMULATION

Three stages of the deformation of the multicrystal have been used for comparing experiment and simulation. They correspond to overall macroscopic axial logarithmic strains of 0.03, 0.075, and 0.33. At each stage, the sample is placed in the SEM and measurements of grid displacements and local orientations are performed. Three grains are analyzed in detail, which present dierent patterns of behavior owing to their initial orientations and position within the sample. One has selected the two largest grains near the ends of the sample (grains 1 and 4) and the central grain (grain 6) completely surrounded by others. For these three grains, the initial values of the ®ve largest Schmid factors and the corresponding slip systems are listed in Table 3. One recalls that the Schmid factor is computed under the assumption of uniaxial tension. Table 3 indicates that:

. Grain 1 is close to a stable double slip orientation with glide on B4 and C1. . Grain 4 has an unstable double slip orientation with glide on B4 and A3. The dierence between the Schmid factors is however large enough to promote mostly single glide on B4. . Grain 6 is oriented for single slip on B4.

The actual stress state applied at the boundaries of each grain is not exactly that of a pure tensile test, owing to interactions between grains and actual (or simulated) end conditions. However, the response of the single crystal, initially oriented as the grain in the aggregate can be used as a reference for the discussion. We also point out at this stage that since we deal with a group of crystallites, one cannot replace an orientation by an equivalent one, as in the case of single crystals treated separately. The ®rst grain may be chosen so that B4 is the primary system for uniaxial tension and then the other grains are oriented using the EBSD technique in the SEM, without use of symmetry operations, this procedure ensures that the grain boundaries retain their relative orientation with respect to the grains. A convenient way to represent the initial orien-tations is to draw the inverse pole ®gure (or its relevant part) which gives the position of the tensile axis with respect to the crystallographic axes (Fig. 4).

Lagrangian strains and slip system activity

The plastic deformation of the multicrystal may be studied experimentally by the measure of the surface grid and by the observation of slip plane traces. The computation yields values of the six components of the Lagrangian strain and access to the cumulated glide on every glide system. 5.1.1. Strains. Strain heterogeneities are already present at an average global strain of 0.03. At larger global strains, they remain qualitatively unchanged, even though their extent and magnitude may have varied. Isovalues of axial strain components ignore grain boundaries. The band of localization that goes through grains 9, 4, and 3 is well predicted by the simulation albeit with a lesser spread than in the experiment.

At a global strain of 0.03, the axial strain E 11 is already close to 0.06 in a band that stretches across grains 4, 3, and 9. Grain 4 that is in single slip initially deforms a little faster than its neighbors, especially grain 9 which is in symmetric double slip. At larger strains (see Fig. 5), the value of E 11 in the band becomes two to three times larger than the average value and the band narrows. The exper-imental or simulated axial strain does not show any jump when crossing a grain boundary almost parallel to the tensile axis.

The transverse strain however presents jumps along the grain boundaries parallel to the tensile axis, namely 4±3 and 4±9. There is a marked dierence between grain 4 which deforms with a very small homogeneous transverse strain and grain 1 which has a negative transverse strain, with a gradient toward and across grain 6.

Crystallographic slip.

A good marker of plastic deformation is provided by the slip plane traces which appear on the polished surface of the sample. Their observation allows in most instances the determination of the corresponding slip system. The ambiguity about glide direction in a given plane is resolved by computation of the Schmid factor in the deformed, i.e. re-oriented state, and in all cases, the experimental data check with the simulations.

The overall picture is given by Fig. 6 which shows the sample at an overall axial strain of 0.075. The grain boundaries are regions where a strong relief appears, particularly between grains 3 and 4. The triple points are also regions of fairly high slopes, in particular the junction 1±5±6. In these regions, it is dicult to measure local orientations and strains. The observation of superimposed systems of slip lines in some domains cannot be readily equated with the simultaneous activity of slip systems, but rather tells about a history of plastic deformation. The computed orientations of the slip plane traces are drawn on Fig. 7 for grains 4, 6, and 1 and a direct comparison with Fig. 6 shows the close agreement. Figure 8 is a sketch deduced from Fig. 6 that outlines the intragranular domains with dierent slip activity. Computed glide quantities agree with these observations and give an insight into the development of plastic deformation in the dierent intragranular regions. For a given material point and a given slip system, the plot of the glide quantity with respect to the macroscopic axial strain (which plays the part of a time increment) allows us to follow the local plastic activation of the dierent slip systems as the overall deformation proceeds. One has selected four points for grain 1 which are representative of the dierent behaviors (see Fig. 9). There is a good agreement between the observed slip plane traces and the most active slip plane, either B4 or C1 depending on the region considered. One notices that near GB 1±2 and GB 1±6, the primary system B4 stops gliding and single glide on C1 then appears. On the contrary it is a situation of multiple slip on B4 and C1 that takes place near GB 1±8 and at larger strains near GB 1±10.

5.1.2.2. Grain 4. This grain is characterized by large axial deformation taking place in a band not far from the boundary with grain 6 and which also extends into its upper and lower neighbors. One can distinguish three zones: a peripheral zone, GB 4±7, 4±3, 4±6, and 4±5 where few slip lines are observed and a strong relief is present, a large domain of mostly B4 activity and a wide band stemming from GB 4±9 where traces of critical system A3 are observed. The simulations (see Fig. 10) however indicate that along the grain boundaries as well as in the grain interior single glide B4 prevails and indeed a band where glide is fairly large appears, just next to the boundary with grain 6, in a section where one ®nds grains 9±4±3 in ``parallel''. A closer analysis of the simulation shows that the axial deformation starts to increase or ``localize'' in grain 9 which has a large initial Schmid factor and is oriented for stable double slip and that subsequently, grain 4 deforms very rapidly. It appears that systems A3 (critical or secondary system) and B5 (tertiary system) play a role in accommodating the conditions imposed by the boundary. The cumulated glides on these two systems remain small but may be critical to ensure compatibility. 5.1.2.3. Grain 6. In this central grain, one may distinguish four regions, namely the top of the grain, the boundary with grain 4, the bottom of the grain and the center region with an extension to the boundary with grain 1. In the ®rst three regions, one notices mostly single slip, although the sharp relief near the GB makes precise observations dicult. In the more central region, A3 and B4 are present.

The computations of glide quantities at several locations within grain 6 support these observations and allow the analysis of the sequence of events leading to the observed slip plane traces. The interior of the grain starts in double slip A3±B4, for axial strains up to 0.1, when the activity of B4 stops and single glide A3 takes place. It is interesting to notice that inside grain 1, we have observed the same situation near the middle of the GB 1±6, where glide on B4 stops for E 11 0:1 and glide on the secondary system C1 appears. But for grain 1, plastic glide had initiated on B4 only.

Orientations

The inverse pole ®gures (Fig. 11) show a good agreement between experiment and simulation at least for the intermediate global axial strain of 0.075.

For the central grain however, there is more dispersion in the measured orientations than expected by the simulation. One should emphasize that the experimental points are surface measurements, since the electron beam interacts with an area of about 7 mm 2 and a depth of 10±50 nm while the computed orientations are evaluated at the Gauss points of the mesh which are at least 300 mm below the surface. The grain that presents the largest re-orientation is grain 4, which is the most deformed, with a band of localization of axial deformation. It is also a grain that deforms mostly by single slip. As expected, most of the grain rotates towards a ``stable'' orientation for uniaxial tension, namely h112i for the tensile axis as one can see on the corresponding inverse pole ®gure. The same tendency is found for grain 1. Not as many points are available for grain 6, experimentally or in the simulations, and for this grain the agreement is the least satisfying. Owing to the more complex state of interaction between neighbors and the relatively smaller size of grain 6, the exact spatial position of the point of measure or of computation may be of greater importance here, and one also notices that this is a grain which presents quite important out of plane deformation, near triple points and boundaries.

Dislocation densities

The dislocation densities are the internal variables of our plasticity model. One should of course exert a certain caution in the discussion of the results and keep in mind that in this respect there may exist a discrepancy between the actual physical dislocation density and what the computations yield. In particular, we have chosen rather arbitrarily an initial value and a rather phenomenological way of accounting for dislocation density evolutions. It appears, however, that at least qualitatively, the evolutions of the computed quantities may be linked to evolutions of dislocation densities. The total dislocation density remains fairly homogeneous through the grains, and in this sense does not show maxima in the band of localized axial deformation. It thus appears that two grains such as grains 1 and 4 present very dierent axial strains and glide activities but have almost the same total dislocation density (see Fig. 12). It may now be interesting to look at the dislocation density on the primary glide system. There, again, for the two grains 1 and 4, we ®nd fairly similar values and this can be understood by analyzing the evolution law for dislocation densities, and more particularly, the evolution of the mean free path. Grain 4 is for the most part of its strain history, in single slip, the mean free path associated with the primary system does not change. On the contrary, grain 1 is almost always in double slip, so that the mean free path associated with these systems decreases almost like the square root of the total dislocation density, hence reducing the corresponding glide quantities.

CONCLUSION

The multicrystal, at an intermediate scale between the single crystal and the polycrystal, is the ®rst relatively complex structure that yields both to experimental and numerical studies, in order to provide a better understanding of the development of plastic heterogeneities in crystalline materials. The comparisons between observation, measurements and simulations have shown a good overall agreement of similar quantities, namely strain and local orientations. The kinetics of plastic glide has also been accurately described, with no discrepancy between methods for the determination of locally active slip systems. The experimental and the numerical scales have been chosen of the same order, that is, slightly less than a millimeter. Since the grains are mostly centimetric, this constitutes the ®rst smaller order necessary for a description of intragranular heterogeneities.

A ®rst striking result is the analysis of the devel-opment of a zone of strain concentration which crosses three grains of the multicrystal. The simulations show that the phenomenon starts at the beginning of the plastic stage, too early to be measured by the grid. At ®rst, a grain along the edge (grain 9) is well oriented for stable double slip and deforms easily. It favors somewhat the deformation of its immediate neighbor (grain 4) which is oriented for single slip and rotates toward a stable orientation. The band of strain concentration presents a maximum that reaches three times the global axial strain and ignores grain boundaries aligned with the tensile axis. Another important result concerns grain interactions and this can be seen very clearly for grain 6 which is surrounded with neighbors and may thus be the grain less likely to behave like a homogeneous single crystal in uniaxial tension. Grain 6 presents a fairly large gradient of re-orientation, with a maximum near the boundary with grain 4. In grain 6, one observes also gradients of axial, transverse and shear components of strain. It is placed ``in series'' between two large grains, grains 1 and 4, which behave very dierently, as can be seen not only in terms of magnitude but also of shape of the local strain tensors and it must also accommodate ``in parallel'' grains 3 and 2 on one side and grain 5 on the other. This accommodation does not seem to be achieved by a layer at the periphery Fig. 11. Inverse pole ®gures (tensile axis) for grains 1, 4, and 6 at an axial strain of 0.075 (the larger dot represents the initial orientation, see also Fig. 4).

of the grain, with the inside of the grain behaving somewhat homogeneously. One tends to see intragranular bands extending from a grain boundary and often bounded by lines originating at a triple point as a virtual extension of a grain boundary. These results were made possible through the combination of the experimental and numerical techniques. The mesh that has been used however limits the accuracy of our analysis. The use of a dierent mesh of the same characteristic coarseness, with two layers of quadratic hexahedral elements, does not change dramatically the simulated values, except for stresses that are more accurately estimated. This shows that signi®cant improvement requires to go at least one of magnitude higher for mesh re®nement, with the corollary that experimental data should also be gathered at a smaller scale. Unfortunately, multicrystals are di-cult to produce and even more dicult to reproduce so that one cannot multiply experiments.

A critical point of the model remains the hardening laws. They are very material, temperature, rate and grain size dependent. Without changing the framework presented here, one could envision introducing a local mean free path, bounded by a distance to the nearest grain boundary, measured along a glide direction. This would introduce more clearly a characteristic length and grain size dependence. It would also be of interest to vary hardening parameters in order to better assess the in¯uence of each hardening mechanism, but this would constitute a whole study by itself, albeit a study made possible by the numerical tool which has been presented here.

Fig. 1 .

 1 Fig. 1. Sketch of the multicrystal and numbering of its 11 grains.

Fig. 2 .

 2 Fig. 2. Deformed sample and grid at 0.30 average axial logarithmic strain.

Fig. 3 .

 3 Fig. 3. The ®nite element mesh: two layers of six-node prismatic elements.

Fig. 4 .

 4 Fig. 4. Inverse pole ®gure: initial position of the tensile axis with respect to the crystallographic axes for the 11 grains of the sample.

Fig. 5 .

 5 Fig. 5. Axial strain distribution.

5. 1 . 2 . 1 .

 121 Grain 1. Slip line patterns limit four intragranular domains. A multiple slip domain in a central position where the two expected systems are active together and three other regions where the interactions with the neighboring grain boundary (GB) or triple point play a part: a region with mostly C1 activity, along the boundary with grain 8; a deep region stretching from the border with grain 2 with B4 activity and a local zone of A3 near the GB; a region with a sharp relief near the triple point 6±5±1.

Fig. 6 .

 6 Fig. 6. Picture of the deformed sample at 0.075 average axial strain.

Fig. 7 .

 7 Fig. 7. Computed orientation of the slip plane traces for grains 1, 4, and 6 at about 0.07 average axial strain.

Fig. 8 . 6 .Fig. 9 .

 869 Fig. 8. Domains of dierent glide activity for grains 1, 4, and 6.

Fig. 10 .

 10 Fig. 10. Slip activity in grain 4 at the overall axial strain of 0.33.

Table 1 .

 1 Schmid and Boas notation

	Slip system	Plane	Direction
	A2	111 0	11
	A3			[101]
	A6			[110]
	B2	(111)	0	11
	B4			101
	B5			1	10
	C1	1	11	[011]
	C3			[101]
	C5			1	10
	D1	1	11	[011]
	D4			101
	D6			[110]

Table 2 .

 2 Values of the parameters entering the model

	c 11 166 Â 10 3 MPa _ g 0 10 À3 =s b 2:57 Â 10 À7 mm isotropic m 45 Â 10 3 MPa	c 12 120 Â 10 3 MPa n 100 y C 0:5 Â 10 À6 mm a ss 0:52	c 44 76 Â 10 3 MPa r 0 1200=mm 2 K 75 a

su 0:72, s T u

Table 3 .

 3 The ®ve highest Schmid factors of the three grains considered

Grain Schmid factor values and associated glide system 1 0.464 B4 0.440 C1 0.438 A3 0.414 A2 0.315 B2 4 0.491 B4 0.461 A3 0.308 B5 0.233 C1 0.203 A2 6 0.493 B4 0.439 A3 À0.319 B5 0.265 C1 0.211 A2


Pedro Salas-Ambrosio

Antoine Tronnet

Marc Since

Sandra Bourgeade-Delmas

Jean-Luc Stigliani

Amelie Vax

Sébastien Lecommandoux

Bruno Dupuy

Pierre Verhaeghe

Colin Bonduelle

Jean-Luc Stiglia- Ni

Cyclic Poly(α-peptoid)s by Lithium bis(trimethylsilyl)amide (LiHMDS)-Mediated Ring-Expansion Polymerization

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

So far, three main routes have been proposed to synthesize cyclic polymers. 7,[START_REF] Laurent | Synthetic Approaches for the Preparation of Cyclic Polymers[END_REF][START_REF] Kricheldorf | Cyclic Polymers: Synthetic Strategies and Physical Properties[END_REF] Two of these routes involve a post-polymerization grafting using tailored extremities of linear polymers and uni-or bimolecular couplings. [START_REF] Iatrou | Synthesis and Characterization of Model Cyclic Block Copolymers of Styrene and Butadiene[END_REF][START_REF] Schappacher | α-Acetal-ω-Bis(Hydroxymethyl) Heterodifunctional Polystyrene: Synthesis, Characterization, and Investigation of Intramolecular End-to-End Ring Closure[END_REF][START_REF] Schappacher | Characterization, and Intramolecular End-to-End Ring Closure of α-Isopropylidene-1,1-Dihydroxymethyl-ω-Diethylacetal Polystyrene-Block-Polyisoprene Block Copolymers[END_REF] In these two options, the synthesis of polymers remains difficult in marked contrast to the third option, named ring-expansion polymerization (REP), that is far more efficient. 1,[START_REF] Jeong | Organic Spirocyclic Initiators for the Ring-Expansion Polymerization of β-Lactones[END_REF] REP involves the insertion of cyclic monomers via the use of specific catalyst/initiator systems and it has been used in metathesis polymerization to produce cyclic olefins, [START_REF] Bielawski | Endless" Route to Cyclic Polymers[END_REF] or in ring-opening polymerization (ROP) to produce cyclic polyesters [START_REF] Jeong | Organic Spirocyclic Initiators for the Ring-Expansion Polymerization of β-Lactones[END_REF][START_REF] Piedra-Arroni | Ring-Opening Polymerization with Zn(C 6 F 5 ) 2 -Based Lewis Pairs: Original and Efficient Approach to Cyclic Polyesters[END_REF] and polypeptides. [START_REF] Zhang | Straightforward Access to Linear and Cyclic Polypeptides[END_REF] N-heterocyclic carbenes (NHCs) have provided spectacular results in this direction by promoting REP with evolving growth control. [START_REF] Jeong | Organic Spirocyclic Initiators for the Ring-Expansion Polymerization of β-Lactones[END_REF][START_REF] Zhang | Straightforward Access to Linear and Cyclic Polypeptides[END_REF][START_REF] Guo | Cyclic Poly(α-Peptoid)s and Their Block Copolymers from N-Heterocyclic Carbene-Mediated Ring-Opening Polymerizations of N-Substituted N -Carboxylanhydrides[END_REF] Cyclic polymers are unique macromolecules that have different physical properties than their linear counterparts due to the "double constraint" related to the cyclization. For instance, cyclic polyesters demonstrated modified solubility, 1,2 crystallinity 2,3 and improved degradation profile against chemical hydrolysis. 4 Theoretically, the radius of gyration of a cyclic polymer is expected to be 1/Ö2 that of the linear one in unperturbed conditions. 5 With cyclic diblock copolymers, the models predict a transition from the disordered to an ordered phase in bulk at (cN)c=17.8 for a symmetric diblock copolymer as compared to linear diblocks for which (cN)c=10.5. 6 In solution, cyclic polymers possess a ring-like topology displaying smaller hydrodynamic volume and it has been shown that cyclization concentrates spatial distribution of the side chains to achieve unique biological properties. 7,8 For these reasons, efficient syntheses of cyclic polymers have attracted great interest in recent years. This work is focused on the synthesis of cyclic poly(α-peptoid)s using the ROP of glycine-based Nalkylated-N-carboxyanhydrides (NNCA). [START_REF] Chan | Polypeptoid Polymers: Synthesis, Characterization, and Properties[END_REF] D. Zhang and her collaborators carried out pioneer works about such cyclic structures: they have shown that NHC [START_REF] Guo | Cyclic Poly(α-Peptoid)s and Their Block Copolymers from N-Heterocyclic Carbene-Mediated Ring-Opening Polymerizations of N-Substituted N -Carboxylanhydrides[END_REF] and 1,8-diazabicycloundec-7-ene (DBU) [START_REF] Li | Amidine-Mediated Zwitterionic Ring-Opening Polymerization of N-Alkyl N-Carboxyanhydride: Mechanism, Kinetics, and Architecture Elucidation[END_REF] can afford cyclic polypeptoids from NNCA. Although being quite efficient, these approaches remain sensitive to solvent effects, limiting the scope of the monomers that can be used for solubility reasons. [START_REF] Guo | Cyclic Poly(α-Peptoid)s and Their Block Copolymers from N-Heterocyclic Carbene-Mediated Ring-Opening Polymerizations of N-Substituted N -Carboxylanhydrides[END_REF][START_REF] Guo | N -Heterocyclic Carbene-Mediated Zwitterionic Polymerization of N-Substituted N -Carboxyanhydrides toward Poly(α-Peptoid)s: Kinetic, Mechanism, and Architectural Control[END_REF] Herein, we report an alternative approach leading to another REP of NNCA that can be performed in a more polar medium (DMF), using lithium bis(trimethylsilyl)amide (LiHMDS). LiHMDS is a strong base already known to promote the superfast ROP of non-alkylated NCA monomers. [START_REF] Wu | Centers for Disease Control and Prevention (CDC). Antibiotic Resistance Threats in the United States. Antibiotic resistance threats in the United States[END_REF] Unexpectedly, in this work, the use of LiHMDS allowed the synthesis of cyclic polypeptoids when mixed with various NNCA: sarcosine-NCA (Sar-NCA), N-benzyl-NCA (N-phe-NCA) or N-Cbzaminobutyl-NCA (Z-N-lys-NCA). Using those monomers, the LiHMDS methodology allows simple access to cationic macromolecules with interesting antibacterial activities (Scheme 1). moiety (5-hydroxy-2,3-dihydrooxazolium enolate = 5-OH-DHE in figure 2).

As part of a catalytic study aiming at synthesizing polysarcosine (poly(sar)), we first implemented the ROP of Sar-NCA at 0.4 M in DMF from hexylamine, with or without a stoichiometric amount of LiHMDS (as compared to hexylamine) at a monomer/LiHMDS ratio (M/B; B stands for base) = 100, leading to poly(sar) 1 and 2, (table 1). Monitoring the disappearance of the NNCA peak by FTIR (figure S3) revealed that the Sar-NCA stretching was disappearing at a faster rate of k = 9.5x10 -3 min -1 in absence of hexylamine (poly(sar) 3). Upon purification, polymer 3 was characterized by 1 H NMR and the SEC analysis revealed a lower elution time and a narrow dispersity (figures 1A, S4 and table 1). MALDI analyses were then carried out to clarify the structure of 3: it revealed the presence of three sets of ions matching Δ ≈ 71 g/mol with an initiation attributable to previously observed spirocyclic structures (figure 1B). [START_REF] Guo | N -Heterocyclic Carbene-Mediated Zwitterionic Polymerization of N-Substituted N -Carboxyanhydrides toward Poly(α-Peptoid)s: Kinetic, Mechanism, and Architectural Control[END_REF] So far, the existence of cyclic poly(α-peptoid)s in solvents with high dielectric constants such as DMF was never reported. In this work, ESI-MS and FTIR spectra run during the early stages of the polymerization (figures S6-7) highlight most of the key intermediates of a possible REP mechanism starting from Sar-NCA deprotonation (scheme S4.4). [START_REF] Guo | N -Heterocyclic Carbene-Mediated Zwitterionic Polymerization of N-Substituted N -Carboxyanhydrides toward Poly(α-Peptoid)s: Kinetic, Mechanism, and Architectural Control[END_REF] Moreover, DFT calculations carried out on a model zwitterionic intermediate (5 monomer unit) show that the cyclic form is energetically more favorable than the linear one (ΔG = -18.6 KJ/mol, figure S8). As depicted in figure 2, the chemical structure of this intermediate includes LiHMDS and involves three stabilizing interactions between the polymer and the Li + cation, allowing spatial proximity between the anionic carbamoyl extremity and the positive charge of the Münchnone By decreasing the M/B ratio to 35 (an increase in dispersity was observed for lower ratios), polymer 4 was synthesized to confirm the presence of a Münchnone ring on the macromolecule. 1 H and [START_REF] Schappacher | Characterization, and Intramolecular End-to-End Ring Closure of α-Isopropylidene-1,1-Dihydroxymethyl-ω-Diethylacetal Polystyrene-Block-Polyisoprene Block Copolymers[END_REF] C NMR spectra of this new polymer corroborated the proposed Münchnone specie (figures S10 and S13) and FTIR analyses showed both the carbonyl stretching of the zwitterionic structure and the Münchnone initiation species at 1568 and 1759 cm -1 as well as the polymer backbone at 1639 cm -1 (figure S11). [START_REF] Boyd | Unstable Mesoionic Oxazolium-5-Oxides[END_REF] It is to note that the mesoionic oxazole moiety is known to ring-open, in acidic conditions 24 (scheme S4.5). Thus, the reactivity of the 5-OH-DHE ring from the cyclic structures was first evaluated by exposing polymer 4 to HBr (2 equiv.) in TFA. After this treatment, MALDI-TOF and 13 C NMR spectra indeed revealed possible ring-opening into an iminium acid derivative (figures S12-S14) without modification of the overall macrocyclic topology, as attested by NMR, MALDI-TOF and SEC analyses (figure S15). On another hand, cyclic poly(α-peptoid) were reported to react with acyl chlorides to afford the linear counterparts. [START_REF] Guo | N -Heterocyclic Carbene-Mediated Zwitterionic Polymerization of N-Substituted N -Carboxyanhydrides toward Poly(α-Peptoid)s: Kinetic, Mechanism, and Architectural Control[END_REF] Accordingly, MALDI-TOF analyses revealed that the treatment of cyclic polymer 4 with propionic anhydride (2 equiv.) afforded a linear analogue of the polymer for which the SEC analysis revealed a higher molar mass (figures S15-18). Similar propylation was repeated with polymer 3 and, in agreement with previous reports, [START_REF] Guo | Cyclic Poly(α-Peptoid)s and Their Block Copolymers from N-Heterocyclic Carbene-Mediated Ring-Opening Polymerizations of N-Substituted N -Carboxylanhydrides[END_REF] the more compacted structure of the cyclic polymer (before propylation) was reflected by a larger elution volume at the same molar masses (figure 3A andS15).

To further study the LiHMDS-induced polymerization, several REP of Sar-NCA were carried out in DMF by varying the M/B ratio from 60 to 400 (yield: 89%-97%, table S5). SEC characterizations were performed and all polymers presented increasing Mw values with controlled dispersities (Ð = 1.03-1.08, table 1 and figure 3B). The lower the amount of LiHMDS added, the higher the weighted-average molecular weight (Mw), and an unexplained deviation was observed, especially for the higher ratios. For all polymers, the Mw cyclic/Mw after propylation from SEC analysis were close to 0.8, the value established with cyclic polypeptoids when formed from NHCs promoted REP (<G>, table 1 and ESI). [START_REF] Alberty | Synthesis and Characterization of Macrocyclic Vinyl-Aromatic Polymers[END_REF][START_REF] Lutz | Solution Properties of Ring-shaped Polystyrenes[END_REF] Ultimately, the living nature of the REP was evaluated. Synthesis of poly(sar) at M/B = 50 was first performed and, at full conversion, another feed of Sar-NCA (M/B = 50) was added to the reaction until it was also consumed (FTIR monitoring). The resulting poly(sar) 9 presented a Mw of 3200 g.mol -1 close to the Mw of polymer 3 and associated with a low dispersity of Ð = 1.1 (table 1). This result indicated that diblocks could be designed using LiHMDS.

Scheme 2. Postulated basic mechanism for LiHMDS-mediated REP of NNCA. phenylalanine-like NNCA (N-phe-NCA). Monomers were prepared through an optimized 3-step synthesis method (ESI). The LiHMDS-mediated REP of Z-Nlys-NCA was carried out in DMF at M/LiHMDS = 50, alone (polymer 10 table 2 and ESI) or in copolymerization with N-phe-NCA, either one-pot (leading to statistical copolymers 11 to 15, table 2) or sequentially (leading to block copolymers 16 to 20, table 2). For all copolymers, acidic deprotection of the Cbz group (HBr 2 equiv. in TFA) afforded side-chain deprotection, allowing producing cationic macrocycles showing molar masses and chemical compositions, in agreement with the feed ratios (table S6 and figure S25).

Poly(sar) 9 50+50 7100 3200 40 1.10 -

The bioactivity of the copolymers was then assessed. Cyclic peptoids are promising scaffolds to design anti-infective agents, [START_REF] Benson | A Comparison of Linear and Cyclic Peptoid Oligomers as Potent Antimicrobial Agents[END_REF][START_REF] Miller | Comparison of the Proteolytic Susceptibilities of Homologous L-Amino Acid, D-Amino Acid and N-Substituted Glycine Peptide and Peptoid Oligomers[END_REF] but antibacterial copolymers have never been developed based on this chemical design. We targeted a pathogenic Grampositive bacterium responsible for severe hospital acquired intestinal infections: Clostridioides difficile (C. difficile) [START_REF] Liu | Synthetic Polymers Active against Clostridium Difficile Vegetative Cell Growth and Spore Outgrowth[END_REF]35 and copolymers 11-21 were evaluated in vitro against this pathogen (table 2). In this study, the bacteria were incubated for 24 h with increasing concentrations of cyclic polypeptoids, to determine the minimum inhibitory concentrations (MICs). In parallel, the in vitro cytotoxicity of the best polymers (15, 19-21) was tested on a human intestinal epithelial cell line (Caco-2), allowing to measure the 50% cytotoxic concentrations (CC50) and calculate the selectivity indices (SI = CC50/MIC). Optimizing the cationic content of copolymers was a key design to optimize the antibacterial activity and copolymer 15 showed a good profile, similar to the ones of LL37 (linear AMP, table 2) or antibiotic controls (table S7). Interestingly, this statistical copolymer 15 was significantly less toxic than copolymer diblock counterparts (19-20) or than the linear counterpart 21, (linearization of 15 by propylation, figure S26). We also observed that compound 15 preserved its activity after trypsin digestion, in marked contrast to LL37 (figure S28). This resistance to proteolysis and the ability to destabilize phospholipid membranes were further confirmed by in vitro assays involving liposomes (figure S29). Overall, this proof-of-concept demonstration shows that LiHMDS-mediated REP enables quick and simple access to libraries of copolymers to design antiinfective backbones. Cyclic structures are used by living systems to achieve specific bioactivities. [START_REF] Sprott | Structures of Archaebacterial Membrane Lipids[END_REF] This encompasses cyclic antimicrobial peptides (cAMP) that are cationic and broad-spectrum anti-infective backbones. [START_REF] Hemu | Total Synthesis of Circular Bacteriocins by Butelase 1[END_REF] The use of a simple REP methodology, working in DMF, opens up interesting prospects to design simplified analogues of such cAMPs. Up to now, only few monomers have been used in REP because of some limitations related to the solvents used (in which, for example, poly(sarcosine) was not soluble). [START_REF] Lee | Crystallization and Melting Behaviors of Cyclic and Linear Polypeptoids with Alkyl Side Chains[END_REF][START_REF] Lahasky | Synthesis and Characterization of Cyclic Brush-Like Polymers by N -Heterocyclic Carbene-Mediated Zwitterionic Polymerization of N -Propargyl N -Carboxyanhydride and the Grafting-to Approach[END_REF] Considering that both the hydrophobic and cationic characters are key structural features to design antimicrobial polymers, [START_REF] Salas-Ambrosio | Synthetic Polypeptide Polymers as Simplified Analogues of Antimicrobial Peptides[END_REF] we consequently prepared copolymers by mixing N-protected-lysine-like NNCA (Z-N-lys-NCA) and Table 2. In vitro antibacterial activity against C. difficile and cytotoxicity over Caco-2 cell line (r: random copolymers and b: block copolymers).

Figure 1 .

 1 Figure 1. Macromolecular characterizations A) SEC characterization in DMF (1% LiBr) of poly(sar) 1 and poly(sar) 3. B) Full MALDI-ToF spectra with assignment of the individual peaks from poly(sar) 1.

Table 1 .Figure 2 .

 12 Figure 2. The most stable zwitterionic intermediate upon DFT calculation (basis M062X/6-31+G(d,p)) in DMF. A) front view and B) top view. The polymer backbone is colored black. Red lines show stabilizing interactions between Li + , the polymer chain, the carbamoyl extremity and the Münchnone moiety (5-OH-DHE).

Figure 3 .

 3 Figure 3. LiHMDS-mediated REP of Sar-NCA. A) Difference of elution profiles of polymer 3 before or after propylation versus the logMw (g/mol). The difference in slope is hypothetically explained by a slight contamination of 3' with cyclic structures. B) Evolution of Mn and ÐM (Mw/Mn) with M/LiHMDS ratio (determined by SEC in DMF using a dn/dc = 0.0978).

  Polymer (r: random & b: block) Cationic content (%) a C. diff. MIC (µg/mL) Caco-2 CC50 (µg/mL) SI b P(N-lys) 10 100 >100 c --P(N-lys-r-N-phe) 11 93 (90) >100 c --

The authors acknowledge Sylvain Bourasseau for assistance with size-exclusion chromatography, Anne-Laure Wirotious for NMR analysis and Joan Vignolle for helpful discussions. Christelle Absalon for MALDI TOF analysis. This work has benefited from the facilities and expertise of the CESAMO platform (Bordeaux University). This work was also granted access to the HPC resources of CALMIP supercomputing center under the allocation 2020 -P19021.


T Hoc
email: hoc@mssmat.ecp.fr

C Rey

J L Raphanel

EXPERIMENTAL AND NUMERICAL ANALYSIS OF LOCALIZATION DURING SEQUENTIAL TEST FOR AN IF-Ti STEEL

Keywords: Steel, Microstructure, Mesostructure, Scanning electron microscopy (SEM), Finite element

Localization of plastic deformation in polycrystals often occurs after a change of loading path. An experimental study using several techniques at different scales (scanning electron microscopy, microgrids, local orientation measurements, transmission electron microscopy) has been conducted on mild steel sheets (interstitial free titanium steel) prestrained by plane tension and then deformed by uniaxial tension along the first tensile axis or the transverse axis. In this way localization has been linked to microstructural anisotropy developed during the first loading path. A model based on crystalline plasticity has then been introduced in a finite element code (Abaqus) with a description of intragranular hardening linked to the evolution of the dislocation densities on each slip system. The simulations make use of a finite element representative pattern based on actual grain distribution in the aggregate. The computations predict a localization in good agreement with the experimental data and show a link between saturation of dislocation densities and concentration of plastic deformation.

Résumé -La localisation de la déformation plastique dans les polycristaux peut apparaître lors de changements de trajets de chargements. Une étude expérimentale utilisant plusieurs techniques à différentes échelles (microscopie électronique à balayage, microextensométrie, mesures locales d'orientations, microscopie électronique à transmission) a été effectuée sur une tôle d'acier doux prédéformée en traction plane soumise à des essais de traction simple dans le sens de prédéformation ou dans la direction orthogonale. La localisation a alors été reliée à l'anisotropie microstructurale formée pendant le premier trajet. Un modèle cristallin basé sur l'évolution des densités de dislocations attachées à chaque système de glissement a été introduit dans le code élement finis Abaqus. Ce modèle utilisant un motif représentatif extrait de l'expérience a permis une simulation de la localisation en bon accord avec les résultats expérimentaux et a confirmé la relation entre la saturation de la densité de dislocation et la concentration de la déformation plastique.

INTRODUCTION

Forming processes of polycrystalline sheets are often hampered by strain localizations which lead to damage or rupture and strongly limit the metal formability. The material is subjected to large strains and an element of volume follows a complex loading path. Sequential loading paths have thus been extensively used to simulate the process and to gain some understanding of strain localization phenomena in polycrystals [1][2][3]. These studies have established that strain localizes within bands and have linked this behavior to two main sources of softening resulting from the evolutions of crystallographic textures and intragranular microstructures [4]. The crystalline structure of the material and the physical mechanisms of plastic deformation by intragranular slip on particular crystallographic slip systems have thus to be accounted for. Along a monotonic load path, there is both a reorientation of the grains to optimize the conditions of plastic slip and an intragranular arrangement of dislocations to form walls with a high dislocation density limiting cells of low dislocation densities. These orientations and structures will interact and evolve when a change of strain path is imposed.

The use of finite element codes in crystalline plasticity has been growing rapidly in the past 15 years and three main applications can be found in the literature. The first is devoted to single and bi-crystals under monotonic and cyclic loadings [5], the second uses crystalline behavior to predict the development of deformation textures [6,7] for polycrystalline aggregates, and the third is focused on the study of heterogeneous local (intra-and intergranular) strain and rotation fields [8][9][10] for special aggregates with a small number of grains or for an area of a real polycrystal. Most of these studies are based on a monotonic or at the most cyclic load path and do not take into account a pre-existing intragranular microstructure. To the best of our knowledge very few simulations have been performed with the aim of studying the strain localization after a change of loading path. The main problem in this case arises in the simulation of the second path for which an adequate description has to be provided for the microstructural anisotropy and hardening resulting from the activation of slip systems during the first path.

The present work, which includes an experimental study and numerical simulations, is an attempt to evaluate the respective roles of the intragranular microstructure and the deformation texture on the strain localization for mild steel sheets submitted to a sequential loading path composed of plane strain followed by uniaxial tension.

The experimental study has focused on the analysis of the localization occurring at the beginning of the second loading path. The characterization of the material at the macroscopic scale is obtained by several mechanical experiments using uniaxial testing machines and by deformation texture measurements with a texture goniometer. Strain localization during the second loading path is analyzed at the scale of the grains via in situ mechanical testing performed by scanning electron microscopy (SEM) with local orientation measurements with electron back-scattered diffraction (EBSD) and local strain measurement with microgrids deposited on the surface of the polished sample. At the smaller scale of the dislocation microstructures, thin foils are observed by transmission electron microscopy (TEM) to characterize dislocation cells (shape, orientation, etc.).

A crystalline model for plasticity and hardening is then proposed and introduced in the Abaqus finite element code. It allows us to simulate the evolution of the local strain field and the density of dislocations on the different slip systems before and at the beginning of the localization during the second loading path. One originality of the simulations is to build a virtual aggregate by the juxtaposition of an actual pattern of 114 grains called hereafter the "finite element representative pattern" (FERP) whose initial geometry, i.e. grain shape and size, position of grain boundaries and initial crystallographic orientations, has been measured on an area of the actual polycrystal.

Experiments and simulations are shown to be in reasonable qualitative agreement. The simulations provide estimates of non-directly measurable quantities such as dislocation densities or glide magnitudes which contribute to the analysis of experimental data.

EXPERIMENTAL PROCEDURE

The material and the experimental procedure have been described elsewhere [11]. In order to make this paper self-contained, we recall only some key elements in this section. The mild steel sheets (0.9 mm thick) have been initially cold-rolled, then annealed and skin-passed so that the initial crystallographic texture is close to a Ͻ111> fiber texture. The average grain size is 20 µm.

The first load path is plane strain along the rolling direction (RD) up to macroscopic logarithmic strain values of about 0.12, 0.18 and 0.24; these three predeformed states are denoted by ⑀=0.12, ⑀=0.18 and ⑀=0.24 respectively. The second load path is uniaxial tension. It is applied to smaller samples taken from a region of uniform prestrain (checked by measures on a1m m ×1 mm grid) with their tensile axis parallel or orthogonal to the RD [12]. A letter L or T respectively is used to denote this second loading path. These tensile tests are performed by SEM at 353 K, and are referred as in situ tests [START_REF] Chiron | Local strain and temperature measurements in non-uniform fields at elev-ated temperatures[END_REF].

Microgrids (area 1 mm 2 ; mesh 5 µm×5 µm) have been deposited on the surface of the polished samples. The measure of the three-dimensional positions of their nodes in the initial and deformed states allows the computation of six components of the transformation gradient tensor F ˜(namely F 11 , F 12 , F 21 , F 22 , F 31 , and F 32 ) between two successive configurations at time t and t+dt (reference and current configurations respectively). The component F 33 can be estimated by the conservation of volume, but F 13 and F 23 cannot be determined by surface measurements. Three components of the Green-Lagrange tensor E ˜(namely E 11 , E 12 and E 22 ) are then computed in the reference configuration. With the assumption of linearity between two successive steps, one can estimate components of the velocity gradient L ˜, hence components of the deformation and rotation rates. More details on the technique and computation can be found in [START_REF] Lineau | Analyse expe ´rimentale de la de ´formation plastique d'un polycristal d'acier comparaison avec les simulations de mode ´les polycristallins[END_REF][START_REF] Lineau | [END_REF][16]. During the same test, the local orientations of several grains were measured using the EBSD technique.

In order to characterize heterogeneities at a smaller scale, some transmission electron microscopy (TEM) observations were carried out after the prestrain and after the second load path. These observations of dislocation cells were performed at 200 kV with a JEOL 200CX electron microscope equipped with a double tilt goniometer. The preparation of the samples required mechanical polishing to a thickness of approximately 150 µm followed by final thinning by a twin-jet electropolisher with an electrolyte composed of 5% perchloric acid in 2-butoxyethanol solution at 55 V and 12°C. Light marks were engraved on the sample edges parallel to the RD direction.

CRYSTAL PLASTICITY MODEL

A crystalline plasticity constitutive model that accounts for deformation by crystallographic slip and for lattice rotation with deformation was used in this analysis and implemented in the Abaqus finite element code. Implementations of the constitutive relations as a User Material subroutine in the Abaqus code have been described by Smelser and Becker [START_REF] Smelser | ABAQUS User Subroutines for Material Modeling[END_REF]. The mechanical modeling follows the framework developed by Peirce et al. [START_REF] Peirce | [END_REF] and used later by Teodosiu et al. [START_REF] Teodosiu | Proc. Int. Seminar MECAMAT'91[END_REF].

The kinetics of large transformations are used which are based on the multiplicative decomposition of the deformation gradient into elastic and plastic parts. The velocity gradient L ˜is decomposed additively into an elastic part L ˜e and a plastic part L ˜p due to glide on crystallographic slip systems. A system (s) is defined by a glide direction g →(s) and a glide plane normal n →(s) . For body-centered cubic (b.c.c.) materials such as α-iron, the directions are the Ͻ111> directions and we consider glide on the {110} and {112} planes which lead to 24 slip systems [START_REF] Chin | [END_REF]21].

The kinetics can be summed up by the following equations:

L ˜= D ˜+ W ˜= L ˜e + s g ˙(s) g →(s) n →(s) (1)
where D ˜and W ˜are respectively symmetric and antisymmetric parts of L ˜and g ˙(s) is the slip rate on system (s).

The rate form of the constitutive equation couples elastic behavior with the flow theory of plasticity in the following way:

t ෂ = K Ϸ :D ˜Ϫ s g ˙(s) R ˜(s) (2)
where t ෂ is the Jaumann rate of Kirchhoff stress, defined by

t ෂ = t • ෂ ϪW ˜t ˜+ t ˜W ˜(3) K Ϸ
is a tensor of elastic moduli, which for small elastic strains identifies with the usual fourth rank tensor of elastic moduli c Ϸ and R ˜(s) is a tensor containing the slip geometry and may be expressed as:

R ˜(s) = K Ϸ :D ˜(s) + W ˜(s) t ˜Ϫt ˜W ˜(s) (4)
The plastic slip rates are determined by a viscoplastic power law with the introduction of a threshold, so that the activation of a slip system (s) is governed by Schmid's law, |t s |Նt s c where t s c is the critical shear stress, which depends on the structural variables and temperature, while t s is the resolved shear stress on slip system (s) given by t s = s ˜:( › g s n →s ). The viscoplastic law gives:

g ˙s = g ˙o| t s t s c | n sgn(t s )i f | t s |>t s c g ˙s = 0 otherwise (5)
where g ˙o is a reference strain rate, the exponent n is close to 100 at room temperature for metallic materials.

Hardening characteristics of the material depend on the interaction mechanisms between the different slip systems (activated and latent slip systems) at the considered temperature and strain rates. Hardening is governed by the stacking fault energy of the metal, the density of dislocations, and the mean free path of the activated dislocations. It is generally expressed by a hardening matrix whose components h su take into account the interactions between systems (s) and (u):

t ˙s c = u h su |g ˙u| (6)
In its simplest form, the hardening matrix is taken with constant coefficients, sometimes reduced to two values, one for diagonal terms expressing self-hardening and one for off-diagonal terms or latent hardening terms. This can hardly be the case for sequential tests, where latent slip systems during the first loading become activated slip systems during the second path.

It is thus important to express the coefficients of the hardening matrix in terms of internal variables whose evolutions may be linked to the history of plastic deformation in the material. In our case we chose to use an average dislocation density on each slip system.

The evolution of the total dislocation density, based on Orowan's relation and an annihilation process of dislocation dipoles [22], is given by:

r ˙s = 1 b ͩ 1 L s ϪG c r s ͪ |g ˙s| (7)
where b is the magnitude of the Burgers vector, G c a parameter proportional to the characteristic length associated with the annihilation process of dislocation dipoles, and L s the mean free path of system (s) which can be expressed by:

L s = K ͩ u s r u ͪ Ϫ1/2 (8)
where r u is the total dislocation density on a latent system (u). In this formulation, K is a material parameter and the dislocation-dislocation interactions (forest interactions) are the only ones taken into account.

The critical shear stress on system (s) can be related to the dislocation densities by the relation: Table 1. Parameters of the model

a 0 K t 0 (MPa) G c (nm) n g ˙0 (s Ϫ1 ) r 0 (m Ϫ2 ) k 1 k 2 k p1 k p2 k s0 0.2 20 22 10 99 0.8×10 Ϫ15 66×10 9 1 1.15 1.05 1.05 1.3 t s c = t o + mb ͩ u a su r u ͪ 1/2 (9)
where t o is the friction stress which depends on temperature and a su characterizes the interactions between two families of dislocations (s) and (u). By differentiation of equation ( 9) with respect to time, the phenomenological law ( 6) is obtained and the hardening matrix [START_REF] Tabourot | Loi de Comportement Elastoviscoplastique du Monocristal en Grandes Transformations[END_REF] is now given by:

h su = m 2 a su ͩ l a sl r l ͪ Ϫ1/2 Ά 1 K ͩ l u r l ͪ 1/2 ϪG c r u · ( 10
)
The hardening matrix depends explicitly on the current values of the dislocation densities. Its form is similar to the one suggested by Franciosi [START_REF] Franciosi | Etude The ´orique et Expe ´rimentale du Comportement Elastoplastique des Monocristaux Me ´talliques se De ´formant par Glissement: Mode ´lisation pour un Chargement Complexe Quasi Statique[END_REF][START_REF] Franciosi | [END_REF] for b.c.c. single crystals and small strains. In his model, the terms h su increase with the strain amplitude and present a sharp variation at the beginning of the plastic deformation, then tend to a constant. In order to determine the different parameters of the hardening law, a polycrystalline self-consistent model using an explicit concentration law has been developed. The identification of the physical parameters is performed via an inverse method by using different loading paths including uniaxial tension, plane strain, cyclic plane shear and plane strain tests followed by uniaxial tensile tests. The homogeneous parts of the responses are the only parts used for identification purposes. The macroscopic stresses and texture evolutions are computed and compared to experimental results. Details of the procedure are given in [26]. The hardening law parameters obtained by this approach have values in good agreement with literature. These values are given in Table 1 where a 0 , k 1 , k 2 , k p1 , k p2 and k s0 are interaction coefficients between families of dislocations (as defined in Table 2).

The relations outlined above require a very small time step to achieve stable and accurate numerical integration. This is mainly due to the highly non-lin-

k s0 a 0 Collinear k 1 a 0 k p1 a 0 k s0 k 1 a 0 Non-collinear k 2 k 1 a 0 k p2 k p1 a 0 k s0 k 2 k 1 a 0
ear nature of the viscoplastic power law used for the computation of the slip rates. In order to increase the time step size, a forward gradient method has been used [27] for the integration of slip rates.

EXPERIMENTAL RESULTS

Macroscopic strain localization

Figure 1 shows typical nominal stress-stretch curves for the prestrained samples tested in tension along the rolling direction (L) and the transverse direction (T). The stretch is given by the ratio of the cross-head displacement to a gauge length L o .

The virgin material presents the same behavior in uniaxial tension along the rolling or the transverse direction. The effect of a prestrain by plane strain is to increase the yield stress and to produce a very specific shape of the reloading curves which can be divided into three distinct stages (after a short initial transition at the very start of the loading):

1. an elasto-plastic transition which extends to about 1% and is parabolic in the transverse direction and sharper in the rolling direction; 2. a sudden load decrease more pronounced in the transverse direction; 3. a linear decrease of the load with stretch.

Analysis of the nominal stress-stretch curves can be made in connection with the surface observation of the macroscopic strain localization (Fig. 2). The maximum of the load corresponds to the onset of the macroscopic strain localization in a macroband at about 54°to the tensile axis irrespective of the tensile direction. A second symmetrical macroband (see Fig. 2) is activated for a small increase in the applied strain for all samples tested in the rolling direction and for the sample ⑀=0.12T. In the case of sample ⑀=0.18T a new symmetrical macroband is activated at 5.7% of stretch and no new macroband occurs for the sample ⑀=0.24T.

The computation of the evolution of the strain and rotation fields within the macrobands during postbifurcation is based on microextensometry with a step the axial strain E 22 and rotation angle W 12 for the sample ⑀=0.12T at several stages of deformation along the new path, namely 4%, 5.7%, 8.2% and 10.6% of macroscopic strain. Since Lagrangian quantities are used, they are plotted on the initial undeformed con-figuration, on a vertical at a given abscissa one finds the values for the same material point.

The axial component of strain E 22 reaches fairly large values within the band (larger than 1, for an average macroscopic strain of 10.6%) and is close to zero elsewhere. Figure 3a shows the propagation mechanism of the macrobands with, for each macroband, one front remaining fixed (the one near the fixed origin of our abscissa) and the other propagating. The total rotation is large within the band, with a maximum absolute value of about 30°. The shear component E 12 is non-zero but remains small, and this is also the case when the shear component is expressed in a reference frame linked to the band [11].

Texture evolution and local orientations: the mesoscopic scale

Deformation texture measurements and computations have been performed on virgin samples and after the prestrain. The initial texture is close to a fiber texture, with a Ͻ111> fiber axis normal to the sheet plane. During plane strain a reinforcement of this fiber texture appears, increasing the average level of the orientation distribution function from 7 to 9. The textural evolution may thus be considered as somewhat limited.

Observations of the slip lines pattern within the macrobands (Fig. 4 of [11]) show two families of wavy (about 50 µm long) quasi-parallel coarse slip bands at ±50°from the tensile axis which appear at the beginning of the localization. We call these small parallel bands "mesobands". This kind of mesostructure is observed for all samples. With increasing deformation, the traces of these mesobands rotate towards the tensile axis.

This may be expressed in a more quantitative way by computing and plotting the distribution of the axial component E 22 of the Green-Lagrange strain tensor. The plot reflects the underlying structure of mesobands (see Fig. 3 of [11]).

The local reorientation field is another interesting physical quantity which may be computed across the macrobands. The EBSD technique allows the meas- ure of the local orientation of the same material point for several levels of straining along the second path. The orientation at the beginning of the appearance of the macroband is considered as reference (average macroscopic strain of 0.6% for ⑀=0.24T). The computation of a reorientation is made possible by using the Rodrigues vector as a metric representation of an orientation [28]. A difference between two Rodrigues vectors thus represents the local reorientation, which may be decomposed into an axis of rotation and an amplitude. In our case, it is ascertained that the reorientation axes are very close to the sheet plane normal, so that only the amplitude needs to be plotted. Three successive levels of strain along the second path are considered (1%, 1.3% and 1.5%) and the material points are taken along two lines, parallel to the tensile axis and crossing the macroband. The principle of the measurement is shown in Fig. 4a. Figure 4b shows the magnitudes of the reorientation of the various material points. Note that the maximum value is near the middle of the band and that outside the band the reorientation is almost negligible. Inside the band the reorientation can vary greatly. For instance, the points labeled Grain A and Grain B in Fig. 4b have very different behaviors which cannot be due only to different initial orientations. These points are located at the same distance from the origin axis, but are on two distinct lines of measurements (see Fig. 4a). The local behavior may thus be linked to the mesoband structure, and may be attributed to different levels of axial strains.

Microscopic scale: evolution of the dislocation microstructures

TEM observations have been made in grains with a normal oriented along a Ͻ111> direction which constitute the majority of the grains owing to the deformation texture. After a small amount of strain, dislocations tend to arrange themselves to form walls delimiting cells of material with low dislocation densities. The shape of these cells depends on the strain path. At the end of plane strain predeformation, the microstructure consists in two families of walls delimiting cells forming parallelograms. Neverthe-less, a better definition of the set of walls at about 90°of the tensile axis is noted.

The microstructure of the sample ⑀=0.12T is studied inside the macrobands during the second load path at 3% and 10% of macroscopic stretch. The microstructure pattern at the smaller stretch level presents large fluctuations inside a grain and from grain to grain. Nevertheless, the dislocation arrangements typical of the preloading tend to disappear. For typical microstructure see, for instance, [START_REF] Hoc | Multiscale Phenomena in Materials-Experiments and Modeling[END_REF]. At the larger stretch level of 10%, the new dislocation cells have an aspect similar to cells which would have developed along the same strain path for a monotonic straining of virgin material.

NUMERICAL SIMULATIONS

The finite element code Abaqus was used, in which a constitutive model was introduced by means of a user subroutine UMAT. The model is based on physical mechanisms of plastic deformation by intracrystalline slip, for b.c.c. metals at a temperature slightly above the transition temperature (in our case the in situ experiments were at 353 K). Its equations have been given in Section 3, one only recalls here that it is a large-deformation, elasto-viscoplastic model with an original description of intragranular hardening. The coefficients of the hardening matrix are expressed in terms of dislocation densities on each slip system, which by their evolutions provide a model of the evolution of the anisotropy of the microstructure in the course of a sequential loading. A saturation level has been introduced, since it has been observed that the large increase of deformation within a macroband does not give rise to a comparable increase of dislocation densities.

Finite element representative pattern

The polycrystalline structure had to be somewhat idealized for the implementation of a finite element code. Between the impossibility of knowing and meshing a three-dimensional structure, without destroying it, and the complete arbitrariness of imagining a virtual aggregate, a median way has been chosen which is to use a part of an actual polycrystal for the definition of a pattern of grains and to build up the complete structure by repeating this pattern. We call this pattern the finite element representative pattern (FERP) and we present hereafter its definition and how we have established its representativity.

The FERP is constructed from the surface of a mild steel sample. An area S e of 190 µm×90 µm is used which contains 114 grains whose orientations are measured by EBSD. A grid of 5µm×5 µm is superimposed which defines 684 elements, chosen as eightnoded brick elements. Each element is assigned a crystallographic orientation so that the grain boundaries are followed approximately (see Fig. 5 where different shades of gray are used to visualize the different grains).

A first check of the representativity of the FERP is that the crystallographic texture as measured by EBSD is close to the texture of the polycrystal, as measured by X-ray diffraction with a goniometer. The next check is by performing two simulations of a mechanical response, one using the FERP texture and the other with the complete experimental texture. The simulations of an uniaxial tensile test and of a cycle of plane shear deformation by a self-consistent model [26] using each data set show almost identical responses which supports the claim of representativity of the FERP.

In order to obtain a sample with adequate dimensions, the FERP is repeated eight times by translation along the tensile axis (see Fig. 5). The orientations of the 114 grains and the dislocation densities in each grain at the beginning of the second loading path correspond to the values computed at the end of the first loading. This method provides a direct construction of (T) samples. For (L) samples, we have chosen to keep the same morphology, that is to translate a FERP eight times along the tensile axis but to rotate each crystalline orientation by 90°about the normal to the sample surface. This avoids a tedious remeshing but has the drawback of introducing new orientation relations at the local level.

The results of computations presented hereafter only concern the simulation of the second loading path, that is, uniaxial tension along the rolling (L) or transverse (T) direction after a prestrain.

Stress-stretch curves

Figure 6 shows nominal stress-stretch curves in the transverse direction and in the rolling direction for any prestrain, as the result of a finite element computation using the sample and mesh discussed above.

All these curves show an elasto-plastic transition and present a maximum followed by softening. Localization, which in a macroscopic sense may be equated with the maximum of a curve, occurs earlier for the L samples than for the T samples. For comparison, Fig. 6 shows the simulation by a self-consistent model [26] of a second path in tension along the transverse direction T of a sample with a 24% prestrain. The two models give very similar results in terms of elastic limit and maximum value (although the maximum is reached faster in the finite element simulation). The discussion will define the field of application of each type of approach.

Figure 6 retains the part of the curves, after the maximum nominal stress, which should be eliminated since the validity of the model in these cases can be questioned, not only for numerical stability reasons but also because the mechanisms of reorganizations of the dislocation substructures are not part of the model.

Local analysis

In order to compare the development of localization for (L) and (T) samples, the longitudinal strain fields is examined at two successive steps of overall deformation for samples prestrained at ⑀=0.18. This analysis of the morphology of the localization process is one of the advantages of the finite element approach which computes the whole field of relevant mechanical quantities over the sample. At the beginning of the second loading, the deformation is mainly concentrated at the center of the (L) sample, while for sample (T) a large deformation occurs near the side of the sample. At a later stage, these two configurations develop localization bands presenting different morphologies, diffuse necking for (L) and a well-defined macroband for (T). These results corre-spond qualitatively to the experimental observations (Fig. 7).

A main advantage of the numerical simulations is to provide the local values of physical parameters inaccessible by experiments. More specifically, the glide quantities and the densities of dislocations can be computed in different grains. Before the beginning of the macroscopic localization and for all samples, the deformation is concentrated in five points forming a symmetrical configuration. For the reloading in the transverse direction (T) the evolution of this configuration is governed by the behavior of a "source" grain in which a large strain appears (see Fig. 7). On the contrary, for the reloading in the same direction (L) this symmetrical configuration is rapidly erased and diffuse necking takes place. The local analysis of glide amplitudes and densities of dislocations in this grain is given in the figure for the case ⑀=0.18T.

Figure 8a shows the initial activation of two systems (S1,S2), then a sharp increase of the glide amplitude for one of the two slip systems previously activated and the rapid activation of a new system (S3). This takes place at about 10% overall deformation and corresponds to a localization of the deformation.

Figure 8b shows that the dislocation densities are constant on three slip systems (P1, P2, P3) close to 9×10 7 mm Ϫ2 . This means that these three slip systems, which have been activated during the first loading, are now latent. Deformation occurs by activation of two new slip systems whose dislocation densities increase then saturate. Saturation close to 6×10 7 mm Ϫ2 corresponds to localization. Note that the activation of the third slip system is revealed by a very fast increase in its density of dislocation.

DISCUSSION

The phenomenon of localization and the behavior immediately following localization in prestrained polycrystals subjected to a new loading path has been investigated experimentally at various scales. This experimental background has been the motivation for modeling the bifurcated behavior in a layer of grains which is obtained by the juxtaposition of an FERP with the texture and mechanical behavior similar to those of the whole polycrystal.

The objective of this modeling scheme is to account for the interactions between neighboring grains by using an actual grain distribution measured by EBSD. This provides a way to include secondorder stresses in the model. At a smaller scale, the dislocation interactions are also included through the evolution laws for dislocation densities, validated in the transitory domain (beginning of a new loading path) and at the start of localization.

Our assumption is that localization results from the saturation, in terms of hardening capabilities or dislocation densities, of the active glide systems along the new loading path. This saturation is strongly influenced by the activity of glide systems during the first loading path which may have generated more or less strong obstacles to glide on the new systems. It is thus very important to have a model where intragranular anisotropy evolutions are well described. This is a reason for choosing a very anisotropic hardening matrix with variable coefficients (see for instance [START_REF] Franciosi | [END_REF]) and performing several experiments to validate the model. For b.c.c. metals it is necessary to consider 24 glide systems and the corresponding dislocation densities.

The finite element computations are consistent with the overall stress strain curves and they also reproduce the main features of the bands of localization: diffuse localization for a longitudinal reloading (weak change of load path) and a well-defined band for a transverse reloading (strong change of load path).

The experimental analysis of the localization with a measure of local crystalline orientations within the macroband complements the one done by microextensometry and presented elsewhere [11]. The strain is mostly localized within macrobands which are made up of two conjugate families of mesobands that may probably be assimilated to coarse slip. The thickness of these bands is difficult to measure because of large strain gradients but it is almost of the order of the grain size. It seems that two families of glide systems, acting on planes more or less parallel across the grains, control the deformation immediately after localization. EBSD measures made at material points inside and outside the bands show, in agreement with microextensometry measurements, that the macrobands present a fixed interface and an interface which moves as the deformation proceeds. This phenomenon has also been observed in single crystals [16]. The propagation of a band takes place by activation of mesobands near the mobile interface. These mesobands are the only sites of crystal rotations which may reach 30°. As strain and rotation saturate in a mesoband, another one is activated. The strain level reached in the bands is very large and exceeds the scope of the constitutive laws which is why the numerical simulations should be limited to the beginning of post-bifurcation. It is also noteworthy that the axes of the bands rotate toward the tensile axis as the strain increases.

The activation of mesobands is accompanied by a macroscopic strain softening. This may be modeled by the evolution law for the densities of dislocations (recall Eq. ( 7)) whose parameters have been identified on experimental data, before localization. This law expresses an equilibrium between creation and annihilation and thus enables the strain to increase without increasing the dislocation density. The anisotropy inherited from the first loading path is also taken into account. This anisotropic effect is clearly demonstrated by curves showing the evolution of dislocation densities for a grain located in the macroband subjected to a weak or a strong change of load path. For instance, Fig. 8 shows that for a strong change of path, the dislocation densities on the systems activated during the first load path remain constant as two new systems become activated during a transition period before reaching saturated values of dislocation densities.

The use of spatially continuous densities of dislocation is somewhat limiting when it comes to modeling an intragranular dislocation microstructure and its evolution. The only characteristic length introduced, albeit without any orientation effect, is the mean free path which can be linked to the distance between dislocation walls. The number of active glide systems (two or three) seems to be predicted well by the simulations. It remains to be checked that the glide plane traces associated with the active systems in various grains are almost parallel. The comparison of the stress-strain curves for the second loading path shows that the effect of the prestrain level on the slope and localization threshold is well accounted for but that the threshold is too high in the computations and that the sudden softening effect is not simulated.

The model accounts for dislocation densities and mean free paths but the strain heterogeneity at the end of the first load path has not been included. The model is not able to describe the undoing of the inherited microstructure. Despite this limitation, the most important features of the anisotropy inherited from the first load path are retained since the morphology of the macrobands is well simulated and since one checks that localization starts in particular groups of grains near the center and near the edges of the sample, as seen on the experiments, and not randomly.

There are indeed simpler models, such as self-consistent models, which can be used for the study of localization. To our knowledge, however, they have not been applied to sequential loading paths with the appearance of softening. A self-consistent scheme has been used (see [26]) to predict the stress-strain curve after prestrain of 24% and a strong change of loading path. The localization is found by the computation but with a very high threshold and followed by a very weak softening compared to experimental data. Moreover, the model cannot yield any information about the morphology of the band. Some very recent work [START_REF] Langlois | [END_REF] attempts to use self-consistent schemes to describe the evolution of microstructures before bifurcation. These schemes consider a two-phase material with moving boundaries. These models may be promising, but require complex mathematical development.

The main result of this study, which is supported by experimental evidence, is that the bifurcation finds its origin in microstructural features rather than textural ones and is strongly dependent upon the intragranular anisotropy inherited from the prestrain. The chosen criterion for localization is based on the saturation of the glide systems which have been active during the first loading path. The transitory phase, just before the bifurcation, is linked to changes in the intragranular microstructures (see for instance [3] or [4]). Two characteristics of this model may be emphasized as its strong points: a good account of intragranular anisotropy and a realistic description of orientation relations between neighboring grains through the FERP. In order to make improvements, better evolution relations for the dislocation densities would be needed, and also an account of microstructural evolutions, in particular the transition between the dislocation microstructure inherited from the prestrain and the building of a new one characteristic of the new loading path. On a somewhat different level, one should be able to consider a FERP consisting of more than one layer of grains but retaining a realistic pattern of orientation relations between neighboring grains.

CONCLUSION

This study is one of the first attempts to compare experiments and numerical simulations based on crystalline plasticity for predeformed samples undergoing a new strain path. The essential characteristics of the response of a prestrained material have been described by using a finite element code in which a constitutive relation is based on a crystalline model relating the hardening of a slip system to the evolutions of dislocation densities in the crystal.

The experimental study has called for several observation techniques at different scales to identify the physical mechanisms which control the plastic deformation of the material. The microstructural evolutions, such as the formation of dislocation cells, have been observed by TEM. Local measurements of strain fields using surface grids and crystalline orientations by EBSD have been performed by SEM. The overall deformation textures have been measured with a texture goniometer by X-ray diffraction. These observations allowed us to determine the pertinent microscale and to formulate hypotheses for the evolutions of dislocation densities on each glide system.

The model, after identification of its key parameters, has been introduced in the finite element code Abaqus, with a description of the macroscopic sample based on a representative pattern taken directly from experimental data. The use of finite elements is a direct way to account for actual grain to grain interactions and to set a characteristic length. The results of the computations are in very good agreement with experiments for the morphology of the localization and prestrain effects; they are less precise for the localization threshold. One of the main advantages of the model is an access to quantities linked to physical variables, such as dislocation densities, which cannot be measured directly. The model thus shows a link between the saturation of dislocation densities on some glide systems and the occurrence of localization at the macroscopic level.

Fig. 1 .

 1 Fig. 1. Nominal stress-stretch curves.

Fig. 2 .

 2 Fig. 2. Localization function of prestrain and reloading direction. The initial width of the sample is 3 mm.of 50 µm. The measure is done along a line parallel to the tensile axis which intersects the macrobands. The origin is fixed somewhat arbitrarily at some distance from the band. Figures3a and bshow respectively the distribution along that line of the values of

Fig. 4 .

 4 Fig. 4. Local rotation in macroband for ⑀=0.24T: (a) measurement procedure; (b) misorientation in macroband, for stretches 1%, 1.3%, 1.5%.

Fig. 5 .

 5 Fig. 5. Description of the finite element representative pattern (FERP).

Fig. 6 .

 6 Fig. 6. Nominal stress-stretch curves (self-consistent and finite element calculations).

Fig. 7 .

 7 Fig. 7. Longitudinal deformation obtained by finite element calculation for ⑀=0.18L and ⑀=0.18T. Reloading in the rolling direction: (a) E=6.3%; (c) E=12.8%. Reloading in the transverse direction: (b) E=7.1%; (d) E=15.7%.

Fig. 8 .

 8 Fig. 8. Local parameters for the "source" grain (⑀=0.18T). (a) Glide quantities. (b) Dislocation density.

Table 2 .

 2 Interaction coefficients

	Plane	{110}ʝ{110}	{110}ʝ{112}	{112}ʝ{112}
	Same	a 0

Acknowledgements-We thank Drs Ph. Pilvin and S. Forest for stimulating discussions, and the laboratory PMTM, Universite Ṕaris 13, and Laboratory LEM, ONERA, France, for help with the experimental work.