Ph.D Lionel Gelebart
email: lionel.gelebart@cea.fr

Aldo Marano
email: aldo.marano@cea.fr

Samuel Forest
email: samuel.forest@mines-paristech.fr

Ricardo Lebensohn
email: lebenso@lanl.gov

Professor, Jeffrey Kysar Chair

David Mcdowell
email: david.mcdowell@me.gatech.edu

Fionn Dunne
email: fionn.dunne@imperial.ac.uk

Lionel Gélébart

Intragranular localization induced by softening crystal plasticity: Analysis of slip and kink bands localization modes from high resolution FFT-simulations results

Keywords: Intragranular localization induced by softening crystal plasticity Crystal plasticity, Intragranular localization, Slip bands, Kink bands, FFT simulations Crystal plasticity, Intragranular localization, Slip bands, Kink bands, FFT simulations

Intragranular localization induced by softening crystal plasticity analysis of slip and kink bands localization modes from high resolutionFFT-simulations results

Introduction

A critical issue to address in bridging the gap between microscopic mechanisms and macroscopic mechanical behavior of crystalline materials is the intrinsically heterogeneous nature of plastic slip. Several authors have provided a precise description of these discrete phenomena [START_REF] Jaoul | Etude de la plasticité et application aux métaux[END_REF][START_REF] Neuhäuser | Dislocation in solids[END_REF] : deformation occurs by formation of discrete surface steps caused by the emergence of dislocations. When intense dislocation glide occurs on a few crystallographic planes, a sharp slip localization band parallel to dislocations glide planes forms called slip band, widely observed in metallic single crystals and polycrystals. Another type of slip localization band observed in deformed crystals, associated with high lattice rotation and orthogonal to the glide direction, is the so-called kink band. Kink bands are reported as a late deformation mode of metallic single crystals [START_REF] Jaoul | Etude de la plasticité et application aux métaux[END_REF][START_REF] Churchman | The yield phenomena, kink bands and geometric softening in titanium crystals[END_REF], in strongly anisotropic hexagonal crystals such as ice or Zinc [START_REF] Wilson | The origin of kinks in polycrystalline ice[END_REF][START_REF] Montagnat | Measurements and fullfield predictions of deformation heterogeneities in ice[END_REF][START_REF] Mansuy | Localization of deformation in polycrystalline ice[END_REF][START_REF] Hagihara | Orientation dependence of the deformation kink band formation behavior in Zn single crystals[END_REF][START_REF] Hagihara | Electron backscatter diffraction pattern analysis of the deformation band formed in the Mg-based long-period stacking ordered phase[END_REF], as a crack-tip localization mode [START_REF] Crone | Experimental study of the deformation near a notch tip in copper and copperberyllium single crystals[END_REF][START_REF] Kysar | Crack tip deformation fields in ductile single crystals[END_REF][START_REF] Patil | Observation of kink shear bands in an aluminium single crystal fracture specimen[END_REF] or for titanium alloys under high strain rate deformation [START_REF] Zheng | Kink deformation in a beta titanium alloy at high strain rate[END_REF]. Asaro and Rice bifurcation analysis [13] showed that in presence of strain softening the constitutive equations of crystal plasticity can predict both localization modes, and a few authors have studied the formation of kink bands in crystal plasticity simulations [START_REF] Forest | Modeling slip, kink and shear banding in classical and generalized single crystal plasticity[END_REF][START_REF] Lebensohn | Modeling viscoplastic behavior and heterogeneous intracrystalline deformation of columnar ice polycrystals[END_REF][START_REF] Kimura | Dislocation-based crystal plasticity FE analysis for kink band formation in Mg-based LPSO phase considering higher-order stress[END_REF][START_REF] Montagnat | Measurements and fullfield predictions of deformation heterogeneities in ice[END_REF].

Theoretical and numerical studies have evidenced the role of local softening mechanisms in the apparition of heterogeneous deformation [START_REF] Estrin | Local strain hardening and nonuniformity of plastic deformation[END_REF][START_REF] Brechet | Static versus propagative strain localisations[END_REF]. These mechanisms result of interactions between dislocations and crystal defects and are particularly intense in several metals such as irradiated or hydrogen charged polycrystals. In that case dislocation sources 20 are locked by a hydrogen atom atmosphere making their activation much harder. Therefore required stress to generate further slip from an already active source is thought to be lower than stress needed to activate a new locked source thus promoting intense slip on a limited number of atomic planes [START_REF] Ulmer | Hydrogen-induced strain localization and failure of austenitic stainless steels at high hydrogen concentrations[END_REF][START_REF] Aubert | Crystal plasticity computation and atomic force microscopy analysis of the internal hydrogen-induced slip localization on polycrystalline stainless steel[END_REF]. Another example is the extensively studied dislocation channeling mechanism observed in a large variety of irradiated metals such as Copper 25 [START_REF] Sharp | Correlation between cleared channels and surface slip steps in neutron irradiated copper crystals[END_REF][START_REF] Edwards | Initiation and propagation of cleared channels in neutronirradiated pure copper and a precipitation hardened CuCrZr alloy[END_REF], Zirconium [START_REF] Onimus | A statistical TEM investigation of dislocation channeling mechanism in neutron irradiated zirconium alloys[END_REF][START_REF] Fournier | Proton irradiation effect on microstruc-385 ture, strain localization and iodine-induced stress corrosion cracking in Zircaloy-4[END_REF], steel [START_REF] Sauzay | TEM observations and finite element modelling of channel deformation in pre-irradiated austenitic stainless steels -interactions with free surfaces and grain boundaries[END_REF][START_REF] Byun | Characteristics of microscopic strain localization in irradiated 316 stainless steels and pure vanadium[END_REF][START_REF] Byun | Deformation mode map of irradiated 316 stainless steel in true stressdose space[END_REF][START_REF] Field | Microstructural characterization of deformation localization at small strains in a neutron-irradiated 304 stainless steel[END_REF] or Vanadium [START_REF] Hashimoto | Deformation microstructure of neutron-irradiated pure polycrystalline vanadium[END_REF], but also quench-hardened aluminum [START_REF] Mori | Plastic deformation of quench-hardened aluminium single crystals[END_REF] and gold [START_REF] Bapna | Deformation of quench-hardened gold crystals[END_REF] single crystals. Those materials are hardened by a high density of nanometer size defects like dislocation loops or stacking fault tetrahedra. Their interaction with gliding dislocations leads to their progressive sweeping/annihilation [START_REF] Onimus | In situ TEM observations of interactions between gliding dislocations and prismatic loops in Zr-ion irradiated zirconium alloys[END_REF] and thus to a strong local softening promoting further slip in the region where it has already 30 occurred. Defect-free channels parallel to dislocation glide planes are indeed observed after deforming those materials, and are strongly correlated to surface slip steps indicating that defect-annihilation based softening is responsible for the formation of intense slip bands.

A similar mechanism is observed in ω-enriched Ti-Nb-based gum metals where ω-particles depleted channel are correlated to surface localization bands [START_REF] Lai | Deformation mechanism of omega-enriched Ti-Nb-based gum metal : Dislocation channeling and deformation induced omega-beta transformation[END_REF]. Conversely correlation of 35 clear channels with kink bands has never been observed and more generally kink bands are not reported as intragranular localization modes in strongly softening crystals.

Numerous efforts have been made aiming at modeling these phenomenon at the polycrystal scale as they have a first order influence on the macroscopic mechanical behavior or on cracking initiation due to localization induced stress concentrations. Relying on molec-40 ular dynamics or dislocations dynamics analyses of the dislocation channeling mechanism [START_REF] Rodney | Atomic-scale modeling of clear band formation in FCC metals[END_REF][START_REF] Nogaret | Clear-band formation simulated by dislocation dynamics: Role of helical turns and pile-ups[END_REF][START_REF] Arsenlis | A dislocation dynamics study of the transition from homogeneous to heterogeneous deformation in irradiated body-centered cubic iron[END_REF], many dislocation-based classical crystal plasticity models have been developed to simulate irradiated metals behavior [START_REF] Arsenlis | Dislocation density-based constitutive model for the mechan[END_REF]38,[START_REF] Onimus | A polycristalline modeling of the mechanical behavior of neutron irradiated zirconium alloys[END_REF][START_REF] Barton | A polycrystal plasticity model of strain localization in irradiated iron[END_REF][START_REF] Patra | Crystal plasticity-based constitutive modelling of irradiated BCC structures[END_REF][START_REF] Xiao | A size-dependent tensorial plasticity model for FCC single crystals with irradiation[END_REF][START_REF] Hure | Intergranular stress distributions in polycrystalline aggregates[END_REF], accounting for the softening through local defect-density dependent critical shear stress and defect-density evolution equation modeling their annihilation with increasing glide. This framework successfully 45 reproduces the main features of macroscopic behavior of irradiated metals. These strain softening based models have also been used to conduct full-field simulations of strain localization. Sauzay et al. [START_REF] Sauzay | TEM observations and finite element modelling of channel deformation in pre-irradiated austenitic stainless steels -interactions with free surfaces and grain boundaries[END_REF] have explicitly modeled one or two predefined soft clear channels embedded in a hard matrix in order to study induced stress concentrations.

Zhang et al. [START_REF] Zhang | Simulation of slip band evolution in duplex Ti-6Al-4V[END_REF] have followed a different approach at the polycrystal scale by prescribing 50 softening only on a predefined network of potential slip bands in 2D polycrystalline FE simulations. This modeling strategy allows localization bands to form but requires prior knowledge of geometric characteristics of the bands network. On the contrary, Erinosho and Dunne [START_REF] Erinosho | Strain localization and failure in irradiated zircaloy with crystal plasticity[END_REF] have modeled a full 3D polycrystal with a strain softening behavior with lar localization bands due to slip softening, however they used cubic grains microstructure and associated mesh resolution (average of 1000 elements per grain) is too coarse to reproduce accurately intragranular slip localization. Patra and McDowell [46,[START_REF] Patra | Crystal plasticity investigation of the microstructural factors influencing dislocation channeling in a model irradiated bcc material[END_REF] have recently published simulations with higher resolution using a dislocation channeling based softening model, and their results predict indeed formation of intragranular localization bands. However they simulated a 2D aggregate under the generalized plane strain hypothesis which is not representative of complex localization patterns that can arise in 3D microstructures.

Most importanty, they did not analyze the slip or kink nature of the simulated bands in order to compare results with experimentally observed cleared channels in irradiated steels.

This work aims at gaining deeper insight into the ability of softening classical crystal plasticity to simulate accurately intense intragranular localization observed in the various metals exhibiting softening mechanisms. To this end, we implemented a generic softening crystal plasticity model within a massively parallel FFT solver allowing to model three dimensional polycrystalline unit cells with higher grid resolutions and number of grains. A systematic analysis of simulated localization bands is carried out on simulations results. For 70 that purpose, we developed a methodology using full-field outputs of the FFT simulations to build localization mode maps that allow to identify and quantify simulated slip and kink bands populations.

Crystal plasticity constitutive model

The constitutive model used in this study reduces to the two main ingredients allowing to simulate strain localization along crystallographic planes: the classical finite deformation crystal plasticity framework combined with a softening flow rule. This simple and generic formulation allows to avoid any material complexity to focus the study on the link between strain softening crystal plasticity and slip localization. Crystal plasticity kinematics is described by the classical multiplicative decomposition of the deformation gradient tensor F into its elastic part F e and its plastic part F p [START_REF] Mandel | Equations constitutives et directeurs dans les milieux plastiques et viscoplastiques[END_REF] : F = F e • F p . where F p maps the reference configuration to the isoclinic stress-free local configuration where crystal lattice orientation is unchanged with respect to the reference configuration and F e maps the intermediate local configuration into the deformed configuration, describing crystal lattice distortion and rotation. It is assumed that plastic deformation takes place through the slip of dislocations on prescribed slip planes with normal n α along slip direction m α . The plastic velocity gradient L p = Ḟp • F -1 p is then determined by the shearing rate of the N s material slip systems through the relation :

L p = Ns s γα µ α (1)
where γ α is the plastic slip on slip system α and µ α = m α ⊗ n α the Schmid tensor for slip system α. Crystal elasticity is defined by a linear relation between the Green-Lagrange elastic strain tensor E e = 1 2 (F T e • F e -1) and Piola elastic stress tensor

Π e = det(F e )F -1 e • σ • F -T e
(defined in the isoclinic configuration) : Π e = Λ : E e . Λ is the fourth-order elastic tensor, σ the Cauchy stress tensor w.r.t. current configuration. Plastic flow is described by a Nortontype visco-plastic flow rule (Eq. 2) and an exponential softening of critical resolved shear stress τ α c with increasing cumulated slip on each slip system γ α cum (Eq. 4).

γα = |τ α | -τ α c K n sgn(τ α ) (2)
τ α = M : µ α (3)
τ α c = τ α c i -∆τ α (1 -exp - γ α cum γ α 0 ) (4)
Resolved shear stress τ α , is calculated by the projection of the Mandel stress tensor M = 75 det(F e )F T e • σ • F -T e on slip system α (Eq. 3). n and K are Norton-law parameters. The model involves three important material coefficients : τ c i , ∆τ α are respectively the initial critical shear stress and the maximum softening that can be reached on slip system α, and γ α 0 is a parameter adjusting the softening rate. Despite its simplicity, we believe it to be representative of most softening models for irradiated metals [START_REF] Arsenlis | Dislocation density-based constitutive model for the mechan[END_REF]38,[START_REF] Onimus | A polycristalline modeling of the mechanical behavior of neutron irradiated zirconium alloys[END_REF][START_REF] Barton | A polycrystal plasticity model of strain localization in irradiated iron[END_REF][START_REF] Patra | Crystal plasticity-based constitutive modelling of irradiated BCC structures[END_REF][START_REF] Xiao | A size-dependent tensorial plasticity model for FCC single crystals with irradiation[END_REF][START_REF] Hure | Intergranular stress distributions in polycrystalline aggregates[END_REF][START_REF] Ling | Void growth and coalescence in triaxial stress fields in irradiated FCC single crystals[END_REF]. They are indeed formulated within the same kinematic framework and rely on softening rules that, when written in the case of single slip, reduce to the form :

τ c = τ 0 + A √ ρ D (5) ρD = -Bρ D | γ| (6)
A and B being material coefficients and ρ D a defect density. Eq. 5 describes hardening due to local defect density and Eq. 6 defect sweeping by gliding dislocations, inducing softening.

When integrated for a constant shear rate these equations yield an exponential decay of the critical resolved shear stress similar to Eq. 4.

Slip/kink localization modes analysis

In this section, we present a post processing methodology designed to identify the nature of localization bands in FFT simulations results. In order to identify localization bands, we use the equivalent plastic strain field defined by Eq. 8 as a measure of local slip intensity. Contrary to slip bands, kink bands involve high lattice rotation that can be used to distinguish kinks from slip bands. Lattice rotation angle is computed using the polar decomposition of the elastic part of the deformation gradient : F e = R e • U e . Neglecting the small elastic distortion described by the right stretch tensor U e , the elastic rotation tensor R e is interpreted as the lattice rotation and the corresponding angle θ can be computed with Eq. 7. p is the cumulative plastic strain.

θ = arccos 1 2 (tr(R e ) -1) (7)
p = t 0 L p : L p dt (8)
These fields are then used to define L and R, indicator functions respectively of slip localization and high lattice rotation areas, by the relations:

L(X) = H(P (X) -pΦ D ) (9) R(X) = H(θ(X) -θΦ R ) ( 10
)
where H is the Heaviside step function and X is the material point coordinate vector.They indicate regions where fields p and θ are above a level defined by their mean value p and θ over the whole unit cell multiplied by suitably chosen relative thresholds, Φ D and Φ R respectively. Therefore L maps plastic strain localization areas that are mainly localization bands. As mentioned above, kink bands can be distinguished from slip bands because they involve large lattice rotation which suggests a natural definition of kink bands as areas exhibiting both intense plastic slip and high lattice rotation, that is to say L(X) = 1 and

R(X) = 1.
We then assume that localization area without intense lattice rotation are slip bands. Hence, kink and slip bands indicator functions, S and K, are given by :

K(X) = L(X) • R(X) (11)
S(X) = L(X) -K(X) (12)
and can be plotted simultaneously to evidence localization modes. Finally, these functions are used to determine for the slip and kink bands population, volume fractions, f S and f K , and mean value of equivalent plastic strain, < P S > and < P K >, as follows :

< P K > = 1 f K V Ω K(X)P (X)dX (13)
< P S > = 1 f S V Ω S(X)P (X)dX (14)

FFT Simulations

85

Simulations are performed using the FFT solver AMITEX FFTP1 which offers two main advantages. First, its massively parallel implementation enables to simulate strain localization in three dimensional and high resolution polycrystalline unit cells in order to overcome limitations observed in the literature. Previous simulations aiming at modeling softening induced strain localization where indeed conducted either on 3D microstructures but with 90 low resolutions [START_REF] Erinosho | Strain localization and failure in irradiated zircaloy with crystal plasticity[END_REF] or with high resolution but 2D microstructures [START_REF] Patra | Crystal plasticity investigation of the microstructural factors influencing dislocation channeling in a model irradiated bcc material[END_REF]. Second, FFT based methods provide mechanical fields in the form of 3D images very well suited for the computation of localization modes indicator functions and associated quantities defined in the former section.

The FFT solver AMITEX is based on the original fixed-point numerical scheme proposed by Moulinec and Suquet [START_REF] Moulinec | A numerical method for computing the overall response of nonlinear composites with complex microstructure[END_REF] but implements a modified discrete Green operator based on a finite difference method to evaluate derivatives in Fourier space, equivalent to the one proposed by Willot [51]. This numerical scheme is shown to be equivalent to the use of hexahedral finite elements with reduced integration [START_REF] Schneider | FFT-based homogenization for microstructures discretized by linear hexahedral elements[END_REF]. In addition, Anderson's convergence 100 acceleration technique is applied to the fixed point algorithm [START_REF] Anderson | Iterative procedures for nonlinear integral equations[END_REF][START_REF] Walker | Anderson acceleration for fixed-point iterations[END_REF][START_REF] Ramière | Iterative residual-based vector methods to accelerate fixed point iterations[END_REF].

Material constitutive law is integrated using a fully implicit θ-method implementation of constitutive equations generated by the MFront code generator [START_REF] Helfer | Introducing the open-source MFront code generator: Application to mechanical behaviours and material knowledge management within the PLEIADES fuel element modelling platform[END_REF]. Internal variables increments {∆E e , ∆γ α } between time t and t + ∆t are computed from their value at time t {∆E t e , γ α,t } by solving with Newton-Raphson algorithm the following system of non linear equations :

∆E e + E t e - 1 2 (F T,t e • F t e -1) = 0 (15) ∆γ α - |τ α | -τ α c K n sgn(τ α )∆t = 0 (16)
Note that here Green-Lagrange elastic strain E e tensor is considered as an additional variable to avoid inaccuracies in numerical evaluation of F e , following the approach presented in Ling et al. [START_REF] Ling | An elastoviscoplastic model for porous single crystals at finite strains and its assessment based on unit cell simulations[END_REF].

Simulations description

105

FFT simulations apply periodic boundary conditions to 3D regular grids of voxels (i.e. 3D images). Two types of polycrystalline microstructures have been generated using voxelized periodic Voronoi tessellations :

• 2D periodic unit cells (1 voxel thick in the X direction) containing 225 grains ( 152 ).

Because of X direction periodicity, this is equivalent to 3D infinite columnar grains 110 in the X direction, also equivalent to a 2D modeling under generalized plane strain hypothesis. Three in plane slip systems with a 60 degree misorientation relative angle are modeled ( Fig. 1), and each grain is assign a random "2D orientation" is assigned to each grain, leaving all slip plane normals and glide directions in the (Y, Z) plane. Unless otherwise stated grid resolution has been chosen so that mean grain size is resolved 120 by 50 elements. Thus, 2D and 3D simulations contain in average respectively 2500 and 125000 voxels per grain.

In addition to periodic boundary conditions, tensile loading is applied in the Z direction by prescribing the mean value of the corresponding component of the displacement gradient A systematic study of simulated bands confirms that this correspondence holds for most detected bands. The proposed methodology provides then an efficient tool to identify plastic set of threshold values to analyze different simulations allows for relative comparison of these quantities that provides qualitative insights on strain localization properties, as presented in the following sections. This rotation induced hardening mechanism opposes to material softening and can explain why kink bands population exhibit a low mesh size dependance despite the strongly softening material behavior.

H = F -

Softening influence on slip and kink bands

Fig. 5 presents evolution of slip/kink volume fractions and mean plastic strain after 1% total elongation when varying the maximum softening level ∆τ for 2D simulations. Again, slip and kink band population evolutions present significant differences. Increasing ∆τ causes slip bands volume fraction to slightly decreases whereas kink bands volume fraction increases.

The mean plastic strain level increases with softening intensity for both populations but 185 increase is steeper for slip bands.

Softening will also promote increased plastic slip in kink bands but because of rotation induced hardening, slip localization will be hindered and additional plastic slip will occur by widening of existing kink bands or formation of new ones. This mechanism explains the smaller increase in kink bands mean plastic strain level as well as their increased volume 190 fraction for a higher ∆τ .

Comparison between 2D and 3D simulations

Sensitivity to simulated Volume Element

To our knowledge, kink bands have never been reported for strongly softening metals.

However, the results presented in previous sections reveal large proportions of kinks in simulated localization band networks. In order to find out if these proportions are due to the specific microstructure of the used unit cell, 9 random realizations of a 64 grain and one 215 512 grain 3D polycrystals have been generated to characterize the variability of simulated localization bands populations with volume element instances, and volume element size. They defined this two modes as slip localization planes respectively orthogonal to glide plane normal direction n (slip bands) and glide direction m (kink bands). In the case of strain-softening crystals, their analysis shows that both modes become simultaneously possible at incipient plasticity. Incipient plasticity always occurs at small strain in metallic materials and in these conditions small strain formulation of crystal plasticity equations are valid to describe material behavior. In the small strain framework, resolved shear stress τ α on slip system α and plastic strain are computed as follow :

τ α =m • σ • n (17) εp = α γα m ⊗ sym n (18)
The Cauchy stress σ being a symmetric tensor and ⊗ sym the symmetric tensor product, m and n play a perfectly symmetric role in Eq. 17: Inverting them leaves the equation unchanged. This consideration implies that kink and slip bifurcation modes are strictly equivalent at incipient plasticity with respect to the constitutive equations. It follows that 230 structural effects, i.e. grain to grain plastic strain incompatibilities will govern the selection of slip or kink localization modes.

In order to provide an illustration of this property, two simulations have been carried out using the same grain geometry, grain orientations and material coefficients, but using is then strongly anisotropic and should be more influenced by structural effects than in more isotropic cases such as cubic crystals. Indeed, the only potential plane for slip band 250 formation in each grain is the basal plane, whereas kink banding, in prismatic planes offer 3 times more planes to accommodate grain to grain plastic incompatibilities. Fig. 11-a shows that kink bands volume fraction is approximately two times higher than slip band volume fraction, whereas the two quantities are much closer for the FCC crystals (Fig. 9). Associated localization map (Fig. 11-b) shows that localization occurs mainly at grain boundary triple 255 lines from which kink bands seem to emerge more often than slip bands.

Triple lines grain boundary induce strong stress concentrations due to grain to grain incompatibilities. They are highly likely to trigger strain localization in some or all of the neighboring grains. Thus this structural effect will promote localization paths that extend over at least 2 grains and cross grain boundaries close to triple lines. Considering 260 that geometrically there are three distinct kink planes against only one slip plane, the probability to form such transgranular localization paths across kink bands is higher. This could explain why the gap between slip and kink bands volume fractions is higher in the case of the simulated HCP crystal.

Lebensohn et al. [START_REF] Lebensohn | Modeling viscoplastic behavior and heterogeneous intracrystalline deformation of columnar ice polycrystals[END_REF] carried out crystal plasticity simulations of ice HCP polycrystals 265 deforming mainly through basal slip. As in our HCP simulation, they note significant occurrence of kink bands in their results which is in agreement with reported observations of kink bands in ice [START_REF] Wilson | The origin of kinks in polycrystalline ice[END_REF][START_REF] Montagnat | Measurements and fullfield predictions of deformation heterogeneities in ice[END_REF]. Flouriot et al. [START_REF] Flouriot | 475 Strain localization at the crack tip in single crystal CT specimens under monotonous loading: 3D finite element analyses and application to nickel-base superalloys[END_REF] have also noted kink bands formation in crack-tip field simulations in a FCC crystal that are similar to those observed with crack-tip field measurements in ductile crystals [START_REF] Kysar | Crack tip deformation fields in ductile single crystals[END_REF]. As in this two situations, kink bands observations are mostly 270 reported where strong strain incompatibilities arise such as crack-tip fields, compression of HCP single crystals [START_REF] Hagihara | Orientation dependence of the deformation kink band formation behavior in Zn single crystals[END_REF] or deformation of strongly anisotropic HCP polycrystals [START_REF] Hagihara | Electron backscatter diffraction pattern analysis of the deformation band formed in the Mg-based long-period stacking ordered phase[END_REF][START_REF] Wilson | The origin of kinks in polycrystalline ice[END_REF][START_REF] Montagnat | Measurements and fullfield predictions of deformation heterogeneities in ice[END_REF] mainly deforming through basal slip. In those cases, crystal plasticity models appears to be well-suited to simulate their formation.

However, the case of irradiated Zirconium polycrystals provides a good example to also 275 highlight their limitations. They are HCP crystals deforming mainly through basal slip and could exhibit formation of kink bands, like ice polycrystals, because of this slip anisotropy.

On the contrary, deformed Zr polycrystal observations reveal only very intense slip bands, associated to dislocation channeling, a strongly softening mechanism, while kink bands have never been reported yet [START_REF] Onimus | A statistical TEM investigation of dislocation channeling mechanism in neutron irradiated zirconium alloys[END_REF][START_REF] Fournier | Proton irradiation effect on microstruc-385 ture, strain localization and iodine-induced stress corrosion cracking in Zircaloy-4[END_REF]. Thus, kink band formation in polycrystals could be the result of a competition between structural effects and microscopic mechanisms that cannot be accounted for within the classical crystal plasticity framework. In agreement with Asaro's and Rice's bifurcation analysis, we showed that slip or kink bands localization modes are strictly equivalent for softening classical crystal plasticity mod-315 els. They only account for strain incompatibility effects on plastic localization mode formation, which translate into a large amount of kink bands in all simulations of softening polycrystals. This feature is in contradiction with experimental observation of localization bands in strongly softening metals which reveals the fundamental physical differences in slip or kink band formation from the dislocation mechanism perspective. Therefore, in order 320 to accurately simulate intragranular strain localization at the continuum scale in softening metals, physical mechanisms that can promote individually kink or slip band formation need to be accounted for with more complex theories such as strain gradient plasticity. For instance, this framework allows to account in the constitutive equations for an energy density associated to the geometrically necessary dislocation (GND) density tensor, which is known to be high in kink bands because of the large lattice curvature that they induce. Such models could be able to reduce kink band formation and will be explored in future works.

Figure 1 :

 1 Figure 1: Three inplane coplanar slip systems used for 2D simulations

1

 1 at a constant strain rate of 10 -5 s -1 . Mean values of all other components of 125 engineering stress (first Piola-Kirchhoff stress) are prescribed to zero.Constitutive model parameters used in all simulations unless otherwise stated are listed in Tab. 1. All simulations feature isotropic linear elasticity. Norton law coefficients n and K values are chosen in order to limit rate dependence without damaging numerical convergence.As simulated crystal systems have only one family of equivalent slip systems, superscript α 130 on flow rule parameter is omitted in the rest of this paper. The single slip system softening behavior and simulated macroscopic mechanical behavior for 2D, 3D FCC and 3D BCC polycristals are plotted on Fig.2for this set of material parameters which involves 20% maximum softening of the critical resolved shear stress on each slip system.

Figure 2 : 1 .

 21 Figure 2: Critical resolved shear stress evolution curve used for all simulations (a), and associated macroscopic stress-strain responses for different polycrystalline simulations (b) (FCC and BCC curves are almost superimposed)

Figure 3 :Figure 4 :Figure 5 :

 345 Figure 3: Equivalent plastic strain (a) and lattice rotation angle (b) fields for the 2D polycrystal (3 inplane slip systems) after 1% overall strain. Associated localization map (c). Slip (red) and kink (blue) bands are always respectively parallel/orthogonal to a slip plane (yellow) as illustrated by the zoom at the green-surrounded region (d). Grain boundaries are represented by black lines. Grid resolution: 750x750 voxels

165 5 . 2 .

 52 Fig.6shows the evolution of slip and kink band volume fractions and mean plastic strain after 1% total elongation for 2D simulations conducted with increasing grid resolution. When increasing grid resolution, slip bands volume fraction decreases and their mean strain level increases. Softening material behavior is known to induce numerical instabilities leading to 170

Fig. 7 and

 7 Fig. 7 and Fig. 8 show the evolution of slip/kink band volume fractions and mean plastic strain level with increasing loading for a 2D simulation and a 3D simulation of a FCC polycrystal. Results highlight a clearly distinct evolution of slip and kink band populations195

2 Figure 6 :Figure 7 :Figure 8 :

 2678 Figure 6: Influence of grid resolution on slip (red) and kink (blue) bands volume fraction (a) and mean equivalent plastic strain (b) for the 3 in-plane slip systems 2D polycristal after 1% overall strain. (c -e): associated localization maps.

Fig. 9 Figure 9 : 6 .

 996 Fig. 9 presents the evolution of slip/kink bands volume fraction with increasing loading. It is found that for the 64 grains volume element the standard deviation of simulated band volume fractions is generally smaller than the difference between slip and kink bands volume220

  two distinct crystal slip systems : the 12 FCC {111} < 110 > slip systems and the 12 235 BCC {110} < 111 > slip systems. The latter are indeed obtained by switching slip planes normals and glide directions of the FCC slip systems, thus according to Asaro and Rice analysis the potential localizations planes are the same in both polycrystals. The two identical microstructures induce identical structural effects and thus should activate the same localization planes at incipient plasticity. Simulations results clearly illustrate this point: In 240 corresponding localization maps, shown on Fig. 10, most of FCC slip (resp. kink) bands have a kink (resp. slip) counterpart for the BCC crystal structure. Then, finite strain kinematics leads to distinct evolutions of slip and kink bands because of rotation induced hardening, discussed in previous sections. It explains why the two localization maps are not strictly equivalent after applying 1% overall tensile strain.245 With a the view to investigating the influence of structural effects on slip/kink band formation a simulation has been carried out for a HCP crystal considering only the 3 basal slip systems {0001} < 1 210 >. In that case the distribution of available localization planes

Figure 10 :

 10 Figure 10: Localization maps of 3D polycrystals after loading to 1% strain with (a) : 12 FCC slip systems {111}[110] and (b) 12 BCC slip systems {110}[111], with identical geometry and grains orientations. Grains marked A,B and C illustrate the almost identical band structure, with FCC slip banding (resp. kink) corresponding to BCC kink banding (resp. slip).

20

 20

Figure 11 :

 11 Figure 11: Slip/kink bands volume fraction evolution for a 64 grains random HCP polycrystal (a). Zones marked A in the associated localization map after 1% overall strain (b) show that localization occurs generally at triple grain boundary junctions mostly through kink banding.

Figure 12 :

 12 Figure 12: Schematics of active dislocation source distributions required to form a slip or a kink band.

  Indeed, slip and kink localization modes are strictly equivalent in these models from the constitutive perspective. Yet considering associated dislocation mechanisms, both modes are very different. Indeed, slip bands formation involves only a few very active dislocation 285 sources located in close slip planes whereas kink band formation involves the activation of a considerably larger number of sources aligned along the direction normal to the slip plane as represented on Fig. 12. Hence, the two modes are not equivalent regarding dislocation mechanisms . Hence for materials exhibiting strong softening mechanisms intensification of dislocation emission from already active sources is promoted over activation of new sources and thus kink bands formation should be highly unlikely. Kink bands are indeed never reported in observations of deformed irradiated or quench-hardened metals, that undergo such softening mechanisms (dislocation channeling), where slip localization seems to occur only through intense slip bands. Consequently classical crystal plasticity models appear to be fundamentally unable to reproduce accurately localization bands formation in these tion maps that where used to accurately identify the slip and kink band populations in the localization network, for both simple bidimensional simulations and realistic three dimensional polycrystals with complex crystal systems. Quantification of band volume fraction and mean plastic strain highlight the distinct evolution of slip and kink bands. Because of rotation induced hardening, kink bands tend to widen with increasing loading, and their 310 volume fraction and mean strain level are less sensitive to mesh dependence and softening magnitude. We conclude that it is essential to identify the nature of localization modes when simulating intragranular strain localization.

  ical behaviour of irradiated Cu, Philosophical Magazine 84:34 (2004) 3617-3635. doi:10.1080/ 14786430412331293531.[38] S. Krishna, A. Zamiri, S. De, Dislocation and defect density-based micromechanical modeling of the mechanical behavior of FCC metals under neutron irradiation, Philosophical Magazine 90:30 (2010)

Table 1 :

 1 Material parameters used for all simulations

http://www.maisondelasimulation.fr/projects/amitex/html/

Material Science & Engineering A 702 (2017) 218-224. doi:10.1016/j.msea.2017.07.015.

materials. [START_REF] Edwards | Initiation and propagation of cleared channels in neutronirradiated pure copper and a precipitation hardened CuCrZr alloy[END_REF]

Conclusions

We have used a massively parallel FFT solver to study intragranular strain localization induced by a strain softening classical crystal plasticity model. This numerical tool allows to compute realistic three dimensional polycrystalline simulations including full description 300 of complex crystal systems and achieve grid resolutions that allow for a fine description of intragranular localization bands. Our simulations show the formation of an intragranular localization network of narrow bands exhibiting high plastic slip.

We proposed a new method to study the nature of simulated localization bands based on the analysis of equivalent plastic strain and lattice rotation fields. We produced localiza-

Supplementary Material

Click here to download Supplementary Material: Film_localisation_3D.avi *Graphical Abstract