École doctorale n 364 : Sciences Fondamentales et Appliquées. Doctorat ParisTech T H È S E. pour obtenir le grade de docteur délivré par

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1 N : 2009 ENAM XXXX École doctorale n 364 : Sciences Fondamentales et Appliquées Doctorat ParisTec T H È S E pour obtenir le grade de docteur délivré par l École nationale supérieure des mines de Paris Spécialité Mécanique Numérique présentée et soutenue publiquement par Ana Laura CRUZ FABIANO le 10 décembre 2013 Modelling of crystal plasticity and grain boundary motion of 304L steel at te mesoscopic scale ~~~ Modélisation de la plasticité cristalline et de la migration des joints de grains de l'acier 304L à l'écelle mésoscopique Directeurs de tèse : Marc BERNACKI et Roland LOGE Jury Mme. Brigitte BACROIX, Directeur de recerce CNRS, LSPM, Université Paris 13 Président M. Laurent DELANNAY, Professeur, immc, Université catolique de Louvain Rapporteur M. Frank MONTHEILLET, Directeur de recerce CNRS, SMS, ENSM-SE Rapporteur M. Rémy BESNARD, Ingénieur de recerce, CEA Valduc Examinateur M. Roland LOGE, Directeur de recerce CNRS, CEMEF, Mines ParisTec Examinateur M. Marc BERNACKI, Cargé de recerce, HDR, CEMEF, Mines ParisTec Examinateur Examinateur T H È S E MINES ParisTec CEMEF UMR CNRS , rue Claude Daunesse - BP Sopia Antipolis Cedex - France

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3 1 Acknowledgments First of all, I would like to convey my deepest gratitude to my advisors Marc Bernacki and Roland Loge. I am very grateful for teir elp and teir support. I appreciate all teir contribution, ideas and remarks. Tanks to teir wise advices and requirement level, my tree years ere in CEMEF were scientifically and personally enricing. It is important to mention Javier Signorelli and Laurent Delannay. Witout teir advices I would not be able to reac all te results presented in tis work. Finally, I want to tank Natalie Bozzolo for te scientific and personal conversations we ad during tis period. I would also like to express my gratitude to all CEMEF staff, Goup EII and everyone from support informatique. Special tanks go to Patrick Coels and Marie-Françoise Guenegan for teir motivation and elp and to te directors Yvan Castel and Elisabet Massoni. I am also grateful for all my friends in CEMEF, Rebecca, Ugo, Fadi, Koffi, Andrea, Micel, Olivier, Tommy, Alexis, Karim, Ala, Cristope, Takao, Alejandro, Ke, Sira, Oscar. Life in CEMEF would not be so pleasant witout tem. Special tanks for Sebastian and Yuan, my two colocs de bureau. It was a great pleasure to sare wit you my day by day life during tese tree years. At last, but not least, to Siam, a great friend tat I found ere. Witout er support it would be more difficult to face all te callenges encountered during tis tesis. During tese tree years I also found good friends outside CEMEF wose support was essential during tese tree years. Stepanie, Fabrice, Eléonore, Mikolay, Benjamin, Nicolas, Angelo, Asutos, Micele, Romain, Silvia, Hue, Sara, Leo, Josette and Raymond. Special tanks for Amir and is patience to read my report. Tanks Ludo for everyting. I will never forget te great moments we all spent togeter. I could not fail to mention my Brazilian friends. Even toug we are far from eac oter, I always knew tat I could count on you. Silas, Vitor, Paulão, Taina, Carlina, Diguino, Gustavo, Bruno, Vini, Rapael, Carol, Lucas, André, David, Daniel, Steil. Tank you for your friendsip! Finally, I would like to tank my family, especially my parents for everyting tey ave given me and all tey give me every day. Tank you for your confidence, for your advices, for being tere every time I needed you. Also, I want to tank my broters and may fake uncle Tio Marcão.

4 2 Table of contents General Introduction Constitutive Beaviour of Polycrystalline Aggregates Introduction Crystal Plasticity Caracteristics of deformed microstructure Components of a single crystal model Hardening Hardening laws witout size effects Hardening laws including size effects Single Crystal Model Formulation Elastic-Viscoplastic Formulation Flow rule and ardening law Time Integration Sceme of te constitutive law Finite Element Formulation Balance laws Variational formulation Time discretization Spatial discretization Te MINI-element Te discrete problem Resolution Non-linear system of equation to be solved Resolution of te non-linear system General solution procedure and numerical implementation Conclusion Statistical generation of polycrystals in a finite element context and a level-set framework Metod and results Introduction Voronoï and Laguerre Voronoï tessellations Microstructure immersion in FE Mes Microstructure Generation - Results Obeying a grain size distribution in 2D microstructures Voronoï Tessellation Metod Laguerre-Voronoï Tessellation Metod Obeying a grain size distribution in 3D microstructures Voronoï Tessellation Metod Laguerre-Voronoï Tessellation Metod Topological study of 2D Laguerre-Voronoï microstructures Morpological study of Laguerre-Voronoï microstructures Conclusion Deformation beaviour of 304L steel polycrystal and tantalum olygocrystal Introduction L Steel Case... 79

5 L stainless steel Cemical composition Mecanical beaviour Microstructural caracterization - initial state Planar compression test (Cannel Die test) Description of te test Material parameters - 304L Effects of mes type in igly resolved polycrystalline simulations Considering only SSDs Considering bot SSDs and GNDs Results Stress - Strain analysis Dislocations density and energy distribution analysis Strain distribution analysis Grain size effects Tantalum Case Tantalum Compression test Test description Digital oligocrystal Material parameters Results Sample sape Dislocations density distribution Deformation distribution Texture analysis Conclusion Grain Boundary Migration - Grain Growt Modelling Introduction Context and Equations Full Field modelling of grain growt assuming isotropic grain boundary energy and mobility Te equations Mobility Te grain boundary energy Finite element model in a level set framework Velocity field expression Numerical treatment for multi-junctions Container level set functions Grain growt algoritm Results Academic Test - Circular grain case Academic test - 3 grains case Von Neumann-Mullins cases L - 2D polycrystal grain growt Influence of te microstructure generation metod on grain growt prediction Assessment of simplified 2D grain growt models from numerical experiments based on a level set framework Grain size distributions

6 4 5.2 Burke and Turnbull study Hillert/Abbruzzese model study Conclusion Grain Boundary Migration - Static Recrystallization Introduction Static recrystallization Nucleation mecanisms Nuclei growt Te Jonson-Mal-Avrami-Kolmogorov (JMAK) model L steel static recrystallization Strain level effect Temperature and strain rate effects Recrystallization simulations using a full field formulation Finite element model and level set framework Velocity field expression Recrystallization algoritm Results Academic Test - Circular grain case Academic Test 2 grains case Polycrystalline 2D simulation Random nucleation sites Necklace-type nucleation sites Coupling crystal plasticity and static recrystallization Critical dislocation density and nucleus radius Number of nucleation sites Coupling wit crystal plasticity results D recrystallization kinetics results in site saturated nucleation conditions Conclusion Conclusions and future work References Appendix

7 5 General Introduction

8 6 Aims Metallic materials exibit a crystalline structure and are in essence eterogeneous materials. Tey are in fact polycrystals, eac crystal presenting a continuous lattice orientation. All mecanical and functional properties of metals are strongly related to teir microstructures, wic are temselves inerited from termal and mecanical processing. In a general view, te pysical penomenon induced by te termo-mecanical processes generates complex microstructure canges and tese microstructure evolutions modify te material properties. Tus, te understanding and modelling of tese microstructure evolutions during termo-mecanical processes are of prime importance concerning te prediction and te control of te material mecanical properties. In tis work, te addressed relevant pysical penomena are recovery, recrystallization and grain growt. Microstructural transformations suc as pase transformation or particles precipitation will not be discussed. Te term recovery refers to canges in a deformed material wic occur prior to recrystallization and wic partially restore te material properties to teir values before deformation. It is known tat recovery is primarily due to modifications in te dislocations structure of te material. A part of te existing dislocations will be anniilated wile te oters are reorganised. Normally, te dislocations reorganize temselves into cells. Dislocation recovery corresponds to a series of events wic are scematically illustrated in figure I.1. Figure I.1: Various stages in te recovery of a plastically deformed material [Humpreys, 2004]. Weter any or all of tese stages occur during te annealing of a particular specimen will depend on a important number of parameters, including te material purity, strain, deformation and annealing temperature. A review of te intensive work done concerning te recovery of deformed metals can be found in [Beck, 1954], [Bever, 1957], [Titcener, 1958] and [Nes, 1995].

9 7 Te recrystallization process involves te formation of new strain-free grains in certain parts of te material and te subsequent growt of tese to consume te deformed microstructure. Te location were tese new grains will be formed depends on te material microstructure before plastic deformation. It is convenient to divide te recrystallization process into two regimes: nucleation wic corresponds to te first appearance of new grains in te microstructure, and growt during wic te new grains replace te deformed material. Te progress of recrystallization wit time is commonly represented by a plot of te volume fraction of material recrystallized (X) as a function of time. Tis plot usually as a caracteristic sigmoidal form. Figure I.2 illustrates static recrystallization steps. Figure I.2: Static recrystallization sceme [images from Humpreys, 2004]. Te grain boundary motion driving forces are te internal energy difference stored during plastic deformation (in te form of crystallograpic defects, especially dislocations) between two adjoint areas (recrystallization penomenon) and te reduction in te energy wic is stored in te form of grain boundaries (grain growt penomenon). In [Doerty, 1997] and [Rios, 2005], te autors present a complete review of te recrystallization process. Recovery, recrystallization and grain growt processes can occur statically (after plastic deformation), or dynamically (during plastic deformation). Te way tese penomena will occur is based on te material properties and te processing conditions (temperature and strain rate). Grain growt is a penomenon occurring during and after a polycrystal full recrystallization and as te effect of increasing te average grain size at te expense of smaller ones tat will tend to disappear. Te grain growt kinetics is slower tan te recrystallization kinetics. Te grain growt driving force is te reduction in te material energy stored in te form of grain boundaries. Figure I.3 illustrates te grain growt penomenon. Figure I.3: Grain growt sceme [images from Humpreys, 2004]. Altoug recrystallization often precedes grain growt, it is of course not a necessary precursor. Te teoretical basis for understanding grain growt was laid in [Smit, 1948], [Smit, 1952], [Burke, 1952], [Hillert, 1965]. It is interesting to igligt tat materials processing, including recrystallization and grain growt, is affected by te grain boundary properties. A grain boundary separates two

10 8 regions of te same crystal structure but of different orientation. Te grain boundary is made of many crystal lattice defects (like dislocations and vacancies) wit a tickness of a few atomic layers (figure I.4). Figure I.4: Sceme of a disordered grain boundary structure [Porter, 2008]. Te nature of any given boundary depends on te misorientation of te two adjoining grains and of te orientation of te boundary plane relative to tem. Te lattices of two grains can be made to coincide by rotating one of tem troug a suitable angle about a single axis. In general te axis of rotation will not be simply oriented wit respect to eiter grain or te grain boundary plane. However, tere are two special types of boundary tat are relatively simple: pure tilt and pure twist (figure I.5). Te difference between tese two simple boundaries is related to te rotation axis direction relative to te grain boundary. Te first grain boundary type, te tilt boundary, occurs wen te axis of rotation is parallel to te grain boundary plane wile a twist boundary is formed wen te rotation axis is perpendicular to te grain boundary. Figure I.5: Te relative orientation of te crystals and te boundary forming (a) a tilt boundary and (b) a twist boundary [Porter, 2008]. Anoter convenient classification of grain boundaries is related to teir misorientation. Te first group corresponds to te grain boundaries wose misorientation is greater tan a certain critical angle - ig angle grain boundaries, and te second corresponds to tose wose misorientation is smaller tan tis critical angle - low angle grain boundaries. Te critical angle at wic te transition from low to ig angle boundaries occurs is typically taken between 10 and 15 for cubic materials. Te difference in structure between low angle and ig angle grains boundaries is illustrated by te bubble-raft model in figure I.6.

11 9 Figure I.6: Raft of soap bubbles sowing several grains varying misorientation. Note tat te boundary wit te smallest misorientation is made up of a row of dislocations, wereas te ig angle boundaries ave a disordered structure in wic individual dislocations cannot be identified [Sewmon, 1969]. Hig angle boundaries contain large areas of poor fit and ave a relatively open structure. Te bonds between te atoms are broken or igly distorted. In low angle boundaries most of te atoms fit very well into bot lattices. Normally tis boundary is formed by a few aligned dislocations. As a consequence, te lattice presents a very little free volume and te interatomic bonds are ligtly distorted. Tus, a precise prediction of grain boundary motion remains also dependent of a precise description of te grain boundary pysical properties according to its nature and to te termal state. Finally, as already underlined, during a deformation, te microstructure of a metal canges in several ways, but, te most important cange for grain boundary motion is te accumulation of dislocations. Te stored dislocations are te main defects responsible for te internal energy increasing during plastic deformation. Te sum of te energy of all dislocations represents te deformation stored energy. Knowing tat every stage of te annealing process involves loss of some of te stored energy, te correct prediction of te stored energy after deformation is essential to correctly predict te microstructure evolution during annealing treatments. Te objective of tis P.D. tesis is te development of a numerical model able to predict microstructure evolution, including te material ardening, recrystallization and grain growt at te mesoscopic scale. Te motivation beind te modelling of polycrystalline metallic material evolutions during termo-mecanical treatments at tis scale are: - from a tecnological point of view, assist te design of termomecanical processes by taking into consideration microstructural features, suc as crystallograpic orientations, in order to account for macroscopic anisotropic beaviour, - te use of full field models, as te one presented in tis work, allows te verification and te improvement of models wic were developed based on teoretical assumptions wic are not easily verified experimentally (mean field models), - from a more fundamental point of view, increase our understanding of te mecanisms operating at te microstructural scale.

12 10 Framework and layout of tis tesis Even if te scope of te present work seems rater fundamental, te development of more accurate numerical models for te grain size predictions in metals during termomecanical treatments is nowadays of prime importance in order to conform to increasingly tigter standards witout excessive development costs, mainly in te nuclear and aeronautic domains. Tis observation seems perfectly illustrated by te fact tat te present work, dedicated to models of polycristalline structures, was supported by an industrial consortium involving five partners. Indeed, tis PD tesis was conducted as a part of te MicroPro project wic gaters te following partners: - ArcelorMittal/Industeel ( ot rolled metallic plates producer. - Areva/Sfarsteel ( world-leading company in nuclear energy. - Ascometal ( special steels pieces producer. - Aubert et Duval ( special steels pieces producer. - CEA ( nuclear power applications developer. Te manuscript is organized as follows: te first capter presents a review of crystal plasticity and work ardening models existing in te literature. Also, a detailed description of te crystal plasticity model used during tis P.D. tesis is presented. Two different ardening laws are presented and discussed. Te first one does not take into account te size effects, contrary to te second one. Te finite element implementation associated to te model wit an implicit integration is exposed. After presenting te crystal plasticity models, te digital microstructure statistical generation togeter wit te grain boundary description metod are presented in capter 2. Two different digital microstructure generation metods are discussed and tested in order to statistically obey te features of te considered microstructures: te Voronoï, and te Laguerre-Voronoï metods. Te grain boundary description is performed wit an implicit metod capturing te grain interfaces and immerging tem in a finite element mes [Bernacki, 2009]. Tis metod is based on a level-set approac allowing to define te different interfaces and to perform mesing adaptation. In capter 3, crystal plasticity numerical results are presented and discussed. Two different test cases are used in order to validate te ardening laws implemented in te crystal plasticity model. Te first one corresponds to a planar compression test (cannel die test) on a 304L steel polycrystal. Te second one corresponds to a simple compression test on a tantalum olygocrystal composed of six different grains. After presenting te crystal plasticity simulations results, an extension of te model developed in [Bernacki, 2008], [Bernacki, 2009] and [Logé, 2008] concerning te modelling of grain boundary migration in a full field context is proposed. In capter 4, te addition of te grain growt penomenon by capillarity to te full field model is detailed. A few academic tests are presented in order to validate te proposed grain growt algoritm, and tis is followed by a 2D study of polycrystalline grain growt. Full field predictions are compared to tose of different grain growt mean field models, and conclusions are drawn on te applicability of te latter. Finally, in capter 5, static recrystallization is investigated. A first study analysing te influence of te capillarity effects during te recrystallization process is presented and, in a second part, te static recrystallization model is coupled to te crystal plasticity model detailed in te first capter. Comparisons between experimental results on

13 11 304L steel and te numerical predictions obtained wit te proposed new formalism are discussed. All te developments and calculations of tis PD tesis were performed wit CimLib, a C++ finite element library developed at CEMEF [Digonnet, 2007]. Te work involved in tis PD tesis as contributed to te following written communications: A.L. Cruz-Fabiano, R. Logé and M. Bernacki, Assessment of simplified 2D grain growt models from numerical experiments based on a level set framework, submitted to Modeling and Simulation in Materials Science and Engineering in and to te following oral communications: M. Bernacki, K. Hitti, A.L. Cruz-Fabiano, A. Agnoli, R. Logé, Génération statistique de VERs et modélisation EF d évolutions microstructurales, Journées tématiques MECAMAT, Mai 2011, CEMEF, Sopia-Antipolis. A.L. Cruz-Fabiano, R. Logé, M. Bernacki, Modelling of static and dynamic recrystallization processes at te mesoscopic scale in 304L stainless steel, Journées tématiques MECAMAT, Mai 2011, CEMEF, Sopia-Antipolis. A.L. Cruz-Fabiano, R. Logé, M. Bernacki, Comparison between different simplified grain growt models using full field modelling metod results, ECCOMAS 2012, Vienna, Austria, September 10-14, 2012 A.L. Cruz-Fabiano, R. Logé, M. Bernacki, I. Poitrault, M. Teaca, A. Gingell, F. Perdriset, E. Guyot, Modèles de recristallisation et de croissance de grains de l acier inoxydable 304L et applications industrielles, Journée "Aciers inoxydables et Industrie Nucléaire, dernières avancées", SF2M, 16 Mai 2013, Saint Etienne. M. Bernacki, N. Bozzolo, R. Logé, Y. Jin, A. Agnoli, A.L. Cruz-Fabiano, A.D. Rollett, G.S. Rorer, J.-M. Francet, J. Laigo, Full field modelling of recrystallization in superalloys tanks to level-set metod, EUROSUPERALLOY 2014, Giens, France, May 13-16, Résumé français Ce capitre constitue l introduction de cette tèse. Les objectifs de ce travail de tèse ainsi que les concepts généraux de joints de grains, restauration, recristallisation et croissance des grains y sont présentés. Finalement, l organisation de ce manuscrit de tèse est détaillée et les contributions liées à ces travaux de tèse sont énumérées.

14 12 Capter 1 Constitutive Beaviour of Polycrystalline Aggregates Contents 1 Introduction Crystal Plasticity Caracteristics of deformed microstructure Components of a single crystal model Hardening Hardening laws witout size effects Hardening laws including size effects Single Crystal Model Formulation Elastic-Viscoplastic Formulation Flow rule and ardening law Time Integration Sceme of te constitutive law Finite Element Formulation Balance laws Variational formulation Time discretization Spatial discretization Te MINI-element Te discrete problem Resolution Non linear system of equation to be solved Resolution of te non linear system General solution procedure and numerical implementation Conclusion... 45

15 13 1 Introduction As it was discussed in te General Introduction, te main objective of tis P.D. tesis is te development of a numerical model able to predict microstructure evolution, including te material ardening, recrystallization and grain growt at te mesoscopic scale. Tese microstructural evolutions are not independent from eac oter. In a simple way, one can say tat recrystallization processes depend on te nature of te deformed state, especially te dislocation density distribution and te crystallograpic misorientations. Te material state after plastic deformation depends itself on te deformation process (rolling, wire drawing, extrusion, compression, tension, torsion, sear, etc.) and on te type of material. Bot aspects play an important role in te recrystallization kinetics taking place during or after plastic deformation. At te same time, grain growt kinetics is linked to te grain size distribution after total recrystallization. Since tese penomena are linked to eac oter, a first step of an accurate modelling of recrystallization and grain growt involves te correct prediction of te microstructural eterogeneities developed during plastic deformation. So, in tis work, some of te penomena involved in plastic deformation of metals are considered, in order to predict important features of a final deformed state. Te simulation of te microstructure transformation during plastic deformation is not a simple problem. Numerous models ave been developed to describe te microstructure evolution taking into account te material texture and te influence of te neigbouring grains (crystal plasticity models). In [Resk, 2010], te autor presents an interesting review of several models. Here, only te main approaces are described. One of te first models proposed to take te texture into account was proposed by Taylor [Taylor, 1938]. Tis model, also known as full-constrained model, is still used in order to study te constitutive response of polycrystalline materials due to its low numerical cost. However, since tis model assumes tat all grains deform in te exact same manner as te polycrystal, violating te stress equilibrium, te interaction between neigbouring grains are not taken into account and, as a result, tis model is not able to describe te features of te deformed state. Anyow, in large scale applications, texture-induced anisotropy is fairly well predicted [Marin, 1998a], especially for materials presenting a ig degree of crystal symmetry (like cubic crystals) and subjected to strains well beyond te elastic limit. To improve te prediction of polycrystalline deformation beaviour, oter models suc as relaxed-constraint [Honeff, 1981], [Kocks, 1982] and multi-grains-relaxed-constraint models [Van Houtte, 2002], [Van Houtte, 2005] ave been developed. Relaxed-constraint models satisfy selected compatibility relations and ignore some intergranular equilibrium components. In oter words, tese models are based on te idea of freeing some degrees of freedom of eac grain in order to improve textures predictions. One of te main conceptual faults of tese relaxed-constraint models is te fact tat te volume average of te velocity gradients over te wole polycrystal migt not be equal to te macroscopic one. As opposed to te Taylor model, te self-consistent models satisfy bot compatibility and equilibrium between grains in an average sense [Eselby, 1957]. In [Eselby, 1957] te autors considered te problem of determining te stress and strain in an elastic ellipsoidal inclusion surrounded by an unbounded elastic medium. Te particle and te medium present te same properties. Wen dealing wit polycrystalline materials, eac grain is considered as an inclusion embedded in a polycrystalline medium composed of all oter grains. Te medium beaviour is determined by considering different representative grains. Tese grains represent all oter grains aving te same crystallograpic orientation. Finally, te overall beaviour is computed as te average response of te representative grains. Later on, te model was extended to elastic-plastic beaviour [Berveiller, 1979], [Hill, 1965], and to viscoplastic beaviour [Hutcinson, 1976], [Lebenson, 1993], [Molinari, 1987].

16 14 Bot Taylor type and self-consistent models assume tat individual grains deform omogeneously. As a consequence, it is not possible to predict te exact positions of nucleation sites during recrystallization. In order to capture te intragranular eterogeneities developed during plastic deformation, two metods can be used. Te first one is te finite element metod. Wit tis metod, bot intra and intergranular eterogeneities can be computed. Te constitutive response of eac finite element integration point is determined using te single crystal constitutive model. Te main advantage of finite element models compared to te oter polycrystal plasticity models is tat morpological effects (grain size, sape and topology) can be taken into account. Te second one is te fast Fourier transform (FFT). Tis metod as been used to describe te mecanical beaviour of polycrystalline aggregates wit periodic boundary conditions for elastic deformations [Brenner, 2009] and for rigid-viscoplastic conditions [Lebenson, 2001], [Lebenson, 2008], [Lee, 2011]. In [Lebenson, 2001], te autors sowed tat te FFT metod is numerically less expensive tan te finite element metod for problems of te same size. However, te need of periodic boundary conditions and of regular grids tat cannot be adapted during te simulation represent te model principal limits. Concerning te crystal plasticity simulations, te work presented in tis report is a continuation of te work developed by H. Resk [Resk, 2010] during er P.D. tesis at CEMEF. Te main objective of er P.D. project was te development of a crystal plasticity finite element model (CPFEM) and its implementation witin te CimLib library. Te ardening model considered during er P.D. tesis was a Voce type saturation law, a model were te dislocation density is implicitly represented. During te present P.D. tesis, due to te interest in modelling recrystallization processes based on crystal plasticity simulation results, te explicit computation of dislocation densities was adopted, i.e. dislocation densities were internal variables of ardening laws. Two kinds of ardening models are studied: te first one does not consider grain size effects, wile te second one does. Wen size effects are considered, two kinds of dislocation density are taken into account: te statistically stored dislocations (SSDs) and te geometrically necessary dislocations (GNDs). In tis capter, a bibliograpic review of te crystal plasticity teory is presented. Different ardening laws tat can be found in te literature are igligted, and te differences between te two dislocation types (SSDs and GNDs) are discussed. In a second part, te single crystal plasticity model implemented in te CimLib library is presented and finally, te finite element implementation of te above constitutive laws is detailed, in te context of an implicit integration. 2 Crystal Plasticity In crystal plasticity teory, te material plastic deformation is modelled using te slip system activity concept. Dislocations are assumed to move across te crystal lattice along specific slip systems, wic are caracterized by specific crystallograpic planes and directions. Wen te material is subjected to loading, te applied stress resolved along te slip direction on te slip plane initiates and controls te extent of dislocation glide. Tis latter as te effect of searing te material wereas te material volume remains uncanged and te crystal lattice remains constant. Moreover, te crystal lattice can deform elastically. However, elastic strains are small compared to plastic strains and are sometimes neglected in crystal plasticity models. Finally, te crystal lattice can also rotate to accommodate te applied loading. Tis lattice rotation (or spin), is responsible for texture development. Te concept of lattice rotation in crystal plasticity is not, at first and, easy to grasp, especially compared to material rotation (or rigid body rotation). In [Peeters, 2001], te autors illustrate well tis fundamental difference wit Figure 1.1.

17 15 Figure 1.1: (a) and (b) A sear γ on a slip plane does not cause te lattice to rotate, altoug a material vector may rotate; (b) and (c) An additional rotation - wic also causes te crystal lattice to rotate - will bring te crystal in a position corresponding to te strain forced upon it: e.g. pure elongation in te direction AC [Peeters, 2001]. Normally, dislocation slip takes place on most densely packed planes and directions. In te case of FCC metals (like te 304L austenitic steels), slip occurs in {111} planes and <110> directions. Terefore, crystallograpic slip is assumed to occur on te 12 octaedral {111}<110> slip systems. Tese considerations form te basis of classical crystal plasticity teory. Oter modes of deformation in polycrystals, like twinning or grain boundary sliding, are neglected in tis discussion. 2.1 Caracteristics of deformed microstructure As it was discussed in te General Introduction, te material microstructure after plastic deformation plays an important role in recovery, recrystallization and grain growt penomena. In tis paragrap, a brief review of te microstructure canges suffered by te material during termo-mecanical process is presented. Wen polycrystalline materials, wit an assumed random crystallograpic texture, plastically deform, deformation is essentially eterogeneous even under simplified condition. Tis is a consequence of te interaction between neigbouring grains leading to te formation of complex microstructures. During te material deformation, its microstructure canges in diverse ways. First, te grains sape canges and, often, an increase in te total grain boundary area takes place. Te new grain boundary areas are created during deformation by te incorporation of some of te dislocations tat are continuously created. Te rate of increase of grain boundary area per unit volume depends on te mode of deformation. Figure 1.2 illustrates te calculated increase in te grain boundary area as a function of strain for several deformation modes.

18 16 Figure 1.2: Rate of growt of grain boundary area per unit volume (S v ) for different modes of deformation assuming an initial cubic grain size D 0, [Humpreys, 2004]. Nucleation of new grains normally occurs on te areas near te grain boundaries. As a consequence, te increase in te grain boundary surface may impact te nucleation of new grains during recrystallization. A second main microstructural cange is te appearance of an internal structure witin te grains. Once again, tis material cange is a result of te accumulation of dislocations during plastic deformation. Figure 1.3 summarises te main features of te deformed state according to teir lengt scale. Figure 1.3: Te ierarcy of microstructure in polycrystalline metal deforming by slip. Te various features are sown at increasing scales: (a) dislocations, (b) dislocation boundaries, (c) deformation and transition bands witin a grain, (d) specimen and grain scale sear bands. [Humpreys, 2004]. Sear bands (Figure 1.4a) are non-crystallograpic in nature and may pass troug several grains. Tey are a result of plastic instability and it can be compared to te necking occurring in a tensile test. Deformation or transition bands (Figure 1.4b) are te consequence of subdivision witin grains into regions of different orientations. Tis is a consequence of eiter eterogeneous stress transmitted by neigbouring grains or te intrinsic instability of te grains during plastic deformation. At lower scale, te deformed polycrystalline aggregate is made of cells or sub-grains (Figure 1.4c and 1.5b). Tese cells are essentially low angle

19 17 grain boundaries wic subdivide te grain, producing eterogeneities inside te different grains. Tis is a consequence of aving non omogeneous dislocations distributions. Eac cell boundary contains a ric density of entangled dislocations and te cells bulk areas are almost dislocation free. Finally, dislocations can also exist in random structures (Figure 1.5a and 1.5c), especially after low strains. In metals wic do not form cells, suc diffuse arrangements of dislocations are found even after large strains. (a) (b) (c) Figure 1.4: (a) Sear bands in Al Zn Mg alloy cold rolled 90%, (b) deformation bands (B) in a grain (A) in Al- 1%Mg, (c) cell structure in 25% cold rolled copper. [Humpreys, 2004]. (a) (b) (c) Figure 1.5: TEM brigt field images of an Al alloy (5005) deformed at (a) 2% and (b) 10% and of anoter Al alloy (3003) deformed at 2%.(c) [Trivedi, 2004]. Finally, during deformation te crystallograpic orientations cange relative to te direction(s) of te applied stress(es). Tese canges are not random and involve rotations wic are related to te crystallograpy of te deformation (as discussed in te introduction of

20 18 section 2). As a consequence te grains acquire a preferred orientation (texture), wic becomes stronger as deformation proceeds. Since annealing processes (recovery, recrystallization and grain growt) involve te loss of some of te energy stored in te material during plastic deformation, first of all, te material microstructure induced by plastic deformation must be correctly caracterized and/or modelled. 2.2 Components of a single crystal model In order to account for te mecanics of grain structure eterogeneous deformation, crystal plasticity models are based on microstructural variables suc as crystallograpic orientations and dislocation densities. Polycrystal models are based on single crystal models as illustrated in Figure 1.6. Figure 1.6: From polycrystal level to slip system level [Resk, 2010]. poly In Figure 1.6, corresponds to te macroscopic stress applied on te material. Using c continuum mecanics, we are able to compute te local stress and, using crystal plasticity teory, we compute te critical resolved sear stress (CRSS) for slip system α. Once te CRSS value is known, te dislocation slip rate is computed and, based on te dislocation c c slip rate, we are able to compute te local strain ( ). Finally, based on te computed, te poly macroscopic strain is calculated ( E ). Tree components are needed in order to describe te mecanical beaviour of a single crystal. Te first one is a kinematic framework describing te motion of te single crystal. Generally, te kinematic decomposition used in crystal plasticity is te multiplicative decomposition (originally developed by [Bilby, 1957], [Kröner, 1971], [Lee, 1967] and [Lee, 1969]) as opposed to an additive decomposition [Sabana, 2008] wic is normally used for small deformations. In classical plasticity teory, if elastic beaviour is considered, te decomposition is composed of a plastic and an elastic term. Secondly, an elastic relation describing te elastic beaviour depending on te crystal structure of te material is needed. Elastic strains are small compared to plastic ones but are sometimes important to consider if te aim of te simulations is to compute residual stress [Marin, 1988a]. Te assumption of small elastic strains enables neverteless simplifications in te governing equations [Marin, 1988b]. In oters applications were elasticity is not of concern, te elastic beaviour is neglected [Beaudoin, 1995].

21 19 Finally, te evolution rules for te intragranular variables of te model, flow rule and ardening rule, need to be establised. Different forms of tese equations can be found in te literature. In te following, we will concentrate on viscoplastic flow rules. In its simplest form, te viscoplastic beaviour is described by a power law, as first introduced by [Hutcinson, 1977]: were 1 m c 0 sign (1.1) T : M is te resolved sear stress, 0 is a reference slip rate, m te sensitivity exponent and c te CRSS for slip system α [Delannay, 2006], [Erieau, 2004], [Marin, 1988b]. A rate sensitive formulation is used ere in order to avoid te non-uniqueness problem associated wit te identification of te active slip systems in rate independent formulations. 2.3 Hardening Te ardening rule represents te strain-induced evolution of te material resistance to plastic deformation. Tis strain-induced evolution is a consequence of te increased number of dislocations present in te material during plastic deformation. Altoug te ardening depends on te dislocation density existing in te material, tis quantity is not always used explicitly in te calculation. In [Resk, 2009], [Resk, 2010], [Logé, 2008] te autors use te Voce type saturation law in order to compute te material ardening: sat c c H, 0 (1.2) sat 0 were 0 and sat are, respectively, te initial and saturation values of CRSS, H 0 is a ardening coefficient. In [Iadicola, 2012] te autors use anoter example of implicit ardening law. In tis case, te increase of te CRSS form primary work-ardening is given by an extended Voce ardening law: 0 c 0 ( 1 1 ) 1 exp, (1.3) 0 were, 1 0, 0 and 1 are te accumulated slip (sear), te back-extrapolated CRSS, te initial ardening rate and te asymptotic ardening rate on eac slip system, respectively. In tis paper te autors incorporate latent ardening of multiple slip planes allowing te model to explain te decrease in flow stress wen canging from equal-biaxial to uniaxial deformation. Hardening models can be more pysically-based if expressed in terms of dislocation densities and if te basic mecanisms of dislocation generation and anniilation are considered. Moreover, strain gradient concepts, wic lead to size-dependent effects, can be introduced directly at te level of te ardening model. In bot cases, te CRSS is often given by:

22 20 c 0 b T, (1.4) were α is a constant, is te elastic sear modulus, b te Burgers vector amplitude, and T te total dislocation density (i.e. considering all types of dislocations). Differences arise mainly from te way of computing te evolution of one or more dislocation densities Hardening laws witout size effects In tis paragrap te size effects are not taken into account, meaning tat te deformation gradient in te material is no considered wen calculating te dislocation densities. In oter words, only te statistically stored dislocations (SSDs) are considered. A common way of writing te dislocation evolution equation is to consider two terms, one due to ardening (+), and te oter due to annealing (-): ( ) ( ). (1.5) Te ardening term usually relates to Frank-Read sources of dislocations, wile te annealing term is connected to te anniilation of parallel dislocations of opposite signs. An exception for tis kind of presentation is te power law [Monteillet, 2009]: K 1, (1.6) wit 0, since it leads to closed form expressions. In tis case, K 1 is a material parameter related to te material ardening. Te dislocation density rate decreases wit strain, even toug te model does not present a specific term for te recovery. Te second ardening law is te Yosie-Laasraoui-Jonas equation [Laasraoui, 1991]: K 1 K 2, (1.7) were K 1 and K 2 are two material parameters, related to te production and te annealing of dislocations, respectively. K 1 is a positive parameter so as te dislocation density increases wit te increase of te plastic deformation by slip over all te slip systems. However, te increase of te number of dislocations favours on te oter and te annealing mecanism as te more dislocations, te iger te probability tey come in contact to one anoter. Tis explains te K 2 term, wit K 2 a positive parameter. In tis case, stress saturation is K1 reaced wen. K 2 A tird widely used model is te Kocks-Mecking equation [Kocks, 1976]: K 1 K 2. (1.8) Here, once again, K 1 and K 2 are two material parameters representing respectively te production and te annealing of dislocations. In tis case we consider tat te dislocation production also increases wit te increase in dislocation density. Te increase of te number of dislocations favours not only te annealing mecanism but also te creation of new

23 21 dislocations by a Frank-Read source mecanism. In tis case, stress saturation is reaced wen K 2 1 K 2. Finally, a more general differential equation for te material ardening can be computed [Monteillet, 2009]: K 1 K 2. (1.9) Te ardening law represented by Equation 1.9, is able to represent bot Yosie- Laasraoui-Jonas ( 0 - Equation 1.7) and Kocks-Mecking ( Equation 1.8) relations. However, it is valuable to observe tat any oter expression can be used. In [Estrin, 1998a], static recovery is taken into account by including an extra term to te above Kocks-Mecking equation: were r K K, 1 2 (1.10) U 0 r r 0 exp sin, (1.11) k BT k BT wit U 0 being te activation energy, β and r 0 constants and k B te Boltzmann constant. Finally, more sopisticated formulations to calculate te dislocation evolution during plastic deformation can be found in te literature. As an example, in [Roters, 2000], Roters et al. propose a model based on tree different kinds of dislocations: represents te immobile dislocations tat we find in te material cell walls; i w represents te immobile dislocations inside te cells, and m te mobile dislocations inside te cells. Figure 1.7 illustrates tese different types of dislocations. Figure 1.7: Scematic drawing of te arrangement of te tree dislocation classes considered in te treeinternal variables model: mobile dislocations (ρ m ), immobile dislocations in te cell interiors (ρ i ) and immobile dislocations in te cell walls (ρ w ) [Roters, 2000]. For eac dislocation type an evolutionary equation is proposed. Tese equations are based on pysical penomena of generation and anniilation of dislocations. Te mobile dislocations carry te plastic strain and it is assumed tat te mobile dislocations density can

24 22 evolve according to tree mecanisms: te formation of dislocation dipoles, te formation of dislocation locks, and anniilation. It is interesting to note tat dislocation dipoles can still move, but since teir motion does not contribute to te net strain, tey are no longer considered in te class of mobile dislocations. Consequently, te evolution Equation 1.5 applied to mobile dislocations becomes: (anniilat ion) (lock) - (dipole). (1.12) m m m m Considering te immobile dislocations ( i ), te rate of increase of te dislocation density inside te cells is equal to te decrease of te mobile dislocation due to te formation of locks ( m(lock) ). Since locks cannot glide, te only process to decrease te immobile dislocation density is anniilation by dislocation climb. Based on tese considerations, te evolutionary equation for te immobile dislocations becomes: (climb) (lock) - (climb). (1.13) i i i m Te tird type of dislocations concerns te immobile dislocations in te cell walls w. Tese dislocations undergo te same processes as tose in te cell interiors, but tere is one additional process, wic contributes to te increase of tis particular dislocation density. It can be assumed tat all dislocation dipoles finally end up and accumulate in te cell walls. As dipoles are created in te wole volume, but stored only in te walls, tey represent te main mecanism of w generation. As a consequence, te evolutionary equation for tese dislocations is: i 1 w wi w (climb) m (dipole) - w (climb). (1.14) f w were f w is te volume fraction of cell walls. All terms of tese different evolutionary equations are detailed wit various expressions in [Roters, 2000]. In [Estrin, 1998b] Estrin proposes a similar model but considering only two dislocation types: immobile dislocations in cells walls (ρ w ) and dislocations inside te cells (ρ c ). Te difficulty wen dealing wit several dislocation types is related to te introduction of several material parameters wic are often difficult to identify. Tis largely explains te success of simpler equations like tose of Yosie-Laasraoui-Jonas [Laasraoui, 1991] or Kocks-Mecking [Kocks, 1976], discussed earlier Hardening laws including size effects Wen a polycrystalline material is subjected to a stress, te dislocation slip systems are activated and te material is plastically deformed. Determining wic slip systems will be activated in eac grain of te polycrystal depends on te grain crystallograpic orientation and on te crystallograpic texture of te material [Sarma, 1996], [Van Houtte, 2005]. In addition, differences in te crystallograpic orientation between neigbouring grains induce a eterogeneous plastic deformation. As we can see in Figure 1.8, if eac grain is caused to undergo its uniform strain, te result is te formation of overlaps in some places and voids in oters. For tis reason, a certain amount of dislocations must be introduced into te material in order to ensure te crystal lattice continuity. m

25 23 Figure 1.8: If eac grain of a polycrystal, sown at (a), deforms in a uniform manner, overlap and voids appear (b). Tese can be corrected by introducing geometrically necessary dislocations, as sown at (c) and (d) [Asby, 1970]. Several examples found in te literature illustrate te importance of tese geometrically necessary dislocations in te mecanical beaviour of te material [Gerken, 2008a], [Gerken, 2008b], [Petc, 1953], [Hall, 1951], [Cen, 1998], [Papadopoulos, 1996]: - Te Hall-Patc relation [Petc, 1953], [Hall, 1951], wic connects te yield strengt wit te grain size. Te increase in te yield strengt wit te decrease of te grain size is explained by te presence of grain boundaries, wic represent obstacles to dislocation slip. Te presence of dislocations due to plastic deformation gradient as an important role explaining tis penomenon. - te initiation and propagation of cracks across grains are igly dependent on deformation gradient effects. Cen [Cen, 1998] sowed a maximum sear stress up to tree times greater tan tat estimated wit te conventional elastic-plastic fracture mecanics. - Papadopoulos and Panoskaltsis [Papadopoulos, 1996] discussed te importance of tese plastic deformation gradients for te fatigue resistance of metals. Te total density of dislocations wic can be found in a slip system α is expressed as:. (1.15) T S G In tis expression, S corresponds to te statistically stored dislocation density (SSD) and G corresponds to te geometrically necessary dislocations (GND) introduced in te model to ensure te crystal lattice continuity. GNDs and SSDs Te concept of GNDs was first proposed by Nye [Nye, 1953] and Asby [Asby, 1970]. Te GNDs appear in areas of strain gradient, tus ensuring te crystal lattice continuity. Tese regions of deformation gradients are a consequence of te existence of te geometric constraints of te crystal lattice, tat is to say, inconsistencies in te crystal lattice due to te presence of non-uniform plastic deformations, [Arsenlis, 1999], [Evers, 2002], [Ma, 2006].

26 24 (a) (b) Figure1.9: Diagram of te accumulation process of geometrically necessary dislocations (GNDs) edge type (a) and screw type (b) [Arsenlis, 1999]. Figure 1.9 illustrates te GNDs formation process. A crystal divided in several blocks is submitted to a searing stress. Eac block of te crystal will suffer a different plastic deformation wic is not necessarily te same as te neigbouring blocks. Te dislocations of te same type but wit different sign will anniilate. Te dislocation excess resulting from tis anniilation step is proportional to te gradient strain in te direction of te dislocation slip [Abrivard, 2009]. In oter words, we can say tat tis dislocation excess is proportional to te plastic strain gradient. During plastic deformation, grain boundaries areas are more susceptible to plastic deformation gradients. As a consequence, te formation of GNDs near te grain boundary areas is favoured. Te GNDs formed in tis area can be identified by transmission microscope images as dislocations rooted in te grain boundaries areas. In [Yu, 2005], Yu performed tis kind of analysis using aluminium samples formed by ultrafine grains initially free of dislocations. Te presence of GNDs results in an increase of te strain energy near te grain boundaries due to te relative rotation between te grains. In oter words, te presence of te GNDs around te grain boundaries affects te material ardening. Te GNDs are immobile dislocations. Tey remain in te areas of strain deformation to ensure te continuity of te material. Since te GNDs are immobile dislocations, tey do not ave a significant influence on plastic deformation. However, due to teir static nature, tese dislocations work as obstacles for te slip of oter dislocations. As a consequence, te GNDs ave an important role in te material ardening. Te SSDs are te dislocations accumulated in te material during omogeneous plastic deformation. Tese dislocations can be divided in two different groups. Te first one is te mobile SSDs, wic are able to glide over activated slip systems. Te second one corresponds to te immobile SSDs. In tis case te dislocations are immobile eiter because tey belong to an inactive slip system, or because tey are ancored dislocations. Pysically tere is no difference between GNDs and SSDs. Tey cannot be distinguised by image analysis metods. Te difference between tese two different types of dislocations is te role tey play during plastic deformation. A second difference is te distribution of teir sign in space. Te SSDs, as a result of teir statistical nature, present a random distribution across te grains. Some of tese dislocations cancel te influence of te oters and, as a consequence, te SSDs do not contribute to te plastic deformation eterogeneity. Unlike te SSDs, te GNDs sign is strongly dependent on te geometry. Terefore, te GNDs present a ig periodicity since tese dislocations exist to ensure te material continuity in areas of deformation gradient, for example at te grain boundaries. Te sceme of Figure 1.10 illustrates te difference between tese two types of dislocations.

27 25 (a) (b) Figure 1.10: Scematic illustration of dislocation sign distribution for (a) SSDs and (b) GNDs. To sum up we can say tat te SSD density is a caracteristic of te material, tat is, of te crystal structure, sear modulus, stacking fault energy, etc. On te oter and, te GND density is a caracteristic of te microstructure, tat is, te geometric arrangement and size of grains and pases. Crystal plasticity models based on plastic deformation gradient Efforts to incorporate inomogeneities of plastic deformation in te standard models of continuum mecanics are multiple, but te proposed teories can be classified into two groups: 1 - Hig-order or top down: te material beaviour is considered to be a function of te material deformation and/or te stress state. In tis case, te determination of te stress and deformation states is considered crucial for determining te mecanical beaviour and te evolution of te state of te material. From a more practical point of view, tis type of model introduces significant canges in te finite element formulation of te classical crystal plasticity model. Ono and Okumura [Ono, 2007], Gerken and Dawson [Gerken, 2008a], [Gerken, 2008b], Gurtin [Gurtin, 2008], [Gurtin, 2002], [Gurtin, 2000] proposed tis kind of approac. 2 - Low-order or bottom up: te material beaviour is considered as dependent on te action of dislocations, and te definition of tis dependence troug time defines te beaviour and material state. Tis kind of approac is often easier to implement, as compared to te iger order models, since te classical crystal plasticity formulation remains uncanged. In order to take into account te GNDs effects, te equation of dislocation density evolution in time is modified. Evers [Evers, 2004], Bassani [Bassani, 2001], Beaudoin [Beaudoin, 2000], Voyiadjis [Abu Al-Rub, 2006], Busso [Busso, 2000], Acarya [Acarya, 2001], [Acarya, 2004], Ma [Ma, 2006], Acarya and Bassani [Acarya, 2000] proposed tis kind of model. Hig-order/topdown models Te model proposed by Gurtin [Gurtin, 2008], [Gurtin, 2002], [Gurtin, 2000] can be summarized by te following: 1 Multiplicative decomposition of deformation gradient tensor (F) in elastic (F e ) and plastic (F p ) portions; 2 Te mecanical model is linked to te work done by eac active slip system. Energy principles account for te work associated wit eac activated slip system. A system of microforces also associated to dissipated work accompanies tat introduced by dislocation

28 26 slip. Te microforces system consists of a stress vector,, and a scalar value representing te internal forces wit a microforce balance tat is supplemental to te classical Newtonian balances of momentum. Denoting te stress projected to te slip plane α, te microforces equilibrium equation is given by: Div 0; (1.16) 3 A mecanical version of te second termodynamic law is proposed. In tis case, te idea tat te internal increase in te material internal energy is not greater tan te work performed by eac slip system is considered. Tus, an energy inequality is establised: J 1 A. grad 0 e CL, (1.17) 1 e e e1 wit te material internal energy, C te Caucy stress tensor, L F F te elastic e p deformation rate tensor, J det F and det F 1 (plastic deformation wit volume conservation). From te inequality 1.17, constitutive equations for and are establised: 1 t H( ) J S T TS G, (1.18) 1 e J F ( m Ts ), (1.19) were α = 1, 2,, A labels te individual slip systems and A is te total number of slip systems existing in te material. Substituting Equations 1.18 and 1.19 into te balance Equation 1.16, te equation used to compute te resolved sear stress on a slip system α is obtained: t 1 e S T TS G divj F ( m Ts ) 1 H( ) J, (1.20) were H( ) caracterizes te stress dependency on te dislocations slip rate, G is te ( G) geometrically necessary dislocations (GNDs) density, and T represents te GNDs G energetic contribution to te material ardness. T is a function of partial derivatives tat do not exist in te classical crystal plasticity model. Moreover, wen determining te constitutive equations of te model, te internal energy of te material ( ) is expressed as te sum of te elastic energy of te material added to an energy value due to te presence of a small amount of crystalline defects. Te model is terefore only valid for small plastic deformation. Te tensor G is written as a function of an edge dislocation density ( E ) and a screw dislocation density ( ), as follows: S G S s s E l s, (1.21) were s is a vector parallel to te Burgers vector ( b ), l m s, wit m te normal to te slip plane. s s and l s are canonical dislocation dyads. If l b is a

29 27 dislocation dyad, for an edge dislocation l b and for a screw dislocation and E are calculated using equations proposed by [Arsenlis, 1999]: l b. S E s. grad and S l. grad. (1.22) Finally, te addition of a microforces tensor requires te need for supplementary boundary conditions beyond te classical macroscopic boundary conditions used. Tese extra boundary conditions are used in order to solve te partial differential equations (PDEs) tat represent te non-local yield conditions wile te standard boundary conditions are used to solve te PDEs tat represent te standard force balance. Two extra boundary conditions are needed: first, if a grain boundary is considered to be rigid, i.e. it does not allow te dislocation passage, te boundary condition is assumed to be 0. However, if te grain boundary does allow te dislocation passage, only sear stress is assumed to operate at te boundary, and te condition is. n 0. Ono and Okumura [Ono, 2007] proposed a model tat takes into account te GNDs energy to analyze te grain size impact on te material initial yield strengt. Te energy value is used to construct a ig order stress tensor. Bot te GNDs energy and te ig stress tensor are incorporated in Gurtin s [Gurtin, 2008], [Gurtin, 2002], [Gurtin, 2000] teory of plastic deformation gradient. Te model as been applied to te determination of te yield strengt dependence on te grain size of 2D and 3D single crystals. Te model is owever not suited for te modeling of large plastic deformation of polycrystalline materials. In [Gerken, 2008a], [Gerken, 2008b], Gerken and Dawson proposed anoter ig-order model example. Tis model is based on te multiplicative decomposition of te gradient deformation tensor into tree parts. Proposing tis multiplicative decomposition, teir idea is to develop a more accurate representation of te mecanism responsible for deformation. In te classical crystal plasticity model, te gradient deformation tensor is usually decomposed into two parts: one related to te reversible or elastic part of te deformation, and te second one to te permanent or plastic part. In te new model, te tird term of te gradient deformation tensor corresponds to te long range strain tat is a deformation of te lattice due to dislocations remaining in te lattice (te GNDs). Tis long range deformation is related to slip gradient wic results in a differential equation relating te total deformation to te tree deformation mecanisms represented by te tree factors of te multiplicative decomposition. Figure 1.11 illustrates te multiplicative decomposition of te gradient deformation tensor used in tis model. Figure 1.11: Multiplicative decomposition of te deformation gradient including terms for crystallograpic slip, long range strain, rotation and elastic strain.

30 28 In Figure 1.11, F e represents te elastic part of te deformation wic can be decomposed into two parts: U el, corresponding to te stretcing of te crystal lattice, and R*, representing te lattice rigid rotation. F d represents te deformation induced by dislocations, wic can also be divided into two parts. Te first one, F p, represents te permanent deformation due to dislocation slip. Te second one, F b, represents a long range deformation due to te GNDs distribution in te crystalline structure. Based in tis assumption, te sear rate along eac active slip system is given by: were m te strain rate sensitivity, ardness of te slip system. g 1 m 0 sgn, (1.23) g te sear stress projected to te slip system, and g depends on te SSDs and GNDs densities and is given by: g te G G, wit, 1 2 (1.24) x i G1 ( ) t 0 g S g, (1.25) g S g0 G 2 GND. (1.26) xi d Te functions G 1( ) and G 2 x i represent te ardness due to te presence of SSDs, and GNDs, respectively. In Equation 1.26 d represents te dislocation slip direction. Unlike te model proposed by Gurtin [Gurtin, 2008], [Gurtin, 2002], [Gurtin, 2000], tis model proposed by Gerken and Dawson [Gerken, 2008a], [Gerken, 2008b] allows te modeling of GNDs effects for large plastic deformation, wic makes its use more interesting in te context of recrystallization penomena. Furtermore, te model evaluates te GNDs effects not only on te material stress state (Equation 1.24), but also on te material deformation (F b ). Te F b tensor allows te estimation of te crystal lattice rotation due to te presence of GNDs. Low-order/bottom up models All low order metods rely on an internal state variable approac to determine te macroscopic response of te material wereby strain gradient effects are introduced directly into te evolutionary laws of te state variables. Te strain gradient effects are incorporated by determining te GND density and distribution based on te Nye s tensor. Te Nye s tensor gives a measure of te plastic deformation incompatibility. In a pysical way, Nye s tensor can be interpreted as a measure for te closure failure of Burger s circuit enclosing an infinitesimal surface. Te inner product of Nye s tensor wit te unit normal vector n of surface S is integrated over te surface.

31 29 G. nds, S (1.27) wit G being te cumulative Burger s vector. Tis vector represents te GNDs density enclosing te surface S. Finally, te Nye s tensor is given by p p F F curl, (1.28) were F p is te plastic deformation gradient. Bassani [Bassani, 2001] and Acarya [Acarya, 2000] propose a model tat takes into account te influence of GNDs and SSDs implicitly. Te CRSS is calculated being proportional to te Nye s tensor. However, dislocation density values are not explicitly calculated. Wen modeling recrystallization penomena, te dislocation density value is usually essential information. Implicit models are terefore not of great interest for applications related to recrystallization penomena. Al-Rub [Al-Rub, 2006] as proposed a model tat represents, explicitly, te GNDs and SSDs contribution to te material ardening. In tis case, te CRSS is given by: 0 b S G. (1.29) An evolution equation is proposed for eac type of dislocation. As indicated above, te GNDs density is linked to te incompatibility of te plastic deformation and te crystal lattice rotation. As a consequence, GNDs are not created or anniilated like te SSDs; eiter tey are transported from/to oter regions were tere is a deformation gradient, or tey are te result of geometrical reactions between oter existing GNDs. Te GNDs accumulate in proportion to te plastic deformation gradient [Asby, 1970], [Nix, 1998], [Arsenlis, 1999], according to Equation Te evolution of SSDs density is calculated using te equation proposed by Beaudoin [Beaudoin, 2000], based on te Kocks- Mecking equation (Equation 1.8): S K b 1 P K0 G S K2S, (1.30) were K 0 G represents te immobile dislocations increase due to te tangle of mobile dislocations wit te GNDs. Busso [Busso, 2000] as also proposed a ardening law tat explicitly represents te GNDs and SSDs contributions. Te description of te flow caracteristics in a generic slip system (α) under a given temperature, microstructural state and resolved sear stress ( ) is based on te dislocation mecanics and stress-activation concepts introduced by Busso [Busso, 1996]. Te particular form of te flow rule exibits an explicit dependence of te activation energy on a driving stress ( termally activated obstacles: l ) wic accounts for lattice friction effects and l S, (1.31) 0

32 30 were S represents te atermal component of te flow stress over slip resistance and 0 te ratio of te sear modulus at te temperature of interest to tat at 0 K. Bot SSDs and GNDs contribute to te total slip resistance S as follows: S b 0 2 S 2 S G G G, (1.32) were λ S and λ G are statistical coefficients accounting for te deviation from regular spatial arrangements of SSDs and GNDs and G is a function representing te interaction between different GNDs populations existing in te material. e SSDs are decomposed into pure edge ( S ) and screw ( ) components in order to account for teir different mobilities, ardening and recovery process. Dislocations generation is assumed to be related to Frank-Read sources and te anniilation is assumed to occur due to sign differences between te same type of parallel dislocations. Te evolutionary laws are written as balance laws between dislocation generation and anniilation: S sw S S e sw C b e C b K sw e K sw 2 T de S, e T S sw K sw d 2d. 2 sw T sw (1.33) In tis expression, d e and d sw are critical distances for spontaneous anniilation of opposite sign edge and screw dislocations respectively. C e and C sw represent te relative contributions of edge and screw dislocations to te slip produced by SSDs wile K e and K sw are related to teir respective mean free pat. For describing te evolution of GNDs, a vector field G related to te GND density is introduced for every slip system. Tis dislocation line vector can be furter discretised into its edge and screw components by solving along axes of te coordinate system defined in terms of te slip direction m, te slip plane normal n and a tird ortogonal direction t m n : G G m G t G n. (1.34) sm et en Here, G sm refers to its screw component parallel to edge components parallel to t and m wile n, respectively (Figure 1.12). and G et are te G en

33 31 Figure 1.12: Local ortogonal coordinate system of reference for a generic geometrically necessary dislocation line in an arbitrary slip system α. Nye s dislocation tensor (Equation 1.28) is used to define a tensorial measure of te GND density wic can be related to te resultant Burger s vector (Equation 1.27) of all GNDs. Te evolution of GNDs is obtained by differentiating Equation 1.27 wit respect to time. As a result of tis matematical treatment, te GNDs density is calculated as: b p m t n curl n F, (1.35) Gsm Get Gen were F p is te plastic deformation gradient (Figure 1.13). Te determination of Equation 1.35 is presented in details in [Busso, 2000], [Abrivard, 2009]. Te curl term translates te dependency of te GND density evolution on te spatial gradient of te slip rate, ence te non-local terminology wic is sometimes used to describe strain-gradient plasticity concepts. Finally, te total GND density value is given by: G 3 Single Crystal Model G sm G et. (1.36) Following te previous considerations, it is cosen ere to adopt a constitutive beaviour at te single crystal level wic is based on an elastic-viscoplastic crystal plasticity formulation, enriced by te introduction of bot SSDs and GNDs, according to te principles of low order / bottom up models. 3.1 Formulation Elastic-Viscoplastic Formulation Te single crystal model described ere relies on an elastic-viscoplastic formulation. Te main concepts are igligted in tis section and more details can be found elsewere [Delannay et al., 2006]. In tis model, plastic deformation is acieved by dislocation slip. For instance, in te case of an FCC crystal, te dislocation slip appens along te {111} <110> crystallograpic systems wile for a BCC crystal, te dislocation slip takes place especially along te {110} <111>. Te kinematics of te single crystal is a combination of dislocation slip, lattice rotation and elastic stretc. Figure 1.13 illustrates te multiplicative decomposition, wic corresponds to tat given in Figure 1.11 but witout te concept of long range strain. G en

34 32 Figure 1.13: Crystal kinematics: initial (C 0 ), intermediate (C i ) and final configuration (C f ) Te deformation gradient tensor F, typically considered in finite strain kinematics, is decomposed as follows: el p F F F R* U el F p (1.37) were R* is te lattice rotation and U el te elastic stretc. Te intermediate configuration C i corresponds to a stress-free configuration or relaxed configuration obtained by elastically unloading te crystal in te final configuration C f and rotating it. In oter words, tis intermediate configuration represents te plastic deformation witout taking into account te elastic deformation. An additional fictitious configuration could be introduced between C i and C f as in [Marin and Dawson, 1998d] were it is obtained by elastically unloading C f to a relaxed state but witout rotation. Te crystal constitutive equations could be written in any of tese configurations and tis coice conditions te procedure and sceme used to integrate tem. Here, te equations are written in te intermediate configuration C i. In tis case, te velocity gradient tensor L is given by: L FF el el 1 T 1 * * el p p el 1 T U * U R R U F F U R T 1 * * *. (1.38) R R R Te plastic velocity gradient L p, accounting for te dislocation slip, is ten written as follows: 1 p p p L F F M. (1.39) In tis expression, M is te Scmid tensor of slip system α. Te Scmid tensor as te same expression in te initial and intermediate configuration as crystallograpic slip does not distort te lattice. Te elastic strain tensor E is calculated wit respect to te intermediate configuration and is terefore given by: 1 E 2 T 1 T el el el el F F 1 U U 1. 2 (1.40) Te work-conjugate measure of stress is te second Piola-Kircoff stress T. Tis latter is related to te Caucy stress σ troug:

35 T det 1 T el el el el F F F det U 1 1 el T el U R* R * U. (1.41) Te fourt order anisotropic elasticity operator C sets te proportionality of T wit regards to E via te relation: T CE. (1.42) In a crystal wit cubic symmetry (suc as FCC or BCC), wit te cartesian axes oriented along te cube edges, te nonzero elements of C are te same ones as for te isotropic solid, but te tree values C 11, C 12 and C 44 are independent. So, for FCC crystals te elasticity matrix takes te following form: 33 C C C C C C C C C C C C 44 (1.43) Tis form of te anisotropic elastic tensor allows for te separation of te deviatoric and sperical components of T, wic as consequences in terms of finite element implementation. For materials subjected to important plastic strains, te elastic strains are small compared to te plastic one. Typically, in metal forming operations, tey never exceed 1%. It is ten assumed tat tey are small compared to unity. Tis assumption yields: U el el I, (1.44) were I is te identity tensor, order terms in el leads to: U el el is a symmetric tensor wit 1 1 el 1 el I I el el el U deti 1 tr e det. Neglecting iger, (1.45) el For te same sake of simplification, for any tensor X, : X is neglected compared to X. Bearing tese assumptions in mind te velocity gradient tensor can be rewritten as: T el T L R * R * R * M R *. (1.46) and Equations 1.40, 1.41 and 1.42 simplify to: T T el * * C R R (1.47)

36 Flow rule and ardening law In order to complete te single crystal model, te viscoplastic power law is used as a flow rule (Equation 1.1). Regarding te ardening, it is assumed tat all slip systems ave te same c and tat tey all arden according to Equation 1.4. Hardening law - SSDs Considering only te SSDs, te Yosie-Laasraoui-Jonas equation (Equation 1.7) can be implemented witin te crystal plasticity model. To transform tis equation, used at te macroscopic scale, to te dislocation slip scale (mesoscale), te strain term is replaced by te total dislocation slip, and K 1 and K 2 values are divided by te polycrystal Taylor factor (M). At te mesoscale (te transformation from macro to mesoscale is discussed in capter 3), te Yosie-Laasraoui-Jonas equation becomes: ' ' K 1 K 2 (1.48) were ' K1 K 1 M and ' K 2 K 2 M. Hardening law - GNDs Considering te GNDs, te Busso [Busso, 2000] model (Equation 1.35) may be implemented in a standard crystal plasticity model. Te gradient terms require a non-local computation, wic can for example be done using a finite element formulation. It is important to igligt tat only te GND evolutionary equation proposed by Busso is implemented ere. Te equations considering te SSD density evolution proposed by Busso can be replaced by Equation Time Integration Sceme of te constitutive law To summarize, te equations describing te elastic-viscoplastic model can be listed as follows: sym 1 T 2 (1.49a) T 1 T el T Lant W R * R * R * M M R * 2 (1.49b) el T C, (1.49c) T : M (1.49d) el T Kinematics: L D R * M M R *, Kinematics:, Elasticity: Scmid Law: 1 Flow: m 0 sign, (1. 49e) Hardening: SSDs evolution: c c c b (1. 49f) T ' ' S K 1 K 2 (1. 49g)

37 35 GNDs evolution: p 1 n F curla 1 G curl. (1. 49) b b Te objective of te integration of te constitutive model is to compute te modeldependent variables at time t + Δt, given tat teir values are known at time t and tat te applied deformation is known for every crystal (L or equivalently D and W). In te previous system, te unknown independent variables are: te crystal stress T t (or equivalently te t el elastic deformation ), te slip resistance c and te lattice orientations R *. tt tt t t Tese constitute te state variables of te problem. In order to simplify te time integration procedure, one can coose to eliminate te lattice rotation from tis system (and ence te ~ evolutionary Equation 1.49b) wic is approximated by R * as follows: ~ R * R* Wt R *. (1.50) tt As mentioned in [Delannay 2006], te impact of suc an approximation in negligible in metal forming simulations suc as tose performed in tis work. Te update of te lattice orientation is performed later, after te integration of te rest of te equations, as it is explained below. Equations 1.49a, 1.49c and 1.49d are combined in order to find te crystal stresses T. A fully implicit time integration sceme yields te following discretized equations: t t t T tt T t ~ C R * T ~ 1 DR * 2 T M M, (1.51) m M : T 0 t (1.52) tt tt c tt tt were t signm : T, 1 and c tt c t b 1 2 tt. (1.53) Concerning te evolution of dislocation density, te SSDs density is calculated implicitly, transforming equation 1.48 into: K'. K ' 1 2 tt (1.54) In order to determine te evolution of te GNDs, te quantity curl(a) (Equation 1.49) as to be calculated. Given tat te spatial variation of te quantity (A) is necessary to estimate its curl, spatial information must be known, for example by using a finite element mes (see section 4). A is ten calculated at eac Gauss point and linearly interpolated to te mes elements nodes using te sape functions of te element. Subsequently, te curl of (A) is calculated. As described in equation 1.36 and recalled ere, te tree components of te dislocation density vector field are added, leading to te total GNDs density, for eac slip system :

38 36 G G sw G et. (1.55) Ten te GNDs densities of eac slip system are added in order to calculate te total GNDs density: G en G G. (1.56) An explicit update is performed using a forward Euler sceme,. (1.57) G, tt G, t tg, tt Since te GNDs are explicitly updated, te accuracy of te solution is ensured only if a sufficiently small time step is used. A two level iterative sceme is used to solve te above system. A first Newton-Rapson procedure solves Equation 1.51, wile a second one computes te slip increments (1.52) based on te stress estimate. Slip resistance is computed subsequently using Equation 1.53 and tese tree operations are performed until convergence. Finally, regarding lattice reorientation, Equation 1.49b is integrated using an exponential map [Simo, 1998]. In practice, all tese equations are expressed in te crystal reference frame, so tat M is identical in crystals belonging to te same pase. At te beginning of te simulation, te orientation of eac crystal is recorded and used to sift from te sample coordinate system to te crystal coordinate system ( T R R * 0 ). 4 Finite Element Formulation Te finite element model is composed of a representation and mesing of te microstructure and of a finite element formulation used to solve te mecanical problem. Representation and mesing of te microstructure are discussed in details in capter 2. In tis part, te solution of te mecanical model in a finite element formulation is presented. Finite element simulations in solid mecanics are dominated eiter by displacement/velocity based formulations or by mixed formulations suc as displacement/pressure or velocity/pressure formulations. In tis work, a mixed velocity/pressure finite element formulation is used wit appropriate combination of interpolation functions. More specifically, te mini-element (P1+/P1) wit linear continuous pressure and linear velocity wit a bubble function added at its centre for tis latter, is used. Specific details of te formulation are found below. 4.1 Balance laws Te resolution of te mecanical problem is based on momentum and mass conservation coupled wit te appropriate boundary conditions and constitutive equations. In local form, te conservation of momentum and mass is written as: dv div g, (1.58) dt

39 37 d.( v) 0. dt (1.59) were v is te velocity vector, σ is te Caucy stress tensor, ρ te density and g te gravity. Appropriate surface traction and velocity boundary conditions are given by: for te velocity vector v, and: app on Γ ν, (1.60) n t t app on Γ t, (1.61) for te traction vector t were t is te boundary of te domain Ω. In te present work, we restrict our attention to isotermal, quasi-static deformation of polycrystalline aggregates. Also neglecting body forces, Equation 1.58 is reduced to: div 0. (1.62) Te Caucy stress tensor σ can be decomposed into its deviatoric and sperical (pressure) components. However, suc decomposition is only meaningful if te constitutive law, tat is to be solved in concert wit tis equation, can give te evolution of te deviatoric and pressure components separately. Besides, for constitutive models tat account for te elastic beaviour, Equation 1.59 is replaced by te volumetric response of te elasticity relations, as te density can be directly determined once te motion of te body is obtained. Bearing tese ypoteses in mind, te final system of equations to be solved may be written as: divs p 0, p tr 0, (1.63) were S and p are respectively te deviatoric and pressure components of σ, te bulk (elastic) modulus and te strain rate tensor defined as: T (1.64) Variational formulation Te formulation of te finite element problem is based on te weak integral form of te system presented in Te procedure for obtaining te weak form consists of first multiplying te equations by test functions v* V and p* P were V and P are appropriate functional spaces given by:

40 38 v 1 d V, v H ( ) v vapp on 1 d V 0 v, v H ( ) v 0 on v 2 P p, p L ( ), v,, (1.65) wit d te space dimension, V te space of kinematically admissible velocity fields and V 0 te space of kinematically admissible velocity fields to zero. Integrating over te volume of te domain Ω and using te Green formula yields te following variational problem: find v, pv, Psuc tat v p * V, P *, 0 : S( v) : ( v*) d p tr p * d 0. p. v* d t t app. v* d 0, (1.66) 4.3 Time discretization Te large deformation of te microstructure is modelled using an updated Lagrangian framework. In tis incremental approac, te total simulation time t tot is discretized into N 1 i0 t t N increments suc tat t t, t t tot i i i. At time t te configuration of te body Ω t is known and te solution v, p satisfying te balance laws at tat time, can be determined based on stresses calculated at time t + Δt. Te new configuration is te updated one, using a finite difference sceme, namely an Euler explicit sceme: t t t t x x v t (1.67) wit x te node coordinates vector and v t te velocity vector solution of te mecanical problem on te current configuration. 4.4 Spatial discretization In order to compute te solution of te variational problem given by Equation 1.66 using te finite element metod, te domain Ω is discretized suc tat: K, K ( ) (1.68) were k a finite element mes of te domain Ω, K a simplex and a parameter denoting te mes size. We introduce te functional vector spaces of finite dimensions V and P close to te continuous spaces V and P of infinite dimension, suc tat te discrete solution v k, pk V k, Pk is close to te real one v, p V, P. Te spaces V and P ave to be cosen in suc a way tat te existence and uniqueness of te solution is guaranteed. For tis is a spatial discretization of te domain Ω,

41 39 purpose, tese spaces cannot be cosen independently. Tey ave to satisfy te Brezzi Babuska stability conditions [Brezzi, 1991]. In tis respect, te MINI-element P1 + /P1 is a convenient and an appropriate coice Te MINI-element Te MINI-element is an isoparametric triangle in 2D and tetraedron in 3D wit a linear interpolation for te pressure field. On te oter and, te velocity field interpolation as a linear component and a nonlinear one, te so-called bubble function, wic is added at te centre of te element as sown in Figure 1.14 for te 3D case. Figure 1.14: MINI-Element P1 + /P1. For suc an element, te finite element spaces can be written as:, 0 : ) ( 0, ), ( : )) ( (,, P(K) :p ) (Ω C p P, ) and b P(K :b ) (Ω C b B, v P(K),v v C v V v K P v v V B V W B V W K d Γ i K d app K d K d K K v K v (1.69) were d is te space dimension, K i, i = 1 d + 1 are te d + 1 sub-simplexes composing K. Denoting N l and N b te linear and bubble sape functions respectively, te discretized velocity and pressure fields can be written as: Nbelt j i b j Nbnode i i l i b x N v x N b v w 1 1, ) ( ) ( (1.70) Nbnode i i l i p x N p 1, ) ( (1.71) were Nbnode and Nbelt are te total number of nodes and elements in te mes respectively. On te local level, for eac element, te velocity and pressure unknowns are written as:

42 40 1 1, ) ( ) ( d i j b i l b x N V x N b v w (1.72) 1 1, ) ( d i i l P x N p (1.73) were N b is a linear function defined on eac of te d + 1 sub-elements. For eac element K, te bubble function as te following fundamental properties: 1 - K K d b 0 : A for any constant tensor A, terefore K K d b v 0 :, 2 - K K K K d b v d b v.., 3 - b = 0 on Γ K Te discrete problem Te discrete problem is formulated as follows: find P W p b w,, suc tat P W p b w,, 0 * * * * : t d p p w tr d w t d w p d w w S app 0. ) ( 0,.. ) ( ) : ( * * * * (1.74) Taking into consideration te decomposition b w and * * * b w te previous system can be rewritten as: t t d p p b v tr d b t d b p d b b v S d v t d v p d v b v S app app 0. ) ( 0,.. ) ( : ) ( 0,.. ) ( : ) ( * * * * * * * (1.75) It as been sown [Jaouen, 1998] tat in te case of an elastic-viscoplastic constitutive law, te deviatoric component S can be decomposed into a linear S(v ) and a bubble part S(b ). Tis yields to:

43 41 t t d p p b v tr d b t d b p d b b S d b v S d v t d v p d v b S d v v S app app 0. ) ( 0,.. ) ( ) : ( ) ( ) : ( 0,.. ) ( ) : ( ) ( ) : ( * * * * * * * * * (1.76) In practice, te system 1.76 is equivalent to te following system: 0. ) ( 0,.. ) ( ) : ( ) ( ) : ( 0,.. ) ( ) : ( ) ( ) : ( ) ( * ) ( * * * * ) ( * * * * K K K K K K K app K K K K K K K K app K K K K K K d p p b v tr d b t d b p d b b S d b v S d v t d v p d v b S d v v S (1.77) Tanks to te bubble function properties, we ave: 0; ) ( : ) ( ) ( : ) ( * * K K K K d v b S d b v S (1.78). * 0. K K app d b t (1.79) Considering tese properties and removing te subscribe for more clarity, te system 1.77 becomes: 0. ) ( 0,. ) ( ) : ( 0,.. ) ( ) : ( ) ( * ) ( * * ) ( * * * K K K K K K K K K K K K K app K K d p p b v tr d b p d b b S d v t d v p d v v S (1.80) Te elastic viscoplastic constitutive beavior is used in te crystal plasticity simulations, wic is a non-linear constitutive beavior. As a consequence, system 1.80 is also non-linear. Te non-linearity related to te material beavior does not require fundamental reformulation of te problem, as opposed to geometric non-linearities. However, te constitutive law and te system of equations are written in incremental form. Small step increments are needed to

44 42 correctly account for pat dependence and obtain pysically sound solution. In incremental form, te problem can be re-written as follows: Given S n, p n, t n and Ω n suc tat equilibrium is satisfied at time t, P W p b, * *, * 0 : 0. ) ( 0,. ) ( ) : ( 0,.. ) ( ) : ( ) ( * ) ( * * ) ( * * * n K n K K n K n K K K K n n K K n K n K K app K n K n d p p b v tr d b p d b b S d v t d v p d v v S (1.81) Find v n+1 kinematically admissible, S n+1, p n+1, t n+1 and Ω n+1 tat satisfy te equilibrium at time t+δt, P W p b, * *, * 0 : 0. ) ( 0,. ) ( ) : ( 0,.. ) ( ) : ( ) ( * 1 1 ) ( * 1 * 1 ) ( * * 1 * n K n K K n K n K K K K n n K K n K n K K app K n K n d p p b v tr d b p d b b S d v t d v p d v v S (1.82) In practice, if te time step is taken sufficiently small so as to enable small strain increments, Ω n+1 is taken equal to Ω n. Te configuration of te body is ten updated via Equation Resolution 5.1 Non-linear system of equation to be solved Te nonlinear algebraic system of Equations 1.77 can be written as global residuals as follows: 0 ),, ( 0 ),, ( 0 ),, ( pp pb pv p b bp bb bv b v vp vb vv v R R R p b v R F R R R p b v R F R R R p b v R R (1.83) Taking into consideration te various simplifications, te actual discrete problem 1.80 yields te following system, decoupled in terms of v and b :

45 43 0 ),, ( ), ( 0 0 ), ( pp pb pv p bp bb b v vp vv v R R R p b v R R R p b R F R R p v R R (1.84) In a more general form, if α stands for a vector containing te discretization parameters (pressure and velocity), te incremental problem consists of solving R(α n+1 ) = 0 starting from a (pseudo) equilibrium solution at R(α n ) = 0 suc tat α n+1 = α n + Δα n. 5.2 Resolution of te non-linear system A Newton-Rapson metod is used to solve te non-linear system Te Newton- Rapson converges quite rapidly, providing tat te initial guess is close to te actual zone containing te solution. If tis is te case, te convergence is quadratic. Oterwise, it may diverge. Te metod consists in linearizing te residual R(α n+1 ) wit respect to te discretization parameter α. Tis yields to:, i n i n i n i n R R R (1.85) were i stands for te iteration counter. Te iterative correction is ten given by:. 1 1 i n i n i n R R (1.86) Once convergence is acieved, te variables are updated as follows: i n n i n i n i n (1.87) Applying tis procedure to te system 1.84 yields, in eac element, te following algebraic system: i n p b v i n p b v i n pp pb pv bp bb vp vv R R R K K K K K K K , (1.88) were K xy are te local stiffness matrix components. Tese are given by: x R K xy xy, (1.89) wit p b v y x,, ), (.

46 44 A condensation of te bubble is used at te local level in order to eliminate te extra degree of freedom δ b associated wit te bubble [Jaouen, 1998]: b bb b bp K R K 1. (1.90) Tis yields to te following local system, were te unknowns are te pressure and te tree components of te velocity field: p K K vv pv K pp K bp K 1 vp ( K bb ) 1 K bp i n1 i v p n R p K bb R t v ( K bb ) 1 R b i n1. (1.91) In effect, te use of te bubble function in te mini-element is equivalent to adding stabilizing terms to te local (and so to global) stiffness matrix of te problem. Tese stabilizing terms are responsible for te versatility of te mini-element, wic can be used in all contexts (compressible or nearly incompressible materials). Te Newton-Rapson (NR) metod is te most widely used metod to solve te equilibrium equations in crystal plasticity models. However, different metods to solve tis equilibrium problem can be found in te literature. In [Hoon Kim, 2013], Hoon Kim proposes a Nelder-Mead (NM) simplex algoritm. Wit tis tecnique it is possible to solve te equilibrium equations witout te first order derivatives of te constitutive equations. Hoon Kim compared te results obtained wit bot tecniques and similar results were obtained. In [Cockalingam, 2013], Cockalingam proposes a Jacobian-Free Newton Krylov (JFNK) metod. In tis case te Jacobian can be approximated wit finite differences, tereby avoiding te calculation of te first order derivatives. In is paper Cockakingam sows tat, wen using a temperature dependent flow rule, te calculation using te JFNK metod can be seven times faster tan te calculation performed using te NR tecnique. However, wen using te simpler power flow rule, it was found tat te JFNK metod was 67% slower tan NR. 5.3 General solution procedure and numerical implementation In order to solve te mecanical problem, te constitutive law as to be coupled to te finite element sceme. It is important to note tat, given te type of element used in tis work, tere is only one Gauss point per element for integration of te constitutive equations. Te solution procedure at a given time step can be summarized as follows: CP1 calculate te initial estimate of te velocity vector (solution of te previous increment), CP2 compute te element velocity gradient used as input to te constitutive model, CP3 iterate at te constitutive law level in order to compute te table variables at time t + Δt and te tangent modulus, CP4 solve te global system of equations for te velocity and pressure fields at time t + Δt using a Newton-Rapson algoritm and go back to point 2 until convergence is acieved, CP5 if convergence occurs, update te velocity and pressure fields and te configuration of te body and move on to te next increment.

47 45 To ensure tat te strain increment remains small so as to ensure optimum convergence of te procedure, an automatic time step adaptation algoritm is used. Te automatic adaptation of time step depends on te strain imposed in tis given time step. If te strain increment in a given time increment is iger tan a critical value, te time step will automatically decrease. On te oter and, if te strain increment is smaller tan a limit value, te time step will increase. Moreover, an automatic subincrementation algoritm is adopted in order to ensure accurate integration of te constitutive equations. Tis subincrementation is performed inside te crystal plasticity code wen te calculation does not converge. In tis case, te time step is divided in a few subincrements and te simulation is repeated for all subincrements. 6 Conclusion As it was discussed in tis capter introduction, te correct simulation of te material microstructure evolution during termo-mecanical process is of prime importance in order to simulate recrystallization and grain growt penomena. In tis capter te process of plastic deformation of FCC metals is briefly reviewed so as to underline te resulting intergranular and intragranular eterogeneities. Te mecanical beaviour of a single crystal is analysed, wit empasis on a finite element implementation. Te slip system concept represents a omogenized way of taking into consideration te motion of individual dislocations gliding under te effect of te critical resolved sear stress. Te flow rule reflects te non-linear relationsip between strain rate and stress. Considering te ardening, pysically-based models were empasized, focusing on te use of one or several dislocation densities as primary variables. Te account of size effects was discussed, distinguising between statistically stored dislocations (SSDs), and geometrically necessary dislocations (GNDs). A simple model (Yosie-Laasraoui-Jonas) is adopted to estimate te SSD density evolution wit plastic deformation, wile te Busso model is cosen to calculate te contribution of te GND density. Finally, te link between te crystal plasticity model and te finite element formulation is presented as well. Te crystal plasticity simulation results are presented and discussed in capter 3. Résumé en Français Lorsqu un matériau polycristallin est soumis à une transformation termomécanique, ses propriétés mécaniques évoluent de manière corrélée à sa microstructure. De plus, ces cangements microstructuraux sont également de première importance vis-à-vis des cinétiques de restauration, recristallisation et croissance de grains pendant les traitements de recuit. Ainsi, afin de bien modéliser les pénomènes de recristallisation et de croissance de grains, la prédiction de l état microstructural d un matériau polycristallin après déformation plastique doit être correctement réalisée. Dans ce capitre, la déformation plastique des matériaux métalliques présentant une structure de type C.F.C. est brièvement présentée. De plus, un état de l art des différents modèles de simulation de la plasticité cristalline existant dans la littérature est présenté : l accent est mis sur les modèles d écrouissage. Deux types de modèle d écrouissage sont présentés : un premier modèle qui considère uniquement les densités de dislocations totales, et un deuxième modèle qui différencie les dislocations statistiquement stockées (SSDs), des dislocations géométriquement nécessaires (GNDs). Dans le premier cas, le modèle de Yosie-Laasraoui-Jonas est implémenté et, dans le deuxième, le modèle de Busso. Finalement le couplage entre le modèle de plasticité cristalline retenu et une

48 46 formulation éléments finis est présenté. Les résultats des simulations de plasticité cristalline sont présentés et discutés dans le capitre 3.

49 47 Capter 2 Statistical generation of polycrystals in a finite element context and a level-set framework Metod and results Contents 1 Introduction Voronoï and Laguerre Voronoï tessellations Microstructure immersion in FE Mes Microstructure Generation - Results Obeying a grain size distribution in 2D microstructures Voronoï Tessellation Metod Laguerre-Voronoï Tessellation Metod Obeying a grain size distribution in 3D microstructures Voronoï Tessellation Metod Laguerre-Voronoï Tessellation Metod Topological study of 2D Laguerre-Voronoï microstructures Morpological study of Laguerre-Voronoï microstructures Conclusion... 76

50 48 1 Introduction Metallurgical penomena occurring in metallic materials during termo-mecanical processing (plastic deformation, recovery, recrystallization, grain growt, pase transformations, etc.) are closely related to te material microstructural condition. Te grain size distribution, te presence or not of precipitates togeter wit teir sape and distribution, te dislocation density distribution after plastic deformation, all tese factors can completely cange te metallurgical penomena kinetics. As a consequence, te material termomecanical beaviour will also cange. Terefore, in order to accurately model te metallurgical penomena occurring during termo-mecanical processing, it as become more and more frequent to build te so called digital microstructures, wic are numerical representations (and idealizations) of real ones. If one looks at te dislocation density distribution induced by plastic deformation, different models can be identified in te literature. Suc models can be used to compute te dislocation density evolution during plastic deformation, and some of tem ave been presented and discussed in capter 1. In capter 3, a comparison between crystal plasticity simulation results, based on two different ardening laws, are discussed. Finally, te influence of te dislocation density distribution on te recrystallization kinetics is analysed in capter 5. In tis work, idealized metallic microstructures will be considered, in particular ignoring te potential presence of precipitates. An example of digital microstructure generated considering te presence of precipitates can be found in [Agnoli, 2012]. In tis capter we will focus on te generation of digital microstructures obeying a given experimental grain size distribution, and on te construction of te associated finite element meses. Te creation of representative digital microstructures based on experimental data is not a simple problem and is not related only to te creation of polycrystalline materials. Wen modelling particle suspensions, porous media or powders, te correct representation of te experimental data is of prime importance. In [Hitti, 2011], [Hitti, 2012] te autors presented and discussed all te callenges tat must be overcome in order to correctly generate a representative microstructure based on experimental data. In te work presented ere, we use te model developed by [Hitti, 2011] and implemented in te CimLib library to generate our digital microstructures. In [Hitti, 2011] two different tecniques to generate representative digital microstructures are implemented. Te first one is te Voronoï tessellation metod, a geometric metod tat partitions a space into convex polyedral cells. Owing to its resemblance to many cellular structures appearing in te nature, Voronoï tessellations are used in a wide range of fields, including biology [Finney, 1978] and zoology [Mallory, 1983]. However, te Voronoï tessellation presents some limitation towards grain size distributions despite its widely use and easy numerical implementation. In fact, wit tis metod, te location of te Voronoï cell nuclei is te only way to define te Voronoï tessellations witout control on te cell caracteristics. Te second tecnique is te Laguerre-Voronoï tessellation metod [Aurenammer, 1987], [Imai, 1985]. A Laguerre-Voronoï diagram corresponds to a Voronoï diagram were te location of te cells faces is constrained by a given nonintersecting sperical packing. Once te digital microstructure is generated, it must be connected to finite element simulations. Once again, several tecniques are available in te literature. Te most widely used metod consists in generating a surface mes for eac grain, and ten generating a volume mes based on tese surface meses [Rollet, 2004]. In tis work, we use instead an implicit metod to capture te grain interfaces. Tis implicit metod is based on a level-set approac to define te different interfaces. Te use of level-set functions in modelling equiaxed polycrystals made of Voronoï cells was introduced by Bernacki [Bernacki, 2009].

51 49 Tis approac was applied successfully for te generation of 2D or 3D polycrystals but for a moderate number of grains. Indeed, one difficulty of tis metod is te numerical cost, wic depends on te number of grains, as well as on te finite element mes used to describe te polycrystal. In tis capter bot models implemented by [Hitti, 2011], Voronoï and Laguerre- Voronoï tessellations, are presented. Te metod used to immerse tese digital microstructures in a finite element mes using a level-set approac is discussed. Moreover, te metodology used to generate adapted anisotropic finite element mes is also described. Finally, te capabilities of te proposed numerical framework are illustrated. More precisely, te caracteristics of te generated digital microstructures are discussed in terms of grain size distribution (in 2D and 3D) and morpological/topological attributes (in 2D). It must be underlined tat all te distributions or mean values discussed in tis capter concerning grain features (grain size, number of sides...) were evaluated as number-weigted (oterwise, it will be indicated in te text). 2 Voronoï and Laguerre Voronoï tessellations Te Voronoï tessellation is te most widely used metod for generating digital polycrystalline microstructures [Rollett, 2004]. In [Logé, 2008], [Bernacki, 2009], [Resk, 2009] and [Quey, 2011] te autors use te Voronoï metod associated wit a level set framework and a finite element formulation to model crystal plasticity and static recrystallization. Anoter example can be found in [Elsey, 2009] were te autors also generate teir digital microstructures using te Voronoï metod. Tey ten simulate te grain growt penomenon using a level set framework associated wit a finite difference formulation. Suc attractiveness of Voronoï tessellation can be explained by its simplicity of implementation. Indeed te Voronoï S,..., tessellation or diagram is fully described by a set of N seeds or Voronoï nuclei i i 1 N. Eac nucleus S i defines a Voronoï cell V i, wic consists of all points closer to S i tan to any oter nucleus (see Figure 2.1): d Vi x R / dx, Si min dx, S j, (2.1) 1 jn were d is te space dimension and d(.,.) is te usual Euclidian distance. Figure 2.1: A Voronoï diagram in 2D. Te Voronoï diagrams can be generated using two different tecniques. Te first one is a direct metod, wic consists in constructing te perpendicular bisectors of te adjacent sites and teir intersections will form te diagram. Te second is based on te construction of

52 50 Delaunay s triangulation (Voronoï s dual) and ten drawing te perpendicular bisectors of its edges. Figure 2.2 illustrates te Voronoï nuclei (left side) used to generate a 3D Voronoï diagram made of 100 cells. Despite its widespread use, te classical Voronoï metod does not allow to obey a given grain size distribution. In fact, only te mean grain size is set. Even toug, in general, tis limit is not discussed; in [Xu, 2009], Xu igligted divergences between statistical properties classically observed in equiaxed polycrystals and te results obtained using Voronoï metod. (a) (b) Figure 2.2: (a) 100 sites of Voronoï in a unit cube and (b) te resulting 3D Voronoï structure. Hence, obeying a specific grain size distribution wile generating a statistical digital microstructure is not easy. To obtain suc microstructures, te Laguerre-Voronoï tessellation Metod can be used [Aurenammer, 1987], [Imai, 1985]. Tis metod consists in using a distribution of non-intersected sperical particles tat serves as a basis for constructing te microstructure. In tis case, te Laguerre-Voronoï tessellation is described by a set of N seeds S, defines a Laguerre-Voronoï cell L i, and weigts S i, r i i 1,..., N. Eac nucleus and weigt i r i wic consists of all points closer to S i, via te power distance (defined below), tan to any oter nucleus: x, S min x S d L i x R / i, 1 jn j, (2.2) 2 2 were x, Si dx, Si ri is te power distance of S i to x. Figure 2.3 illustrates a set of sperical particles used as a base to generate te corresponding Laguerre-Voronoï cells. Figure 2.3: A 3D Laguerre-Voronoï tessellation and te dense spere packing used to generate it.

53 51 Tis metod was successfully used to model polycrystalline structures [Fan, 2004], [Hitti, 2011], [Hitti, 2012] and nanostructured materials [Benabbou, 2010]. In [Lavergne, 2013], using te Laguerre-Voronoï metod, te autors generate not only microstructures wit equiaxed grains but also more complex microstructures wit elongated or flat grains. It is important to underline tat te simplicity of Laguerre-Voronoï metodology is only apparent. Indeed, in order to obey statistically a given grain size distribution, te principal difficulty of te Laguerre-Voronoï metodology is in te generation of te dense spere packing wic must obey te size distribution wit te igest possible density. Once te microstructure is generated, te digital microstructure must be linked to te finite elements mes. In te context of unstructured meses, as used in tis work, different metods can be found in te literature. Te most widely used metod consists in generating a surface mes for eac cell (coincident wit te neigboring grain) and ten generating a volume mes based on tese surface meses (Figure 2.4) [Rollet, 2004]. Figure 2.4: Tree-dimensional mes of an equiaxed single-pase microstructure containing 134 grains (Voronoï cells) generated using a surface mes [Rollet, 2004]. Tis metod is appropriate in te case of polycrystals deformation simulation. However, wen modeling recrystallization and grain growt, te grain boundaries move wit nucleation and disappearance of certain grains. In tese situations, te remesing operations needed to take into account tese topological events are extremely complex. So, te use of tis metod is not straigtforward in tose conditions. Tat is te reason wy finite element metods empasizing an explicit description of grains boundaries (for example: Vertex metods [Barrales, 2008]) are confronted to problems of mes management wic are mainly inextricable in 3D, and tis explains also te current craze for metods favoring an implicit grain boundaries description suc as te pase field [Suwa, 2008], [Takaki, 2010] or level set metods. Since 2005, CEMEF works on te development of a formalism to generate polycrystals immerged into finite element mes. Tis formalism, described in te following, is based on an optimized level set functions calculation from Voronoï or Laguerre-Voronoï sites, and a mes adaptation in te context of a large number of level set functions forming a partition of te study area [Loge, 2008], [Hitti, 2012], [Bernacki, 2009], [Resk, 2009].

54 52 3 Microstructure immersion in FE Mes In our metodology, te locations of te grain boundaries are defined implicitly using a level-set framework. A level-set function, defined over a domain Ω, is called distance function of an interface Γ of a sub-domain Ω S if, at any point x of Ω, it corresponds to te distance from Γ. In turn, te interface Γ is given by te zero isovalue of te function : ( x) ( x) d( x, ) S x, ( x) 0. S ( x) d( x, ), x (2.3) wit S te caracteristic function of Ω S equal to 1 in Ω S and 0 elsewere. Wit te previous proposed equation: ( x) 0 inside te domain defined by te interface Γ and ( x) 0 outside tis domain. Assuming tat te domain is a polycrystal, for eac existing grain in te domain, a level-set function is created. So, if domain Ω is formed by N grains, we will ave N level set functions i,1 i N. In tis case, a global unsigned level-set function can be defined as: x, glob ( x) max i ( x). (2.4) Tis function is positive everywere and tends to zero on te grain boundaries network. Figures 2.5 a, b, c illustrates tree level set functions for tree different grains forming a triple junction in 2D (grain boundaries correspond to te black lines) and Figure 2.5d illustrates te global level set function (Equation 2.4) for tese tree same grains. 1i N Figure 2.5: (a), (b) and (c) Tree level-set functions defining tree grains and (d) te global level-set function. In te following, we explain ow te level-set functions of te considered cells (in Voronoï or Laguerre-Voronoï formalisms) can be defined in an unstructured FE mes. Concerning Voronoï cells, te idea is to consider for eac FE mes integration point X of coordinates x and two Voronoï nuclei s i and s j, te function:

55 53 1 sis j si x ij ( x) sis j, 1 i, j N, j i, 2 s s i j (2.5) wic corresponds to te signed distance of X to te perpendicular bisector of [s i s j ]. Te levelset function (x), defining te Voronoï cell of te nucleus s i, is ten given by: i ( x) min i 1 jn ji ( x), 1 i N. ij (2.6) Moreover, for te above level-set calculation, te Delaunay triangulation [Frey, 1999], [Hitti, 2012] can be elpful. Te Delaunay triangulation gives te grap of eac nucleus s i, wic is te set of te nuclei s j, s i tat sare an edge of te Delaunay triangulation wit s i. As a consequence, te grap of eac nucleus s i, wic corresponds to te set of its neigbours in te Delaunay triangulation, is sufficient to determine te level-set functions and Equation 2.6 can be rewritten: ( x), 1 i N. i ( x) min ij (2.7) j Grap( si ) Figure 2.6 illustrates ow useful te Delaunay triangulation is for calculating level-set functions. Witout it, to calculate te level-set defining te Voronoï cell of nucleus S i, all te nuclei S j ( j i ) must be taken into consideration in te algoritm. Tanks to te Delaunay triangulation, only te points S j belonging to te grap (S i ) are considered. Te computation time decrease depends on te number of grains and te number of nodes existing in te domain. In [Hitti, 2011] te computation time decrease for different meses and number of grains is presented. Figure 2.6: FE Mes (red), Delaunay triangulation (wite), s i site (black), its grap (wite dots) and te global level-set function [Hitti, 2011]. Concerning Laguerre-Voronoï cells, te level-set functions are also given by Equation 2.7 (wit grap(s i ) given by te weigted Delaunay triangulation [Hitti, 2011], [Hitti, 2012]) but te α ij functions (Equation 2.5) are modified as follows:

56 r r s s s x ( x) i j i j i ij sis j, 1 i, j N, j i, (2.8) 2 sis j sis j were r i and r j are, respectively, te radii of s i and s j. Until now, only te microstructure generation and its immersion into a finite element mes ave been discussed witout dealing wit te problem of te mes refinement required for a good accuracy of te calculations. Te use of a monolitic approac in association wit a level set description of te digital microstructure leads to te need of using a fine mes or a ig interpolation degree at te interfaces to ensure te correct description of te microstructure. It is important to igligt tat te grain boundary velocity during grain growt and recrystallization depends on te grain morpology (grain boundary curvature see capter 4). So, te correct description of te microstructure is of prime importance in order to ensure te correct simulation of grain boundary motion. Moreover, an accurate description of te grain boundaries is also needed wen dealing wit possible pysical properties discontinuities appearing at interfaces or multiple junctions. In our numerical strategy, mesing adaption wit a P1 interpolation order was preferred. If a global isotropic remesing of te mes can be used to reac te desired accuracy in te interface description, tis strategy leads to a significant increase in computational resources. Terefore, an adaptive anisotropic remesing tecnique is favored. It is valuable to underline tat te use of anisotropic mes elements generates no damage to recrystallization or grain growt simulations since grain boundary velocity is always perpendicular to te grain boundary, i.e. in te direction of te mes refinement. Different ways to generate an adapted anisotropic mes are found in te literature. In tis P.D. work, two metods ave been used to build anisotropic mes metrics, from wic anisotropic meses were built using te MTC topological meser-remeser developed by Coupez et al. [Coupez, 2000]. Te first metod, called a priori, consists in refining te mes in a small tickness zone around te interfaces. Te refinement operates only in te direction perpendicular to te interface, leading to an anisotropic mes. In order to create an anisotropic mes, in a first step, te tickness (e) of te re-mesing zone is defined. Next, te mes size in te direction perpendicular to te interface ( f ) inside te re-mesed zone is imposed. Finally, a coarser mes size () is imposed far from te interface. Te value is also generally used as te mes size in te direction tangential to te interface, inside te re-mesed zone. Wen dealing wit polycrystalline aggregates and multiple interfaces, te above strategy is repeated for eac grain. Combining all information, te number of refinement directions is ten evaluated at eac mes node. For te nodes outside te remesed zone, tere is no refinement direction and, as a consequence, te mes size is isotropic wit a mes size equal to. As te number of refinement directions increases, te mes size is reduced in one or several directions. Tis appens wen te node is placed on te remesing zone of two different level-set functions. As a consequence, at te triple or multiple junctions, te refinement may become isotropic wit a mes size equal to f [Resk, 2009], [Bernacki, 2009]. Wen using tis metod, te number of elements existing in te final mes cannot be easily controlled, wic can lead to a dramatically increase of te computation time, especially in 3D. Te second metod, called a posteriori metod, consists in using an error analysis in order to obtain an optimal mes for a given pysical field and number of elements [Almeida, 2000], [Mesri, 2008]. In our particular case, as we seek to adapt our mes in te grain boundary areas extending perpendicularly to te interfaces, we use a function based on te global level set function of our microstructure to generate te adapted mes. Tis anisotropic

57 55 (re-)mesing metod leads to a very ig accuracy near te interfaces witout increasing dramatically te computation resources [Coupez, 2000] since te number of elements is imposed. [Almeida, 2000], [Mesri, 2008]. Figure 2.7 illustrates meses obtained wit te two metods, a priori and a posteriori. (a) (b) (c) Figure 2.7: (a) Zoom of a 2D microstructure immerged in a finite element mes adapted using an a priori metod. Te mes is formed by elements. (b) Zoom of a 2D microstructure immerged in a finite element mes adapted using an a posteriori metod. Tis mes is formed by elements. (c) 3D polycrystal immerged in a finite element mes adapted using an a posteriori tecnique. Tis mes is composed of elements. 4. Microstructure Generation - Results In te above topics of tis capter, all te numerical tools needed to generate te digital microstructures used in all simulations performed during tis project ave been presented. Once te basic concepts are well understood, we are now interested in study te capabilities and te limitations of te digital microstructure generation metods. In te following topics, we will discuss te relevance of coosing te Laguerre-Voronoï metod instead of te Voronoï metod, based on te ability of obeying a given grain size distribution (2D and 3D). Finally, a topological and morpological analysis of 2D digital microstructures generated using te Laguerre-Voronoï metod is presented.

58 Obeying a grain size distribution in 2D microstructures A 2D 304L experimental grain size distribution was obtained using optical microscope image analysis (Fig. 2.8). Surface preparation included a standard metallograpic metod consisting in successive wet grinding from 600 grit paper to 4000 grit. Te sample was ten polised using a 1 µm diamond paste. A final procedure using colloidal silica (OPS) was used to obtain a good finis of te surface. Te polised samples were etced wit a cemical solution composed of a 50 ml ydrocloric acid (HCl) solution (100 ml of HCl for 1 litre of water) and 15 drops of H2O2 (~5ml). To obtain a representative result, 21 images ave been combined. Figure 2.8 sows a part of te total image. Grain boundaries were identified and are drawn wit wite lines. Te tin and elongated twin boundaries were not taken into account wen determining te number-weigted grain size distributions. Figure 2.8: Optical microscope image of 304L stainless steel wit grain boundaries drawn in wite lines. Image analysis of te surface was done wit te Visilog 6.3 software, and involved te calculation of individual grain areas. Equivalent circles wit te same areas were used to build te statistical distributions. Only te grains wit a radius larger tan 30µm were considered, for te following reasons: Te total volume fraction of small grains is limited (volume fraction = 0.025) and terefore does not influence very muc te macroscopic mecanical beaviour; in grain growt regime, te smallest grains disappear fast, witin a first initial transient; from a mes adaptation point of view, it is preferable to limit te grain size difference between smaller and larger grains Voronoï Tessellation Metod Tanks to te previously described experimental work, te mean grain radius of 304L steel samples was estimated at 67.4 µm. In order to study te Voronoï metod, two digital microstructures obeying tis mean grain size ave been randomly generated (ten times). Te first one is composed of 280 grains, in a 2 mm x 2 mm square domain, and te second one exibits grains, in a 12 mm x 12 mm square domain (see Figure 2.9). Te average of

59 57 te numerical grain size distributions obtained was ten compared to te experimental 304L grain size distribution as illustrated in Figure We can observe tat te generated distributions are really different from te experimental one. Wile te experimental distribution sows a log-normal beavior, bot numerical microstructures are closer to a Gaussian number-weigted grain size distribution, independently of te number of grains in te aggregate. (a) (b) Figure 2.9: Virtual generation of a 304L polycrystal using te Voronoï metod (10080 grains in a 12 mm x 12 mm square domain): (a) complete image of te digital microstructure, (b) zoom of te same microstructure.

60 58 (a) (b) Figure 2.10: Comparison between experimental 304L data and digital microstructures generated (average of ten generations) using a Voronoï metod in a 2 mm x 2 mm domain and in a 12 mm x 12 mm domain: (a) grain size distribution and (b) cumulative fraction. Te L2 errors (Equation wit f ex te discrete experimental fractions and f num te discrete numerical ones) were evaluated for te two Voronoï tessellations in terms of bot grain size distribution and cumulative fraction distribution. Te results are summarized in Table f ex f num EL 2(%).100, (2.9) 2 f Table 2.1 Probability in number and cumulative fraction L2 error for Voronoï microstructures (average of ten generations). Voronoi Histogram Cumulative fraction Number of R 67.4 m L2 Error (%) L2 Error (%) grains 2 mm x 2 mm mm x 12 mm ex

61 59 Tese results illustrate tat te Voronoï metod is not appropriated to generate a log normal grain size distribution and tat, more precisely, its use sould be restricted to Gaussian grain size distributions. Considering tat log normal grain size distributions are extremely common, te Voronoï metod is not, in general, te appropriate tool to generate a numerical aggregate obeying an experimental grain size distribution. Xu, in [Xu, 2009], igligted similar divergences between statistical properties classically observed in equiaxed polycrystals, and te results obtained using te Voronoï metod Laguerre-Voronoï Tessellation Metod Regarding te Laguerre-Voronoï metod, first results concern te experimental 304L steel grain size distribution: 8 different digital microstructures are generated using 8 different domain sizes: x mm x x mm, x {1, 2,, 6, 8, 12}. Figure 2.11 illustrates te average results obtained for ten generations of microstructures in te domains 4 mm x 4 mm, 8 mm x 8 mm and 12 mm x 12 mm. Te L2 error between te experimental data and te numerical results for te considered microstructures are presented in Figure (a) (b) Figure 2.11: (a) Comparison between numerical grain size istograms obtained using a Laguerre-Voronoï metod (average of ten generations in domains 4 mm x 4mm, 8 mm x 8 mm and 12 mm x 12 mm) and te experimental one (for 304L steel) and (b) comparison between te cumulative fraction distributions for te same cases.

62 60 Analysing Figure 2.12, we observe tat te L2 error decreases wit te increase of te domain size. Tis beaviour is expected since increasing te domain size implies generating a larger number of grains and ence a better representation of te microstructure. For domains larger tan 4 mm x 4 mm, te error stabilises around 12 %. In te istogram of Figure 2.11, te grain size distribution error is concentrated in te small grain size families. Tis is due to a sligt mismatc between te size of te Laguerre-Voronoï spere and te grain size obtained after te creation of te Laguerre-Voronoï cells. Tereby, a few grains tat sould be placed in te range [30, 40] are placed in te [40, 50] range. Considering te cumulative fraction curve of Figure 2.11, te error is smaller tan 3.5% for all investigated cases. Neverteless, wen compared to te results obtained for Voronoï tessellations (Table 2.1), and to te state of art [Hitti, 2012], te errors obtained ere correspond to very good descriptions of te experimental distribution. Tere is owever a need to continue improving te algoritm of digital microstructure generation, especially for te small grain size range. Figure 2.12: Error L2 between experimental data and numerical results (average of ten generations) of digital microstructures obtained using a Laguerre-Voronoï metod - Histogram error (blue line) and cumulative fraction error (pink line). Figure 2.13 illustrates one of te digital microstructures generated using te Laguerre- Voronoï metod in te 12 mm x 12 mm domain.

63 61 (a) (b) Figure 2.13: A 2D digital microstructure made of 8294 grains obtained using a Laguerre-Voronoï metod, based on a 304L steel experimental grain size distribution (domain size: 12 mm x 12 mm): (a) complete image of te digital microstructure, (b)zoom of te same microstructure. After analysing te possibility of generating a digital microstructure wit a grain size distribution close to tat of te experimental 304L steel, a more general study can be done about te capacity to generate various grain size distributions. Six different 2D digital microstructures are studied, wose features are presented in Table 2.2.

64 62 Table 2.2: Features of 6 grain size distributions. Mean Radius - μ (µm) Standard Deviation - σ (µm) σ/ μ Initial number of grains Domain Size (mm) Log x 11 Log x 12 Log x 13 Log x 14 Log x 14 Bimodal x 12 Te first five distributions are syntetic log normal, meaning tat tey are not based on real experimental data. In tese cases, te mean grain size is more or less te same, wit a standard deviation ranging between 7µm and 40µm. Te last distribution corresponds to a syntetic bimodal distribution. Te number of grains is around 10,000 for all microstructures, ensuring good microstructure representativity. Figure 2.14 describes comparisons between imposed distributions and te obtained numerical results, and between te teoretical and numerical cumulative fractions. Distributions and cumulative fractions are given as a function of te normalized equivalent radius R R, were <R> corresponds to te mean grain size. Table 2.3 gives te L2 errors between te teoretical and te numerical distributions and cumulative fractions. (a)

65 63 (b) Figure 2.14: Grain size distributions istograms (a) Log1, Log2, Log3, Log4, Log5 and Bimodal and cumulative fraction (b) Log1, Log2, Log3, Log4, Log5 and Bimodal. Table 2.3 Distribution and cumulative fraction L2 errors for te 6 microstructures of Table 2.2. Histogram L2 Error (%) Cumulative fraction L2 Error (%) Number of grains Bimodal Log Log Log Log Log Errors are iger for more complex distributions, like te Bimodal distribution. However, as mentioned earlier, wen compared to te state of art in tis domain [Hitti, 2012], te errors obtained ere correspond to very good descriptions of all distributions. 4.2 Obeying a grain size distribution in 3D microstructures In order to generate a 3D digital microstructure close to tat of 304L, te 3D experimental grain size distribution was estimated using te Saltykov metod [Underwood, 1970]. Tis metod allows converting a 2D grain size distribution (bar-plots), for an equiaxed microstructure, to a 3D discrete grain size distribution. Tis metod is based on te idea tat all grains are speres and te 3D distributions prediction is based on te probability of a spere being intersected by a plane section. In [Tucker, 2012] te autors ave sown, for a Ni-Based superalloy, tat te Saltykov metod corrects te disparity between te 2D and 3D grain size distribution mainly for te mean and te upper grain size ranges. Te metod is explained in details in Appendix 1. Figure 2.15 describes te 3D extrapolated discrete grain size distribution of te 304L steel.

66 64 Figure 2.15: 3D extrapolated grain size distribution obtained using te Saltykov metod, for te 304L steel. Te predicted distribution exibits a large number of small grains. Te mean grain radius is equal to 48.8 µm in 3D (in 2D, mean grain radius is equal to 67.4µm) Voronoï Tessellation Metod Figure 2.16 illustrates one of te digital microstructures obtained using te Voronoï metod, considering grains randomly generated in a 2 mm x 2 mm x 2 mm RVE (i.e. obeying te prescribed 3D mean grain size of 48.8 µm for 304L). As for te 2D discussions, ten distinct digital microstructures ave been generated and Figure 2.17 gives a comparison between te average of tose numerical distributions, and te experimental one. Figure 2.16: A 3D microstructure of grains generated using a Voronoï metod in te domain 2 mm x 2 mm x 2mm for te 304L steel.

67 65 Figure 2.17: Comparison between te numerical 3D grain size distribution (average of ten calculations barplots) obtained using a Voronoï metod and te 3D extrapolated 304L one ( teoretical discrete distribution). Te L2 error between te experimental and numerical distributions is equal to 137.8% (eac set of te numerical distributions being replaced by its mean value for tis calculation); sowing tat te digital microstructure obtained using te Voronoï metod is completely inadequate Laguerre-Voronoï Tessellation Metod Regarding te Laguerre-Voronoï metod, ten digital microstructures were also generated based on te extrapolated 304L steel 3D grain size distribution obtained wit te Saltykov metod. Figure 2.18 illustrates a comparison between te experimental distribution and te average of te numerical grain size distributions. Figure 2.18: Comparison between te 3D numerical grain size distribution (average of ten calculations barplots) obtained using a Laguerre-Voronoï metod and te extrapolated 304L one ( teoretical discrete distribution).

68 66 Te average of te number of grains generated in a 2 mm x 2 mm x 2 mm domain is equal to Te L2 error between te experimental distribution and te average of te numerical ones (eac bin of te numerical distributions being replaced by its mean value) is equal to 53%. In Figure 2.18, once again, te error is concentrated in families of smaller radii. Knowing tat te smallest grains are te first to disappear during grain growt modelling and also tat te reduced volume fraction of smaller grains (volume fraction = 0.02) leads to a minor influence on te global mecanical beaviour of te RVE, one can decide to compare te experimental and numerical volume-weigted grain size distributions (Figure 2.19). In tis case, te L2 error is reduced to 36.3%. Te results become acceptable, altoug some improvements could be done in te future. Te same analysis was performed for te microstructure generated using te Voronoï metod, and te error for te volume-weigted distribution, 212.2%, is even larger tan te error for te number-weigted distribution (137.8%). Figure 2.19: Comparison between te 3D numerical volume weigted grain size distribution (average of ten calculations bar-plots) obtained using a Laguerre-Voronoï metod and te extrapolated 304L one ( teoretical discrete distribution). Figure 2.20 illustrates one of te 3D digital microstructures obtained using te Laguerre-Voronoï metod. Figure 2.20: A 3D microstructure made of 6400 grains generated using a Laguerre-Voronoï metod in te domain 2 mm x 2 mm x 2mm for te 304L steel.

69 Topological study of 2D Laguerre-Voronoï microstructures In tis work, all digital microstructures will be used in order to simulate plastic deformation wit crystal plasticity (capter 3), static recrystallization (capter 5) and grain growt (capter 4) penomena. Previous discussions in tis capter ave illustrated te capability of te proposed numerical strategy to generate polycrystals in a FE context, wic obey grain size distributions in 2D and in 3D. However, all te above cited penomena are also dependent on te material structure and on te microstructural topology. For example, in capter 5, a discussion about te topological distribution of new (recrystallized) grains and its influence on te recrystallization kinetics is presented. In te case of grain growt, te associated kinetics depends on te microstructure topology, especially te number of sides of eac grain, as discussed in [Hillert, 1965], [Abbruzzese, 1992a], [Abbruzzese, 1992b]. In addition, in [Abbruzzese, 1992a], [Abbruzzese, 1992b] te autors sow tat teir grain growt equations, commonly used in grain growt mean field models [Bernard, 2011], are valid only if te microstructure obeys a Special Linear Relationsip (SLR) between te grain number of sides, and te radius size. In tis subsection, te topological caracteristics of seven 2D different grain size distributions (te six distributions presented in Table 2.2 plus te 304L distribution) are presented. In fact, a similar topological study as te one presented in [Abbruzzese, 1992a] is performed and te SLR is studied in order to verify if te Abbruzzese grain growt model ypotesis is valid for te initial microstructures generated using our Laguerre-Voronoï algoritm. Considering te 3D microstructures, despite our capability to generate tem numerically, it is currently estimated tat te accuracy of te grain size distribution still needs improvements, in order to generate realistic results, from wic topological and morpological studies could be performed. Table 2.4 sows tat all seven digital microstructures present a mean number of sides around 6. However, from Figure 2.21 it is observed tat distributions of sides number (n) vary significantly from one case to anoter. For example, te Log1 distribution is narrower tan te oters, wile te Bimodal distribution as two peaks. Table 2.4 Average number of sides for te seven microstructures generated numerically. Mean Initial Domain number number of Size of sides grains (mm) (n) Log x 11 Log x 12 Log x 13 Log x 14 Log x 14 Bimodal x L x 13

70 68 Figure 2.21: Number of sides distributions for all seven 2D digital microstructures: 304L, Bimodal, Log1, Log2, Log3, Log4 and Log5. One can notice also a correlation between te number of sides distribution and te grain size distribution. For te same microstructure, bot curves present more or less te same sape. Based on tis idea, one can imagine a correlation between te sides number and te grain radius. Tis intuition is confirmed in Figure 2.22, were radius values are normalized according to: r R R. (2.10) 304L Bimodal Log1 Log2

71 69 Log3 Log4 Log5 Figure 2.22: Normalized radius vs. number of sides for all considered digital microstructures. Since, in general, te larger grains ave a larger number of sides, te two parameters r (normalized radius) and n (number of sides) of te distribution are not independent of eac oter. But a unique matematical relationsip between te values r and n wic would be valid for individual grains cannot exist, i.e. grains wit te same equivalent radius may ave different number of sides. Only statistical relationsips considering grain families (eac family being labeled wit a subscript i) can be expressed. In [Abbruzzese, 1992a], Abbruzzese proposes te following two linear relationsips between r i (grain radius of grain family i) and n i (number of sides of grain family i). n a 0, (2.11) i ar i n b0 b r, (2.12) i wit a 0, a, b 0 and b being constants. Te curve n f r ) is obtained by averaging over te n at a given r i wile te curve ni f ( ri ) is calculated by averaging over te r i at a given n. Also, in [Abbruzzese, 1992a], it is sown tat: a0a bo b 6. (2.13) Verifications of tese relations ave been performed on te seven digital microstructures generated wit te Laguerre-Voronoï tessellations. Figure 2.23 illustrates te relationsips n i f ( ri ) and Table 2.3 gives te corresponding b and b 0 values obtained by linear regression. i i ( i

72 Number of sides Figure 2.23: n f ( r ) relationsip for all seven digital microstructures. i i 304L Bimodal Log1 Log2 Log3 Log4 Log Mean Equivant Radius Table 2.5: Values of b 0 and b (Equation 2.13) obtained by linear regression. b b 0 b 0 +b 304L Bimodal Log Log Log Log Log In [Abbruzzese, 1992a], it was found b = 4.14 and b 0 = Except for te Log1 distribution, te b and b 0 values measured in te digital microstructures are similar. Interestingly, te study performed by Abbruzzese is based on an experimental distribution of an Al-3%Mg alloy wit a log normal grain size distribution. Log1 distribution is close to a sarp normal distribution wic may be te reason for te different beaviour presented by tis distribution. However, te summations of b 0 + b for all digital microstructures are around 6, wic agrees well wit Now, considering relationsip 2.12, Figure 2.24 illustrates te relationsip between n i f ( r i ) obtained for te seven digital microstructures, and Table 2.4 gives a and a 0 values obtained by linear regression.

73 Mean number of faces Figure 2.24: n f r ) relationsip for all seven digital microstructures studied. i ( i 304L Bimodal Log1 Log2 Log3 Log4 Log Equivant Radius Table 2.6: Values of a 0 and a (Equation 2.12) obtained by linear regression. a 0 a a 0 +a 304L Bimodal Log Log Log Log Log In [Abbruzzese, 1992a], it was found a = 2.99 and a 0 = 2.98, wit terefore a good agreement wit Table 2.6. Te summations of a 0 + a are all very close to 6, wic again validates relation Relation 2.12 is te Special Linear Relationsip proposed by Abbruzzese and used to determine te Abbruzzese mean field grain growt model presented in [Abbruzzese, 1992b]. Hence, te results obtained ere illustrate tat topological caracteristics of te 2D Laguerre-Voronoï tessellations are in agreement wit te topological assumptions of te famous Hillert and Abbruzzese grain growt model. More global analysis concerning grain growt modelling of te considered microstructures in te full field FE/level set context will be detailed in capter Morpological study of Laguerre-Voronoï microstructures Te Laguerre-Voronoï metod is generally used to generate microstructures composed of equiaxed grains. Since te Laguerre-Voronoï metod consists in using a distribution of non-intersected sperical (in 3D, circular in 2D) particles tat serve as a basis for constructing te microstructure, one can indeed expect to generate equiaxed grains wen using tis metod. However, if tese sperical (in 3D, circular in 2D) particles compaction is not as dense as possible, te generated digital microstructure may present non-equiaxed grains. In oter words, te generation of equiaxed digital grains is ensured by a good dense spere packing algoritm in te Laguerre-Voronoï tessellation. In tis subsection, te digital morpology of te seven 2D different grain structures numerically generated is analyzed. Te objective is to verify te absence of a morpological texture in te digital microstructures.

74 72 One way to verify in 2D if a grain is equiaxed is to compare its equivalent radius wose calculation is based on te grain surface (R), wit te equivalent radius calculated using te grain perimeter (R eqs Equation 2.14, were L is te grain perimeter). L R eqs. (2.14) 2 Te smallest te difference between R and R eqs, te closest to a circular sape te grain is and consequently te more equiaxed. In Figure 2.25, tis analysis was performed for all grains of te considered digital Laguerre-Voronoï microstructures (one of te ten generations). It can be recognized tat, rougly, R eqs R for all microstructures. 304L Bimodal Log1 Log2 Log3 Log4

75 73 Log5 Figure 2.25: R eqs vs. R for all seven digital microstructures: 304L, Bimodal, Log1, Log2, Log3, Log4 and Log5 For nearly equiaxed grains, te difference between R and R eqs is rater small, amounting to mean value smaller tan 12% [Abbruzzese, 1992a]. From Table 2.7 we observe tat all digital microstructures exibit a mean error between R and R eqs wic is smaller tan 12%. However, analysing te graps from Figure 2.26, it can be concluded tat tis difference is concentrated on grains wit smaller equivalent radii. Statistically speaking, te smaller grains are less equiaxed tan te largest ones. Tis means tat wit te Laguerre-Voronoï metod, compaction problems occur mainly on small circles areas. Table 2.7: Mean error between R eqs and R for all digital microstructures. Mean error between R eqs and R (%) 304L 8.8 Bimodal 9.8 Log1 7.5 Log2 8.1 Log3 8.3 Log4 9.9 Log L Bimodal

76 74 Log1 Log2 Log3 Log4 Log5 Figure 2.26: Difference between R eqs and R vs. radius dispersion for all seven digital microstructures studied. Finally, te relationsip between te grain perimeter S and te grain area A is studied. Analyzing graps from Figure 2.27, an approximate equation S f (A) was obtained for all te considered microstructures, wic are summarized in Table 2.8. Beaviours are very similar.

77 75 304L Bimodal Log1 Log2 Log3 Log4 Log5 Figure 2.27: Grain perimeter as a function of grain area dispersion for all digital microstructures: 304L, Bimodal, Log1, Log2, Log3, Log4, and Log5.

78 76 Table 2.8: S f (A) equations for all digital microstructures: 304L, Bimodal, Log1, Log2, Log3, Log4 and Log5. S is te mean perimeter and A te mean area of grains. S f (A) 304L 0. 5 S 3.81A Bimodal 0. 5 S 3.85A Log S 3.83A Log S 3.84A Log S 3.82A Log S 3.85A Log S 3.86A Based on an average of te equations from Table 2.8, a statistical equation for te relationsip between te grain perimeter and te grain surface for a 2D microstructure generated using te Laguerre-Voronoï metod can be proposed: 0.5 S 3.84A. (2.15) Comparing Equation 2.15 wit te one expected for a perfect circular sape 0.5 S 3.5A, it is concluded tat te Laguerre-Voronoï metod not only generates microstructures obeying a given grain size distributions, but also leads to equiaxed microstructures. 5 Conclusion Tis capter presents te main matematical tools used to generate and to immerge digital microstructures into a finite element mes. Initially, wit respect to te microstructure generation, te Voronoï and te Laguerre-Voronoï metods were presented. Te conclusions are: For 2D digital microstructures, te use of a Laguerre-Voronoï metod allows te generation of a microstructure obeying an experimental grain size distribution wit a very low L2 error, and tis does not require a very ig number of grains in te digital microstructure. Te use of te Voronoï metod in 2D does not allow obeying a given grain size distribution. Even toug te mean grain size can be set properly, te error between te numerical and te experimental grain size distributions can be very ig, even wen generating a large number of grains. Te above results were also verified for 3D digital microstructures. In a second part, a topological study of seven different 2D digital microstructures was presented. In all microstructures, te mean number of grain sides was around six, but te distribution of te number of sides differed from one microstructure to anoter. Two statistical linear relationsips between te grain size and te number of sides, proposed by [Abbruzzese, 1992a], were critically analysed. For all investigated 2D digital microstructures, tese linear relationsips were verified. Finally, a morpological study of te same seven 2D digital microstructures was presented. Te objective of tis study was to verify if te digital microstructures generated using te Laguerre-Voronoï algoritm are equiaxed. Based on te results discussed in tis capter, one can conclude tat, even toug te smallest grains are less equiaxed tan te

79 77 larger ones, te digital microstructures generated wit our Laguerre-Voronoï algoritm can be considered, overall, as very close to perfectly equiaxed microstructures. Concerning te last two topics, recommended future work would be to perform te same study in 3D. Résumé en français Dans ce capitre, les principaux outils numériques utilisés dans la génération statistique et l immersion des microstructures digitales dans un maillage éléments finis sont présentés. Concernant la génération statistique des microstructures digitales, deux métodes sont discutées : l approce de type Voronoï et l approce de type Laguerre-Voronoï. Il est tout particulièrement illustré que la métode de Laguerre-Voronoï permet de respecter une distribution expérimentale de taille de grains contrairement à l approce de type Voronoï. Dans une deuxième partie, une étude topologique de sept microstructures digitales 2D générées par l approce de type Laguerre-Voronoï retenue est présentée. Il est mis en lumière que les relations de type SLR proposées par Abbruzzese [Abbruzzese, 1992a] sont vérifiées pour l ensemble des microstructures. Finalement, une étude morpologique des mêmes microstructures digitales en 2D est présentée afin de vérifier le caractère equiaxe ou non des microstructures générées par cette approce.

80 78 Capter 3 Deformation beaviour of 304L steel polycrystal and tantalum oligocrystal Contents 1 Introduction L Steel Case L stainless steel Cemical composition Mecanical beaviour Microstructural caracterization - initial state Planar compression test (Cannel Die test) Description of te test Material parameters - 304L Effects of mes type in igly resolved polycrystalline simulations Considering only SSDs Considering bot SSDs and GNDs Results Stress - Strain analysis Dislocations density and energy distribution analysis Strain distribution analysis Grain size effects Tantalum Case Tantalum Compression test Test description Digital oligocrystal Material parameters Results Sample sape Dislocations density distribution Deformation distribution Texture analysis Conclusion

81 79 1 Introduction Recrystallization processes depend on te nature of te deformed state. In order to accurately model recrystallization, microstructural eterogeneities, and te associated distribution of defects, must be understood, and even predicted. Tis explains te close connection between recrystallization and plasticity penomena. In tis capter, a crystal plasticity model, based on te concept of dislocation slip (presented in details in capter 1) is tested and calibrated using two different test cases. Te first one corresponds to a planar compression test (cannel die test) on a 304L steel polycrystal, taking place at ig temperature. Te second one corresponds to a simple compression test on a tantalum oligocrystal composed of six different grains. In bot materials, empasis is given on te use of statistically stored dislocations (SSD), and geometrically necessary dislocations (GND). Te latter introduce grain size effects in plasticity, and reorganize te spatial distribution of dislocation densities, wic is sown to influence te potential location of recrystallization nuclei L Steel Case L stainless steel Cemical composition 304L steel is an F.C.C. (face centred cubic) stainless steel. Te major feature of a stainless steel is its resistance to corrosion. Tis property is reaced by adding more tan 10.5% of cromium to te alloy. Except for cromium, te principal added element is nickel, wic elps stabilizing austenite at low temperatures. Austenitic stainless steels do not undergo pase transformation upon cooling or eating. Tey are terefore good candidates for te investigation of recrystallization and grain growt processes. Te cemical composition of 304L steel is presented in Table 3.1. Table 3.1: 304L stainless steel cemical composition [Roucoules, 1994]. Component C Mn Si P S Cr Ni N Weigt percentage <0.03 <2.00 <0.75 <0.045 < ~ ~12.0 <0.10 Sometimes, residual ferrite is found in 304L steel (as sown in Figure 3.1), but it usually represents less tan 1% of te total volume. Te presence of residual ferrite may be a consequence of eterogeneous nickel distribution, and/or solidification conditions. Figure 3.1: EBSD analysis sowing residual ferrite in a 304L austenitic stainless steel. Te red zones correspond to te austenitic pase wile te green zones correspond to te ferrite pase.

82 Stress (MPa) 80 Since te residual ferrite represents a small amount of te material, it will not be considered in te modelling work, i.e. 100% austenite will be assumed. Te 304L alloy is widely used in equipment and utensils for food processing and andling, beverages and dairy products. It is also found in eat excangers, piping, tanks and oter equipment wic are in contact wit fres water Mecanical beaviour Te mecanical properties and beaviour of metallic materials depend mostly on te dislocation content and structure, te grain size and te texture. Te dislocation density of a typical annealed state is ~10 11 m -2 (altoug tis may significantly vary from one material to anoter), wic may increase to as muc as ~10 16 m -2 wen te material is eavily deformed. Te increase of material dislocation density is responsible for te material work ardening. In 1934, Taylor [Taylor, 1934] put forward te idea of ardening: some dislocations become stuck inside te crystal and act as sources of internal stress, wic oppose te motion of oter gliding dislocations [Asby, 2009]. Also, dislocations multiplication and anniilation depend on te temperature and strain rate. It is terefore expected tat te material mecanical beaviour will depend on temperature and strain rate. It is interesting to igligt tat wen deformation at sufficiently ig temperature exceeds a critical strain, dynamic recrystallization is initiated and increases wit furter deformation until a steady state were te microstructure is stable due to a dynamic equilibrium between softening by dynamic nucleation and work-ardening. Te onset of dynamic recrystallization is identified as a peak on te stress-strain curve followed by a material softening. 304L steel recrystallizes dynamically under te analysed conditions so, its stress-strain curves will present te typical dynamic recrystallization beaviour. All experiments presented in tis topic were performed by Ke HUANG during is PD Tesis [Huang, 2011]. Temperature influence In order to study te temperature influence on te 304L steel mecanical beaviour, torsion tests (Appendix 3) at 900 C, 1000 C, 1100 C and 1150 C were conducted wit a constant strain rate of 0.01 s -1. Te smooted stress-strain curves are presented in Figure Strain 900 C 1000 C 1100 C 1150 C Figure 3.2: Stress - strain curves for 304L steel torsion test for different temperatures: 900 C, 1000 C, 1100 C and 1150 C - =0.01 s -1 [Huang, 2011].

83 81 At a deformation temperature of 900 C, te flow stress increases gradually and no apparent stress peak is found before fracture. Te dynamic recrystallization of 304L steel in tese conditions is very slow, wic may explain te difficulty in identifying a peak. At a moderate temperature of 1000 C, some canges of stress-strain curves are observed. Te stress sows more softening compared to te curve at 900 C. Considering te microstructure evolution, te flat grain boundaries become irregular after a small deformation of 0.3 (Figure 3.3.a), and start to serrate at a deformation level equal to 0.5 (Figure 3.3.b). Some small nuclei (new recrystallized grains) can be found among te original grain boundaries wile oter initial grain boundaries remain undecorated at ε = 1.0 (Figure 3.3.c). One can conclude tat dynamic recrystallization starts at a strain between 0.5 and 1.0 at tis temperature. Wit furter deformation (Figures 3.3.d, 3.3.e and 3.3.f), te dynamically recrystallized grains volume fraction increases gradually and a large fraction of te initial microstructure is replaced by smaller grains. (a) (b) (c) (d) (e) Figure 3.3: Microstructure of samples deformed a 1000 C, =0.01 s -1 for (a) ε = 0.3, (b) ε = 0.5, (c) ε = 1.0, (d) ε = 1.5, (e) ε = 2.0 and (f) ε = 2.5 [Huang, 2011]. Te stress-strain curves at iger temperature (1100 C and 1150 C) exibit typical dynamic recrystallization beaviour, wit a distinct stress peak followed by a mecanical steady state, wic continues until fracture of te sample. Comparing all stress-strain curves, te usual trend is confirmed, i.e. (i) te flow stress decreases wit increasing temperature, and (ii) te peak stress appears at lower strain wit increasing temperature. (f)

84 Stress (MPa) 82 Strain rate influence In order to study te strain rate influence on te 304L steel mecanical beaviour, tests wit = 0.01 s -1, = 0.1 s -1 and = 1 s -1 were conducted at a constant temperature of 1000 C. Te smooted stress-strain curves are presented in Figure s s-1 1 s Strain Figure 3.4: Stress - strain curves for 304L steel torsion test for different strain rates; 0.01 s -1, 0.1 s -1 and 1 s -1 at 1000 C. Analysing Figure 3.4 te expected trend of a stress increase wit increasing strain rate is found. For example, increasing te strain rate from = 0.01 s -1 to = 1 s -1 leads to an increase of flow stress of about 80 MPa. At lower strain rate ( = 0.01 s -1 ) te typical flow stress curve induced by dynamic recrystallization is obtained (as already mentioned in paragrap A.1.2.1). As strain rate increases to = 0.1 s -1 and = 1 s -1, te flow curves sow a broader peak followed by softening. Tis is likely due to deformation eating, as documented in te literature [Mataya, 1990]. In [McQueen, 1995], [Mirzade, 2013], [El-Waabi, 2003], [Degan-Mansadi, 2008] te autors present 304L steel experimental results for different test conditions. All results sow similar tendencies to te ones presented ere. In tis capter, only plastic deformation is studied, and terefore only te first part of te stress-strain curves, before te stress peak, is considered in te comparisons between te numerical and experimental results. Even toug it is possible to simulate large simulations, as Resk describes in [Resk, 2009], [Resk, 2010], in te case of 304L steel, te numerical results ave no pysical meaning for strain values iger tan te strain peak, due to te occurrence of dynamic recrystallization transforming te microstructure. In te case of a deformation at 1000 C and 0.01 s -1, stress-strain curves are analysed for strain values smaller tan Microstructural caracterization - initial state Te initial state of 304L steel samples is caracterized using EBSD maps. From Figure 3.5, we observe tat te initial sample present a completely recrystallized microstructure. Also, we identify a large number of twin boundaries: 48% of all grain boundaries are, in fact, annealing twins.

85 83 Figure 3.5: 304L steel EBSD map As discussed in capter 2, in te initial state, te material presents a log-normal grain size distribution (capter 2, paragrap 4). From te pole figure given in Figure 3.6 and te orientation distribution functions (ODF) in Figure 3.7, it is concluded tat te material as a quasi-random crystallograpic texture. Figure 3.6: 304L steel pole Figure - initial state. Figure 3.7: Orientation distribution function of 304L steel at te initial state.

86 84 In te description of te initial microstructure of te investigated 304L steel, and in particular for te crystal plasticity calculations, te discrete set of grains crystallograpic orientations is terefore selected in a random way. Te corresponding Euler angles are given in Appendix Planar compression test (Cannel Die test) Description of te test Te planar compression test is a compression test were deformation proceeds along two directions, as te case in te rolling process. In te simulations, a normal force is applied in te z 0 -direction and te material tinning results in extension along te x 0 -direction, as illustrated in Figure 3.8.a. (a) (b) Figure3.8: (a) Cannel die test description. (b) Homogeneous strain rate boundary conditions for a cannel die test. Homogeneous strain rate boundary conditions are applied: at every node belonging to te boundary of te domain, te tree components of te velocity v are prescribed according to te following equation: v LX on, (3.1) were L represents te velocity gradient tensor and X te position vector in te current configuration. For te cannel die example, L is given by: 0 0 L (3.2) 0 0 wit β being a constant. Figure 3.8.b illustrates te boundary conditions for te cannel die test.

87 Material parameters - 304L As discussed in capter 1, te material ardening is computed according to te following rule: 0 Mb, (3.3) c T were 0 is te dislocation free yield stress, M te Taylor factor, α a constant, μ te sear modulus, b te magnitude of te burgers vector, and T te total dislocation density wic is often expressed as T S G. In tis case S is te statistically stored dislocation (SSD) density and G te geometrically necessary dislocation (GND) density. Considering only te SSDs, te Yosie-Laasraoui-Jonas [Laasraoui, 1991] equation was implemented in our crystal plasticity model: S K, (3.4) K 1 2 T were K 1 and K 2 are material parameters. Since tese parameters are identified from macroscopic mecanical tests, Equation 3.4 needs to be adjusted to te context of crystal plasticity, describing mesoscopic scale penomena. Tis adjustment is explained in details in te following topics. Considering te GNDs, te Busso [Busso, 2000] model was implemented in te crystal plasticity model (capter 1, paragrap 2). b p b t n curl n F G sm. (3.5) G et G en Six parameters involved in Equations 3.3 to 3.5 can be found in te literature: te sear modulus μ; te constant α; te magnitude of te burgers vector b ; te components of te anisotropic elastic tensor modulus C 11, C 12 and C 44. On te oter and, K 1, K 2 and 0 must be identified from mecanical testing at different temperatures and strain rates. Sear modulus μ Following te metod proposed by Frost and Asby [Frost, 1982], Gavard [Gavard, 2001] analysed te 304L steel sear modulus evolution wit temperature using experimental data presented in Table 3.2 Table 3.2: Sear modulus of 304L [ASM, 1982]. T( C) µ (GPa) 79/77 73/72 70/68 66/64 55/61 59/58 55/55 50/52 Linear dependence wit temperature was found between 24 and 816 C, as sown in Figure 3.9.

88 86 Figure 3.9: Sear modulus evolution of 304L steel wit temperature [Gavard, 2001] Te curve of Figure 3.9 is ten extrapolated until 1200 o C, wic leads to sear modulus values presented in Table 3.3. Table 3.3: Sear modulus from room temperature to 1200 C for 304L steel [ASM, 1982] T( C) µ (GPa) K 1, K 2 and τ 0 At te macroscopic scale, te material ardening is computed using Equations 3.3 and 3.4. Te K 1 and K 2 identification procedure at tis scale is explained in details in [Jonas, 2009]. To compute te ardening at te mesoscopic scale, Equations 3.3 and 3.4 are modified according to Equations 3.6 and 3.7 by considering te meaning of te Taylor factor (M). Te Taylor Factor is a geometric factor wic describes te propensity of a crystal to slip (or not slip) based on te orientation of te crystal relative to te sample reference frame. Macroscopic Mesoscopic c 0 Mb c 0 b (3.6) K ) ( K ' ' ) 1 K 2 (3.7) ( 1 K2 0 ' K1 Equation 3.3 is divided by M, i.e. 0. In Equation 3.4, K M 1 M, ' K 2 K 2 due to M te fact tat M. Te Taylor factor value depends on te mecanical solicitation and on te grain orientation [Humpreys, 2004]. As a consequence, parameters K 1, K 2 and 0 are, at te grain scale, also dependent on te Taylor factor. On te contrary, K 1, K 2 and 0 are independent of te Taylor factor value; tey are intrinsic to te material, at a given temperature and strain rate. As a consequence, once K 1, K 2 and 0 are correctly identified, te crystal plasticity model can be used to model any mecanical condition, and sould account for te loading pat influence. Te identification procedure of tese parameters is detailed below. First of all, K 1, K 2 and 0 are identified using experimental curves of 304L steel torsion tests. Figure 3.10 illustrates a comparison between te experimental (1000 C and 0. 01)

89 87 results (green line) and te teoretical curve (red line) obtained wit K 1 = m -2, K 2 = 3.3 and 53 0 MPa (red line). Figure 3.10: Comparison between experimental results and teoretical stress - strain curves for 304L steel at 1000 C and deformation rate equal to 0.01 s -1. Secondly, te 304L steel mean Taylor factor during a torsion test is identified using EBSD analysis. As sown in paragrap 2.1.3, te 304L steel exibits a quasi-random texture. To furter verify te material isotropy, te mean Taylor factor of a torsion test is calculated for 3 different solicitation directions, using te TSL OIM Analysis software. According to Table 3.4, te mean M is 3.3. In order to obtain K 1, K 2 and 0 values, we terefore need to divide K 1, K 2 and 0 by tis value. For a planar compression test, te mean Taylor factor is in average 3.33, but no experimental data is available for tis test. As te Taylor factor depends on te grains crystallograpic orientations, te particular set of representative grains used in simulations may influence te mean Taylor factor value and, as a result, influence te polycrystal mecanical beaviour. In tis work we perform simulations wit 50-grain domains and 100-grain domains. Te corresponding mean Taylor factor values for our 50-grain and 100-grain polycrystal are given for torsion and planar compression tests in Table 3.4. Te crystallograpic orientations of all grains are given in Appendix 2. Table 3.4: 304L steel mean Taylor factor for planar compression and torsion solicitations for a 50-grain, 100- grain polycrystal, and for te total domain measured experimentally. Planar compression test Torsion test Velocity gradient tensor (L) Taylor factor Mean Velocity gradient tensor (L) Taylor factor Mean grains grains grains grains 100 grains 3.34 = grains 3.31 = Total domain grains = grains grains grains 3.36 Total 100 grains domain = Total Total 3.33 domain domain Total domain grains = 3.30 Total domain = 3.30

90 grains Total domain grains Total domain 100 grains grains 3.30 Based on Table 3.4, we conclude tat, in all cases, te Taylor factor value is iger for a planar compression solicitation tan for a torsion solicitation. Also, for bot cases, te 100- grain domain sows te same mean Taylor factor value as tat of te total measured domain. In te case of planar compression solicitation, we observe tat te mean Taylor factor value for a 50-grain domain is larger tan te one for a 100-grain domain. For te torsion solicitation, te inverse is observed. Knowing te K 1, K 2 and 0 intrinsic parameters of 304L, and te mean Taylor factor values of 304L for different mecanical solicitations and domain size, teoretical stress-strain curves can be predicted from numerical simulations. Tis reasoning is owever valid wen considering only SSDs. Wen considering bot SSDs and GNDs, an extra amount of dislocation is added to te material. As a consequence K 1, K 2 and 0 values must be re-adjusted to keep a good matc wit experimental results. 304L steel parameters Table 3.5 presents te material parameters used in simulations considering only SSDs and simulations considering bot SSDs and GNDs. Due to confidentiality, tese parameters are represented using capital letters. Tese parameters are valid for deformations at 1000 C and wit a strain rate of 0.01 s -1. For different conditions, new parameters need to be identified, using te same procedure as tat explained in te above paragraps. Table 3.5: 304L steel parameters used in te crystal plasticity model. SSDs SSDs + SSDs + SSDs GNDs GNDs K 1 (m -2 ) A A 0 (s -1 ) K 2 B 2.9B α D D (MPa) C C μ (GPa) E E 0 C 11 (GPa) b (m) C 12 (GPa) m F F C 44 (GPa) Te only different parameter between te SSDs and te SSDs+GNDs case is te K 2 value. As discussed in capter 1, wen te GNDs are taken into account, an extra amount of dislocations is introduced. As a consequence, in order to matc te macroscopic mecanical beaviour, dislocation recovery is cosen to increase, leading to a iger value of K 2. It is important to note tat oter combinations of parameter values for te SSDs+GNDs case would probably allow to correctly predict te experimental stress-strain curve. It is cosen ere to keep te same K 1 value, and to adapt te K 2 value, but additional tests wit variable initial grain sizes would potentially provide oter solutions. We also keep in mind tat te presence of GNDs introduces strong intragranular gradients of dislocation densities. Tose are not easily captured by suc a simple model, wic ignores, for example, dislocation transport [Arsenlis, 2004].

91 Effects of mes type in igly resolved polycrystalline simulations Te study of mes size effect on te material mecanical beaviour or on te texture predictions is a subject discussed by several autors. In fact, meses can be structured ( regular ) or unstructured ( free ), wit various degrees of mes refinement. In order to study te mes influence on te mecanical beaviour of 304L steel, wen considering only SSDs, a 50-grain polycrystal is generated in a 0.4 mm x 0.4 mm x 0.4 mm domain. Te crystallograpic orientations of all grains are based on experimental EBSD measurements, and given in Appendix 2. Four different meses are studied: 2 of tem are isotropic witout local mes adaptation, and te oter 2 meses are also isotropic but wit mes adaptation around te grain boundaries. Table 3.6 sums up all meses features and Figure 3.11 gives a grapical illustration. A planar compression test is simulated using material parameters from Table 3.5. In Table 3.6, te mes size is normalized by te mean grain radius. Table 3.6: Mes features - considering only SSDs. Normalized mes Normalized mes size on te grain size inside te grain boundary Number of mes elements Mes 1 Non adapted ,000 Mes 2 Non adapted ,000 Mes 3 Adapted ,000 Mes 4 Adapted ,000 Mes 1 Mes 2 Mes 3 Mes 4 Figure 3.11: Te four meses used to test te mes influence on crystal plasticity numerical results, considering only SSDs colour scale represents te distance function Considering only SSDs In [Sarma, 1996] te autors sow tat te use of a coarse discretization, in conjunction wit regular meses and wit only one element per grain (but wit more tan one integration point), is relatively adequate for texture evolution predictions. Oter autors ave sown tat increasing mes refinement as very little impact on te global stress-strain response of te polycrystal wen using conventional crystal plasticity [Barbe, 2001a], [Barbe, 2001b], [Buceit, 2005], [Diard, 2005], [Resk, 2009]. However, all agree tat finely discretized meses are needed to capture local details of microstructure evolution and local gradients, regardless of te mes type. In tis case, te mes influence on te stress-strain curve is studied. Te macroscopic stress is computed based on te equivalent Von-Mises stress (at eac mes element) as follows:

92 macro eqd and eq tr( S²), (3.8) V 2 were V is te total volume of te representative volume element (RVE). Similarly, te macroscopic strain is computed based on te equivalent strain as: 1 macro eq d V and eq t 2 eqdt : dt. (3.9) t 3 Te obtained stress-strain curves for meses 1, 2, 3 and 4 are ten compared to te teoretical curve calculated using parameters from Table 3.5 and te Taylor factor value for a 50-grain domain under a planar compression solicitation (M = 3.36, Table 3.4). Figure 3.12 presents te obtained results. (a) (b) Figure3.12: Stress - strain curves comparison between te teoretical result and te numerical result obtained wit four different finite element meses: (a) complete curve; (b) zoom for strains from 0.1 to Comparing mes 1 (red line) and mes 2 (green line) curves (bot meses are isotropic and non-adapted), it is concluded tat accuracy is improved wen decreasing te mes size. Comparing mes 3 (orange line) and mes 4 (ligt blue line) - bot meses are isotropic and adapted wit te same mes size near te grain boundaries leads to te influence of te intragranular mes size, wic is small. In oter words, te mes size near te grain boundaries matters more tan te intragranular one, even toug we only deal ere wit SSDs. Tis is furter confirmed by comparing mes 1 (red) to mes 3 (orange): te sole refinement of mes size near te grain boundaries is enoug to significantly improve te results Considering bot SSDs and GNDs Once again, te stress-strain curves are plotted using Equations 3.8 and 3.9 in order to study te mes influence. Figure 3.13.a presents te stress-strain curves for all four meses compared to te teoretical curve (wic considers only SSDs).

93 91 (a) (b) Figure 3.13: Stress - strain curves comparison between te numerical results obtained wit four different finite element meses and te teoretical curve calculated wen considering only SSDs: (a) complete curve; (b) zoom for strains from 0 to From Figure 3.13.b representing te stress-strain curves for low strain levels, it is seen tat all meses present a similar beaviour as te one obtained previously wit only SSDs. However, for strains larger tan 0.1, we observe tat simulations using mes 3 (orange line) and mes 4 (ligt blue line) predict a more important ardening. It is interesting to igligt tat tis mes influence takes place only after te GND density reaces a critical value. For small deformations, dislocation density is mostly controlled by SSD density, and te extra ardening beaviour is not observed. As GNDs build-ups are associated wit slip gradients developing at te grain boundaries, in tis case, te mes sensitive response is linked to a significant increase in te number of GNDs relative to te SSD population. Tus mes sensitivity increases as te element size around te boundaries decreases. Figure 3.14 illustrates te increased GND density in mes 3 and 4 along grain boundaries, as compared to mes 1 and mes 2. Te dislocation density distribution will be discussed in more details in paragrap Mes 1 Mes 2

94 92 Mes 3 Mes 4 Figure 3.14: GND density distribution for four different meses for ε = 0.3. In [Abrivard, 2009] te autors also sow te effect of te mes refinement on te stress-strain curves of a 2D 20-grain Al polycrystal. Teir model presents mes sensitivity as soon as plastic deformation occurs. Also, tis effect as been underlined by Ceong in [Ceong, 2005] for Cu polycrystal from grains smaller tan 30 µm during tensile loadings. In order to avoid tis mes sensitivity, [Abrivard, 2009] proposes tat te strain gradient sould be calculated not for all Gauss points presented in te mes, but only for Gauss points wit a critical distance from eac oter. Te critical distance would be linked to te GNDs spread at te grain boundary [Liang, 2009]. In te present study, since te stress difference between te different meses is small (around 10 MPa), te metod was not implemented. It may be done in future work owever. 2.4 Results Te planar compression test presented earlier is used to simulate 304L steel rolling. A 0.5 mm x 0.5 mm x 0.5 mm cubic 100-grain RVE is generated Stress - Strain analysis Te macroscopic stress-strain numerical curve is plotted considering te equivalent Von-Mises stress (at eac mes element) using Equation 3.8, and te equivalent strain using Equation 3.9. Figure 3.15 illustrates a comparison between te numerical results, te teoretical result and te experimental torsion-test result. Figure 3.15: Stress - strain curves: comparison between numerical, teoretical and experimental results.

95 93 Bot SSDs and SSDs+GNDs models predict well te experimental mecanical beaviour, even toug te simulation is in plane strain compression, wile te experimental curve comes from a torsion test. Tis is due to te fact tat mean Taylor factors are close to eac oter, as sown previously in Table 3.4. Tis can be visualized by comparing te teoretical curve, wic takes into account te correct mean Taylor factor, and te experimental curve: tey are very close to eac oter. Looking at te stress distribution trougout te RVE (Figure 3.16), it is seen tat in taking into account SSDs + GNDs, te stress concentration near te grain boundaries becomes more evident. Consequently, even toug te macroscopic stress-strain beaviour is similar for bot ardening models, te stress distribution in te microstructure is different. SSD SSD+GND Figure 3.16: 304L steel stress distribution for ε = 0.5, for strain rate equal to 0.01 s -1 and T = 1000 o C. A possible outcome of tese crystal plasticity simulations is te possibility to predict nucleation events in recrystallization regime. Tis motivates furter a detailed study of dislocation densities and strain distribution inerited from plastic deformation Dislocations density and energy distribution analysis Figure 3.17 sows te dislocation density distribution considering te two ardening models, for ε = 0.5. SSD SSD+GND Figure 3.17: Dislocation density distribution for ε = 0.5, for strain rate equal to 0.01 s -1 and T = 1000 o C.

96 94 Analysing Figure 3.17, we observe tat te model taking into account bot SSDs and GNDs presents a dislocation density concentration near te grain boundaries wile te model considering only te SSDs densities present a more omogeneous dislocation density distribution. Te average dislocation density over te wole RVE is computed as: 1 moy d, (3.10) V were ρ is te dislocation density in a given mes element. Figure 3.18 sows tat dislocation density is an increasing and non-linear function of te plastic deformation. For simulations considering only SSDs, saturation of ardening leads K' 1 14 to a maximum dislocation density given by m -2. K' 2 For te model considering SSD + GND, for strain levels between 0 and 0.4, te dislocation variation beaviour is similar to previous case. However, for strain levels iger tan 0.4, an extra increase of te dislocation variation is observed. At tis stage, te material ardening near grain boundaries becomes ig, and continued ardening in tese zones migt be overestimated. SSD SSD+GND Figure 3.18: Global dislocation density evolution wit macroscopic strain. Te dislocation energy is computed using te dislocation density from: 2 b E disloc, (3.11) 2 were μ is te sear modulus. Te global dislocation energy evolution terefore follows te same trend as tat of te global dislocation density given in Fig Instead of looking at te global dislocation density evolution (RVE scale), it is interesting to focus on te dislocation density evolution at te grain scale. To do so, ten randomly cosen grains are analysed. We compute teir dislocation density evolution using: 1 grain d. (3.12) V grain grain Figure 3.19 illustrates te calculated evolutions for bot ardening models.

97 95 SSD SSD+GND Figure3.19: Dislocation density evolution for ten randomly cosen grains. Te mecanical response of te selected grains is different from eac oter, wit dislocation densities spreading around te RVE (global) value. A first reason comes from te grain crystallograpic orientations: te distribution of slip rates along te different slip systems results in dissimilar dislocation density evolutions. A second reason comes from te location of te grains, weter tey interact wit neigbours, wit te fixed walls of te cannel die, or wit te zone of free boundaries. It is also noticed tat te grain beaviour may cange during deformation. For example, wen considering only SSDs, grain 80 initially ardens faster tan grain 28, but after a wile, tis trend is inversed. Tis is of course te consequence of crystal lattice rotation, keeping in mind tat tis rotation is non omogeneous. Anoter example can be found in comparing grains 27 and 47. Comparing te two ardening models, similar trends are observed for small strain levels, since ardening is ten mostly controlled by SSDs multiplication. For iger strain levels, te introduction of GNDs does make a difference, as discussed already wen analysing te global RVE beaviour. For example, at a strain level of 0.5, te igest dislocation density is not reaced for te same grains wit te two ardening models (see grain 28). Tis can be connected to te saturation beaviour, wic automatically leads to dislocation

98 96 ' ' densities of K 1 K2 for all grains in te SSDs model, wereas no saturation is reaced wen introducing GNDs. At te intra-granular scale, te dislocation density distribution can be analysed as a function of te distance to te grain boundary. All mes elements are ten taken into account and te mean dislocation density (taken over te entire polycrystal) is computed only as a function of te distance to te closest grain boundary. Figure 3.20 sows te results for different deformation levels. Te distance to te closest grain boundary is normalized by te polycrystal mean grain size. SSD SSD+GND Figure 3.20: Dislocation density distribution as a function of te normalized distance to te closest grain boundary for ε=0.1, ε=0.2, ε=0.3, ε=0.4, ε=0.5 and ε=0.6. In te SSD ardening model, te dislocation density is, in average, sligtly iger near te grain boundaries. However te difference between te grain boundaries zone and te grain bulk is small. SSDs ave a statistical nature, and only te local increase of te amount of strain near te boundaries can explain tis result. In average, te dislocation distribution trougout te grain is quasi uniform. On te oter and, wen considering te SSD + GND ardening model a clear increase of te dislocation density is noticed near te grain boundaries, as expected. Since te nucleation of new grains is often considered to predominantly take place in zones of igest dislocation densities, it is concluded ere tat te coice of ardening model will ave a strong impact on te predicted location of recrystallization nuclei. One way of normalizing te above results is to write: ( )/ R, ( x)/ R, ( ) x, (3.13) were ( )/ R, RVE mean dislocation density, ( )/ R, is te mean grain size. Figure 3.21 illustrates te ( )/ R, x is te mean dislocation density at position (x), ( ) is te x is te proportionality coefficient and R x values evolution obtained for bot ardening models.

99 97 SSD SSD+GND Figure 3.21: Beta value as a function of te normalized distance to te closest grain boundary for ε=0.1, ε=0.2, ε=0.3, ε=0.4, ε=0.5 and ε=0.6. In te SSD model, te dislocation density is sown to be more and more uniform wen te strain level increases. Te opposite olds for te SSD + GND model. Instead of looking at mean values, one can also detail te spread of dislocation densities as a function to te normalized distance to te closest grain boundary (Figure 3.22), and for a given strain level. SSD a SSD + GND a SSD b SSD + GND b Figure3.22: Dislocation density spread for (a) ε=0.3 and (b) ε=0.5. Even wit te SSD ardening model, te spread of dislocation densities is igest near te grain boundaries. As a consequence, dislocation energy gradients are also expected to be te igest near te grain boundaries, and tis is confirmed by Figure Tis, again, promotes a preferential positioning of recrystallization nuclei near te grain boundaries. However, in te SSD model, Figure 3.24 sows tat not all te grain boundaries would be potential nucleation sites.

100 98 SSD SSD+GND Figure 3.23: Norm of dislocation energy gradient as a function of te normalized distance to te closest grain boundary for ε=0.1, ε=0.2, ε=0.3, ε=0.4, ε=0.5dand ε=0.6. SSD SSD + GND Figure 3.24: Norm of dislocation energy gradient distribution for ε = 0.5, for strain rate equals to 0.01 s -1 and T = 1000 o C. Coming back to Figure 3.22, wit te SSD+GND ardening model, it is observed tat not only te mean dislocation density is iger near te grain boundaries, but also te deviation from te mean. Tis dislocation density spread increases wit increasing strain levels. Tis leads, as in te SSD case, to an increase of dislocation energy gradient near te grain boundaries, but wit iger amplitude (Figure 3.23). Te norm of dislocation energy gradient globally increases near te grain boundaries as illustrated in Figure Te sligt decrease observed in Figure 3.23.SSD+GND very close to te grain boundaries is due to a mes size effect. Indeed, tis is a purely interpolation effect and it sould not be considered in te pysical analysis. Also, te increase is more systematic (i.e. it olds for all grain boundaries), as can be seen in Figure Figure 3.25 describes te dislocation density distribution for five different normalized distances to te grain boundary (d = 0.037, d = 0.113, d = 0.414, d = and d = 1.128) and for two different deformation levels (ε = 0.3, ε = 0.5).

101 99 SSD a SSD + GND a SSD b SSD + GND b Figure 3.25: Dislocation density distribution for five different normalized distances to te grain boundaries: d = 0.037, d = 0.113, d = 0.414, d = and d = Results for two strain levels (a) ε=0.3 and (b) ε=0.5 are presented. Te distributions depend on te distance to te grain boundary. In te SSD model, te distribution is wider wen approacing te grain boundary. Reversely, it becomes narrower wit increased deformation levels. Te igest dislocation densities always remain close to te grain boundary, altoug te mean value is almost uniform (as noticed earlier). In te SSD+GND model, one can summarize tat (i) bot te distribution widt and te mean dislocation density are increased wen approacing te grain boundaries, and (ii) wit increasing strain levels, distributions become wider and spatially more eterogeneous. Based on te above results, one can conclude tat nucleation processes during recrystallization will be predicted to take place mostly near te grain boundaries, in bot ardening models, since in tis area we find te igest levels of dislocation density and dislocation gradient values. A striking difference between te SSD and te SSD+GND ardening models, is tat te former predicts omogenization of ardening wit increased levels of strain, wereas te latter predicts more and more eterogeneity Strain distribution analysis From Figure 3.26, te introduction of GNDs does not ave a significant influence on strain distribution. A few areas of iger strain levels can be identified, but te difference wit te SSD model is minor. At te intra granular scale, Figure 3.27 describes te mean strain as a function of te distance to te closest grain boundary, following te idea used already for dislocation densities. Here again, distances are normalized by te mean grain size of te polycrystal, and results are given for different macroscopic deformation levels.

102 100 SSD SSD+GND Figure 3.26: Strain distribution for ε = 0.5, for strain rate equals to 0.01 and T = 1000 o C. SSD SSD+GND Figure 3.27: Strain distribution in function of te normalized distance to te grain boundary for ε=0.1, ε=0.2, ε=0.3, ε=0.4, ε=0.5 and ε=0.6. Te strain distributions sow sligtly iger values near te grain boundaries, for bot ardening models. In Figure 3.28, te spread of tese values is, like for dislocation densities, increased near te grain boundaries. SSD a SSD + GND a

103 101 SSD b SSD + GND b Figure 3.28: Strain dispersion for (a) ε=0.3 and (b) ε=0.5. Te deviation of strain values wit respect to te mean increases wit an increase of te macroscopic strain, for bot ardening models. Te spread in te SSD+GND model is a little iger Grain size effects Te grain size effect on te 304L steel mecanical beaviour as been investigated at T = 1000 o C. Te simulated strain rate is 0.01 s -1. Te simulations predictions for tree grain sizes (0.2< R >,< R > and 3.2 < R >, wit < R > being te experimental mean grain size) are presented in Figure 3.29.a. It is observed tat an initial grain size reduction results in te polycrystal strengtening. For eac grain size, te build-up of strain gradients leads to te accumulation of GNDs during deformation, affecting significantly te ardening beaviour of polycrystalline aggregates. (a) Figure 3.29: (a) Stress-strain numerical curves for tree different initial grain sizes: 0.2< R >, < R > and 3.2 < R > wit < R > being te mean grain size measured on 304L samples. (b) Flow stress dependency on grain size at various strain levels (black dots) wit te fitted Hall-Petc relation (blue lines). It was found tat te grain size dependence of te flow stress, σ, follows a modified Hall-Petc equation of te type: (b)

104 102 0 ( ) k( ) 1 2 d, (3.14) were 0 ( ) and k(ε) are parameters varying wit te strain level. In Figure 3.29.b, te linear interpolation of te numerical results for different grain sizes is done in a Hall-Petc grap. Te stress variation wit te grain size follows well relation (3.14), for all investigated macroscopic deformation levels. In [Mirzade, 2013], te effect of 304L steel grain size influence on te stress-strain curve is studied, and a similar beaviour is observed. Also, it is important to igligt tat, for smaller grain sizes, te dynamic recrystallization onset is at smaller strain levels. As a consequence, for a polycrystal wit 0.2<R>, results for strain levels larger tan 0.3 may not ave a pysical meaning. Predicted stresses probably overestimate te experimental results, as was already noticed by [Abrivard, 2009] in te small grain sizes range. Tis is probably a consequence of te GND evolution model, wic does not account well for dislocation transport and/or te anniilation of dislocations. Dislocation anniilation is computed only by te SSD evolution equation. Te SSD+GND model owever predicts pysical results in terms of grain size effects. In order to improve tis study, complementary experimental tests, wit samples presenting different initial mean grain sizes, sould be done. Also, an improvement in te GND evolution equation formalism is needed. 3 Tantalum Case Tis second study relates to uniaxial compression tests of tantalum olygocrystals. Olygocrystals are macroscopic samples containing only a few grains, i.e. te grains ave mm to cm sizes. Te experimental compression tests were performed by Cristope Kerisit during is P.D. [Kerisit, 2012]. Tantalum was cosen ere for two main reasons: (i) te tecnical possibility of producing olygocrystals, wit columnar structures and (ii) its ig ductility at room temperature. Wit suc samples, it is possible to perform a numerical study on: - te evolution of te macroscopic sape of te samples, wic is a consequence of te plastic anisotropy of single crystals; - te local distributions of dislocation densities and strain (as was done for 304L steel); - global and local texture evolutions. Some of tese quantities can be measured experimentally, and can serve as validations of te crystal plasticity models. In future work, eat treatments of te deformed samples can ten be used to identify te spatial positioning of recrystallization nuclei, and attempt te development of appropriate nucleation criteria. 3.1 Tantalum Tantalum is a rare material presenting a B.C.C. (body-centred cubic) crystalline structure, wit a lattice parameter equal to 3.3 x m. Tis material as a ig fusion temperature (2996 C) and also a ig density (16650 kg.m -3 ). Te 3 active slip systems

105 103 families accommodating plastic deformation are {110} <111>, {112} <111> and {123} <111>, leading, in total, to 48 different slip systems. Pure tantalum is known to be very ductile at room temperature, but wen temperature increases, ydrogen (for T > 200 C), oxygen (for T > 300 C), nitrogen (for T > 1100 C) and carbon (for T > 1200 C) diffuse into te material. Te introduction of tese elements results in a strengt increase, and a ductility decrease. As a consequence, all tantalum eat treatments sould be performed under good vacuum conditions. Te mecanical beaviour of tantalum is identified and studied in details in [Kerisit, 2012]. Te ardening model used to describe tantalum ardening is te equivalent to te Laasraoui-Jonas implemented in te crystal plasticity code. 3.2 Compression test Test description Figure 3.30 illustrates te principle of uniaxial compression tests performed on te Ta samples. (a) (b) Figure 3.30: Simple compression test sketc: (a) test description, (b) imposed boundary conditions. Te mean digital microstructure deformation (over te volume Ω) is controlled by imposing te velocity on te z = z a and z = z b planes (Fig b). In tis case, = 0.01 s Digital oligocrystal Te studied oligocrystal is a 28.2 mm x 5.9 mm x 5 mm parallelepiped composed of 6 grains wic are assumed to be quasi columnar. Te oligocrystal is digitally generated and immerged into a finite element mes. In tis case, neiter te Voronoï nor Laguerre-Voronoï metods are used. Eac grain is generated individually and ten, te six different grain meses are immerged in a unique mes. Eac grain is represented using a different level set function (capter 2). Figure 3.31 illustrates te real and te digital olygocrystals. (a) (b)

106 104 (c) Figure 3.31: (a) Real tantalum oligocrystal, (b) digital oligocrystal and (c) te same digital olygocrystal immerged into a finite element mes Material parameters Tantalum material parameters used in te simulations are presented in Table 3.8. Table 3.8: Tantalum parameters at room temperature [Frénois, 2001]. C 11 (GPa) C 12 (GPa) C 44 (GPa) 0 (s -1 ) μ (GPa) b (m) m , , is a reference slip rate and m te sensitivity exponent. K 1, K 2, 0 and α parameters ave been identified as well, but, for confidentiality reasons, te numerical values are not given. Te same material parameters are used for bot SSD and SSD + GND ardening models, because te GND influence on te macroscopic mecanical beaviour is negligible wen dealing wit very large grain sizes, as it is observed in figure Figure 3.32: Stress-strain numerical curves compared to te teoretical curve. Te Euler crystallograpic orientation angles and Taylor factors are given for eac of te 6 grains in Table 3.9. Table 3.9: Crystallograpic orientation (Euler angles) and Taylor factor of all six oligocrystal grains. Φ 1 Ф φ 2 Taylor Factor for a simple compression test Grain 0 76,6 o 43,3 o 304,2 o Grain 1 8,0 o 4,2 o 343,8 o Grain 2 276,0 o 36,0 o 92,0 o Grain 3 227,5 o 51,8 o 138,9 o Grain 4 18,1 o 40,3 o 354,4 o Grain 5 300,5 o 35,0 o 49,2 o 3.058

107 105 Altoug te BCC structure exibits 48 different slip systems, it was cosen ere to restrict te description to 24 slip systems ({110} <111>, {112} <111>). 3.3 Results Sample sape Te first ting to be examined is te oligocrystal sape after deformation. Figure 3.33 compares te experimental and te two numerical oligocrystals obtained after a deformation of ZZ Te two numerical simulations correspond to te two usual ardening models (SSDs, SSDs + GNDs). Experimental SSDs SSDs + GNDs Figure 3.33: Comparison between experimental and numerical (bot SSDs and GNDs+SSDs simulations) oligocrystal sapes after a deformation of 0.5 (ε ZZ = 0.5). A first observation is tat te ardening model does not influence te predicted final sape. Comparing wit experiments, it is noticed tat grains 2, 3, 4 and 5 ave a final sape wic is well predicted. Grain 3 in particular: te material sear a consequence of te anisotropy goes in te good direction, wit appropriate curvatures of te external boundaries. Similar comments apply to grains 2 and 4. However, numerical results of grains 0 and 1 do not agree well wit te experimental results. Experimentally, tese two grains bend downwards, wile te prediction goes upwards. Te searing of grain 0 seems to follow a good trend, but tere is clearly an experimental cange wic cannot be captured by te model. One possible explanation would be te existence of anoter grain underneat te surface, before deformation, wic ten came

108 106 onto te surface during deformation (wic would mean tat te initial microstructure was not, as assumed, columnar in tis region). Despite some differences between te experimental and numerical results one can conclude tat te numerical model as a good ability in predicting final sapes induced by plastic deformation Dislocations density distribution Figure 3.34 gives te spatial distributions of dislocation densities for bot ardening models (only SSD and SSD+GND), in te final deformed configuration. SSDs SSDs + GNDs Figure 3.34: Dislocation density distribution for ε ZZ = 0.5, for two numerical olygocrystals: considering only SSD and considering bot SSD + GND. As usual, te introduction of GNDs increases te dislocation density near te grain boundaries. Once again, te average dislocation density over te wole sample is computed using Equation 3.10, and te evolution wit te macroscopic strain is presented in Figure Figure 3.35: Sample average dislocation density evolution for ardening models considering only SSDs and bot SSD + GND.

109 107 Differences between te two ardening models start appearing around a deformation of 0.2. However, even after a deformation of 0.5, te predicted average dislocation densities remain very close (and ence lead to a similar macroscopic mecanical beaviour), due to te very large grain sizes in te sample. A grain-by-grain analysis is done in Figure (a) SSDs (a) SSDs+GNDs Figure 3.36: (a) Dislocation density evolution as a function of strain for six tantalum grains. For bot ardening models, grain 3 exibits te igest dislocation density during te total material deformation. Differences may appear for oter grains. In capter 5, tese type of numerical results will be used to simulate recrystallization processes. In tis context, te above differences can result in a difference in te nucleation site positions and, as a consequence, in differences on te predicted recrystallization kinetics. Te intra granular scale analysis can be performed as well, looking at te mean dislocation density as a function of te distance to te closest grain boundary. Tis is done in Figure 3.37 for different deformation levels, and wit distances expressed in cm. SSDs SSDs+GNDs Figure 3.37: Dislocation density distribution as a function of te distance to te closest grain boundary for ε=0.1, ε=0.2, ε=0.3, ε=0.4, ε=0.5, ε=0.6 and ε=0.7. For distances between 0.2 cm and 0.4 cm, te dislocation density is iger tan for distance around 0.1 cm. In te SSDs model, te dislocation density in tis area is actually even iger tan near te grain boundary. Tis is a consequence of a geometrical effect in te oligocrystal, wic can be understood from Figure A ig density dislocation area is observed far from te grain boundaries, represented by yellow and green zones, due to strong local curvatures. Note tat in Figure 3.38, te oligocrystal is viewed from beind, wen compared to te views given previously (e.g. Fig. 3.34).

110 108 SSDs SSDs + GNDs Figure3.38: Dislocation density distribution for ε ZZ = 0.5 (view from beind). Tis beaviour is terefore particular to te large grains oligocrystals: it is not expected to appen in standard polycrystals, at least not in uniaxial compression. From Figure 3.37, SSD results would indicate a preferential nucleation of recrystallization between 0.2 and 0.4 cm from te closest grain boundary. On te contrary, wen introducing GNDs, grain boundaries remain te preferential sites. In order to understand in more details te dislocation density distribution for eac grain, te spread, or dispersion of values is displayed in Figure All grains SSDs All grains - GNDs + SSDs Grain 0 SSDs Grain 0 - GNDs + SSDs

111 109 Grain 1 SSDs Grain 1 - GNDs + SSDs Grain 2 SSDs Grain 2 - GNDs + SSDs Grain 3 SSDs Grain 3 - GNDs + SSDs Grain 4 SSDs Grain 4 - GNDs + SSDs

112 Dislocation density (m -2 ) Dislocation density (m -2 ) 110 Grain 5 SSDs Grain 5 - GNDs + SSDs Figure 3.39: Dislocation density distribution at ε ZZ = 0.5 for te wole oligocrystal and for all six individual grains. Comparing te dislocation density dispersion obtained wit bot models, we observe tat, for all grains, te dispersion near te grain boundary is more important wen te GND density is taken into account. Also, it is interesting to observe tat eac grain beaves differently. For example, grain 0, 1, 4 and 5 present a narrower distribution. On te oter and, grain 2 is very eterogeneous. Finally, grain 3, wit its five neigbouring grains, as an intermediate beaviour. In some regions, te dislocation density is relatively omogeneous (similar to grains 0, 1, 4 and 5), and in oters it is muc more eterogeneous. Te dislocation density profiles at interfaces between neigbouring grains (Figure 3.40) vary also, indicating more or less compatible deformations between te neigbours. SSDs 8.E+14 SSDs+GNDs 1.E+15 9.E+14 7.E+14 8.E+14 6.E+14 7.E+14 6.E+14 5.E Distance to grain 3 (cm) Grain 0-3 Grain 1-3 Grain 2-3 Grain 4-3 Grain E Distance to grain 3 (cm) Grain 0-3 Grain 1-3 Grain 2-3 Grain 4-3 Grain 5-3 SSDs SSDs+GNDs Figure 3.40: Dislocation density distribution for ε ZZ = 0.5 for te neigbouring zones of grain 3. Coming back to criteria for te nucleation of new grains, not only a critical dislocation density is necessary, but tere is also a role played by dislocation density gradients (grain boundary motion and driving forces are discussed in details in capters 4 and 5). Dislocation energy is calculated using Equation Figure 3.41 presents te norm of dislocation energy gradient distribution for ε ZZ = 0.5, and Figure 3.42 presents te intra granular dislocation energy gradient distribution.

113 111 SSDs SSDs + GNDs Figure 3.41: Norm of dislocation energy gradient distribution for ε ZZ = 0.5. SSDs SSDs+GNDs Figure 3.42: Norm of dislocation energy gradient as a function of te distance to te closest grain boundary for ε=0.1, ε=0.2, ε=0.3, ε=0.4, ε=0.5, ε=0.6 and ε=0.7. Dislocation energy gradients, especially near te grain boundaries, are sligtly increased wit te SSD+GND ardening model, owever profiles look similar. Two areas of ig gradients are near te grain boundaries, and around 0.4 cm from tem. Te second one is related to te geometrical effects already discussed for Fig Deformation distribution As already noticed in te 304L plane strain test case, te ardening model does not influence muc te spatial distribution of strains in te sample (Fig. 3.43).

114 112 SSDs SSDs + GNDs Figure 3.43: strain distribution in te oligocrystal for ε ZZ = 0.5. SSDs SSDs+GNDs Figure 3.44: (a) Strain evolution as a function of te macroscopic strain for te six tantalum grains. Te grain-by-grain evolution of strain given in Figure 3.44 exibits tat grain 0 is te one tat eventually deforms te most. Te initial Taylor factor is not te igest (M = Table 3.9), but crystal reorientation wit plastic deformation is responsible for te progressive increase of strain. Te connection between Taylor factor and amount of strain is quite systematic, wit owever some neigbourood effects wic can influence te strain distribution (e.g. grain 5 wic is te 2 nd most deformed, but wit a Taylor factor wic is not te 2 nd ). Analysing te intragranular strain distribution (Figure 3.45), it is observed tat bot ardening models present almost te same curves.

115 113 SSDs SSDs+GNDs Figure 3.45: Strain evolution in function of te distance to te grain boundary for ε=0.1, ε=0.2, ε=0.3, ε=0.4, ε=0.5, ε=0.6 and ε=0.7. Te strain dispersion witin individual grains of te oligocrystal (Figure 3.46) sows once again contrasted beaviours, wit a distinction between grains 0, 1, 4 and 5 (omogeneous), and grains 2 and 3 (more eterogeneous). Wit te SSD+GND model, strain dispersion near te grain boundaries is increased. All grains SSDs All grains - GNDs + SSDs Grain 0 SSDs Grain 0 - GNDs + SSDs

116 114 Grain 1 SSDs Grain 1 - GNDs + SSDs Grain 2 SSDs Grain 2 - GNDs + SSDs Grain 3 SSDs Grain 3 - GNDs + SSDs Grain 4 SSDs Grain 4 - GNDs + SSDs

117 115 Grain 5 SSDs Grain 5 - GNDs + SSDs Figure 3.46: Strain dispersion for ε ZZ = 0.5 for te wole oligocrystal and for all six individual grain Texture analysis In tis section, predicted crystallograpic textures are analysed and compared to experimental results in Figure Te numerical textures are gatered from all finite elements on te surface of te mes. Experimentally, owever, textures are measured from local scans witin te different grains. Tis explains a general trend in aving more dispersed crystallograpic orientations from te model. Experimental SSDs SSDs+GNDs Grain 0 Experimental SSDs SSDs+GNDs Grain 1 Experimental SSDs SSDs+GNDs Grain 2

118 116 Experimental SSDs SSDs+GNDs Grain 3 Experimental SSDs SSDs+GNDs Grain 4 Experimental SSDs SSDs+GNDs Grain 5 Figure 3.47: Comparison between experimental and numerical pole figures. Te black dots correspond to te initial crystallograpic orientation and te coloured dots represent te crystallograpic orientation for ε ZZ =0.5 for direction (001). Numerical predictions do not exactly matc te experimental results, at least for some grains. In [Resk, 2010], te texture prediction of an Al (FCC) polycrystal was successfully predicted, using te same model. In te Ta case, te bcc structure differs. Also, it was noticed earlier tat some geometrical evolutions of te oligocrystal were not well predicted. Additional measurements, covering te wole surface of te oligocrystal, sould elp in understanding better te origins of te discrepancies. For example, grain 2 is very well predicted by te model, including te crystallograpic fragmentation of te grain, and it is also te smallest grain in size, wic means tat experimental and numerical surfaces are probably te closest. One way of analysing intragranular texture evolutions is to look at te rotation angle from te initial state (Figure 3.48). Experimentally, te fragmentation of grain 2 is significant, and tis is reflected by te wider dispersion of rotation angles, as compared to te oter grains. Wen te distributions of rotation angles are narrow, it indicates a global crystallograpic rotation, wit no significant fragmentation.

119 117 All grains SSDs All grains - GNDs + SSDs Grain 0 SSDs Grain 0 - GNDs + SSDs Grain 1 SSDs Grain 1 - GNDs + SSDs Grain 2 SSDs Grain 2 - GNDs + SSDs

120 118 Grain 3 SSDs Grain 3 - GNDs + SSDs Grain 4 SSDs Grain 4 - GNDs + SSDs Grain 5 SSDs Grain 5 - GNDs + SSDs Figure 3.48: rotation angle from initial state at ε ZZ = 0.5 for te wole olygocrystal and for all six individual grains. It is interesting to igligt tat te increased dispersion of grain 2 is not only observed for te rotation angle from te initial state, but also for te dislocation density and te local strain. Tis makes im a good candidate for early initiation of recrystallization. 4 Conclusion In tis capter, numerical microstructures induced by plastic deformation are investigated. Te Crystal Plasticity Finite Element (CPFEM) model used is tis capter is explained in details in capter 1. Two different ardening models are studied: te first one computes te total dislocation density, and te second one distinguises two different dislocation types statistically stored dislocations (SSD) and geometrically necessary dislocations (GND). Test cases are performed based on (i) a plane strain compression of a 304L steel at 1000 o C and 0. 01s -1, and (ii) a uniaxial compression of a 6-grains oligocrystal of Ta, at room temperature, wit 0. 01s -1.

121 119 Considering te first test case, bot ardening models can correctly predict te stressstrain experimental beaviour. Te dislocation density distribution calculated wit te SSD ardening model is more omogeneous tan tat obtained wit te SSD+GND model. In te latter case, te dislocation density is larger near te grain boundaries. Even toug te dislocation distribution is different, for bot models, te dislocation density dispersion is larger near te grain boundaries. It is in tis area tat te igest dislocation density values are found. As a consequence, te norm of te dislocation energy gradient is also larger near te grain boundaries. Wen te local strain distribution is analysed, bot ardening models present similar results. Te strain intragranular distribution is rater omogeneous, wit owever, once again, an increased dispersion near te grain boundaries. Based on tese results, nucleation of new grains during a recrystallization process sould take place near te grain boundaries, as most often observed in experiments. For te second test case (tantalum oligocrystal), te prediction of final macroscopic sape is well predicted for four grains, out of six. Once again, te dislocation density distribution sows a concentration near te grain boundaries, especially wen using te SSD+GND ardening model. Oter maxima of dislocation density (and its gradient) are found away from te grain boundaries, as a consequence of macroscopic geometrical effects. Te predicted material texture of te oligocrystal at ZZ 0. 5 tends to overestimate te spread of crystallograpic orientations, and sometimes does not predict well te mean orientation. However, it is noticed tat wen te numerical and experimental areas are very close to eac oter (as te case for grain 2), bot te mean orientation and te grain fragmentation are well predicted. Based on tree different criteria on te dislocation densities and teir gradients, te distribution of local strain, and te grain fragmentation, one grain of te polycrystal grain 2 is a good candidate for early recrystallization, if te olygocrystal is to be subjected to a eat treatment. Te numerical results obtained is tis capter will be used in capter 5 in order to predict te location of nucleation sites, and te kinetics of static recrystallization. Résumé en Français Dans ce capitre, la déformation plastique de microstructures digitales, générées grâce aux outils numériques présentés dans le capitre précédent à partir de données expérimentales, est étudiée en utilisant le formalisme de plasticité cristalline introduit dans le premier capitre. Différents cas tests sont considérés (description différenciée ou non des SSDs et GNDs) pour différents essais : le premier correspond à un essai de compression plane o 1 sur un polycristal d acier 304L T 1000 C, 0.01s tandis que le deuxième correspond à un essai de compression simple à froid d un oligocristal de tantale composé de six grains avec une vitesse de déformation égale à 0.01 s -1. Concernant l acier 304L, il est mis en évidence, entre autres, que les deux lois d écrouissage permettent de prédire correctement son comportement et que les résultats sont similaires en terme de distribution de déformation. La distribution de densité de dislocations obtenue est cependant plus étérogène lorsque les deux types de dislocation sont considérés avec principalement une plus forte dispersion au niveau des joints de grains. Concernant l oligocristal de tantale, une confrontation directe entre les formes des grains obtenues après essai et les formes prédites numériquement est réalisée. La distribution de densité de dislocations est également plus élevée dans les zones plus proces des joints de grains, ce pénomène étant, comme pour le 304L, exacerbé lorsque la distinction entre SSDs et GNDs est faite. L aspect prédiction de texture est également abordé.

122 120 Capter 4 Grain Boundary Migration - Grain Growt Modelling Contents 1 Introduction Context and Equations Full Field modelling of grain growt assuming isotropic grain boundary energy and mobility Te equations Mobility Te grain boundary energy Finite element model in a level set framework Velocity field expression Numerical treatment for multi-junctions Container level set functions Grain growt algoritm Results Academic Test - Circular grain case Academic test - 3 grains case Von Neumann-Mullins case D polycrystal grain growt Influence of te microstructure generation metod on grain growt prediction Assessment of simplified 2D grain growt models from numerical experiments based on a level set framework Grain size distributions Burke and Turnbull study Hillert/Abbruzzese model study Conclusion

123 121 1 Introduction Grain growt penomena in polycrystalline metals occur during and after full recrystallization, and ave te effect of increasing te average grain size at te expense of smaller ones tat will tend to disappear. Even if tis penomenon of capillarity is always present, it is generally neglected during primary recrystallization comparatively to te predominant driving force induced by te inomogeneous spatial distribution of dislocations stored energies. Grain growt is owever known to be of primary importance wen dealing wit long annealing treatments, were capillarity effects become predominant. It ten largely dictates te final grain size of te material. In capter 5 it will be sown tat, depending on te spatial position of new grains (nuclei), capillarity effects may ave a significant impact during recrystallization on te evolving microstructures and on te recrystallization kinetics. Understanding and modelling capillarity forces and teir interactions wit oter driving forces appears terefore to be an important callenge, besides te better understanding of pure grain growt itself. In tis capter, after a brief overview of te state of art concerning full field modelling of grain growt, a new full field grain growt model is proposed, in te context of isotropic grain boundary energy and mobility. Te model is based on a finite element formulation combined wit a level set framework. It is initially tested wit some academic cases presenting known results. In a second part, polycrystalline grain growt is simulated and te assessment of two 2D mean field grain growt models is done based on te full field simulation results. Te two mean field grain growt models discussed in tis capter are: te Burke and Turnbull model and te Abbruzzese/Hillert model. Te mean field models, wic were developed based on teoretical assumptions, are not easily verified experimentally. Since te full field metod presented in tis capter allows te control of te initial microstructure and of te material properties, te verification of te mean field models under different conditions can be done in details. Also, based on te results obtained using a full field model, mean field models can be corrected or improved. All te simulations presented in tis capter are 2D simulations and for all systems solved (convective and/or diffusive equations), a stabilized P1 solver as SUPG or RFB metod was used. In te next capter, te proposed full field metod will be extended to a primary recrystallization model, and applied to 2D and 3D static recrystallization cases. As in capter 2, all te distributions or mean values discussed in tis capter concerning grain features were evaluated as number-weigted. 2 Context and Equations 2.1 Full Field modelling of grain growt assuming isotropic grain boundary energy and mobility For te last tree decades, under simplified conditions of isotropic grain boundary energy and mobility, several grain growt modelling metods at te mesoscopic scale ave been developed [Miodownik, 2002]. Te improvement of te pysical understanding of grain growt penomena associated to an explosion of numerical resources, explain te improvement in te field of grain growt modelling. Te most famous and still widely used grain growt modelling tecnique is te probabilistic Monte-Carlo metod (MC). Tis metod is based on two main points: first, te use of a pixilated or voxelated description of te granular microstructure and, secondly, te construction of probabilistic evolution rules [Rollet, 2001]. In te case of grain growt

124 122 modelling, a Hamiltonian energy minimization, defined by summing te interfacial energies, is used as an evolution rule. Te interfaces are represented by te faces of te elementary cells, favouring te modelling of te topological evolutions. Tis is te first main advantage of tis metod. Moreover, it is noticeable tat te use of regular grids associated wit a simple evolution law, results in a negligible numerical cost for any simulation [Hassold, 1993]. However, te simple nature of tis approac is also synonymous of defects. Indeed, te absence of time scale complicates any comparison wit experimental data [Rollet, 1997]. Moreover, tis metod is only statistically representative, wic means tat a large number of simulations, aving a great number of cells could be necessary to ensure a good statistics of te penomenon. Tis is even more important wen te material presents a eterogeneous microstructure. Te cellular automaton (CA) approac is anoter probabilistic metod. Several pysical rules are used to locally determine te cell propagation in relation to te neigboring cells and all te cells can be updated at te same time solving te first MC negative aspect. However, as in te MC approac, te stocastic nature can lead to te same problem of representativity. In a finite element context, tree main metods are developed nowadays: te Vertex metod (VM), te Pase Field metod (PF) and te Level-Set metod (LS). Only te VM metod presents an explicit grain boundary description. Surface elements of te considered mes are used to describe te grain boundaries. Tis approac allows a simple representation of te initial microstructure in a finite element mes and te microstructure evolution is modeled by calculating te new position of te nodes belonging to te grain boundaries. A velocity field normal to te grain boundary is used to calculate te new nodes position [Nagai, 1990]. However, te main weakness of tis approac is tat topological events are controlled by mes adaptation rules. As a consequence, te management of topological events (like grains disappearance) is not straigtforward. Even toug a set of well calibrated canging mes rules exists in 2D, its extension in 3D is very complex and expensive [Weygand, 2001], [Sia, 2010], [Barrales, 2008]. In bot PF [Cen, 1995] and LS [Zao, 1996] metods, te grains boundaries are implicitly described. For te PF metod, tis representation is obtained from a continuous approximation of te Heaviside function at te interface (ence te notion of fuzzy or diffuse interfaces). For te LS metod, as explained in capter 2, te zero isovalue of a distance function represents te grain boundary. As a consequence, tese two approaces can automatically andle topological canges. For te PF metod, eac granular orientation is used as an order parameter field and te grain free energy is expressed as a Landau development of te structural order parameters. Te grain boundary energy is introduced as te gradient of structural order parameters and te interfaces are described by an isovalue (arbitrary witin te diffuse zone) of te order parameter fields. An important number of publications ave illustrated te potential of tis approac in modeling te grain growt penomenon in 2D [Cen, 2002] and, more recently, in 3D [Moelans, 2009]. Te main difficulties of tis approac remain: - te construction of te free energy density expression (using a Landau development) wic must reflect as closely as possible te material pysical properties and microstructure; - te abrupt pase functions transition witin te interfaces can be synonymous of intensive and expensive calculations, especially in 3D (degree of interpolation, mes refinement ); Finally, grain growt can also be modelled using a level set description of interfaces in te context of uniform grids [Elsey, 2009] or finite elements (FE) [Bernacki, 2011], [Fabiano, 2013]. As for te PF approac, in te LS metod, topological canges are treated automatically. From a purely matematical point of view, a grain disappears wen te corresponding level set function becomes negative over te entire domain. Te level set tecnique was used in

125 123 [Elsey, 2009] to model 2-D and 3-D isotropic grain growt in te context of uniform grids wit a finite-difference formulation. In tis work grain growt simulations are performed using level set tecniques combined wit a finite element formulation, not only to avoid all te disadvantages presented by te oter metods, but also, as it will be illustrated in te next capter, to be able to extend tese developments to a primary recrystallization model, and to couple tem to te CPFEM calculations developed witin te same formalism [Logé, 2008], [Bernacki, 2009]. 2.2 Te equations Te mecanism of boundary migration depends on several parameters including te boundary structure wic, in a given material, is a function of te misorientation between two neigbouring grains, and of te boundary plane. It also depends on te experimental conditions, in particular te eat treatment temperature. Finally, te grain boundary migration is strongly influenced by point defects suc as solutes and vacancies. It is usually assumed tat grain boundary displacement can be approximated by [Humpreys, 2004], [Bernacki, 2009], [Kugler, 2004]: v M( E ) n, (4.1) were M is te grain boundary mobility; E is te material internal energy gradient, wic depends on te dislocation density; is te grain boundary energy, is te grain boundary mean curvature and n is te outward unit normal to te grain boundary. Te grain boundary driving force can terefore be divided into two parts: te first one is related to te internal material energy field ( En ), and te second one is related to te grain boundary capillarity effects ( n ). In pure grain growt, a omogeneous dislocation density may be assumed, wit E =0, and terefore only te capillarity term is taken into account. It will be considered tat grain boundary mobility and energy are uniform trougout te microstructure, owever a brief recall of te meaning of tese pysical parameters is done below Mobility Low angle grain boundaries migrate troug climb and glide of dislocations. As a consequence, many aspects of low angle grain boundary migration may be interpreted in terms of te teory of dislocations. In general, low angle boundary mobilities are lower tan tose of ig angle boundaries. In te case of ig angle grain boundaries, te basic process for boundary migration is te atoms transfer to and from neigbouring grains. Te activation energy for boundary migration was found to be a function of misorientation up to, approximately, 14, above wic it remained constant. As te transfer of atoms togeter wit dislocations climb and glide are temperature dependent, we can also say tat te mobility of grain boundaries is temperature dependent. Actually, it is usually found tat te grain boundary mobility evolution as a function of temperature obeys an Arrenius type relationsip of te form [Humpreys, 2004], [Bernacki, 2009], [Kugler, 2004]: M Q b m0 ( T)exp, (4.2) RT

126 124 were Qb te boundary diffusion activation energy. Figure 4.1 illustrates te mobility dependence on temperature and misorientation in Al-0.05%Si [Humpreys, 2004]. (a) (b) Figure 4.1: (a) Te effect of misorientation and temperature on boundary mobility in Al-0.05%Si as a function of misorientation, measured from subgrain growt in crystals of Goss and Cube orientation [Huang, 2000]. (b) Te effect of misorientation on activation energies for low angle boundary migration in Al-0.05%Si measured from subgrain growt in single crystals [Humpreys, 2004]. In te context of tis PD work, te misorientation impact on te mobility is not considered and te value of te mobility at ig misorientation is used uniformly. In te case of 304L steel, El Waabi [El Waabi, 2003] reports Q b = 280 kj.mol -1 and m 0 ( T ) = Table 4.1 summarizes te 304L steel mobility values for temperatures ranging between 1000 C and 1200 C. Table 4.1: 304L Steel grain boundary mobility in function of temperature and in isotropic mobility context. Temperature ( C) Mobility (m 4 /J.s) All grain growt simulations in tis capter were performed using an Arrenius interpolation of te data coming from table Te grain boundary energy Low angle boundary energy Grain boundary energy also depends on te misorientation. Low angle grain boundaries can be represented by an array of dislocations. In tis case, eac dislocation accommodates te mismatc between te two lattices on eiter side of te boundary. For small values of misorientation between two neigbouring grains, te dislocation spacing is large and te grain boundary energy is approximately proportional to te density of dislocations in te boundary. Te free energy of a low angle grain boundary can be calculated from te Read-Sockley teory, resulting in te following equation [Sockley, 1949], [Humpreys, 2004], [Bernacki, 2009]:

127 125 m m 1 ln, (4.3) m were is te grain boundary misorientation, m and m are, respectively, te grain boundary energy and te misorientation for a ig-angle boundary. According to Equation 4.3, te grain boundary energy increases wit te misorientation, as sown in Figure 4.2 for various metals. Figure 4.2: Te measured (symbols) and calculated (solid line) energy of low angle tilt boundaries as a function of misorientation, for various metals [Sockley, 1949]. Hig angle boundary energy For low angle boundaries, te Read-Sockley model as been well establised. However, for misorientations iger tan 15, measurements of grain boundary energy reveal no furter cange wit increasing rotation angle. In tis case, te dislocation spacing is so small tat te dislocation cores overlap and it is ten impossible to pysically identify te individual dislocations. As a consequence te dislocation model fails and te grain boundary energy is almost independent of misorientation. However, not all ig angle boundaries ave an open disordered structure as presented in te general introduction. Tere are some special ig angle grain boundaries wic ave significantly lower grain boundary energy. Tese boundaries only occur at particular misorientations and boundary planes wic allow te two adjoining lattices to fit togeter wit relatively little distortion of te interatomic bonds. One example of special grain boundary is te twin boundary. A twin boundary is a symmetry plane between two areas of te crystal. In te case of F.C.C. crystal, planes {111} are close packed wit a stacking sequence ABCABCABC A cange in tis stacking sequence tat divides te crystal in two parts, one being te reflection of te oter, ABCABACBA is a twin boundary. One may classify twin boundaries as eiter coerent or partially coerent. Figure 4.3.a illustrates a complete coerent twin boundary, wic is obtained witout any deformation of te lattices since a perfect lattice is automatically obtained. If te twin boundary rotates off te symmetry plane (figure 4.3.b) we obtain a partially or non coerent twin boundary. Te energy of a twin boundary is terefore very sensitive to te orientation of te boundary plane.

128 126 (a) (b) Figure 4.3: (a) a coerent twin boundary and (b) a partially coerent twin boundary [Veroeven, 1975]. Figure 4.4 describes measured grain boundary energies for several symmetric tilt boundaries in aluminium. Wen te two grains are related by a rotation about (100) axis (figure 4.4.a), we observe tat te grain boundary energy is relatively constant for ig angle grain boundaries. So, te ig angle grain boundaries in tis case ave a relatively disordered structure caracteristic of random boundaries. However, wen two grains are related by a rotation about a (110) axis (figure 4.4.b), several ig angle orientations wic ave significantly lower grain boundary energy tan te random boundaries are observed. Te misorientation equal to 70.5 corresponds to a F.C.C. coerent twin boundary. Figure 4.4: Measured grain boundary energies for symmetric tilt boundaries in Al (a) wen te rotation axis is parallel to (100) and (b) wen te rotation axis is parallel to (110) [Hasson, 1971]. Finally, if eterogeneity of te energy boundary is directly responsible of local fluctuations of te grain growt kinetics (see Equation 4.1), anoter impact of te energy boundary, less visible, concerns te equilibrium position of te multiple junctions. Te triple junction angles are determined based on te grain boundary energies by [Humpreys, 2004], [Garcke, 1991]: sin1 sin sin , (4.4) were ij represents te energy boundary between grain G i and G j and te angle formed by te grain G i at te triple junction. Based on te relationsip 4.4, if te boundary energy is omogeneous, te triple junction angles are ten equal to 120.

129 127 3 Finite element model in a level set framework In order to properly model te grain growt penomenon, we must, initially, generate a digital microstructure tat is representative of te material of interest. We come back ere to te 304L steel discussed previously. Te grain size distribution is determined based on optical microscope image analysis. Te corresponding digital microstructure is generated using a Voronoï-Laguerre metod (capter 2, paragrap 2). Te grains are implicitly identified using level set functions and te mes is adapted around te grain boundaries. All numerical tools used were already described in details in capter Velocity field expression Te model presented ere is an extension of te recrystallization model proposed by [Loge, 2008], [Bernacki, 2009]. Tis extension includes te capillarity term of te grain boundary kinetics Equation 4.1 into te recrystallization model, as was briefly described in [Bernacki, 2011]. In recrystallization regime, te grain boundary velocity between two neigbouring grains (G i and G j ) is defined as te addition of a primary recrystallization term (PR) wit a grain growt term (GG): vij ( x, t) vpr ( x, t) v ( x, t), ij GGi v ( x, t) M ( e ( x, t) e ( x, t)) n ( x, t), PR ij j i i v GG i ( x, t) M ( x, t) n ( x, t). i i (4.5) In tis capter, only grain growt is taken into account, so 0 v PR ij and v ij v GGi. Te GG term depends on te grains curvature wic can be computed from te level set description of te granular structure. Indeed, by considering i te level set function of grain G i, we ave: (, ) (, ). i x t i x t. (, ) (4.6) i x t Knowing tat in our FE formalism, a linear approximation (P1) for te level set functions is used and tat a P1 description of te curvature is necessary (since velocity interpolation must be P1), te direct use of Equation 4.6 to calculate v is not recommended. Indeed, suc approac will imply a two-step P0-P1 interpolation and terefore a very poor accuracy. Anoter difficulty in te grain curvature calculation is related to its non regular beaviour in polygonal grains. At te multiple junctions, te curvature as extremely ig values (teoretically infinite), making any attempt to solve te convective problem very difficult. So, to avoid te direct grain curvature calculation (Equation 4.6), te following grain curvature equation was considered and solved in a weak sense: (, ) (, ) (, ) i x t i x t i x t, (, ) (4.7) i x t wit a diffusive term allowing te regularization of te grain curvature calculation. Tis equation also allows avoiding te two-step P0-P1 interpolation tanks to te integration of its weak form. Even toug tis tecnique is effective wen considering a circular grain GG i

130 128 srinking study (paragrap 4.1), it as not improved te grain curvature calculation in multiple junctions in te case of polyedral structures (paragrap 4.2). So, an alternative approac, considering te level set functions metric properties is used. If level set functions remain, near te grain interfaces, distance functions, ten, for 1 i NG (N G being te number of grains): ( x, t) i so: v GG i 1 i ( x, t) and n ( x, t). ( x, t) ( x, t) ² 1, i i i i i 1 (4.8) ( x, t). ( x, t) M ( x, t) n ( x, t) ( x, t) M ( x, t). (4.9) i i i Terefore, te convection term of te grain boundary motion problem becomes a simple diffusion term. In [Merriman, 1994], [Elsey, 2009], te same approac was used in a finitedifference metod context. Finally, te problem solved in order to model te pure grain growt process boils down to a diffusion problem: i i i ( x) Mi ( x) 0 t, i 1,..., NG. 0 i( t 0, x) i ( x) (4.10) Te system is solved for all N G grains present in te domain Ω. Solving te weak form of Equation 4.10 avoids te resolution of a difficult curvature-dependent convection problem. Te assumption tat te capillarity effects can be described by a level set diffusion equation is true if and only if te level set functions satisfy a distance function property near te boundaries. So, to ensure tis condition, a re-initialization treatment must be done at eac time step [Oser, 1988]. Also te diffusion of te level set function can generate some kinematic incompatibilities: vacuums regions or level set overlapping [Merriman, 1994]. To avoid tis kind of problem, a treatment at multiple junctions is performed [Bernacki, 2011]. Tis treatment is presented in details in te following paragrap. 3.2 Numerical treatment for multi-junctions Compatibility problems, as vacuum or overlapping regions, can occur wen te diffusion approac is used [Merriman, 1994]. To avoid tis problem, a multiple junction treatment is performed at eac time step, after solving Equation Tis metod consists in removing all incompatibilities by modifying all level set functions according to: ~ 1 i ( x, t) i ( x, t) max( j ( x, t)), 2 ji i 1,..., N. G (4.11) In oter words, tis metod calculates te bisector between te debonding level set functions (red dased lines in Figure 4.5) and replace te zero isovalue of tese level set functions to te bisector position. Figure 4.5 illustrates te metod.

131 129 Figure 4.5: Sceme of ow te numerical treatment of multiple junctions works: calculation of te bisector between te debonding level set functions (red dased lines) and replace te zero isovalue of tese level set functions to te bisector position. However, tis metod presents te disadvantage of being valid only in case of omogeneous grain boundary energy, i.e. wen te triple junction equilibrium angles are equal to 120 (according to Eq. 4.4). Also, it is important to igligt tat tis treatment is performed only in a tin layer around te zero isovalues, and always before te re-initialisation treatment (since it modifies te distance functions metric properties). 3.3 Container level set functions All te presented procedures (diffusion of level set functions, re-initialisation tecnique and numerical treatment at multiple junctions) are solved for eac level set function existing in te domain. So, te principal weakness for te grain growt algoritm remains te numerical cost, particularly in 3D. A first waste of computational resources can be identified in te fact tat we work wit one-level set per grain. Even for te recrystallization modelling (wic will be detailed in capter 5), a global velocity depending on local properties can be defined witout te knowledge of individual grains topology. Finally wen modelling recrystallization or grain growt penomena, knowing te individual level set function of eac grain is not necessary to perform numerical simulation. In order to limit te number of required level-set functions needed in our simulations, we use a classical tecnique of grap colouring [Kubale, 2004]. Te idea is to colour te vertices of a grap suc tat no two adjacent vertices sare te same colour wit a minimal number of colours. Te most famous result from te grap colouring field of researc is te four colour teorem [Kubale, 2004]. Here we use te algoritm implemented by Hitti [Hitti, 2012]. Te idea developed by Hitti is to use a simple grap coloring algoritm on te Delaunay (or weigted Delaunay) triangulation used on level set functions calculation (see capter 2). Several non-neigbouring grains are grouped in only one level set function (called container level set function). As a consequence, te wole microstructure can be represented by a few level set functions corresponding to a set of strictly disjoint grains. Figure 4.6a illustrates a 2D equiaxial polycrystal made of 2000 grains represented using only five level set functions (eac colour representing one container level set function). Figure 4.6b illustrates a 3D polycrystal of 440 grains represented using only 15 container level set functions.

132 130 (a) (b) Figure 4.6: (a) A 2000-grain 2D equiaxial polycrystal described using five level set functions and (b) a 440- grain 3D equiaxial polycrystal described using 15 level set functions. Te above tecnique owever presents te limitation tat wen two grains belonging to te same container level set function start toucing eac oter, tey coalesce. In [Elsey, 2009], a metod is proposed to avoid tis problem: wen two different grains belonging to te same container level set function get closer tan a critical distance, one of te two grains is removed from te container level set function, and placed in anoter one. However, if te proposed metodology can be used in te context of regular grids were te connected components (te individual grains) of eac container level set function can be easily extracted, te problem becomes muc more complex for unstructured FE meses. Terefore, in tis work, in order to delay te onset of grains coalescence, a constraint is introduced in te grap colouring suc tat only te n t (n = 2, 3, 4 ) nearest neigbour can belong to te same level set function. Te simulation is ten stopped as soon as two grains begin to coalesce. Te coice of n value depends on ow long we want to simulate te penomenon before coalescence appears. 3.4 Grain growt algoritm Using all te numerical tools presented above, we propose te following grain growt algoritm: GG1 Resolution of Equation 4.10 for all active container level set functions. GG2 Numerical treatment (Equation 4.11) for all active container level set functions. GG3 Re-initialization step for all activate container level set functions. GG4 Deactivation of all container level set functions tat are negative all over te domain (meaning tat all grains of te container level set function ave disappeared). GG5 Re-mesing operation. 4 Results 4.1 Academic Test - Circular grain case In order to verify te grain growt algoritm proposed in paragrap 3, one first simple test case is performed. Te interest is to compare a pure diffusive approac (paragrap 3) wit a convective approac associated wit te reinitialization procedure [Coupez, 2007], [Bernacki, 2009]

133 131 i ( x) vi ( x) s t 0 i ( t 0, x) i ( x) 1 s, i 1,..., N, G (4.12) were s is a function of and a coefficient defined tanks to te mes size and te time step. We refer te reader to [Coupez, 2007] for a more precise description of te metod and its extension (regular truncation of ). In te convective approac (Eq. 4.12), te grain curvature is calculated using te resolution of te weak form of Equation 4.7 in our context of P1 interpolation. To compare tese two approaces, we consider a test case defined by a grain embedded in anoter large grain. Te domain size is 1 mm x 1 mm and an a priori remesing tecnique is used to adapt te mes around te grain boundary (see capter 2). Bot grain boundary mobility and energy are considered isotropic and equal to one. In te test case, te grain srinks wile remaining a circle, and te GG kinetics could be summarized as a simple differential equation on te circle radius: 1 2 r ( t) ( t) r( t) r(0) 2t. (4.13) r( t) Te initial grain radius is set to 0.25 mm. In a first stage, bot metods are tested under te same mes and time step conditions. In tis case, te mes size outside te adapted zone is equal to 0.01 mm. Witin te adapted zone, te mes size in te direction perpendicular to te boundary is equal to 2 µm, and remains equal to 0.01 mm in te tangential direction. Te time step is set to 0.1ms. Remesing operations are performed at eac time step wit te same parameters. Figure 4.7 compares te grain srinking kinetics of bot approaces wit te analytical solution. Figure 4.7: Comparison between numerical and teoretical results for te srinking of a circular grain wit a mes size equal to 0.01 mm outside te remesed zone and mm on te anisotropic adapted zone (in te normal direction to te interface) Grap from figure 4.7 illustrates tat, for te mes and time step conditions tested in tis case, te convective approac is not stable and te simulation of a srinking circular grain is not possible. On te oter and, wen using te diffusive approac, te grain srinking simulation was possible. So, a second test of te convective approac, using a finer mes and a smaller time step is performed. In tis case, te mes size outside te adapted area and witin te adapted area in te tangential direction remains equal to 0.01 mm, wereas te

134 Radius (mm) Radius (mm) 132 mes size near te interface in te normal direction is fixed to 0.8 µm. Te time step is reduced to 50 µs. Figure 4.8 presents te comparison between te convective and diffusive approaces. For tis simple case presenting a regular grain curvature, bot approaces ensure a good description of te grain srinking kinetics. For bot cases, te L2 error (Equation 2.9 applied to te equivalent radius) is smaller tan 1% Teoretical Convective approac Diffusive approac Teoretical Convective approac Diffusive approac Time (s) (a) (b) Figure 4.8: Comparison between numerical and teoretical result for te srinking of a circular grain: (a) complete evolution in function of time; (b) zoom of grain srinking end. Te number of mes elements at te simulation beginning is equal to for te diffusive approac and for te convective approac. For te diffusive approac, te computation time is about 40 minutes (1 CPU, 2.3 GHz, 2 Go) wile for te convective approac, te computation time is equal to 1 our and 10 minutes (1CPU, 2.3 GHz, 2 Go). Tese results sow tat for a simple grain growt case, wen te grain curvature is regular all over te domain, bot diffusive and convective approaces can describe wit good accuracy te grain srinking kinetics. However, wit te convective approac a finer mes around te boundaries (for tis case, 2.5 times smaller) and a smaller time step (for tis case, 2 times smaller) is needed in order to obtain te same accuracy as te diffusive metod. Finally, since a finer mes and a smaller time step is needed, te calculation time for te convective approac is iger tan for te diffusive approac. 4.2 Academic test - 3 grains case Time (s) In tis part, two different tests were performed, bot wit 3 polygonal grains. In bot cases te grain boundary mobility and energy are isotropic and uniform, equal to one. Te first test represents a triple junction in te equilibrium state (angles between te grain boundaries are equal to 120 ) in te middle of a 1mm x 1 mm domain. As te triple junction is already in te equilibrium state, te triple junction sould not deform or move during te simulation. An a priori anisotropic remesing tecnique is used. Te number of mes elements present in te domain is equal to Te time step of te simulation is equal to 2.5 ms. Figure 4.9 illustrates te configuration of tis test.

135 133 Figure 4.9: Initial configuration of te triple junction test - triple junction in equilibrium conditions respecting te 120 equilibrium angle. It was not possible wit te considered mes adaptation and time step to model tis triple grain configuration using a convective approac eiter by direct (Equation 4.6) or indirect (Equation 4.7) calculation of te grain curvature. On te oter and, te diffusive approac allows te verification tat te equilibrium state from figure 4.9 does not evolve during te simulation (simulation was performed until t = 1.75 s). So te diffusive approac leads to te equilibrium angle of 120 at te triple junctions. Te second test case is te «T» configuration. In [Garcke, 1991], Garcke as proved te existence of an exact solution for te grains configuration wen pure grain growt takes place. In tis case we use an a posteriori mes adaptation and te number of mes elements is equal to Figure 4.10a illustrates te initial configuration and figure 4.10b represents te obtained result. (a) (b) Figure 4.10: Garcke case: (a) initial configuration, (b) equilibrium configuration (black line) compared wit te teoretical result (red line) for t = 47 ms zoom of triple junction. Red line in figure 4.10b represents te teoretical result wile te black line is te numerical result obtained at t = 47 ms (time step is equal to 1 ms). Te L2 error concerning te final positions of te interfaces is smaller tan 1%. Tese two tests illustrate tat te diffusive algoritm associated wit an anisotropic mes adaptation allows an accurate simulation of pure grain growt penomena, even wit a non regular grain curvature. 4.3 Von Neumann-Mullins cases From Equation 4.4 we note tat in a two dimensional grain structure, wen te grain boundary energy is isotropic, te only stable arrangement wic can fulfil te boundary

136 134 tension equilibrium requirements is an array of regular exagons. Tis is true because regular exagons angles are equal to 120 (figure 4.11) wic is te equilibrium angle for grains triple junctions. Any oter topological arrangement will inevitably lead to grain growt. From figure 4.11 we observe tat for grains wit a number of sides different from 6, in order to maintain te 120 angles at te vertices, te sides of te grains must become curved. Grain boundary migration ten tends to occur in order to reduce te boundary area, and te boundaries migrate towards teir centre of curvature. Any grain wit more tan six sides will tend to grow because its boundaries are concave. Similarly, any grain wit less tan six sides will tend to srink since its boundaries are convex. Figure 4.11: 2D grain boundary configurations. Te arrows indicate te directions boundaries will migrate during grain growt [Porter, 2008]. Based on tis idea, von Neumann [vonneumann, 1952] and Mullins [Mullins, 1956] proposed tat te growt of a 2D grain of area A and N sides is given by te following relation: da ( N 6) C. (4.14) dt were C is a constant tat depends on te material parameters. In order to verify if te grain growt algoritm obeys tis model, we simulate te grain growt of five different regular polygons: a square, a pentagon, a exagon, a eptagon and an octagon. In all cases te grain boundary mobility and energy are considered isotropic and uniform (equal to one). Also, an adapted mes around te boundaries is used and te simulation time step is fixed to 0.1 ms. Te adapted mes is generated using an a posteriori remesing tecnique. Te number of mes elements depends on te polygon. For example, for te exagon, te initial number of mes elements existing in te domain is equal to 50,000. A remesing operation is performed at eac time increment. Figure 4.12 illustrates te different sape evolutions trougout te simulations (wit te surface ratio), and te grap in figure 4.13 sows te corresponding grain growt kinetics. t = 0 s t = s t = s t = s S r = 1.0 S r = 0.75 S r = 0.5 S r = 0.25

137 Area/Initial Area 135 t = 0 s t = s t = s t = s S r = 1.0 S r = 0.75 S r = 0.5 S r = 0.25 t = 0s t = 0.05s t = 0.1s t = 0.15s S r = 1.0 S r = 1.0 S r = 1.0 S r = 1.0 t = 0 s t = s t = s t = s S r = 1.0 S r = 1.25 S r = 1.5 S r = 1.75 t = 0 s t = s t = s t = s S r = 1.0 S r = 1.5 S r = 2.0 S r = 2.5 Figure 4.12: Polygonal grains evolution in function of time: square, pentagon, exagon, eptagon and octagonsape grains Time (s) square pentagon exagon eptagon octagon Figure 4.13: Normalized area evolution for all five different polygons studied: square, pentagon, exagon, eptagon and octagon.

138 alpa 136 From Figure 4.12, we observe te cange in te grain boundaries sapes in order to conform to te 120 equilibrium angle at te vertices. Consequently te grains wit a number of sides smaller tan 6 srink, a grain wit 6 sides remain uncanged and grains wit more tan 6 sides grow, following te von Neumann-Mullins teory. Finally, analysing te grap in figure 4.13, we observe tat te area mainly evolves linearly wit time. Te non linearity in te end of te simulation for grains wic disappear, i.e. wit 4 and 5 sides, is a consequence of numerical accuracy issues wen te ratio "grain size/ finite element mes size near te grain boundaries" becomes too low. Indeed, wile te mes evolves spatially wit time it does maintain constant parameters concerning te mes size. Wen te ratio becomes low, te grain interfaces become poorly described and te numerical simulation accuracy drops. One way to avoid tis problem is to decrease te mes size near te grain boundaries (at least in te normal direction) during te simulation. Now, to verify if te kinetic beaviour of te grains obeys Equation 4.14, we analysed te linear part of all curves, and calculated te alpa value from Equation da C (4.15) dt Te grap in Figure 4.14 compares te numerical results obtained from te simulations wit te teoretical results from wic N 6 (Eq. 4.14) Numerical Teoretical Number of sides Figure 4.14: Comparison between te von Neumann Mullins model and te numerical results of polygonal grains: square, pentagon, exagon, eptagon and octagon-sape grains. Te L2 error between te two curves is less tan 7% for all considered sapes, wic represents a clear agreement wit te von Neumann-Mullins model, and terefore a good validation of te numerical approac L - 2D polycrystal grain growt After testing academic configurations wit well-known teoretical results, a polycrystal grain growt simulation is performed. Te digital microstructure is generated using te Laguerre-Voronoï tecnique (capter 2, paragrap 2) based on an experimental 304L steel grain size distribution. Te eat treatment is performed at 1050 C. Grain boundary energy and mobility are isotropic and uniform equal to 0.6 J/m² and 1.37x10-12 m 4 J -1 s -1, respectively.

139 137 Te size of te square domain is set to 4 mm x 4 mm and approximately one tousand grains are generated. An a posteriori remesing adaptation is used and te number of mes elements existing initially in te domain is equal to Te time step is set to 120 seconds. Figure 4.15 illustrates te resulting microstructure evolution as a function of time. (a) (b) (c) (d) Figure 4.15: Grain growt of 304L steel at 1050 C after: (a) t = 0; (b) t= 20 min; (c) t = 1 40 min; (d) t = 5. Again, te grains wit more tan 6 sides grow wile tose wit less tan 6 sides srink, in agreement wit te von Neumann Mullins teory. Figure 4.16 describes te number of grains and te mean grain surface evolution as a function of time.

140 Normalized Area Nb of sides Normalized Area Nb of sides Normalized Area Nb of sides 138 Figure 4.16: Evolution of number of grains (green line) and mean area (blue line) in function of time. Te mean surface per grain evolves quasi linearly wit time, wile te evolution of te number of grains is proportional to 1 t. A more detailed study of grain growt kinetics, wit comparisons wit te Burke and Turnbull model [Burke, 1952] and wit te Hillert/Abbruzzese model [Hillert, 1965], [Lücke, 1992], [Lücke, 1998] and [Abbruzzese, 1992] is presented in paragrap 5. A selection of four individual grains is done in order to analyse te normalized area evolution (normalized by te initial grain surface) and te number of sides evolution wit time (figure 4.17). Te area evolution wit time is clearly not linear, and te type of evolution is related to te interaction wit neigbours. However, te number of sides of eac grain does dictate te tendency to grow, srink, or remain stable, in agreement wit te von Neumann Mullins teory Time (s) (a) Area Nb of sides Time (s) (b) Area Nb of sides Area Nb of sides Time (s) 0 (c) (d) Figure 4.17: Evolution of grain normalized area (full red lines) and number of sider (dased red lines) in function of time.

141 139 In figure 4.17c, we clearly see te connexion between te grain growt rate and te number of sides. At te beginning te grain grows fast since te number of sides is around 10. At t = 4500 s, te number of sides decreases to 8, and te grain area evolution becomes slower. In te end te number of sides becomes equal to 6 and, once again, te grain growt rate decreases, becoming negligible. Te same kind of analysis can be done wit figures 4.17a, 4.17b and 4.17d and all of tem will sow te same type of connexion between te two monitored quantities. Tis first pure grain growt simulation of a 304L polycrystal illustrates te capability of te proposed formalism. We now come back to te issues already discussed in capter 2 concerning te impact of te metod used to generate digital microstructures. In capter 2, te geometrical aspects of polycrystalline aggregates were analysed for two different generation metods (Voronoï, and Laguerre-Voronoï). In wat follows, a similar analysis is done regarding te grain growt prediction. Section 5 will ten be dedicated to te impact of te grain size distribution, and comparisons wit grain growt mean field models predictions. 4.5 Influence of te microstructure generation metod on grain growt prediction As it was discussed in capter 2, te Voronoï metod is te most widely used tecnique to generate a digital microstructure. Te Voronoï approac allows to define te mean grain size by setting te number of grains (i.e. te number of Voronoï sites) in a given volume. However, using tis metod it is not possible to obey a given grain size distribution. On te oter and, wit te Laguerre-Voronoï tecnique (also discussed in details in capter 2), a given grain size distribution can be obeyed. In [Xu, 2009], Xu igligted divergences between statistical properties classically observed in equiaxed polycrystals and tose obtained using te Voronoï metod. However te influence of te grain size distribution on te grain growt kinetics and te importance of obeying an experimental grain size distribution are not well discussed in te literature. So, in tis section, a comparison between te grain growt kinetics of two digital microstructures wit te same mean grain size but generated wit different metods (Voronoï and Laguerre-Voronoï) is performed. Te digital Laguerre- Voronoï microstructure was already used in te previous paragrap (304L grain size distribution, square domain of 4 mm x 4 mm, 1033 grains). In te digital Voronoï microstructure, te mean grain size of te 304L microstructure is set (67.4µm) using 1120 grains randomly generated in te 4 mm x 4 mm square domain. For bot cases te grain boundary mobility and energy are isotropic and uniform (T=1050 C), equal to 0.6 J/m² and m 4 J -1 s -1, respectively. Figure 4.18 illustrates te two digital microstructures (a similar comparison as already been done in a 12 mm x 12 mm square domain from figures 2.9 and 2.13 in capter 2) and figure 4.19 presents te grain size and te number of sides distribution for bot of tem.

142 140 (a) (b) Figure 4.18: 304L Polycrystal in a 4 mm x 4 mm square domain generated using a (a) Laguerre-Voronoï metod (1033 grains) and (b) Voronoï metod (1120 grains) (a) (b) Figure 4.19: Comparison of (a) grain size distribution and (b) number of sides distribution for bot digital microstructures, one generated wit te Voronoï metod and te oter generated using a Laguerre-Voronoï metod. Comparing Laguerre-Voronoï and Voronoï grain size distributions, we observe, as already discussed in capter 2, tat Laguerre-Voronoï is a log normal distribution wile Voronoï is a Gaussian distribution. Considering te number of sides distribution, even toug bot microstructures present a mean number of sides around 6 (5.95 for Voronoï and 5.97 for Laguerre-Voronoï), teir distributions are different. Once again, te number of sides distribution is closer to a Gaussian sape for te Voronoï case, and closer to a log-normal sape for te Laguerre-Voronoï case. In Table 4.2, we observe tat even toug bot grain structures present a majority of grains wit less tan 6 sides, tis proportion in almost 10% iger for te Laguerre-Voronoï structure. Te proportion of grains wit more tan 6 sides is almost te same for bot microstructures, but te Laguerre-Voronoï distribution presents larger grains, wit more sides. Finally, te Voronoï microstructure sows a iger proportion of grains wit 6 sides. Table 4.2: Proportion of grain wit less tan 6 sides, exactly 6 sides and more tan 6 sides for te Laguerre- Voronoï and Voronoï grain structure of 304L generated in a 4mm x 4mm square domain. Laguerre-Voronoï Voronoï n < % 39.2% n = % 30.1% n > % 30.7%

143 141 Analysing te grap of te normalized average grain surface evolution wit time (Figure 4.20), we observe tat te Laguerre-Voronoï grain growt is faster tan te Voronoï one. Tis beaviour is related to te grain size distribution. Te Laguerre-Voronoï distribution is more spread and as a consequence, te difference between te equivalent radii of neigbouring grains is, in average, larger for te Laguerre-Voronoï distribution. Tis beaviour is also observed in Figure For te Laguerre-Voronoï microstructure (Figure 4.18a) a few bigger grains are surrounded by small grains. In tese areas te local curvatures will be increased and te grain growt kinetics will be faster. For te Voronoï microstructure, since te grain size difference between neigbouring grains is smaller, te local curvatures are not as ig. Figure 4.20: Comparison for 304L in a 4mm x 4mm square domain of te normalized number of grains evolution during pure grain growt regime for te Laguerre-Voronoï metod (blue dased line) wit te one for te Voronoï metod (red dased line); comparison of te normalized mean grain area evolution for te Laguerre-Voronoï metods (blue full line) wit te one for te Voronoï metod (red full line). It is confirmed in tis capter tat taking into account te neigbourood effects is of fundamental importance in determining te kinetics of grain growt. Generating a digital microstructure using a Voronoï metod cannot ensure a good prediction of experimental grain growt wen dealing wit log-normal grain size distributions. On te contrary te Laguerre- Voronoï metod is appropriate. In te next section a more detailed study of te influence of te grain size distribution on grain growt kinetics is performed. 5 Assessment of simplified 2D grain growt models from numerical experiments based on a level set framework Te study presented in tis paragrap as two main purposes. Te first one is to test, in 2D, two mean field grain growt models existing in te literature - Burke and Turnbull [Burke, 1952] and Hillert/Abbruzzese [Hillert, 1965], [Lücke, 1992], [Lücke, 1998], [Abbruzzese, 1992] under simplified conditions of isotropic grain boundary energy and mobility, constant temperature, and absence of precipitates, using te results obtained wit our full field modelling metod for different initial grain size distributions. Tese models were developed based on teoretical assumptions and tey are not easily verified

144 142 experimentally. So, te idea is to use te full field simulations results, tus verifying if and wen te mean field model predictions are acceptable. Te second main purpose of tis section is to study weter or not a full field model is needed under tese simple conditions. In [Kamacali, 2012], te autors present a similar study using a pase field framework combined wit a finite difference modelling tecnique. Kamacali discusses bot statistical and topological perspectives of ideal grain growt using te results of 3D simulations. Te paper sows tat, despite a few discrepancies wit te mean field teory (Burke and Turnbull model and Hillert model) te parabolic kinetics of grain growt remains valid for te entire process. Te simulation reaces a steady state and te grain size distribution in tis steady state agrees wit te Hillert distribution [Hillert, 1965]. Te volumetric growt rate obtained compares well wit te mean field assumptions. However, te paper analyses only one initial grain size distribution, putting aside te influence of te initial grain size distribution on te grain growt kinetics. In tis section, te influence of te initial grain size distribution is studied by considering seven different cases in 2D configurations. 5.1 Grain size distributions To ceck te validity of different grain growt models, we now study seven different grain size distributions. Tese seven grain size distributions were described in details in capter 2. Table 4.3 sums up teir principal features. Table 4.3: Caracteristics of grain size distributions and error concerning te digital generation of te seven microstructures Mean Radius - μ Standard Deviation - σ/ μ Initial number of Domain Size Error (%) (µm) σ (µm) grains (mm) Log x Log x Log x Log x Log x L x Bimodal x In all cases, te number of grains is around 10000, wic ensures a good statistics during te simulation. Figure 4.21 recalls te comparison between te targeted microstructures and tose built numerically (see Figures 2.14 and 2.11b). In tis figure, te x-axis corresponds to te normalized radius R R, were <R> corresponds to te mean grain size. Table 4.3 presents te L2 errors between te teoretical and te obtained numerical distributions.

145 143 (a) (b) Figure 4.21: Numerical initial grain size distributions compared wit te targeted one: (a) Bimodal and 304L distribution; (b) Log1, Log2, Log3, Log4 and Log5 distributions. For all simulations presented ere and referring to te 304L steel properties, te time step is 120 seconds, te termal treatment is applied 5 and te temperature is constant and equal to 1050 o C (M = m 4 /(J.s) and γ = 0.6 J/m 2 ). Te finite elements meses are adapted tanks to te a posteriori metod in a square domain of 13 mm x 13 mm, leading to an average of 3,200,000 mes elements for te initial microstructures. Te computation time is around 24 ours using 128 cpu for all calculations. A grap colouring tecnique (paragrap 3.4) is used for limiting te number of level set functions required to describe te wole microstructure. In oter words, one level set function will describe several non-neigbouring grains instead of describing only one grain. So, instead of dealing wit N G level set functions in te domain, we will ave N C container level set functions, knowing tat N C is significantly smaller tan N G. In order to delay te onset of grains coalescence, we impose tat only te 4 t nearest neigbour can belong to te same level set function. Terefore, using tis coloration tecnique, te wole aggregate (around for te considered microstructures) can be fully described using only around 30 container level set functions. Figure 4.22 illustrates te grain growt simulation of te 304L grain structure.

146 144 (a) (b)

147 145 (c) (d)

148 146 (e) Figure 4.22: 304L steel grain growt at 1050 C : (a) to (d) grain interfaces after, respectively, 0 min, 20min, 140min and 5; and (e) zoom on a few grains after 130min were an anisotropic FE mes appears in wite. Grey levels of te grains relate to te container level-set functions. All te studied distributions except for te Bimodal distribution (wic will be discussed later) reac at some point a quasi stationary steady state. Te final grain size distributions calculated from te six distributions are compared wit Hillert [Hillert, 1965] and Rayleig distributions. Te Rayleig distribution was originally derived in te 2D grain growt context by Louat [Louat, 1974] and corrected by Mullins in [Mullins, 1998]. Table 4.4 presents te equations for tese teoretical stationary distributions in 2D. Table 4.4: Hillert and Rayleig distribution equations. Hillert distribution R R R R R R e R R P H 2 4 exp Rayleig distribution 4 exp 2 2 R R R R R R P R Te Hillert distribution as a non-analytic cut off at R R = 2 wile te Rayleig distribution presents an infinite tail. In Figure 4.23 te obtained numerical quasi-stationary distributions are in good agreement wit te Rayleig distribution, wic is coerent wit Mullins work (te Bimodal distribution did not reaced a quasi-stationary state so te corresponding results will be discussed later).

149 147 Figure 4.23: Quasi-stationary state and comparison wit Hillert and Rayleig predictions. According to [Mullins, 1998], te reason for te discrepancy between te Hillert teoretical distribution and te experimental and/or numerical results comes from topological reasons: in 2D, te Hillert distribution is reaced only if te average number of sides relates linearly to te grain size. Te relationsip between te number of sides and te grain size in te results presented in Figure 4.23 sould terefore be te subject of future work. 5.2 Burke and Turnbull study Te first mean field model tested is te Burke and Turnbull model [Burke, 1952]. Tis model is based on tree main ypoteses: te driving force for te penomenon is proportional to te grain boundary mean curvature 1 R, were R is te equivalent radius. So te grain boundary migrates toward te centre of its curvature, wic in turns reduces te interfacial area as well as its associated energy; te mobility and grain boundary energy are isotropic and uniform. As a consequence te angles in te triple junction are equal to 120 o C; te eat treatment temperature is constant. Tese ypoteses lead to te following grain growt kinetics equation in 2D: R 1 ² 2 R 0 Mt 2, (4.16) were R (resp. R 0 ) corresponds to te average grain radius (resp. at t = 0s). Wit tis equation, neiter te topological nor te neigbouring effects are taken into account; te grain growt kinetics is caracterized only by te average grain size. To ceck te consistency between te full field simulation results and tis model, te 2 2 curves log R R 0 f log( t) ave been plotted in Figure 4.24 for eac initial distribution described in Table 1, except for te Bimodal distribution, wic will be treated separately. For tese distributions, Equation 4.17 is generalized according to: R 2 n ² R 0 Mt. (4.17)

150 148 From Equation 4.17, te validity of te Burke and Turnbull model can be verified if te 2 2 slope of log R R 0 f log( t), n, is equal to 1 and if te fitted curve leads to an α value around 0.5. Computed curves and teir linear fits are given in Figure 4.24 and lead to te α and n parameters summarized in Table 4.5. Table 4.5: Burke and Turnbull model analysis tanks to full field simulations results for different grain size distributions. Distribution Slope (n) α Number of grains in te end σ/ μ Log Log Log Log Log L Figure 4.24: Computed evolutions of grain structures starting wit different initial grain size distributions, using te full field model. Linear approximations are used for comparison wit te Burke and Turnbull model. As illustrated in Table 4.5 and Figure 4.24, n and α are igly dependent on te initial grain size distribution. Only te Log4 and 304L distributions lead to te values of n and α expected by te Burke and Turnbull model. Bot distributions are log-normal wit a value of 0.45 (see Table 4.3). For te oter distributions, it can be noticed in Figure 4.25 tat n values decrease wit te increase of, wile α values increase wit. Tese evolutions can be approximated by te following simple relationsips: ln 11.15ln 8.62, (4.18) n 1.08.ln 0.1. (4.19)

151 149 Figure 4.25: Relationsip between α and n fitted parameters and σ/μ of te initial grain size distributions (given in Table 4.3) For a log-normal distribution, α presents an power law dependence on and n a logaritmic dependence. Equations 4.18 and 4.19 confirm tat grain growt kinetics significantly depends on grain structure and neigbourood effects. Figure 4.26 sows tat te Bimodal kinetics can be divided in two linear parts A between t = 0s and t = 4400s and B, between t = 4400s and t = 18000s. In an attempt to understand tis beaviour, te grain size distribution at t = 0s, t = 4400s and t = 10800s (figure 4.27) are analysed. Figure 4.26: Burke and Turnbull model study for Bimodal grain structure. (a) (b)

152 150 (c) Figure 4.27: Grain size distribution at (a) t=240s, (b) t=4400s and (c) t=10800s for te Bimodal grain structure. Figure 4.27 sows tat for t = 0 s, we ave two different grains populations. Te number of grains belonging to te smaller radius family is bigger tan te one belonging to te larger radius family. Knowing tat te average grain curvature is equal to te inverse of te equivalent radius, grains aving a smaller radius will ave a bigger curvature and, as a consequence, will be quickly consumed. At t = 4400 s, te grain size distribution as canged, and starts switcing to a single peak distribution. Te average grain curvature decreases, and te grain growt kinetics consequently becomes slower. Finally, t = s, a single peak distribution is reaced. One can conclude tat te slope cange in te kinetics of grain growt coincides wit te quasi disappearance of te smallest grains population, i.e. wit te switcing from a bimodal distribution to a single peak distribution. It is terefore concluded tat te Burke and Turnbull model is not valid for most of te investigated grain size distributions. Te model only beaves well for lognormal grain size distributions, wit a value close to Hillert/Abbruzzese model study In a second part, bot Hillert [Hillert, 1965] and Abbruzzese [Lücke, 1992], [Lücke, 1998], [Abbruzzese, 1992] mean field models were studied. Bot models are derived from von Neumann-Mullins [von Neumann, 1952], [Mullins, 1956] model. Even toug te development of te model proposed by Hillert is different from te one proposed by Abbruzzese, bot autors finally propose te same grain growt model equation (Equation 4.20). In te development of te Abbruzzese model, te autors use te Special Linear Ri Relationsip (SLR) ni 3. 1 [Abbruzzese, 1992] discussed in capter 2. In tis R work, bot models are analysed as one. Te central equation is: V i 1 1 M, (4.20) R Ri were te grain structure is assumed to be described by a discrete set of grain families (eac family being labelled wit a subscript i), V i corresponds to te average grain boundary velocity of te grain family i, R corresponds to te average grain size and is a constant generally assumed to be 0.5 in 2D, and 1 in 3D, even toug oter values can be found in te literature. For example, in [Kamacali, 2012], Kamacali found a value of 1.25 in 3D, using a pase field framework combined wit a finite-difference modelling tecnique. In [Rios,

153 ], Rios and Glicksman calculated a 3D value of 0.81, using te average N-edra metod (ANH) [Glicksman, 2005]. In tis model, te neigbouring effects are taken into account by te addition of te term 1 R. Tis term represents te mean curvature of all te grains belonging to te domain. Also, in te Hillert/Abbruzzese model, te grain growt beaviour of a istogram wit several grain families is analysed. As a consequence, te interaction between te different families is also taken into account (in te Burke and Turnbull model tis interaction is ignored). Figure 4.28 illustrates te interaction between te different families. Figure 4.28: Grain size distribution: probability (φi) vs. radius (Ri). Examples for te grain size flux during grain growt are indicated for te cases R i < R cr and R i > R cr ) [Lücke, 1998]. To ceck te consistency between te full field simulation and te Hillert/Abbruzzese model, we compared te grain size distributions obtained wit te full field model and te mean field model described by Equation 4.20, and implemented witin a more general model described in [Bernard, 2011]. Tree different time steps (t = 2 min, t = 76 min and t = 150 min) were analysed. Figure 4.29 illustrates te comparison and table 4.6 numerically quantifies te comparisons for te seven grain size distributions. Table 4.6: Error between istograms obtained wit full field model and Hillert/Abbruzzese mean field model. Error L2 (%) t = 2 min t = 76 min t = 150 min Log Log Log Log Log L Bimodal

154 Cumulative Fraction Cumulative Fraction Cumulative Fraction Normalized Radius 304L t=120s Mean Field t=120s FullField t=4560s Mean Field t=4560s FullField t=9000s Mean Field t=9000s FullField Normalized Radius Bimodal t=120s Mean Field t=120s FullField t=4560s Mean Field t=4560s FullField t=9000s Mean Field t=9000s FullField Normalized Radius Log1 t=120s Mean Field t=120s Full Field t=4560s Mean Field t=4560s Full Field t=9000s Mean Field t=9000s Full Field

155 Cumulative Fraction Cumulative Fraction Cumulative Fraction Normalized Radius Log2 t=120s Mean Field t=120s Full Field t=4560s Mean Field t=4560s Full Field t=9000s Mean Field t=9000s Full Field Normalized Radius Log3 t=120s Mean Field t=120s Full Field t=4560s Mean Field t=4560s Full Field t=9000s Mean Field t=9000s Full Field Normalized Radius Log4 t=120s Mean Field t=120s Full Field t=4560s Mean Field t=4560s Full Field t=9000s Mean Field t=9000s Full Field

156 Cumulative Fraction Log5 Figure 4.29: Comparison between full field results and Hillert/Abbruzzese mean field results for all grains structures studied: 304L, Bimodal, Log1, Log2, Log3, Log4, and Log5. From Table 4.6 it can be concluded tat te Hillert/Abbruzzese mean field model is in good agreement wit te computations. For all considered grain size distributions, te error between te grain size cumulative istograms obtained wit te two different approaces is smaller tan 11%, even for te Bimodal distribution. A closer look at te distributions illustrated in Figure 4.29, sows tat errors increase in te lowest grain sizes range. A ypotesis was initially made to explain tis localization. Te idea was to relate tis error to volume conservation issues discussed in [Bernard, 2011]: volume canges of te srinking grains are computed by redistributing te total volume canges of growing grains. Consequently, te rates of volume cange for small grain sizes are sligtly modified from te values strictly deriving from Equation Comparing te results obtained using te mean field model discussed in [Bernard, 2011] and te direct use of Equation 4.20, we observe tat te effect of tis volume conservation treatment is more important for a small number of representative grains (around 50). In fact, wen using 40 representative grain families (as in te comparisons presented in Figure 4.29), te direct use of Equation 4.20 gives better results tan te use of te model presented in [Bernard, 2011]. However, for a number of representative grains around 200, bot metods present te same results, and te volume conservation treatment does not affect te results anymore. Terefore, in te simple considered configurations, i.e. wen grain boundary mobility and energy are isotropic trougout te polycrystal, and no Zener pinning forces are taken into account (absence of second pase particles), te Hillert/Abbruzzese mean field models describe quite accurately te grain growt penomenon for any initial grain size distribution. 6 Conclusion t=120s Mean Field t=120s Full Field t=4560s Mean Field t=4560s Full Field t=9000s Mean Field t=9000s Full Field Normalized Radius In tis capter a finite element model was proposed in order to simulate te grain growt penomenon. Grain boundaries are implicitly represented using level set functions. Two different metods to simulate grain boundary motion are tested: a convective approac and a diffusive approac. Even toug bot approaces are appropriate to simulate grain growt wit regular grain curvature (for example, a circular grain immerged in an infinite medium), te diffusive approac was te only metod able to simulate polyedral grain growt wit no regular grain curvature.

157 155 It was also sown tat te diffusive approac associated wit a multiple junction treatment and isotropic boundary energy leads to te expected 120 equilibrium angle at triple junctions. Finally, te simulation results of different polyedra (square, pentagon, exagon, eptagon and octagon) growt rates is sown to agree well wit te von Neumann-Mullins teory. A comparison between te grain growt kinetics of two different digital microstructures, one based on setting te mean grain size (Voronoï approac), and te oter based on obeying a grain size distribution (Laguerre-Voronoï), as been performed for an isoterm termal treatment. Te kinetics were sown to be significantly different: te Laguerre-Voronoï one is faster tan te Voronoï. Tis result igligts te importance of te size distribution on global grain growt kinetics. Finally, in tis capter we discussed te validity of two grain growt models (Burke and Turnbull, Hillert/Abbruzzese) using full field modelling results. Seven initial grain size distributions ave been considered in order to test tese mean field metods. Te Burke and Turnbull model, wic does not take into account te topology or te neigbourood effects, is valid only for two distributions and bot of tem are lognormal and present a standard deviation value rougly equal to 0.45 of te mean grain size. Coincidentally, one of tese two distributions is te one based on experimental 304L data. Also, tis kind of distribution is classically observed in polycrystalline metals. On te oter and te Hillert/ Abbruzzese model is valid for all tested distributions, even for te bimodal one. Consequently, if te development of full field models is justified for te description of grain growt in complex conditions, teir use appears disproportionate in te simple configurations investigated ere, were te Hillert/ Abbruzzese approac is sufficient. Tis assessment test was not performed in 3D because of lack of time. 3D simulations wit a representative number of grains (around grains) are time and computational resources consuming. During tis P.D. tesis we did not ave te time to perform all tese simulations. Comparisons between 3D grain growt simulations and mean field models will be done in te near future. In capter 5, te grain growt model introduced in tis capter is extended to a recrystallization model. Te influence of capillarity effects is investigated. Te coupling between te recrystallization grain growt model wit te crystal plasticity model is ten detailed. Te capter ends wit an analysis of nucleation criteria in recrystallization. Résumé en français Un modèle de croissance de grain à l'écelle mésoscopique basé sur une formulation éléments finis dans un cadre level-set est proposé dans ce capitre. Dans une première partie, deux approces pour simuler la capillarité des joints de grains sont testées: une approce convective et une approce diffusive. L approce diffusive développée est finalement retenue par sa capacité à simuler précisément la croissance de grains initialement polyédriques en respectant les angles d équilibre de 120 aux joints multiples imposés pysiquement par l ypotèse d isotropie de l énergie d interface. Finalement, grâce au formalisme développé, un travail d analyse de modèles de croissance des grains à camp moyen existant dans la littérature est réalisé. Deux modèles particuliers ont été étudiés : celui de Burke et Turnbull et celui de Hillert/Abbruzzese. En comparant ces modèles avec les résultats obtenus par notre approce en camp complet, il est mis en évidence que le modèle simple de Burke et Turnbull, qui ne prend pas en compte les caractéristiques topologiques de la microstructure, n est pas approprié pour décrire la croissance de grains pour tout type de distribution initiale de taille de grains. Pour le modèle de Hillert/Abbruzzese, nous prouvons que le modèle est valide pour toutes les distributions

158 156 analysées. Dans le capitre 5, le modèle de croissance de grains présenté dans ce capitre est couplé à un modèle de recristallisation statique.

159 157 Capter 5 Grain Boundary Migration - Static Recrystallization Contents 1 Introduction Static recrystallization Nucleation mecanisms Nuclei growt Te Jonson-Mel-Avrami-Kolmogorov (JMAK) model L steel static recrystallization Strain level effect Temperature and strain rate effect Recrystallization simulations using a full field formulation Finite element model on a level set framework Velocity field expression Recrystallization algoritm Results Academic Test - Circular grain case Academic Test 2 grains case Polycrystalline 2D simulation Random nucleation sites Necklace-type nucleation sites Coupling of crystal plasticity and static recrystallization Critical dislocation density and nucleus radius Number of nucleation sites Coupling wit crystal plasticity results D recrystallization kinetics results in site saturated nucleation context Conclusion

160 158 1 Introduction A driving force for grain boundary displacement arises wen a boundary displacement leads to a reduction of te total energy. In recrystallization penomena, two kinds of energy are responsible for te grain boundary motion: (1) an excess of energy due to grain boundary itself (grain boundary curvature) and (2) a free energy difference between te adjacent grains due to energy stored during deformation. In capter 4, grain boundary migration due to te grain boundary curvature was presented and discussed in details. In tis capter, grain boundary motion based on bot driving forces (recrystallization regime) is studied. A first study analysing te influence of te grain boundary curvature effects during te recrystallization process is presented, and te results sow tat te capillarity effects cannot be neglected in all cases, in contrast wit wat is often accepted in te literature. In fact, te importance of te grain boundary curvature effects on recrystallization kinetics depends on te nuclei spatial distribution trougout te material. Tese results igligt te importance of te developments performed in tis work, for te improvement of te formalism developed in [Bernacki, 2009] and [Logé, 2008]. In a second part, te static recrystallization model is coupled to te crystal plasticity model discussed in capter 1. An analysis of nucleation criteria for static recrystallization is presented, distinguising te two crystal plasticity ardening laws discussed in capters 1 and 3 (one model considering te total dislocation density and a second model considering two dislocation types, SSDs and GNDs). 1.1 Static recrystallization Te recrystallization process involves te formation of new strain-free grains in certain parts of te material and te subsequent growt of tese to consume te deformed microstructure. Te microstructure at any point is divided into recrystallized or nonrecrystallized regions (Figure 5.1). Te recrystallized fraction increases from 0 to 1 during te recrystallization process. Figure 5.1: An SEM cannelling contrast micrograp of aluminium sowing recrystallized grains growing into te recovered sub grain structure [Humpreys, 2004]. It is convenient to divide te recrystallization process into two regimes: (i) nucleation wic corresponds to te appearance of new grains and (ii) growt of tese new grains, wic replace te deformed material. Te extent of recrystallization is often described by te recrystallized fraction (X v ). However, it is also possible to follow te progress of recrystallization by measurement of various pysical or mecanical properties, like te material ardening.

161 Nucleation mecanisms Nucleation is te key concept in te understanding of microstructure recrystallization. Te classical omogeneous nucleation teories associated wit solidification or pase transformation do not old true for recrystallization due to its low driving force and ig grain boundary energies [Humpreys, 2004]. Dislocation free regions are not formed by termal fluctuations, as in te solidification process. It is now well-known tat a critical stored energy must be reaced before te onset of recrystallization. Also, nuclei mostly originate at or near te pre-existing grain boundaries, since tis is were te igest dislocation density is found. Even toug extensive investigation as been made, significant disagreements on te mecanisms of nucleation still exist. Te mostly accepted mecanisms are summarized below. Strain induced boundary migration (SIBM) SIBM is te most common nucleation mecanism and it is also known as bulging or migration of pre-existing grains boundaries. Tis mecanism was originally proposed by Beck and Sperry [Beck, 1950] based on Al optical microscopy observations. Tis mecanism considers te migration of a pre-existing grain boundary toward te interior of a more igly deformed grain, leaving a dislocation free region beind te migrating grain boundary. Figure 5.2 illustrates tis penomenon. In tis case, te nuclei ave a similar orientation to te old grain, from wic tey ave grown. Figure 5.2: (a) SIBM of a boundary separating a grain of low stored energy (E1) from one of a iger energy (E2), (b) dragging of te dislocation structure beind te migrating boundary, (c) te migrating boundary is free from te dislocation structure, (d) SIBM originating at a single large subgrain. Te SIBM mecanism occurs wen te process is energetically favourable. In oter words, te decrease of stored energy due to te elimination of dislocations in te area beind te migrating grain boundary sould be iger tan te total grain boundary surface increase due to boundary bulging. Te growt condition is given by: b L 2, (5.1) E

162 160 were b is te grain boundary surface energy per unit of area, E is te released energy associated wit te decrease in defects, and 2L is te initial bulging boundary lengt (Figure 5.3.a). Terefore, a critical stored energy difference is necessary for te nucleation onset. Tis mecanism is believed to be particularly important for strains up to 40%. Recent researc as sown tat te SIBM mecanism is also very important during recrystallization after ig temperature deformation of steels. In tis case, te material presents a deformed microstructure more omogeneous tan wen te material is deformed at low temperature [Teyssier, 1999], [Hutcinson, 1999]. Oter types of nucleation also seem possible in association wit existing grain boundaries. Beck and Sperry [Beck, 1950] also presented evidence for new recrystallized grains wic did not sare eiter of te parent grains orientation. Te same penomenon as been observed in bicrystals of iron by Hutcinson [Hutcinson, 1989]. Nucleation by low angle boundary migration Tis model as been presented independently by bot Beck [Beck, 1949] and Can [Can, 1950] in te 1940s. Te dislocation density around low angle boundaries maybe relatively ig, promoting teir migration. During te migration of te sub-boundaries, te dislocations are continuously absorbed increasing te low angle boundary orientation until it is finally transformed into a ig angle boundary. Te microstructural defects beind te moving subgrain boundaries are removed or rearranged ence decreasing te stored energy. In [Rios, 2005], te autors summarized te experimental evidences for tis nucleation mecanism. In tis paper te autors also exibit tat tis mecanism usually occurs at ig strains, largely spread subgrain size distribution, ig annealing temperatures. Finally tis mecanism occurs specially in low SFE (stacking fault energy) metals. Figure 5.3 illustrates te sub boundary migration nucleation mecanism. Figure 5.3: Te sequence sows te nucleation of a recrystallized grain starting from a subgrain: (a) initial substructure; (b) te larger (middle) subgrain growt over te oter (smaller) ones and (c) an area free of defects associated to te large angle boundary tat is being formed [Rios, 2005]. Figure 5.4 presents an experimental analysis of a low angle boundary migration. In tis case, te recrystallization of a Zr-2Hf alloy deformed by planar compression test at room temperature is studied.

163 161 Figure 5.4: HVEM brigt field images of te same area (a) in te as-deformed condition (55% strain) and (b) after in situ annealing at 700 C for 5 min, revealing te rearrangement of dislocations into a subgrain structure and low angle boundary migration (wite arrows) [Zu, 2005]. Nucleation by subgrains coalescence Te mecanism of subgrain coalescence is based upon te coalescence of two adjacent subgrains. Tis coalescence is equivalent to te rotation of te grains causing te crystal lattices to coincide, as illustrated in Figure 5.5. Figure 5.5: Coalescence of two subgrains by rotation of one of tem: (a) original structure prior to coalescence; (b) rotation of te CDEFGH grain; (c) subgrain structure subsequent to coalescence and (d) final structure after sub-boundaries migration. Tis mecanism seems to be associated wit transition bands (wic consists of a cluster of long narrow cells or subgrains wit a cumulative misorientation from one side of te cluster to te oter [Humpreys, 2004]), large spread in te distribution of subgrains angles, moderate strains, relatively low annealing temperature and metals wit ig SFE. Figure 5.6 presents an experimental analysis of a low angle boundary migration. In tis case, a Zr-2Hf alloy deformed by planar compression test at room temperature is used in order to study static recrystallization.

164 162 Figure 5.6: HVEM brigt field images of te same area after in situ annealing at 700 C for 5 min (a) and 30 min (b), revealing te coalescence of subgrains. Subgrains A, B and C form a larger subgrain D [Zu, 2005]. Table 5.1 sums up some conditions for wic te different nucleation mecanisms occur. Table 5.1: Summary for nucleation mecanism. Strain induced grain boundary Sub-boundary migration migration small strains (up to 40%) ot working Nuclei growt ig strains ig temperatures eterogeneous subgrain size distribution low SFE metals and alloys Subgrain coalescence moderate strains relatively low temperatures large spread in te distribution of subgrain misorientation transition bands ig SFE metals and alloys Te expansion of new grains follows te grain boundary motion equations already discussed in capter 4. It is generally accepted tat te velocity of a ig angle grain boundary can be approximated by [Humpreys, 2004], [Bernacki, 2009], [Kugler, 2004]: v ME n (5.2) were M is te grain boundary mobility; E is te material internal energy difference, wic depends on te dislocation density; is te grain boundary energy; is te grain boundary mean curvature and n is te outward unit normal to te grain boundary. As already explained, te grain boundary driving force can be divided in two parts: te first one is related to te internal material energy gradient ( En ) and te second one is related to te grain boundary capillarity effects ( n ). Experimentally, te driving force of grain boundary motion, and terefore te boundary velocity, may not remain constant during recrystallization. In particular, te driving force may be lowered by concurrent recovery and bot te driving force and te mobility may vary troug te specimen. As a consequence, te grain boundary velocity, wic is an important parameter in any recrystallization model, is a complex function of te material and te deformation and annealing conditions. In tis work, te material recovery is not considered during te recrystallization simulations. As a consequence, te internal energy stored by dislocations during te plastic deformation will not evolve during te recrystallization simulations.

165 Te Jonson-Mal-Avrami-Kolmogorov (JMAK) model Te first and still widely used model to describe te recrystallization process based on nucleation and growt of new grains was developed in te 30 s by [Jonson, 1939], [Avrami, 1939], [Kolmogorov, 1937]. Tis model, also known as te Jonson-Mel-Avrami- Kolmogorov (JMAK) model is largely discussed in te literature. For example, in [Co, 2001], [Barracloug, 1979], [Towle, 1979], te autors use te JMAK approac to model 304L steel static recrystallization kinetics. Tis model is based on tree main ypoteses: (i) termal treatment is performed in isotermal conditions; (ii) te material presents a omogeneous state after plastic deformation; (iii) nucleation sites are assumed to be randomly distributed. Wen recrystallization begins, new grains nucleate wit a nucleation rate N and grow wit a growt rate G. Te resulting recrystallized fraction X v is given by: X ( t) 1 exp( Bt v ), (5.3) were B is a constant depending on a sape factor f, N and G (Equation 5.4), and n is te JMAK or Avrami exponent. n B fng 4 3. (5.4) Tis JMAK model proposed above assumes tat te rate of nucleation and growt remains constant during te recrystallization process. In [Avrami, 1939], te autors also considered te case in wic te nucleation rate is not constant, but a decreasing function of time, N aving a simple power law dependence on time. In tis situation, te obtained n value is different from te case were N is constant. Also, te exact value of n depends on te form of N functions. Two limiting cases are important: te first one in wic N is constant wit a tree dimensional growt and te second one, were te nucleation rate decreases so rapidly tat all nucleation events occur at te recrystallization start. Tis is termed site saturated nucleation. In tis case, N is replaced by N (number of nuclei) in Equation 5.4. Once again te n value obtained in tese conditions will be different from te n value obtained wit constant N. Finally, if te grains are constrained eiter by te sample geometry or by some internal microstructural constraint, te nuclei will grow only in one or two dimensions. As a consequence, in tese situations, te JMAK exponent will present different values summarized in Table 5.2. Table 5.2: Ideal JMAK exponents for different nucleation conditions. Site Growt Constant saturated dimensionality nucleation rate nucleation 3-D D D 1 2 In [Can, 1956], te autor as extended te teory to include nucleation at random sites near te grain boundaries and found n ranging from 4 at te start of recrystallization to 1 at te end. However, no general analytical treatment of non-randomly distributed nucleation

166 164 sites is available. In tis work, for all simulations, a site saturation nucleation condition is considered. Tis means tat for all simulations, all nuclei are created at te same time, at te beginning of te simulation L steel static recrystallization Te material studied in tis capter is 304L steel. In tis section, a brief review of te 304L steel static recrystallization beaviour is presented. All experiments presented ere were performed by Ke HUANG during is PD Tesis [Huang, 2011]. Samples ave been submitted to torsion tests (Appendix 3), and eat treated afterwards for different times. Recrystallized fractions ave been measured using te EBSD tecnique (Appendix 4) Strain level effect Besides te processing conditions (temperature and strain rate, wic will be discussed in te next subsection), te amount of applied deformation plays a role on te static recrystallization kinetics. Wen te material is deformed until iger strain levels, te amount of stored energy is iger. As a consequence, te static recrystallization kinetics will cange. Figure 5.7 presents two different static recrystallization tests, bot of tem were performed at 1000 C wit a strain rate equal to 0.01 s -1 but one sample was deformed until ε = 0.3 (blue line) and te oter until ε = 0.5 (red line). Figure 5.7: Effect of applied strain on te static recrystallization kinetics in 304L steel deformed at 1000 C to ε = 0.3 and ε = 0.5 wit a strain rate of =0.01 s -1. Comparing bot kinetics curves, we observe tat larger strain levels can decrease te incubation time needed to start te static recrystallization process and, at te same time, accelerate recrystallization kinetics. Also, as it was discussed in paragrap 1.1.1, te type of nucleation mecanism is also a function of strain. If te nucleation type canges wen te strain level increases, new spatial distributions of grains may impact te recrystallization kinetics as well Temperature and strain rate effects At temperatures were termally activated restoration processes suc as dislocation climb occur during deformation, te microstructure will be dependent on te deformation temperature and strain rate in addition to te strain. After deformation at ig temperatures te stored energy will be reduced and recrystallization will occur less readily tan after

167 165 deformation to a similar strain at low temperature. On te oter and, since te grain boundary motion velocity depends on te grain boundary mobility (Equation 4.2 capter 4) and, as it was discussed is capter 4, te mobility increases wit temperature, one can conclude tat te grain boundary motion also increases wen increasing te eat treatment temperature. So, wen deformation and static recrystallization are performed at iger temperatures, a competition between te material recovery, wic decreases te internal energy, and te grain boundary mobility, wic increases te grain boundary motion will take place. In te case of 304L steel, from Figure 5.8 (bot samples are deformed wit = 0.1 s -1 until ε = 0.3 for two different temperatures) we observe tat wen increasing te deformation and annealing temperature, te recrystallization kinetics becomes faster. So, in tis case, te effect of te grain boundary mobility is more important tan te material recovery. Figure 5.8: Effect of temperature on te static recrystallization kinetics in 304L steel deformed at 1000 C and 1100 C to ε = 0.3 wit a strain rate of =0.1s -1. Considering te strain rate effect, after deformation wit lower strain rate, te stored energy and, as a consequence, te recrystallization driving force will be reduced. Figure 5.9 describes te recrystallization kinetics for two 304L steel samples, bot of tem deformed (ε = 0.3) and annealed at 1000 C, but wit different strain rates. Figure 5.9: Effect of strain rate on te static recrystallization kinetics in 304L steel deformed at 1000 C to ε = 0.3, strain rates of =0.01 s -1 and =0.1 s -1. Te recrystallization kinetics is faster wen te material is deformed wit a iger strain rate, wic agrees wit te expectations [Humpreys, 2004]. In [Co, 2001] and [McQueen, 1995], a similar analysis of 304L steel recrystallization beaviour is presented and te results agree wit tose reported above.

168 Recrystallization simulations using a full field formulation All full field metods developed for grain growt modelling were described in details in capter 4. In tis capter a brief recall of tese metods wit teir applications in recrystallization simulations is presented. Te first metod is te probabilistic Monte-Carlo metod (MC). Wen used to model recrystallization penomena, tis metod presented te following disadvantages: - te linear relationsip between te grain boundaries velocity and te stored energy cannot be verified wen using te standard form of tese approaces; - te absence of time scale complicates any comparison wit experimental data [Rollet, 1997]; - tese metod principles are only statistically representative, meaning tat a large number of simulations need to be done in order to ensure te simulation validity. Te cellular automaton (CA) approac is anoter probabilistic metod, wic is usually used to model te primary recrystallization penomenon [Raabe, 1999]. Several pysical rules are used to locally determine te cell propagation in relation to te neigbouring cells, solving te first MC negative aspect. However, tis metod still presents a few negative points wen modelling te recrystallization process: - at present, tere is not an effective CA metod to manage new grains nucleation during recrystallization [Rollet, 1997]. In a finite element context, tree main metods are developed nowadays: Vertex (VM) Pase Field (PF) and Level-Set (LS). Altoug te VM metod was initially developed for te grain growt modelling [Nagai, 1990], tis approac was recently extended to primary recrystallization modelling, taking into account an early nucleation pase [Piekos, 2008a; Piekos, 2008b]. As for te grain growt modelling, topological events are controlled by mes adaptation rules. As a consequence, te management of topological events (like grains nucleation or grains disappearance) is not always easy. Nowadays, a VM tecnique taking into account a non-early nucleation pase does not exist. For te PF metod, despite recent developments [Takaki, 2008], modelling te nucleation pase during primary recrystallization, remains an open problem. In LS metods [Zao, 1996], [Oser, 1988], [Sussman, 1994], [Merriman, 1994], te grains boundaries are implicitly represented. Tis metod as already been used to model 2D or 3D primary recrystallization, including te nucleation stage [Bernacki, 2008], [Bernacki, 2009], [Loge, 2008]. Tis metod was first briefly described in [Bernacki, 2008] for very simple microstructures. It was ten improved to deal wit more realistic 2D and 3D microstructures and to make te link wit stored energies induced by large plastic deformations [Bernacki, 2009], [Loge, 2008]. As it was already discussed in capter 4, one of te prominent advantages of level set approaces is tat topological canges are automatically taken into account. In te case of primary recrystallization, te introduction of new nuclei during te simulation can be easily done. A simple metod to create a nucleation site is to build a new signed distance function at a desired time increment and at a given spatial position. Te new distance function can be initialized as te distance to a spere (3D) or to a circle (2D), centred around one node of te mes. Te signed distance functions of grains intersecting tese new nuclei are accordingly modified.

169 167 In tis capter, a level set metod associated wit a finite element formulation is cosen to model recrystallization and grain growt penomena not only to avoid all te limitations presented by te oter metods, but also in order to make an efficient link wit te crystal plasticity model discussed in capters 1 and 3. 2 Finite element model and level set framework In order to model te recrystallization penomenon, we must, initially, generate a digital microstructure and a set of nuclei trougout te microstructure. Te grains and nuclei are implicitly identified using level set functions and te mes is adapted around te grain boundaries. All numerical tools are described in details in capter Velocity field expression Te model presented ere is an extension of te recrystallization model proposed by [Loge, 2008], [Bernacki, 2009]. Tis extension includes te capillarity term [Fabiano, 2013] of te grain boundary displacement (Equation 5.2) into te recrystallization model, and was briefly described in [Bernacki, 2011]. Te impact of te capillarity term on static recrystallization modelling is one of te aspects wic ave not been discussed in dept, and for wic te literature is relatively quiet. Considering te global recrystallization model, te grain boundary velocity between two neigbouring grains (G i and G j ) is defined as te addition of a primary recrystallization term (PR) and a grain growt term (GG): v v ij ( x, t) v PR ij PR ( x, t) M e ij ( x, t) v ( x, t) i ( x, t) e ( x, t) j GG i n ( x, t), i v GG i ( x, t) M ( x, t) n ( x, t). i i (5.5) Until now, as described in capter 4 by te Equation 4.12 and recalled ere wit te following Equation 5.6, interfaces motion during recrystallization was modelled using a set of convection-reinitialization equations [Bernacki, 2008], [Bernacki, 2009], [Loge, 2008], [Coupez, 2000]: i ( x, t) vpr. i ( x, t) si i( x, t) 1 0 ij t, i 1,..., N 0 i ( t 0, x) i ( x) G (5.6) were vpr ij ( x, t) was null in capter 4 and now defined by Equation 5.5. At any time t, te interface Γ i of grain G i is implicitly given by te equation i ( x, t) 0. Te tird term of Equation 5.6 is a reinitialization term necessary to keep te property of te distance function 1, wic is crucial for te accuracy of te resolution. It is important to igligt tat Equation 5.6 is solved, in our formalism, wit one single velocity value at eac mes node and a given time t, in order to avoid kinematic incompatibilities due to te convection part of our formulation. Indeed, tese incompatibilities would result from Equation 5.5, were v is built according to parameters wic are specific to local features (grain G i and its neigbours). In [Bernacki, 2008], te autors propose instead a velocity expression, denoted PRij

170 168 v PR ( x, t), allowing to use a common global velocity for all grains of te microstructure. Tis velocity expression is given by: v PR ( x, t) N G G i1 N j1 j1 ( x, t) M exp( ( x, t) )( e G i j i e j ) n j ( x, t), (5.7) were α is a positive parameter [Bernacki, 2008], ( x, t) te unit outside normal of grain G j calculated as te opposite of te function gradient (wit te considered sign convention of j level-set generation), and ( x, t) is te caracteristic function of grain G i. During tis P.D. tesis, te capillarity term G i v GG i n j was introduced into te recrystallization formulation. As it was discussed in details in capter 4, te GG term depends on te grains curvature and is treated as a diffusion term in our formalism (see Eqs. 4.8 and 4.9). Finally, te problem solved in order to model te recrystallization process taking into account te capillarity effects boils down to a convection-diffusion problem: i ( x, t) vpr ( x, t). i ( x, t) Mi ( x, t) 0 t, i 1,..., NG. 0 i ( x, t 0) i ( x) (5.8) Te system is solved for all N G grains present in domain Ω. Of course, as for pure grain growt problems discussed in te previous capter, te translation of te capillarity term into a diffusive term is only true if te level set functions satisfy a distance function property 1 near te boundaries. So, to ensure tis condition, a re-initialization treatment must be done at eac time step [Oser, 1988]. Moreover te treatment in multiple junctions, detailed in capter 4 paragrap 3.2, is also performed at eac time step. Finally, a few simulations presented in tis capter were performed using te colouring tecniques (capter 4, paragrap 3.3). Te use of tis tecnique in te context of recrystallization is less systematic and more restrictive tan for pure grain growt. Tis is due to te fact tat te energy of eac container level set function must be assumed omogeneous to be used in Equation 5.7. Wen tis strategy is used, it implies tat all te grains of a considered container level-set function ave te same stored energy. It will specify for eac described simulation if tis ypotesis is verified. 2.2 Recrystallization algoritm Using all te numerical tools presented above and considering te use of te container level-set functions, we propose te following recrystallization algoritm (at eac time step): RX1 Calculation of v PR ( x, t) from Equation 5.7 (wit a double summation over te container level-set functions). RX2 Resolution of system 5.8 for all active container level set functions. RX3 Numerical treatment (Equation 4.11, capter 4) for all active container level set functions. RX4 Re-initialization step for all active container level set functions [Oser, 1988], [Coupez, 2000].

171 169 RX5 Deactivation of all container level set functions tat are negative all over te domain (meaning tat all grains of te considered container level set functions ave disappeared). RX6 Nucleation criteria: Probabilistic or deterministic rule to coose te new nucleation sites (ignoring te part of te domain tat is already recrystallized). A nucleus is assumed circular in 2D (sperical in 3D) and centred around one node of te FE mes. If at least one nucleation site is activated, te corresponding new signed distance functions are built and te container level-set functions of te existing grains, wic ave an intersection wit tese new grains, are modified accordingly. More precisely by denoting, at te considered time step, (x) a container level-set function and n (x) te level-set function corresponding to te distance to te union of te new nuclei, te modified container level-set function (x m ) is obtained, at eac integration point, tanks to te following equation: RX7 Re-mesing operation. ( x) min ( x), ( x). (5.9) m Two main remarks: wen eac grain is considered separately (no container level-set functions), te same algoritm is used by replacing te term container level-set function by level-set function, te previous algoritm is in fact adapted in te more global context of non site saturated nucleation, wic is not considered in te simulations presented in tis capter. Finally, wen only site saturated nucleation is considered, te step RX6 can be ignored and te generation of te new grains is performed only at te beginning of te simulation. 3 Results 3.1 Academic Test - Circular grain case Analysing te grain boundary motion equation (Equation 5.2) we observe tat a competition between te two driving forces (driving force due to internal energy gradients, M En, and driving force due to grain boundary capillarity effects, M n ) can take place during te recrystallization process. A first simple test case was performed in order to verify if te recrystallization algoritm correctly describes te grain boundary motion wen te two driving forces are taken into account. To do so, we ave considered a grain embedded in a media were te capillarity effect promotes te grain srinking wile te internal energy difference promotes te grain growt. Te domain size is 1 mm x 1 mm and an a priori remesing tecnique is used to adapt te mes around te grain boundary. Te initial grain radius is set to 0.2 mm and te grain is centred in te middle of te domain. Te mes size outside te adapted zone is 10 µm. In te adapted zone, te mes size in te direction perpendicular to te grain interface is 2 µm, and 10 µm in te tangential direction. Te number of elements at te simulation beginning is Bot grain boundary mobility and energy are considered isotropic and equal to one. Finally, an internal energy difference between te circular grain and te domain is imposed. Six different values of internal energy difference are tested: 2, 3, 4, 5, 6 and 7 J/mm². Figure 5.10 illustrates te test sceme and table 5.3 sums up all tested n

172 170 cases. In Figure 5.10 te black arrows represent te grain boundary motion due to internal energy difference, wile te orange arrows represent te grain boundary motion due to te grain capillarity forces. Figure 5.10: Circular grain case sceme. Te black arrows represent te grain boundary motion due to internal energy difference wile te orange arrows represent te grain boundary motion due to te grain capillarity forces. Wite lines represent te mes used. Table 5.3: Internal energy difference values tested. ΔEnergy (J/mm²) Case 1 2 Case 2 3 Case 3 4 Case 4 5 Case 5 6 Case 6 7 Knowing tat te initial grain radius is set to 0.2 mm, te initial grain curvature is equal to 5 mm -1. As a consequence, for cases 1, 2 and 3 we expect tat te circular grain will srink wit different srinking rates. For test case 4, te grain will remain uncanged and, finally, for test cases 5 and 6, we expect te grain to grow, wit different growing rates. As for te configuration of te srinking of one circular grain by pure capillarity grain growt considered in te previous capter, te problem could be summarized as a simple differential equation in time for te circle radius, but ere witout analytical solution. In order to compare te 2D-FE predictions wit a teoretical radius value, te following first-order sceme, explicit in time, was used: R tt R t te 1 t, (5.10) R were R t+δt and R t are, respectively, te grain radius at t+δt and t, ΔE is te internal energy difference and Δt is te time step. Figure 5.11 illustrates te kinetics curves for all te tested cases. Concerning te radius of te FE simulations, it was obtained, at eac time step, as an average of te distance of positions of te grain interface to te centre of te domain (eac position being separated by an angle of 45 ).

173 171 Figure 5.11: Comparison between numerical and teoretical result for te static recrystallization of a circular grain embedded in an infinite medium. For all test cases, te model correctly simulates te beaviour of te circular grain. L2 errors on te radius prediction remain smaller tan 2% (table 5.4). Table 5.4: L2 errors for te six circular grains test cases. ΔEnergy (J/mm²) L2 Error (%) Case 1 2 1,6 Case 2 3 1,4 Case 3 4 0,9 Case 4 5 0,2 Case 5 6 0,8 Case 6 7 0,9 Tese results illustrate tat te diffusive-convective approac developed ere can represent wit a very good accuracy te grain motion kinetics resulting from te competition between two driving forces: internal energy difference and grain boundary curvature. 3.2 Academic Test 2 grains case Te second test case corresponds to a bicrystal recrystallization. Once again, te grain boundary mobility and energy are isotropic and equal to one. An energy difference is imposed between te two grains. Te domain size is 1 mm x 1 mm and an a priori remesing tecnique is used to adapt te mes around te grain boundary wit te same features as tose described for te previous test case. Te number of elements at te simulation beginning is 12,300. Figure 5.12 illustrates tis second test case setup.

174 172 E = 0 E = 10 Figure 5.12: Bicrystal test case: te black vectors correspond to te recrystallization velocity at te grain interface. Te internal energy of te blue grain is null wile te internal energy of te red grain is 10 J/mm². As a consequence, te grain boundary will move towards te red grain and te blue grain will grow wile te red grain will srink. Since te grain boundary is flat, te grain boundary curvature is equal to zero: only te internal energy difference will drive te grain boundary motion, and te grain boundary will remain flat. Figure 5.13 describes te grain boundary motion obtained by solving te system 5.6. (a) (b) (c) (d) (e) (f) Figure 5.13: Bicrystal test results. Te black line represents te numerical result wile te red line corresponds to te teoretical result. (a) t = 0 s ; (b) t = 2 ms ; (c) t = 4 ms ; (d) t = 6 ms ; (e) t = 8 ms and (f) t = 10 ms. Errors between te numerical and teoretical results are smaller tan 3% for all cases (obtained by te comparison of te teoretical vertical position of te grain interface wit an average of te vertical coordinates of ten positions of te simulated grain interface). One can conclude tat te convective-diffusive approac associated wit an anisotropic mes adaptation allows ere an accurate simulation of te static recrystallization penomenon.

175 Polycrystalline 2D simulation Usually, wen modelling te recrystallization penomena, te capillarity effects are not taken into account and only te internal energy difference is considered [Monteillet, 2009], [Bernacki, 2009]. So, te first interesting point to test wit our modelling metodology is weter or not tis assumption is valid. In oter words, we want to estimate te impact of te capillarity term on te recrystallization kinetics and on te microstructure after total recrystallization. Simulations wit random nucleation sites and wit necklace-type nucleation are studied, and are restricted for te moment to 2D configurations. For all tested cases, grain boundary mobility and energy were considered isotropic trougout te domain and are equal to, respectively, m 4 /(J.s) and 0.6 J/m². Tese values correspond to 304L steel data at 1000 C Random nucleation sites In tis first recrystallization test, 636 randomly distributed nuclei are generated in a polycrystal wit 76 grains. Te energy difference between te matrix and te nuclei is omogeneous trougout te domain and it is equal to J/mm 2. In tis case, te colouring tecnique was used separately for te initial unrecrystallized grains and for te initial nuclei. Indeed, as initial unrecrystallized grains are assumed to ave te same stored energy, te grap colouring tecnique can be used witout any difficulty. Te same procedure is used for te initial nuclei (wic sare te same null energy). Finally, te initial "unrecrystallized grain - container level-set functions", modified tanks to Eq. (5.9) in order to take into account te presence of te nuclei, are concatenated wit te "nuclei - container level-set functions" to generate te initial set of container level-set functions used for te simulation. It must be noticed tat for te nuclei, te colouring tecnique is applied to te Delaunay triangulation of te sites formed by te centres of te nuclei, in contrast wit te initial unrecrystallized grains were te Delaunay triangulation is performed wit te Laguerre-Voronoï sites. Figure 5.14 illustrates a comparison of te microstructure evolution, as a function of time, between recrystallization simulations wit and witout capillarity effects. Te grap in Figure 5.15 presents a kinetics comparison of tese two cases and also te Avrami exponent analysis. Time (s) ReX wit capillarity effects ReX witout capillarity effects 0 Recrystallized fraction: 5% Recrystallized fraction: 5%

176 Recrystallized fraction: 26% Recrystallized fraction: 42% 100 Recrystallized fraction: 69% Recrystallized fraction: 80% 250 Recrystallized fraction: 100% Recrystallized fraction: 100% Figure 5.14: Microstructure evolution comparison, in function of time, between te recrystallization simulation wit capillarity effects and witout capillarity effects (304L, T= 1000 C)

177 175 (a) (b) Figure 5.15: Recrystallization kinetics comparison between a case taking into account te capillarity effects (red line) and anoter case tat do not take into account te capillarity effects (green line). Comparing te kinetics of bot cases (Figure 5.15.a) we observe tat, at te beginning of recrystallization, te kinetics of te simulation wic does take into account te capillarity effects is slower tan te one wic does not. Tis is a consequence of te competition eld between te two types of driving forces, as discussed in paragrap 3.1. Analysing a zoom of te microstructure for t = 40 s (Figure 5.16), few differences can be observed between te two simulations. (a) (b) Figure 5.16: Zoom of te microstructure for t = 40s: (a) simulation wit capillarity effects, (b) simulation witout capillarity effects. Considering te nuclei witin te grains in Fig. 5.16, we observe tat capillarity effects reduce teir size. As we discussed above, tis difference is again a consequence of te competition between te two types of driving forces. Looking now at te nuclei localized on te grain boundaries, we observe a difference between te two cases concerning teir sape. Capillarity effects lead to nuclei wic are no longer circular, but ovalized. Tis is a consequence of te equilibrium angle of 120 at a triple junction (Figure 5.17) [Bernacki, 2009], [Humpreys, 2004].

178 176 Figure 5.17: A triple junction for a nucleus on a grain boundary - equilibrium contact angle of 120. Te competition between te two driving forces is more apparent for small nuclei as teir curvature is ig. Wen te nuclei become larger, te capillarity forces are reduced, but te rate at wic recrystallization progresses remains different from te case were capillarity forces are ignored (see Fig. 5.15). In te end, owever, bot simulations reac 100% of recrystallization almost at te same time, due to an asymptotic beaviour wic appears to be faster wen including capillarity. Analysing a zoom of te microstructure for t = 250s (Figure 5.18), we observe tat generally te microstructures obtained in te two simulations are very similar. A difference comes from te existence of equilibrium angles at multiple junctions wen introducing capillarity, but in te presence of stored energy forces, tese equilibrium angles may not be satisfied completely during recrystallization (Fig. 5.14). At te end of recrystallization, we still find a few quadruple junctions, or grains wit only 3 or 4 faces (Figure 5.18), but tese features disappear progressively wit increasing annealing time, wic is not te case wen only stored energy forces act on te microstructure. (a) (b) Figure 5.18: Zoom of te microstructure for t = 250s. (a) Simulation wit capillarity effects, (b) simulation witout capillarity effects. Concerning results of Fig 5.15(b), tey are in agreement wit te JMAK model (see Eq. 5.3) as te function ln ln1/ 1 X v is a linear function of ln t for bot cases. Moreover te Avrami exponent obtained for bot cases is very close of its teoretical value of 2 for tis topological configuration (site saturated nucleation wit a growt dimensionality equal to 2 as te nucleation is random, see Table 5.2). As a conclusion, for randomly distributed nuclei, taking into account te grain curvature canges te sape of te recrystallization kinetics curve but does not cange significantly te time needed to fully recrystallize te material. Ignoring capillarity effects lead to multiple

179 177 junctions wic do not form 120 equilibrium angles, and te corresponding microstructures are terefore not very realistic Necklace-type nucleation sites It is widely recognized tat te nucleation sites in recrystallization are non-randomly distributed. Tey are often located in te grain boundaries areas. So, to verify te influence of te grain curvature on te recrystallization kinetics, 1236 nuclei distributed along te grain boundaries were generated in a polycrystal wit 100 grains. Once again, te energy difference between te matrix and te nuclei is omogeneous trougout te domain and it is equal to J/mm 2. Te same strategy used in te previous test case concerning te unrecrystallized grains and nuclei colouring can be used. Figure 5.19 illustrates a comparison of te microstructure evolution, as a function of time, between te recrystallization simulation wit capillarity effects and witout capillarity effects. Te grap in Figure 5.20.a presents a kinetics comparison of between te two cases, and Figure 5.20.b describes te Avrami exponents for bot simulations. Time (s) ReX wit capillarity effects ReX witout capillarity effects 0 Recrystallized fraction: 10% Recrystallized fraction: 10% 50 Recrystallized fraction: 33% Recrystallized fraction: 46%

180 Recrystallized fraction: 62% Recrystallized fraction: 79% 400 Recrystallized fraction: 93% Recrystallized fraction: 100% 650 Recrystallized fraction: 100% Recrystallized fraction: 100% Figure 5.19: Microstructure evolution comparison, as a function of time, between te recrystallization simulation wit and witout capillarity effects.

181 179 (a) (b) Figure 5.20: Recrystallization kinetics comparison between a case wit capillarity effects (red line) and anoter case witout te capillarity effects (green line). From Figures 5.19 and 5.20 we see tat te effects of te grain curvature on te recrystallization kinetics and geometry are more important tan in te first case (wit randomly distributed nucleation sites). Here, wen te capillarity effects are not taken into account, total recrystallization is reaced after about 400 seconds. Wen te capillarity effects are introduced, total recrystallization is reaced after 600 seconds. Tis is a consequence, once again, of te competition between te driving forces. (a) (b) Figure 5.21: Microstructure after 400s wen taking into account te capillarity effects: (a) general view and zoom of an area wit unrecrystallized and recrystallized grains. In Figure 5.21, we observe tat te local radii of recrystallized grains are small as a consequence of te initial position of te nuclei. Terefore, te capillarity effects cannot be neglected and tey continue to be significant during all te recrystallization process. Interestingly, as expected for tis topological configuration (site saturated nucleation wit a growt dimensionality equal to 1 due to te necklace-type configuration), bot simulations present an Avrami exponent around 1 (see Fig. 5.20b). In comparison to te previous test case, te capillarity effects ere seem to affect only te coefficient B of Eq. 5.3, but not te Avrami exponent. Looking at microstructures in te fully recrystallized state (Figure 5.22), we observe again te issue on te triple junction equilibrium angle of 120. Tis leads to very unrealistic sapes, as tose igligted wit te red squares in Fig. 5.22b.

182 180 (a) (b) Figure 5.22: Zoom of te microstructure for 100% recrystallization fraction: (a) simulation wit capillarity effects, (b) simulation witout capillarity effects. Te red squares igligt grains wit unrealistic sape obtained wen capillarity effects are not taken into account. As a conclusion, for necklace-type nucleation, taking into account te grain curvature affects bot te recrystallization kinetics and te fully recrystallized microstructure. Neglecting capillarity effects not only results in non equilibrium angles at triple junctions, but also to very unrealistic grain sapes. In order to study tis competition between te grain curvature and te internal energy gradients in more details, we carried out five different tests. In all cases we ave te same initial distribution as for te previous necklace-type configuration. Te grain boundary mobility and grain boundary energy are te same for all cases. Stored energy is assumed to be constant in te recrystallized part (equal to 0 J/mm 2 ), and also in te unrecrystallized part. Te internal energy value for te unrecrystallized part is canged from one test to anoter, as sown in Table 5.5. Table 5.5: Tests performed in order to study te competition between te internal energy gradient and te curvature effects in te recrystallization penomenon for a necklace-type site saturated nucleation. ΔEnergy (J/mm²) γ ( J/mm) ΔEnergy/ γ (mm -1 ) For all cases we simulate te recrystallization wit and witout te capillarity effects. Figure 5.23 describes all te recrystallization kinetics curves obtained and Figure 5.24 presents te corresponding L2 errors (or difference) between te simulations wit and witout te capillarity effect.

183 181 (a) (b) Figure 5.23: (a) Recrystallization kinetics comparison between a case taking into account te capillarity effects (Rex+GG continuous lines) and anoter case tat does not take into account te capillarity effects (Rex - dased lines). (b) Avrami exponent analysis for all tested cases. Figure 5.24: Error between te recrystallization kinetics curves obtained wit and witout te capillarity effect for te 5 studied cases.

184 182 In Figures 5.23.a and 5.24, it is interesting to observe tat te impact of te grain curvature effects decreases wen te ratio E / increases and tat te L2 error converges toward zero. Based on tese results we can estimate ow te capillarity forces will affect te recrystallization kinetics. Also, for any E /, we ave confirmed tat te capillarity effects canges only te coefficient B of Eq. 5.3, but not te Avrami exponent (Figure 5.23.b) in te considered necklace-type sites saturated nucleation. Te next step for te static recrystallization simulation consists in coupling te crystal plasticity simulation results to te recrystallization simulation. Tis is te topic of te next section. 4 Coupling crystal plasticity and static recrystallization After analysing te influence of te capillarity effects on static recrystallization penomena, te recrystallization algoritm is now coupled to te crystal plasticity model. A 100-grain digital microstructure is subjected to a plane strain compression up to ε = 0.3. For tis deformation level, te appearance of new grains as not started. So, only static recrystallization is simulated. Te calculated dislocation density distribution is used to define a set of potential nucleation sites. Ten, te dislocation density average of eac grain is computed in order to calculate te grain boundary motion rate (Equation 5.7). In te next sections, te definition of te nucleation sites is explained. Finally, te obtained numerical results are compared to te available experimental results. 4.1 Critical dislocation density and nucleus radius Static recrystallization may occur wen a deformed material is subsequently annealed. In [Kerisit, 2012], te critical dislocation density for static recrystallization is identified based on experimental tests. Tese experimental tests correspond to mecanical tests followed by an annealing treatment. However, a large number of tests, wit different strain rates at different annealing temperatures need to be performed in order to correctly estimate te critical dislocation density needed to trigger recrystallization, after a given time. In tis current work, we use te same equations used as tose found in te numerical model proposed in [Huang, 2011]. Te framework of tis model is te same as tat of te dynamic recrystallization (DRX) mean field model, wit owever some modifications due to te different nature of static recrystallization (SRX) as compared to DRX. For te DRX model, te critical dislocation density depends on deformation conditions (temperature and strain rate), and can be determined from te energy canges in relation to te formation of a nucleus on a pre-existing grain boundary. In [Sandstrom, 1975], te autors proposed a semi-quantitative approac by assuming tat DRX is only possible wen te rate of boundary migration of te potential nucleus is ig in relation to te rate of reaccumulation of dislocations beind it. Following teir model, [Roberts, 1978] furter analysed te critical condition for DRX. Based on te results presented in [Roberts, 1978], te following Equation 5.11 is proposed, using parameters of te mean field model developed in [Huang, 2011], to calculate te critical dislocation density value for nucleation in dynamic recrystallization: DRX cr 20. K K 1 3, (5.11)

185 183 were is te dislocation line energy and K M. It is observed from Equation 5.11 tat 3 DRX cr is indirectly influenced by temperature and strain rate troug te parameters K 1 and K 3. As it was discussed in capter 3 (Yosie-Laasraoui-Jonas equation), K 1 is a material DRX parameter representing te material ardening. Temperature variation of cr is mainly dictated by K 3 troug te temperature dependence of M (Equation 4.2, capter 4). Te influence of strain rate is incorporated only by te variations of K 1. In general, Equation 5.11 expresses tat te critical dislocation density increases wit decreasing temperature and increasing strain rate, wic is pysically justified. However, te critical dislocation density obtained using Equation 5.11 neglects te DRX material recovery. So, in order to take it into account, cr is finally defined as a solution of te following equation: DRX cr K2 2 K3 K2 ln 1 cr K1 DRX 1 2, (5.12) were K 2 is a material parameter representing te material recovery (as it was discussed in DRX details in capter 3 - Yosie-Laasraoui-Jonas equation). Te initial guess value of cr is obtained directly from Equation 5.11 and iterative calculation by Equation 5.12 leads to a DRX converged value of cr wic is ten used in te model. In te current work, te critical dislocation density for static recrystallization process, DRX, is related to te using te following equation: SRX cr cr. SRX cr DRX cr, (5.13) were is a constant cosen equal to 0.3 according to te analysis presented in [Huang, 2011]. A nucleus becomes viable wen its radius reaces a critical value. Tis SRX R cr corresponds to te condition wen te stored energy of te material is large enoug to overcome te capillarity force of te nucleus, so we ave: SRX cr. 2 R SRX cr SRX 2 i.e. Rcr. SRX (5.14). cr To overcome te problem tat a nucleus may srink soon after its creation, all te nuclei SRX created in te REV ave a radius bigger tan R cr, tereby ensuring tat a created nucleus as te needed driving force to grow. 4.2 Number of nucleation sites It is assumed tat new grains appear in areas were te dislocation density is greater tan a critical value. As a consequence, in te mean field teory, te nucleation of new cr

186 184 grains will appen in grains presenting a dislocation density value iger tan te critical dislocation density. Since te nucleation rate is difficult to evaluate experimentally [Humpreys, 2004], it is presumed tat a certain percentage per unit time of te potential nucleating sites actually nucleates. If nucleation appens in te bulk, te number of potential nucleating sites N p will be proportional to te total volume of grains wit a dislocation density SRX iger tan cr. If nucleation appens mainly at grain boundaries, te total surface of te same grains will be considered instead of te volume. In te general situation, te number of potential nucleation sites N p is terefore written as N p i cr N pi i cr R q i, (5.15) were q = 2 in case of necklace-type nucleation and q = 3 in case of bulk nucleation. R i is te grain radius of te i t grain family presenting a dislocation density value iger tan te critical dislocation density. Based on experimental 304L analysis [Huang, 2011], we observe tat, during static recrystallization, te 304L steel new grains appear near te grain boundaries, like a necklace-type nucleation. So, te value q = 2 is cosen. Equation 5.16 is ten applied to eac representative grain presenting a dislocation density value iger tan te critical dislocation density in order to determine te number of nucleation sites: N i, nucl K S g cr k cr k k k bg q Niri i cr q N r cr bg. (5.16) were bg is a constant wic is cosen to be 3, as found in [Monteillet, 2009], and K K ( T, ) a probability constant depending on te processing conditions, S cr te total g g surface area of grains wit, and N k te number of grains composing te k t cr family of SRX grains. All nuclei are assumed to start wit te same initial radius R cr. Te K g value used in tis work was identified by [Huang, 2011]. 4.3 Coupling wit crystal plasticity results Previous equations allow evaluating te critical dislocation density and te volume of new grains appearing during te static recrystallization process. Te next step is ten to couple te static recrystallization simulations to te crystal plasticity results. As it was discussed in capter 3, in tis work, two different ardening models ave been studied: te first one considering te material total dislocation density, and a second one considering two dislocations types: statistically stored dislocations (SSD) and geometrically necessary dislocations (GND). In tis paragrap, te analysis of nucleation sites is done for bot ardening models and te influence of te results on te recrystallization kinetics is presented. For bot cases te applied deformation is 0.3 and te annealing treatment temperature is 1000 o C (still referring to 304L steel). Initially, te critical dislocation density for te mean field model and te number of new grains are calculated. Using te parameters presented on table 5.6 (for confidentiality, te parameters are presented as letters) te critical dislocation density calculated using Equations 5.11, 5.12 and 5.13 is equal to m -2 and te number of grains to be generated, calculated using Equation 5.16, is 2060.

187 185 Table 5.6: 304L steel parameters. K 1 (m -2 ) A K 2 B K 3 C K g D τ (J/m) E (ms -1 ) 1 It is important to igligt tat te critical dislocation density calculated wit Equations 5.11, 5.12 and 5.13 are valid for te mean field models. Te value needs to be adapted to te mesoscopic scale, based on te crystal plasticity formulation. As it was discussed in details in capter 3, it is possible to assume tat te dislocation density, at any point of te grain is defined wit te following equation: ( )/ R, ( x)/ R, x, (5.17) were ( x )/ R, a function of te deformation and of te distance to te closest grain boundary normalized by R, te polycrystal mean grain size. Tis function is described in Fig for ε = 0.3 wit and witout te introduction of GNDs. is te mean dislocation density in te RVE and Figure 5.25: 304L steel β function as a function of te normalized distance to te grain boundary for ε = 0.3 at T = 1000 o C. Knowing tat for te 304L steel, nucleation takes place mainly near te grain boundaries, we can calculate te mesoscopic critical dislocation density using Equation 5.17 from te knowledge of te function at te grain boundary. Based on Figure 5.26, we obtain 0, and 0, Using Equation 5.17 and bot β values, we find 03 SSD SSDGND 14 SSD cr ( 0.3) m -2 GND 14 and cr ( 0.3) m -2. To coose te nucleation sites in te mesoscopic simulation, we coose only te nodes presenting a dislocation density iger tan tese values. Figure 5.26 presents te dislocation density dispersion for te two ardening models, togeter wit te mean field critical dislocation density and te mesoscopic critical dislocation density.

188 186 (a) SSD (b) SSD+GND Figure 5.26: Dislocation density dispersion compared to te mean field and mesoscale critical dislocation densities. Te number and distribution of possible new nucleation sites based on te crystal plasticity simulations results are different for te two ardening models. Considering only te SSDs dislocations (Figure 5.26.a), te possible nucleation sites are found all over te material and not only near te grain boundary. Also, alf of te material sows a dislocation density iger tan te critical mesoscopic dislocation density. On te oter and, in Figure 5.26.b we observe tat te possible nucleation sites are only concentrated near te grain boundaries as only tis zone presents a dislocation density iger tan te critical mesoscopic dislocation density. For bot cases, te number of nodes presenting a dislocation density iger tan te mesoscopic critical dislocation density is larger tan te 2060 nuclei expected in te RVE. So, two different criteria are set in order to coose te nucleation sites. Te first one takes te dislocation density and te distance to te grain boundary into account and te second one considers only te dislocation density value. Table 5.7 sums up tese criteria. Case 1 Case 2 Table 5.7: Nucleation test cases description. - te first 2060 nodes wit a normalized distance to te grain boundary smaller tan 0.075; and wit te igest dislocation density values. - te first 2060 nodes presenting te igest dislocation density values. Figure 5.27 sows te first 2060 nodes taken into account for eac case and for eac ardening model. Eac node corresponds to a new nucleus.

189 187 (a) (b) Figure 5.27: Dislocation density dispersion grap wit te description of te nucleation sites for two types of ardening models: (a) only SSDs and (b) bot SSDs+GNDs. Wen only SSDs are taken into account, te new grains positions are quite different for bot tested cases. For case 1, te new grains mean normalized distance to te grain boundary is equal to 0.057, wile for case 2, te normalized distance becomes Wen bot SSDs and GNDs are taken into account, te new grains mean normalized distances are and 0.075, respectively for cases 1 and 2. Tey are terefore closer to eac oter. In te next paragrap, te influence of te coice of nucleation sites on te recrystallization kinetics is studied D recrystallization kinetics results in site saturated nucleation conditions Te static recrystallization simulation was performed for te two test cases defined in te previous section and for bot ardening models. In tese simulations, all te new grains are represented initially using te same level set function. As a consequence, only te recrystallization front is simulated (not te individual nuclei). Figures 5.28 and 5.29 present te recrystallization evolution wen only SSDs are taken into account and Figures 5.30 and 5.31 present te recrystallization evolution wen bot SSDs and GNDs are taken into account. X v Internal Energy Recrystallization front 0%

190 188 50% 75% Figure 5.28: 3D recrystallization simulation based on te crystal plasticity simulation results considering only SSDs and for Case 1 concerning te nuclei positions. X v Internal Energy Recrystallization front 0% 50%

191 189 75% Figure 5.29: 3D recrystallization simulation based on te crystal plasticity simulation results considering only SSDs and for Case 2 concerning te nuclei positions. X v Internal Energy Recrystallization front 0% 50% 75% Figure 5.30: 3D recrystallization simulation based on te crystal plasticity simulation results considering bot SSDs+GNDs and for Case 1 concerning te nuclei positions.

192 190 X v Internal Energy Recrystallization front 0% 50% 75% Figure 5.31: 3D recrystallization simulation based on te crystal plasticity simulation results considering bot SSDs+GNDs and for Case 2 concerning te nuclei positions. Te recrystallization kinetics summarizing te above figures are compared to te experimental data in Figure (a) (b) Figure 5.32: Recrystallization kinetics simulation results compared to te experimental results wen (a) only SSDs are taken into account and wen (b) bot SSDs + GNDs are taken into account.

193 191 In Figure 5.32.a, wen considering only SSDs, we observe tat te recrystallization kinetics is not te same for case 1 and case 2. Comparing bot cases wit te experimental data, we observe tat only case 1 correctly predicts te experimental static recrystallization kinetics. For case 2, te recrystallization kinetics is slower tan te experimental data. From Figures 5.27.a, 5.28 and 5.29 we observe tat te nucleation sites distribution is different for cases 1 and 2. Wile for case 1 all nuclei are localized near te grain boundaries; for case 2, te nuclei are more spread inside te grain. So, tese results sow, once again, tat te recrystallization kinetics is strongly dependent on te spatial distribution of nuclei. Based on tese results, one can conclude tat, wen only SSDs are taken into account, not only te dislocation density but also te distance to te closest grain boundary must be taken into account in order to define a nucleation model allowing to predict correctly te recrystallization kinetics. In Figure 5.32.b, considering te SSDs+GNDs ardening model, we observe tat bot cases results correctly reproduce te experimental recrystallization kinetics. Analysing Figures 5.27.b, 5.30 and 5.31 we observe tat for bot cases, te nucleation sites are localized near te grain boundaries. One can conclude tat, wen a static recrystallization model is based on crystal plasticity calculations considering bot SSDs and GNDs, a simple dislocation density criterion is sufficient to correctly predict te nucleation sites positions, and te corresponding recrystallization kinetics. As seen in Figure 5.27, te mesoscale critical dislocation density calculated from crystal plasticity simulations considering only te SSDs is muc smaller tan te nucleation sites dislocation density. Wen considering bot SSDs and GNDs, te mesoscale critical dislocation density tis time corresponds to te minimum value found in te selected nucleation sites. It can terefore be reasonably concluded tat taking into account bot dislocation densities is important to correctly predict nucleation in static recrystallization. In te above simulations, only te recrystallization front is followed and, as a consequence, only te recrystallization kinetics is studied. In order to study te final grain size distribution after te recrystallization process, several level set functions are needed to represent te evolution of te individual grains. Te study of te grain size distribution after te recrystallization process is left for future work. It would also be interesting to perform similar analyses for different deformation levels in order to validate te current conclusions. 5 Conclusion In tis capter, a finite element model was proposed in order to simulate te static recrystallization penomenon. Grain boundaries are implicitly represented using level set functions. A convective-diffusive approac is used to simulate te grain boundary motion taking two driving forces into account: (1) an excess of energy due to grain boundary itself (grain boundary curvature) and (2) a free energy difference between te adjacent grains due to energy stored during deformation. Te convective part of te model represents te grain boundary motion due to te free internal energy gradients and te diffusive part represents te grain boundary motion due to te grain boundary curvature. Two academic test cases wit well-known results ave been presented in order to illustrate te capability of te proposed convective-diffusive approac to finely model te considered penomena. In a second part, te effect of introducing capillarity forces in site saturated nucleation conditions is studied. Considering randomly distributed nucleation sites, it is observed tat taking into account te grain curvature impacts te sape of te recrystallization kinetics curve (even toug te JMAK exponent are te same for bot simulations) but do not cange significantly te time needed to fully recrystallize te material. However, te microstructure

194 192 obtained wen ignoring te capillarity effects fails to impose te 120 equilibrium angle at te triple junctions. As a consequence, te microstructures contain grains wit unrealistic and/or unstable sapes. In necklace-type nucleation conditions, taking into account te grain curvature driving force affects bot te recrystallization kinetics and te microstructure, but te Avrami exponent of te JMAK model is, tis time, not affected. From a global perspective, it is demonstrated tat te influence of te grain curvature decreases wen increasing te ratio between te magnitude of te internal energy gradients, and te boundary energy. In a tird part, te static recrystallization model is coupled to te crystal plasticity simulation results from capter 3. A nucleation criterion is borrowed from tat introduced in te mean field model developed in [Huang, 2011]. Te influence of te dislocation density distributions obtained wit two ardening models is evaluated based on te capacity to determine nucleation sites. Te corresponding static recrystallization kinetics are compared to experimental data obtained in previous work [Huang, 2011]. It is concluded tat considering a ardening model wic introduces geometrically necessary dislocations is important to correctly predict nucleation in static recrystallization, as well as te recrystallization kinetics. In order to study te final grain size distributions after te recrystallization process, several level set functions are needed in order to represent te evolution of te individual grains. Tis is left to future work. Oter extensions of te present work include te modelling of dynamic recrystallization. Résumé en français Dans ce capitre, un modèle de recristallisation à l'écelle mésoscopique basé sur une formulation éléments finis dans un cadre level-set est proposé. Les deux forces motrices du pénomène de recristallisation sont prises en compte. Une approce convective-diffusive est utilisée afin de modéliser le déplacement des joints de grains. La partie convective correspond à la migration des joints de grains liée aux gradients d énergie stockée sous la forme de dislocations tandis que la partie diffusive, développée dans le capitre précédent, correspond à celle liée à la capillarité des joints de grains. Plusieurs cas tests académiques sont présentés afin de valider l algoritme proposé. Ensuite, l influence de la prise en compte des effets capillaires est étudiée à partir de la simulation de la recristallisation statique de microstructures présentant différentes distributions de germes en site saturé. Il est par exemple illustré que pour une distribution aléatoire des germes, la prise en compte des effets capillaires a peu d influence sur le temps total de recristallisation mais modifie de manière significative les paramètres du modèle JMAK. Pour une distribution en collier, les effets capillaires influencent alors le temps total de recristallisation et l exposant d Avrami du modèle JMAK. Finalement, l algoritme de recristallisation statique est couplé aux résultats des simulations de plasticité cristalline (capitre 3). La recristallisation d un écantillon déformé de 304L jusqu à 0.3 à 1000 C est étudiée et les prédictions numériques sont confrontées aux résultats expérimentaux. L influence du type de modèle d écrouissage utilisé dans le modèle de plasticité cristalline sur la cinétique de recristallisation statique est discutée principalement pour les critères de germination qui en découlent.

195 193 Conclusions and future work

196 194 Te full field modeling of metallurgical penomena wic occur during termal and mecanical processes is nowadays of prime importance in order to provide correct predictions of crystallograpic orientations (textures) and grain size distributions. Indeed, metallurgical transformations are complex penomena and, generally, teir experimental study is not easily done. Te numerical work performed in tis PD tesis, dedicated to te development of a new full field FE formalism for recrystallization and grain growt, can be summarized in tree main parts: (i) te generation and te immersion of statistical digital microstructures in a finite element context (capter 2); (ii) te crystal plasticity modeling (capters 1 and 3) and (iii) te grain boundary motion modeling (capters 4 and 5). Considering te digital microstructures statistical generation, two different generation metods were presented: Voronoï and Laguerre-Voronoï. Based on te results discussed in tis document, we can conclude tat te use of a Laguerre-Voronoï metod allows te generation of a microstructure tat obeys te experimental grain size distribution wit a very small L2 error, even for a digital microstructure generated wit a reasonable number of grains. Moreover, it was also proved tat te well-known and widely used Voronoï metod does not allow to obey a given grain size distribution. Even toug te mean grain size is well defined, te error between te numerical and te experimental grain size distribution can be very large, even for a microstructure wit a ig number of grains. Tese observations were verified for 2D and 3D microstructures. In a second part, a topological and morpological study of seven different digital microstructures, generated using a Laguerre-Voronoï metod, was also presented. Te ability of tis approac to generate, in 2D, equiaxed microstructures and to obey te well-known SLR (special linear relationsip) of Abbruzzese [Abbruzzese, 1992a] was described. Future work will consist in extending tis study to 3D Laguerre- Voronoï digital microstructures. Te process of plastic deformation of FCC metals was briefly reviewed so as to underline te resulting intergranular and intragranular eterogeneities, from crystal plasticity simulations. Te crystal plasticity finite element model used in tis work was presented and analysed. Considering te material ardening, pysically-based models were empasized, focusing on te use of one or several dislocations densities as primary variables. Te account of size effects was discussed, distinguising between statistically stored dislocations (SSDs), and geometrically necessary dislocations (GNDs). A simple model (Yosie-Laasraoui-Jonas [Laasraoui, 2009]) was adopted to estimate te SSD density evolution wit plastic deformation, wile te Busso model [Busso, 2000] was cosen to evaluate te contribution of te GND density. Two different mecanical tests were used to validate te crystal plasticity model: a cannel-die ot compression test of a 304L steel for a 100-grains polycrystal, and a simple uniaxial compression test of a tantalum 6-grains oligocrystal. For bot mecanical tests, te two investigated ardening laws were compared. Considering te first test case (304L polycrystal), te ardening models correctly predict te stress-strain experimental beaviour. Te dislocation density distribution calculated wit te first ardening law (only considering te total dislocation density) is more omogeneous tan te dislocation density distribution computed wit te model wic takes into account bot SSDs and GNDs. Even toug te dislocation distribution is different, for bot cases te dislocation density dispersion is increased near te grain boundaries, were te igest and lowest dislocation density values are found. Wen te strain distribution is analysed, we observe tat bot models present similar results. Te strain intragranular distribution as a function of te distance to te closest grain boundary is rater omogeneous. Once again, te strain dispersion is more important near te grain boundaries. Based on tese results, nucleation of new grains during te

197 195 recrystallization process is expected to take place near te grain boundaries, as experimental results sow. For te second test case (tantalum oligocrystal), we observe tat te oligocrystal sape induced by plastic deformation is correctly predicted for four grains, out of six. Once again, te dislocation density distribution exibits concentration near te grain boundaries. Wen comparing te two investigated ardening models, dislocation density near te grain boundaries is iger wen bot SSDs and GNDs are taken into account. Te material texture canges for zz 0. 5 predicted wit te crystal plasticity model overestimate te experimental results, but tis can be partly attributed to te difficulty in comparing similar areas. In one grain were numerical and experimental areas are expected to be close to eac oter, te texture canges are well predicted, including te extent of grain fragmentation. Te fragmentation is observed not only for te crystallograpic orientations, but also for dislocation density and strain. Finally, considering grain boundary migration, two penomena were studied. Te first one is te grain growt penomenon were te decrease of te interfacial energy is te driving force of te grain boundary migration. An improvement of te level-set formalism developed at CEMEF for primary recrystallization modelling [Bernacki, 2009], [Logé, 2008], was proposed in order to take into account te capillarity effects. Te ability, in te considered isotropic mobility and interface energy context, of te proposed diffusive approac to simulate very accurately academic test cases, to deal wit complex 2D configurations and to be connected wit primary recrystallization simulations was illustrated. Tis numerical development was used to discuss te impact of an initial grain size distribution on te grain growt penomenon in a single pase material. Based on full field simulations, te validity of two grain growt mean field models was discussed. It was illustrated tat, in general, te simple model of Burke and Turnbull is not predictive, contrary to te Hillert/ Abbruzzese model. Te latter approac was sown to be predictive for all considered grain size distributions, even for te more complex ones. Consequently, if te development of full field models is justified for te description of grain growt in complex conditions, teir use appears disproportionate in te simple configurations investigated ere, were te Hillert/ Abbruzzese approac is sufficient. By lack of time, te same assessment was not performed in 3D but suc a study represents a sort-term extension of tis work. As explained above, te diffusion formulation accounting for capillarity effects was added to te pre-existing primary recrystallization algoritm developed in [Bernacki, 2009], [Logé, 2008]. Once again, tis new model was validated for academic and complex test cases. Wit tese developments, it was mainly igligted tat te classical ypotesis wic consists in neglecting te capillarity term wen stored energy gradients and nucleation of new grains are present, can be very armful concerning te predictions of te grain sapes, but also for statistical results suc as te grain size distribution or te recrystallized fraction. Tis result is amplified wen necklace-type site saturated nucleation is considered. It was also proved tat te influence of te capillarity effects decreases wit te increase of te stored energy gradient/boundary energy ratio. Te static recrystallization model as been finally coupled to te crystal plasticity simulation results. A nucleation criterion is borrowed from tat introduced in te mean field model developed in [Huang, 2011]. Te influence of te dislocation density distributions obtained wit two ardening models is evaluated based on te capacity to determine nucleation sites. Te corresponding static recrystallization kinetics are compared to experimental data. It is concluded tat considering a ardening model wic introduces geometrically necessary dislocations is important to correctly predict nucleation in static recrystallization, as well as te recrystallization kinetics.

198 196 Finally, if te developments performed in tis work ave allowed focusing on different fine metallurgical questions, tey ave also permitted to build a robust and precise FE framework concerning te modelling of recrystallization. Besides te improvement of te numerical cost, wic is still a limitation of te proposed metodology for 3D simulations wit a large number of grains, current oter developments concerning tese numerical tools are dedicated to Zener pinning penomenon, anisotropy of mobility and boundary energy, twinning appearance and disappearance/impact of te twins in te recrystallization kinetics, and dynamic recrystallization modelling.

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209 207 Appendix 1: Saltykov metod Considering tat te grains are represented by teir equivalent speres, te Saltykov metod is dedicated to te question of ow to work backward from a grain size distribution (bar-plots) of two-dimensional sections of a material to a tree-dimensional grain size distribution (discrete) of it. Te problem is furter complicated by te fact tat for any positive value d, all te speres wit a diameter superior to d can contribute to te circles of diameter d obtained in a section of te considered 3D particle. Now, considering te following notations in te discretization of te enunciated problem: - Δ is defined as te ratio of te maximum diameter, Dmax 2Rmax, to te total number of grains families considered, m. Ten, te diameter of te 3D-particles belonging to te j t group is equal to jδ wit D max m. - N A (i), i 1,...,m, is defined as te number, per unit area, of 2D-grains wit a diameter between (i-1)δ and iδ. N A denotes te corresponding vector. - ( j) j 1,...,m, is defined as te number of 3D-grains, per unit volume, wit N V, a diameter equal to jδ,. N V denotes te corresponding vector. - N A ( i, j), i, j1,, j1,, m, is te contribution of te 3D-grains from 3Dclass j to te 2D-class i. Te number of grains in 2D-class i can be written as: N A m ( i) N ( i, j). (A.1) j1 A Now, wit te notations of te Figure A.1 wic describes a 3D-grain of radius j i 1,..., j if tis grain is intercepted 2 r j : tis grain will contribute to te 2D class i 2 2 between i-1 and i wit r r k 1 j. k j k,.., Figure A.1: Geometry involved in te intersection of a spere of radius r j by a plane witin te slice.

210 208 Tus, te contribution of grains from 3D-class j to te 2D-class i is given by: N A( V i 1 i i, j) 2N ( j)( ), (A.2) N A were te factor 2 is a consequence of considering bot grain emisperes. Ten, ( i, j) 2N ( j) wit teconvention k V rj ri 1 rj ri 2NV ( j) j ( i 1) j i ij j 2 ( i 1) 2 j 2 i 2, ( i, j) 1,..., j1,..., m. N V ( j) k ij (A.3) It is interesting to igligt tat k ij coefficients depend only on te families size index i and j. By combining Eqs.(A.1) and (A.3), te upper triangular matrix, K, composed by te k coefficients allows to exibit a 2D grain size distribution (bar-plots) based on te real 3D ij grain size distribution (discrete): N A i.e. N ( i) A m N ( i, j) V m k N ( j) ( K. N A ij V V i j 1 j1 (A.5) K. N ) As K is an invertible matrix, indeed, m m (2m)! det( K ) kii 2i 1 0 (A.6) m 2 m! i1 i1 te purpose of tis annex, i.e. to be able to build a tree-dimensional grain size distribution (discrete) tanks to a grain size distribution (bar-plots) of a two-dimensional section, is obtained by an inversion of Eq A.5: N 1 1 V K N A. (A.7) 1 Te Table A.1 illustrates te values of te matrix K for m 15. If te use of te Saltykov metodology is based, of course, on an important number of ypoteses as te coice of m, te representativeness of te 2D section used to generate te 3D distribution and te approximation of te grain sape by sperical particles, it remains owever a very useful tool to build a first approximation of a 3D distribution tanks to 2D data.

211 209 Table A.1: Coefficients of te Saltykov matrix 1 K for m= e e-2-1.3e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-1

212 210 Appendix 2: 304L crystallograpic orientation of grains Te grains crystallograpic orientation in a polycrystalline sample can be defined in different ways. Te most common orientation representation is based on te tree Euler angles and is expressed as g g 1,, 2 (symbol g is traditionally used for denoting te orientation). Te tree angles correspond to te successive rotations according to a specific sequence (Bunge, Roe, symmetric). In tis work, te Bunge convention is used to represent te grains crystallograpic orientations. φ 1 ( o ) Ф( o ) φ 2 ( o ) φ 1 ( o ) Ф( o ) φ 2 ( o ) φ 1 ( o ) Ф( o ) φ 2 ( o )

213 211 Appendix 3: Torsion test In te torsion test, te samples are submitted to a rotating load, as illustrated in Figure A.2. Te experimental results presented in tis report were obtained using samples presenting a cylindrical sape. Figure A.2: Torsion test sceme. Using torsion tests we are able to reac large strains before plastic instability occurs. In addition to tis, te samples do not undergo significant sape cange as tey are deformed as long as te gage section is restrained to a fixed lengt. On te oter and, te interpretation of torsion test data is more complex tan tat used for axial testing metods due to te fact tat te strain and strain rate vary linearly wit te sample radius. In torsion tests, te stress-strain curves are derived from te torque-time data recorded automatically by te torsion test system. Assuming tat te deformation along te rotation axes is omogeneous, te equivalent strain is calculated as: ( R) t 0 2 NR dt 3 L 2 NR, (A.8) 3 L were N is te number of laps, N is te rotation speed (Rad/s), L is te gauge lengt (mm) and R is te gauge radius (mm). Te equivalent tensile stress (R) is obtained using te Fields and Backofen [Fields, 1957] analysis: R 3T ( R) n~ m~ 3 3, (A.9) 2R were T is te torque (N.m), m ~ is te slope of te straigt line T f lnn ln, for a given lnt f ln N, for a given rotation speed N. In oter words, m ~ and n ~ correspond respectively to: number of laps N, n ~ is te slope of te straigt line T m~ ln ln N N and T n~ ln. (A.10) ln N N

214 212 It is interesting to notice te ypoteses of te Fields and Backofen [Fields, 1957] analysis: te materials are omogeneous and isotropic (material ardening is represented by an isotropic ardening law). In oter words, te sample cylindrical sape does not cange during deformation; plastic deformation is omogeneous trougout te sample; te deformation is uniform along te specimen (no flow localization); te cross sections are straigt and move as a rigid body motion wit speed ω.

215 213 Appendix 4: EBSD test Electron backscattered diffraction (EBSD) is based on te acquisition of diffraction patterns from bulk samples in te scanning electron microscope (SEM). Te EBSD acquisition ardware comprises a sensitive CCD camera and an image processing system for pattern averaging and background subtraction. Figure A.3 presents a scematic diagram illustrating te main components of an EBSD system. Figure A.3: Scematic diagram of a typical EBSD installation in an SEM [Humpreys, 2004]. Te EBSD acquisition software (TSL OIM data collection 5) is used to control te data acquisition, to solve te diffraction patterns and to store te data. In order to analyse, manipulate and display all te stored data, te TSL OIM Analysis version 5.0 software is used in tis work. EBSD is carried out on a specimen wic is tilted between 60 and 70 from te orizontal and a series of data points are obtained by scanning te electron beam across te sample.