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Micromechanical modeling and simulationsof transformation-induced plasticity

in multiphase carbon steels

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This work is part of the research program of the Netherlands Institute for Met-als Research (NIMR) and the Stichting voor Fundamenteel Onderzoek der Ma-terie (FOM), nancially supported by the Nederlandse organisatie voor Weten-schappelijk Onderzoek (NWO). The research is carried out under project number02EMM20 of the FOM/NIMR program Evolution of the Microstructure of Ma-terials (P-33).

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Micromechanical modeling and simulationsof transformation-induced plasticity

in multiphase carbon steels

PROEFSCHRIFT

ter verkrijging van de graad van doctoraan de Technische Universiteit Delft,

op gezag van de Rector Magnicus prof. dr. ir. J.T. Fokkema,voorzitter van het College voor Promoties,

in het openbaar te verdedigen op dinsdag 29 januari 2008 om 10 uur

door

Denny Dharmawan TJAHJANTOingenieur toegepaste wiskundegeboren te Cirebon, Indonesi e

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Dit proefschrift is goedgekeurd door de promotor:Prof. dr. ir. S. van der Zwaag

Toegevoegd promotor:Dr. S.R. Turteltaub

Samenstelling promotiecommissie:Rector Magnicus, VoorzitterProf. dr. ir. S. van der Zwaag, Technische Universiteit Delft, promotorDr. S.R. Turteltaub, Technische Universiteit Delft, toegevoegd promotorProf. dr.-ing. D. Raabe, Max-Planck-Insitut f ur EisenforschungProf. dr. ir. M.G.D. Geers, Technische Universiteit EindhovenProf. dr. ir. T. Pardoen, Universit e Catholique de LouvainProf. dr. ir. L.J. Sluys, Technische Universiteit DelftDr. ir. A.S.J. Suiker, Technische Universiteit Delft

Dr. ir. A.S.J. Suiker heeft als begelieder in belangrijke mate aan de totstandkomingvan het proefschrift bijgedragen.

Trefwoorden:Martensitic transformation, Crystal plasticity, Transformation-induced plasticity,Thermo-mechanical framework, Consistent stress-update algorithm, Finite ele-ment method, Homogenization scheme, Microstructural properties

Copyright c 2007 by D.D. Tjahjanto

Printed in the Netherlands by PrintPartner IpskampISBN-13: 978-90-9022499-2

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To my parents and my brothers

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Foreword

This thesis summarizes the four-year research project I have done on the designof optimized multiphase transformation-induced plasticity (TRIP)-assisted steels.The work is part of a joint research program between the Netherlands Institutefor Metals Research (NIMR) and the Stichting Fundamenteel Onderzoek der Ma-terie (FOM), which is nancially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO). The research is performed under projectnumber 02EMM20 of the FOM/NIMR program Evolution of the Microstructureof Materials (P-33).

First of all, I would like to gratefully acknowledge Prof. Sybrand van derZwaag as the promotor for the effective support and guidance during this re-search, and Dr. Sergio Turteltaub and Dr. Akke Suiker, who have provided anexcellent day-to-day supervision and many inspirations. In addition, I would liketo acknowledge Dr. Pedro Rivera for all discussions and feedbacks on the thermo-dynamical and metallurgical aspects of the models, and Prof. Ren e de Borst forthe opportunity to use the research facilities in the Engineering Mechanics (EM)group. Furthermore, I would like to thank Prof. Dierk Raabe, Dr. Franz Rotersand Dr. Philip Eisenlohr for offering me a wonderful place during a three-monthvisit to the Max-Plank-Institut f ur Eisenforschung (MPI-E) in Dusseldorf, and fortheir assistance during this visit.

Next, I would like to express my gratitude to Prof. Marc Geers (Eindhoven

University of Technology), Prof. Thomas Pardoen (Universit e Catholique de Lou-vain) and Prof. Bert Sluys (Delft University of Technology) as the members of thedoctoral committee, as well as to Prof. Gijs Ooms (Delft University of Technol-ogy) as the reserve member. Furthermore, I would like to acknowledge the discus-sions with Prof. John Bassani (University of Pennsylvania) on the basic conceptand the implementation of the non-glide stress effect in BCC crystals. In addition,I address my gratitute to the fellow researchers in the NIMR Cluster 5 and to theCorus Research Development and Technology (RD&T) team for the discussions

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FOREWORD

and feedback during this research.I owe many thanks to Carla Roovers, Harold Thung and Laura Chant for the

wonderful assistance to solve administrative and technical issues. In addition, I amindebted to all colleagues and former colleagues at the EM group (Prof. MiguelGuti errez, Dr. Steven Hulshoff, Dr. Harald van Brummelen, Dr. Christian Mich-ler, Dr. Edwin Munts, Dr. DooBo Chung, Dr. Olaf Herbst, Thomas Hille, ClemensVerhoosel, Andr e Vaders, Marcela Cid, Juliana Lopez, Wijnand Hoitinga, Gertjanvan Zwieten, Jingyi Shi, Kris van der Zee and Ido Akkerman) and at the Funda-mentals of Advanced Materials (FAM) group (Dr. David San Martin and Dr. DotyRisanti) for creating a pleasant atmosphere and interesting discussions.

Personally, I would like to deeply thank Angelica Tanisia, Fr. Ben Engel-

bertink, Rev. Waltraut Stroh and Kasia my virtual sister Wac for all support,courage and motivation that were given during the last couple of years. I amalso grateful to the Indonesian community in Delft (particularly, Julius Sumihar,Ferry Permana, Sinar Juliana, Dwi Riyanti, Xander Campman, Nelson Silitonga,Iwan Kurniawan, Henri Ismail, Sandy Wirawan and Yuli Tanyadji) and friends inthe International Student Chaplaincy Delft (especially, Ruben Abellon, Fr. AvinKunnekkadan, Francesca Mietta, Carmen Lai, Ludvik Lidicky, Anna DallAcqua,Maria Parra, Henk van der Vaart and Mieke and Reini Knoppers) for sharing a lotof fun during my stay in Delft. Last but not least, I would like to thank my family

and friends in Indonesia, for their long-distance support and prayers.All in all, I wish that this thesis gives valuable knowledge and insight to all

people interested in studying the TRIP effect in steels. Enjoy reading!

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Contents

Contents ix

1 Introduction 11.1 Background: Multiphase TRIP-assisted steels . . . . . . . . . . . 2

1.1.1 Two-stage heat-treatment process for TRIP steels . . . . . 31.1.2 Martensitic transformation in low-alloyed carbon steels . . 41.1.3 Microstructural parameters inuencing the stability of austen-

ite against transformation . . . . . . . . . . . . . . . . . . 51.1.4 Modeling of TRIP effect in steels: State of the art . . . . . 6

1.2 Objectives and scope . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 General scheme of notation . . . . . . . . . . . . . . . . . . . . . 9

2 Elasto-plastic deformation of single-crystalline ferrite 112.1 Single crystal elasto-plastic model for ferrite . . . . . . . . . . . . 13

2.1.1 Kinematics and congurations . . . . . . . . . . . . . . . 142.1.2 Thermodynamic formulations . . . . . . . . . . . . . . . 162.1.3 Constitutive relations and Helmholtz energy density . . . 202.1.4 Driving force, non-glide stress and kinetic law . . . . . . 242.1.5 Hardening and evolution of microstrain . . . . . . . . . . 25

2.2 Simulations of elasto-plastic deformation of single-crystalline ferrite 292.2.1 Material parameters and validation . . . . . . . . . . . . . 292.2.2 Sample geometry and boundary conditions . . . . . . . . 312.2.3 Stress-strain response of single-crystalline ferrite . . . . . 33

3 Elasto-plastic-transformation behavior of single-crystalline austenite 473.1 Single crystal elasto-plastic-transformation model for austenite . . 49

3.1.1 Kinematics and congurations . . . . . . . . . . . . . . . 49

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CONTENTS

3.1.2 Thermodynamic formulations . . . . . . . . . . . . . . . 533.1.3 Constitutive relations and Helmholtz energy density . . . 57

3.1.4 Driving forces, nucleation criteria and kinetic laws . . . . 643.1.5 Hardening and evolution of microstrain . . . . . . . . . . 67

3.2 Simulations of elasto-plastic-transformation behavior of single-crystalline austenite . . . . . . . . . . . . . . . . . . . . . . . . . 713.2.1 Material parameters and validation . . . . . . . . . . . . . 713.2.2 Sample geometry and boundary conditions . . . . . . . . 753.2.3 Stress-strain response of single-crystalline austenite . . . . 77

4 Numerical solution algorithm for transformation-plasticity model 89

4.1 Stress-update algorithm for coupled transformation-plasticity model 904.1.1 Discretization of model equations . . . . . . . . . . . . . 914.1.2 Newton-Raphson iteration procedure (return-mapping) . . 964.1.3 Consistency checks for slip and transformation systems . . 1004.1.4 Sub-stepping procedure . . . . . . . . . . . . . . . . . . 102

4.2 Tangent operator . . . . . . . . . . . . . . . . . . . . . . . . . . 1044.2.1 Finite difference approximation for tangent operator . . . 1054.2.2 Tangent operator in the Eulerian setting . . . . . . . . . . 105

4.3 Validation of the numerical solution algorithm . . . . . . . . . . . 1074.3.1 Sample geometry and nite element meshes . . . . . . . . 1074.3.2 Simulation results (mesh renement analysis) . . . . . . . 108

5 Micromechanical simulation of TRIP-assisted steel 1135.1 Simulation of multiphase TRIP steel at single grain level . . . . . 114

5.1.1 Microstructural sample geometry and boundary conditions 1145.1.2 Strain-strain response of TRIP steel microstructure . . . . 117

5.2 Parametric study of polycrystalline TRIP steel behavior as a func-

tion of microstructural properties . . . . . . . . . . . . . . . . . . 1275.2.1 Sample geometry and boundary conditions . . . . . . . . 1285.2.2 Microstructural conguration and model parameters . . . 1285.2.3 Simulation results . . . . . . . . . . . . . . . . . . . . . 133

6 Macroscale simulation of multiphase TRIP-assisted steels 1416.1 Homogenization scheme for multiphase microstructure . . . . . . 143

6.1.1 Weighted-Taylor scheme with iso-work-rate criteria . . . . 143

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6.1.2 Preliminary analysis and comparison to direct FEM sim-ulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

6.2 Deep-drawing simulation of multiphase TRIP-aided steel . . . . . 1496.2.1 Sample geometry and boundary conditions . . . . . . . . 1496.2.2 Sample crystallographic orientation distribution function . 1516.2.3 Simulation results and analysis . . . . . . . . . . . . . . . 155

7 Simulation of thermal behavior of multiphase TRIP-assisted steel 1617.1 Single-crystalline thermo-mechanical models for multiphase TRIP-

assisted steel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1627.1.1 Thermo-elasto-plastic-transformation model for austenite . 163

7.1.2 Thermo-elasto-plasticity model for ferrite . . . . . . . . . 1667.2 Simulation of TRIP steel behavior under cooling . . . . . . . . . 1677.2.1 Boundary conditions and model parameters . . . . . . . . 1677.2.2 Analysis of TRIP steel behavior under cooling . . . . . . 1707.2.3 Comparison with experimental results . . . . . . . . . . . 177

A Kinematics of martensitic transformation at lower length-scales 183

B Effective elastic stiffness for martensitic transformation systems 187

C Plastic slip systems for FCC austenite and BCC ferrite 189

References 193

Summary 209

Samenvatting 213

Intisari 217

Curriculum vitae 221

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CONTENTS

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1IntroductionThe improvement of strength in carbon steels is often obtained at the expense of ductility, and vice-versa . A long-standing ambition has been to develop a class of steels where both ductility and strength can be simultaneously improved. For thispurpose, transformation-induced plasticity (TRIP)-assisted steels are particularyappealing. TRIP-assisted steels are a class of multiphase steels that exhibit a goodcombination of strength and ductility characteristics. This unique characteristicis attributed to the presence of a metastable austenitic phase in the microstruc-ture at room temperature. Upon applied thermal and/or mechanical loadings,the metastable retained austenite may transform into a harder martensitic phase,

which may increase the effective strength of the material. In addition, transforma-tion from austenite to martensite is accompanied by shape and volume changes,which are accommodated by local plastic deformations in the surrounding phases,creating the so-called TRIP-effect [61]. The additional plastic deformation dueto the transformation increases the effective work-hardening of the material. Incomparison to similar steels that contain no retained austenite in their microstruc-ture, e.g., dual-phase (DP) steels, TRIP-assisted steels have a similar (ultimate)strength, but exhibit a signicantly higher ductility.

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CHAPTER 1. INTRODUCTION

1.1 Background: Multiphase TRIP-assisted steels

At room temperature, a typical microstructure of TRIP steel consists of severalphases, i.e., the intercritical ferrite as the most dominant phase, bainite, retainedaustenite and occasionally a small fraction of thermal martensite [8, 34, 42, 64,102, 112, 133]. Intercritical ferrite (sometimes also referred to as pro-eutectoidferrite) occupies up to 75 % volume of the microstructure. Ferrite has a body-centered cubic (BCC) lattice and, compared to other constituent phases, is thesoftest phase. Nano-indentation tests performed by Furn emont et al . [42] showedthat the hardness of ferrite in a typical multiphase steel is about 5 GPa. As re-ported in the literature [42, 64], the size of ferritic grains in a typical TRIP steelmicrostructure ranges from 5 to 10 m. Unlike ferrite, bainite does not have asingle-phase structure. The microstructure of bainite consists of an assembly of layers of iron carbide (cementite) and bainitic ferrite. Bainite is formed duringan isothermal bainitic holding at a temperature between 600 and 700 K. In gen-eral, bainite is harder than intercritical ferrite due to its smaller grain size and thepresence of carbide precipitations. The typical size of bainitic grains ranges from1 up to 6 m. In addition, initial bainite can possess a higher dislocation den-sity [64]. In the case of TRIP steels, the chemical composition is chosen such that

the formation of carbides is restricted (or postponed), which results in a bainitein TRIP steels that is essentially carbon-free, but still has the characteristics of a ne plate-like structure [42, 61, 64]. The next constituent phase in TRIP steelmicrostructure is retained austenite. In contrast to other constituent phases thatare stable, retained austenite is a metastable phase. In general, austenite is a hightemperature phase, which has a face-centered cubic (FCC) structure. Stabilizationof austenite at room temperature is due to local carbon enrichment and the con-straining effect from neighboring grains. Upon the application of thermal and/ormechanical loads, metastable austenite may transform into martensite and gener-

ate the TRIP effect. In some cases, the initial TRIP steel microstructure may alsocontain a small fraction of thermal martensite. Thermal martensite is obtainedwhen austenite is rapidly cooled (or quenched) such that diffusion of carbon is pre-vented during transformation. Martensite has a body-centered tetragonal (BCT)structure that contains supersaturated interstitial carbon atoms, which can createstrain elds that restrict the movement of dislocations in the lattice [23]. Marten-site can also have a high dislocation density resulting from a displacive (or diffu-sionless) transformation mechanism. In comparison to other constituent phases,

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1.1. BACKGROUND: MULTIPHASE TRIP-ASSISTED STEELS

T e m p e r a

t u r e

t 2

Time t

t 1

1

2

Intercriticalannealing

Bainiticholding

Quenching

AF

AF

B

ABF

= Austenite= Bainite= Ferrite

Figure 1.1: Schematic representation of temperature prole of the two-stage heattreatment typically used in low-alloyed TRIP steels processing and the corre-sponding microstructural phases obtained at the end of each stage.

martensite shows the highest hardness level. Nano-indentation tests by Furn emontet al . [42] indicated that the hardness of martensite can exceed 17 GPa. In TRIPsteel microstructures, martensite appears in platelets or needle-shaped laths.

1.1.1 Two-stage heat-treatment process for TRIP steels

In many cases, the microstructure of multiphase TRIP-assisted steels are producedthrough a two-stage heat treatment process [5961, 64, 92, 102, 133]. Similar tothe processing route for dual-phase (DP) steels, the rst stage of the heat treat-ment process is the intercritical annealing, in which the material is brought to atemperature 1 between the intercritical temperatures A1 and A3 . This processtransforms some parts of the initial microstructure into the austenitic phase. As aresult, the microstructure after the intercritical annealing process consists of two

phases, i.e., (pro-eutectoid) ferrite and austenite, as schematized in Figure 1.1.While for dual-phase steels the microstructure resulting from the intercritical an-nealing is directly quenched to a room temperature, the intermediate TRIP steelmicrostructure is brought to a bainitic temperature 2 for isothermal holding overa period t2. During this isothermal holding, a fraction of the austenite formedduring intercritical annealing transforms into bainite, whereas the remaining partof the austenite is further stabilized by the enrichment of carbon expelled fromthe bainite formed. At this stage, the size of the retained austenite grains in the

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Austenite

Martensite

T h e r m a l

( G i b b s )

f r e e e n e r g y G

GAM ( ) < G barrier

Temperature T

GAM (M s) = G barrier

GA

GM

M s

Gmech

Figure 1.2: Schematic representation of thermal (or chemical) free energy of theaustenitic and martensitic phases as functions of temperature.

resulting microstructure depends on the bainitic holding temperature T 2 and theholding time t2. After the isothermal bainitic holding, the steel is quenched toroom temperature. In general, the phase composition created during the bainiticholding is preserved during the nal quenching. Occasionally, a small fractionof retained austenite further transforms into thermal martensite during the nalquenching, particularly in the austenitic regions in which the carbon enrichmentwas not sufcient.

1.1.2 Martensitic transformation in low-alloyed carbon steels

Martensitic transformations occur as a consequence of energy minimization amongthe phases in the microstructure. At high temperatures, the austenitic phase pos-sesses a lower free-energy level than the martensitic phase and, therefore, is a

stable phase. Conversely, at low temperature, the martensitic structure becomesmore favorable since it has a lower free-energy level. Figure 1.2 schematicallyillustrates the free-energy of the austenitic and martensitic phases as functions of temperature. Transformation from austenite to martensite in carbon steels can betriggered either by thermal loading (cooling) or through the application of externalmechanical loading. Upon cooling and in the absence of stress, the transformationfrom austenite to martensite starts to occur at the transformation temperature, M s ,where the difference in thermal energy (sometimes referred to as chemical en-

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1.1. BACKGROUND: MULTIPHASE TRIP-ASSISTED STEELS

ergy) between the austenite and martensite, GA M , is sufcient to overcomethe transformation energy barrier ( Gbarrier ).

At temperatures higher than M s , transformation may occur with the assistanceof mechanical stress, such that the mechanical strain energy ( Gmech ) added to thethermal energy difference is sufcient to overcome the transformation barrier, asshown in Figure 1.2. However, in carbon steels, stress-assisted martensitic trans-formation are irreversible, i.e., reverse transformations (from martensite to austen-ite) cannot occur upon reversal of loading. Transformation from martensite toaustenite can only be realized by re-heating. This is in contrast to shape-memoryalloys, where stress-assisted transformations are crystallographically reversible.

1.1.3 Microstructural parameters inuencing the stability of austen-ite against transformation

In the nal microstructure, the stability of retained austenite grains against trans-formation plays an important role in characterizing the overall performance of TRIP-assisted steels. Experimental investigations have shown that the stabil-ity of the austenitic grains is inuenced by various microstructural parameters,such as (i) the carbon concentration in the retained austenite [8, 34, 64, 102, 112],(ii) the size and shape of the austenitic grains [11, 68, 147], (iii) the morphol-ogy of microstructural phases [63, 64, 151], (iv) the crystallographic orientationof grains (microstructural texture) [72, 91] and (v) the stiffness of the surroundingphases [102, 133].

In TRIP steel microstructures, the volume fraction of the phases, the sizeand shape of the austenitic grains, as well as their local carbon concentrationobtained in the two-stage heat treatment process depend upon the intercriticalannealing and bainitic transformation process conditions, e.g., temperature andholding time [59, 60, 64, 84, 102, 112, 147]. For example, a longer bainitic hold-

ing time results in a nal microstructure with smaller grains of retained austenite,but with a higher carbon content. Furthermore, the carbon concentration in theretained austenite grains is controlled through the presence of alloying elements,such as silicon, aluminum and phosphor [12]. These elements effectively pre-vent carbide precipitation during the bainitic holding stage and, thus, enhance thecarbon enrichment in the austenite. For typical TRIP steel microstructures, thecarbon concentration in the retained austenite reportedly varies from 0.6 wt.% upto 2.3 wt.% [59, 60, 64, 68, 84, 112, 121]. It should be pointed out that the deter-

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mination of the carbon concentration in the austenitic grains is rather complexand there is no generally accepted method for determining the austenite carbon

concentration. It is generally accepted that the real range of austenite carbon con-centrations in multiphase TRIP steels is probably smaller than rst indicated.

Besides carbon enrichment, the stability of retained austenite is inuencedby the mechanical properties of the surrounding phases. Under external thermo-mechanical loading the stresses experienced by the austenitic grains depend onthe elasto-plastic properties of the surrounding ferritic grains (e.g., yield stressand strain hardening behavior), which can be controlled by the addition of ele-ments, such as manganese and molybdenum, as well as by changing the ferriticgrains size [43, 66]. In addition, the crystallographic orientation of the austenitic

grains with respect to the loading direction plays an important role in the austen-ite stability against transformation [72, 91]. This information is relevant for themacroscopic behavior of TRIP steels, particulary if the steel is produced througha rolling process, which may induce a microstructural texture, where a large num-ber of grains are oriented in a specic crystallographic direction [144, 149].

1.1.4 Modeling of TRIP effect in steels: State of the art

The modeling of the TRIP effect involves two key aspects [39, 64], namely (i)the elasto-plastic deformation in the transforming austenitic region as well as inthe neighboring phases to accommodate shape and volume changes associatedwith the martensitic transformation, which is often referred to as the Greenwood- Johnson effect [45] and (ii) the strong dependency of the martensitic formationupon the crystallographic orientation with respect to the loading axis, also knownas the Magee effect [83]. From a historical point of view, the modeling of marten-sitic phase transformations can be traced back to the pioneering work of Wechsleret al . [148] in 1953, where a crystallographically-based model was proposed to de-

scribe the kinematics for a martensitic transformation. This concept was renedby Ball and James [7] by formulating the model within an energy minimizationframework. During the last decades, various constitutive models for martensitictransformations have been developed for describing the TRIP effect, such as theone-dimensional phase transformation model of Olson and Cohen [93], whichwas extended into a three-dimensional model by Stringfellow et al . [120] andBhattacharyya and Weng [16]. Furthermore, models based on a more complexmicromechanical framework were also proposed. These can be found in the work

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1.2. OBJECTIVES AND SCOPE

of, e.g., Leblond et al . [75, 76] and Levitas et al . [79, 80]. Following the classicalcrystallographic model of Wechsler et al . [148], Marketz and Fischer [85, 86] pro-

posed a model for stress-assisted martensitic transformation for single-crystallineand polycrystalline austenite, see also Tomita and Iwamoto [138, 139]. Further,Diani et al . [32, 33] proposed a model that takes into account the effect of thecrystallographic orientations of grains on the elasto-plastic response using a smallstrain formulation. Similar models can be also found in Cherkaoui et al . [24, 25]and Taleb and Sidoroff [127].

Most of the models mentioned above were developed within a small-strainframework, which can lead to inaccurate predictions since martensitic transfor-mations can locally induce large elasto-plastic deformations, even if the effec-

tive macroscopic deformation is relatively small. In addition, an isotropic elasto-plastic response is often assumed, which is a strong simplication, particulary foranalyses at smaller length scales (e.g., at the single-crystal level), where the effectof anisotropy due to crystallographic orientations cannot be neglected [41, 72, 91].Within the context of a large deformation framework, Turteltaub and Suiker [124,141, 143] have developed a crystallography-based model for martensitic phasetransformations in carbon steels. The model is derived following a multiscale ap-proach, where material parameters, e.g., transformation deformation kinematicsand effective elastic stiffness, at higher length-scales are calculated from lower

scale quantities by means of averaging schemes. In addition, the model is con-structed within a thermo-mechanically consistent framework, where the thermalquantities are derived analogous to the mechanical counterparts. The model of Turteltaub and Suiker [124, 141, 143] lays the foundation for the work to be pre-sented in this thesis.

1.2 Objectives and scope

Despite of its superior characteristics, there is room for further improvement in theoverall performance of a TRIP-assisted steel. However, this can only be achievedby developing a thorough understanding of the TRIP mechanism. The presentwork is aimed at developing crystallographically-based computational models forsimulating the behavior of TRIP-assisted steels. The underlying goal of the sim-ulations is to study systematically the mechanism of TRIP, particularly the mech-anism of the stress-assisted martensitic transformation in the austenitic grains,as well as the elasto-plastic interactions between the transforming grains and the

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neighboring phases. The models developed to study these effects are numericallyimplemented within a nite-element framework.

Within a parametric analysis setting, various sets of simulations are performedin order to identify the role of every microstructural property on the austenite sta-bility and, thus, the overall response of the TRIP-assisted steels under mechanicaland thermal loadings. The analyses cover various length-scales, i.e., from simu-lations at the level of single crystal up to simulations of forming processes at themacroscopic scales. On the whole, the present work will provide a good insightfor further improvement of the performance of TRIP-assisted steels, as well as forthe optimization of the TRIP steel processing parameters.

1.3 Thesis outline

The outline of this thesis is as follows: The elasto-plastic responses of single-crystalline ferrite are simulated and studied in Chapter 2 . For this purpose, asingle crystal elasto-plasticity model is adopted. In order to mimic the asym-metric behavior of slip in twinning and anti-twinning senses typically found inBCC metals, the model incorporates the effect of the non-glide stress into thekinetic formulation. The model is derived within a large deformation frame-

work. In order to demonstrate the basic features of the model, several simula-tions are performed for various types of elementary deformation modes. Sub-sequently, a crystallography-based model for simulating the behavior of single-crystalline austenite is presented in Chapter 3 . This model is derived throughcoupling the multiscale martensitic phase transformation model of Turteltaub andSuiker [141, 143] to an FCC single-crystal elasto-plasticity model. The couplingbetween the transformation and the plasticity terms is derived systematically usinga thermodynamically-consistent formulation. The model is used to study the re-sponse of single-crystalline austenite, in particular the interaction between phase

transformation and plastic deformation mechanisms in the austenite under variousloading conditions.Key aspects of the numerical implementation of the models are presented in

Chapter 4 . The discussion is thereby mainly focussed on the numerical imple-mentation of the elasto-plastic-transformation model for the austenitic phase. Thenumerical algorithm for the ferrite elasto-plasticity model can be performed anal-ogously through eliminating the terms related to transformation. In addition, anumber of simulations are presented to show the numerical stability and conver-

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1.4. GENERAL SCHEME OF NOTATION

gence of the implemented algorithm.In Chapter 5 , the single-crystalline models presented in Chapters 2 and 3 are

combined to simulate the response of multiphase TRIP-assisted steel microstruc-tural samples. The simulations focus on the interaction between the transformingaustenitic grain and the surrounding ferritic matrix for different combinations of crystallographic orientations. In addition, the role of microstructural properties,such as local carbon concentration and austenitic grain size on the overall re-sponses of the TRIP steels are studied in a parametric analysis. In Chapter 6 ,simulations of TRIP steel behavior at the macroscopic scale, e.g., during deep-drawing process, are shown. For this purpose, the present single-crystalline mod-els for ferrite and austenite are employed in combination with a simple averaging

scheme, namely the iso-work-rate weighted-Taylor scheme. A direct reconstruc-tion of orientation distribution functions (ODF) by means of a probabilistic ap-proach is performed in order to replicate the crystalline texture of the samplesduring simulations. Finally, the behavior of multiphase TRIP-assisted steels dur-ing thermal loading is simulated and analyzed in Chapter 7 . The analyses coverthe thermal behavior of TRIP-assisted steels as a function of microstructural prop-erties, similar to the analyses of the mechanical loading presented in Chapter 5 .Moreover, the transformation behavior under thermal loading as predicted by thepresent models is compared to experimental observations.

1.4 General scheme of notation

As a general scheme of notation, scalars are written as lightface italic letters,vectors as boldface lowercase letters (e.g., a , b), second-order tensors as boldfacecapital letters (e.g., A , B ) and fourth-order tensors as blackboard bold capitalletters (e.g., A , B ). For vectors and tensors, Cartesian components are denoted asa i , Aij and Aijkl . The action of a second-order tensor upon a vector is denoted

as Ab (in components Aij b j , with implicit summation on repeated indices) andthe action of a fourth-order tensor upon a second order tensor is designated asA B (i.e., A ijkl Bkl ). The composition of two second-order tensors is denoted asAB (i.e., Aij B jl ). The tensor product (dyadic product) between two vectors isdenoted as a b (i.e., ai b j ). All inner products are indicated by a single dotbetween the tensorial quantities of the same order, e.g., a b for vectors and A B for second-order tensors (in components, respectively, ai bi and Aij B ij ). Thetranspose of a tensor is denoted by a superscript T and its inverse by a superscript

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1. A superimposed dot represents a material time derivative. Subscripts A, M ,and F indicate that the quantities correspond to material properties of austenite,

martensite and ferrite, respectively.

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2Elasto-plastic deformation of single-crystalline ferriteIn multiphase steels assisted by the TRIP effect, ferrite is the most dominantphase in terms of its volume. Therefore, although it is not considered as themost important ingredient of TRIP steel microstructure, the elasto-plastic behav-ior of the ferrite-based matrix determines to a large extent the overall behavior of the multiphase steel. Despite of this, little attention has been given to the mod-eling of ferrite in many models for TRIP-assisted steels available in the litera-ture [57, 58, 109, 110, 120], where relatively simple elasto-plasticity models wereused for the non-transforming phase.

Continuum models used to simulate the elasto-plastic behavior at the level of asingle crystal are often based on crystal plasticity theory. The plastic deformationis kinematically described as the result of slip on specic crystallographic planesand in specic directions (i.e., slip systems). The foundations of crystal plastic-ity theory were laid down in the works of Taylor and Elam [129, 130] and Tay-lor [128]. The concept was further developed by Rice [108], Hill and Rice [53],Asaro and Rice [6], Asaro and Needleman [5], Peirce et al . [100, 101] Bassaniand Wu [10], Cuiti no and Ortiz [30], Gurtin [46] and Gurtin and Anand [48, 49].

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CHAPTER 2. ELASTO-PLASTIC DEFORMATION OF SINGLE-CRYSTALLINE FERRITE

Slip-rate and hardening constitutive relations in some of these models, particu-larly the early ones, have a strong phenomenological nature, whereas more recent

models attempt to connect the slip mechanism to smaller length scale phenomenathrough dislocation-based constitutive relations.

In general, crystal plasticity models based on the classical Schmid stress (re-solved shear stress) provide satisfactory predictions for crystalline materials withclose-packed structures, such as face-centered cubic lattices (FCC). However,phases with body-centered cubic lattices (BCC), such as ferrite, require additionalattention for the following reasons: (i) Due to the absence of close-packed planes,there is no clear denition of slip systems in BCC structures. The lack of ex-

perimental observations with sufcient resolution has precluded reaching consen-sus among researchers regarding the crystallographic slip planes that are activeduring plastic ow. This controversy has been aggravated since experimental ob-servations indicate that the trace of slips in BCC metals also depends on tem-perature [99, 113]; (ii) as opposed to FCC lattices, slip in BCC lattices behavesasymmetrically in the twinning and antitwinning directions [36, 54, 55], which, atmacroscopic scales, results in an asymmetric response in tension and compres-sion.

Several crystal plasticity-based models have been proposed for BCC metalswithin single-crystalline and polycrystalline contexts, for example, Nemat-Nasseret al . [90], Stainier et al . [118], Peeters et al . [97, 98] and Ma et al . [82]. Thosemodels shared some similarities, e.g., the above models include the families of

{211} and {321} planes as potential slip planes in order to solve the ambiguityof slip traces in BCC metals. On the other hand, the issues of asymmetric behav-ior of slip in the twinning and antitwinning directions is, unfortunately, not welladdressed. Since the model for BCC single crystals proposed here is part of abigger framework of TRIP steel modeling, the accuracy on the prediction of the

ferritic stress-strain behavior, particularly the asymmetry in tension-compression,is important. In this chapter, a thermodynamically-consistent elasto-plastic modelfor BCC ferrite is developed. The present model is based on the non-glide stressformulation proposed by Bassani et al . [9], which allows to predict the twinning-antitwinning asymmetric behavior in BCC crystals. The formulation of the elasto-plasticity model for ferrite single crystal is discussed in Section 2.1 . Simulationsof single-crystalline ferrite with elementary loading modes are presented in Sec-tion 2.2 in order to illustrate the key features of the present model.

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2.1. SINGLE CRYSTAL ELASTO-PLASTIC MODEL FOR FERRITE

2.1 Single crystal elasto-plastic model for ferrite

Plastic slip in BCC crystals operates through propagation of screw dislocationsalong the 111 directions. Due to the lack of close-packed planes in BCC struc-tures, there is no clear consensus among researchers on which planes crystal-lographic slip occurs. Experimental observations often indicate slip traces along

{110}, {211}and {321}planes, and sometimes along non-crystallographic planes.However, higher resolution micrographs show that slip on {321}and higher indexplanes appear in small wavy patterns, which can be interpreted as slip composedof alternating glide contributions along lower index planes [40, 54, 106]. Althoughslip traces along

{211

} planes in BCC ferrite have been reported, particularly at

and above room temperature, with the addition of silicon, which is the case forthe ferritic matrix in TRIP-assisted steels, it is observed that slip occurs predom-inantly along the {110} planes [113]. Furthermore, atomistic simulations per-formed by Vitek and co-workers [145, 146] indicate that slip along {211} planescan be constructed of equal segments of slip along alternating {110} planes. Ac-cordingly, it will be assumed in the present formulation that the systems corre-sponding to the {110}111 family are sufcient to describe slip in ferrite.

The classical approach in crystal plasticity theory is to assume that gliding

along an individual slip system is solely determined by the Schmid law, in whichthe resolved shear stress is equated with the corresponding critical value represent-ing resistance against slip. Although this assumption works reasonably well forFCC metals, it cannot be directly applied to BCC crystals. Atomistic simulationsof BCC crystals performed by Duesbery and Vitek [36] have shown that for met-als with a BCC structure, the cores of 12 111 screw dislocations spread into three

{110} planes intersecting along the 111 directions. The non-planar spreadingof a dislocation core causes the slip along an individual plane of the {110} classto become dependent on resolved stresses acting on or normal to another {110}plane of the [111] zone, called the non-glide plane. The resolved stress act-ing on or normal to the non-glide plane is referred to as non-glide stress. In thepresent model, the effect of non-glide stress is incorporated following the ap-proach developed by Bassani et al . [9, 145]. Although the non-glide stress modelwas originally derived based on atomistic simulations of BCC molybdenum andtantalum, it is assumed that the plastic slip in BCC ferrite can be described by asimilar mechanism. This assumption is reasonable since the ferrite lattice sharesthe same generic features with the lattice of molybdenum and tantalum, e.g. the

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X

mF

n F

x

F p F eReferenceconfiguration

Currentconfiguration

Intermediate (or relaxed)configuration

F ( x )

Infinitesimalneighborhood of x

y( x )

Figure 2.1: Schematic representation of the decomposition of deformation gradi-ent F . Vectors m F and n F are, respectively, the slip direction and the slip planenormal of BCC ferrite in the intermediate (relaxed) conguration [47].

asymmetry of slip in twinning and antitwinning directions [55]. The validationof this assumption would require atomistic simulations of deformation of BCCferrite; however these fall outside of the scope of the present work.

2.1.1 Kinematics and congurations

Based on a large deformation theory, the total deformation gradient F is decom-posed as [6, 53, 77]

F = F e F p , (2.1)

where F e is the elastic part of the total deformation gradient F , describing thedeformation due to elastic distortion of the lattice, and F p is the plastic part of the

total deformation gradient representing deformation related to cumulative crys-tallographic slips. It is assumed that the plastic part of the deformation gradient,F p, does not change the lattice structure and that the elastic properties of the ma-terial remain unaltered during a deformation process. As shown in Figure 2.1,the decomposition of the total deformation gradient can be illustrated through theintroduction of a reference conguration, an intermediate (relaxed) congurationand a current conguration. The plastic deformation gradient F p maps a materialpoint from the reference conguration to the relaxed intermediate conguration.

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In turn, the elastic deformation gradient F e projects the point from the intermedi-ate conguration to the current conguration. It is worth noting that, in general,

the order of the decomposition does not correspond to the actual deformation se-quence and that the elastic and plastic parts do not correspond to the gradientsof globally-dened functions [47]. Furthermore, although the present model isderived to study coupled thermo-mechanical problems, the decomposition of thetotal deformation gradient in (2.1) does not include the effect of thermal expan-sion/contraction. In this chapter, it is assumed that thermal expansion/contractionis relatively small, and thus, may be neglected. Nonetheless, the incorporation of the thermal expansion/contraction in the total deformation gradient will be dis-cussed in Chapter 7 .

The velocity gradient in the current conguration, denoted as L , can be writtenas the sum of the elastic part L e and the plastic part L p, i.e.,

L = F F 1 = F eF 1e + F e F pF

1 p F

1e := L e + L p . (2.2)

Note that the velocity gradients L e and L p are measured in the current congura-tion. In the intermediate (relaxed) conguration, the plastic velocity gradient L pis determined by the cumulative slip rates on all possible slip systems as

L p := F pF 1 p =N F

i=1 (i)F m (i)F n (i)F , (2.3)

where (i)F is the rate of slip on a system i and the vectors m(i)F and n

(i)F are,

respectively, unit vectors describing the slip direction and the normal to the slipplane of the corresponding system in the intermediate conguration. In view of (2.3), the rate of change in volume due to plastic slip is given by

d(det F p)dt = det F p tr L p = det F p

N F

i=1

(i)F m

(i)F n

(i)F = 0 , (2.4)

where the last relation follows from the fact that the vectors m (i)F and n(i)F are

orthogonal to each other for all slip systems, i.e., m (i)F n(i)F = 0 . Consequently,

if the initial plastic deformation gradient is such that det F p(0) = 1 , it followsthat det F p(t) = 1 for all t [0, T ], i.e., the plastic deformation is isochoric. Fur-thermore, the total velocity gradient in the current conguration L can be written

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as

L =

L e +

N F

i=1 (i)

F m(i)

F n(i)

F , (2.5)

where m (i)F and n(i)F are, respectively, the slip direction vector and the vector

normal to the slip plane measured in the current conguration, dened by

m (i)F = F e m(i)F and n

(i)F = F

T e n

(i)F . (2.6)

Clearly, from (2.6), the vectors m (i)F and n(i)F are not unit vectors.

2.1.2 Thermodynamic formulations

Decomposition of entropy density

The objective of formulating the model in a thermo-mechanical framework is toderive a consistent expression of the driving force for plastic slip. In thermo-mechanical processes, the entropy and temperature may be viewed as the thermalanalogues of deformation and stress, respectively [22, 143]. Hence, in analogy

to the decomposition of the total deformation gradient in (2.1), the total entropydensity per unit mass, , is decomposed as

= e + p , (2.7)

where e represents the conservative (reversible) part of the entropy density and p is the entropy density related to the plastic deformation process. Similar tothe entropy decomposition in the isotropic elasto-plasticity model of Simo andMiehe [116], the rate of change of the plastic entropy is assumed to be propor-

tional to the rate of change of the plastic deformation, which here is measured bythe rate of slip (i)F ,

p =N F

i=1 (i)F

(i)F , (2.8)

where (i)F is interpreted as the entropy density related to plastic deformation perunit slip in system i.

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Balance principles and dissipation

Let P be the rst Piola-Kirchhoff stress in the reference conguration, bf thebody force per unit reference volume and a the acceleration of a material point x .Assuming that all variables are continuously differentiable, the balance of linearmomentum per unit volume in the reference conguration is given by

div P + bf = 0a , (2.9)

with 0 the ferrite mass density in the reference conguration.Furthermore, let be the internal energy density per unit mass, q the heat ux

per unit reference area and r the body heat source per unit reference volume. The

balance of total energy, combined with the balance of linear momentum per unitreference volume can be expressed as

0 + ( div q r ) P F = 0 , (2.10)where the term P F is known as the internal power .

The rate of change of entropy per unit volume in the reference conguration,, is dened by

:= 0 + div s , (2.11)where and s are, respectively, the entropy ux per unit area and the entropysource per unit volume in the reference conguration,

= q

and s = r

, (2.12)

with the (absolute) temperature. Dening the dissipation density per unit refer-ence volume as D := and invoking equations (2.10)-(2.12), the total dissipa-tion can be written as

D = 0 + 0 + P F . (2.13)Taking the time derivative of the total deformation gradient in (2.1) and combiningit with the expression for the plastic velocity gradient given in (2.3), provides thefollowing expression for the internal power:

P F = P F T p F e +N F

i=1

(i)F (i)F , (2.14)

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where (i)F is referred to as the resolved shear stress (or Schmid stress) for slipsystem i, given by

(i)F := F T e P F T p m (i)F n (i)F . (2.15)From expressions (2.7) and (2.8), the contribution of the term 0 to the totaldissipation can be obtained as

0 = 0e +N F

i=1

(i)F (i)F , (2.16)

where the quantity (i)F in (2.16) can be interpreted as the thermal analogue of the

resolved shear stress, as given by (i)F := 0

(i)F . (2.17)

Subsequently, the rate of change of the internal energy density that appearsin (2.13) needs to be determined. The internal energy density in the present modelis decomposed into various mechanical and thermal contributions: The bulk strainenergy density is characterized by the elastic deformation gradient F e while thethermal energy density is dependent of the conservative entropy e . Furthermore,a scalar variable F is introduced to represent the local strains (or distortions) of

the BCC ferrite lattice associated with the presence of dislocations. Correspond-ingly, a lower length scale strain energy, called the lattice defect energy, can beexpressed as a function of the scalar microstrain F . The internal energy density is assumed to be dependent of the state variables F e , e and F . In accordance

with the Coleman and Noll procedure [29], it is momentarily assumed that theinternal energy density also depends on the uxes F and , i.e.,

= (F e , e , F ; F , ) . (2.18)

Using (2.14), (2.16) and (2.18), the expression for the total dissipation in (2.13)

can be rewritten as

D = P F T p 0

F e F e + 0 e

e

+N F

i=1 (i)F +

(i)F

(i)F 0

F

F 0

F F

0 .

(2.19)

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The second law of thermodynamics requires that the local entropy rate must benon-negative during any thermo-mechanical process, i.e., 0. This restrictionleads to a non-negative energy dissipation, i.e. D 0, since the (absolute) tem-perature is strictly positive. Furthermore, the terms in (2.19) that are multipliedwith the rates F e, e , F and must vanish since otherwise a process could bespecied for which the dissipation is negative. This requirement results in

P = 0

F eF T p and =

e

, (2.20)

and that the internal energy density does not depend on the uxes F and .In anticipation of a constitutive model for hardening and in order to simplify

the presentation, the rate of change of the scalar microstrain, F , is taken to belinearly dependent of the rate of change of the plastic slip, (i)F , as follows:

F =N F

i=1w(i)F

(i)F , (2.21)

where the functions w(i)F depend on the slip resistance, as will be discussed inSection 2.1.5 . From (2.20) and (2.21), the remaining non-zero terms of the total

dissipation in (2.13) can be decomposed into the dissipation related to the plasticdeformation, D p, and the dissipation due to the heat conduction process, Dq, i.e.

D = D p + Dq , (2.22)where D p and Dq are, respectively, given by

D p :=N F

i=1 (i)F +

(i)F 0

F

w(i)F (i)F and Dq := . (2.23)

Following the formalism proposed by Onsager for irreversible thermodynamics(see e.g., [2, 22]), for each physical phenomenon where the energy is dissipated,a pair of conjugate quantities can be identied as an afnity (or driving force) andthe corresponding ux. In relation to the dissipation due to plastic deformation,

D p, the driving force for plastic slip along a system i is dened byg(i)F :=

(i)F +

(i)F 0

F

w(i)F , (2.24)

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whereas the rate of plastic slip (i)F is viewed as the corresponding ux. In re-lation to the dissipation due to heat conduction, Dq, the afnity is given by thetemperature gradient, , with as the corresponding ux. In view of thedecomposition of the total dissipation (2.22), the dissipation inequality can bewritten as

D = D p + Dq 0 . (2.25)In the present model, it is assumed that the dissipation inequality holds for theplastic deformation and heat conduction processes independently, which resultsin

D p 0 and Dq 0 . (2.26)2.1.3 Constitutive relations and Helmholtz energy density

It is common practice to work with the Helmholtz energy density instead of the internal energy density in order to use the temperature as an independentvariable instead of the entropy. The Helmholtz energy density, which depends onthe state variables F e , and F , can be obtained from the internal energy densityusing the following Legendre transformation 1 :

(F e, , F ) = (F e , e(F e, , F ), F ) e(F e, , F ) . (2.27)Relations between partial derivatives of the Helmholtz and the internal energydensities can be obtained by taking derivatives in (2.27) while holding the corre-sponding natural variables xed, which results in

F e

= F e

,

= e and

F

= F

, (2.28)

where the relation (2.20) 2 was used to obtain (2.28) 2 .In order to full the principle of material frame indifference, the Helmholtz

energy density (and also the internal energy density) cannot be dependent of thefull elastic deformation gradient F e . Alternatively, the elastic Green-Lagrangestrain E e, which is based on the elastic stretch part only, is used instead. Theelastic Green-Lagrange strain is dened by

E e := 12

F T e F e I , (2.29)1Note that on the right hand side of (2.27), it uses the product e instead of the more classical

expression , in accordance with the choice of state variables for and .

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with I the second-order identity tensor. Correspondingly, an alternative expres-sion for the Helmholtz energy density in terms of the elastic Green-Lagrange

strain E e is introduced such that

(E e , , F ) = (F e , , F ) . (2.30)

From the denition of the elastic Green-Lagrange strain in (2.29) and using thechain rule, the relation (2.28) 1 becomes

F e

E e=

F e

, (2.31)

where the symmetry of the elastic Green-Lagrange strain E e was used. Hence-forth, it is assumed that the Helmholtz energy density can be written as

(E e , , F ) = m (E e) + th () + d( F ) , (2.32)

where m , th and d represent the contribution of the bulk strain energy, ther-mal energy density and the lattice defect energy, respectively. Note that in thedecomposition of the Helmholtz energy density (2.32), the terms m , th and dare fully decoupled.

Stress-elastic strain constitutive relation

Let S be the second Piola-Kirchhoff stress in the intermediate (or relaxed) con-guration, which is related to the rst Piola-Kirchhoff stress P measured in thereference conguration by

S = F 1e P F T

p . (2.33)

Note that the relation in (2.33) is derived by taking into account the fact thatJ p := det F p = 1 . From (2.20) 1 , (2.28), (2.31) and (2.33), the partial derivative

of the Helmholtz energy density with respect to the elastic Green-Lagrange strainis obtained as

E e=

10

S . (2.34)

The second Piola-Kirchhoff stress in the intermediate conguration, S , andits work-conjugated strain measure, the elastic Green-Lagrange E e, are relatedconstitutively by

S = C F E e , (2.35)

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where C F is the fourth-order elasticity tensor of the BCC ferrite. In terms of acommonly used 66 matrix representation (Voigts notation), the components of the elasticity tensor C F can be written as

[C F ]F =

F 1 F 2 F 2F 2 F 1 F 2F 2 F 2 F 1

F 3F 3

F 3 F

, (2.36)

where F 1 , F 2 and F 3 are the elastic moduli of the BCC ferrite. The subindex F in (2.36) indicates that the stiffness components of C F are referred to the BCCferrite lattice basis. In the present model it is assumed that the elasticity tensorC F does not depend on the elastic strain E e nor on the temperature . Hence,using the stress-strain constitutive relation (2.35) and through integrating the par-tial derivative (2.34) with respect to E e, the expression of the bulk strain energydensity m (E e) can be written as

m (E e) = 1

20C F E e

E e . (2.37)

Reversible entropy-temperature constitutive relation

Similar to the stress and elastic strain relation, the reversible part of the entropydensity e is related constitutively to the temperature as follows [136, 143]:

e = hF ln F

+ F , (2.38)

where hF is the specic heat of the BCC ferrite, which, in this case, is assumed tobe a constant (temperature-invariant), and F and F are, respectively, a referencetemperature and a reference entropy density. In view of (2.38), the derivative of the Helmholtz energy density with respect to temperature in (2.28) 2 is given by

= hF ln F F . (2.39)

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Through integrating (2.39) with respect to temperature, the thermal energy densityth () can be written as

th () = hF ln F

+ ( hF F ) . (2.40)

Lattice defect energy density (cold work)

In addition to the bulk strain energy density, a lower scale elastic strain energydensity is introduced that accounts for the (elastic) distortion of the lattice dueto the presence of dislocations. This energy density term is called lattice defect energy or cold work . The present model does not explicitly resolve the kinematicsand kinetics at the length scale of individual dislocations. Instead, an isotropicphenomenological model that is commonly used in the materials science literatureis adopted. According to this model, the elastic strain energy associated witha single dislocation is proportional to b2, where is an equivalent (isotropic)shear modulus and b is the magnitude of the Burgers vector (see e.g., Hull andBacon [55]). Further, the expression for the defect energy per unit volume is givenby 12 b

2d , where d measures the total dislocation line per unit volume and is a scaling factor for strain energy of an assembly of dislocations. For notationalconvenience, it is useful to introduce a strain-like internal variable, i.e. := b d(see also [27]).

Adopting the above model, the lattice defect energy per unit mass, d , isdened as a function that depends quadratically on the microstrain F , i.e.,

d( F ) := 120

F F 2F , (2.41)

where F and F are, respectively, the scaling factor that accounts for an assem-bly of dislocations and the equivalent isotropic shear modulus of the BCC ferritelattice. The equivalent shear modulus F can be determined in terms of the elasticmoduli F j , with j = 1 , 2, 3, following the averaging procedure outlined in [141],which gives

F = 110

2 F 1 F 2 + 3 F 3 . (2.42)Hence, from the decomposition of the Helmholtz energy density (2.32) and using

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(2.37), (2.40) and (2.41), the Helmholtz energy density per unit mass is given by

(E e, , F ) = 120 C F E e E e +

120 F F

2F

hF ln F

+ ( hF F ) .(2.43)

2.1.4 Driving force, non-glide stress and kinetic law

Using the expression of the Helmholtz energy density given in (2.43) and in viewof (2.28) 3 , the driving force for plastic slip in system i in (2.24) can be reformu-lated as

g(i)F = (i)F + (i)F F F F w(i)F , (2.44)with (i)F and

(i)F given by (2.15) and (2.17), respectively. Using (2.33),

(i)F can

be written in terms of the second Piola-Kirchhoff stress as

(i)F = F T e F eS m

(i)F n

(i)F . (2.45)

As mentioned earlier in this chapter, the non-planar spreading of the cores of 12 111 screw dislocations causes the slip along an individual plane of the {110}class to become dependent on resolved stresses acting on or normal to the non-glide plane, i.e. another {110} plane of the [111] zone. In accordance with themodel proposed by Bassani et al . [9] (see also, Vitek et al . [145]), the non-glidestress (i)F corresponding to a slip system i is dened as a resolved shear stressparallel to the slip direction acting on the non-glide plane, i.e.,

(i)F = F T e F eS m

(i)F n

(i)F , (2.46)

where n (i)F is the unit vector perpendicular to the corresponding non-glide plane.

The choice of the non-glide plane for each slip system i follows from the resultsof atomistic simulations [36, 146], which determines the asymmetry of slips. Forexample, the slip system [111](011) corresponds to the non-glide plane (110)whereas the opposite slip system [111](011) relates to the non-glide plane (101).The expression (2.46) for the non-glide stress is formally similar to the expression(2.45) for the Schmid stress, where n (i)F plays an equivalent role as n

(i)F . In accor-

dance with the model of Bassani et al . [9], the contributions of non-glide stressesacting perpendicular to the slip direction are not accounted for.

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The effect of the non-glide stress on the evolution of plastic slip can be mod-eled by incorporating this term into the effective resistance against plastic slip,

s (i)F = s(i)F a (i)

(i)F , (2.47)

which includes the slip resistance s(i)F and the effect of the non-glide stress (i)F ,

with a(i) a factor that determines the net contribution of the non-glide stress to theeffective slip resistance.

In a rate-dependent crystal plasticity formulation, the evolution of plastic slipin a slip system i is described using a kinetic law, which relates the driving forceg(i)F to the rate of slip

(i)F . The kinetic law must be dened such that it satises

the requirement of non-negative energy dissipation. In the present model, thepower law kinetic relation proposed by Cuiti no and Ortiz [30] is adopted, i.e.,(see also [88])

(i)F = F 0

g(i)F s (i)F

(1/p F )

1 if g(i)F > s

(i)F ,

0 otherwise ,

(2.48)

where F 0 and pF are, respectively, the reference slip rate and the rate-sensitivityexponent. Both parameters have positive values. The kinetic law (2.48) will re-duce to a rate-independent model as F 0 and/or pF 0 (see Figure 2.2).

Notice that the above kinetic relation gives a distinction between the elasticand plastic regimes explicitly. Furthermore, the power law equation (2.48) alwaysleads to a non-negative plastic slip rate so that positive and negative senses of slipare accounted for separately. Plastic slip is initiated as soon as the driving forceof a slip system exceeds a critical value, i.e.,

g(i)F s (i)F . (2.49)

2.1.5 Hardening and evolution of microstrain

In general, the magnitude of the slip resistance s(i)F evolves during plastic de-formations, which is dened through a hardening model. In the present work,

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(b)(a)Plastic driving force g F(i)

R a t e o f p l a s

t i c s l i p F

( i ) .

R a t e o f p l a s

t i c s l i p F

( i ) .

Plastic driving force g F(i) g F(i) = sF(i) g F(i) = sF(i)

increasing 0 .

decreasing pFF

0

. F p F

0

Figure 2.2: Rate of change of plastic slip as a function of plastic driving forceaccording to the kinetic relation (2.48) with variations of (a) reference slip rateand (b) rate-sensitivity exponent.

the evolution of the slip resistance s(i)F is computed using the phenomenologicalmodel proposed by Peirce et al . [101],

s (i)F =N F

j =1

H (i,j )F ( j )F , (2.50)

where H (i,j )F is a matrix containing the hardening moduli with the diagonal termsreferring to self-hardening and the off-diagonal terms referring to cross-hardening,i.e.,

H (i,j )F =k( j )F for i = j ,q F k

( j )F for i = j .

(2.51)

Here, q F denes the ratio between cross- and self-hardening moduli on each slipsystem, called the latent hardening ratio , and k( j )F is the single-slip hardeningmodulus of a slip system j . The evolution law for the single-slip hardening mod-ulus is described by a power law equation proposed by Brown et al . [20], i.e.,

k( j )F = kF 0 1

s( j )F sF

u F

, (2.52)

with kF 0 a reference hardening modulus, sF the saturation value of the slip resis-tance, assumed to be identical for all slip systems, and uF the hardening exponent.

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In addition, the initial value of the slip resistance s(i)F is given by

s(i)F (t = 0) = sF 0 . (2.53)

In the present model, the initial value for the slip resistance, sF 0 , is assumed to bethe equal for all slip systems.

When introducing the microstrain variable F in Section 2.1.2 , it was as-sumed beforehand that the rate of change of the microstrain parameter was con-nected to the rate of change of plastic slip through the functions w(i)F (c.f., equation(2.21)). In line with the model proposed by Clayton [27], the state variable F is constitutively related to the average value of the slip resistance, which in a rateform can be written as

cF F F = 1N F

N F

i=1s (i)F , (2.54)

with cF a scaling factor that accounts for average hardening. The assumption of isotropy in (2.54) is adopted for reasons of simplicity. Substituting (2.50) into(2.54) leads to the following expression for the rate of change of the microstrain F :

F = 1cF F N F

N F

i=1

N F

j =1H (i,j )F ( j )F . (2.55)

The functions w(i)F can be related to the hardening moduli matrix H ( j,i )F by com-

paring the expressions (2.55) and (2.21), which results in

w(i)F = 1

cF F N F

N F

j =1H ( j,i )F . (2.56)

Summary of single crystal elasto-plasticity model for ferrite

For convenience, the main ingredients of the elasto-plasticity model for single-crystalline ferrite are summarized as follows: The decompositions of the defor-mation gradient (2.1) and the entropy density (2.7) are, respectively,

F = F eF p and = e + p .

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The evolution of the plastic parts of the deformation gradient, F p, and of theentropy density, p, are, respectively, described by

L p = F pF 1

p =N F

i=1 (i)F m

(i)F n

(i)F and p =

N F

i=1 (i)F

(i)F .

The constitutive relations between conjugated variables, i.e., stress-elastic strain(2.35) and temperature-reversible entropy (2.38), are

S = C F E e and e = hF ln F

+ F .

The relation between plasticity driving force and the rate of plastic slip (kineticrelation) is given by

(i)F = F 0

g(i)F s (i)F

(1/p F )

1 if g(i)F > s

(i)F ,

0 otherwise ,

where the driving force for plastic slip g(i)F includes the contributions of the re-solved shear stress (Schmid stress), the plastic entropy density and the defect en-ergy, i.e,

g(i)F = F T e F eS m

(i)F n

(i)F + 0

(i)F F F F w

(i)F ,

and the effective slip resistance s (i)F = s(i)F a (i)

(i)F , which accounts for the

contribution of the classical slip resistance and the effect of the non-glide stress

(i)F = F T e F eS m

(i)F n

(i)F .

Finally, the evolution laws for the slip resistance s(i)F (hardening model) and themicrostrain F are, respectively, given by

s (i)F =N F

j =1H (i,j )F

( j )F and F =

N F

i=1w(i)F

(i)F ,

where H (i,j )F and w(i)F are, respectively given by (2.51) and (2.56).

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2.2. SIMULATIONS OF ELASTO-PLASTIC DEFORMATION OF SINGLE-CRYSTALLINE FERRITE

[100] F-loaded [110] F-loaded [111] F-loaded

e1F

e2F e3

F

e1F

e2F

e3F

e1F

e2F

e3F f 1

f 2 f 3 Global axis(0,0,0) (45,0,0) (45,35.26,0)

Figure 2.3: Schematic representation of the crystallographic orientation of theferrite single crystal samples with respect to the global basis

{f 1, f 2, f 3

}.

2.2 Simulations of elasto-plastic deformation of single-crystalline ferrite

In order to illustrate the basic features of the crystal plasticity model for BCCferrite, the mechanical behavior of a single crystal ferritic sample is studied bymeans of numerical simulations. In the present work, three elementary loadingmodes are considered, namely (i) uniaxial tension and compression, (ii) simple

shear and (iii) plane-stress equibiaxial stretch. Furthermore, the analyses arecarried out considering three different crystallographic orientations, which, ex-pressed in terms of the 323-Euler rotation (about the global basis with cartesianunit vectors {f 1, f 2, f 3}), are (0 , 0 , 0 ), (45 , 0 , 0 ) and (45 , 35.26 , 0 ), re-spectively, The above orientations are chosen such that the global f 1-axis corre-sponds to, respectively, the [100]F , [110]F and [111]F directions, where the Millerindices refer to the basis of the BCC lattice, as illustrated in Figure 2.3.

2.2.1 Material parameters and validation

The parameters used in the crystal plasticity model with the non-glide stress ef-fect for the BCC ferrite are discussed in this section. The elastic moduli forthe BCC ferrite used in (2.36) are obtained from the data reported in Kurdju-mov and Khachaturyan [73], i.e., F 1 = 233.5, F 2 = 135.5 and F 3 = 118 .0[GPa]. With these data, the equivalent (isotropic) shear modulus is obtained from(2.42) as F = 55.0 GPa. The mass density of the ferrite (in the reference con-guration) is assumed to be equal to the characteristic density of a typical car-

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0

200

400

600

800

1200

1000

A x i a l

C a u c h y s t r e s s

T 1 1 [

M P a ]

0 0.025 0.050 0.075 0.100 0.125 0.150Axial nominal strain 11

Polycrystal model (Taylor average)Experimental data

(75% ferrite + 25% bainite)

Figure 2.4: Axial stress-strain response of a Taylor-type polycrystalline sampletted to experimental data of Jacques et al . [63] for ferrite-based material.

bon steel, 0 = 7800 kgm 3 . For simplicity, it is assumed that the values of the weight parameters for the non-glide stress contribution used in (2.47) are thesame for all slip systems, i.e, a (i) = a . The value for a is calibrated from the dataof uniaxial tensile tests on single crystal BCC ferrite presented in [40], particu-larly, the data on the asymmetry of the resolved shear stress on the

{110

} and

{211} (twinning and antitwinning) planes. Following the procedure highlightedin Bassani et al . [9], the magnitude of a is obtained by tting the following curve: = cr / [cos + a cos( + 60 )] to the experimental data [40]. In this case, cris the critical resolved shear stress on the maximum resolved shear stress plane(MRSSP), the angle denes the orientation of the MRSSP with respect to theslip plane and a is a parameter that characterizes the asymmetry of in the twin-ning and antitwinning senses. This calibration procedure results in a = 0 .12.

Furthermore, the parameters for the power-law kinetic model (2.48), i.e., F 0

and pF , are chosen such that the overall response under quasi-static loading con-ditions is close to a rate-independent response. The purpose of introducing a smallrate-dependency is to avoid numerical singularity problems often encountered inrate-independent crystal plasticity models [88]. For this reason, the parametersfor the kinetic model are taken as F 0 = 0 .001 s

1 and pF = 0 .02, which fallwithin the typical range of values used in rate-dependent crystal plasticity models(see, e.g., [30, 88]).

The hardening parameters used in (2.52) and the initial slip resistance are

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calibrated from the experimental uniaxial stress-strain data of a polycrystallineferrite-based material reported in [63]. Note that in this case the material used as

a reference for the polycrystalline ferrite is a dual-phase material that consists of 75% ferrite and 25% bainite, which is representative of a typical (ferrite-based)matrix of multiphase TRIP-assisted steels. The inuence of the bainite, whichitself is a ne mixture of ferrite and cementite, is implicitly lumped into the fer-rite. Ideally, a separate model for the bainite would be required, however this fallsoutside the scope of the present work. In the calibration procedure, the polycrys-talline response is simulated by combining the present crystal plasticity modelwith a Taylor averaging procedure (see Figure 2.4), which results in kF 0 = 1 .9GPa, sF = 412 MPa, uF = 2 .8 and s

(i)F, 0 := s

(i)F (t = 0) = sF, 0 = 154 MPa (the

initial value for slip resistance is taken to be identical for all systems). In addition,the (isotropic) latent hardening ratio, q F = 1 , is used.

The thermal part of the driving force is assumed the same for all slip sys-tems and its value is set to (i)F = 0

(i)F = 10 MPa at the ambient temperature

( = 300 K). The values for parameters F in (2.44) and cF in (2.54), which arerelated to the defect energy contribution, are obtained from the following heuristicapproach: The value for F is chosen such that in the inelastic regime the orderof magnitude of the defect energy remains a relatively small fraction of the bulk strain energy, which leads to F = 7 .

The terms cF is chosen such that the contribution of the defect energy to thedriving force is about 10 % of the initial value of the critical resistance againstslip, which gives cF = 5 . In addition, it is assumed that the initial value for themicrostrain parameter, F, 0 = F (t = 0) , is related to the (common) initial slipresistance through sF, 0 = cF F F, 0, which gives F, 0 = 5 .610 4 . The summaryof the parameters used in the BCC ferrite crystal plasticity model is presented inTable 2.1 and the list of the vectors of the slip directions, the vector normal to theslip planes and the vector normal to the non-glide planes for the BCC ferrite canbe found in Appendix C .

2.2.2 Sample geometry and boundary conditions

The three elementary deformation (or loading) modes of the single-crystallineferritic samples studied in this section (i.e., uniaxial, simple shear and plane-stressequibiaxial stretch) are illustrated in Figure 2.5.

The single crystal ferrite is represented by a cubic sample with sides of length

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Table 2.1: Summary of the material parameters used in the crystal plasticity modelfor the BCC ferrite.

Parameter(s) Value(s) Equation(s)Elastic moduli F 1 = 233.5, F 2 = 135.5, (2.36)

F 3 = 118 .0 [GPa]Non-glide parameter a = 0 .12 (2.47)Plastic kinetic law F 0 = 0 .001 s

1 , pF = 0 .02 (2.48)Hardening law sF = 412 , sF, 0 = 154 [MPa] (2.51)-(2.52)

kF 0 = 1 .9 GPa, uF = 2 .8, q F = 1Thermal driving force ( i )F = 0

( i )F = 10 MPa (2.44)

Defect energy F, 0 = 5 .6

10 4 , F = 7 (2.44),(2.55)

cF = 5 , F = 55 .0 GPaMass density 0 = 7800 kgm 3

l (see Figure 2.5). In the rst loading case, this cubical sample is subjected toa uniaxial tensile loading up to a nominal strain 2 in the axial direction of 11 =

0.25 along the f 1-axis using a straining rate of 10 4 s 1 , which is obtained byimposing the following boundary conditions: (i) the displacement normal to the

face is set to zero on three mutually perpendicular faces of the cubic sample; (ii)the normal displacement u1 is applied to the top surface, which is prescribed as

u1 =10 4lt for tension ,

10 4lt for compression , (2.57)

with time t running from 0 to 2500 s; (iii) the two remaining faces and the di-rections not specied above are traction-free. Due to anisotropy in the materialproperties of the sample, the above loading condition does not exactly correspondto a uniaxial tension (or compression). Nevertheless, the deviation from an aver-

age uniaxial stress state is found to be negligible.In the second loading case, the simple shear deformation is obtained by ap-

plying a deformation z = z (x ), which, with respect to the global basis, can bewritten in components as

z1(x ) = x1 + x2 , z2(x ) = x2 , z3(x ) = x3 , (2.58)2The nominal strain tensor is dened as := V I , where V is the left stretch tensor in the

polar decomposition of the total deformation gradient F .

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Plane-stress biaxial stretchSimple shear deformationUniaxial tensile loading f 1

2 f 3

f 1

f 2 f 3

f 1

f 2 f 3

l

l

u1 u1

u2

Figure 2.5: Boundary conditions (loading modes) applied to the single crystalferrite samples, i.e., uniaxial tensile loading, simple shear deformation and plane-stress equibiaxial stretch.

where represents the amount of shear applied to the sample, described by =10 4t , with time t running from 0 to 2500 s.

In the last loading case, the sample is subjected to plane-stress equibiaxialstretch boundary conditions, which are dened as follows: (i) on three mutuallyperpendicular faces intersecting at the origin, the displacement normal to eachof these faces is set to zero; (ii) normal displacements u1 and u2 are applied,respectively, to the top face and to the front face (that is perpendicular to the f 2-

axis), as follows:

u1 = u2 =10 4lt for tension ,

10 4lt for compression , (2.59)

with time t running from 0 to 1200 s (up to nominal strains 11 = 22 = 0 .12);(iii) the remaining face and the directions not specied above are traction-free. Foreach of the above cases, three different crystallographic orientations described inthe beginning of this section are simulated and analyzed.

2.2.3 Stress-strain response of single-crystalline ferrite

Prior to the discussion of the results, it is useful to introduce the total accumulatedplastic slip F , dened as the sum of the accumulated plastic slip of all systems i,i.e,

F :=N

i=1

(i)F . (2.60)

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(1)

(3)

(2)

(3)

(1)

(2)

(4)

(4)

-1200

-900

-600

-300

0

1200

900

600

300


T 1 1

[ M P a ]

(1) [100] F-loaded(2) [110] F-loaded(3) [111] F-loaded(4) Polycrystal

-0.20 -0.15 -0.10 -0.05 0 0.05 0.10 0.15 0.20Logarithmic strain e11

Figure 2.6: Cauchy stress T 11 depicted against logarithmic strain e11 of single-and polycrystalline ferrite samples undergoing uniaxial tension and compression.

and the logarithmic strain measure, e , dened as

e := ln V , (2.61)

where V is the left stretch tensor in the polar decomposition of the total deforma-

tion gradient F , i.e., F = V R , with R the rotation tensor. The logarithmic strainis a convenient measure in large deformation simulations due to the followingreasons [69]: (i) it preserves tension-compression symmetry, (ii) its volumetric-deviatoric decomposition is additive, and (iii) the logarithmic strains of two sub-sequent deformations with the same principal stretch directions are additive.

Uniaxial tension and compression

Figure 2.6 shows the Cauchy stress component T 11 in the axial direction plotted

against the corresponding axial logarithmic strain e11 of the single-crystalline fer-ritic samples loaded under uniaxial tension and compression. It can be observedthat the sample loaded in the [111]F direction gives the highest level of stress of the three crystallographic orientations (about 980 MPa in tension and 930 MPain compression measured at an axial logarithmic strain of 0.2). In uniaxial ten-sion, the sample loaded in the [110]F direction shows a higher level of stress thanthe [100]F -loaded sample whereas an opposite trend is observed in uniaxial com-pression. These results were obtained including the effect of the non-glide stress

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(2)

(3)

(1)

(3)

200

0

1200

1000

800

600

400


T 1 1 [

M P a ]

0 0.05 0.10 0.15 0.20Logarithmic strain e11

-1000

-1200

0

-200

-400

-600

-800


T 1 1 [

M P a ]

-0.20 -0.15 -0.10 -0.05 0Logarithmic strain e11

(1) [100] F-loaded(2) [110] F-loaded(3) [111] F-loaded


(1)* (2)* (1)

(3)*

(3)*

(1)* (2)* (2)

(b)(a)

Figure 2.7: Comparison of the axial stress-strain responses between the results of simulation with non-glide stress effect and the classical Schmid law (plotted inthinner lines and their labels are indicated with an asterisk) in (a) uniaxial com-pression and (b) uniaxial tension.

in (2.47). In comparison, the classical crystal plasticity model (Schmid law, witha set to zero in (2.47)) predicts the same stress-strain behavior for the [100]F -and [110]F -loaded samples, both in tension and compression, as shown in Fig-ures 2.7a and b (curves with an asterisk). The difference between the predictionsof the present BCC crystal plasticity model with the non-glide effect and those of the classical Schmid law for the [100]F -loaded sample in tension and the [110]F -and [111]F -loaded samples in compression is due to the fact that in these threecases the non-glide stresses for the active slip systems are positive (and equal forall active systems). Consequently, the magnitude of the effective slip resistances (i)F is smaller than the magnitude of s

(i)F that solely determines the resistance

against slip in the classical Schmid law. Furthermore, Figure 2.6 shows that forthe (Taylor-type) polycrystalline sample, the magnitude of the stress in uniaxial

tension is about 5 % higher than the magnitude of stress in uniaxial compression.The asymmetry of the stress-strain response in tension and compression for thepolycrystalline sample is in agreement with the yield surface reported in [145].

Figure 2.8 shows the evolution of the amount of plastic slip F as a functionof the axial logarithmic strain e11 . The results of the present model with non-glide stress effect incorporated are plotted in thick lines, whereas the results of the corresponding samples simulated using the classical Schmid law are printedin thinner lines. It can be observed from Figure 2.8 that the simulations using

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(1) (2)(1)* (2)*

(1) (2)(1)* (2)*

(3) (3)*(3) (3)*

0

0.150

0.300

0.450

0.900

0.750

0.600

-0.20 -0.15 -0.10 -0.05 0 0.05 0.10 0.15 0.20Logarithmic strain e11

A m o u n t o f p l a s

t i c s l i p F


Figure 2.8: Evolution of the accumulated plastic slip F as a function of logarith-mic strain e11 of single-crystalline ferrite during uniaxial tension and compres-sion, simulated using the non-glide stress model and the classical Schmid law.

the classical Schmid law and the model with the non-glide stress included re-sult in a virtually similar prediction in terms of the total amount of plastic slip(constructed from the same number of active slip systems with equal contribution

from each system). This result conrms that upon uniaxial tension and compres-sion, the asymmetry in the axial stress-strain response is only related to the effectof the non-glide stress. Furthermore, the highest amount of plastic slip is ob-served for the [111]F -loaded sample, both in tension and compression. For thesamples loaded in the [100]F and [110]F directions, the total amounts of slip arenearly identical, despite of the difference in the stress-strain curve, see Figure 2.6.The simulations further indicate that there are eight equally active slip systems inthe [100]F -loaded sample during uniaxial tension and compression, whereas the[110]F -loaded sample shows the trace of only four equally active slip systems,

but with twice the amount of slip per system generated in comparison to the othersamples. For the [111]F -loaded sample, six equally active slip systems are ob-served durin

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