Relation Between Microstructure and Mechanical Properties of a Low-Alloyed TRIP Steel

Relation between Microstructure and Mechanical Properties of a Low-alloyed TRIP steel

Von der Fakultat¨ fur¨ Georessourcen und Materialtechnik der Rheinisch-Westfalischen¨ Technischen Hochschule Aachen

zur Erlangung des akademischen Grades eines

Doktors der Ingenieurwissenschaften

genehmigte Dissertation

vorgelegt von M.Sc.

Kemal DAVUT

aus Kadikoy,¨ Turkei¨

Berichter: PD Dr.-Ing. Stefan Zaefferer Professor Dr.-Ing. Dierk Raabe Univ.-Prof. Dr.-Ing. Wolfgang Bleck

Tag der mundlichen¨ Prufung:¨ 12. Februar 2013

Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfugbar.¨

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Kemal Davut

Relation between Microstructure and Mechanical Properties of a Low-alloyed TRIP steel

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Zugl.: D 82 (Diss. RWTH Aachen University, 2013)

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ISBN 978-3-8440-1782-3 ISSN 1618-5722

Shaker Verlag GmbH • P.O. BOX 101818 • D-52018 Aachen Phone: 0049/2407/9596-0 • Telefax: 0049/2407/9596-9 Internet: www.shaker.de • e-mail: [email protected] Preface

This thesis is submitted in partial fulﬁllment of a Dr.-Ing degree from the RWTH Aachen University. The work presented in the thesis was carried out during the pe- riod April 2008 to September 2012 at the Max-Planck-Institut fuer Eisenforschung (MPIE), in the Department of Microstructure Physics and Alloy Design, under the supervisions of Associate Professor Dr.-Ing Stefan Zaefferer and Professor Dr.-Ing Dierk Raabe. The work was a part of the joint research project between MPIE and RWTH Aachen University, having the title ”The relation between microstructure and damage mechanisms in multi-phase steels (Zusammenhang zwischen Gefuge¨ und Schadigungsmechanismen¨ bei mehrphasigen Stahlen)”¨ and ﬁnancially supported by the German Research Foundation (DFG - Deutsche Forschungsgemeinschaft) under reference numbers BL 402/19-1 and ZA 287/5-1

Acknowledgements

This doctoral thesis come into its final form with the help and support of kind people around me, to only some of whom it is possible to give particular mention here. First and foremost, I gratefully thank Prof.Dr.-Ing. D. Raabe, the head of the department and the director of the institute, for giving me the opportunity to carry out this project. Valuable and inspiring discussions together with the given scientific freedom have made the results possible. This work has reached this stage with the strong support, constant encouragement and day-to-day supervision of P.D.Dr.-Ing. Stefan Zaefferer. His comments have often set me on the right track towards the solution. Besides introducing me the world of EBSD and texture analysis, our stimulating discussions over the years have always inspired and helped me to improve my scientific formation. Thank you for sparing time for me, for the really great work together, and for your friendship. I am grateful to the defense committee members, Prof.Dr.-Ing. D. Raabe and Prof.Dr.- Ing. W. Bleck for their valuable comments on this thesis. My working days (and nights and also conference trips) have become enjoyable due to the friendly and cheerful colleagues in the institute, particularly to mention Tom, David, Dr. A. Khorashadizadeh, Pradeep, Darius, Dr. O. Dmitrieva, Dr. C. Tas¸an, Igor, Xia, Farangis, Dr. Duancheng Ma, Helene, Dr. P. Romano, Dr. D. Tjahjanto, Dr. H. Fabritius and also Dr. I. Gutierrez. Eralp, Mutlu, Bekir, Ceylan and Onur have formed a small Turkey inside the Institude and they all deserve a separate thank. Special thanks to Claudio for sharing his Schmidt factor calculation code and LATEX header files. I also thank Dr.-Ing. D. Ponge and Dr.-Ing. P. Eisenlohr for the valuable discussions. The technical assistance and support for the experimental set-ups by M. Nellessen, K. Angenendt, H.Bogershausen,¨ M. Adamek, H. Faul, F. Schluter,¨ G. Bialkowski and B. Breitbach are gratefully acknowledged. C. Regner and P. Siegmund are acknowledged for the fast and reliable library services. A. Kuhl and B. Beckschafer¨ are thanked for their continuous support in the areas of computational hardware and software. I greatly acknowledge the financial support of the project by German Research Founda- tion (DFG). Department of Ferrous Metallurgy (IEHK, Institut fur¨ Eisenhuttenkunde)¨ in RWTH Aachen University is specifically thanked for providing the steel sheets used in this study. Prof. A.D. Rollet is thanked for a very helpful discussion during a GLADD meeting at UniGhent. This specific discussion brought the idea of applying a logistic regression vi model to austenite stability. I have appreciated the expertise of Peggy Schmidt from Thyssen & Krupp Stahl AG on determining the phase fractions of TRIP steels via XRD. Last, but by no means least, I express my deepest gratitude to my family, especially to my parents Lale and Haluk for their endless love and long-distance support. My aunt Oya Araslı needs to be mentioned separately, as she helped me to make the deci- sion of proceeding an academic carreer. I also thank my friends in my home-country, especially Serhat, Anıl, Çagrı,ˇ Barıs¸, Yudum, Aslı and Eylem for keeping the social enviroment warm during my visits. Special thanks to Dr. Caner S¸ims¸ir for the friendship, stimulating discussions and also for showing me an example life-cycle between Germany and Turkey. My final thanks go to Ozlem¨ who not only helped me with the statistical analysis but more important than that she brought more joy and happiness to my life. A single word thanks is not enough for the care and support I received. Contents

Preface iii

Acknowledgements v

1 Introduction 1 1.1 Background: Low-alloyed TRIP steels ...... 1 1.1.1 How are the TRIP steels produced ? ...... 4 1.1.2 Alloying concepts ...... 6 1.1.3 Martensitic transformation in TRIP steels ...... 7 1.2 Objectives and Scope ...... 9 1.3 Thesis Outline ...... 10

2 Experimental 13 2.1 Material ...... 13 2.2 Microstructure characterization ...... 13 2.3 Mechanical tests ...... 15 2.4 FEM model ...... 15

3 The reliability of EBSD-based microstructure description 19 3.1 Introduction and motivation ...... 19 3.2 EBSD-based phase fraction determination ...... 20 3.2.1 Estimation of the required number of measurement points for statistical reliability ...... 20 3.2.2 The effect of step-size and observed area size on reliability . . 22 3.2.3 Determination of the size of a representative area ...... 23 3.2.4 Error estimation ...... 28 3.2.5 EBSD vs. XRD ...... 30 viii CONTENTS

3.3 EBSD-based macro-texture analysis ...... 32 3.3.1 Measuring the statistical reliability of a texture analysis .... 33 3.3.2 Comparing the results of conventional EBSD mapping techniques ...... 34 3.4 A new mapping technique ...... 36 3.4.1 The need for a new technique ...... 36 3.4.2 The application of the technique ...... 36 3.4.3 Comparing the results of the new and conventional EBSD mapping techniques ...... 38 3.4.4 EBSD vs. XRD based texture analysis ...... 43 3.5 Conclusions ...... 44

4 The effect of microstructure on macroscopic work-hardening rate and stress-strain partitioning 49 4.1 Introduction and motivation ...... 49 4.2 Martensitic transformation and the work hardening rate ...... 49 4.3 The effects of grain size and morphology on the deformation of austenite 53 4.4 The effect of grain shape on stress partitioning ...... 55 4.5 Conclusions ...... 61

5 Comparative study of the texture evolution of TRIP microstructure constituents during room temperature deformation 63 5.1 Introduction and motivation ...... 63 5.2 Texture development during bending ...... 63 5.2.1 Changes in BCC texture components ...... 64 5.2.2 Changes in FCC texture components ...... 64 5.3 Relation between BCC and FCC shear ...... 67 5.4 Comparing the changes in FCC and BCC texture components ..... 68 5.5 Conclusions ...... 69

6 Microstructural parameters influencing the stability of retained austenite 71 6.1 Introduction and motivation ...... 71 6.2 The detection of deformation and transformation of meta-stable austenite 72 6.3 Identification of stable and un-stable austenite grains ...... 80 6.4 Factors influencing the stability of austenite ...... 82 CONTENTS ix

6.5 Relative contributions of the microstructural parameters ...... 84 6.5.1 Grain size and composition ...... 84 6.5.2 Crystallographic orientation ...... 86 6.5.3 Grain morphology ...... 93 6.5.4 Statistical analysis of the contribution of the microstructural parameters ...... 93 6.6 Why the transformation is not continuous ? ...... 100 6.7 Remarks ...... 101 6.8 Conclusions ...... 101

Summary and Outlook 103

Zusammenfassung (German summary) 107

A Considere` Criterion 111

B Common BCC Texture Components 113

C Common FCC Texture Components 115

D Precision and Accuracy 117

Bibliography 119

Curriculum Vitae 131

CHAPTER 1

Introduction

It is of greatest importance for the industry, specifically for the automotive industry, to develop steels with a better combination of strength and formability. Such steels will allow weight reduction, improved fuel-efficiency and environmental friendliness as well as higher passenger safety. The importance of steels with better combination of mechanical properties has been reflected as an increase in the tensile strength (TS) of each body component and also as an increasing ratio of high-strength-steels (HSS) in automobile bodies (Oliver et al., 2007, Kuziak et al., 2008, Takechi, 2008, Takita and Ohashi, 2001, Takahashi, 2003). Figure 1.1 compares the tensile strength and total elongation of common automotive steels. Conventional steel types such as interstitial free (IF) steels, low-carbon steels or high-strength-steels (HSS) follow an inverse ex- ponential relation in the figure, indicating that the strength can only be improved at the expense of ductility, and vice-versa. The advanced high strength steel (AHSS) grades and specifically the steels with transformation induced plasticity (TRIP), on the other hand, possess a better combination of high strength and good ductility.

1.1 Background: Low-alloyed TRIP steels

The term ”TRIP steel” actually covers a wide range of steels containing a meta-stable austenite phase, which can be present in room temperature microstructure of both high and low alloyed steels. In this contribution, only low-alloyed TRIP steels will be considered due to their increasing industrial importance. Typical microstructure of an undeformed low-alloyed TRIP steel is composed of ferrite, bainite, metastable retained austenite and potentially martensite. Intercritical ferrite (also referred to as pro-eutectoid ferrite) is the most dominant phase since it occu- pies up to 80% of the microstructure. Ferrite has a body-centered cubic (BCC) lattice and it is the softest phase among the constituent phases. The size of ferrite grains in a typical TRIP steel are in the range of 5 − 10 μm (Furnemont et al., 2002, Jacques et al., 1999). Unlike ferrite, a typical bainite is not a single-phase constituent. The microstructure of bainite consists of ﬁne platelets of ferrite that form via a displacive (diffusionless) mechanism and cementite (Fe3C) that precipitates in-between and/or inside the ferrite platelets. In general, bainite is harder than intercritical ferrite due to its ﬁner structure as well as the presence of carbide precipitations. The bainite in TRIP steels is generally referred to as ”bainitic ferrite” and it is not exactly the same as the above explained typical bainite. The bainitic ferrite in TRIP steels is essentially 2 1.1 Background: Low-alloyed TRIP steels n (%) elongatio Total

Tensile strength ( MPa)

Fig. 1.1: Comparing the strength and elongation of common automotive steels including interstitial free (IF), high strength low alloy (HSLA), bake hardening (BH), martensitic (MART) as well as dual phase (DP), complex phase (CP) and TRIP steels (USS-webpage, 2010). carbide-free due to the presence of Al or Si, which restricts (or postpones) carbide precipitation. The key constituent of the TRIP steel microstructure is retained austenite, which is a meta-stable phase at room temperature. Austenite is normally stable at high temperatures and has a face-centered cubic (FCC) lattice. Carbon enrichment and the con- straining effect from neighbouring grains can stabilize austenite at room temperature. The strength of retained austenite in TRIP steels is generally higher than that of the ferritic constituents due to the strengthening effect of carbon. Occasionally, the initial TRIP steel microstructure can contain a small fraction of thermal martensite, that forms when the austenite is rapidly cooled (i.e. quenched). Martensite has a body- centered tetragonal (BCT) lattice containing supersaturated interstitial carbon atoms. It is interesting to note that carbon in interstitial solid solution expands the FCC-iron lattice uniformly, but with BCC-iron the expansion is non-symmetrical giving rise to tetragonal distortion (Honeycombe and Bhadeshia, 1995). Martensite has a high dislocation density due to the displacive (diffusionless) transformation mechanism and it is generally the strongest phase in steels. The unique mechanical properties of TRIP steels are mainly attributed to the presence of the metastable austenite phase in the room temperature microstructure. Upon application of mechanical and/or thermal loads, the meta-stable retained austenite can transform into a harder martensitic phase, which would increase the effective strength of the material. Moreover, the co-existence of hard and soft phases (composite effect) could also explain the enhanced mechanical properties of TRIP-steels. During mechanical CHAPTER 1INTRODUCTION 3

loading, the plastic strain will be focussed ﬁrstly in the soft phase and the harder phase can reserve its ductility until the later stages of overall deformation. Many publications on TRIP steels have attributed the enhanced ductility of TRIP steels to transformation strain. However, calculations of Bhadeshia (2002) has shown that only 0.75 − 2.25 % strain can be contributed from transformation strain for a TRIP steel containing 5−15 % retained austenite; assuming the formation of the most favoured variant and complete transformation into that variant. If variants form randomly then the contribution will be even smaller. Quite possibly, the role of transformation plasticity has been exag- gerated. On the other hand, the martensitic transformation in TRIP steels brings an additional strain hardening mechanism and therefore contributes to an increase in plasticity in line with the Considere` criterion (see Appendix A). The Considere` criterion states that the necking starts when

dσt = σt (1.1) dt here σt and t denote the true stress and true strain, respectively. ε ε /d t

t dσ t

rate,d high dε Stress, dening r e dσ low t Tru dε ork ha W

low high uniformeltilongation uniformeltilongation True Strain,ε

Fig. 1.2: Schematic representation of dependence of ductility on hardening rate. The ’s in the ﬁgure represent the points at which Equation 1.1 namely is satisﬁed; meaning necking starts according to Considere` criterion. (Courtesy of D. Ponge).

Figure 1.2 compares the uniform elongations (ductility) of a high strength and a low dσ strength steel, both having a low hardening rate ( d ). If one tries to improve the strength of a steel without changing the hardening rate, then the Equation 1.1 would be satisﬁed at a lower t value. On the other hand, simultaneous improvement of both strength and ductility can be realized via improving the hardening rates as shown in dσ Figure 1.2 for the case of the steel with high hardening rate ( d ). Therefore, increas- 4 1.1 Background: Low-alloyed TRIP steels

ing the strain hardening rate postpones necking and hence improves ductility according to the Considere` criterion.

4000 ε /d

t 3500

d 3000 t 2500 without TRIP effect ess, grate,

r 2000 present TRIP steel n

1500 True St ardeni 1000 h 500

Work 0 0.0 0.1 0.2 0.3 0.4 True Strain,ε

Fig. 1.3: The true stress vs. true strain and the corresponding hardening rate curves of the present TRIP steel compared to a hypothetical high strength steel exhibiting no TRIP effect (or any other extra-ordinary hardening mechanism). The ’s in the ﬁgure represent the points at which necking starts according to Considere` criterion.

The austenite to martensite transformation in TRIP steels significantly increases the hardening rate and therefore improves the ductility compared to similar high strength steels (without TRIP effect) as shown in Figure 1.3. Moreover, increasing the martensite content during deformation increases the effective strength of TRIP steels. In comparison to similar high strength steels that does not contain retained austenite (i.e. DP-steels), TRIP steels have comparable (or even identical) strength but significantly higher ductility. 1.1.1 How are the TRIP steels produced ? The earlier TRIP steels were having meta-stable austenite in their room temperature microstructure due to higher amount of austenite stabilizing alloying elements. On the other hand, modern low alloy TRIP steels require a special heat treatment in order to stabilize the austenite at room temperature. Figure 1.4 shows the time-temperature profile of the two-stage heat treatment that is typically used in the processing of low alloyed TRIP steels. The first stage of this heat treatment process is the intercritical annealing, in which the steel is brought to a temperature between the intercritical temperatures A1 and A3; similar to the processing route of dual-phase (DP) steels. At the end of the intercritical annealing the microstructure consists of (pro-eutectoid) ferrite and austenite. Since the CHAPTER 1INTRODUCTION 5

1000 γ 900 α ) ) 800 C

o 700 600 TRIP ure ( t t bainite 500 γ α r b 400 Ms α 300 mpera

e 200 DP T 100 α´ 0 α 1 10 100 1000 10000 100000 1000000 Time (s)

Fig. 1.4: Schematic representation of the time & temperature profile of the two-stage heat treatment used in low-alloyed TRIP steels and DP- steels; superimposed into a schematic continuous cooling (CCT) dia- gram. The corresponding microstructural phases obtained at the end of each step are also shown. Note that, the TRIP-steel microstructure at the end of bainitic holding step is frozen to room temperature via quenching. solubility of carbon in ferrite is extremely low (< 0.02 wt %), the remaining carbon dissolves and increases its concentration in the austenite which in turn decreases the Ms temperature of the steel slightly. The resulting microstructure at the end of this step is quenched to room temperature in case of DP-steels as schematically shown in Figure 1.4. In case of TRIP steels, the microstructure is rapidly cooled to the bainite region. The second step of the heat treatment of TRIP steels is bainite holding. During this isothermal holding, a part of the austenite transforms into bainitic ferrite. As the bainite transformation proceeds the remaining austenite is enriched with carbon that is ex- pelled from the newly formed bainite. The bainite in TRIP steels is essentially carbide- free, since the selected chemical composition inhibits (or postpones) the formation of carbides. After the bainite holding the microstructure is composed of carbon depleted ferrite (both bainitic ferrite and intercritical ferrite) and austenite that is stabilized by carbon enrichment. This phase composition is preserved during the final quenching to room temperature. A small fraction of retained austenite can further transform into thermal martensite during the final quenching, specifically in the austenitic regions where the carbon enrichment is not sufficient. An alternative way of producing room temperature stabilized austenite has been proposed by Speer et al. (2003). The proposed ”quenching and partitioning” (Q & P) process can create a mixture of carbon depleted martensite and carbon enriched retained 6 1.1 Background: Low-alloyed TRIP steels austenite (Clarke et al., 2008, 2009, Moor et al., 2008). The proposed processing concept involves formation of a controlled fraction of martensite upon quenching to a temperature between Ms and M f ; followed by partitioning of carbon into austenite. This process differs from conventional quenching and tempering, in which the carbon supersaturation of martensite is relieved by precipitation of carbides and retained austenite decomposes into ferrite and cementite. Here, alloying additions should be chosen to suppress carbide formations; as in the case of the previously explained bainitic holding step. 1.1.2 Alloying concepts The key factor in the selection of alloying elements is to improve the stability of retained austenite by using an economical low-alloy concept. Unlike the earlier high alloyed TRIP steels, the newer low-alloyed TRIP steels contain no nickel and only 1.5−2 wt. % of manganese. Mass production of TRIP steels, without changing the current steel-production lines has been realized by the employment of low alloy concepts. The role of common alloying elements in low alloyed TRIP steels can be summarized as follows: Carbon is the most effective austenite stabilizer and it improves the strength of austenite and martensite. Carbon contents higher than 0.2 wt.% are not preferred for weld- ability requirements. Silicon is not soluble in cementite, therefore it prevents (or delays) carbide precipitation during the bainitic reaction which in turn helps to stabilize retained austenite via carbon enrichment. In addition, silicon improves the strength of ferrite due to solid solution strengthening. On the other hand, it causes formation of a strong oxide layer during hot-rolling. This layer can hardly be removed which in turn deteriorates the surface quality and constricts the subsequent galvanizing treatment of cold-rolled steels. Aluminum is used instead of silicon in order to improve the surface quality and coata- bility of steels. However, aluminum is less effective in suppressing carbide precipitation and it has a lower solid solution strengthening potential. Aluminum is generally used in combination with phosphorus in order to compensate these shortcomings. In comparison to Si-TRIP steels, Al-TRIP steels exhibit lower strength and higher elongation. Manganese is an austenite stabilizer and it retards the pearlite and bainite formations; therefore, it extends the applicable range of cooling rates. In addition, it strength- ens ferrite by solid solution strengthening. Too much manganese additions should be avoided because it promotes carbide formation. Phosphorus is a ferrite stabilizer and has the same effects as silicon. Less than 0.1 wt% additions was proven to be sufficient to retard carbide precipitation and to strengthen ferrite. Micro-alloying elements such as niobium, titanium and vanadium are occasionally used in small amounts in order to improve the mechanical properties by grain refinement and precipitation hardening. In general, micro-alloying elements impede the pro- cesses that need the movement of dislocations and/or grain boundaries. CHAPTER 1INTRODUCTION 7

1.1.3 Martensitic transformation in TRIP steels The name martensite was ﬁrst used by Osmond in 1895, in honor of the German met- allurgist Adolf Martens, in order to identify the very hard, platelike or lenticular microstructure constituent in many steels that are rapidly quenched from the austenite state (Sinha, 2003). Later, this name was extended to include a number of solid state transformations in other metals and alloys, including shape memory alloys. Ideally, the martensite transformation is a diffusionless shear transformation and highly crystallographic in character (Honeycombe and Bhadeshia, 1995). Rapid quenching of austenite to room temperature often results in the formation of martensite, in which the carbon, formerly in the solid solution in the austenite, remains in solution in the newly formed phase. This entrapped carbon gives rise to tetragonality; therefore martensite in steels is a metastable body centered tetragonal (BCT) phase. Martensite transformation is driven by energy minimization among the phases, like any other phase transformations. At high temperatures the austenite has a lower free- energy; whereas at lower temperatures body-centered cubic (BCC) ferrite has lower free energy and is, therefore, the stable phase. The free energy difference between the FCC and BCC structures is the driving force for martensitic transformation. Figure 1.5 schematically shows the free energy of austenite and martensite phases as a function of temperature. ) G ( y ΔG G A M = barrier

e Energ ΔG G A M < barrier e

G mech G Gibbs Fr M

G A

Ms RT Temperature

Fig. 1.5: Schematic representation of free energy of the austenite and martensite phases as a function of temperature

The transformation of austenite into martensite starts at the Ms temperature, at which the free energy difference between austenite and martensite (ΔGA→M) is sufﬁcient to overcome the transformation energy barrier. The Ms temperature depends on the chemical composition of the steel. The Ms temperature for the nominal composition of a 8 1.1 Background: Low-alloyed TRIP steels

◦ low-alloyed TRIP steel is about 350 C. On the other hand the Ms temperature for the austenite in TRIP steels is well below room temperature since the retained austenite has been enriched in carbon during the bainitic holding process.

At temperatures higher than Ms, the martensitic transformation can be triggered by the application of external mechanical loading. Here, the mechanical energy term (Gmech) is added to the free energy difference (ΔGA→M) in order to overcome the transformation barrier (ΔGbarrier) as shown in Figure 1.5. It should be noted that martensitic transformation in carbon steels is irreversible in contrast to shape memory alloys, where stress assisted transformations are crystallographically reversible and are associated with a considerable temperature hysteresis. The reverse reaction from martensite to austenite in carbon steels can only be realized by re-heating; however this reverse reaction is intervened by tempering. The martensitic transformation triggered by deformation can be classiﬁed into two modes, namely, stress-induced and strain-induced transformations; based on the ori- gin of the nucleation sites. The interrelations between applied stress, plastic strain, testing temperature and martensite transformation could result in different nucleation mechanisms. These interrelations were studied extensively by Bolling and Richman (Bolling and Richman, 1969, 1970a,b, Richman and Bolling, 1971), and a temperature σ Ms (which is normally above Ms) was deﬁned to distinguish between stress-induced and strain-induced transformation. ress σ γ y lied st p Ap

σ Ms Ms Md Temperature

Fig. 1.6: Schematic representation of the interrelation between stress, temperature and martensite nucleation. The critical stress required for stress or strain induced martensitic transformation is shown as a function σγ σ of temperature. The regular yield strength of austenite y and the Ms temperature are also indicated. (based on Brandt and Olson (1993)) CHAPTER 1INTRODUCTION 9

Figure 1.6 compares the critical stress required for stress or strain-induced transformation as a function of temperature. The yield strength of austenite and the Ms temper- σ ature are also indicated. Below the Ms temperature martensitic transformation occurs below the regular yield stress (σy) of austenite and it causes plastic flow of the sample. The transformation is believed to be predominantly initiated by nucleation at the same pre-existing sites responsible for the spontaneous transformation upon cooling, but assisted by elastic stresses (Olson and Azrin, 1978, Olson and Cohen, 1976), there- σ fore defined as stress-induced. Above the Ms the transformation is initiated after the applied stress reaches or even exceeds the yield strength of austenite (Olson and Co- hen, 1972, 1975b). New nucleation sites can be introduced by plastic deformation and this mechanism is defined as strain-induced nucleation since plastic deformation of austenite precedes the transformation. The plastic deformation of austenite could continue until fracture as represented by the point • in Figure 1.6. This point also defines the highest temperature Md, at which the chemical driving force is so small that it is practically impossible to nucleate martensite, even by external mechanical loading. Figure 1.7 illustrates the volume fraction of mechanically induced transformation as < σ > σ a function of strain for stress-induced (T Ms ) and for strain-induced (T Ms ) transformation modes. ction ction a a site fr site fr n n Marte Marte

Strain Strain < < σ σ < < (a) for Ms T Ms (b) Ms T Md

Fig. 1.7: Schematic representation of transformation behavior showing the increasing martensite fraction as a function of strain for (a) stress- induced, and (b) strain induced modes. (based on Olson (1986))

1.2 Objectives and Scope

TRIP steels exhibit superior mechanical properties; however, there is still a demand and possibility for further improvements. These improvements can only be realized after a thorough understanding of the relation between the microstructure and mechanical properties. The present work, therefore, aims at addressing the following research objectives:

• First, obtaining an accurate and reliable description of the microstructure. 10 1.3 Thesis Outline

• Second, correlating the high resolution microstructure description to the macroscopic mechanical properties, speciﬁcally to the high strain hardening rate and to the related martensitic transformation. • Third, identifying the effect of each microstructural parameter separately on the mechanical properties and ﬁnd out the relative contribution of each parameter particularly on the martensitic transformation and the corresponding austenite stability.

These objectives could provide a good insight for further improvement of the performance of TRIP steels, as well as for the optimization of the processing parameters. 1.3 Thesis Outline The experimental details will be given at Chapter 2 and the results of this work will be presented in the following 4 chapters. In Chapter 3 the statistical reliability and the representativeness of the EBSD technique, which has been extensively used in this study for the characterization of the microstructure is discussed. The rather heterogenous, fine and multiphase microstructure of the present TRIP steels makes this task extremely important. Optimum sampling conditions for phase fraction determination and texture analysis are determined in a systematic analysis. Errors that may arise due to insufficient sampling or false assignment of phases are also discussed. Moreover, a new mapping technique, that improves the reliability and representativeness of EBSD-datasets without large sacrifices in measurement time or data set sizes, is presented. The results of this new mapping technique are compared to the results of conventional EBSD-mapping schemes and also to the XRD technique. The macroscopic mechanical properties will be related to the microstructure in Chap- ter 4. Here, the deformation and transformation of austenite grains are investigated by combining ex-situ bending tests with EBSD analysis. A finite element model (FEM) is used to relate the EBSD based results obtained in the bending experiments to the hardening behavior obtained from tensile tests. Particularly the effects of size and shape of austenite grains on the macroscopic work-hardening behaviour are discussed. More- over, the effect of microstructure on load partitioning is interpreted via a simple rule of mixture and a short fiber reinforced composite model. The texture development during tension and compression of TRIP steels is presented in Chapter 5. Based on a simple analysis using the relation between FCC and BCC shear geometries, theoretically expected changes in texture components due to deformation are proposed. By comparing the results of this theoretical analysis to experimental observations the changes in austenite texture due to deformation are distinguished from those due to transformation. Moreover, selective transformation of certain austenite texture components is identified. Chapter 6 focusses on the microstructural parameters influencing the stability of retained austenite. The transformation of individual austenite grains into martensite during deformation was observed by combining ex-situ bending tests and EBSD investigations in the same manner as in Chapter 4. The morphological, chemical and CHAPTER 1INTRODUCTION 11 crystallographic orientation variations of the austenite grains in the same material are discussed. The influence of each these microstructural parameters as well as the interplay among them are studied both qualitatively and quantitatively. Particular attention is given to the distributions of these parameters, which plays an important role due to the heterogeneity of the microstructure.

CHAPTER 2

Experimental

2.1 Material

The material used in this study was a 0.8 mm thick, cold rolled and TRIP-annealed sheet of a low alloyed Al-TRIP steel whose composition is given in Table 2.1. The TRIP-treatment was composed of two steps; an intercritical annealing step at 890 ◦C for 4 min, followed by cooling down to 400 ◦C with a rate of 50 K\s and holding there for 420 s.

Table 2.1: Chemical composition (wt.%) of the TRIP-steel used

CSiMnP S CrMoNiAlNbFe 0.2 0.04 1.6 0.02 < 0.001 0.02 < 0.008 0.02 1.4 0.001 Bal.

2.2 Microstructure characterization

The microstructure of the present TRIP steel was characterized mainly by using orientation microscopy based on electron backscatter diffraction (EBSD). The EBSD technique allows a reliable distinction and quantitative description of austenite, ferrite, bainite and martensite (Gourgues et al., 2000, Petrov et al., 2007). The shallow interaction volume of EBSD (Figure 2.1), necessitates the specimen surfaces to be free from any artifacts due to preparation. The following procedure was used to obtain relief-free, artifact-free surfaces, which was also very important for the subsequent phase determination. Firstly, the samples were ground mechanically by conventional SiC grinding papers followed by mechanical polishing sequentially by 3 μm and 1 μm diamond pastes, 0.05 μm-diameter colloidal silica particles. After- wards, the samples were chemically polished by immersing them in a solution of 10 % HF in H2O2. Finally, electropolishing was performed using Struers A2 electrolyte at 20 V at 20 ◦C for 20 sec. Orientation microscopy was performed using a JEOL JSM 6500 F ﬁeld emission gun (FEG) scanning electron microscope, equipped with an EDAX/TSL EBSD system and a Digiview EBSD camera. The acceleration voltage was 15 kV and the working distance 15 mm; maps were measured on a hexagonal grid with step sizes of 100 nm, 14 2.2 Microstructure characterization

0.5 μm,1μm and 5 μm. These measurment parameters provide a specimen - electron beam interaction volume, whose size is schematically shown in Figure 2.1, according to Zaefferer (2007). The EBSD camera was run at 40 frames s−1, with a 0.03 exposure time in 4 × 4 binning mode (312 × 234 pixels). For the scans with 0.5 μm,1μm and 5 μm step sizes, the ”combo-scan” mode of the EBSD software, in which a stage and a beam movement are combined in order to scan larger areas, was used. Each of the individual beam-scan ﬁelds was measured at a magniﬁcation of 1000 × to ensure that all points were in focus.

primary electron energy: 15 kV

half width: depth: 20… 35 nm 5…10 nm Fig. 2.1: Schematical representation of the size and shape of the electron beam - specimen interaction volume from which EBSD patterns originate. The interaction volume was determined experimentally by Zaefferer (2007) for Fe-based alloys. (Courtesy of S. Zaefferer)

During post-processing of the raw EBSD data, firstly the grain confidence index standardization (CIS) method was used. The CIS method checks the confidence index of all points within a grain and then assigns the highest found value to all points in that grain. Afterwards, a minimum confidence index filter of 0.1 was used to exclude only the cer- tainly falsely indexed points. Note that, this clean-up procedure does not change the measured orientation of any of the pixels. The grains were defined and reconstructed from sets of neighbouring pixels having a misorienation less than 5 ◦ between each other. The retained austenite fraction and the macro-texture of the TRIP steel used in this CHAPTER 2EXPERIMENTAL 15

study was determined also by X-ray diffraction (XRD). The measurements were performed by a Bruker D8 Discover diffractometer equipped with a laser-video alignment system and an area detector. Monochromated Co Kα,40kV,40mA radiation was used to determine the diffraction peak intensities and incomplete pole figures in a reflection geometry. For phase fraction determination, {110}, {200}, {211} peaks of ferrite and {111}, {200}, {220}, {311} peaks of austenite were used. Moreover, the whole diffraction pattern fitting procedure, which is known as the method of Rietveld (1967, 1969), was used to determine the peak intensities. The generalized spherical harmonic series expansion method of Bunge (1982) was used for texture analysis. The harmonic series were expanded to a rank (L) of 22, and a Gaussian smoothing with a half-width of 5◦ was used. For XRD-based texture analysis the incomplete pole figures {110}, {200}, {211} of ferrite and {111}, {200}, {220} of austenite were used to calculate the three-dimensional oriention distribution functions (ODF) via the mentioned series expansion method. The recalculated pole figures (from the ODF) were used to estimate the quality of the results by symmetry analysis.

2.3 Mechanical tests

The deformation and transformation of austenite was followed by interrupted ex-situ bending tests. Here, bending tests were preferred because sample size can be smaller than a tension test, it can be realized easily and approximately uniaxial deformation can be conﬁned to a smaller area than in a tensile test. The microstructure evolution during bending was then observed by EBSD. The dimensions of the samples and the bending set-up are schematically shown in Figure 2.2. All bending tests were performed on a special miniaturized tensile stage with two moveable crossheads, which is produced by Kammrath & Weiss GmbH to be used inside scanning electron microscopes. The hardening curve and the true stress vs. true strain curve of the present TRIP steel were obtained via a simple tension test of small dog-bone type samples. For the tension test the digital image correlation (DIC) technique was employed using a commercial system from GOM mbH. The DIC technique (also known as photogrammetry) allows obtaining both the macroscopic and local variations of the displacement ﬁeld in order to calculate some components of the plastic strain tensor at each deformation step. The technique is based on recognition of geometrical changes in the grayscale distribution of a surface pattern before and after straining. The light optical images of the surface patterns, which were produced by applying a color spray on the surface of the tension test specimen, were recorded by high resolution CCD cameras.

2.4 FEM model

In order to relate the EBSD based results of the bending experiment to the hardening and true stress-true strain curves of the tensile experiment, a 3-dimensional finite element model (FEM) based on the commercial program MSC Marc-Mentat was employed. This model was based on finite strain J2 flow theory with isotropic hardening. The true stress-true strain curve of the tensile experiment was used as an input 16 2.4 FEM model

ND 0.8 mm RD TD 8 mm 16 mm

2 mm

10 mm

Fig. 2.2: The dimensions of the samples and bending set-up

to the model for deﬁning the material behavior. Due to symmetry, only one quarter of the specimen was considered to build up the FE-mesh composed of eight-noded 3-dimensional solid elements. It is worth mentioning that the correspondence between the true stress-true strain curve plus the hardening curve and the microstructure quantiﬁcation via EBSD during bending tests was made using only the strains calculated by the FEM model. Due to the extra-ordinary hardening behaviour of the present TRIP steel, the actual hardening during the bending test may not be captured by the present FEM model. On the other hand, the bending strains are mainly determined by the geometry of the testing set-up, (i.e. distance between base supports, diameter of the center cylindrical support, thickness of the test specimen). Therefore, this procedure provides a reliable distinction of the deformation stages that are explained in detail in the following chapters. CHAPTER 2EXPERIMENTAL 17

Fig. 2.3: Undeformed and deformed mesh of the bending simulation.

CHAPTER 3

The reliability of EBSD-based microstructure description

3.1 Introduction and motivation

The EBSD technique is an ideal tool to describe the multiphase nature of materials in a quantitative manner provided that the diffraction patterns of different phases can be distinguished clearly. EBSD has been widely used in past years and it still increases its popularity which is reﬂected as an progressive increase in the number of publications per year as shown in Figure 3.1.

800

ICOTOM 15

600

blications ICOTOM 14 u 400

ICOTOM 13

200 Number of P

0 1990 1995 2000 2005 2010 Year

Fig. 3.1: The number of EBSD-related papers published per year since 1990. A remarkable increase, apart from the regular increase trend, is observed in certain years in which ICOTOM conferences were orga- nized. The number of papers were determined from ISI Web of Knowl- edgeISI-webpage (2012).

The statistical reliability of EBSD based results, despite its popularity and widely usage, remains an open question since typical observation volumes for EBSD are usually 20 3.2 EBSD-based phase fraction determination small compared to other techniques such as XRD. In particular, the tendency to use smaller step sizes for materials with a finer microstructure limits the investigation to relatively small areas in order to complete the measurements within the available (or a reasonable) time. Therefore, generalized conclusions are frequently drawn based on information obtained from a relatively small volume of the material. Particularly for heterogenous materials, scans of smaller areas quite possibly give rise to inaccuracies. With regard to texture analysis, EBSD has been compared to XRD-based techniques in determining the required number of individual orientation measurements (Baudin and Penelle, 1993, Wright et al., 2007). In those studies the symmetry of pole figures, which actually reflects the symmetry of the material’s production process, was used as a measure of representativeness. Moreover, Wagner et al. (2002) and Ubhi (2005) have used a multi-map acquisition technique in order to determine ODF precisely. With regard to austenite volume fraction determination in TRIP steels, different techniques including EBSD have been compared and the determined phase fraction showed a rather large variability among those techniques (Jacques et al., 2009). In that particular study, the EBSD was found to be suffering from insufficient size of the probed area. In few other publications the size of a representative area was quantified based on microstructural statistics in order to determine the volume fraction plus the distribution of inclusions in steel (Graham and Yang, 2003) and of fibers in composite laminates (Grufman and Ellyin, 2007). The aim of this study is to investigate the EBSD measurement parameters (i.e. step size, size of the probed and scanned areas) on the validity of phase volume fraction determination and of texture analysis. For that purpose the low-alloyed TRIP steel (described in the previous section), that contains about 10 % metastable austenite in a matrix of ferrite and bainite, is used. The austenite is very heterogeneously distributed in the form of bands, mainly due to Mn segregation. This particular arrangement of austenite deteriorates the optimum mechanical properties and also makes the characterization of phase composition particularly important and difficult. In addition, the accompanying martensitic transformation in TRIP steels (as explained in Chapter 1) is highly crystallographic and makes the texture analysis critical. Therefore, the present TRIP steel is an ideal material to investigate the representativeness and reliability of EBSD datasets. Moreover, since EBSD technique has been extensively used as the main microstructure characterization tool in this work, this chapter will determine the reliability of the results of the coming chapters.

3.2 EBSD-based phase fraction determination

3.2.1 Estimation of the required number of measurement points for statistical reliability In an ideal case, the size of a statistical sample (which means the amount of data taken out from a population of values in order to determine a certain parameter) can be calculated based on statistical equations. The parameter to be determined in the present case is the phase fraction of the investigated material. The measurement value obtained at every EBSD map point is an indicator ”gamma”or”not gamma”. The phase fraction CHAPTER 3THE RELIABILITY OF EBSD-BASED MICROSTRUCTURE DESCRIPTION 21

is calculated as the arithmetic mean of these values across all measurement points. Assuming that the γ-phase fraction obtained from different statistical samples of the same material is normally distributed around the correct value, the required sample size n can be determined using the formulation of Cochran (1963) as follows

Z2 × (p) × (1 − p) n = (3.1) e2

Z is calculated based on the required confidence level of sampling (indicating the allowed spread or confidence interval of the sampled data set). It is that parameter of the error function that leads to the following desired confidence level, lc(Z): Z lc(Z) = er f ( √ ) (3.2) 2

The values of Z which result in the desired lc(Z) are tabulated elsewhere (Wikipedia, 2012, Sachs, 2004). For the frequently used conﬁdence level of 95 %, Z is about 1.96. The value e is the sampling error or level of precision. It indicates the allowed tolerance of the value to be determined by sampling. In the present case of phase fraction determination, a reasonable value might be 1 %, meaning that the phase fraction may be determined on 1 % precision.

Fig. 3.2: The required sample size n calculated by Cochran’s formula and shown with respect to sampling error e and to the conﬁdence level for an austenite volume fraction of 10 % (p = 0.1).

The variable p, ﬁnally, denominates the degree of variability of the investigated parameter. In the present case, it indicates the approximately expected phase fraction; if the 22 3.2 EBSD-based phase fraction determination

phase fraction of γ is 10 %, then the variability of the investigated parameter phase, which can be gamma or not gamma is 0.1. If this number is not known beforehand, then a value of 0.5, which is the largest possible variability, gives an upper bound for the sample size (Bartlett et al., 2001) Using Equation 3.1 with a variability of 0.5, a confidence level of 95 %, and a level of precision of 1 %, one obtains a sample size of roughly 9600 points. These points have to be measured independently (i.e., the measurement step size must be larger than the beam interaction distance [Section 3.3.2]). For an austenite fraction of p = 0.1, the sample size is even smaller. The n is calculated for various confidence levels and sampling errors as shown in Figure 3.2. The graph shows that only a few 100 to a few 1000 measurement points are required to obtain the phase fraction with good confidence and precision. It must be noted, however, that the so-calculated values are ideal values that assume a perfectly random distribution of the phases and, of course, no measurement errors. In reality, phases are clustered in grains and bands, and a significant number of measurement errors occur. The actual numbers to be measured and the measurement errors that might occur are the subject of the remainder of this part. 3.2.2 The effect of step-size and observed area size on reliability When performing EBSD maps, the user selects an area of interest, which will be referred to as the covered area, and defines a step size used to scan that area. It is a general misconception to assume that the measured information comes from that area. Rather, because the spot size of the electron beam is small, the information comes from the probed area. The probed area is equal to the number of points in the EBSD scan times the specimen–electron beam interaction surface. The beam interaction volume in a FEG-SEM for a typical accelerating voltage of 15 kV measured on a sample of medium atomic number (e.g., iron) is about 35 nm in a horizontal (parallel to the tilt axis) direction and 90 nm in a vertical (perpendicular to tilt axis) direction. The depth resolution is 5 nm to 10 nm. These values may vary in dependence of accelerating voltage, material type, type of microscope, software used for pattern analysis, and detector geometry (Zaefferer, 2007). Yet, any second- phase particle with dimensions bigger than these resolution values can be identified. For phase fraction determination on a TRIP steel, the probability that the beam hits the α or γ phase is the key parameter. Naturally, this probability depends on the area fraction of the phases on the observation surface and on the beam interaction volume. Later it will also be shown that this probability also depends on the grain size of the γ phase. To obtain a representative value for the phase fraction, it is important that the EBSD data set covers a large enough area and includes enough points. To check the influence of the measurement step size on the phase fraction result, several measurements were carried out with very different step sizes. For every measurement, efforts were made to ensure that the total measurement area was large enough (see later for further explanations on that size) to obtain a representative result. Measuring a sufficiently large area, however, is difficult for the smallest step sizes as the data set (i.e., the number of measured points) becomes very large in these cases. The austenite content detected in the measurements with different step sizes is shown in Table 3.1. All scans give nearly the same austenite volume fraction. The difference CHAPTER 3THE RELIABILITY OF EBSD-BASED MICROSTRUCTURE DESCRIPTION 23

Table 3.1: The details of the scans performed using different step sizes. Note that the detected austenite fraction is almost the same for those different scans.

step size covered area number of points γ- fraction (μm)(μm × μm) (n.a.) (%) 5 3000 × 490 59 499 10.2 1 585 × 380 222 680 10.8 0.5 575 × 580 1 336 320 10.5 0.1 160 × 100 1 848 578 10.8

between the four measurements is within the range of the calculated error estimation. The step size, therefore, has no direct inﬂuence on the detected phase content. How- ever, the step size s inﬂuences the measurement because it determines the size of the probed area Ap for a given covered area Ac. For a measurement on a square grid with a beam-interaction area Ab, the probed area is given as A A = A × c (3.3) p b s2

Equation 3.3 means that increasing step size decreases the probed area (i.e., the area that actually contributes to information obtained), for a given EBSD map size (i.e. a given covered area). 3.2.3 Determination of the size of a representative area To determine the covered area required to obtain a representative measure of the phase fraction, sections of increasing number of points and covered area were cropped in a systematic manner from large original scans performed with 5 μm and 0.5 μm step sizes. Figure 3.3 shows these sections schematically. The horizontal and vertical section sizes were increased by keeping the proportions constant. The detected austenite content of each section is then plotted against the number of points and covered area. Figures 3.4 and 3.5 show such plots for the scans with 5 μm and 0.5 μm step size, respectively. For the scan with a 5 μm step size, the detected austenite content stabilizes at 10.2%, just after 30 000 points, which corresponds to a covered area of 0.75 mm2. However, the stabilized austenite content starts to decrease signiﬁcantly after the covered area reaches 1.5 mm2 because the extreme areas of the scan are slightly out of focus, which leads to more frequent misindexing of the small austenite grains. Figure 3.4(b) shows the corresponding pattern quality and conﬁdence index of the sections for the 5 μm analysis. When the covered area exceeds 1.5 mm2, both the pattern 24 3.2 EBSD-based phase fraction determination

bdband reg ions

regions slightly out of focus

analysissectionsanalysis sections

Fig. 3.3: Schematic representation of the selection of analysis sections from measured large area maps. Gray areas indicate the regions that go slightly out of focus during a large area measurement and elongated black lines symbolize the bands rich in austenite. quality and confidence index decrease progressively. Generally, the average pattern quality and, consequently, the confidence index of austenite are lower than that of ferrite for the present material. Consequently, when the focusing conditions deteriorate, austenite is less frequently indexed correctly, whereas ferrite still is indexed well; thus, the austenite volume fraction decreases. The combo scan mode of the EBSD software used here does not allow for correcting the changing focus across large areas, which results from imperfect sample flatness and sample mounting. Redefining the focusing parameters at each stage position would improve the quality of data at the borders of the covered areas. In the scan with the 0.5 μm step size whose plots are shown in Figures 3.5, all points are in focus. As shown in Figure 3.5(b), the confidence index remains the same throughout the sections. The pattern quality, however, starts to drop as the sections come close to the borders of the sample. This effect originates from the sample preparation, specifically from the electropolishing step, which produces curved surfaces close to the sample edges. Nevertheless, the complete 0.5 μm step size analysis maintains its quality much better than the 5 μm step size one. The detected austenite content for the 0.5 μm step size analysis converges at 10.5% when the covered area reaches 0.275 mm2, which corresponds to 1.1 million points, as shown in Figure 3.5. However, just 30 000 points were enough for reliability for the scan with 5 μm step size. The reason for such a variance is the differences in the covered areas. A scan with 30 000 points, using a 0.5 μm step size does not cover an area large enough to be representative. Similarly, in the scan with a 5 μm step size, the required 30 000 points for representativeness cover an area that is larger than required. Thus, for the present steel, a representative scan must include at least 30 000 points and cover an area of 0.275 mm2. In this way, the total austenite content of the entire sheet CHAPTER 3THE RELIABILITY OF EBSD-BASED MICROSTRUCTURE DESCRIPTION 25

(a) Detected austenite content plotted against number of points and covered area.

(b) The pattern quality and conﬁdence index plotted against number of points and covered area.

Fig. 3.4: The results of the analysis sections cut from 5 μm step sized map. The detected austenite content (a); and the conﬁdence index together with the pattern quality (b) is plotted against the covered area Ac and number of measurement points. The error bars in (a) indicate the error due to mis-indexing. The shaded area shows the error bound due to insufﬁcient sampling. 26 3.2 EBSD-based phase fraction determination

(a) Detected austenite content plotted against number of points and covered area.

(b) The pattern quality and conﬁdence index plotted against number of points and covered area.

Fig. 3.5: The results of the analysis sections cut from 0.5 μm step sized map. The detected austenite content (a); and the conﬁdence index together with the pattern quality (b) is plotted against the covered area Ac and number of measurement points. The error bars in (a) indicate the error due to mis-indexing. CHAPTER 3THE RELIABILITY OF EBSD-BASED MICROSTRUCTURE DESCRIPTION 27

is speciﬁed correctly. It should be noted, however, that TRIP steels may show a systematic variation of γ content depending on the depth in the sheet, for example, because of surface decar- burization. To check that another scan was made in the center of the sheet using a 1 μm step size, and the sectioning was made in the same manner as described in Figure 3.3. The results of this analysis are shown in Figure 3.6. Just after 30 000 points, the detected phase fraction stabilizes at 11 % when the covered area is between 0.03 mm2 − 0.07 mm2. The detected austenite content ﬂuctuates slightly as the covered area increases further. Thus, the 30 000 points are enough to describe the γ content in the sheet center. When the covered area exceeds 0.25 mm2, the detected austenite content remains constant at 10.8 %, which again, is the value for the complete sheet.

Fig. 3.6: The results of the analysis sections cut from 0.1 μm step sized map. Here only the detected austenite content is plotted against the covered area Ac and number of measurement points. The error bars in indicate the error due to mis-indexing

A sample size of 30 000 points (which, at the same time, is the experimentally determined number for representative sampling) corresponds to a sampling error of less than 0.5 % and a conﬁdence level converging to 100 % according to the Cochran’s formula (Equation 3.1. On the other hand, the experimentally determined standard deviation (as shown in Figure 3.4 as error an bound) is still higher than 1 %. This deviation from the values proposed by Cochran’s formula is attributed to the heterogeneity of the austenite distribution, which causes a deviation from the assumed standard normal distribution. 28 3.2 EBSD-based phase fraction determination

3.2.4 Error estimation

Figure 3.7 shows the pattern quality map of the investigated steel. White areas indicate austenite regions. Two very clear austenite-rich bands, lying parallel to the rolling direction, can be observed easily in the upper part of the figure. The presence of such bands makes austenite distribution inhomogeneous and results in errors in volume fraction determination when the data set does not cover a large enough area or include enough points. To quantify the error resulting from insufficient sampling, the standard deviation of the average austenite content was calculated for sections with the same number of points and covered area. For each standard deviation calculation, the analyzed sections were selected from different regions of the total covered area to avoid oversampling. The shaded area in Figure 3.4(a) shows the deviations from the correct austenite content. These deviations can be as large as 40 % when only a few hundred points are measured, and they become insignificant as the number of points reaches 30 000 for the 5 μm step sized analysis.

pattern quality in austenite martensite 260…5230 ferrite andbainite

Fig. 3.7: Pattern quality map of the present TRIP steel measured with a 1 μm step size. The austenite and martensite are indicated by white and red areas in map, respectively.

There is another type of error inherent to EBSD measurements, which occurs when the electron beam hits grain or phase boundaries. In such cases, the interaction area (from which the Kikuchi pattern originates) covers the grains or phases on both sides of the boundary. The resulting overlapping pattern of two individual phases or grains causes misindexing. Although the actual width of boundaries is on the order of only 1 nm (Fultz et al., 1995, Fultz and Frase, 2000), the width of the misindexed region is about equal to the mean diameter of the beam interaction area. Thus, the total area of CHAPTER 3THE RELIABILITY OF EBSD-BASED MICROSTRUCTURE DESCRIPTION 29

the boundary regions can be calculated simply by multiplying the total length of the boundaries with the diameter of the beam interaction area. The length of phase boundaries on the studied TRIP steel can be obtained directly from a high-resolution EBSD scan, like the 0.1 μm step size scan shown in Figure 3.7. The area fraction of phase boundaries, in other words, the probability of misindexing, is then equal to the area fraction of these boundary regions with respect to the total covered area. This type of error is independent of the number of points or covered area of the EBSD scan, but it depends on the grain size and the volume fraction of the second phase and on the size of the beam interaction volume. In the present case, austenite has an average grain size of 0.41 μm. Using the total phase boundary length measured from the EBSD map with the finest step size (shown in Figure 3.7) and a mean beam interaction length of 50 nm, the error resulting from misindexing was determined to be 4.8 % of the measured value. It is shown as error bars at all data points in Figures 3.4, 3.5 and 3.6. For the present TRIP steel, the error resulting from insufficient sampling is more significant than this second error type.

db,min db,max Ap dp

Ap Ap At

Fig. 3.8: Schematic representation of the determination of the grain boundary - electron beam interaction area. White discs indicate the second phase grains. The gray areas indicate the area, where the EBSD pattern is generated from both the matrix and second phase. The area in elliptic due to anisotropic interaction volume of the electron beam with specimen. The arithmetic mean of db,max and db,min is used as the beam interaction length in the calculations.

More generally, the error resulting from boundary misindexing can be calculated by assuming, in a ﬁrst approach, that the second phase is distributed on the investigated 30 3.2 EBSD-based phase fraction determination

surface in the form of circular disks of similar diameter dp and making up a phase volume fraction fp, that is equal to the area fraction of the disks on the surface, as displayed in Figure 3.8. The total length of the phase boundaries is then

4 fp At lp = (3.4) dp where At is the total investigated surface. The area on which the beam interacts with both phases, indicated in Figure 3.8 as gray areas, then is calculated by

Ae = lp db (3.5) where db is the mean beam interaction length (the subscript e indicates the error in phase assignment). The fraction of the area where misindexing and erroneous phase assignment occurs is expressed as follows

4 fp db fe = (3.6) dp For the TRIP steel sample investigated here, Equation 3.6 results in the same outcome as the directly calculated one ( fe = 0.05) by selecting fp = 0.1, db = 0.05 μm, and dp = 0.4 μm. In general, Equation 3.6 shows that the error resulting from misindexing increases with increasing volume fraction and decreasing grain size of the second phase. The relatively simple Equation 3.6 is no longer valid if the grain size ap- proaches the beam interaction length, i.e., the resolution limit of the technique. An additional error can be introduced by the resolution limit of the technique. Austen- ite or any other microstructural feature smaller than the beam interaction volume may not be detected and introduces a certain error. For the present material, however, transmission electron microscopy studies showed that no such small islands of austenite exist (Zaefferer et al., 2004). Lastly, some artifacts may occur in EBSD measurements due to shallow interaction depth of back-scattered electrons. The EBSD technique can only measure the phase fraction at the surface which can be different from the bulk volume fraction. More- over, some austenite may transform into martensite during preparation; for example, electropolishing at a low temperature (T < Ms) may lead to thermal transformation, and grinding may lead to mechanical transformation. Nevertheless, martensite formation is easily visible in the EBSD maps, and this error therefore can be estimated correctly. Using the kernel average misorientation mapping function ((Zaefferer et al., 2008) for more information), 1.3 % martensite was detected in the present material. This martensite could have formed during the thermal treatment of the material or during metallographic preparation. If the latter is true, then this martensite fraction should be added to the determined austenite fraction. 3.2.5 EBSD vs. XRD The results of phase fraction determination based on EBSD have been compared with other methods such as optical microscopy, XRD, or magnetic methods in the works CHAPTER 3THE RELIABILITY OF EBSD-BASED MICROSTRUCTURE DESCRIPTION 31

of Zaefferer et al. (2008), Petrov et al. (2007), Wilson et al. (2001). These works proved that EBSD usually delivers results comparable to other techniques. However, the detailed comparison of Jacques et al. (2009) revealed that, for TRIP steels, the γ phase fraction values determined via EBSD were generally smaller than those given by other techniques. This behavior was attributed to the relatively small observation areas of the EBSD. This explanation seems to be incomplete, as an extremely small sampling area should lead to a large scatter of the values (as seen in Figure 3.5), including too large and too small values. Another probable reason can be the artifacts coming from sample preparation as previously explained. In this work the TRIP steels were prepared via chemical and electropolishing routes. This procedure minimizes, if not completely eliminates, the artifacts due to metallographic preparation. Moreover, the size of a representative area and the optimum measurement parameters for EBSD (number of measurement points, step-size) scanning has been determined. Those attempts have shown the improvement of the precision, which indicates that EBSD-results are now reproducible. Although being reproducible (precise), the EBSD-based results may suffer from systematic errors (as claimed by Jacques et al. (2009)), indicating inaccuracy. (please refer to Appendix D for the deﬁ- nitions of precision and accuracy)

RD EBSD (up) XRD (i) XRD (ii) EBSD (low)

EBSD interaction depth ~ 5 – 10 nm XRD interaction depth ~ 20 μm

Fig. 3.9: Schematic representation of the measured volumes for determination of phase volume fraction by EBSD and XRD. Note that the graph shows only a small part of the thickness of the sample. The interaction depths of both methods are also indicated.

The results of EBSD-based phase fraction determination is compared to XRD in an attempt to find the accuracy. For this comparison (in contrast to the remaining parts of this work) the phase fraction was measured from RD - TD plane as schematically shown in Figure 3.9. The compared EBSD and XRD techniques have different interaction depths, which could alter the comparison, especially in case of a strong deviation of austenite content through the thickness of the sample. In order to compensate this effect, after finishing the XRD measurement and the first EBSD scan (EBSD up), the upmost layer was removed via electropolishing. After the removal, a second EBSD scan (EBSD low) was performed. Assuming that the austenite fraction changes linearly 32 3.3 EBSD-based macro-texture analysis

16 16

14 14

12 γ 12 ( -fraction

10 10

8 8 % 6 6 % fraction (%) - - ) γ 4 4

2 2

0 0 EBSDup EBSDlow EBSD(AVG) XRD(AVG) XRD(i) XRD(ii)

Fig. 3.10: The austenite volume fraction (at the RD-TD surface) of the present TRIP steel obtained by EBSD and XRD techniques. The EBSD side error bars indicate the error due to possible mis-indexing. The XRD side error bars are estimated to be 1 %, coming from selection of ﬁtting parameters for Rietfeld reﬁnement as well as texture effects. through the thickness, the two results of the two EBSD scans were averaged (EBSD AVG). This procedure allows the comparison the two techniques to be performed on exactly the same volume. Figure 3.10 compares the austenite fractions determined and it clearly shows that EBSD based phase fraction determination can be as accurate and as precise as XRD technique provided that sampling and sample preparation is properly done. 3.3 EBSD-based macro-texture analysis

One of the most important application areas of the EBSD technique is texture analysis of multiphase materials. In comparison to well-known and frequently used XRD techniques the EBSD technique has the following advantages

• XRD only measures pole ﬁgures, whereas EBSD measures the orientations (and hence texture) directly without a need for pole ﬁgure inversion. This allows calculating the odd part of the texture without any further assumptions. In addition, EBSD also allows describing very weak textures. • There are no peak overlaps in EBSD scans of multiphase materials. Moreover, no corrections are needed for absorption and defocusing. • EBSD has a better spatial resolution combined with a comparable angular resolution; therefore it can deliver valuable information about misorientations within or between grains and phases and about grain boundary character. • The texture obtained via EBSD can be correlated with microscopic and macroscopic features since EBSD gives the exact location of features having an orientation of interest in the microstructure.

One major concern, despite its advantages, is the statistical representativeness of the EBSD based texture analysis. Other than that, the texture of heavily deformed and CHAPTER 3THE RELIABILITY OF EBSD-BASED MICROSTRUCTURE DESCRIPTION 33

ultra-ﬁne grained materials may not be measured accurately by EBSD since the diffraction patterns coming from those materials can not be indexed correctly due to the spatial resolution.

The present TRIP steel contains metastable retained austenite having an average grain size of 0.5 μm and a phase volume fraction of about 10 %, which is heterogeneously dispersed in a ferritic-bainitic matrix. On a macroscopic level the texture of this material presents a well developed orthotropic symmetry due to its large degree of rolling. On the microscopic level, in contrast, the texture and microstructure is rather heterogeneous due to the occurrence of intense austenite banding caused by Mn-segregations. Therefore, the present TRIP steel is an ideal candidate to test statistical representativeness of the EBSD based texture analysis.

3.3.1 Measuring the statistical reliability of a texture analysis

The TRIP steel sheet used in this study was rolled during the production stage. As the rolling process has orthotropic symmetry, the same symmetry is expected to be seen in pole ﬁgures as schematically shown in Figure 3.11. Note that, the symmetry here

(a) (b)

Fig. 3.11: Schematic representation of (a) grain orientations as cubes that appear with the same frequency in rolled sheet materials exhibiting orthotropic sample symmetry; and (b) a pole ﬁgure showing symmetric locations. ((Wright et al., 2007) concerns the sample symmetry and not the crystal symmetry. Sample symmetry is a statistical symmetry in a sense that crystal orientations that have the same properties (e.g. slip system activation) with respect to the symmetry in the production process should appear in equal volume fractions. The orthotropic symmetry of the rolling process will therefore appear in pole ﬁgures when enough grains from a representative area are included in the analysis. In order to measure this statistical symmetry, S ,asa measure of reliability, Wright et al. (2007) used the following formulation of Baudin 34 3.3 EBSD-based macro-texture analysis

and Penelle (1993) ⎧ ⎫ ◦ 2 1/2 ◦ ◦ ⎪ [P (α, β) − P (α, 180 − β)] ⎪ 90 85 ⎨ ⎬ = 1 + α, β − α, ◦ + β 2 S . ⎪ [P ( ) P ( 180 )] ⎪ (3.7) 18 19 α= . β= . ⎩ ⎭ 0 5 0 5 +[P (α, β) − P (α, 360◦ − β)]2

Here α and β are the azimuth and polar angles respectively and P(α, β) is the intensity of the pole figure at a given angular location. Equation 3.7 compares the intensities at symmetric positions in the pole figure, assuming that the pole figure was measured on a5◦ × 5◦ grid. For a pole figure exhibiting a perfect orthotropic symmetry S = 0. 3.3.2 Comparing the results of conventional EBSD mapping techniques In order to quantify the statistical symmetry, S , of the EBSD maps, center regions of the scans shown in Table 3.1 with 5 μm and 0.5 μm step sizes, were cropped out for analysis. The scan with 0.1 μm step size was in the center region itself; therefore no cropping was needed for that one. Table 3.2 shows the details of these regions, from which pole figures of ferrite (BCC) were re-calculated and compared.

Table 3.2: The details of the analysis sections, from which ferrite pole ﬁgures were calculated.

step size covered area number of points number of α-grains points / grain (μm)(μm × μm) (n.a.) (n.a.) (n.a.) 5 5000 × 100 18 269 13 596 1 0.5 575 × 100 209 016 7 275 29 0.1 160 × 100 1 651 253 5 334 310

Numerous researchers compared the textures measured by EBSD to the ones measured with XRD to test the statistical reliability (Engler, 2009, Engler et al., 1994a,b, Boz- zolo et al., 2007, Baudin et al., 1995, Paillard et al., 1994, Pospiech et al., 1994, Wagner et al., 1998, Wright and Kocks, 1996). These studies suggested the number of individual orientation measurements to be in the range of 200 - 3000 for EBSD based texture analysis. All the measurements listed in Table 3.2 satisfy these proposed conditions. On the other hand, in another comparison by Wright et al. (2007), 10 000 measurement points were proposed for materials having moderate texture strengths. The differences in the comparison methods, texture strengths of the test materials as well as the differences in the covered areas are the possible reasons for such a variance in the proposed number of measurement points. An EBSD scan of a ﬁne grained material using a smaller step size may contain a few 1000 individual orientation measurements, but it would cover a smaller area; thus only the local texture can be obtained. The EBSD scan of the present TRIP steel using 0.1 μm step size is a good example for that case. CHAPTER 3THE RELIABILITY OF EBSD-BASED MICROSTRUCTURE DESCRIPTION 35

(a) 5 μm step size. S(0 0 1) = 0.01163

(b) 0.5 μm step size. S(0 0 1) = 0.01742

Fig. 3.12: Re-calculated BCC pole ﬁgures of the analysis sections of different step sizes. The statistical symmetries of (0 0 1) BCC pole ﬁgures, S(0 0 1), are calculated according to Equation 3.7 36 3.4 A new mapping technique

Both the calculated S values and the pole figures shown in Figure 3.12 reveal that the symmetries in the pole figures of the 5 μm step size analysis are better than the rest. This particular analysis includes only 18 269 points, which is significantly lower than the rest. On the other hand, with the help of such a large step size (that is about the average size of a ferrite grain), this analysis covers a larger area and probes the highest number of grains in comparison. Although the other analyses with smaller step sizes contain significantly more points, these points probe less grains, and hence reduce the expected symmetry perfection. Using a larger step size improves the statistical reliability of texture analysis in the case of determining ferrite texture. Large step sizes would also decrease the number of points required to cover a representative area and make the scan quicker. 3.4 A new mapping technique 3.4.1 The need for a new technique Using a large step size (on the order of average grain size) is beneficial for texture quantification of ferrite (BCC) of the present TRIP steel as shown in the previous section. Large step size allows covering larger areas and probing many more grains which in turn improves the statistical reliability of both, phase fraction determination and of texture analysis of ferrite. In addition, the scans can be finished relatively quick. However, no misorientation information can be gathered and none of the grains can be reconstructed in post-processing. The most commonly used EBSD data clean-up method is to filter out all the points below a certain confidence index. This method, however, leads to a preferred removal of austenite pixels and consequently to a lower detected austenite fraction because the austenite in the present TRIP steel generally has a lower confidence index than ferrite. For the measurements with a smaller step size this problem can be solved by applying the grain confidence index standardization (GCIS) clean-up method. The GCIS method checks the confidence index of all points within a grain and assigns the highest confidence index to all points in that grain. Afterwards, a minimum confidence index filter could be used to exclude only the falsely indexed points. Unfortunately, this procedure cannot be used for scans with a larger step size since only one point per austenite grain is available and, therefore, no γ-grains can be reconstructed. To make use of the GCIS clean-up method, a step size significantly smaller than the austenite grain diameter (but larger than the beam interaction diameter) should be used. A smaller step size allows reconstructing the grains and obtaining valuable information about misorientations and grain boundaries’ character. On the other hand, smaller step size leads either to covered areas being too small to be representative or to very large datasets and corresponding long measurement times. Figure 3.13 compares the inverse pole figure maps of two EBSD scans, one with a very large and the other with a significantly smaller step size. 3.4.2 The application of the technique A new scanning scheme, as shown in Figure 3.14, is proposed in order to keep the advantages of a small step size while retaining the statistical reliability. This new method CHAPTER 3THE RELIABILITY OF EBSD-BASED MICROSTRUCTURE DESCRIPTION 37

(a) coarse step size (b) ﬁne step size

Fig. 3.13: Comparing the IPF’s of two EBSD scans obtained using (a) coarse (Courtesy of S. Zaefferer) and (b) ﬁne step sizes.

consists of a two-step size mode, a large step size (principal step size) is used to cover a sufficiently large area and at every principal step position a small map is measured with a much smaller step size (substep size). The proposed method allows the reconstruction of grains within the small maps and the application of the GCIS clean-up. The new method can be realized through minor modifications in the commercial measurement software. Wagner et al. (2002) and Ubhi (2005) also suggested using similar so-called multi- mapping techniques in order to improve the reliability of EBSD based texture analysis. However, the usage of conventional mapping techniques, including those multi- mapping ones, causes a dramatic loss of both the pattern quality and the confidence index; specifically when the covered area gets extremely large and the material has a fine and complex microstructure. The present TRIP steel is actually a very good example for such a scenario, as shown in Figure 3.4, the analysis of 5 μm step sized scan covering an extremely large area. The Figure 3.15 shows the pattern quality map of that particular scan and it clearly shows that the pattern quality is very low at extreme ends of the map. Those border regions are slightly out of focus which in turn causes the retained austenite to be mis-indexed, due to its finer grain size. The coarser ferritic grains, on the other hand, can still be indexed correctly even the focussing is not perfect. A new measurement software has been designed to keep the electron beam in focus even on extremely large surfaces. Here, the user defines the appropriate focussing at the end points of the covered area, Ac, as schematically shown in (Figure 3.14); and the software calculates and sets the microscope to correctly focus the electron beam at each principal step position. The software is able to control not only the focussing but also the stigmation. Apart from the refocussing improvement, this new software can control the stage movements as well as the commercial EBSD measurement software; 38 3.4 A new mapping technique

Fig. 3.14: Schematic representation of the new mapping technique which allows high resolution EBSD scans over an extremely large covered area. Small high resolution maps of size xsub ×ysub are measured by a ﬁner step size, stepsub, at every step of principal step size, stepprincipal.

Fig. 3.15: The pattern quality map of the EBSD scan with 5 μm step size. The detailed analysis of this scan can be found in Figure 3.4.

therefore this proposed new mapping technique can be realized fully automatically. Figure 3.16 shows the detected austenite content, pattern quality, confidence index and Figure 3.17 shows the pattern quality maps of the EBSD scan made with the new mapping technique using the mentioned focus control software. These results clearly show that the refocussing improvement allows, for the first time, to measure large area mappings even on materials with a fine structure (grains, dislocation cells etc.) without loosing data due to defocusing of the electron beam. 3.4.3 Comparing the results of the new and conventional EBSD mapping techniques The new multi-mapping technique with a focus control can gather high quality patterns and hence EBSD maps even on extremely large areas, as shown in the previous section. In order to find out the advantages that this new technique brings on the reliability of EBSD-based texture analysis, the pole figure symmetries of 4 different EBSD maps, CHAPTER 3THE RELIABILITY OF EBSD-BASED MICROSTRUCTURE DESCRIPTION 39

(a) Detected austenite content plotted against number of points and covered area.

(b) The pattern quality and conﬁdence index plotted against number of points and covered area.

Fig. 3.16: The results of the analysis sections cut from the map obtained by the new technique. The detected austenite content (a); and the con- ﬁdence index together with the pattern quality (b) is plotted against the covered area Ac and number of measurement points. The error bars in (a) indicate the error due to mis-indexing. 40 3.4 A new mapping technique

Fig. 3.17: The pattern quality maps obtained using the new mapping technique. Note that the small maps were taken 200 μm apart from each other. For ease of post-processing and visualization these smaller high resolution maps are combined into a single larger one.

obtained by various techniques (including the new one) were compared. The details of the EBSD scans in comparison are shown in Table 3.3. The first map in this comparison is a fine step sized, high resolution scan having the smallest covered area. The second and third maps were obtained using the so-called combo-scan mode of the commercial software. In this mode the beam and stage movements are combined to cover larger areas. However, it should be noted that the commercial measurement software does not allow refocussing at different stage positions. Due to the fact that the step size is on the order of grain size, the grains cannot be reconstructed and GCIS clean-up cannot be used in the second and third EBSD maps. On the other hand, the duration of EBSD measurements is considerably shorter when combo-scan mode is used with a large step size. The last EBSD technique in the comparison is the new mapping technique, which combines the usage of a fine and a coarse step size as explained. The new technique covers the largest area with measurement duration identical to the 1st high resolution scan. The grains can be reconstructed and hence the GCIS clean-up method can be used with this new mapping.

Table 3.3: The details of the EBSD maps, which were compared in terms of pole ﬁgure symmetries for both ferrite and austenite.

map number step size covered area post processing comments (n.a.) (n.a.) (μm × μm) (n.a.) (n.a.) 1 0.1 160 × 100 GCIS +CI > 0.1 high-reso scan 2 0.5 575 × 100 CI > 0.1 combo scan 3 5 6250 × 100 CI > 0.1 combo scan 4 0.2 − 120 6080 × 140 GCIS +CI > 0.1 new technique

The symmetries of the ferrite (BCC) pole ﬁgures re-calculated from the EBSD maps listed in Table 3.3 are shown comparatively in Figure 3.18. The 1st ﬁne step sized CHAPTER 3THE RELIABILITY OF EBSD-BASED MICROSTRUCTURE DESCRIPTION 41

Ferrite (BCC)

Fig. 3.18: The symmetry parameter of re-calculated pole ﬁgures, number of probed grains and the average conﬁdence index (CI) for ferrite (BCC) compared. The x-axis of all the graphs indicates the EBSD map number listed in Table 3.3. The yellow symbols indicate the results obtained with the new method. 42 3.4 A new mapping technique

Austenite (FCC)

Fig. 3.19: The symmetry parameter of re-calculated pole ﬁgures, number of probed grains and the average conﬁdence index (CI) for austenite (FCC) compared. The x-axis of all the graphs indicates the EBSD map number listed in Table 3.3. The yellow symbols indicate the results obtained with the new method. CHAPTER 3THE RELIABILITY OF EBSD-BASED MICROSTRUCTURE DESCRIPTION 43

EBSD scan has the worst symmetry (higher S value according to Equation 3.7); whereas the remaining EBSD scans have identical statistical symmetries. The 1st EBSD scan probes the least number of grains whereas the average confidence index is very high. Using just a larger step size to probe more grains and cover a significantly larger area significantly improves the pole figure symmetries, as for the 2nd and 3rd scans. Those scans use the combo-scan mode of the commercial software, which does not allow refocussing. The slight defocusing of the electron beam while covering an extremely large area results in a comparatively lower average confidence index. Never- theless, the average confidence index for these combo-scanned maps is still very high. The new mapping technique, 4th map, allows probing over 10 000 grains which is as high as the combo-scanned maps. At the same time; the average confidence index of the ferrite grains probed using the new mapping technique is as high as a smaller area high resolution scan. The performance of the EBSD mapping techniques in comparison is different for the case of the austenite (FCC) pole figures, as shown in Figure 3.19. In this case, using the combo-mapping technique with a larger step size does not improve the statistical symmetries. The average confidence index of the 2nd and 3rd maps are considerably low; and after the application of the CI filtering the number of correctly indexed grains does not increase but it even decreases. The disadvantage of fewer correctly indexed austenite grains for the 2nd and 3rd is compensated by the larger area covered; so that the pole figure symmetries are similar to the 1st map. The 4th map covers an area as large as the maps of the combo-scan mode. At the same time, the new mapping technique keeps the average confidence index of austenite as high as the high resolution (1st) map, and even after CI filtering the number of austenite grains probed remains the highest. As a result, the pole figure symmetries of the finer austenite phase are the best for the new mapping technique. 3.4.4 EBSD vs. XRD based texture analysis In the previous section, it has been shown that the reliability of EBSD based texture analysis can be improved via a new mapping technique. Although the EBSD pole figures exhibit a quite well orthotropic symmetry (rolling symmetry), the improvement of EBSD based macro-texture measurement can be better understood by comparing it to the most popular and the most extensively used macro-texture measurement tool (Wenk, 2000), namely XRD. The re-calculated pole figures of austenite and ferrite are shown in Figures 3.20 and 3.21, respectively, in order to compare EBSD and XRD based texture analysis. The XRD-based pole figures agree quite well with the EBSD-based ones. However it should also be noted that the XRD-based pole figures {200} and {011} of ferrite plus {200} of austenite do not agree to the EBSD-based ones, specifically at higher azimuth angles (α>80). The XRD pole figures show a very high intensity at α = 90 for {200} pole figures which does not exist in EBSD ones. Also the {011} ferrite pole figure of XRD does not show the high intensity regions visible in the EBSD pole figure, again at α = 90. The original XRD pole figures were collected in a reflection geometry. Although reflection pole figures are accurate in the center (α = 0) and are reasonable up to 50−70; 44 3.5 Conclusions

large angles (α>75) can not be reached in a reflection scan (Cullity, 1978). The complete pole figures shown in Figures 3.20 and 3.21 were actually calculated from the orientation distribution function (ODF), which has been calculated using 3 different incomplete pole figures measured. Moreover, the XRD pole figures suffer from defocusing of the incident X-ray beam and peak broadening as the tilting angle (α) increases. These effects are corrected using empirical or theoretical formulas. For the EBSD technique each individual orientation can be obtained at almost the same accuracy, whereas for the XRD technique, the α>75 parts of the pole figures may suffer from the effects ofdefocusing and broadening. Since EBSD measures all the existing orientations directly (without inversions or corrections), the EBSD pole figures are more likely to be correct at these extreme angles. Excluding the extreme angles (α>80) the EBSD based pole figures agree perfectly with the XRD based ones. Figures 3.22 compares the pole figure symmetries (S) for the EBSD and XRD techniques. The symmetry is better in the XRD pole figures of austenite; however, it should also be noted that the intensity corrections and inversions used in the calculation of XRD pole figures may inherently force an orthotropic symmetry. For the ferrite the symmetry of EBSD based pole figures are almost the same as XRD based ones. The results show that EBSD gives macro-texture measurements comparable to XRD.

3.5 Conclusions

The phase volume fraction and texture can be determined correctly and accurately via EBSD technique provided that sampling (also mapping) and sample preparation are properly done. Since it is a direct measurement method, EBSD has some potential advantages over XRD techniques. However, the statistical reliability of EBSD results may suffer from the comparatively small areas usually measured, speciﬁcally in case of materials with heterogeneously distributed phases. The following points should be considered for the reliability and representativeness of EBSD based results:

• In contrast to XRD techniques, the scanned area and the probed area are not identical for the EBSD technique. However, both of these areas should be considered for representativeness.

• The step size has an indirect effect on representativeness as it relates the covered area to the probed area (Equation 3.3). On a large enough covered area scanned with enough number of points, the step size does not inﬂuence the detected phase content. For texture analysis the step size should be considered relative to the grain size, as it will determine the total number grains probed.

• For the present TRIP steel a minimum area of 0.275 mm2 should be covered with at least 30 000 points in order to obtain representative datasets. For texture analysis using a larger step size to probe more grains and cover a larger area improves the reliability, speciﬁcally for the easily indexed ferrite. CHAPTER 3THE RELIABILITY OF EBSD-BASED MICROSTRUCTURE DESCRIPTION 45

XRD-based EBSD-based 200

RD 220

RD 111

Fig. 3.20: The austenite (FCC) pole ﬁgures obtained via EBSD and XRD are compared. 46 3.5 Conclusions

XRD-based EBSD-based 200

RD 011

RD 111

Fig. 3.21: The ferrite (BCC) pole ﬁgures obtained via EBSD and XRD are compared. CHAPTER 3THE RELIABILITY OF EBSD-BASED MICROSTRUCTURE DESCRIPTION 47

0,015 6 Max. Inten (200) (011) (111) 5

s y 0,010 4 in Ploe Fig ity

0,005 2 u sitcal Symmetr sitcal res (MRD)

Stati 1

0,000 0 XRD EBSD (a) Ferrite (BCC).

0,015 4 Max. Inten (200) (011) (111) 3 s y 0, 010 in Ploe Fig ity

0,005 u sitcal Symmetr sitcal res (MRD) i 1 Stat

0,000 0 XRD EBSD (b) Austenite (FCC).

Fig. 3.22: The symmetry parameter of the re-calculated pole ﬁgures of (a) ferrite (BCC) and (b) austenite (FCC), obtained via EBSD and XRD are compared. 48 3.5 Conclusions

• Using Cochran’s equation (3.1) one can calculate the necessary sampling size (i.e. the number of measurement points) required for a statistically representative data set. This calculated value, however, is valid only for materials with homogeneously distributed phases. For the present TRIP steel, which has strong heterogeneities, the experimentally determined sampling size is about 3 times larger than what is calculated according to Cochran’s equation. • The phase volume fraction determination by EBSD is prone to two different types of errors. One originates from insufficient sampling. The second type comes from mis-indexing of phase or grain boundaries and is related to the grain size and the volume fraction of minor phase as well as to the physical resolution of the microscope used. For the present TRIP steel the first type of error is more significant. • The reliability of EBSD based texture analysis of easily indexed phases can be improved by covering larger areas and probing more grains with the help of a larger step size. On the other hand, a smaller step size is more beneficial for grain reconstruction and for using appropriate data clean-up methods. • A new large-area mapping technique is presented here, that significantly improves the statistical representativeness while keeping the particular advantages of smaller step sizes. In addition, this new technique keeps the measurement time short and the datasets small with minor modifications in the commercial measurement software. The technique is fully automated and it is capable of keeping the electron beam in focus even on extremely large (and even on potentially mis-aligned) surfaces. This is crucial for correct indexing, which in turn improves the reliability of texture analysis of hardly indexed austenite in the present TRIP steel.

It should be noted that the minimum requirements (to obtain a representative dataset) given here are only valid for the TRIP steel used in this work. The indicated numbers may change with the homogeneity of specimen as well as the grain size and with the type of microscope used. In principle, every study using EBSD method should check the representativeness of the obtained data. The equations given here have general validity and can be used to check whether the number of data points being enough and to estimate possible error(s). CHAPTER 4

The effect of microstructure on macroscopic work-hardening rate and stress-strain partitioning

4.1 Introduction and motivation

The low alloyed TRIP steels have complex microstructures composed of ferrite, bainite, metastable austenite and potentially martensite. The transformation of metastable austenite into martensite during straining leads to strong local hardening and prevents early localization of strain. The unique mechanical properties of TRIP steels, therefore, are mainly governed by the extraordinary strain hardening mechanism due to martensitic transformation as also explained in Chapter 1. Previously, the effects of microstructure (Timokhina et al., 2004, Takahashi and Bhadeshia, 1991) and the inﬂuence of the interactions between phases (Jacques et al., 2001, 2007) on the stability and transformation of retained austenite have been studied in detail. Those studies made comparisons of alloys having different composition and processing routes. However, little has been reported on the effect of size and shape variations of the austenite grains in the same material. The present study aims in closing this gap by investigating the effect of size and shape variations of austenite grains on the strain hardening rate. Moreover, the effect of austenite grain size and shape on the load partitioning between ferrite and austenite is studied. The microstructure observations were carried out using orientation microscopy based on EBSD. The EBSD technique allows a reliable distinction and quantitative description of austenite, ferrite and martensite (Zaefferer et al., 2008, Petrov et al., 2007, Gourgues et al., 2000, Gourgues-Lorenzon, 2009). Particularly, the error of EBSD based phase fraction determination has been determined to be as low as 0.5 % for the present TRIP steel, provided that sampling and sample preparation is properly done, as shown in Chapter 3.

4.2 Martensitic transformation and the work hardening rate

Figure 4.1 shows the change of the volume fraction of austenite grains and the evolution of their internal strain. For the latter the grain average misorientation (GAM) function was employed. To calculate this value ﬁrstly the misorientation between each neighboring pair of points within the grain is calculated. Second these values are averaged over all points belonging to the corresponding grain. The GAM value shows the misorientation variations which are caused by the presence of dislocations (see also Section 6.2). The GAM (and also the KAM) values, in ﬁrst instance, indicate the 50 4.2 Martensitic transformation and the work hardening rate

heterogeneity of plastic flow. Usually, as the plastic flow continues its spatial heterogeneity increases. Therefore, the GAM value can be regarded as a good indicator of plastic strain. The austenite fraction stays almost constant up to a macroscopic strain of 0.02, but the GAM value increases smoothly. From a strain of 0.02 to 0.05 the austenite transforms into martensite at a high rate. The transformation stops between a strain of 0.05 and 0.07. Upon further deformation the GAM value progressively increases and austenite continues to transform into martensite, however the rate of transformation decreases. The shaded area indicates the spread of the GAM, which increases uniformly at the initial stages. The misorientation spread was estimated first by fitting a Gaussian curve to each of the GAM-distribution curves. After fitting the peak width at half maximum indicated the misorientation spread.

12 1.2 Austenite fraction Grain average misorientation (GAM) 111.1 Misorientation spread 10

1.0 A u s )

090.9 ͚͘ tenite ͇ ƔSTSU 8 ͘ 0.8 grees fractio e 0.7 6

M (d 0.6 n A 4 (%) G 0.5 ͚͘ ͇ ƔRTXX 0.4 ͘ 2 0.3 Stage0 Stage1 Stage2 Stage3

0000.00 0020.02 0040.04 0060.06 0080.08 0100.10 0120.12 True strain

Fig. 4.1: Evolution of austenite volume fraction, GAM and misorientation spread (shaded area) during the interrupted bending tests. The martensite formation rate indicated by the triangles is calculated from the slope of the γ volume fraction curve. See text about the details of the stages.

Figure 4.2 shows the true-stress-true strain curve and the corresponding hardening curve obtained in the tensile test. The correspondence between these data and the data observed in the EBSD mapped areas obtained from the bending tests was made using the mentioned FEM model. The corresponding stages are marked in the graph. Embury and Bouaziz (2010) proposed the following formulation of the ﬂow stress (σF ) CHAPTER 4THE EFFECT OF MICROSTRUCTURE ON MACROSCOPIC WORK-HARDENING RATE AND STRESS-STRAIN PARTITIONING 51

4000 Stage0 Ǝ Stage1 0050.05 Norm ͘ ̀ ́ 3500 Stage2 ͘

a) ͘ ͘ Ȟ ͚ ƍ ͚ 0.04 a lised t lised P 3000 ͚͇͘ Ƴ Ǝ Ʒ ͘ ͘ ͘ ͇ 2500 Stage3 0.03 r ess (M ue stre r 2000

0.02 1500 ͚͇͘ Ƴ͇ Ǝ Ʒ

rue st ͘ s T

1000 s 0.01 ( σ

500 /G)

0 0000.00 0.00 0.02 0.04 0.06 0.08 0.10 0.12 True strain

Fig. 4.2: The true stress vs. true strain curve obtained from simple tension test and the corresponding hardening curve. The dotted dark blue lines indicate the estimated hardening rate due to strain hardening of ferrite and austenite. The difference between dotted dark blue and gray curves indicate the net hardening rate due to martensitic transformation. Stage deﬁnitions are same as in Figure 4.1. according to an isostrain rule of mixture:

σF = fασα + fγσγ + fMσM (4.1)

The work hardening rate would then be:

dσF dσα dσγ dσM d fM = fα + fγ + f + (σ − σγ) (4.2) d d d M d d M

In Equations 4.1 and 4.2 the terms σα, σγ and σM are the ﬂow stresses and fα, fγ and fM are the volume fractions of ferrite, austenite and martensite, respectively. The terms fα (dσα/d) and fγ (dσγ/d) indicate the strain hardening contributions of ferrite and austenite, respectively, to the total strain hardening. The strain hardening of martensite, fM (dσM/d) is assumed to be zero (Jacques et al., 2006) and the contribution of the austenite to martensite transformation to the total strain hardening is thus given exclusively by (d fM/d)(σM − σγ). The correlation of the EBSD-based results with the hardening curve in Figure 4.2 allows a further description of the 3 stages that were distinguished in Figure 4.1:

• Stage 1 (0.02 <<0.05): The martensite fraction increases progressively in the soft ferritic matrix. The highest hardening rate is achieved at this zone. The GAM of the austenite grains slightly increases. 52 4.2 Martensitic transformation and the work hardening rate

• Stage 2 (0.05 <<0.07): The work hardening rate starts to decline. The martensite transformation rate becomes almost zero, as indicated by the stable austenite fraction. The work hardening rate at this stage is given by the strain hardening of ferrite and austenite as follows:

dσF dσα dσγ ≈ fα + fγ + (4.3) d d d

• Stage 3 (0.07 <<0.11): The work hardening rate continues to decline, however the rate of decline is not as high as that in the 2nd stage. Both the GAM and misorientation spread of austenite grains increase at a higher rate at this stage.

The hardening rates (dσ/d) are quite different in all stages; speciﬁcally the difference between the 1st and 3rd stage is quite high although the martensitic transformation is observed in both stages. The hardening rate during stage 2 is only due to strain hardening of ferrite fα (dσα/d), and of austenite fγ (dσγ/d). In an attempt to obtain information on the effect of martensitic transformation on the total strain hardening it is assumed that the contributions of austenite and ferrite strain hardening in the stages 1 and 3 are roughly identical to those in stage 2. In fact, the fraction of austenite and its hardening rate change in between the stages. However, the austenite fraction ( fγ) and hence its contribution to the overall strain hardening is signiﬁcantly smaller than that of ferrite. Therefore, the hardening rate in stage 2 can be used as an approximation to the contributions of ferrite and austenite strain hardening in stages 1 and 3. The difference between the extrapolation and the hardening curve then gives the contribution of the austenite - martensite transformation, (d fM/d)(σM − σγ). The extrapolations are shown in Figure 4.2 and the estimated contribution of the martensitic transformation is given in Table 4.1.

Table 4.1: Flow stress of austenite, martensite and the martensite transformation rate as well as the contribution of austenite to martensite transformation compared for all of the deformation stages. All of the tabulated values (except d fM/d) are in MPa.

Stage d fM/dσγ σM σM − σγ (d fM/d)(σM − σγ)dσ/d Calculation Extrapolation

1 1.13 720 2200 1480 1672 1700 3708 2 ≈ 0 >720 2200 <1480 ≈ 0 ≈ 0 2688 3 0.66 970 2200 1230 812 750 1609

Figure 4.1 also shows that the transformation rate of martensitic transformation (d fM/ d) is quite different in stages 1 and 3. Usually one would expect that austenite grains have all possible degrees of stability and transform one after another in a continuous manner. However, the size, shape and crystallographic orientation of austenite grains affect their stability (Timokhina et al., 2004, Takahashi and Bhadeshia, 1991, CHAPTER 4THE EFFECT OF MICROSTRUCTURE ON MACROSCOPIC WORK-HARDENING RATE AND STRESS-STRAIN PARTITIONING 53

Jacques et al., 2001, Rao and Rashid, 1997, Jimenez-Melero et al., 2007b, Creuziger and Foecke, 2010) and not all of these factors are homogeneously distributed. Es- pecially, the crystallographic orientation distribution of austenite is not homogenous due to texture. The variations among these parameters affecting the austenite stability cause a varying transformation rate. A detailed study of these variations are presented in Chapter 6. In addition, the size and morphology of austenite grains have been reported to influence the carbon saturation inside austenite (Bhadeshia and Edmonds, 1980, Chang and Bhadeshia, 1995). Blonde´ et al. (2012) showed that the carbon content of the austenite grains may vary from 1.3to1.55 wt.%. As measured by Jacques et al. (2006) by in-situ X-ray diffraction, these variations may cause flow stresses varying between 720 to 970 MPa for austenite and a constant flow stress of 2200 MPa for martensite. Using these values for flow stress of austenite (σγ), martensite (σM) and martensite transformation rate (d fM/d), the contribution of the martensite transformation (d fM/d)(σM − σγ) to the overall hardening of the TRIP steel is calculated according to the last term of equation 2. All values used for the calculation are tabulated in Table 4.1. The calculations displayed in Table 4.1 show that 45 − 50 % of the total strain- hardening comes from the austenite to martensite transformation. However, the magnitude of this contribution significantly changes among the different stages of deformation. Reasons for this are (i) the variation of the carbon concentration in different austenite grains which causes a varying austenite flow stress at different deformation stages and (ii) the varying transformation rate which itself depends on the stability of the austenite grains. The calculated hardening rates agree quite well with the ones obtained via extrapolation of experimental results.

4.3 The effects of grain size and morphology on the deformation of austenite

The obtained EBSD datasets allow calculation of the average misorientation inside each grain (GAM). Misorientation between neighboring points in a grain may be used to estimate the density of geometrically necessary dislocations in deformed metals (Demir et al., 2009, Calcagnotto et al., 2010). In a first approximation one may claim that increasing strain leads to increasing GAM values. In order to follow the strain evolution as a function of grain size, the change in the number of grains having a particular combination of grain size and GAM value was calculated and plotted in Figure 4.3. Here, the overall increase of the GAM can be clearly observed since the number of grains having a lower GAM decreases (blue areas in the figures) and the number of grains having a higher GAM (red areas in the figures) increases. Figure 4.3 actually allows the comparison of the GAM increase among differently sized retained austenite grains. The arrows placed in the graphs indicate the changes as explained in the following:

(i) During stage 0, the GAM of grains larger than 0.8 μm changes from 0.35 to 0.55.

(ii) During stage 1, grains smaller than 0.8 μm start to increase their GAM. 54 4.3 The effects of grain size and morphology on the deformation of austenite

1.5 36 27 18 9 1 -8 -17 1.0 -26 -35

0.5 GAM (degrees)

0.1 1 Grain size (μm) (a) during stage 0 1.5 56 43 30 17 5 -8 -21 1.0 -34 -47

0.5 GAM (degrees)

0.1 1 Grain size (μm) (b) during the beginning of stage 1 1.5 62 45 28 11 -7 -24 -41 1.0 -58 -75

0.5 GAM (degrees)

iii iv

0.1 1 Grain size (μm) (c) during the end of stage 1

Fig. 4.3: The change in the number of grains as a function of GAM and grain size during: a) stage 0 (0 <<0.02); b) the beginning of stage 1 (0 <<0.03); c) the end of stage 1 (0 <<0.04). Note that the ﬁgure does not show the absolute numbers of grains. CHAPTER 4THE EFFECT OF MICROSTRUCTURE ON MACROSCOPIC WORK-HARDENING RATE AND STRESS-STRAIN PARTITIONING 55

(iii) Towards the end of stage 1, the smaller grains signiﬁcantly increase their GAM. The GAM of grains smaller than 0.5 μm reaches up to 1.

(iv) The grains larger than 1 μm do not show a signiﬁcant GAM increase at stage 1, although the number of those grains decreases. This means that those larger grains, most probably, transform into martensite before reaching to a higher GAM value by deformation. Previous studies have shown that larger austenite grains transform earlier than the rest (Jimenez-Melero et al., 2007a, Blonde´ et al., 2012).

These results also show that, during the early stages of deformation, mainly larger grains deform, whereas smaller grains start deforming at later stages. This behaviour indicates that the size and the related carbon concentration of austenite grains play an important role in the stress and strain partitioning between the phases and also between different austenite grains. As already shown; in the beginning of straining only larger grains deform whereas after reaching a higher strain almost all grains deform considerably. A further discrimination of the deformation and transformation behavior during the 2nd and 3rd stages of bending is based on the grain shape. Here the grain shape was evaluated by fitting an ellipse to the grain determined through EBSD data. The aspect ratio is defined as the ratio of the minor axis to the major axis of the fitted ellipse and thus ranges from 0 to 1. Figure 4.4 shows the change in the number of grains having the indicated aspect ratio and GAM value as the steel deforms in the same manner as in Figure 4.3. The following changes were observed in Figure 4.4:

(v) At stage 2 the GAM of grains having an aspect ratio of 0.3 (more elongated in shape) increases from 0.55 up to 1.

(vi) The GAM of more equiaxed shaped grains (i.e. aspect ration > 0.5) increases less (to a value of 0.75).

(vii) During the 3rd stage of deformation, the GAM of grains of low aspect ratio increases to very high values i.e. up to 1.4).

(viii) Grains with higher ratio have lower GAM values reaching at most 1.

Figure 4.4 shows that higher GAM values develop in grains having a lower aspect ratio, meaning that elongated grains deform more strongly than equiaxed grains.

4.4 The effect of grain shape on stress partitioning

The effect of grain shape on the deformation and possible transformation can be explained using the short fiber composite model of Cox (1952) considering the elongated austenite grains as short strong fibers in a softer ferrite matrix. The analysis of Cox assumes that the matrix deformation is perturbed locally near the fiber due to load transfer between fiber and matrix (Kelly, 1973). The load is transferred between fiber 56 4.4 The effect of grain shape on stress partitioning

19 14 1,5 9 4 -2 -7 -12 -17 1,0 -22 rees) g

GAM (de 0,5

v vi

0,20,40,60,8 Aspect ratio (a) During stage 2.

2 16 10 1.5 4 -2 -8 -13 -19 -25 1.0 -31 rees) g

GAM (de 0.5 vii viii

020.2 040.4 060.6 080.8 Aspect ratio (b) During stage 3.

Fig. 4.4: The change in the number of grains as a function of GAM and aspect ratio during: a) stage 2 (0.05 <<0.06); b) stage 3 (0.07 < <0.099). Note that the ﬁgure does not show the absolute numbers of grains. See the text for the details about the arrows ”v - viii”. CHAPTER 4THE EFFECT OF MICROSTRUCTURE ON MACROSCOPIC WORK-HARDENING RATE AND STRESS-STRAIN PARTITIONING 57

dP x dx

r r0

Fig. 4.5: Schematic representation of a short fiber and the surrounding matrix. x and r are spatial variables indicating the distance from fiber end along fiber-axis and along the radial-axis, respectively (based on (Lilholt, 1993)).

and matrix by the action of shear strains and corresponding shear stresses on planes and in directions parallel to the fibers (Lilholt, 1993). When the composite is stressed in the fiber direction, the elastic displacements in fiber and in matrix are different because the elastic moduli are different. Normally matrix displacements are larger than the fiber displacements since E f > Em.

Figure 4.5 schematically shows that the load in the ﬁber is P at the position x from the ﬁber end. All the following equations, that are related to the model of Cox (1952), are from (Lilholt, 1993). Here, P can be expressed as

P = σ f A f = E f f A f (4.4) du = E A f f dx where σ f , f , A f and E f are the stress, strain, cross-sectional area and the Young’s modulus of the fiber. P can further be expressed as dP = H(u − v) (4.5) dx where u is the longitudinal fiber displacement and v is the matrix displacement at the same point if fiber was absent. H is a constant which depends on fiber arrangement and moduli of fiber and matrix. Considering e = dv/dx as the strain of the composite 58 4.4 The effect of grain shape on stress partitioning as a whole leads to 2 d P = P − 2 H e (4.6) dx E f A f

Under conditions of P = 0atx = 0 and x = l (l is the ﬁber length), Equation 4.6 has the solution P cosh β(l/2 − x) σ f = = E f e 1 − (4.7) A f cosh β l/2 with H β = (4.8) E f A f

The tensile load in the ﬁber increases by dP over the distance dx due to shear in the matrix and the equilibrium of forces gives

dP + (−2π r0 τi) dx = 0 (4.9) combining Equation 4.4 by 4.9 gives dP 1 τi = dx 2 π r0 dσ A = f f (4.10) dx 2πr0 σ = d f r0 dx 2

From Equation 4.7 the relation for the interfacial stress τi can be expressed as r E e l τ = 0 f − −sinh β ( − x β (4.11) i 2 cosh β l/2 2

r sinh β(l/2 − x) τ = β 0 E e (4.12) i 2 f cosh β l/2

Equations 4.7 and 4.12 give the general form of the tensile stress in the fiber and the shear stress in the fiber-matrix interface. Parameter H (and thus β) can be evaluated for a given geometrical arrangement. For a unit element (as shown in Figure 4.5) of a cylindrical fiber in a cylindrical matrix block; where the radii are r0 and R respectively, the volume fraction of fibers is

r2 V = 0 (4.13) f R2 With no slip between matrix and ﬁber (perfect bonding at the interface), the actual matrix displacements are wm = u at r = r0 (4.14) CHAPTER 4THE EFFECT OF MICROSTRUCTURE ON MACROSCOPIC WORK-HARDENING RATE AND STRESS-STRAIN PARTITIONING 59

wm = v at r = R (4.15) and by deﬁnition the matrix shear strain is

dw γ = m (4.16) m dr

The stress equilibrium for the matrix under shear deformation for positions r in the matrix gives constant = 2π r τm = 2π r0 τi (4.17) with the constitutive equation τm = Gm γm, the relation for wm is

dwm τm τi r0 = γm = = (4.18) dr Gm Gm r and after integration τ |R = i r0 |R wm r0 ln r r0 (4.19) Gm τ r R v − u = i 0 ln (4.20) Gm r0

Combining Equations 4.5 and 4.10 gives 2π r τ H = − 0 i (4.21) u − v and further combination with Equation 4.20 gives 2π G H = m (4.22) ln R/r0

G 2π β = m (4.23) E f A f ln R/r0

= π 2 Combining the relation A f r0 with Equation 4.13 gives

2 G β = m (4.24) r0 E f (− ln V f )

Considering the simple relation 2r0 = d, the combined parameter β l/2 in the equations for σ f and τi can be written as

l l 2 G l G β = m = 2 m (4.25) 2 2 r0 E f (− ln V f ) d E f (− ln V f ) 60 4.4 The effect of grain shape on stress partitioning

By inserting Equation 4.24 into Equation 4.12 τi can be expressed as

Gm sinh β(l/2 − x) τi = E f e (4.26) E f (− ln V f ) cosh β l/2

The tensile fiber stress σ f has its maximum in the middle of the fiber (at x = l/2) and it reaches the maximum value of E f e for an infinitely long fiber. The interfacial shear stress τi has its maximum at the fiber ends and it is zero at the midpoint. Using the Equations 4.7 and 4.26 the tensile stress inside austenite grains and the interfacial shear stress can be calculated as a function of aspect ratio. Here the aspect ratio is defined as d/l for l > d in the same manner as in Figure 4.4. The austenite grains are considered short fibers with an elastic modulus E f = 195 GPa inside the ferritic matrix of shear modulus Gm = 78 GPa. Figure 4.6 shows the results of the composite model for 10 % volume fraction of austenite grains. In the figure the maximum stress represents the case for the infinitely long fibers (σ f = E f e).

1.0 τ max/E e (in terf aci a l s hear s tress ) i f σ max/E e (tensile stress inside fiber) 0.9 f f

0.8

0.7 ax. m 0.6

tio to to tio 0.5 a R 0.4

0.3

0.2 000.0 020.2 040.4 060.6 080.8 101.0 Aspect ratio

Fig. 4.6: The calculated tensile stress inside the fiber and the interfacial shear stress as a function of fiber aspect ratio, according to Cox (1952) model. e is the strain of the composite as a whole and the maximum case represents infinitely long fibers making σ f = E f e .

The composite model indicates that the tensile stress is minimized for spherical austenite grains and the tensile stress increases as grain shape becomes more elongated (smaller aspect ratio). Figure 4.6 shows that the tensile stress inside austenite grains with an aspect ratio of 0.3 would be 50 % higher than that in grains with an aspect ratio larger than 0.5. These results are consistent with the experimental observations shown in Figure 4.4. CHAPTER 4THE EFFECT OF MICROSTRUCTURE ON MACROSCOPIC WORK-HARDENING RATE AND STRESS-STRAIN PARTITIONING 61

4.5 Conclusions

• The austenite transformation rate (d fM/d) and the ﬂow stress difference between martensite and austenite (σM − σγ) affect the extra-ordinary hardening of TRIP steels. Both of these parameters change as the deformation proceeds. The austenite to martensite transformation accounts for 45 to 50 % of the total hardening rate.

• Before stage 1 of deformation (<0.02), mainly grains larger than 0.8 μm deform. Smaller grains deform at the 2nd and 3rd stages (0.05 <<0.11) and reach high local strain values. Larger grains transform into martensite before reaching a strain level comparable to that of the smaller grains.

• Elongated austenite grains having an aspect ratio of about 0.3 deform more than the grains that are more spherical in shape. A short ﬁber reinforced composite model shows that the tensile stress inside the elongated grains is 50 % higher than that in the more spherical grains. Moreover, the GAM analysis of the EBSD results show that elongated grains partition more strain before they transform.

CHAPTER 5

Comparative study of the texture evolution of TRIP microstructure constituents during room temperature deformation

5.1 Introduction and motivation In the previous Chapter 4 the contribution of martensitic transformation on work hardening rate, and hence, on the advanced mechanical properties of TRIP steels was discussed. The overall mechanical properties of TRIP steels, therefore, mainly depend on the deformation and/or transformation of metastable retained austenite. The aim of is this part is to study the effect of texture on the stability of retained austenite. Texture development due to the transformation of austenite into martensite in TRIP-assisted steels has been subject of research over the last 15 years. Park et al. (2007) showed that the measured BCC texture is quite similar to the transformation texture predicted without any variant selection mechanism and using a Kurdjumov-Sachs (KS) orientation relationship. Verlinden et al. (2001) found similar results that no variant selection occurs during transformation of austenite into bainitic ferrite. In contrast, Hutchinson et al. (1998) found indications of variant selection in hot rolled TRIP steels. He et al. (2006) observed variant selection only in undeformed austenite grains; already deformed austenite grains showed no variant selection during transformation into bainitic ferrite. Raabe (1997) reported selective transformation of the Brass (B) {011}211 component during cold rolling of an austenitic stainless steel. These studies mainly fo- cused on the transformation of austenite at higher temperatures; during processing and treatment of these steels. Only Wasilkowska et al. (2006) and Petrov et al. (2007) studied the transformation of retained austenite in Si-TRIP steels during room temperature uni-axial deformation. However, these last two studies rely on EBSD scans of rather small areas containing only few grains. A precise knowledge of the effect of texture on the transformation of austenite into martensite, especially in more commonly used low alloyed Al-TRIP steels, is still missing. The present study aims at closing this gap by evaluating the changes in austenite and ferrite texture due to both tension and compression via 3 point bending. 5.2 Texture development during bending The present TRIP steel was deformed in a 3 point bending test using a special miniaturized stage (see Chapter 2 for further details about the stage). The FEM calculations showed that a maximum true strain of 0.14 was developed during these bending experiments. The rolling direction (RD) of the TRIP steel sheet was parallel to the tension 64 5.2 Texture development during bending

and compression axes of bending. A total area of 575 × 580 μm was examined via EBSD technique before and after the bending tests. The size of that area was proven to be representative in Chapter 3. EBSD maps of the outermost regions, where the tensile or compressive strain was maximum, were cut out from the mentioned total mapped area. These smaller cut out analysis regions contain more than 7500 grains, from which results comparable to XRD based texture analysis can be obtained (Wright et al., 2007, Baudin and Penelle, 1993). A detailed texture analysis of these regions revealed the texture changes in FCC and BCC constituents of the microstructure separately. 5.2.1 Changes in BCC texture components Figure 5.1 compares the changes in the BCC texture components of tension and compression regions to the undeformed state. The present steel was produced via cold- rolling followed by TRIP-treatment and the sample shows a combination of typical cold rolling and recrystallization texture components (Raabe and Lucke,¨ 1994). The undeformed state shows a clear γ-fiber with {111}112 and {111}110 components, which stem from recrystallized ferrite grains. A less pronounced α-fiber including the rotated cube (RC) component, that is inherited from previous cold-rolling, is also present. In the compression region the γ-fiber including the RC component completely disap- pears whereas the {111}1 21 component intensifies and becomes the major component. In contrast, all the other components of the α-fiber decrease. At the tension side, on the other hand, all the components of γ-fiber disappear. The α- fiber, with the {112}110 intensifies strongly, indicating the rotation of recrystallized crystals from the preferred 111||ND to 110||RD. In addition, the RC component intensifies. 5.2.2 Changes in FCC texture components The changes in the FCC texture components upon bending are shown in Figure 5.2. The undeformed sample shows a medium-strength α-fiber extending from Goss (G) {110}100 to Brass (B) {011}211 component and an β-fiber which extends from ◦ ◦ φ2 = 45 to φ2 = 90 including Brass (B) {011}211, Copper and S components. These fibers are typical for cold-rolled samples Engler and Randle (2010).

The strongest texture component in the undeformed sample is the Brass (B) {011}211 component, which is very effectively reduced in both tension and compression zones after bending. The total strength of the FCC texture components is also reduced in both zones. In the tension zone the Goss (G) {110}100 component is slightly reduced whereas in the compression zone cube and rotated cube components intensify a little. The α-ﬁber becomes more clearly visible in the compression zone, because the β-ﬁber components are effectively reduced. CHAPTER 5COMPARATIVE STUDY OF THE TEXTURE EVOLUTION OF TRIP MICROSTRUCTURE CONSTITUENTS DURING ROOM TEMPERATURE DEFORMATION 65

{112}<110>

RC φ ο β 2 = 45 -fiber (a) tension side

{111}<1-21>

RC φ o γ-fiber 2 = 45 (b) compression side

{111}<1-21>

{111}<0-11>

RC {111}<-1-12> φ o γ 2 = 45 -fiber (c) undeformed state

Fig. 5.1: The texture of BCC phases of the TRIP steel after bending (a) at tension, and (b) at compression sides compared to the (c) undeformed state. The plots show the φ2 = 45 sections of the Euler space. 66 5.2 Texture development during bending

(a) tension side

G B

(b) compression side

Fig. 5.2: The texture of FCC phases of the TRIP steel after bending (a) at tension, and (b) at compression sides compared to the (c) undeformed state (continued at the next page). The plots show the constant φ2 sections of the Euler space. CHAPTER 5COMPARATIVE STUDY OF THE TEXTURE EVOLUTION OF TRIP MICROSTRUCTURE CONSTITUENTS DURING ROOM TEMPERATURE DEFORMATION 67

Fig. 5.2: The texture of FCC phases of the TRIP steel after bending (a) at tension, and (b) at compression sides compared to the (c) undeformed state (continued). The plots show the constant φ2 sections of the Euler space.

5.3 Relation between BCC and FCC shear The TRIP steels studied here contains metastable austenite with a FCC lattice and ferrite (either bainitic or intercritical ferrite) having a BCC lattice and both of them deform together under the same global deformation gradient Fij which can be deﬁned as: ∂ui Fij = (5.1) ∂x j where ui denote the displacements and x j deﬁne a Cartesian coordinate system. The deformation gradient can be written as a sum of two sensors (additive decomposition):

F = + ω (5.2)

The symmetric part of this tensor is the strain () and it is given by:

∂u ∂u 1 1 i j ij = /2 (Fij + F ji) = /2 + (5.3) ∂x j ∂xi

The spins or rotations (ω) form the skew-symmetric part of this tensor as follows:

∂u ∂u 1 1 i j ωij = /2 (Fij − F ji) = /2 − (5.4) ∂x j ∂xi 68 5.4 Comparing the changes in FCC and BCC texture components

In the same manner as in the model of Taylor (1938), the strain in each crystal (c)is assumed to be equal to the global strain

= c (5.5) and the crystals deform plastically by dislocation slip as follows: c (s) (s) (s) (s) (s) = 1/2 γ b × n + n × b (5.6) s here b(s), n(s) and γ(s) denote the slip direction, slip plane normal and the magnitude of slip on the plane (s), respectively. The spins and rotations can be expressed in a similar manner: c (s) (s) (s) (s) (s) ω = 1/2 γ b × n − n × b (5.7) s Holscher,¨ Raabe, and Lucke¨ (1994) have shown that the common FCC and BCC texture components are identical if the Miller indices are interchanged from {hkl}uvwFCC into {uvw}hklBCC . Similarly the ordinary slip systems of FCC and BCC crystals can be interchanged as: bFCC(s) = nBCC(s) (5.8) nFCC(s) = bBCC(s) (5.9)

Interchanging the slip geometries of FCC and BCC crystals by implementing the above relations 5.8 and 5.9 into the Equations 5.6 and 5.7, one obtains: FCC (s) FCC(s) FCC(s) FCC(s) FCC(s) BCC = 1/2 γ b × n + n × b = (5.10) s FCC FCC(s) FCC(s) FCC(s) FCC(s) FCC(s) BCC ω = 1/2 γ b × n − n × b = −ω (5.11) s A more detailed analysis of these relations can be found in (Holscher¨ et al., 1994) and in (Kocks, 2000). Regarding the Equations 5.10 and 5.11 the following points can be made:

(i) the rotations or plastic spins of FCC and BCC crystals are opposite when the strains are same ( same, ω opposite). (ii) for uni-axial deformation the rotations (and hence the developed texture) of a FCC component during tension should be the same for the respective BCC component in compression ( opposite, ω same).

5.4 Comparing the changes in FCC and BCC texture components The previously shown changes in the austenite (FCC) and ferrite (BCC) textures are summarized for comparison in Table 5.1. All the texture components are shown in a CHAPTER 5COMPARATIVE STUDY OF THE TEXTURE EVOLUTION OF TRIP MICROSTRUCTURE CONSTITUENTS DURING ROOM TEMPERATURE DEFORMATION 69

Table 5.1: Comparison of the changes in FCC and BCC texture components during bending. Arrow directions indicate an increase(upward) or decrease(downward) of the intensity of the respective texture components. The number of arrows indicate the magnitude of change.

Austenite FCC Ferrite BCC {112}111 ↑↑↑ {111}112 ↓↓↓ {011}211 (B) ↓↓↓ {112}011↓↓ {111}110 ↓↓↓

Tension {100}110 (RC) ↑ {011}100 (G) ↑ {110}100 (G) ↓↓↓ {001}110 (RC) ↑↑↑

{112}111 ↓↓↓ {111}112 ↑↑↑ {011}211 (B) ↓↓↓ {112}011 ↓↓↓ {4411}11 11 8 ↓↓↓ {100}110 (RC) ↑↑ {011}100 (G) ↑ Compression {110}100 (G) {001}110 (RC) ↓↓

way that in each line of Table 5.1, the relations 5.8 and 5.9 are satisﬁed. The number of the arrows in the table indicates the magnitude of the change in the respective texture component. The experimentally detected changes in the FCC texture components are compared to the theoretically expected changes due to shear (namely Equations 5.10 and 5.11)in Table 5.2. It is important to realize that the changes in FCC texture components are not only due to deformation but also due to the transformation of metastable austenite into martensite. Moreover, some texture components satisfy the relations (i) and (ii) partially due to the fact that austenite grains plastically deform before transforming into austenite (as shown in Chapters 4 and 6). The Brass (B) {011}211 component, which was the strongest component in the undeformed state, is very effectively reduced, both in the tension and in the compression regions. In addition, the Goss (G) {110}100 component is reduced remarkably in the tension region. These components do not satisfy or only partially satisfy the previously explained theoretically expected shear relation. Therefore, the strong reduction in those components is expected to be mainly due to transformation of austenite. 5.5 Conclusions

The EBSD technique has been used to study the changes in the textures of the austenite (FCC) and ferrite (BCC) of a low alloyed Al-TRIP steel during simultaneous tension and compression in a bending experiment. A theoretical relation between the deformation of FCC and BCC crystals was derived by replacing simple FCC shear geometry ({111}110) with simple BCC shear geometry ({110}111) in a phenomenological 70 5.5 Conclusions

Table 5.2: The experimentally determined changes in FCC and BCC texture components during bending are compared to the theoretically expected conditions: (i) same, ω opposite and (ii) opposite, ω same. Here, YES denotes that both, the direction and magnitude of the change of the component are same; partially denotes that the direction of change is same but not the magnitude; and NO denotes that the direction of change in the component is opposite of the expected change due to slip.

Condition (i) condition (ii) tension / compression tension / compression {112}111 YES / YES YES / YES {011}211 (B) NO / NO partially / partially {100}110 (RC) NO / partially partially / partially {110}100 (G) YES / NO partially / NO constitutive deformation model. By comparing this derived relationship to the experimental observations, it is possible to ﬁnd out which texture components of the retained austenite change due to slip and which due to transformation into martensite. These comparisons have shown that the strong decrease in the Brass (B) {011}211 and Goss (G) {110}100 components of austenite are mainly due to the transformation of austenite. These experimental ﬁndings are also consistent with other studies in the literature. CHAPTER 6

Microstructural parameters inﬂuencing the stability of retained austenite

6.1 Introduction and motivation The enhanced mechanical properties of TRIP steels are attributed to co-existence of hard and soft phases (composite strengthening effect) and to the transformation of austenite (Bhadeshia, 2002, Jacques et al., 2007, Dan et al., 2008). The transformation of metastable austenite into martensite during deformation brings an additional extraordinary strain hardening mechanism (Embury and Bouaziz, 2010) and therefore contributes to an increase in plasticity in line with the Considere` criterion. Thus, the enhanced mechanical properties (i.e. high strength, good formability) of TRIP steels depend to a significant extend on the stability of the retained austenite (Gibbs et al., 2011, Sugimoto et al., 1995, 1999, 1992b, Nagasaka et al., 2001). If the austenite is too stable, the TRIP effect may not be initiated and if it is very unstable the additional strain hardening is exhausted very early and the protection against necking instability is lost. Optimum stability of austenite leads to the activation of the TRIP effect over a wide strain regime (Sugimoto et al., 1992a), and preferentially at higher strains as shown in the works of Herrera et al. (2011) and Suh et al. (2010). A wide range of retained austenite characteristics (i.e. size, composition, morphology and distribution) can be observed in low alloyed TRIP steels (Hanzaki et al., 1997). Among these characteristics the grain size (Chen et al., 1989, Brandt and Olson, 1993, Jimenez-Melero et al., 2007b, Turteltaub and Suiker, 2006), composition (Brandt and Olson, 1993, Jacques et al., 2006, Jimenez-Melero et al., 2007a, Kruijver et al., 2003), morphology and distribution (Bhadeshia and Edmonds, 1983, Chen et al., 1989, Mi- ihkinen and Edmonds, 1987, Sugimoto et al., 1995, Tomota et al., 2004) of retained austenite as well as the strength of neighbouring phases (Jacques et al., 2001) were reported to influence the stability. Almost all the studies referred to here compare the stability differences of austenite through sets of alloys having different composition and processing routes. In these comparisons, the stability of austenite and the related parameters were averaged for a particular specimen. Averaging the parameters for a particular alloy, however, disregards the heterogeneity of the microstructure which plays an important role in the structure-property relations in TRIP steels. TRIP steels have a rather heterogeneous microstructure which is inherited from earlier stages in the steel production process. Especially, Mn segregation causes the austenite to be formed as bands, which in turn causes variations in the austenite size and composition. More- over, most of the previous studies concentrate on only one or two parameters related to 72 6.2 The detection of deformation and transformation of meta-stable austenite the microstructure. However, the austenite stability is affected by a diverse whole set of parameters, whose contributions may also differ from each other. Neither the differences causing the stability variations on the microscale, nor the complex interplay of different parameters have been investigated in detail yet. The present study aims in closing this gap by revealing the morphological, chemical and crystallographic differences of the austenite grains on the microscale (from one single, heterogeneous material) and determining the relative contributions of these differences to the austenite stability. For that purpose, the deformation and transformation of austenite was followed by interrupted ex-situ bending tests using electron backscatter diffraction (EBSD) in a scanning electron microscope (SEM). The influence of individual microstructural parameters such as the morphology, composition and crystallographic orientation of individual grains and of combinations of these parameters on the stability of austenite are studied both qualitatively and quantitatively. In-situ neutron diffraction (Muransky´ et al., 2008, Huang et al., 2007, Oliver et al., 2002a,b, Zrn´ık et al., 2006) and in-situ synchrotron X-ray diffraction (Jimenez-Melero et al., 2007b, Jia et al., 2009, Zickler et al., 2009, Brauser et al., 2010, Pereloma et al., 2011, Kruijver et al., 2003) techniques have been used intensively to study the deformation and transformation mechanisms in TRIP steels. The high angular resolution of these techniques allowed the investigation of lattice strains of each phase, strain partitioning, volume fraction and morphology of phases. These in-situ experiments provide valuable insights into the micromechanics of TRIP steels. However, these neutron and/or high energy X-ray diffraction techniques are limited either by the small size of the 3D-observable volume or, in case of larger observed volumes, by the limited spatial resolution which is at best in the order of 1 μm for synchrotron based diffraction experiments (Ludwig et al., 2009a,b, 2010). The grain size of the majority of the γ-grains is well below that resolution limit. Moreover, considering the observation volumes, the number of smaller γ-grains that can be probed by synchrotron radiation experiments is rather small. In this study, the electron back-scatter diffraction (EBSD) technique, which allows a reliable distinction and quantitative description of austenite, ferrite, bainite and martensite (Zaefferer et al., 2008, Petrov et al., 2007, Gourgues et al., 2000, Gourgues- Lorenzon, 2009), is used to characterize the microstructure of the present TRIP-steel. The spatial resolution of EBSD technique is less than 100 nm for Fe-based samples (Zaefferer, 2007), which is enough to resolve also the smaller γ-grains. TEM observations of Zaefferer et al. (2004) have also shown that austenite grains smaller than the resolution limits of EBSD do not exist in the undeformed TRIP steel. Furthermore, significantly more grains can be probed by increasing the scan areas, which also improves the statistical reliability as shown in Davut and Zaefferer (2010) and in Chapter 3.

6.2 The detection of deformation and transformation of metastable austenite

The EBSD maps shown in Figure 6.1 were taken at the same position at different stages of deformation; they allow the observation of the transformation of individual austenite grains into martensite during the bending of the present TRIP steel. As indicated by CHAPTER 6MICROSTRUCTURAL PARAMETERS INFLUENCING THE STABILITY OF RETAINED AUSTENITE 73

the arrows in the figure, the transformed austenite grains are replaced by pixels having very low pattern quality and low confidence index. The pattern quality of martensitic regions is lower than that of bainitic and ferritic regions due to the higher density of lattice defects. Moreover, the kernel average misorientation (KAM) increases in the vicinity of regions that are transforming into martensite. KAM indicates the average misorientation of a given point relative to its neighbors. Those characteristics of EBSD maps allow unambiguous identification of regions transforming into martensite. It should also be noted that for ease of visualization only a small portion of the actual EBSD maps is shown in Figure 6.1. The total area observed was exceeding 200 × 100 μm2 containing more than 3000 detected austenite grains. The observed microstructure evolution shown in Figure 6.1 is quantified in Figure 6.2, which shows the austenite fraction as a function of true strain. All the 3 tests show identical characteristics, which can be described by 3 deformation stages. During the 1st and 3rd stage the austenite volume fraction decreases continuously, whereas during the 2nd stage the austenite fraction does not change. A similar 3-stage behavior was also observed in ultra-fine grained 0.2C – 5Mn TRIP steels (Shi et al., 2010). More- over, the results presented in Chapter 4, wherein a similar bending test (but with finer deformation steps) was performed, are also included in Figure 6.2 as Test 1. It is worth mentioning that the 3 tests shown in Figure 6.2 slightly differ from each other. Firstly, the initial austenite fractions are not the same, which is a result of the heterogeneity of the microstructure of the present TRIP steel. Secondly, the highest transformation rate is observed at different strains, presumably due to the differences in austenite stability. The austenite fraction does not change much up to a strain of 0.02 as shown in Fig- ure 6.2. This typical behavior has been attributed to stress-strain partitioning (Jacques et al., 2007, Tomota et al., 2004). During the early stages of deformation the softer ferrite phase yields and deforms plastically earlier than the harder retained austenite. The stress, on the other hand, is partitioned into austenite. The martensitic transformation starts when the stress partitioned to austenite reaches a critical level (stress induced transformation) and/or after the austenite deforms plastically (strain induced transformation). The distribution of KAM values of austenite and ferrite at the respective deformation steps shows a typical standard normal distribution which is shown in Figure 6.3.For quantification, a Gaussian was fitted to each of these distributions. The peak position and peak width at half maximum of those fits indicate the mean KAM value and the spread of misorientations, respectively, and are plotted in Figure 6.4 for both, ferrite and austenite. Both the mean value and the spread of GAM of ferrite grains increase almost monotonously during the deformation. For the austenite grains, on the other hand, a significant change of slope is observed for both, the spread and mean value of GAM, between 0 − 200 μm and between 540 − 820 μm deflections. After the second change of slope, which corresponds to the 2nd stage (as indicated in Figure 6.2), the 74 6.2 The detection of deformation and transformation of meta-stable austenite

Pattern quality maps KAM maps 0 = 047 . 0 = 113 . 0 = pattern quality austenite KAM

260………5230 0..…………5

Fig. 6.1: EBSD maps showing the diffraction pattern quality (left column) and kernel average misorientations (right column) during the bending (tension side) of the TRIP steel. The austenite grains are highlighted in the pattern quality maps. The arrows in the middle row show the transformation of less stable austenite grains an early stage of deformation. CHAPTER 6MICROSTRUCTURAL PARAMETERS INFLUENCING THE STABILITY OF RETAINED AUSTENITE 75

12 Tt1Test 1 10 Test 2

%) Test 3 8 ction ( a 6 nite fr e 4 Aust 2 Stage1 Stage2 Stage3

0.00 0.02 0.04 0.06 0.08 0.10 0.12 True strain Fig. 6.2: Retained austenite fraction as a function of true strain during the interrupted bending tests. See text for the details of the stage deﬁni- tions. 76 6.2 The detection of deformation and transformation of meta-stable austenite

0.5 a) ferrite

0.4

0.3 action r ε = 0 0.2 ε = 0.019 ε = 0.033 ε = 0.047

Number F Number 0.1 ε = 0.065 ε = 0.088 ε = 0.113 0.0

012345 KAM (degrees)

0.5 b) austenite

0.4

0.3 ction a ε = 0 0.2 ε = 0.019 ε = 0.033 ε = 0.047

Number Fr 010.1 ε = 00650.065 ε = 0.088 ε = 0.113 0.0

012345 KAM (degrees)

Fig. 6.3: The kernel average misorientation (KAM) distributions of (a) ferrite and (b) austenite at different strains. CHAPTER 6MICROSTRUCTURAL PARAMETERS INFLUENCING THE STABILITY OF RETAINED AUSTENITE 77

1.8 a) ferrite 1.6

KAM spread 1.4 Mean KAM value 1.2 es) e 1.0

0.8

KAM (degr 0.6

0.4

0.2 0 200 400 600 800 1000 1200 1400 1600 Deflection (μm) μ

1.8 b) austenite 1.6

KAM spread 1.4 Mean KAM value 1.2 es) e 1.0

0.8

KAM (degr 0.6

0.4

0.2 0 200 400 600 800 1000 1200 1400 1600 μ Deflection (μm)

Fig. 6.4: The variation of the mean KAM value and the spread of misorientations (shaded region) inside (a) ferrite and (b) austenite during bending. 78 6.2 The detection of deformation and transformation of meta-stable austenite

austenite GAM values increase slower than those of ferrite. Close to end of the bending test both the spread and the mean GAM values of ferrite and austenite come closer. The kernel average misorientation (KAM) parameter actually shows the misorientation variations which are caused by the presence of dislocations. Nye (1953) has shown that the net density of dislocations that leads to measurable lattice rotations (so called geometrically necessary dislocations, GND) can be quantiﬁed on the basis of measured crystal orientations. Higher orientation gradients (i.e. higher misorientations per distance) correspond to higher GND densities. GND densities of deformed and in- dented copper have been successfully calculated from EBSD data (Demir et al., 2009, Dmitrieva et al., 2010) and as shown by Calcagnotto et al. (2010) the GND densities can be estimated from the KAM parameter. Therefore, the KAM as well as the GAM parameter can be regarded as an indicator of the amount of plastic strain; although, of course, the total dislocation activity is not measured in that way (Wright et al., 2011, Wagner et al., 2011).

300

250

200

(N) 150 Step 1 Step 2 Force 100 Step 3 Step 4 50 Step 5 Step 6 FEM model 0 0 200 400 600 800 1000 1200 1400 1600 Deflection (μm)

Fig. 6.5: Experimental force vs. deﬂection curves of the interrupted bending test compared to the FEM model.

The 3-point bending test was simulated via a 3-dimensional finite element model (FEM) in order to relate the EBSD based results of the bending experiment to the hardening and true stress – true strain curves of the tensile experiment. The validity of the FEM-model was checked by comparing the experimental force-deflection curves (in color) to the one obtained from the simulation (dotted black curve), as shown in Figure 6.5, and an excellent agreement was found. Note that Figure 6.5 apparently shows that the yield stress during reloading is higher than the flow stress at the point of CHAPTER 6MICROSTRUCTURAL PARAMETERS INFLUENCING THE STABILITY OF RETAINED AUSTENITE 79

unloading. The same phenomena has already been observed in TRIP steels by Samek et al. (2008) and was attributed to the internal stresses generated by the martensitic transformation and to the formation of Cottrell atmospheres (Cottrell and Bilby, 1949).

4000 Ǝ

3500 ̀͘ ͘ 3000

(MPa) 2500

2000 stress 1500

True True 1000 Stage1 Stage2 Stage3

500

0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 True strain Fig. 6.6: Corresponding portion of the true stress vs. true strain curve obtained from simple tension test via DIC technique together with the hardening curve. Stage deﬁnitions are the same as in ﬁgure 6.2. The correspondence between the tensile curves and bending tests were made using only the strains calculated via FEM.

Figure 6.6 shows the true stress-true strain curve and the corresponding hardening curve obtained from the tensile test together with the three stages of microstructure evolution that were observed in bending tests. The correspondence between the tensile curves and the microstructural stages obtained in bending tests was made via the mentioned FEM model (using strains only). By comparing the EBSD results to the hardening curve, the 3 stages that were distinguished already in Figure 6.2 can be further detailed as follows: (i) Stage 1 (0.02 <<0.048): The martensite fraction increases progressively in a soft ferritic matrix. The highest hardening rate is achieved at this stage. The rate of austenite to martensite transformation is high, whereas the average GAM of the austenite grains increases only moderately. (ii) Stage 2 (0.048 <<0.065): The work hardening rate starts to decline. The martensite transformation rate becomes almost zero, as indicated by the stable austenite fraction. On the other hand, the GAM of austenite signiﬁcantly increases. This sudden increase is interpreted as an increasing deformation of the austenite grains, since GAM is related to the GND density. 80 6.3 Identiﬁcation of stable and un-stable austenite grains

(iii) Stage 3 (0.065 <<0.113): The work hardening rate continues to decline, however; the rate of decline is not as high as that of the 2nd stage. The GAM values increase at a lower rate than in stage 2 and the remaining austenite transforms into martensite. 6.3 Identification of stable and un-stable austenite grains The stage definitions in Figure 6.2 show that each retained austenite grain transforms into martensite at a different deformation degree. This indicates that austenite grains have different stabilities. The stability differences between austenite grains also influence the strain hardening behavior as shown in Figure 6.6. Based on the stage definitions (in Figure 6.2) and the strain hardening behavior (in Figure 6.6) austenite grains transforming in the 1st stage are considered as early transforming grains and the remaining austenite grains are considered as stable grains. Figure 6.7 compares the grain size distributions of early transforming and stable austenite grains. Almost all of the early transforming grains are larger than 0.7 μm. The average grain size of the early transforming grains is about 2 times larger than that of the stable grains. Figure 6.8 shows the crystallographic orientation of each of the early transforming austenite grains discretely.

0.3

stable γ-grains transforming γ-grains

0.2 tion c

0.1 rea fra A

0.0 0.1 1 4 Grain diameter (μm)

Fig. 6.7: The grain size distributions (given as area fraction) of early transforming and stable austenite grains.

The following section will concentrate on the factors affecting the stability of austenite grains in order to explain differences between early transforming and more stable austenite grains. CHAPTER 6MICROSTRUCTURAL PARAMETERS INFLUENCING THE STABILITY OF RETAINED AUSTENITE 81

ϕ 1 Φ ϕ 2

Fig. 6.8: The orientations of austenite grains transforming at stage 1 (early transforming) shown discretely at constant φ2 sections of reduced Euler space. 82 6.4 Factors inﬂuencing the stability of austenite

6.4 Factors inﬂuencing the stability of austenite

The presented results indicate that the 3-stage hardening behavior of the present TRIP steel is closely related to the transformation rate of the austenite grains. The rate of austenite to martensite transformation is quite different in each one of the stages, which is a consequence of the variation in the stability of individual austenite grains. The stability of austenite has been subject of thorough research and the following parameters were considered:

(a) Grain size: The grain size of austenite inﬂuences both, the kinetics and the energy balance of the austenite to martensite transformation in 3 different ways:

• Shortage of nuclei:Grain size affects the probability of finding nucleation sites in the particle. In case of heterogeneous nucleation (i.e. at grain or phase boundaries), the probability of nucleation of martensite in an austenite grain is proportional to the surface area (and hence size) of the grain. The droplet experiments of Turnbull (1952), Turnbull and Vonnegut (1952) have shown that the nucleation frequency per droplet is proportional to the droplet surface area. Thus, subdividing the system into smaller droplets could cause a shortage of nuclei for most of the droplets. In a similar manner, the smaller austenite grains are stabilized due to the lack of sufficient nuclei. Decreas- ing the austenite grain size would also increase the total number of nuclei required to form one unit volume of martensite. • Interface energy: The interfacial energy of martensite is directly related to thickness and length of martensite plates, or indirectly, to the initial austenite size since the growth of martensite plates stops at grain boundaries Furuhara et al. (2008). The calculations performed by Wang and van der Zwaag (2001) have shown that the Ms temperature and hence the stability of austenite depends exponentially on the this interfacial energy. The contribution of this interfacial energy term to the total energy balance is very significant if the austenite grain size is smaller than 10 μm. • Chemical homogeneity: Grain size may influence the homogeneity of the alloying elements dissolved in the austenite. This influence will be discussed in more detail in the next section.

(b) Chemical composition: The chemical composition of the austenite grains affects the chemical driving force for the martensitic transformation (Haidemenopoulos et al., 1995) and inﬂuences the mechanical properties (i.e. yield strength, elastic stiffness) of the phases. Higher concentration of the austenite stabilizing alloying elements such as carbon and manganese lowers the Ms temperature and hence stabilizes the austenite. Interstitial elements (i.e. carbon) have been reported to be an order of magnitude more effective than substitutional elements in lowering the Ms temperature (Entwisle, 1971). Dissolution of strong solid solution hardening elements either in austenite or in the matrix would also inﬂuence the transformation characteristics (Hanzaki et al., 1997). CHAPTER 6MICROSTRUCTURAL PARAMETERS INFLUENCING THE STABILITY OF RETAINED AUSTENITE 83

(c) Crystallographic orientation: Both, experimental and modeling studies indicate a significant influence of the crystallographic orientation of the austenite grains on their stability. Magee et al. (1971) showed that certain variants can be more active due to their preferred orientations with respect to the applied stress state. More- over, for a stress-induced transformation, the transformation potential is related to the resolved shear stress, which varies among different orientations (Blonde´ et al., 2012, Creuziger and Foecke, 2010, Turteltaub and Suiker, 2005). The stability of the austenite grains for the strain-induced transformation mode would also be influenced by the crystal orientations, since the plastic deformation is strongly dependent on crystallography. These dependences will be discussed thoroughly in the next section. (d) Morphology and distribution: Morphology and distribution of austenite grains were reported to influence the stability of austenite by affecting the local stress concentration (Timokhina et al., 2004, Tomota et al., 2004, Jacques et al., 2001). In addition, the austenite films entrapped between neighbouring bainite sub-units were reported to have a higher carbon content than the islands of residual austenite resulting from the geometrical partitioning of the parent austenite grains by crystallographically different variants of bainite sheaves (Bhadeshia and Edmonds, 1980). Nevertheless, the effect of the morphology has not been studied quantitatively yet. A quantified description of the shape and morphology of the retained austenite grains of the present TRIP steel will be provided in the next section. (e) Accommodation effect: Since martensitic transformation is accompanied by a volume increase, the surrounding regions should accommodate this volume change. This effect is also called as Greenwood and Johnson (1965) effect. An austenite grain could be extrinsically stabilized if the surrounding region consists of stronger constituents, for instance, bainite or martensite. (f) Stress state: The magnitudes of deviatoric and hydrostatic stress components as well as the stress triaxility have been reported to influence the austenite transformation rate (Oh et al., 2002, Jacques et al., 2007). Hydrostatic compressive stresses suppress the transformation since it is associated with a volume increase (Patel and Cohen, 1953). (g) Temperature: : The austenite is more stable at higher temperatures due to the reduction of the driving force and to the increase of stacking fault energy (Saeed- Akbari et al., 2009). (h) Strain rate: A higher strain rate inhibits the deformation induced transformation due to heating effects (Olson and Cohen, 1982).

In the present study, the TRIP-steel was deformed at almost constant room temperature, constant strain rate and only the tension side of the bended specimen was observed to study the stability of austenite. Therefore, the effects of the above mentioned parameters (f), (g) and (h) were disregarded. In addition, the effect of parameter (e) was not signiﬁcant for the present TRIP steel, because it has a relatively soft ferritic-bainitic matrix surrounding the austenite grains. The remaining factors are directly related to 84 6.5 Relative contributions of the microstructural parameters

the microstructure and their effects on the stability variations among austenite grains are discussed in the following section. 6.5 Relative contributions of the microstructural parameters 6.5.1 Grain size and composition The microstructure of TRIP steels develops far from thermodynamic equilibrium. This implies the occurrence of chemical gradients, the strength of which is related to the grain size. Experimental observations on bainitic steels showed that carbon is not uniformly distributed throughout the residual austenite (Bhadeshia and Edmonds, 1979, Bhadeshia and Waugh, 1982). Here the effect of the austenite grain size on local carbon enrichment is analyzed since carbon is the most effective austenite stabilizer (Honey- combe and Bhadeshia, 1995). The present TRIP steel was produced by a two-step heat treatment, ﬁrst an intercritical annealing step, during which austenite with a grain size of 2 ri is formed. The next step is the bainitic holding, during which a part of the austenite grain is transformed into bainitic ferrite. As the bainitic transformation proceeds, the remaining austenite is enriched in carbon. Figure 6.9 sketches the carbon concentration proﬁle along the grain diameter, before and after the bainitic holding. From this principle sketch it is possible to estimate the size of austenite that is stabilized to a maximum by carbon supersaturation. The nominal C concentration of the present TRIP steel is 0.2 wt-%. After intercritical annealing, the austenite fraction is 50 %, making the carbon saturation inside austenite to be roughly 0.4 wt-%. The metastable equilibrium condition during the quenching and bainitic holding (or partitioning) process creates mixtures of carbon depleted ferrite (either bainitic ferrite and/or intercritical ferrite) and carbon enriched austenite (and potentially martensite). Atom probe measurements of Pereloma et al. (2007) and X-ray diffraction studies by Caballero et al. (2007) revealed a carbon concentration of about 6 − 7 at-% (≈ 1.6 wt-%) inside the meta-stable austenite of TRIP steels. Modeling of the heat treatment and transformation process by Speer et al. (2003, 2004) revealed a maximum solubility of 2.1 wt−% for carbon inside austenite. Our own atom probe measurements revealed a maximum concentration of carbon to be 1.6 wt-%.

A ratio between the radius of a single spherical austenite grain before (ri) and after (r f ) bainitic holding, can be derived by taking the conservation of the total carbon content into account: 4/ π 3 = 4/ π 3 3 ri Ci 3 r f C f (6.1) here Ci and C f are the concentration of carbon inside an austenite grain, before and after the bainitic holding, respectively. As explained above, using the carbon concentration values of 0.4 and 1.6 for Ci and C f , respectively, the following ratio can be obtained: r i = 0.63 (6.2) r f The carbon should be able to diffuse from the regions transforming (into bainitic ferrite) to the closest remaining austenite; so that the remaining austenite grain can be CHAPTER 6MICROSTRUCTURAL PARAMETERS INFLUENCING THE STABILITY OF RETAINED AUSTENITE 85

before bainitic holding γ α r b after bainitic holding

% C 1.6 – 2.1 wt-%

0.4 wt-%

2rf

2ri

Fig. 6.9: Schematic representation of carbon concentration proﬁle inside austenite and the radius the corresponding spherical austenite grain just before (ri) and after (r f ) bainitic holding. r f is also the grain size of the undeformed TRIP steel at room temperature.

enriched in carbon. Thus, for a high and homogenous carbon enrichment of a single austenite grain the diffusion distance for carbon should be at least: √ 2 Dt = ri − r f (6.3) and D = D0 exp (−Q/RT) (6.4)

The diffusion distance for carbon can be calculated via the random walk theory, using the diffusion coefﬁcients from literature (Nakajima et al., 2006), as shown in Table 6.1:

Combining the Equations 6.2 and 6.3 gives a theoretical maximum grain size for the optimum stabilization of austenite. This simple analysis shows that austenite grains, whose ﬁnal grain size is larger than 0.8 μm after the TRIP treatment, can not be ideally stabilized due to the limited diffusion distance of carbon. 86 6.5 Relative contributions of the microstructural parameters

Table 6.1: Diffusion coefﬁcients of carbon and calculated diffusion distance during the bainitic holding

Carbon in γ Carbon in α

−5 2 −4 2 D0 1.5 × 10 (m /s)2.2 × 10 (m /s) Q 142.1(kJ/mol) 142.1(kJ/mol) D(400 ◦C) 1.402 × 10−16 (m2/s)6.830 × 10−14 (m2/s) √ at 400 ◦C 2 Dt 0.486 μm 10.1 μm for 420 s

It should be noted that the effects of alloying elements which could influence the diffusion rate of carbon are not taken into account during the calculation of diffusion coefficients. Moreover, austenite grains were assumed to be spherical in shape, whereas the austenite is frequently present in the form of elongated bands in the present microstructure. That would necessitate a longer diffusion distance along the band direction to ensure a high and homogeneous C saturation inside elongated austenite grains. Nevertheless, our calculations give an upper bound for the size of stable grains and illustrate the principal influence of carbon diffusion on that grain size. The above calculated critical grain size value for stable grains of 0.8 μm can be compared to the measured grain size distribution of austenite in the present TRIP steel which is displayed in Figure 6.7. Almost all of the early transforming grains are larger than the calculated critical grain size for optimum stabilization. Based on our simple diffusion model the carbon saturation in those early transforming grains would then be inhomogeneous and possibly less compared to the rest of the austenite grains. In addition, the inhomogeneous carbon saturation could also influence the mechanical behavior of the residual austenite since the yield strength of austenite increases as carbon concentration increases (Jacques et al., 2006). As a consequence the flow stress of austenite σy would be different for grains having different concentrations of carbon. Thus, under a given global stress larger austenite grains containing less carbon may yield and start to deform plastically earlier than the remaining carbon rich austenite grains. 6.5.2 Crystallographic orientation In-situ diffraction studies report that austenite grains with 100 parallel to the loading direction transform preferentially (Jimenez-Melero et al., 2011, Oliver et al., 2002b, Tao et al., 2007). This preference was attributed to higher resolved shear stress along this direction. Among the common austenite texture components, Cube (C) {100}001 and Goss (G) {110}001 components are satisfying this reported relation. On the other hand, the present TRIP steel shows a typical cold-rolling texture with the Brass (B) component {011}211 being the strongest. A detailed texture analysis of the present TRIP steel presented in Chapter 5 has shown a strong decrease in the Brass (B) and Goss (G) components due to martensitic transformation during CHAPTER 6MICROSTRUCTURAL PARAMETERS INFLUENCING THE STABILITY OF RETAINED AUSTENITE 87

bending. Selective transformation of grains with Brass (B) component orientations was also reported for austenitic stainless steels (Raabe, 1997).

0o 5o 10o 15o 20o

25o 30o 35o 40o 45o SlipSystem (111)<110>

50o 55o 60o 65o 70o ϕ 1 Φ ϕ 2

75o 80o 85o 90o

Fig. 6.10: The calculated Schmidt factors (SF) for the simple tension of a FCC-crystal using ordinary slip systems shown at constant φ2 sections of reduced Euler space. Here Schmidt factor of a given orientation corresponds to the maximum value of the Schmidt factor calculated among all the members of the (1 1 1)110 slip system. The dots show the orientation of each early transforming γ-grain discretely.

The austenite grains, whose orientations corresponds to a higher resolved shear stress, would preferentially transform for a stress-induced transformation assumption. Within this frame, the crystallographic orientation-dependent stability of austenite can be quantiﬁed via Schmidt factor. The Schmidt factor relates the level of shear stress (τr) along the respective slip-system to the crystallographic orientation for a given global stress. Figure 6.10 shows the sections of Euler space for constant φ2 where the colour indicates the magnitude of the Schmidt factor calculated for a simple uniaxial tension stress state. The orientations of early transforming grains from Figure 6.8 are superim- τmax posed onto this graphic. Here, the Schmidt factor actually gives the r , which is the highest of all resolved shear stresses on the possible slip systems obtained for the given stress tensor. A considerable fraction of these sections are occupied by high Schmidt factor values. In order to understand the effect of each of the parameters listed above more clearly and quantitatively, it is sensible to plot them in a combined manner. Figure 6.11 shows the number density of grains having corresponding grain size and Schmidt factor. Here 88 6.5 Relative contributions of the microstructural parameters

Fig. 6.11: Distribution plots showing the number of grains having corresponding Schmidt factor and grain size, to compare (a) stable and (b) early transforming austenite grains. The dotted vertical line indicates the critical grain size for optimum stabilization, which was calculated in the previous section. the Schmidt factor of each single grain was calculated by averaging the orientation of τmax that particular grain. The Schmidt factor (and hence the r ) of almost all the stable and early transforming grains is similarly high. Therefore, the resolved shear stress does not distinguish clearly between early transforming and non-transforming crystals of the present TRIP steel. The Schmidt factor and the related resolved shear stress dependence of austenite stability actually rely on the stress induced nucleation of martensite. On the other hand, the interrelations between applied stress, plastic strain, testing temperature and martensite transformation could result in different nucleation mechanisms as explained in σ Chapter 1. The Ms temperature, which indicates the tendency towards stress or strain induced transformation, is reported to be well below room temperature for the present steel composition according to the calculations of Samek et al. (2006). All the exper- σ iments were carried out at room temperature (above Ms ), which would normally lead to a strain induced mode. In addition, the detected increase of the austenite GAM can be regarded as plastic deformation of γ. Therefore, a strain induced transformation mode will be assumed in the coming parts. The strain induced nucleation of martensite was found predominantly at intersections of various types of shear localizations, including the intersection with high angle grain boundaries (Olson and Cohen, 1981). These shear localizations can be mechanical twins, epsilon martensite, bundles of stacking faults and also slip bands or dislocation pile-ups on active slip planes (Brooks et al., 1979b,a, Lagneborg, 1964, Lecroisey and Pineau, 1972, Mangonon and Thomas, 1970, Suzuki et al., 1977, Suezawa and Cook, 1980, Venables, 1962). The volume fraction of intersections of shear localizations is directly proportional to the volume fraction of shear localizations, which in turn is CHAPTER 6MICROSTRUCTURAL PARAMETERS INFLUENCING THE STABILITY OF RETAINED AUSTENITE 89

proportional to the plastic strain accumulated in austenite (Stringfellow et al., 1992, Tomita and Iwamoto, 1995). Therefore, one can simply assume that the number of nucleation sites will increase with increasing plastic deformation for the strain-induced martensitic transformation. Moreover, the rate of formation of nucleation sites is greatest during multiple shear-system operations (Olson and Cohen, 1975a, 1981).

The propensity of multiple slip at the onset of plastic deformation can be estimated in a first, simplified approach, via a resolved shear stress analysis, since it indicates which and how many of the slip system(s) will be activated provided the stress tensor is known (for this it is necessary to assume that the macroscopic stress tensor is equal to the microscopic one). In polycrystalline materials the deformation proceeds generally by the activation of more than one slip system in order to achieve continuity. In order to calculate the number of potentially activated slip systems, a threshold Schmidt factor of 0.27 which gives an averageτ ¯r value calculated over all possible grain orientations, is used. It is assumed that the slip system(s) having a τr higher than this threshold value will be activated. Figure 6.12 shows the number fraction of clusters having a corresponding number of potentially activated slip systems and grain size (in the same manner as in Figure 6.11). Figure 6.12 indicates that almost all of the austenite grains could have 5 potentially activated slip systems, however no significant difference is observed between stable and early transforming austenite grains. Although 5 slip systems can be potentially activated, these slip systems may not be able to accumulate the same strain at the same efficiency.

Fig. 6.12: Distribution plots showing the number of grains having corresponding number of potentially active slip systems and grain size, to compare (a) stable and (b) early transforming austenite grains. The number of active slip systems were estimated as the slip systems whose Schmidt factor (SF) is larger than 0.27. The dotted vertical line indicates the critical grain size for optimum stabilization, which was calculated in the previous section. 90 6.5 Relative contributions of the microstructural parameters

A more appropriate approach to analyze the plastic deformation of a grains as a function of crystallographic orientation is given by the full constraint model of Taylor (1938). This model assumes that every point in the microstructure is deformed by the same strain tensor that is prescribed by the macroscopic deformation process. This requires activation of at least 5 independent slip systems out of the 12 ordinary slip systems for FCC materials. Taylor introduced the hypothesis that a set of 5 slip systems has to be selected out of all possible combinations by a minimum microscopic plastic work criterion. Therefore, the selected 5 slip systems for Taylor factor analysis can be different from those calculated during Schmidt factor analysis. The microscopic plastic work (δW) is given as: δ = γ(s) τ(s) W c (6.5) s

τ(s) γ(s) here c denotes the critical resolved shear stress on slip system (s) and is the τ(s) slip magnitude. The Taylor model assumes c the to be same among the all slip systems, making the minimum plastic work criterion equivalent to a minimum microscopic shear principle. The strain tensor component can be expressed in terms of the microscopic shear rates (dγs) as:

= (s) · γ(s) d ij mij d (6.6) and (s) = (s) × (s) mij bij nij (6.7)

(s) (s) where bij and nij denote the unit vectors of the slip direction and slip plane normal respectively. The resolved shear stress can be calculated as the sum over all the contributions from external stress components:

τ = mij σij (6.8)

Among the possible combinations of 5 shears according to Equation 6.5, 96 sets are independent for a FCC crystal. Those 96 sets of 5 independent slip systems can be short-listed to only 28 via the stress states computed from them, as found byBishop and Hill (1951a); therefore this approach is much more convenient from a numeri- cal perspective. Bishop and Hill (1951b) also introduced a maximum work principle, which states that among the possible stress states that can activate a minimum of 5 slip systems, the operative stress state is the one that maximizes the macroscopic work done. The macroscopic work is expressed as:

δW = σij dij (6.9)

In the present study more generalized states of deformation (i.e. simple tension) are of interest and the FEM model described previously indicates that the strain components other than the principal ones are comparatively quite insigniﬁcant. Therefore, it is CHAPTER 6MICROSTRUCTURAL PARAMETERS INFLUENCING THE STABILITY OF RETAINED AUSTENITE 91

reasonable to use only the principal components of the strain as:

|11|≥|22|≥|33| (6.10) and 22 = λ11 (6.11)

33 = −(1 + λ) 11 (6.12) due to volume conservation and λ is the Lamee’s´ parameter. The maximum work principle of Bishop and Hill (1951b) is actually equivalent to Taylor’s minimum shear (microscopic work) principle. Equation 6.5 can now be rewritten as: δ = τ(s) γ(s) = σ + λσ − + λ σ W c 11( 11 22 (1 ) 33) (6.13) s

An important output of this analysis is the Taylor factor (TF), which describes the propensity of a crystal to slip based on the crystallographic orientation. The Taylor factor is the sum over all the shear strains in the individual slip systems with respect to the prescribed macroscopic strain. γ(s) |σ + λσ − (1 + λ) σ | δW TF = s = 11 22 33 = (6.14) |11| τc |11|τc

Equation 6.14 deﬁnes a generalized Taylor factor which is a function of the orientation of the crystal as well as the deformation tensor. A large value of TF means more work is required to deform the crystal indicating that the slip geometry is less efﬁcient to produce the prescribed strain. Since the EBSD maps were taken from the tension side of bending the deformation gradient tensor (Fij) was calculated for the case of simple tension where the extension occurs in one direction and equal contraction in the other two perpendicular directions as: ⎡ ⎤ 10 0 ⎢ ⎥ Fij = ⎣ 0 −0.50⎦ (6.15) 00−0.5 and the strain is the symmetric part of the deformation gradient tensor given as: 1 ij = /2 Fij + F ji (6.16)

The FEM calculations described in the previous sections also indicate that the other strain components are quite insigniﬁcant compared to tension. Figure 6.13 shows the generalized Taylor factors calculated using the equations above for the whole reduced Euler space at 5 ◦ intervals and for FCC crystals with the ordinary slip of the {111}110 family. 92 6.5 Relative contributions of the microstructural parameters

0o 5o 10o 15o 20o

25o 30o 35o 40o 455o Slipsystem (111)<110>

50o 55o 60o 65o 70o ϕ 1 Φ ϕ 2

75o 80o 85o 90o

Fig. 6.13: The calculated Taylor factors (SF) for the simple tension of a FCC-crystal using ordinary slip systems shown at constant φ2 sections of reduced Euler space. A lower TF value indicates that the orientation of the crystal minimizes the plastic work to produce the given strain, and hence allows efﬁcient slip.

Fig. 6.14: Distribution plots showing the number of grains having corresponding Taylor factor and grain size, to compare (a) stable and (b) early transforming austenite grains. The dotted vertical line indicates the critical grain size for optimum stabilization, which was calculated in the previous section. CHAPTER 6MICROSTRUCTURAL PARAMETERS INFLUENCING THE STABILITY OF RETAINED AUSTENITE 93

The number density distributions for the Taylor factor are plotted in exactly the same manner as for the Schmidt factor and are shown in Figure 6.14. The Taylor factor plot clearly shows a distinction among early transforming and more stable γ-grains. The majority of the stable γ’s have higher Taylor factors. Some of the more stable austenite grains have low Taylor factors but these grains are much smaller and therefore better stabilized. Almost all of the early transforming austenite grains are larger than the previously calculated critical size of 0.8 μm and have lower Taylor factors. If possible flow mechanisms are considered competitively, the one that requires lower driving force will occur according to minimum work principle (Kocks, 2000). Within this regard low-Taylor-factor-grains would partition more strain since the orientations of those grains minimize the microscopic mechanical work, δW, thus allowing efficient slip. Therefore, a higher density of nucleation sites would be available for the early transforming austenite grains. In addition, the mechanical work done by the application of external forces, would alter the free energy balance of the austenite to martensite transformation, and hence influence the austenite stability (Patel and Cohen, 1953). Thus, the ease of deformation can also provide a mechanical driving force for transformation. Nevertheless, the stability difference of retained austenite was reported to be more due to the formation of nucleation sites (kinetics) than to the driving force (Samek et al., 2006). 6.5.3 Grain morphology In order to study the effect of shape and spatial arrangement of the austenite grains the grain shape was evaluated by fitting an ellipse to the grains determined in the orientation maps. The major and minor axes of the fitted ellipse were used to describe its aspect ratio and spatial orientation. The aspect ratio is defined as the length of the mi- nor axis divided by the length of the major axis and thus ranges from 0 to 1. The grain shape orientation (β) is defined as the angle between the major axis of an the ellipse fit to a grain and the tensile and/or rolling direction. Figure 6.15 compares the aspect ratio and ellipse orientation of early transforming and more stable austenite grains. The early transforming austenite grains have lower aspect ratio (≈ 0.3) and their major axis is oriented parallel to the loading axis. More stable γ-grains are more circular in shape and do not have a particular orientation. The fact that strongly elongated structures transform more easily than spherical ones can be understood by considering the elongated grains as short hard fibers in a softer ferritic matrix. Using a short fiber reinforced composite model (Cox, 1952, Lilholt, 1993) it has been shown in Chapter 4 that the smaller the aspect ratio of a grain is (i.e. when the grain is longer, narrower and oriented parallel to the tensile axis) the more the load transferred to it. As a result, early transforming γ-grains can yield and deform plastically earlier than the stable ones. 6.5.4 Statistical analysis of the contribution of the microstructural parameters The stability of retained austenite depends mainly on grain size, Taylor factor and aspect ratio as shown in the previous section. So far the importance of those microstruc- 94 6.5 Relative contributions of the microstructural parameters

Fig. 6.15: Distribution plots showing the number of grains having corresponding aspect ratio and grain shape orientation (β); to compare (a) stable and (b) early transforming austenite grains. Common grain shapes are shown schematically inside the white rectangles. β is the angle between major axis of the ellipse and the loading axis. tural parameters was shown in a rather qualitative fashion only. The relative contribution of each of those studied parameters was not quantified, also not in the current literature. Each of the mentioned microstructural parameters has been reported to influence the stability; however, the austenite stability depends on the combined effect of those parameters. In order to quantify the significance of each of those microstructural parameters and to investigate the combined effect of them, the current results were analyzed by logistic regression. A regression analysis allows to predict the outcome variable from one or more predictor variables. The most simple type of regression, namely simple linear regression, seeks to predict the outcome from only one predictor variable, using the equation of a straight line (Field, 2005). When there is more than one predictor, then the outcome is predicted using a linear combination of predictors as follows:

Y = b0 + b1X1 + b2X2 + ···+ bnXn (6.17) in which Y is the outcome variable, b0 is a constant and bn is the coefﬁcient of the th n predictor (Xn). Equation 6.17 shows basically the regression model for multiple regression which can be used if the outcome variable (Y) is continuous. It is the aim of the analysis to determine the weights, bi, of the different inﬂuencing factors. Logistic regression analysis is a type of multiple regression analysis used for predicting the outcome of a categorical variable based on one or more predictor variables (Miles and Shevlin, 2001). The categorical variable can be bi- or multi-nomial. The binomial (or binary) logistic regression refers to the case that the observed outcome can have only two possible types (e.g. ”dead vs. alive” or ”success vs. failure” or ”yes vs. no”). CHAPTER 6MICROSTRUCTURAL PARAMETERS INFLUENCING THE STABILITY OF RETAINED AUSTENITE 95

The outcome is commonly coded as ”0” and ”1”. In this study the categorical variable refers to the stability of austenite at the 1st stage of deformation and can be coded as:

0 austenite does not transform 1 austenite transforms to martensite

Here, the observed categorical outcome variable (0 or 1), is predicted from the probability (P) of the outcome (i.e. probability of austenite transformation). Logistic regression serves to transform the limited range of a probability, which is restricted to the range [0,1], into the full range [−∞, + ∞]. This makes the transformed values more suitable for fitting using a linear function such as Equation 6.17. The probability range is first extended to [0, +∞] using the odds ratio (OR), which is defined as: P (event) OR = (6.18) P (no event)

The OR provides an improvement over probability, but it still has some limitations. Firstly, OR has a ﬁxed lower limit of zero; secondly its values do not tend to be normally distributed and lastly it does not tend to be related to the predictors linearly. To overcome those limitations, the natural logarithm of the OR is taken.

ln OR = b0 + b1X1 + b2X2 + ···+ bnXn (6.19) P(1) ln = b + b X + b X + ···+ b X (6.20) P(0) 0 1 1 2 2 n n and P(0) = 1 − P(1) (6.21) and combining these equations gives

= 1 P(1) − + + +···+ (6.22) 1 + e (b0 b1 X1 b2 X2 bn Xn)

In linear regression analysis, the regression coefficients (bn) are found using minimization of the sum of squared residuals (the difference between the observed and predicted values). On the other hand, the logistic regression uses the maximum likelihood procedure to estimate the coefficients. The maximum likelihood estimation selects the coefficients that make the observed values most likely to have occurred, that is making P (1) close to 1 and P (0) close to 0. Unlike analytical solutions wherein it is possible to solve directly for the coefficients, the maximum likelihood solution is an iterative process that begins with a tentative solution, revises it slightly to see if it can be improved, and repeats this process until improvement is minute, at which the model is said to have converged. The measure used to check the improvement of a logistic regression 96 6.5 Relative contributions of the microstructural parameters

model is log-likelihood (LL) which is given as:

N LL = Yi ln (P (Yi)) + (1 − Yi)ln(1− P (Yi)) (6.23) i=1 Here N is total number of observations (total number of grains in the present TRIP steel th case) and Yi is the outcome for the i grain. Yi can only be 0 or 1 and the P (Yi) is the probability that is calculated for the outcome of the ith grain via the regression model. The LL is therefore based on summing the probabilities associated with the predicted and actual outcomes (Tabachnick and Fidell, 2001). The log-likelihood statistic is analogous to the residual sum of squares in multiple regression in the sense that it is an indicator of how much unexplained information there is after the model has been fitted. Therefore, large values of LL indicate poor fitting. The log-likelihood statistic allows comparing different models, however since the logistic regression is an iterative process a baseline state is required for comparison. The baseline state actually refers to the model in which only the constant (b0) is included (Field, 2005). The baseline model actually predicts the outcome to be the one that occurs most often. Table 6.2 shows the classification table for the baseline model.

Table 6.2: The classiﬁcation table for the baseline model for the austenite stability

predicted % correct 01 0 274 0 100 observed 1 138 0 0 overall % 66.5

It is worth mentioning that, the ratio of the event:no event observations should be ideally 1:1 and should not be more than 1:3 for a successful application of the logistic regression analysis, due to the iterative nature of it (Agresti, 1996). In case of rare events, wherein the number of event observations are very small compared to no events,even the baseline model would classify a very high fraction of total number of observations correctly. When new predictor values are included, the correct classification percent- age would then remain almost the same and could cause convergence problems. For the present case, the total number of austenite grains observed was more than 3000; however, grains smaller than 0.3 μm were found to be not transforming during almost the entire bending experiment. Moreover, the area fraction of those grains, and therefore their influence on the macroscopic properties are quite insignificant, despite their higher number fraction as shown in Figure 6.7. For this reason, the austenite grains that are smaller than 0.3 μm were not included in the regression analysis. How- ever, after this filtering the number of non-transforming stable austenite grains is 1374, CHAPTER 6MICROSTRUCTURAL PARAMETERS INFLUENCING THE STABILITY OF RETAINED AUSTENITE 97 which is still high. In order to bring the ratio of transforming / non-transforming grains to the suggested level of 1:3, systematic sampling is employed on non-transforming grains. Systematic sampling involves the selection of elements from an ordered sampling frame. The sampling starts by selecting a grain from the list randomly and then every kth grain in the list selected. k is the sampling interval and calculated as: N k = (6.24) n here N is the population size (total number of stable grains) and n is the sample size. The sample size can be calculated using the formulation of Cochran (1977) (Equation 3.1), which has been introduced in Chapter 3. It is worth mentioning that this mentioned formula is valid for unknown (or infinite) populations. An adjust- ment is required for known, finite populations (Cochran, 1946). Accordingly, the n in Equation 6.24 can be calculated as:

= n0 n − (6.25) + n0 1 1 N n0 is the sample size calculated according to Cochran’s equation. The sample size n0 was then calculated for a maximum variability (0.5) for a confidence level of 95 % and an error tolerance of 5 %. k was then determined to be 4.58, therefore, every 5th grain from the list of stable grains were selected after defining a random start point. This procedure allows each grain in the population to have an equal probability of selection (Hannan, 1962). In short, using Equations 6.24 and 6.25 n = 274 stable grains were selected. This selection brings the ratio of the number of transforming grains to the number of stable ones to an acceptable level. Afterwards, the improvement of the model can easily be quantified and a converging solution can be found when the predictor variables are included in the model. The improvement of the model when one or more predictors are added can be computed as follows:

χ2 = 2[LL(New model) − LL(Baseline model)] (6.26)

The multiplication by 2 in Equation 6.26 gives a result obeying a chi-square distribution, and therefore makes it easy to calculate the significance of the value. Significance (p) of a statistical model actually refers to the probability that the obtained results have occurred by chance or coincidence. If p <= 0.05, then the findings are accepted as true and statistically significant (Fisher, 1925). The level of correspondence between the predicted and actual values of the outcome is successfully given by R2 value in linear regression. In logistic regression 3 similar methods were proposed to make a good analogue to the R2 value. These 3 methods are: 2 • Hosmer and Lemeshow (1989) RL: −2LL(Model) R2 = (6.27) L −2LL(Baseline) 98 6.5 Relative contributions of the microstructural parameters

2 • Cox and Snell (1989) RCS :

− 2 − 2 = − n (LL(New) LL(Baseline)) RCS 1 e (6.28)

2 • Nagelkerke (1991) RN : R2 R2 = CS (6.29) N [2LL(Baseline) 1 − e /N]

In terms of interpretation these measures are similar to the R2 value in linear regression; their value varies between 0 (indicating the predictors are useless) and 1 (indicating that the model predicts the outcome perfectly). Another crucial interpretation of logistic regression is the value of exp b, which is an indicator of the change in odds resulting from a unit change in the predictor. exp b is given by: odds ratio after a unit change in the predictor exp b =ΔOR = (6.30) original odds ratio

An exp b parameter equal to 1 indicates that changing the values of the predictor has no effect on the outcome. An exp b value smaller than 1 indicates that increasing the predictor value increases the odds of the outcome occurring; and vice versa. In statistics, the true value for a population is estimated from a sample. In the present TRIP steel example hundreds to thousands of grains were investigated, however the sheet metal used contains billions or even more grains. A different group of grains may have given different values for austenite stability. The accuracy of the values obtained from a sample is assessed by calculating the boundaries within which the true value of the population will fall. These boundaries are called confidence intervals (C.I.); and are expressed for a certain level of confidence. For example 95 % C.I. indicates that the confidence intervals constructed will contain the true value of the population for 95 out of 100 collected samples. Table 6.3 summarizes the results of the logistic regression analysis for the stability of austenite grains. In the table sig. (p) values indicate the significance of the coefficients (b) of predictors, namely the probability that the coefficients are zero. The logistic regression model, the b and exp (b) values, both state that increasing the grain size plus decreasing the Taylor factor and aspect ratio all increase the probability of a austenite grain to transform into martensite. Those findings are consistent with the experimental observations. However, it should be noted the significance (p) of the coefficient of aspect ratio is not acceptable, since there is more than 30 % chance for the constant being zero. Moreover the 95 % confidence intervals for exp b passes through the value of 1 which indicates that the change in the aspect ratio would not change the odds ratio. On the other hand the grain size has the largest exp b value and the best significance, making it the most important parameter among the studied ones. Although its influence is considerably smaller compared to grain size, the second most important CHAPTER 6MICROSTRUCTURAL PARAMETERS INFLUENCING THE STABILITY OF RETAINED AUSTENITE 99

Table 6.3: The summary of the logistic regression model1 for the stability of the austenite during the 1st stage of deformation

95 % CI for exp b b sig.(p) Lower expb Upper Included predictors Constant -1.235 Grain size 2.736 0.001 8.253 15.431 28.849 Taylor factor -0.640 0.030 0.297 0.527 0.938 Aspect ratio -1.029 0.313 0.049 0.357 2.633 1 R2 = 0.789 (Hosmer & Lemeshow), 0.235 (Cox & Snell), 0.327 (Nagelkerke). Model χ2 = 7.032

parameter is the Taylor factor. The other parameters, namely the Schmidt factor and grain shape orientation, does not improve the model when included, therefore their values are not shown. When the constants (b)inTable 6.3 are inserted to the Equation 6.22, one calculate the probability that a certain austenite grain transforms at the 1st stage of deformation based on microstructural parameters. Using a cut off value of 0.5 for the probability, which means if the calculated probability for a certain grain is more than 0.5 then that grain is considered to be transforming. Table 6.4 classiﬁes the observed and predicted behaviours of the austenite grains. This table indicates how well the model predicts group membership.

Table 6.4: The classiﬁcation table for the baseline model for the austenite stability

predicted % correct 01 0 250 24 91.2 observed 1 75 63 45.7 overall % 76.0

Table 6.4 shows that the logistic regression model classiﬁes 91 % of the non- transforming grains correctly and misclassiﬁes only 24 grains. For the transforming grains the model can classify only 45.6 % correctly. The overall accuracy of the model is improved from 66.5 % (for the baseline model as shown in Table 6.2 ) to 76 %, with the grain size, Taylor factor and aspect ratio included as predictors to the model. De- spite the improvement, the accuracy and the R2 values listed in Table 6.3 indicates the 100 6.6 Why the transformation is not continuous ?

model is unable to explain the stability of austenite grains perfectly. Thus, the effect of unknown or not studied parameters should also have an influence. Particularly, the local composition differences (specifically carbon), the strength of the matrix in the vicinity of retained austenite (accommodation effect) and the local stress state are the potential parameters that would improve the model s accuracy when included. Those parameters were reported to influence the retained austenite stability, as explained previously. Apart from the parameters that were not studied, it should also be noted that the current developed model equation can not be generalized for all TRIP steels and for all different testing schemes. Any change in the chemical composition, testing temperature, strain rate and the stress-strain state would alter the stability of retained austenite, and hence the calculated coefficients. Nevertheless, the developed model exhibits the influence of each studied microstructural parameter and their combined effect quantitatively. Therefore, the derived logistic regression model can be used as an estimation of austenite stability based on microstructural parameters which would then help to optimize the stability and hence to improve the overall mechanical properties.

0.08 ns 0.06 n of grai n of o 0.04 r fracti e 0.02 Numb

0.00 2.42.62.83.03.23.43.6 Taylor factor

Fig. 6.16: Taylor factor distribution of the austenite grains. Note that the distribution is not homogeneous but it is rather bi (or even tri-) modal and the fraction of grains with intermediate TF (i.e. 2.5 < TF < 2.9) is considerably low.

6.6 Why the transformation is not continuous ? One would usually expect that the austenite grains have all possible degrees of stability and transform one after another in a continuous manner. In contrast, for the CHAPTER 6MICROSTRUCTURAL PARAMETERS INFLUENCING THE STABILITY OF RETAINED AUSTENITE 101 present TRIP steel the martensite transformation almost stops during the 2nd stage of deformation. The rate of transformation is affected from not only the average (homogenized) values of the microstructural parameters but also from their distributions of the microstructural parameters affecting the stability of austenite. Among these microstructural factors the grain size (and possibly the chemical composition) shows a continuous, normal distribution. On the other hand, the crystallographic orientation distribution of austenite is not homogenous due to texture. This austenite texture results in a bi-modal distribution of Taylor factors as shown in Figure 6.16. Moreover, Figure 6.14a shows that the number of grains having a combination of an intermediate grain size (i.e. between 0.5 μm−0.8 μm) and a moderate Taylor factor (i.e. between 2.6 - 2.8) is signiﬁcantly low. The lack of grains having such a combination, which actually indicates an intermediate degree of stability, is the reason for the low transformation kinetics at stage 2.

6.7 Remarks

It should be noted that in the present investigations the contribution of only surface grains were observed due to the shallow interaction depth of EBSD. The behavior of these grains could be inﬂuenced by stress relaxations at the free surface. Nevertheless, in-situ synchrotron studies on identical alloys revealed a similar gradual transformation of austenite due to texture (Jimenez-Melero et al., 2011) and almost the same grain size and composition (speciﬁcally local carbon concentration) dependence of austenite stability (Blonde´ et al., 2012).

6.8 Conclusions

• The transformation of individual austenite grains into martensite during deformation was observed by combining ex-situ bending tests and EBSD investigations. This measurement procedure allows studying the microscale differences between the austenite grains, which causes a 3-stage transformation rate. At the 1st stage the transformation rate and the corresponding strain hardening rate is highest. At the 2nd stage the austenite transformation almost stops and the corresponding hardening rate declines dramatically. The transformation continues at a lower rate during the later deformation stages.

• The austenite grains transforming within the 1st stage were identiﬁed as early transforming grains. It was found that the early transforming grains are usually of large size (> 0.7 μm) and show all possible Taylor factor values. In most cases they have a strongly elongated shape where the long axis is oriented in the direction of straining. In contrast, the more stable austenite grains are usually small and have either very small or very large Taylor factors but no medium values. Their shape is more spherical. These differences between the early transforming and more stable austenite grains can be rationalized as follows:

(i) Grain size and chemical homogeneity: Limited diffusion distances during the TRIP-treatment of these steels causes lower carbon saturation inside austenite grains larger than a certain threshold size. Based on a simple 102 6.8 Conclusions

diffusion model the threshold was determined in the present case to be 0.8 μm by assessing the maximum path length of carbon diffusion during the bainitic holding process. All of the early transforming γ-grains are larger than the indicated critical size. (ii) Crystallographic orientation: Schmidt factor analysis shows that the maximum resolved shear stress and the number of potentially activated slip systems are almost the same for both stable and unstable austenite grains. Taylor factor analysis, on the other hand, shows that early transforming austenite grains have lower Taylor factors, which indicates that plastic deformation proceeds easily in these grains. A higher density of nucleation sites would then be available for early transforming grains, for a strain- induced transformation assumption. In addition, the ease of deformation would also increase the mechanical driving force for the transformation. (iii) Grain shape and morphology: All the early transforming austenite grains were found to have a lower aspect ratio and be elongated along the loading axis. This specific grain shape results in the development of higher stresses, which in turn cause early yielding of less stable austenite grains. • The effect of each one of the microstructural parameter on the austenite stability was quantified via logistic regression analysis. The results of this analysis revealed that the influence of grain size is about an order of magnitude larger than the second important parameter, the Taylor factor. On the other hand, the effects of Schmidt factor, aspect ratio and grain shape orientation were all found to be statistically insignifant. However, the goodness of the fit (R2) values of this model are not perfect. Obviously, the other related parameters, that were not investigated, need to play a role in austenite stability. • The models and explanations for austenite stability may come short when they are based on single and/or homogenized parameters only because of the microstructural and texture heterogeneities. The distributions of grain size, local carbon content, crystallographic orientation and morphology of the austenite grains, which significantly affect the austenite transformation rate, should also be considered. Especially the bi-modal (or even tri-modal) distribution of the Taylor factor (which is a result of austenite texture) is the reason for the lack of austenite grains with intermediate stability. This results in an intermediate deformation stage during which almost none of the austenite grains transform. Summary and Outlook

Steels with transformation induced plasticity (TRIP) offer a good balance of ductility and strength, which is the result of the gradual transformation of the metastable austenite into martensite during straining. The transformation of metastable austenite into martensite during deformation brings an additional extraordinary strain hardening mechanism and, therefore, contributes to an increase in plasticity in line with the Con- sidere` criterion. In this regard, the enhanced mechanical properties (i.e. high strength, good formability) of TRIP steels depend to a significant extend on the stability of the retained austenite. A thorough understanding of the relation between microstructure and martensitic transformation and also the related austenite stability is the key for a successful improvement of the mechanical properties. The present low-alloyed TRIP steel has a rather heterogenous texture and microstructure on the microscopic level due to the occurence of intense austenite banding caused by Mn-segregations. This makes the microstructure characterization particuarly important and difficult. The statistical reliability of EBSD-based results, specifically for phase fraction determination and texture analysis, has been thorougly discussed in Chapter 3. EBSD scans of various area and step sizes were carried-out and analyzed in order to determine the optimum sampling conditions. Here, the step size has only an indirect influence on the results, as it determines the size of the investigated area if the number of measurement points is kept constant. The experimentally determined sample sizes were compared to the ones obtained from Cochran’s formula for predefined levels of precision and confidence. A significant deviation of theoretical optimum sample size from the experimentally determined one was found. This deviation, for the most part, is a result of the heterogeneous distribution of the austenite phase. False assignment of phases at phase boundaries can also introduce a significant error. This type of error depends on the grain size, volume fraction of the second phase and also on the resolution of the microscope used. A general formula, that allows the estimation of the error caused by these parameters, is introduced. In addition, a new mapping technique, that helps to cover larger areas and probe more grains while keeping the particular advantages of a small step size, is presented. This new technique can keep the measurement time short and datasets small with minor modifications in the commercial data-acqusition software. This technique can be fully automated and it keeps the electron beam in focus even on extremely large surfaces, which is crucial for correct indexing and hence for improving the the reliability of texture analysis of hardly indexed austenite in the present TRIP steel. After identifying the optimum sampling conditions for a reliable and representative EBSD scan of the present TRIP steel, the deformation and transformation of austenite 104 Summary during interrupted ex-situ bending tests were studied by EBSD investigations. Further- more, a FEM model was used to relate the EBSD based results obtained during bending experiments to the hardening behavior obtained from simple tension experiments. The results of these bending experiments were presented in the following chapters: In Chapter 4 the effects of size and shape of austenite grains on the strain-hardening of the present TRIP steel have been shown. The results were interpreted using a simple rule of mixture for stress-partitioning and a short fiber reinforced composite model. It has been shown that both, the martensite transformation rate and the flow stress difference between austenite and martensite significantly influence the hardening rate. Moreover, the grain average misorientation (GAM) analysis of the EBSD results revealed that at the initial stage of deformation mainly larger grains deform. These larger grains, however, do not reach the same strain level as the smaller grains because they transform into martensite at an early stage of deformation. A composite model of Cox was used to investigate the effect of grain shape on load partitioning. The results of this composite model showed that higher stresses develop in more elongated grains. Moreover, the GAM analysis showed that elongated grains partition more strain before they transform. The texture development during simultaneous tension and compression of the present TRIP steel is shown in Chapter 5. Here, the change in the retained austenite (FCC) texture was due to both deformation and transformation into martensite, whereas the ferrite (BCC) texture changed due to deformation only. A theoretical relation between the deformation of FCC and BCC crystals was derived by replacing simple FCC shear geometry ({111}110) with simple BCC shear geometry ({110}111) in a phenomenological constitutive deformation model. By comparing the experimental observations to the theoretical relation, the changes in the austenite texture due to slip were distinguished from the changes due to transformation. These comparisons have shown that the strong decrease in the Brass (B) {011}211 and Goss (G) {110}100 components of austenite are mainly due to the transformation of austenite. In Chapter 6 a detailed analysis and quantification of the effects of grain size, grain shape and crystallographic orientation on austenite stability is presented. The investigations revealed that the retained austenite is not transforming continuously, but rather in a distinct 3-stage behaviour. Accordingly, the grains can be grouped into 3 cate- gories; grains that are transforming with a high rate at low strain, those that transform at high strain with a low rate and lastly a strain gap between these two types where the austenite only deforms but does not transform. Grains that are transforming at small strains tend to be large, have low Taylor factor and a strongly elongated shape, while the remaining more stable grains are small, more spherical in shape and have a high Taylor factor. The grain size effect is quantitatively explained by a diffusion model that shows that larger grains are not fully stabilized due to insufficient carbon diffusion. The grain shape effect is related to the strain distribution that is found in short-fiber composite materials. The Taylor factor indicates the ease of plastic deformation and a low Taylor factor, therefore indicates a large abundance of possible nuclei for martensitic transformation according to strain-induced transformation mode. The strain-gap between the early and late transforming grain fractions is explained by the lack of grains with medium Taylor factor due to texture. Lastly, a logistic regression model was used Summary 105 to quantify the relative contribution of each of the microstructure-related parameter. The results of the model showed that the grain size is the most important parameter as its significance is about an order better than the second most important paramerer, the Taylor factor. The effect of other studied parameters, the Schmidt factor, aspect ratio and grains shape orientation were found to be statistically insignificant for the present case. This thesis shows, firstly, how to obtain a reliable and representative description of rather heterogeneous microstructures using the EBSD technique. The presented concepts can be used to improve the statistical reliability of EBSD-based microstructure analysis, including macro-texture determination, phase fraction analysis, grain size and distribution determination, grain boundary character analysis, orientation relationships between phases and also misorientation analysis. Secondly, a comprehensive analysis of the relation between microstructure and macroscopic mechanical properties of a low alloyed TRIP steel is presented. These results provide a useful basis for the further development and improvement of TRIP-assisted steels.

Zusammenfassung

Stahle¨ mit transformations-induzierter Plastizitat¨ (TRIP) bieten eine gute Balance aus Duktilitat¨ und Festigkeit, die das Ergebnis einer zunehmenden Umwandlung von metastabilem Restaustenit in Martensit wahrend¨ der Verformung ist. Die Transfor- mation des metastabilen Austenits in Martensit wahrend¨ der Verformung stellt einen zusatzlichen¨ außergewohnlichen¨ Verfestigungsmechanismus dar, der entsprechend dem Considere-Kriterium` zu einer Steigerung der Duktilitat¨ fuhrt.¨ Die verbesserten mechanischen Eigenschaften von TRIP-Stahlen¨ hangen¨ deshalb wesentlich von der Stabilitat¨ des Restaustenits ab. Ein tiefes Verstandnis¨ des Zusammenhangs zwischen Mikrostruktur und martensitischer Umwandlung sowie der Stabilitat¨ des Restaustenits ist der Schlussel¨ zu einer Verbesserung der mechanischen Eigenschaften. Der hier untersuchte niedrig legierte TRIP Stahl hat auf mikroskopischer Skala eine relativ heterogene Textur und Mikrostruktur, die durch die Prasenz¨ von ausgepragten¨ Austenitbandern¨ deutlich wird, die ihrerseits durch Mangan-Segregationen verursacht werden. Die Heterogenitat¨ des Werkstoffes macht eine Mikrostrukturcharak- terisierung zugleich wichtig und anspruchsvoll. Der erste Teil der Arbeit konzen- triert sich deshalb auf die statistische Zuverlassigkeit¨ von EBSD-basierten Ergeb- nissen, da EBSD die wesentliche Meßmethode fur¨ die durchgefuhrten¨ Untersuchun- gen war. EBSD-Scans verschiedener Große¨ wurden systematisch im Hinblick auf den Volumenanteil des Restaustenits ausgewertet um eine optimale Stichprobengroße¨ zu ermitteln. Zusatzlich¨ wurde die Formel nach Cochran benutzt, um die optimale Stichprobengroße¨ fur¨ vordefinierte Werte von Genauigkeit und Vertrauensbere- ich zu ermitteln. Dabei wurde eine große Abweichung der theoretischen und exper- imentell ermittelten Stichprobengroßen¨ festgestellt, die auf die heterogene Verteilung des Restaustenits zuruckzuf¨ uhren¨ ist. Fehler, die durch eine zu kleine Stichprobe oder durch Missindizierung von EBSD-Pattern an Korngrenzen entstanden, wurden eben- falls diskutiert. Daneben wurde eine neue Messstrategie entwickelt, mit der große Probenbereiche gemessen werden konnen¨ ohne den Vorteil einer hohen Genauigkeit durch kleine Messschrittweiten aufzugeben. Diese neue Messtechnik, die durch eine unwesentliche Modifikation der kommerziellen Messsoftware ermoglicht¨ wurde, halt¨ die Messzeit und die Datenmengen klein. Sie kann vollautomatisiert werden und halt¨ den Elektronenstrahl auch auf sehr großen Probenoberflachen¨ im Fokus, wodurch die Zuverlassigkeit¨ der Messung von Textur und Phasenanteilen signifikant erhoht¨ werden. Schließlich wurde durch einen Vergleich von Phasenanteilen, die mittels EBSD und XRD gemessen wurden gezeigt, dass EBSD eine verlassliche¨ und reprasentative¨ Messung ermoglicht,¨ vorausgesetzt, dass die Stichprobe groß genug und die Probenpraparation¨ angemessen durchgefuhrt¨ wurden. Nach Bestimmung der optimalen Bedingungen fur¨ eine zuverlassige¨ und reprasentative¨ 108 Zusammenfassung

EBSD Messung des TRIP Stahls wurde diese Methode verwendet, um das Verformung- und Transformationsverhalten des Austenits wahrend¨ unterbrochener ex- situ Biegeversuche zu untersuchen. Zugleich wurde ein FEM Model verwendet, um die EBSD-basierten Ergebnisse aus den Biegeexperimenten mit dem Verfestigungsverhal- ten aus einfachen Zugversuchen zu korrelieren. Die Einflusse¨ von Große¨ und Form der Austenitkorner¨ auf die Verfestigung des un- tersuchten TRIP-Stahls wurden durch eine einfache Mischungsregel fur¨ die Span- nungsverteilung und ein Kurzfaser-Kompositmodell erklart.¨ Es wurde gezeigt, dass sowohl die martensitische Umwandlungsrate als auch die unterschiedlichen Werte der Fließspannungen von Austenit und Martensit die Verfestigungsrate wesentlich bee- influssen. Außerdem zeigten die Ergebnisse der Korn-gemittelten Missorientierung, die aus den EBSD-Scans berechnet wurden, dass wahrend¨ des Beginns der Verfor- mung hauptsachlich¨ große Korner¨ verformen. Sie erreichen dabei aber nicht dasselbe Dehnungsniveau wie die kleinere Korner,¨ da Sie schon im Fruhstadium¨ der Verfor- mung in Martensit umwandeln. Ein Kompositmodell nach Cox wurde benutzt, um den Einfluss der Kornform auf die Lastverteilung zwischen den Kristalliten zu erklaren.¨ Die Ergebnisse dieses Modells zeigen, dass langere¨ Korner¨ hohere¨ Spannungen er- leiden. Die Missorientierungsdaten zeigen außerdem, dass diese Korner¨ auch hohere¨ Dehnungsanteile aufnehmen. Die Texturentwicklung unter Zug und Druck wurde durch Analyse der EBSD Ergeb- nisse des Biegeversuchs bestimmt. Die Entwicklung der Textur des Austenits (kfz- Struktur) wird durch die Verformung und die Transformation der Restaustenitkorner¨ bestimmt, wahrend¨ die Texturentwicklung des Ferrits (krz-Struktur) nur durch die Verformung der Korner¨ bestimmt ist. Es wurde ein theoretischer Zusammenhang zwischen der Texturentwicklung von kfz und krz Phase entwickelt, indem in einem phanomenologischen,¨ konstitutiven Verformungsmodel die einfache Scherung auf {111}110 fur¨ kfz durch eine einfache Scherung auf {110}111 fur¨ krz ersetzt wurde. Durch Vergleich der experimentellen und theoretischen Ergebnisse konnte gezeigt werden, welche Texturkomponenten des Austenits durch die Verformung und welche durch die Transformation verursacht wurden. Die Vergleiche zeigen, dass der starke Ruckgang¨ der Messinglage {011}211 und der Gosslage {110}100 auf die martensitische Umwandlung der so orientierten Austenitkorner¨ zuruckzuf¨ uhren¨ ist. Im letzten Teil der Arbeit wird eine detaillierte Analyse der Einflusse¨ von Korngroße,¨ Kornform und kristallographischer Textur auf die Austenitstabilitat¨ vorgestellt. Die Experimente ergaben, dass die Austenitkristalle nicht kontinuierlich in Martensit umwandeln sondern in einem 3-Stufen-Prozess. Dementsprechend kann das Verhalten der Korner¨ in 3 Gruppen kategorisiert werden, Korner¨ die bei kleinen Dehnungen mit hoher Rate umwandeln, solche, die bei hohen Dehnungen mit niedriger Rate umwandeln und schließlich ein umwandlungsfreier Bereich, bei dem die Korner¨ lediglich verformen und nicht umwandeln. Kristallite die bei kleinen Dehnungen umwandeln sind ublicherweise¨ relativ groß, haben einen kleinen Taylorfaktor und ein stark gelangte¨ Form. Die stabileren Korner¨ sind klein, mehr aquiaxial¨ in ihrer Form und haben einen hohen Taylorfaktor. Der Einfluß der Korngroße¨ kann durch ein Diffusionsmodell quan- titativ erklart¨ werden. Es zeigt, dass ungenugende¨ Kohlenstoffdiffusion dafur¨ verant- wortlich ist, dass große Korner¨ nicht vollstandig¨ stabilisiert werden. Der Einfluss der Zusammenfassung 109

Kornform wird durch die Dehnungsverteilung in einem Kurzfaser-Kompositmaterial erklart.¨ Der Taylorfaktor ist ein Indikator dafur,¨ wie leicht ein Korn verformt werden kann. Ein kleiner Taylorfaktor zeigt an, dass ein Korn leicht verformt und deshalb eine hohe Tendenz zur Umwandlung nach einem Dehnungs-induzierten Modus besitzt. Der Dehnungsbereich zwischen fruh¨ und spat¨ umwandelnden Kristalliten, in dem keine Umwandlung stattfindet, kann dadurch erklart¨ werden, dass es aufgrund der Textur des Werkstoffes nur wenige Kristalle mit mittleren Taylorfaktoren gibt. Schließlich wurde eine logische Regressionsanalyse durchgefuhrt,¨ um den relativen Beitrag der verschiedenen Einflussfaktoren auf die Austenitstabilitat¨ zu quantifizieren. Die Ergeb- nisse dieses Modells zeigten, dass die Korngroße¨ der wichtigste Parameter ist. Seine Signifikanz ist ungefahr¨ zehnmal hoher¨ als die des Taylorfaktors. Der Einfluss an- derer Parameter, etwa des Schmidfaktors, Aspektverhaltnisses¨ und Kornform scheint im vorliegenden Fall gering zu sein. Die vorgelegte Arbeit zeigt an erster Stelle, wie eine zuverlassige¨ und reprasentative¨ Beschreibung der der Mikrostruktur mittels EBSD Technik durchgefuhrt¨ werden kann. Die hier vorgestellten Konzepte konnen¨ dazu verwendet werden, eine statistisch zuverlassige¨ EBSD-basierte Beschreibung von Makrotextur, Phasenanteilen, Korngroßen¨ und –Verteilungen, Korngrenzencharakter, Orientierungsbeziehungen und Missorien- tierungen zu erhalten. Zum zweiten wurde eine umfassende Analyse des Zusam- menhangs von Mikrostruktur und makroskopischen mechanischen Eigenschaften eines niedrig legierten TRIP-Stahls vorgestellt. Die Ergebnisse stellen eine gute Basis fur¨ die weitere Entwicklung von TRIP-Stahlen¨ da.

APPENDIX A

Considere` Criterion

A condition for the onset of necking was originally proposed by Considere` based on the assumption that necking begins at the point of maximum load. At this point, the decrease of cross-section area (A) is larger than the increase in strength due to strain hardening and deformation gets localized. A decreases more rapidly than the load due to strain hardening. Further elongation occurs with decreasing load. At the maximum load: dP = Adσt + σtdA = 0 (A.1) here, P is the load, A is the cross-sectional area and σt is the true stress. Equation A.1 can be re-arranged as: dA dσ = −σ (A.2) t t A It is reasonable to assume that volume is preserved during plastic deformation; therefore: dV = d(Al) = ldA + Adl = 0 (A.3) or dA dl = − (A.4) A l where l is the specimen gage length and dl/l is t, the true strain. Combining equations A.2 and A.4 gives: dσt = σt (A.5) dt Equation A.5 is Considere’s` criterion, which states that necking begins when the strain hardening rate (dσt/dt) is equal to the true stress (σt). In other words, ”necking begins when the increase in stress due to decrease in the cross-sectional area is greater than the increase in load bearing capacity of the specimen due to work hardening.”

APPENDIX B

Common BCC Texture Components

ο ϕ1 + ϕ2 = 90 Cube

β-fibre ο ϕ1 + ϕ2 = 45 Rotated cube

Fig. B.1: Common texture components as well as texture ﬁbers found in typical BCC phases of steels (Courtesy of S. Zaefferer).

APPENDIX C

Common FCC Texture Components

Fig. C.1: Common texture components as well as texture ﬁbers found in austenitic (FCC) steels (from Engler and Randle (2010)).

APPENDIX D

Precision and Accuracy

Fig. D.1: Schematical representation of the difference between accuracy and precision (from www.wellesley.edu).

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KemalDAVUT

"No man should escape our universities without knowing how little he knows." – J. Robert Oppenheimer

Personal Birth date 12.12.1982 Birth place Kadıköy - İstanbul, Turkey Languages Turkish(native), English(advanced), German(intermediate)

Education 2008–present PhD in Materials Engineering, Max Planck Institute for Iron Research, Düsseldorf. "Relation between microstructure and damage mechanisms in TRIP-steels" Supervisors: P.D.Dr-Ing. Zaeﬀerer & Prof.Dr-Ing. D. Raabe 2004–2006 M.Sc. in Metallurgical & Materials Engineering, METU, Ankara. "Characterisation of steel microstructures by magnetic Barkhausen noise technique" Supervisor: Prof.Dr. C.Hakan Gür 2000–2004 B.Sc. in Metallurgical & Materials Engineering, METU, Ankara.

Academic Interests Advanced high strength steels (TRIP, DP, Q&P), in-situ & micro-mechanical testing, SEM, EBSD, microstructure & micro-texture analysis, damage mechanisms, magnetic Barkhausen noise, non-destructive testing.

Experience Academic 2004–2008 Teaching Assistant, METU, Metallurgical & Materials Eng., Ankara. Basic concepts in materials science, Materials Characterisation, Residual stresses in Materials processing, Materials Laboratory (metallography & heat treatment) 2004–2008 Laboratory Assistant, METU, Metallurgical & Materials Eng., Ankara. SEM laboratory, Metallography laboratory 2008–present Project Assistant, Max Planck Institute for Iron Research, Düsseldorf. Department: Microstructure Physics & Alloy Design Other 07–08.2002 Internship, BMC Foundry, Pınarbaşı, İzmir, Turkey. 07–08.2003 Internship, Aegen Ceramic Industry, Kemalpaşa, İzmir, Turkey.

Research Output

Peer Reviewed Journal Articles { Monitoring the microstructural changes during tempering of quenched SAE 5140 steel by magnetic Barkhausen noise K.Davut & C.H.Gür, J. Nondestruct. Eval., 26 (2007) p107 { Statistical reliability of phase fraction determination based on electron backscatter diffraction (EBSD) investigations on the example of an Al-TRIP steel K.Davut & S.Zaefferer, Metall. Mater. Trans. A, 41A (2010) p2187 { The effect of texture on the stability of retained austenite in Al-alloyed TRIP steels K.Davut & S.Zaefferer, MRS Proceedings, 1296 (2010), doi:10.1557/opl.2011.1447 { Improving the reliability of EBSD-based texture analysis by a new large area mapping technique K.Davut & S.Zaefferer, Mater. Sci. Forum, 702-703 (2012) p566 { The effect of size and shape of austenite grains on the mechanical properties of a low alloyed TRIP steel K.Davut & S.Zaefferer, Steel Research International, 83 (2012) p584

Publications in Preparation { Microstructural factors inﬂuencing the mechanically induced transformation of metastable residual austenite in a low alloyed TRIP steel K.Davut & S.Zaeﬀerer, submitted to Acta Materialia, (2012)

Selected Conference Talks { Investigating the eﬃciency of magnetic Barkhausen noise method for determining average grain size of steels K.Davut & C.H.Gür,6th ICBM, Valenciennes - France (2007) { Non-destructive characterization of pearlite spheroidization by MBN method K.Davut & C.H.Gür,17th World Conf. on NDT , Shangai - China (2008)

{ Phase fraction and texture quantiﬁcation of Al-TRIP steel from EBSD data K.Davut & S.Zaeﬀerer,3rd ITAP, Göttingen - Germany (2009)

{ Relation between damage nucleation and microstructure in TRIP steels K.Davut & S.Zaeﬀerer, DFG Congress MSE-2010, Darmstadt - Germany (2010)

{ Factors inﬂuencing the strain-induced transformation of residual austenite in a low-alloyed TRIP steel K.Davut & S.Zaeﬀerer, Euromat 2011, Montpellier - France (2011)