Martensitic Transformations in Steels – A 3D Phase-field Study HEMANTHA KUMAR YEDDU Doctoral Thesis Stockholm, Sweden 2012 ISRN KTH/MSE–12/12–SE+METO/AVH ISBN 978-91-7501-388-6 Materialvetenskap KTH SE-100 44 Stockholm SWEDEN Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framlägges till offentlig granskning för avläggande av teknologie doktorsexamen i materialvetenskap fredagen den 15 juni 2012 klockan 10.00 i sal B2, Brinellvägen 23, Materialvetenskap, Kungl Tekniska högskolan, 10044 Stockholm. © Hemantha Kumar Yeddu, June 2012 Tryck: Universitetsservice US AB iii The highest education is that which does not merely give us information but makes our life in harmony with all existence. – Rabindranath Tagore, Nobel Laureate (1913). v Dedicated to the most wonderful persons in my life – Mom and Aunty vii Abstract Martensite is considered to be the backbone of the high strength of many commercial steels. Martensite is formed by a rapid diffusionless phase transformation, which has been the subject of extensive research studies for more than a century. Despite such extensive studies, martensitic transformation is still considered to be intriguing due to its complex nature. Phase-field method, a computational technique used to simulate phase transformations, could be an aid in understanding the transformation. Moreover, due to the growing interest in the field of “Integrated computational materials engineering (ICME)”, the possibilities to couple the phase-field method with other computational techniques need to be explored. In the present work a three dimensional elastoplastic phase-field model, based on the works of Khachaturyan et al. and Yamanaka et al., is developed to study the athermal and the stress-assisted martensitic transformations occurring in single crystal and polycrystalline steels. The material parameters corresponding to the carbon steels and stainless steels are considered as input data for the simulations. The input data for the simulations is acquired from computational as well as from experimental works. Thus an attempt is made to create a multi-length scale model by coupling the ab-initio method, phasefield method, CALPHAD method, as well as experimental works. The model is used to simulate the microstructure evolution as well as to study various physical concepts associated with the martensitic transformation. The simulation results depict several experimentally observed aspects associated with the martensitic transformation, such as twinned microstructure and autocatalysis. The results indicate that plastic deformation and autocatalysis play a significant role in the martensitic microstructure evolution. The results indicate that the phase-field simulations can be used as tools to study some of the physical concepts associated with martensitic transformation, e.g. embryo potency, driving forces, plastic deformation as well as some aspects of crystallography. The results obtained are in agreement with the experimental results. The effect of stress-states on the stress-assisted martensitic microstructure evolution is studied by performing different simulations under different loading conditions. The results indicate that the microstructure is significantly affected by the loading conditions. The simulations are also used to study several important aspects, such as TRIP effect and Magee effect. The model is also used to predict some of the practically important parameters such as Msσ temperature as well as the volume fraction of martensite formed. The results also indicate that it is feasible to build physically based multilength scale model to study the martensitic transformation. Finally, it is concluded that the phase-field method can be used as a qualitative aid in understanding the complex, yet intriguing, martensitic transformations. Key words: Phase-field method, Martensitic transformations, Plastic deformation, Multi-length scale modeling, Microstructure, Stress states, Steels. Preface The growing interest in the field of computational materials engineering has lead to a successful exploitation of computational tools to model several phase transformations, especially the complex and intriguing martensitic transformation. The present thesis deals with the study of martensitic transformations in steels using three dimensional phase-field simulations. Some of the important physical concepts of martensite are explained in the introduction part and the readers are urged to refer to the cited works for more details. A summary of the mathematical framework of the 3D phase-field model as well as the modifications made to it to study different aspects of the martensitic transformation is presented. Finally, a summary of the results obtained from the above mentioned works is also presented. The full details can be found in the following appended supplements that will be referred to, throughout the thesis, with the respective supplement numbers. 1. H.K. Yeddu, A. Malik, J. Ågren, G. Amberg, A. Borgenstam, Threedimensional phase-field modeling of martensitic microstructure evolution in steels, Acta Materialia 60 (2012) 1538-1547. 2. H.K. Yeddu, A. Borgenstam, J. Ågren, Stress-assisted martensitic transformations in steels – A 3D phase-field study, Submitted manuscript. 3. H.K. Yeddu, V.I. Razumovskiy, A. Borgenstam, P.A. Korzhavyi, A.V. Ruban, J. Ågren, Multi-length scale modeling of martensitic transformations in stainless steels, Submitted manuscript. 4. A. Malik, H.K. Yeddu, G. Amberg, A. Borgenstam, J. Ågren, Three dimensional elasto-plastic phase-field simulation of martensitic transformation in polycrystal, Submitted manuscript. 5. H.K. Yeddu, A. Borgenstam, J. Ågren, Effect of martensite embryo potency on the martensitic transformations in steels – A 3D phase-field study, Journal of Alloys and Compounds, In Press, doi:10.1016/j.jallcom.2012.01.087. 6. H.K. Yeddu, A. Borgenstam, P. Hedström, J. Ågren, A phase-field study of the physical concepts of martensitic transformations in steels, Materials Science and Engineering A 538 (2012) 173-181. ix x PREFACE Respondents’ contribution to the papers • Supplement-1: The respondent performed all the work and wrote the entire paper with inputs from J. Ågren (J.Å.), A. Borgenstam (A.B.), A. Malik (A.M.) and G. Amberg (G.A.). • Supplement-2: The respondent performed all the work and wrote the entire paper with inputs from J.Å. and A.B. • Supplement-3: The respondent performed all the phase-field part of the work as well as wrote major part of the paper with inputs from J.Å. and A.B. The ab initio part of the work is performed by V.I. Razumovskiy (V.I.R.) with inputs from P.A. Korzhavyi (P.A.K.) and A.V. Ruban (A.V.R.). • Supplement-4: The respondent contributed to the model development as well as to drafting the paper with inputs from J.Å., A.B., A.M. and G.A. • Supplement-5: The respondent performed all the work and wrote the entire paper with inputs from J.Å. and A.B. • Supplement-6: The respondent performed all the work and wrote the entire paper with inputs from J.Å., A.B. and P. Hedström (P.H.). This work has also resulted in the following presentations • Invited talk on “Role of Plasticity during martensitic microstructure evolution in steels: A 3D phase field study”, presented at Plasticity 2012, San Juan, PR, USA, January 2012. • Invited talk on “Phase Transformations in Materials - A mesoscale study on steels using phase-field approaches”, presented at Los Alamos National Laboratory, Los Alamos, USA, January 2012. • Oral presentation on “3D Phase Field Modeling of Martensitic Microstructure Evolution in Steels”, presented at Phase Transformations Conference (PTM 2010), Avignon, France, June 2010. • Poster presentation on “Physical Parameters Determination From 3D Phase Field Simulations of Martensitic Transformations in Steels”, presented at TMS Annual Meeting, San Diego, USA, February 2011. • Poster presentation on “3D Modeling of Martensitic Transformations in Steels by Integrating Thermodynamics, Phase Field Modeling and Experiments”, at 1st World Congress on Integrated Computational Materials Engineering (ICME), Seven Springs, PA, USA, July 2011. • Poster presentation on “Effect of martensite nucleus potency on the martensitic transformation in steels: A Phase Field study”, presented at ICOMAT 2011, Osaka, Japan, September 2011. Contents Preface ix Contents xi 1 Introduction 1.1 Aim of the present work . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 2 Martensite 2.1 Morphology . . . . . . . . . 2.2 Transformation . . . . . . . 2.3 Plastic deformation aspects 2.4 Nucleation and growth . . . 2.5 Thermodynamic aspects . . 2.6 Crystallography . . . . . . . 3 3 4 6 6 7 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Phase-field method 13 4 Phase-field modeling of martensitic 4.1 Morphology and transformation . . 4.2 Plastic deformation aspects . . . . 4.3 Nucleation and growth aspects . . 4.4 Thermodynamic aspects . . . . . . 4.5 Crystallography . . . . . . . . . . . transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 18 24 24 25 25 5 Summary of results 5.1 Morphology and transformation 5.2 Plastic deformation aspects . . 5.3 Nucleation and growth aspects 5.4 Thermodynamic aspects . . . . 5.5 Crystallography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 27 36 38 41 43 . . . . . . . . . . . . . . . . . . . . . . . . . 6 Concluding remarks and future prospects xi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 xii CONTENTS Acknowledgements 51 Bibliography 53 Chapter 1 Introduction Steels, due to their superior mechanical properties such as good formability and high strength, have become a vital class of materials that are employed in a wide range of applications, ranging from kitchen cutlery to spacecraft. Based on the alloying composition, steels can be broadly classified as: carbon steels, low alloy steels and stainless steels. Carbon steels, whose basic alloying elements are iron and carbon, are the most commonly used steels for general applications and hence contribute to almost 90% of the world steel production. Stainless steels, whose basic alloying elements are iron, chromium and nickel, are employed for special type of applications that demand good corrosion resistance along with the above mentioned mechanical properties. Based on the main constituent phases, i.e. austenite, ferrite, martensite and mixture of austenite and ferrite, steels can be broadly classified as: Austenitic, Ferritic, Martensitic and Duplex steels, respectively. The mechanical properties of steels are determined by their microstructure and the constituent phases and hence the study of structure-property relations of steels is essential for both the industry as well as the academia. The discovery that the hardest steels contained an acicular microstructural constituent had been a pioneering effort, probably by Sorby [1], in explaining the cause for the widely differing mechanical properties of steels. The acicular structure was named as martensite, in honour of the German metallographer Adolf Martens. Martensite has since then been inevitably associated with the high strength of many commercial steels. Martensite (α) forms from austenite (γ) in a diffusionless manner, i.e. there is no variation in composition of the two solid phases during the transformation, and hence the phase transformation can be classified as a solid state diffusionless phase transformation, named martensitic transformation. Due to its practical importance, the martensitic transformation has undergone extensive studies that have lead to coining the term “martensitic transformation” as a generic name to describe a solid state diffusionless phase transformation, with specific features, that occurs in a wide variety of materials, such as steels, shape memory alloys as well as in ceramics. 1 2 CHAPTER 1. INTRODUCTION Based on the mode of formation, martensitic transformation in steels can be classified as: athermal, i.e. rapid transformation during quenching, isothermal, i.e. slower transformation while holding the steel at a constant temperature, stressassisted, i.e. by application of stress below the yield limit of the steel, and straininduced, i.e. by plastic deformation of the steel. Thus martensite can form in the steel products during their application. Some special steels, e.g. TRIP steels, are known for their ability to enhance their mechanical properties during “in-service” by means of martensite formation under applied stress. Hence in order to better understand the microstructure-property relations of steels, it is essential to thoroughly understand the martensite formation under various thermo-mechanical conditions. Though the martensite formation has been studied quite extensively over the past century, it is still very intriguing due to its complex nature that leaves some questions unanswered. The progress made over the past two decades in the field of electron microscopy as well as in the field of in situ studies using synchrotron radiation seems promising for performing in situ studies of the martensitic transformation. However the rapidity of the transformation makes it very difficult to be studied by in situ experiments. Phase-field method, a computational technique, has proved to be successful to simulate the microstructure evolution during various phase transformations [2–4], including martensitic transformation [5–14]. Hence an in-depth study of martensitic transformation using phase-field method, especially in 3D, could aid in understanding the transformation. Moreover, due to the growing interest in the “Integrated computational materials engineering (ICME)”, i.e. materials design and optimization by means of a combination of multi-length scale modeling approaches, the possibilites to couple the phase-field technique with other computational techniques need to be explored. 1.1 Aim of the present work In the present work, a three dimensional elastoplastic phase-field model is developed to study the athermal martensitic transformation, occurring in single crystal and polycrystalline steels, as well as the stress-assisted martensitic transformation occurring in single crystal steels. 3D phase-field simulations are performed by considering the material parameters corresponding to the carbon steels and stainless steels as the input data, which is acquired from computational as well as from experimental works. Thus an attempt is made to create a physically based multilength scale model by coupling ab-initio method at the atomistic level, phase-field method at the mesoscopic level, experimental works at the macro level and CALPHAD method, which is based on the fundamental principles of thermodynamics. The model is used to study the microstructure evolution as well as to study various physical aspects, such as morphology and transformation, plastic deformation, nucleation and growth, thermodynamics as well as some aspects of the crystallography, associated with the martensitic transformation. Chapter 2 Martensite 2.1 Morphology Depending on the alloy composition, martensite forms in the shape of laths, i.e. ruler shaped units, or plates, i.e. lenticular shaped units, as shown in Fig. 2.1. Carbon content of the alloy plays a major role in determining the martensite morphology. It has been experimentally observed that martensite forms in the shape of laths in low carbon steels and in the shape of plates in high carbon steels [15]. Figure 2.1: Light optical microscope images of the microstructure of (a) lath martensite in an IF steel with very low carbon content of 0.0049 wt% C [16] (b) plate martensite in Fe–1.86 wt% C alloy [17]. In the case of lath martensite, several parallel martensite units are formed adjacent to each other [15]. It has been reported that lath martensite forms in blocks and packets in an austenite grain as shown in Fig. 2.1 [16, 18]. The strength and toughness of martensitic steels, with lath morphology, are strongly related to packet 3 4 CHAPTER 2. MARTENSITE and block sizes [18, 19]. In the case of plate martensite, non-parallel martensite units are formed [15]. Moreover, a distinct feature in the case of plate martensite is the midrib, which is reported to be the first forming unit consisting of many transformation twins [20,21]. On either side of the midrib there exist twinned regions, with some twins that extend from the midrib, and untwinned regions, with dislocations [20, 21]. 2.2 Transformation During rapid cooling (quenching) of austenite, athermal martensitic transformation begins at the martensite start temperature (Ms ). Thereafter, the volume fraction of martensite increases with decreasing temperature and finally, the transformation is completed on reaching the martensite finish temperature (Mf ). In the case of stress assisted martensite, the martensite start temperature at a given stress level σ is termed Msσ . Fig. 2.2 shows the microstructural images obtained during an in situ observation, by using laser scanning confocal microscope, of plate martensite formation by means of athermal transformation during rapid quenching of a high carbon steel with a composition of Fe-0.88%C-4.12 %Cr, expressed in mass % [22]. The real time and temperature can be seen in the upper left corner. Slightly above the Ms , one can see untransformed austenite grains in Fig. 2.2a. Slightly below the Ms , martensitic transformation initiates in a heterogeneous manner, i.e. nucleation occurs at different sites as can be seen in Fig. 2.2b. One can see, in Fig. 2.2b, the untransformed austenite grains as well as partly transformed austenite grains with martensite plates. At a temperature close to Mf , one can see an almost completely transformed steel with complex martensitic microstructure containing several martensite plates oriented in different directions, as shown in Fig. 2.2c. Thus it can be understood that the martensite volume fraction increases with decreasing temperature. During rapid cooling, a diffusion controlled transformation of austenite to ferrite, with a very low carbon content, does not occur due to lack of time. However as the temperature is reduced below the T0 temperature, i.e. the temperature where the Gibbs energies of ferrite and austenite are the same, there exists a thermodynamic driving force available for the formation of ferrite (martensite) with the same composition as that of austenite, i.e. occurrence of a diffusionless phase transformation. Martensitic transformation leads to the crystallographic transformation of face centered cubic (FCC) austenite in to body centered cubic (BCC) martensite. The carbon atoms that are randomly distributed on the interstitial sites in FCC do not have time to migrate to the BCC in a random manner and hence move in a coordinated motion. This increases the tetragonality of the BCC lattice and thus the carbon containing martensite is of body centered tetragonal (BCT) structure. The tetragonality of martensite increases with increasing carbon content [23]. 2.2. TRANSFORMATION 5 Figure 2.2: In situ observation, by using laser scanning confocal microscope, of plate martensite formation in a high carbon steel, where the real transformation time and temperature are marked in the upper left corner [22]. (a) Microstructure at a temperature slightly above Ms . (b) Microstructure at slightly below Ms . (c) Final martensitic microstructure at a temperature close to Mf . In addition to the increased tetragonality, the increase in carbon content also leads to a volume expansion, i.e. dilatation. Moreover, the solid state nature of the transformation, that takes place by a cooperative movement of atoms, requires the two phases to be highly coherent and hence gives rise to a large amount of internal stresses inside the material. The internal stresses are relaxed by means of plastic deformation, i.e. by shear deformation. Moreover, the shear deformation needs to be a lattice-invariant shear such that it does not give rise to a macroscopically inhomogeneous crystal structure. Hence the lattice-invariant shear should only be microscopically inhomogeneous [23]. Theoretically, it has also been shown that the dilatation and shear needs to be associated with the rotation of martensite such that the combination of all three mechanisms gives rise to the experimentally observed crystallographic orientation 6 CHAPTER 2. MARTENSITE relationships of martensite [23]. Thus based on the above discussion, it can be understood that the martensitic transformation occurs by a cooperative short range movement of atoms such that the crystal lattice coherency (continuity) is maintained. 2.3 Plastic deformation aspects As mentioned in Section 2.2, plastic (shear) deformation in the surrounding austenite and internally in the martensite facilitates the relaxation (accommodation) of stresses, which are generated due to the volume and shape changes caused by the athermal transformation. In the case of stress-assisted transformation, the plastic accommodation process is popularly known as the Greenwood-Johnson effect [24–26], which can be considered to be at the origin of TRIP (Transformation induced plasticity) phenomenon. The TRIP phenomenon has been found to be advantageous in increasing the strength, imparted by martensite, and in increasing the ductility, caused due to the dislocations generated by the plastic deformation during martensitic transformation. Thus plastic deformation plays an important role in the martensitic transformation. The platic (shear) deformation should not disturb the overall crystal structure and hence needs to be a lattice-invariant shear, as explained in Section 2.2. The lattice-invariant shear deformation can be generated by the movement of perfect dislocations or partial dislocations that gives rise to slip or twinning respectively [23], as shown in Fig. 2.3. Fig. 2.3a corresponds to a case when there is no plastic deformation inside the martensite and hence in this case the shape changes are completely accommodated by deformation of the surrounding austenite. Figs. 2.3b and 2.3c show two different cases of plastic deformation, i.e. slip and twinning respectively, inside martensite. It has been reported that the type of shearing mechanism also affects the morphology of martensite [15,23]. In low carbon steels, where lath martensite is formed, slip mechanism is preferred and in high carbon steels, where plate martensite is formed, twinning mechanism is preferred [23]. Thus it can be understood that lath martensite is a dislocation-rich structure [15, 23, 27], whereas plate martensite contains several internal twins [15, 23], as mentioned in Section 2.1. The high dislocation density of lath martensite also indicates that the major plastic deformation occurs inside martensite, despite its higher yield strength compared to that of austenite. Dislocations also play a dominant role during nucleation and growth of martensite, as explained in the following section. 2.4 Nucleation and growth Martensite nucleation occurs in a heterogeneous manner [28], as can be seen in Fig. 2.2b. The nucleation initiates at a point in the austenite where there is a dense stacking of dislocation arrays [28–30]. The heterogeneous nature of nucleation can 2.5. THERMODYNAMIC ASPECTS 7 (a) (b) (c) Figure 2.3: Schematic of shear deformation mechanism in martensite [23]. (a) No lattice-invariant shear in martensite except the shape change caused by the transformation. (b) Slip. (c) Twinning. be attributed to the potency of the martensite embryos, which exist in a frozen-in state and attain potency on reaching the martensite start temperature Ms [31]. It should also be emphasized that only the most potent embryos transform at Ms , whereas the less potent embryos will transform at a temperature lower than Ms , as and when they attain potency. The potency of the embryos depends on their size, shape, orientation and the dislocation density at that site [28, 31]. Thus the embryos become potent once they reach a critical size and attain a favorably oriented shape that minimizes the combined interfacial and strain energies, i.e. ellipsoidal shape [31], and also when there is a critical number of dislocations available at the nucleation site [28]. It has also been reported that the stress-assisted transformation initiates at the same nucleation sites where the athermal transformation initiates [32]. After attaining potency, i.e. after a successful event of nucleation, the martensite embryos spontaneously grow with the aid of dislocation glide, in the case of lath martensite [33, 34]. The growth of the martensite units gives rise to higher level of internal stresses, which trigger autocatalysis, i.e. self nucleation of various martensite domains in order to relax the internal stresses, as observed in the case of lath martensite [35], plate martensite [36–38] as well as in the case of isothermal martensite [39]. Thus the transformation proceeds in three different stages, i.e. nucleation, growth and autocatalysis. 2.5 Thermodynamic aspects From a thermodynamic point of view, the reason for the embryos attaining potency at the Ms temperature, as explained above, can be attributed to the thermodynamic driving force ∆Gm , i.e. the difference between the Gibbs energies of austenite and 8 CHAPTER 2. MARTENSITE martensite. At T0 temperature, where there is no chemical driving force available as the Gibbs energies of the two phases are equal, the Gibbs energy barrier ∆G∗ opposes the phase transformation. As the temperature is reduced to Ms , during quenching, the driving force becomes large enough such that it can overcome the Gibbs energy barrier and hence the phase transformation takes place. Apart from the chemical energy Gchem , aroused due to quenching, the austenite-martensite interface gives rise to an interfacial energy, i.e. the gradient energy Ggrad , and the coherency of the two phases gives rise to internal stresses that in turn give rise to elastic strain energy Gel . Based on the studies on nucleation and growth mentioned in Section 2.4, it can be understood that there exist different thermodynamic driving forces that govern the different stages of the martensitic transformation. Some works have showed that a less driving force is needed for growth [40, 41] and autocatalysis [37] compared to that required for nucleation. Thus martensite, once nucleated, could grow and also give rise to autocatalysis above Ms temperature. However Borgenstam et al. [40] reported that the driving force for nucleation and growth are very close to each other, although a less driving force is needed for growth compared to that for nucleation. In the case of stress-assisted martensite formation the externally applied mechanical energy Gappl , depending on the nature of the load, either contributes to or detracts to the thermodynamic driving force. Thus if the nature of load is favorable for martensite formation, one can expect that Msσ is higher than Ms , whereas Msσ is lower than Ms if the nature of load is not favorable for martensite formation. Moreover, for a given stress state, only those martensite units with the most favorable orientation with respect to the applied stress are formed. This effect is known as the Magee effect [24–26]. The thermodynamic aspects of both athermal and stress-assisted transformations, in terms of an internal variable η that locally represents the extent of the transformation, is schematically shown in Fig. 2.4. The internal variable can be considered as a parameter that corresponds to the crystal structure of the two phases, i.e. austenite (η = 0) and martensite (η = 1), as explained in the following section. 2.6 Crystallography The common plane, between the austenite and martensite, on which martensite nucleates is called the habit plane [42]. Experimental results show that the habit γ γ plane in the case of lath martensite is close to {111} [43, 44] as well as {557} [5]. As martensitic transformation occurs by a cooperative movement of atoms that transforms the FCC structure in to the BCC or BCT structure, the habit plane, i.e. the interface between the product and parent phases must be highly coherent. This is also supported by the experimental observations that the habit plane is welldefined crystallographically, i.e. there exist specific crystallographic orientation 2.6. CRYSTALLOGRAPHY 9 Figure 2.4: Schematic showing the thermodynamic aspects of athermal and stressassisted martensitic transformation. Gibbs energy curves in the absence of chemical driving force, in the presence of chemical driving force only, as well as in the presence of both chemical and mechanical driving forces. relationships between austenite and martensite. In order to understand the crystallography of martensite, phenomenological theories were developed independently by Wechsler, Lieberman and Read and by Bowles and Mackenzie [23]. Such phenomenological theories are based on the concept that the habit plane should be an invariant plane, i.e. undistorted and unrotated plane [23]. Thus a deformation along the invariant plane that also satisfies the coherency requirement is called an invariant plane strain, where twinning is an example of such a deformation. An austenite-martensite interface could be a coherent invariant habit plane if the FCC and BCT are oriented in such a way that their dense-packed planes and dense-packed directions are parallel to each other. Thus there exist specific crystallographic orientation relationships between austenite and martensite, i.e. orientation (rotation) of the habit plane of the martensite crystal with respect to a particular crystallographic plane in the austenite crystal. There exist four main austenite-martensite orientation relationships (OR), namely Bain OR, Nishiyama-Wassermann (N-W) OR, Kurdjumov-Sachs (K-S) OR and Greninger-Troiano (G-T) OR [23, 45, 46]. The K-S and N-W OR describe the experimentally observed OR of martensite, whereas the G-T OR is intermediate between the K-S and N-W OR [46]. The Bain OR is considered to be the basic OR 10 CHAPTER 2. MARTENSITE that can be used to explain the K-S and N-W OR, as will be explained below. According to the Bain OR, an FCC austenite crystal (η = 0) can be transformed to BCT martensite crystal (η = 1) by applying the Bain distortion (Bain strain), i.e. a set of a compressive strain along one of the coordinate axes and two equal tensile strains along the other two coordinate axes [23]. There exist three possibilities to apply such a Bain strain on an FCC crystal and hence there exist three Bain strains that give rise to three Bain variants (η1 , η2 , η3 ), as shown in Fig. 2.5. The OR between FCC and BCT crystals, according to Bain correspondence is shown in Fig. 2.6. Figure 2.5: Schematic showing the three different Bain variants [4]. Figure 2.6: Bain orientation relation showing the correspondence between FCC and BCT crystals [23]. 2.6. CRYSTALLOGRAPHY 11 In an FCC crystal of austenite, there exist four close-packed planes, i.e. (111)γ , (−111)γ , (1 − 11)γ , (11 − 1)γ . Thus there exist four independent choices of the γ {111} plane. Both the K-S and N-W orientation relations are in agreement that γ α α {111} // {110} . There exist three {110} planes that could lie parallel to each of γ α the four {111} planes. Therefore there exist 12 independent choices, i.e. 3 {110} γ x 4 {111} , of parallel planes of FCC and BCC along which martensite could form. Figure 2.7: Illustration showing the 2 twin related K-S variants and 1 variant accoriding to N-W OR. The three variants are examples of each of the three Bain variants [47]. According to N-W OR, only one martensite variant could form along each of these 12 parallel planes, as shown in Fig. 2.7, thereby giving rise to 12 N-W γ martensite variants in total, i.e. 3 N-W martensite variants per each of the {111} close-packed plane. According to K-S OR, two twin related martensite variants, as shown in Fig. 2.7, could form along each of the 12 parallel planes, thereby giving γ rise to 24 K-S variants in total, i.e. 6 K-S variants per each of the {111} closepacked plane, as shown in Fig. 2.8. More details regarding the above explanations can be found in Refs. [45, 46]. It has been theoretically explained that the Bain strains (dilatation) coupled with the shear and a rigid body rotation could give rise to the experimentally observed habit planes [9, 45]. Thus the K-S and N-W OR, which are the experimentally observed OR, are composed of Bain strains, shear and rotation. This implies that each of the 24 K-S and 12 N-W martensite variants could be obtained by applying Bain strain on the FCC lattice and thereafter by applying shear and rotation. Thus it can be understood that all the above mentioned martensite variants could be grouped in to three distinct groups based on the applied Bain strains (Bain distortion) and could be called Bain groups or Bain variants [45–47]. 12 CHAPTER 2. MARTENSITE Figure 2.8: A schematic of the K-S orientation relation [18]. Chapter 3 Phase-field method The modeling, performed more than a century ago by van der Waals, of a liquidgas system by means of density function that continuously varies over the interface can be considered as the first application of the phase-field method. The Ginzburg and Landau’s work, approximately 50 years ago, to model superconductivity using a complex-valued order parameter can be considered to be the origin of the concept of the order parameters, i.e. phase-field variables. The microscopic (continuous) theories proposed by Landau to describe the structural changes occurring during a order-disorder transition by using mathematical expressions, which are called Landau polynomials, are widely used in the modeling of ordering transformations. The work by Cahn and Hilliard that proposed a thermodynamic formulation that accounts for the gradients in thermodynamic properties in heterogeneous systems with diffuse interfaces is usually regarded as the basis for the present phase-field models. Khachaturyan and his co-workers, in the 80’s, have developed Phase-field Microelasticity theory, i.e. a phase-field theory coupled with elastic strain calculations based on the Eshelby approach, and by using Landau polynomials to model martensitic transformations. Thereafter the order-disorder transitions have been further explored by several researchers and their co-workers, e.g. L.Q.Chen, Y.Wang, A.Artemev, A.Yamanaka, A.Finel, A.Saxena, T.Lookman, S.Shenoy, K.Bhattacharya and V.Levitas. Another significant contribution to the phase-field approach can be attributed to Langer, who developed a phenomenological single phase-field model that is mainly used to model solidification. Thereafter the solidification-type of models are well studied by several researchers, e.g. J.Warren, B.Boettinger, R.Kobayashi, A.Karma, I.Loginova, J.Agren, G.Amberg and N.Moelans. Phase-field models are now widely used for simulating a wide range of phenomena, e.g. microstructure evolution, solidification, precipitate growth and coarsening, martensitic transformations, grain growth, solid state phase transformations, crack propagation and nucleation, dislocation motion [2–4]. More details about the above mentioned history and the corresponding references can be found in Ref. [4]. 13 14 CHAPTER 3. PHASE-FIELD METHOD Figure 3.1: A schematic of the diffuse interface approach, i.e. continuous variation in the internal properties across the interface [48]. Phase-field method is a diffuse interface approach, i.e. the internal properties of a system undergoing a phase transformation vary continuously over the interface between the two phases. The diffuse interface approach can be explained with the help of schematic Fig. 3.1 that shows the advancement of a domain wall at three instances [48]. Each vertical rod represents the direction of magnetization of a plane of atoms, whereas each row of rods represent the domain wall at an instance. On either side of the center, indicated by an arrow in the middle, one can see that the magnetic spin of atoms is different. It can also be seen that the spin direction varies continuously over the interface. From time to time, it can be seen that the domain wall (arrow) advances and also that the spin characteristics change continuously over the interface. Hence the phase-field method, which is based on the diffuse interface approach, is capable of predicting the continuous variation in the internal properties over the interface between the two phases of a system that is undergoing a phase transformation. Thus the advantage of the diffuse interface approach is that it avoids the need to explicitly track the position of the moving phase interface at each instance. Thus phase-field models are able to predict complex microstructural evolutions. During a phase transformation occurring in a material, the microstructure evolves due to minimization of the Gibbs energy of the system. The Gibbs energy of the system under consideration is described based on the underlying physics of the phenomenon that has to be studied. Thus depending on the physical description of the system, the Gibbs energy G could consist of different parts, e.g. bulk (Gbulk ), interfacial (Gint ), elastic (Gel ), or any other applied external energy (Gappl ). G = Gbulk + Gint + Gel + Gappl (3.1) The bulk part of the Gibbs energy Gbulk corresponds to the bulk chemical properties of the two phases. The interfacial part of the Gibbs energy Gint corresponds to the properties of the interface. The elastic part of the Gibbs energy Gel corresponds to the elastic strain energy that is generated inside the material due to 15 Figure 3.2: A schematic of the phase-field approach. the phase transformation, e.g. martensitic transformation, spinodal decomposition. The applied external energy Gappl corresponds to the energy generated due to an externally applied force, e.g. mechanical force due to externally applied stresses, magnetic forces due to an externally applied magnetic field. The Gibbs energy G is dependent on several internal variables (properties) that describe the material characteristics, such as grain characteristics as well as the composition of the material. In order to model the microstructure evolution, the Gibbs energy G(η, c) of the system is expressed as a mathematical function that depends on the internal variables, η and c. η, a non-conserved quantity, governs the characteristics of the grain (crystal) and c, a conserved quantity, is the composition of the system. Such internal variables that describe the material characteristics are field variables that are continuous mathematical functions in time and space, i.e. η(x, y, z, t) and c(x, y, z, t), and are called phase-field variables. The values of the phase-field variables vary between 0 and 1. In the case of a single phase-field approach used to study solidification, η = 0 or 1 corresponds to either of the two bulk phases, whereas a deviation from these two values corresponds to the interface region between the two bulk phases. In the case of multi-phase-field approach used to study an order-disorder phase transformation, which gives rise to an asymmetry in the resultant crystal structure such that each crystal orientation of the product phase is distinct, one has to consider different phase-field variables that represent each of the above mentioned crystal orientations. Thus the number of phase-field variables will depend on the number of crystal orientations. Each phase-field variable will be equal to 0 in the product phase and equal to 1 in their respective product phases. The temporal, i.e. time-dependent, evolution of the microstructure is tracked by solving a set of partial differential equations (PDE’s). Thus the solutions of 16 CHAPTER 3. PHASE-FIELD METHOD the two PDE’s yield the values of the internal variables at each time step and at each point in the material and thereby predicts the changes in the microstructure. A schematic of the phase-field approach is presented in Fig. 3.2. The PDE that tracks the changes in the non-conserved quantities, such as grain characteristics, is usually referred as Cahn-Allen equation, whereas the PDE that tracks the changes in the conserved quantities, such as composition, is usually referred as Cahn-Hilliard equation [2–4]. Moreover both the equations contain several kinetic coefficients that govern the kinetics of the respective phase-field variables. In the schematic Fig. 3.2, the kinetic coefficients are not presented in order to give a rather simple picture of the phase-field modeling concept. More detailed derivations and explanations can be found in some of the reviews on phase-field modeling [2–4]. Chapter 4 Phase-field modeling of martensitic transformations The pioneering efforts in simulating the martensitic microstructure evolution could be attributed to Khachaturyan [5] and his co-workers [6, 8, 9, 12]. The other significant contributions to this field could be attributed to Bhattacharya [11], Saxena and his co-workers [7, 49, 50], Levitas and his co-workers [10, 51–53]. The effect of the plastic deformation on the transformation, by coupling the phase-field method with the plasticity equation, has been studied by Tomita and his co-workers [13,14], based on the work of Guo et al. [54]. The present work is based on the work by Khachaturyan et al. [5, 8, 9]. The plastic deformation is also included, based on the work by Yamanaka et al. [13]. The modeling of the transformation in polycrystals is based on the work by Yamanaka et al. [14]. The phase-field as well as the continuum mechanics equations are solved by using the finite element method, which allows a straight forward and transparent formulation of the equations. The finite element method (FEM) is well established for the modeling of thermomechanical deformation. The basis of the method is to solve for the response of a complicated body to imposed constraints by dividing it into smaller elements, whose responses are more readily evaluated. Each element has a simple shape, such as a triangle or quadrilateral for a 2D problem, or a tetrahedron for a 3D analysis. The vertices of the elements are called nodes. The complete body is formed by a mesh of elements, joined at nodes. Associated with each element is a stiffness, which is dependent upon the constitutive law of the material within the element. By subjecting the element to a force, the overall response of the body is determined by simultaneously solving for the node displacements such that the conditions of equilibrium and boundary constraints are met. A detailed explanation about the FEM can be found in Ref. [55]. All the single crystal computations are performed on a mesh with 50x50x50 grid points and by using tetrahedral finite elements, whereas the polycrystal computa17 18 CHAPTER 4. PHASE-FIELD MODELING OF MARTENSITIC TRANSFORMATIONS tions are performed on a mesh with 150x150x150 grid points. The entire mathematical framework is solved by using femLego software, a symbolic computational tool [56]. femLego is a set of Maple procedures and fortran subroutines that can be used to build complete fortran simulation codes for partial differential equations, with the entire problem definition done in Maple. OpenDX software is used for plotting the images. In the following sections, a brief overview of the phase-field model developed to study the morphology and transformation aspects, associated with athermal martensitic transformation occurring in a single crystal as well as in a polycrystalline material and stress-assisted martensitic transformation occurring in a single crystal, is presented. Einstein summation convention, i.e. summation implied over repeated indices, is used to express all the equations except for the indices representing different martensitic domains. A brief procedure explaining the multi-length scale modeling, i.e. by coupling CALPHAD, ab-initio, phase-field and experimental data, of martensitic transformation is also presented. A summary of the simulation procedures to study various physical concepts, such as plastic deformation, nucleation and growth, thermodynamic aspects and crystallography, associated with martensitic transformation is also presented. More details can be found in the respective supplements. 4.1 Morphology and transformation Athermal martensitic transformation, Supplement-1 In this part of the work, athermal martensitic transformation occurring in a single crystal of Fe–0.3%C steel is considered. A diffusionless phase transformation, such as martensitic transformation, can be modeled using the Cahn-Allen equation [57], also known as the Time-Dependent Ginzburg-Landau (TDGL) kinetic equation [5]. The microstructure evolution can be simulated by predicting the time-dependent variation of the phase-field variable, which in turn is related to the minimization of the Gibbs energy G of the system with respect to phase-field variable ηp as: q=v X δG ∂ηp =− Lpq ∂t δηq q=1 (4.1) δG is a variational derivative that serves as a driving force for the formawhere δη q tion of martensite denoted by the phase-field variable ηq that is dependent on the position vector r, v is the total number of martensite variants and Lpq is a kinetic parameter. In the present work the Bain orientation relation is considered, as this is regarded as the basic orientation relation that can be used to explain the K-S and N-W orientation relations as explained in Section 2.6. According to Bain orientation relation, explained in Section 2.6, the phase transformation of FCC austenite to BCT martensite gives rise to three martensite variants (orientations). Thus in 4.1. MORPHOLOGY AND TRANSFORMATION 19 order to represent the three possible orientations, three phase-field variables, i.e. η1 , η2 , η3 , need to be considered in the model and hence three phase-field equations, i.e. for p = 1, 2, 3 and v = 3 in Eq. (4.1), need to be solved at each time step. Thus the three phase-field variables correspond to the three Bain groups (Bain variants), mentioned in Section 2.6. From a thermodynamic point of view, as explained in Section 2.5, the Gibbs energy consists of three parts: Z G= (4.2) Gchem + Ggrad + Gel v v v dV V Gchem corresponds to the chemical part of the Gibbs energy density of an unv is the extra Gibbs stressed system at the temperature under consideration. Ggrad v energy density caused by the interfaces. The transformation of cubic austenite into tetragonal martensite induces elastic strain energy density Gel v , into the material. Chemical energy The chemical part of the Gibbs energy density Gchem , expressed as a Landau-type v polynomial [9, 58], is given by: Gchem (η1 , η2 , η3 ) v 1 1 1 A η12 + η22 + η32 − B η13 + η23 + η33 = Vm 2 3 1 2 2 2 2 + C η1 + η2 + η3 4 (4.3) By considering the driving force ∆Gm , i.e. the difference in the Gibbs energies of austenite and martensite, and the Gibbs energy barrier ∆G∗ terms in the above equation, the coefficients are modified [58] as: A = 32∆G∗ , B = 3A − 12∆Gm and mβ C = 2A − 12∆Gm . ∆G∗ is expressed as ∆G∗ = V2δ 2 . Vm is molar volume, δ is thickness of the interface and β relates to interfacial energy as explained below. Gradient Energy as presented in [9,58] can be expressed as: The gradient energy density term, Ggrad v Ggrad = v p=v X p=1 βij (p) ∂ηp ∂ηp ∂ri ∂rj (4.4) where r(x,y,z) is the position vector expressed in Cartesian coordinates. βij is the isotropic gradient coefficient matrix, i.e. diagonal tensor with all the elements equal 9γ 2 Vm ∗ to β = 16∆G ∗ . γ is interfacial energy, Vm is molar volume and ∆G is Gibbs energy barrier. 20 CHAPTER 4. PHASE-FIELD MODELING OF MARTENSITIC TRANSFORMATIONS Elastic energy The internal elastic stresses developed in the material, as explained in Sections 2.2 and 2.5, give rise to elastic strain energy density Gel v . The elastic stress can be calculated using the Microelasticity theory [5] as explained below. As mentioned in Section 2.6, martensitic transformation gives rise to a change in the crystal structure caused by Bain strains ǫ00 ij (r), which in turn induce stress-free transformation strains ǫ0ij (r) in the material. The surrounding austenite matrix exerts strain ǫij (r) on the martensite to resist the stress-free transformation strains and thereby induces elastic strain ǫel ij (r), which in turn gives rise to stress σij (r), in the material. The actual strain tensor ǫij (r) is linearly related to the local displacement vector u(r), which is obtained by solving the mechanical equilibrium condition. The elastic strain energy density term Gel v can be expressed as: Gel v = Z ǫij (r) σij (r)dǫij (r) (4.5) ǫ0ij (r) where ǫ0ij (r) is the stress-free transformation strain, ǫij (r) is the actual strain and σij (r) is the stress, given by: σij (r) = cijkl ǫel kl (r) (4.6) where cijkl is the tensor of elastic modulii and ǫel kl (r) is the elastic strain, given by: pl 0 ǫel kl (r) = ǫkl (r) − ǫkl (r) − ǫkl (r) (4.7) where ǫpl kl (r) is the plastic strain. The actual strain ǫij (r) is given by: ǫij (r) = 1 2 ∂ui (r) ∂uj (r) + ∂rj ∂ri (4.8) where u(r) is the local displacement vector obtained by solving the mechanical equilibrium condition, expressed as: ∂σij (r) =0 ∂rj (4.9) The displacement vector can then be used to calculate the actual strain ǫij (r), which in turn can be used in calculating the local elastic strain energy density Gel v . The stress-free transformation strain is given by: ǫ0ij (r) = p=v X p=1 ηp (r)ǫ00 ij (p) (4.10) 21 4.1. MORPHOLOGY AND TRANSFORMATION where ǫ00 ij (p) is the Bain strain (Bain distortion) tensor, i.e. a set of a compressive strain along one of the coordinate axes and two equal tensile strains along the other two coordinate axes as explained in Section 2.6, given by: ǫ3 0 0 ǫ1 0 0 ǫ1 0 0 0 ǫ1 0 , ǫ00 0 ǫ3 0 , ǫ00 0 ǫ1 0 (4.11) ǫ00 ij (1) = ij (2) = ij (3) = 0 0 ǫ1 0 0 ǫ1 0 0 ǫ3 c c is a compressive transformation strain, ǫ1 = ata−a is a tensile where ǫ3 = cta−a c c transformation strain and are defined based on the lattice constants of austenite (ac ) and martensite (at , ct ). Thus, as shown in Fig. 2.5, martensite (Bain) variants1, 2 and 3 can be transformed from cubic austenite by applying Bain strains ǫ00 ij (1), 00 ǫ00 (2) and ǫ (3) respectively. ij ij The plastic strain comes into existence only when the stress exceeds the yield limit, thereby initiating plastic deformation that acts as a relaxation of the stress, as explained in Sections 2.2 and 2.3. von Mises criterion is employed to check whether the stress has exceeded the yield stress σy . According to von Mises criterion, plastic deformation, i.e. yielding, occurs if Eq. (4.12) is satisfied. o 1n 2 2 2 (σxx − σyy ) + (σyy − σzz ) + (σzz − σxx ) σ̄ 2 − σy2 = 2 (4.12) 2 2 2 − σy2 ≥ 0 + 3 σxy + σyz + σzx where σ̄ is von Mises equivalent stress. In the absence of strain hardening effect, the plastic strain evolution equation [13, 54, 58] can be expressed as: ∂ǫpl ∂Gshear ij (r) v = −kijkl pl ∂t ∂ǫkl (r) (4.13) shear where ǫpl is shear energy density and kijkl is ij (r) is the local plastic strain, Gv plastic kinetic coefficient expressed as: kijkl = kc−1 ijkl (4.14) where c−1 ijkl denotes the compliance tensor and k is a parameter, which controls the rate at which the stresses are relaxed by means of plastic deformation and is called plastic relaxation rate. The shear energy density Gshear is given by: v Gshear v = Z eij (r) Z eij (r) σij (r)deij (r) e0ij (r) = e0ij (r) (4.15) cijkl ekl (r) − e0kl (r) deij (r) 22 CHAPTER 4. PHASE-FIELD MODELING OF MARTENSITIC TRANSFORMATIONS where eij (r) and e0ij (r) are the deviatoric actual strain tensor and deviatoric stressfree transformation strain tensors given by: 1 eij (r) = ǫij (r) − ǫkk (r)δij 3 (4.16a) 1 0tot e0ij (r) = ǫ0tot ij (r) − ǫkk (r)δij 3 (4.16b) where ǫ0tot ij (r) is given by: pl 0 ǫ0tot ij (r) = ǫij (r) + ǫij (r) (4.17) The anisotropic elastic properties of different phases are taken in to account by considering different tensors of elastic modulii cijkl for different phases. In order to consider different cijkl an expression as shown in Eq. (4.18) is employed such that for a given phase, i.e. ηp = 0 or ηp = 1, the corresponding tensors of elastic tetragonal modulii ccubic are considered respectively whereas in the interface a ijkl or cijkl weighted cijkl that depends on the weightage yielded by the phase-field variable is tetragonal considered. The expressions of ccubic are presented in Supplement-1. ijkl and cijkl tetragonal cijkl = ccubic ηp ijkl (1 − ηp ) + cijkl (4.18) In all the figures, iso-surfaces (ηp=1,2,3 = 0.5) of the three phase-field variables, i.e. η1 , η2 , η3 , that represent the three martensite (Bain) variants are plotted in different colors, i.e. red, blue and green respectively. All the simulations in this work are started with a pre-existing spherical martensite embryo of variant-1. Stress-assisted martensitic transformation, Supplement-2 In this part of the work the above model in Supplement-1 is modified to study the stress-assisted martensitic transformation occurring in a single crystal of Fe– 0.3%C steel. In order to model the stress-assisted martensitic transformation, the extra Gibbs energy density aroused due to the externally applied stress needs to be included in the Gibbs energy of the system [8], as explained in Section 2.5. Hence from a thermodynamic point of view, the Gibbs energy of a system undergoing stress-assisted martensitic transformation consists of the following parts: Z appl G= dV (4.19) Gchem + Ggrad + Gel v v v + Gv V where Gchem , Ggrad and Gel v v v are the chemical, gradient and elastic parts of the is the extra Gibbs Gibbs energy density, as explained in the above section. Gappl v energy density aroused due to the externally applied stress, expressed as: appl 0 Gappl = −σij ǫij (r) v (4.20) 4.1. MORPHOLOGY AND TRANSFORMATION 23 appl where σij is the externally applied stress tensor, expressed by the Cauchy stress tensor as: σxx σxy σxz appl (4.21) σij = σyx σyy σyz σzx σzy σzz Eq. (4.2) is replaced with Eq. (4.19) and the entire model framework explained in the above section corresponding to Supplement-1 is updated accordingly to model the stress-assited martensite formation. Multi-length scale modeling of martensitic transformations, Supplement-3 In this part of the work, the above models in Supplements-1 and 2 are employed to study the athermal as well as the stress-assisted martensitic transformations occurring in a single crystal of stainless steel of type 301 with a composition of Fe– 17%Cr–7%Ni. Moreover an attempt is made to develop a multi-length scale model by coupling the CALPHAD technique that is based on the fundamental principles of thermodynamics, ab-initio at the atomistic level, phase-field at the mesoscopic level and the experimental works at the macro level. The elastic constants of martensite (BCC) phase are obtained from ab-initio calculations, the details of which are presented in Supplement-3. The rest of the input parameters such as lattice constants, elastic constants of FCC phase, Ms temperature and interfacial energy are acquired from experimental data. The chemical driving force available at Ms is calculated using the CALPHAD technique. Athermal martensitic transformation in polycrystalline material, Supplement-4 In this part of the work, the model explained in Supplement-1 is modified to study the athermal martensitic transformation occurring in a polycrystalline Fe–0.3%C steel. In order to simulate martensitic transformation in a polycrystalline material that consists of several grains having random orientations, one needs to define a coordinate system in each grain. Each grain is represented by a local coordinate system that is oriented at an angle with respect to the global coordinate system. Thus each martensite variant needs to be rotated according to the grain orientation. As each martensite variant is governed by the transformation strain tensor, the rotation matrix that defines the grain orientation needs to be multiplied by the stress-free transformation tensor. To simulate the polycrystal in 3D, three Euler angles, φ, θ and ψ are introduced to describe the orientation of the grain in the global coordinate system based on x-y-z convention. For the 3D case, Eq. (4.10) is modified as following: 24 CHAPTER 4. PHASE-FIELD MODELING OF MARTENSITIC TRANSFORMATIONS ǫ0ij (r) = p=v X Qik (φ, θ, ψ)Qjl (φ, θ, ψ)ηp (r)ǫ00 kl (p) (4.22) p=1 where the rotation matrix Q(φ, θ, ψ) is given by: Q(φ, θ, ψ) = cosθcosψ cosθsinψ −sinθ −cosφsinψ + sinφsinθcosψ cosφcosψ + sinφsinθsinψ sinφcosθ sinφsinψ + cosφsinθcosψ −sinφcosψ + cosφsinθsinψ cosφcosθ (4.23) Eq. (4.10) is replaced with Eq. (4.22) and the entire model framework explained in the above section corresponding to Supplement-1 is updated accordingly to model the athermal martensite formation in polycrystalline materials. 4.2 Plastic deformation aspects, Supplements-5 and 6 The plastic deformation aspects associated with the transformation can be studied by studying the effect of plastic relaxation rate k on the transformation. Hence different simulations are performed with different plastic relaxation rates k and by considering the driving force for nucleation at Ms . The model presented in Supplement-1 is employed to perform the simulations by considering the athermal martensitic transformation occurring in a single crystal of Fe–0.3%C steel. It can be observed from Eqs. (4.13) and (4.14) that k controls the plastic strain rate and thereby quantitatively controls the extent of plastic deformation of the material. Orowan equation states that the strain rate is proportional to the dislocation density, Burgers’ vector and the dislocation velocity, which in turn is proportional to the shear stress [59]. Thus by comparing Eq. (4.13) with the Orowan equation, it can be understood that k is directly proportional to the dislocation mobility as well as to the dislocation density. Hence by studying the effect of plastic relaxation rate, the effect of dislocation density and dislocation mobility on the embryo potency, morphology as well as on the overall transformation can be studied. 4.3 Nucleation and growth aspects, Supplement-5 Although nucleation is not simulated in the present work, several aspects associated with the martensite nucleation are investigated by studying the effect of the martensite embryo potency on the transformation. The model presented in Supplement-1 is employed to perform the simulations by considering the athermal martensitic transformation occurring in a single crystal of Fe–0.3%C steel. As mentioned in Section 2.4, the potency of the embryos depends on their size, shape, orientation and the dislocation density at that nucleation site. 4.4. THERMODYNAMIC ASPECTS 25 In order to study the effect of the dislocation density on the embryo potency, the effect of the model parameter, i.e. plastic relaxation rate k in Eq. (4.14), on the transformation is studied. Thus different simulations are performed with different values of k. The relation between k and the dislocation density is explained above in Section 4.2. In order to study the favorable size of the martensitic embryo, different simulations are performed with different embryo sizes. In order to study the favorable shape of the martensitic embryo, different simulations are performed by initiating the transformation (simulation) with a pre-existing spherical martensite embryo and with an ellipsoidal embryo. In order to study the effect of the embryo orientation on the transformation, three different simulations are performed by considering a pre-existing ellipsoidal shaped embryo with different orientations, i.e. the longest axis of the ellipsoid being oriented along the three different coordinate axes. 4.4 Thermodynamic aspects, Supplement-6 As explained in Section 2.5, there exist different thermodynamic driving forces that govern different stages of the transformation, such as driving forces required for nucleation, growth and autocatalysis. The driving force for nucleation (∆Gnucl. ) at Ms is obtained by using Thermo-Calc software and TCFE6 database [60], which is based on the CALPHAD method. For a given driving force ∆Gnucl. , the value of the plastic relaxation rate k is determined, as explained above in Section 4.3, such that the transformation occurs at the experimental Ms , i.e. the embryo attains potency at Ms . The model presented in Supplement-1 is employed to perform the simulations by considering the athermal martensitic transformation occurring in a single crystal of Fe–0.3%C steel. The critical driving force required for growth (∆Ggrowth ) of martensite is determined by starting the simulation with ∆Gnucl. and thereafter by reducing the driving force in a step-wise manner until the growth completely stops. In order to study the critical driving force required for autocatalysis (∆Gautocat. ), a similar procedure as mentioned above is employed by ensuring that autocatalysis is avoided at each step. 4.5 Crystallography, Supplement-6 The habit plane of martensite is determined by plotting the plane along which the infinitesimally small unit of martensite forms during the simulations. The rotations of martensite variants, during the simulations, are studied by observing the deviation of the martensite units from the above mentioned habit plane. The concept of the invariant habit plane, as explained in Section 2.6, is also studied by plotting a 2D section image of the 3D microstructure image and the corresponding displacement plot. The model presented in Supplement-1 is employed to perform the simulations by considering the athermal martensitic transformation occurring in a single crystal of Fe–0.3%C steel. Chapter 5 Summary of results 5.1 Morphology and transformation Athermal martensitic transformation, Supplement-1 The simulation results indicate that the martensitic microstructure consists of several morphological mirrors, i.e. twinned variants marked by ‘1’ and ‘2’ as shown in Fig. 5.1, as also observed in the experimental works [15]. It is observed from the simulations, as shown in Fig. 5.2, that nucleation of several martensite variants, i.e. autocatalysis, occurs as a consequence of the materials’ pursuit to minimize the internal stresses. It is also observed that in the absence of plastic deformation and dilatation, the martensitic microstructure formation is completed earlier compared to that in the presence of plastic deformation and dilatation, as shown in Fig. 5.3. It can be understood that plastic deformation facilitates relaxation Figure 5.1: Martensitic microstructure consisting of twinned martensite variants [58]. 27 28 CHAPTER 5. SUMMARY OF RESULTS (a) (b) (c) Figure 5.2: 3D simulation results with elastoplastic material and having anisotropic elastic properties [58]. Snapshots at (a) t*=0 (b) t*=40 (c) t*=230. Figure 5.3: Top view of the simulation results of athermal martensitic transformation in single crystal. (a) Without dilatation and plastic deformation, at non-dimensionless time t*=170. (b) With dilatation and plastic deformation, at t*=230 [58]. of the stresses and hence reduces the need for minimization of stresses by means of new martensite variant formation, i.e. autocatalysis. Thus it can be concluded that both autocatalysis and plastic deformation are different modes of relaxing the internal stresses generated due to the martensitic transformation. From Fig. 5.3, it can be seen that martensite variants form in pairs, i.e. variant pairs (1, 2), (1, 3) and (2, 3), in different directions. This pattern is observed in both pure elastic material, i.e. Fig. 5.3a, as well as in the elastoplastic material, i.e. Fig. 5.3b, and hence it can be understood that the formation of pairs of domains is 5.1. MORPHOLOGY AND TRANSFORMATION 29 Figure 5.4: Results from athermal martensitic transformation in single crystal with non-clamped boundary conditions, at t*=140 [58]. due to the tendency of the system to minimize its elastic strain energy. It is likely that in a given direction, a variant combination is preferred in order to maximize the energy minimization. The simulation results indicate that the boundary conditions on the grain also play a major role in the martensitic microstructure evolution. A clamped grain with constrained boundary conditions gives rise to a muti-variant martensitic microstructure, whereas a non-clamped grain with unconstrained boundary conditions gives rise to formation of a single martensite variant over the entire grain, as shown in Fig. 5.4. In the case of a non-clamped system, the unconstrained boundaries of the simulation domain accommodate the elastic stresses and hence there is no “urgent” need for the material to form the autocatalysed martensite variants. Stress-assisted martensitic transformation, Supplement-2 The results indicate that the role of stress states in martensitic microstructure evolution is very crucial. The microstructure evolution is significantly affected due to the anisotropic loading conditions, as shown in Fig. 5.5. The different anisotropic loading conditions significantly affect the microstructures, as shown in Fig. 5.6. Based on the microstructures obtained under different loading conditions, it can be summarized that the variants that minimize the externally applied strain energy and maximize the net driving force are favored, i.e. only variants with a favorable orientation under a given stress state are formed, viz. Magee effect [24–26]. The comparison of the von Mises equivalent plastic strain plots with the von Mises equivalent stress plots indicates that the transformation induced plasticity (TRIP) is an accomodation process to minimize the stresses aroused due to the phase transformation, viz. Greenwood-Johnson effect [24–26]. Fig. 5.7 shows a stage during the microstructure evolution where the martensite variants have not yet hit the (111)γ plane, whereas Fig. 5.8 shows a subsequent evolution stage when 30 CHAPTER 5. SUMMARY OF RESULTS Figure 5.5: Top view of microstructure evolution under anisotropic loading condition-8c, mentioned in Section 4.2 of Supplement-2, at different time steps (a) t*=10 (b) t*=30 (c) t*=120. Figure 5.6: Results from the simulations of stress-assisted martensitic transformation in single crystal, under anisotropic loading conditions (8a - 8f) mentioned in Section 4.2 of Supplement-2. Snapshots at t*=120. 5.1. MORPHOLOGY AND TRANSFORMATION 31 Figure 5.7: Microstructure, equivalent stress and equivalent plastic strain plots obtained under uni-axial tensile loading at t*=85. (a) Side view of microstructure. (b) Top view of the microstructure plotted on (111) plane. (c) Equivalent stress corresponding to the microstructure plotted on (111) plane. (d) Equivalent plastic strain corresponding to the microstructure plotted on (111) plane. the martensite variants have hit the (111)γ plane. It can be seen from Figs. 5.7 and 5.8 that as the martensite variants hit the (111)γ plane from both sides, the equivalent stress inside the plane increases, as can be observed in the corresponding equivalent stress plot. Once the equivalent stress exceeds the yield limit, it intiates plastic deformation as shown in the corresponding equivalent plastic strain plot, thus indicating the initiation of the TRIP effect. The simulation results, as shown in Fig. 5.9, indicate that the compressive stresses applied by the martensite on austenite, due to the volume expansion caused by the transformation, leads to stabilization of austenite and thereby gives rise to higher content of retained austenite in between the martensite variants. From the stress plot it can be seen that the boundaries of the martensite variants are in 32 CHAPTER 5. SUMMARY OF RESULTS Figure 5.8: Microstructure, equivalent stress and equivalent plastic strain plots obtained under uni- axial tensile loading at t*=88. (a) Side view of microstructure. (b) Top view of the microstructure plotted on (111) plane. (c) Equivalent stress corresponding to the microstructure plotted on (111) plane. (d) Equivalent plastic strain corresponding to the microstructure plotted on (111) plane. 5.1. MORPHOLOGY AND TRANSFORMATION 33 Figure 5.9: Top view of microstructure and the corresponding equivalent stress plotted on (111)γ plane, obtained under a tri-axial anisotropic loading condition. (a) Microstructure at t*=110. (b)von Mises equivalent stress of corresponding microstructure at t*=110. dark blue color, indicating that they exert a compressive stress on the surrounding austenite, which is in accordance with Refs. [23, 61]. It has been observed in the simulations that the interaction between different martensitic variants could also lead to higher internal stresses in the martensitic interfaces and thereby gives rise to plastic accommodation, which is in good agreement with the results of Marketz and Fischer [62]. By comparing the microstructure with the stress plot in Fig. 5.10, it can be observed that the interfaces between different martensitic variants are highly stressed regions and thereby lead to plastic deformation in the interface regions, as can be seen in the plastic strain plot. Multi-length scale modeling of martensitic transformations, Supplement-3 The results indicate that a physically based multi-length scale model to simulate the martensitic transformation in practically important materials such as stainless steels is possible. The ab initio calculations of the elastic properties indicate that stainless steels are elastically anisotropic, as shown in Fig. 5.11. The simulations predict the athermal martensite formed in stainless steels to be of lath-type morphology, which is in good agreement with experimental results [63]. It is observed that, most often, all three martensite variants, i.e. all three Bain variants (groups), form in pairs as 34 CHAPTER 5. SUMMARY OF RESULTS Figure 5.10: Microstructure, equivalent stress and equivalent plastic strain plots obtained under bi-axial compressive loading, plotted on (001) plane. (a) Microstructure at t*=37. (b) Equivalent stress corresponding to the microstructure at t*=37. (c) Equivalent plastic strain corresponding to the microstructure at t*=37. 5.1. MORPHOLOGY AND TRANSFORMATION 35 Figure 5.11: Calculated characteristic surfaces showing the Young’s modulus as a function of crystallographic direction in a random FM Fe–17%Cr–7%Ni alloy. The values on the color scale and on the axes are in GPa. shown in Fig. 5.12. This result is different compared to the result obtained in the case of carbon steel, where only two out of the three Bain variants are formed in any given direction, as shown in Fig. 5.3b. The simulation results of the stress-assisted martensite formation in stainless steels under different stress states are similar to those obtained in the case of carbon steels, as presented in Supplement-2. Athermal martensitic transformation in polycrystalline material, Supplement-4 The simulation results show that impingement of martensite variants on the grain boundaries gives rise to higher stresses along the grain boundary. Such high stresses are relaxed by martensite nucleation in the adjacent grain, viz. autocatalysis. The results are in good agreement with the experimental observations that grain boundaries are probable sites for martensite nucleation [64], as shown in Fig. 5.13. It is also observed that the orientation of a martensitic variant is governed by the orientation of the grain, which is also in agreement with experimental results [64]. It can be seen from Fig. 5.13 that the martensite variants, that grow in one grain, hit the 36 CHAPTER 5. SUMMARY OF RESULTS Figure 5.12: Results from the multi-length scale simulations of athermal martensitic transformation in a single crystal of 301-type stainless steel, at t*=75. grain boundaries and give rise to growth of martensite variants oriented differently in the adjacent grains. This phenomenon can be observed in the grains marked as ’a’ and ’b’ in the simulated microstructure as well as in the grains marked as ’1’ and ’2’ in the experiementally obtained martensitic microstructure. As observed in the case of athermal martensitic transformation in single crystal explained above, plastic accommodation in polycrystalline material also reduces the strain energy. Moreover, a coarser microstructure is obtained when plastic deformation is considered in a polycrystalline material. Fig. 5.14 shows the 3D simulation results of a clamped elastoplastic polycrystalline material. In the case of a polycrystalline material, both clamped and non-clamped systems resulted in a multi-variant martensitic microstructure. This is in contrast to that of the results obtained in the case of a single crystal explained above, where clamped grain resulted in a multi-variant microstructure and a non-clamped grain resulted in the formation of a single martensite variant over the entire grain. 5.2 Plastic deformation aspects, Supplements-5 and 6 Based on the above results, it can be understood that plastic deformation plays a significant role in the martensitic transformation. Moreover, the simulations performed to study the effect of plastic relaxation rate k on the transformation, as mentioned in Section 4.2, indicate that the plastic relaxation rate affects the morphology as well as the overall transformation. 5.2. PLASTIC DEFORMATION ASPECTS 37 Figure 5.13: Comparison of 2D simulation result from (a) an elastoplastic phasefield simulation of athermal martensitic transformation with that of (b) a grain orientation map obtained from SEM-EBSD technique on a 301 steel sample that has been subjected to athermal martensitic transformation [64]. Figure 5.14: Results from the simulations of athermal martensitic transformation in an elastoplastic clamped polycrystalline material, at (a) t*=0 (b) t*=25 (c) t*=300. 38 CHAPTER 5. SUMMARY OF RESULTS (a) (b) Figure 5.15: Results from simulations with (a) low plastic relaxation rate (b) high plastic relaxation rate [65]. A lower plastic relaxation rate results in a microstructure containing of many martensite variants inside a single grain as shown in Fig. 5.15a. A higher plastic relaxation rate results in a microstructure consisiting of a single martensite domain growing across the entire grain as shown in Fig. 5.15b [65]. The above results are in good agreement with the findings of Levitas et al. [35]. The plastic relaxation rate k also affects the growth rate of martensite, as shown in Fig. 5.16. Each sub-figure corresponds to snap shots, at t∗ = 50, from different simulations with different k. It can be seen that under given conditions, Viz. driving force and dimensionless time (t∗ = 50), the growth rate of a martensite domain increases with increasing k. As k is directly proportional to the dislocation mobility, as explained in Section 4.2, it can be understood that a higher dislocation mobility gives rise to a higher growth rate of the domain [65]. This is in good agreement with earlier observations that lath martensite grows with the aid of dislocation glide [33, 34]. The von Mises equivalent plastic strain plot, as shown in Fig. 5.17, indicates that the major plastic deformation occurs in the martensite compared to that of austenite [65]. As a higher plastic strain implies a higher dislocation density, it can be understood that martensite is a dislocation-rich structure, which is in good agreement with the experimental observations of Morito et al. [27]. The plastic relaxation rate also affects the transformation by affecting the embryo potency, as explained in the following section. 5.3 Nucleation and growth aspects, Supplement-5 The effect of plastic relaxation rate on the embryo potency is shown in Fig. 5.18. The results indicate that for a given driving force there exists a critical plastic 5.3. NUCLEATION AND GROWTH ASPECTS 39 Figure 5.16: Results, at t∗ = 50 and mapped on (0 1 0) plane, from different simulations with different k values, as mentioned below each sub-figure [65]. relaxation rate k crit , below which the embryo is impotent, i.e. reverse transformation from martensite to austenite. It can also be seen from Fig. 5.18 that as the plastic relaxation rate is higher than the critical value, the embryo becomes potent and grows. As the plastic relaxation rate is directly proportional to the dislocation density, it can be construed that in order for the embryo to become potent, a critical dislocation density is needed at the nucleation site [66]. The dependence of embryo potency on the dislocation density is in good agreement with earlier report by Cohen [28]. The simulations performed to study the effect of embryo potency on the transformation, as mentioned in Section 4.3, indicate that the size, shape and orientation of the martensite embryo affect the transformation. It becomes very difficult for a small embryo to sustain the very high elastic stresses generated inside the material and hence it shrinks and disappears completely. If the embryo is considerably big such that it can sustain the elastic stresses, it becomes potent and starts to grow. However the comparison of the critical size of the nucleus obtained from the sim- 40 CHAPTER 5. SUMMARY OF RESULTS (a) (b) Figure 5.17: Illustration showing the maximum plastic strain concentrated in martensite [65]. (a) Iso-surface of phase-field variable (= 0.5) that corresponds to a martensite domain, at t∗ = 35, mapped on (0 -1 1) plane with high plastic relaxation rate, i.e. k = 16.6k crit (b) The corresponding von Mises equivalent plastic strain plot of the iso-surface shown in (a). (a) (b) (c) Figure 5.18: Iso-surfaces of phase-field variable (= 0.5) obtained from 3D simulations to study the effect of plastic relaxation rate (k) on the embryo potency [66]. Simulations performed with plastic relaxation rate (a) less than the critical value (k < kcrit ) (b) equal to the critical value (k = kcrit ) (c) greater than the critical value (k > kcrit ) 41 5.4. THERMODYNAMIC ASPECTS (a) (b) (c) Figure 5.19: Iso-surfaces of phase-field variable (= 0.5) obtained at t* = 80 from three different simulations started with a pre-existing ellipsoidal embryo whose major axis is aligned along (a) X-axis (b) Y-axis (c) Z-axis [66]. t* denotes dimensionless time. ulations with that of the size determined from earlier theoretical works [31] still needs an in-depth analysis. The simulations predict the favorable shape of the embryo as an ellipsoidal shape, as can be seen in Fig. 5.18c, which minimizes the combined interfacial and elastic strain energy of the embryo [31]. The results also indicate that a favorably oriented embryo gives rise to a higher martensite volume fraction. The results of three different simulations started with an ellipsoidal martensite embryo oriented in different directions is shown in Fig. 5.19. 5.4 Thermodynamic aspects, Supplement-6 As mentioned in Section 4.4, the procedures to determine the different driving forces required for growth and autocatalysis are shown in Figs. 5.20 and 5.21 respectively. The results indicate that there exist three different critical (minimum) driving forces, Viz. driving force required for nucleation (∆Gnucl. ), driving force required for growth (∆Ggrowth ) and driving force required for autocatalysis (∆Gautocat. ), which govern the martensitic transformation as well as the microstructure evolution. The simulations predict that ∆Ggrowth < ∆Gautocat. < ∆Gnucl. [65]. Thus the results indicate that martensite, once nucleated, can grow above the Ms temperature. This is in agreement with the findings of Borgenstam et al. [40]. However the difference in the temperatures corresponding to the nucleation and growth determined from the present simulations is about 85 K, which is large compared to that of 8 K reported by Borgenstam et al. It can be concluded that, regardless of the absolute value of the temperature difference, martensite growth could continue above Ms . 42 CHAPTER 5. SUMMARY OF RESULTS Figure 5.20: Illustration of the approach to predict the critical driving force for growth [65]. Figure 5.21: Illustration of the approach to predict the critical driving force for autocatalysis [65]. 5.5. CRYSTALLOGRAPHY 43 Figure 5.22: Prediction of initial habit plane [65]. 5.5 Crystallography, Supplement-6 The model predicts the initial habit plane of the infinitesimally small martensite unit to be (−111)γ as shown in Fig. 5.22. In order to minimize the Gibbs energy of the system, this martensite domain rotates away from the initial habit plane as the transformation progresses, as shown in Fig. 5.23. On the formation of self nucleated martensite units, the rate of rotation decreases as the energy minimization is now achieved by means of autocatalysis. Thus the model predicts the final habit plane of the initially formed martensite domain, mentioned above, to be (−211)γ [65]. The results are in agreement with the theoretical explanation that martensite variants tend to rotate in order to minimize the strain energy [5, 9]. The rotations of martensite variants are studied in order to verify the theoretical explanation that dilatation coupled with shear and rotation could give rise to the experimentally predicted habit planes [9], as explained in Section 2.6. The dilatation is governed by the Bain strains, that are supplied as input parameters for the simulations. The shear deformation is governed by the plasticity equation (4.13). The rotation is governed by the minimization of the strain energy that yields displacement, which in turn gives rise to the rotation of the variants [5, 9]. Fig. 5.24 shows a 2D section of the 3D athermal martensitic microstructure as well as the corresponding displacement plot. In the displacement plot, red color indicates maximum displacement and dark blue color indicates zero displacement. By comparing the two images, it can be seen that the dark blue regions in the displacement plot are the centers of the martensitic domains shown in the corresponding microstructure image. This shows that the centers of the martensitic domains are the regions where displacement is zero. Thus the habit plane, where infinitesimally small unit of martensite forms, is undistorted and unrotated as the 44 CHAPTER 5. SUMMARY OF RESULTS Figure 5.23: Simulation results showing the martensite domain rotating away from the initial habit plane of (-1 1 1) [65]. (a) Initial habit plane at t∗ = 7 (b) Rotation at t∗ = 45 (c) Rotation at t∗ = 150. Figure 5.24: Illustration to explain the invariant habit plane concept. (a) 2D section of the 3D athermal martensitic microstructure (b) Displacement plot of the corresponding microstructure shown in (a). Red color indicates maximum displacement and dark blue color indicates zero displacement. displacement is zero. This is in agreement with the phenomenological theories of martensite [23]. Both the phase-field method and the Microelasticity theory [5] are based on the continuum principles, i.e. principles that are used to describe a physical phenomenon by using mathematical functions that are continuous functions in space and time. Thus the phase-field variables and the displacement field, obtained by the minimization of the elastic strain energy by using the Microelasticity theory as explained in Eqs. (4.5) - (4.9) in Section 4.1, are continuous functions of the coordinates in the lattice space. This ensures the crystal lattice continuity, i.e. 5.5. CRYSTALLOGRAPHY 45 compatibility (coherency) of the austenite and martensite, by means of martensite variant rotations as well as by means of the invariant habit plane [5, 9]. In simple words, the martensite variants rotate and orient in such a way that the coherency with austenite is maintained and also such that the strain energy is minimized. Chapter 6 Concluding remarks and future prospects The results obtained in this work indicate that it is possible to model and to study several aspects of the martensitic transformation using the phase-field method in a satisfactory way. Although the nucleation is not simulated in this work, the simulations performed to study the effect of embryo potency on the transformation indicate that the size, shape and orientation of the embryo as well as the dislocation density at the nucleation site govern the martensitic transformation. The simulations predict that martensite, once nucleated, could grow as well as lead to autocatalysis, above Ms . Although this result is in agreement with experimental results [40], the experiments indicate that martensite could only grow up to a temperature that is slightly above Ms , whereas the simulations predict that martensite could grow well above Ms and hence needs to be studied in detail by performing more experiments as well as simulations. The experimental observations that the martensitic microstructures consist of several twinned martensite variants are observed in the simulated microstructures as well. Autocatalysis, i.e. nucleation of several martensite variants, plastic deformation as well as the non-clamped boundary conditions act as relaxation measures of the stresses developed, due to the martensitic transformation, inside the material. The microstructures obtained in a polycrystalline material indicate that the martensite variants growing in one grain, when impinge up on the adjacent grains, could initiate martensite nucleation in the adjacent grains, where the grain boundaries act as nucleation sites. The study of plastic deformation aspects show that the martensitic microstructure is significantly affected by the plastic relaxation rate, which corresponds to the dislocation mobility and density. The model predicts the initial habit plane of martensite in agreement with the experimental results and predicts that the variants rotate in order to minimize the strain energy. If the rotations of various variants could also be plotted along with the Bain variants shown in the present figures, it might be possible to predict some 47 48 CHAPTER 6. CONCLUDING REMARKS AND FUTURE PROSPECTS Figure 6.1: Variation of Msσ with externally applied uni-axial tensile stress. more aspects of martensite crystallography and hence this aspect needs an in-depth study. The simulations of the stress-assisted martensitic microstructure evolution under different loading conditions are appealing and indicate that the microstructure evolution is significantly affected by the stress states and also that the anisotropic loading conditions give rise to a higher level of anisotropy in the microstructure. The results clearly depict the Magee effect. In future, the simulations can be supplemented by experimental works. Moreover, some practically important parameters such as Msσ temperature and the volume fraction of the martensite formed can also be predicted from the simulations, as shown in Figs. 6.1 and 6.2 respectively. The study of the von Mises plastic strain plots of the microstructures enables to understand the TRIP effect as well as the interaction between martensite variants. The TRIP effect can also be understood by studying the variation of von Mises equivalent plastic strain with respect to the martensite volume fraction, as shown in Fig. 6.3. It can be seen that the plastic strain drastically increases once the martensitic transformation starts and thereafter increases with the increasing martensite volume fraction, indicating the initiation of the transformation induced plasticity (TRIP) effect. As a continuation of the present work, the model can be improved further by including the strain hardening effect in the plasticity equations. The plastic de- 49 Figure 6.2: Variation of martensite volume fraction as the microstructure evolves under applied uni-axial tensile stress. formation model can also be further improved by considering the crystal plasticity framework, which might make it feasible to study the martensite formation along different slip planes. Based on the multi-length scale modeling results, it can also be concluded that it is possible to build a physically based multi-length scale model, by coupling different computational methods as well as experimental works, to study the martensitic transformations. However in order to further improve the multi-length scale model, the macroscopic properties of the material also need to be studied by investigating the response of the different simulated martensitic microstructures to various external loading conditions. Such attempts might also contribute to the Materials Genome initiative, which is now one of the most widely discussed applied research topics among the scientific community. The experimental studies on martensite morphology and crystallography using advanced characterization techniques, such as EBSD, FIB etc., seem very interesting. Moreover the recent studies of 3D reconstruction of martensitic microstructures [67] are exciting. Hence it could be interesting to compare such 3D experimental results with the 3D phase-field modeling results obtained in the present work, in order to bridge the gaps between the two realms, viz. experiments and models. 50 CHAPTER 6. CONCLUDING REMARKS AND FUTURE PROSPECTS Figure 6.3: Variation of mean equivalent plastic strain with martensite volume fraction obtained under applied uni-axial tensile stress. Thus with a tone of cautious optimism, it can be concluded by saying that phase-field method can be a useful qualitative tool in understanding several aspects associated with the martensitic transformations, provided the underlying physics is taken care of in an appropriate manner. Acknowledgements First of all I would like to express my sincere gratitude to my principal supervisor Prof. John Ågren for giving me this opportunity to work with one of the intriguing and exciting research topics in the area of Physical Metallurgy, for his supervision and for all the interesting discussions. I would also like to thank my another supervisor Assoc. Prof. Annika Borgenstam for the critical assessment of my work that has greatly added value to the work. I want to express my thanks to all the other collaborators, Prof. Gustav Amberg, Mr. Amer Malik, Dr. Peter Hedström, Dr. Vsevolod Razumovskiy, Dr. Pavel Korzhavyi and Prof. Andrei Ruban. My special thanks to Prof. Emer. Mats Hillert for the interesting discussions and ideas. Thanks to Dr. Lars Höglund and Dr. Minh Do Quang for all the help with the computer related aspects. I would also like to thank Prof. Hasse Fredriksson, whom I always consider as my first teacher in the field of Materials research. Thanks to Prof. Staffan Hertzman for the motivation to work on stainless steels. My thanks to all the other colleagues at MSE that have made all these years pleasant and enjoyable. I would like to thank the VINN Excellence Center Hero-m, Vinnova, Swedish Industry and KTH Royal Institute of Technology for providing finance for performing this work. Thanks to SNIC for providing the computer resources as well as to the STT foundation, Wallenberg foundation and Jernkontoret for providing the travel grants to attend various conferences. I would also like to thank Prof. A.G. Khachaturyan, Dr. Avadh Saxena, Dr. Turab Lookman, Dr. Alfonse Finel, Prof. Greg Olson, Prof. S.Q. 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