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Linkage between mechanical properties and phase transformations in a 301LN austenitic stainless steel Maréchal, David 2011-12-31

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Linkage between mechanicalproperties and phase transformationsin a 301LN austenitic stainless steelbyDavid Mar echalB. Eng., INP Grenoble, France, 2004A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate Studies(Materials Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)April 2011a169 David Mar echal 2011AbstractIn this work, the deformation mechanisms of an austenitic stainless steel(grade 301LN) have been investigated with particular attention on the strain-induced phase transformations from austenite to  and  0 martensites. Theaverage grain size of this alloy was varied in the range 0.5{28 a181m, and twostrain paths, namely uniaxial tension and simple shear, were analyzed. Atthe macroscopic level, the work-hardening response was examined in re-lation to the formation of  and  0 martensites, followed by X-ray phasequanti cation and Feritscope measurements. At a microscopic level, the mi-crostructures after deformation were investigated using electron back-scatterdi raction, energy-dispersive X-ray spectroscopy and transmission electronmicroscopy. It was found that the grain size re nement was responsiblefor a change in nucleation mechanisms of  0-martensite, thereby a ectingthe macroscopic volume fraction of  0-martensite. The switch from ten-sion to shear was not found to a ect the mechanisms of formation of  and 0 martensites, but signi cantly reduced the work-hardening, an e ect toolarge to be attributed to the slight reduction of the kinetics of  0 volumefraction. The stresses borne in the  0-martensite were quanti ed using anovel method based on the magnetomechanical e ect. These stresses, to-gether with the determination of the intrinsic constitutive laws of austeniteand  0-martensite, were used to design a one-dimensional physically-basedmodel of the work-hardening in this alloy. This model, based on the \dy-namic composite" e ect of the formation of fresh  0-martensite in austenite,successfully predicted the measured stress-strain behaviour in tension, aswell as the tensile instabilities encountered in this class of materials.iiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . 42.1 Austenitic Stainless Steels: Stable and Metastable Phases . . 42.2 Deformation-Induced Martensitic Phase Transformations inAustenitic Stainless Steels . . . . . . . . . . . . . . . . . . . 92.2.1 Ferrous Martensites in Austenitic Stainless Steels . . 92.2.2 Techniques Used to Measure Martensite Content inAustenitic Steels . . . . . . . . . . . . . . . . . . . . . 142.2.3 Mechanisms of Formation of Strain-Induced Marten-site in Austenitic Stainless Steels . . . . . . . . . . . 152.3 Factors In uencing the Rate of Strain-Induced MartensiticTransformation in Austenitic Stainless Steels . . . . . . . . . 242.4 Modelling of the Kinetics of the Strain-Induced Phase Trans-formations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.4.1 Review of the Olson-Cohen Model for TransformationKinetics . . . . . . . . . . . . . . . . . . . . . . . . . 342.4.2 In uence of External Parameters on the Olson-CohenModel . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.5 Mechanical Response of Austenitic Stainless Steels . . . . . . 422.5.1 Bulk Mechanical Response . . . . . . . . . . . . . . . 422.5.2 The Intrinsic Mechanical Response of Austenite andMartensite . . . . . . . . . . . . . . . . . . . . . . . . 47iiiTable of Contents2.5.3 Modelling of the Overall Mechanical Response of Aus-tenitic Stainless Steels . . . . . . . . . . . . . . . . . . 532.6 Summary of the Literature Review . . . . . . . . . . . . . . . 623 Scope and Objectives . . . . . . . . . . . . . . . . . . . . . . . 634 Processing and Characterization of 301LN Sheet to DevelopGrain Sizes in the Micrometer to Nanometer Range . . . 654.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.2 Experimental Methodology: Materials Characterization . . . 664.2.1 Quanti cation of 0-Martensite Content via FeritscopeMeasurements . . . . . . . . . . . . . . . . . . . . . . 664.2.2 Materials Characterization by Electron Microscopy . 684.3 Experimental Methodology: Materials Processing . . . . . . 704.3.1 Cold Rolling of As-Received Sheet . . . . . . . . . . . 704.3.2 Post-Rolling Annealing Treatments . . . . . . . . . . 714.4 As-Received Material . . . . . . . . . . . . . . . . . . . . . . 724.5 Generation of Materials With Varying Grain Sizes . . . . . . 724.6 Presence of Other Phases in the Recrystallized Microstruc-tures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.7 Solute Segregation . . . . . . . . . . . . . . . . . . . . . . . . 804.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815 Macroscopic Characterization of the Mechanical Propertiesand Phase Fraction . . . . . . . . . . . . . . . . . . . . . . . . . 825.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.2 Experimental Methods . . . . . . . . . . . . . . . . . . . . . 825.2.1 Uniaxial Tensile Testing . . . . . . . . . . . . . . . . 825.2.2 Testing in Simple Shear . . . . . . . . . . . . . . . . . 845.2.3 Phase Quanti cation by X-Ray Di raction . . . . . . 865.3 Mechanical Properties of 301LN in Uniaxial Tension . . . . . 875.4 Mechanical Properties of 301LN in Simple Shear . . . . . . . 915.5 Quanti cation of the Volume Fractions of Strain-Induced Mar-tensitic Phases . . . . . . . . . . . . . . . . . . . . . . . . . . 945.5.1 Quanti cation of  martensite . . . . . . . . . . . . . 945.5.2 Quanti cation of  0 martensite . . . . . . . . . . . . . 945.6 Relationship between Mechanical Response and Volume Frac-tion of Phases . . . . . . . . . . . . . . . . . . . . . . . . . . 995.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102ivTable of Contents6 Characterization of the Deformed Microstructures . . . . 1046.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046.2 Experimental Techniques and Representation Convention . . 1056.3 Microstructure Evolution in Uniaxial Tension: Large GrainSize Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076.3.1 General Overview of Microstructure Evolution as aFunction of Strain . . . . . . . . . . . . . . . . . . . . 1086.3.2 Relationship between Martensite Morphology and Crys-tallography . . . . . . . . . . . . . . . . . . . . . . . . 1116.3.3 Schmid Analysis of f111g Planes Associated withTrace Analysis . . . . . . . . . . . . . . . . . . . . . . 1146.3.4 Formation of  0-Martensite and Variant Selection . . 1196.4 The E ect of Grain Size on the Strain-Induced Formation ofMartensite . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1276.5 Microstructure Evolution in Simple Shear . . . . . . . . . . . 1416.6 The Link Between Macroscopic Transformation Kinetics andMicrostructure . . . . . . . . . . . . . . . . . . . . . . . . . . 1426.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1487 A Novel Method of Estimating the Stresses in  0-Martensite 1497.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1497.2 Magnetostriction and the Magnetomechanical E ect . . . . . 1507.3 Experimental Techniques . . . . . . . . . . . . . . . . . . . . 1547.4 Estimation of Stress Carried by  0-Martensite via the Mag-netomechanical E ect . . . . . . . . . . . . . . . . . . . . . . 1567.4.1 Comparison of Results with Other Estimates for Stressesin Martensite . . . . . . . . . . . . . . . . . . . . . . . 1607.5 Measurement of Stresses in Samples of Di erent Grains Sizesand the Impact on Overall Mechanical Response . . . . . . . 1667.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1698 Modelling of the Mechanical Response of 301LN . . . . . . 1718.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1718.2 Review of Mechanical Response and Previous MicrostructuralBased Models . . . . . . . . . . . . . . . . . . . . . . . . . . 1728.3 A Dynamic Composite Model for 301LN Stainless Steel . . . 1748.3.1 Behaviour of Austenite . . . . . . . . . . . . . . . . . 1748.3.2 Behaviour of  0-Martensite . . . . . . . . . . . . . . . 1758.3.3 Choice of the Parameters . . . . . . . . . . . . . . . . 177vTable of Contents8.3.4 Discussion of Model Results for D=28 a181m in UniaxialTension . . . . . . . . . . . . . . . . . . . . . . . . . . 1798.3.5 Application of Model to the Grain Size Dependenceof Mechanical Response . . . . . . . . . . . . . . . . . 1818.3.6 Application of Model to the Mechanical Response inShear . . . . . . . . . . . . . . . . . . . . . . . . . . . 1858.4 Application of Model to Literature Data . . . . . . . . . . . 1878.5 De ning an Average  0-Martensite Behaviour in the DynamicComposite Model . . . . . . . . . . . . . . . . . . . . . . . . 1948.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1959 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1979.1 Summary and Key Results . . . . . . . . . . . . . . . . . . . 1979.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 201Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203AppendicesA Calibration of the Feritscope in Grade 301LN . . . . . . . . 223B The Patel-Cohen Model for Variant Selection . . . . . . . . 226viList of Tables2.1 Experimentally-determined stacking fault energies of variousfcc materials at room temperature. . . . . . . . . . . . . . . . 72.2 Review of the techniques used to quantify phase fractions of 0-martensite. . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3 Review of the mechanisms of  0 nucleation observed in aus-tenitic stainless steels. . . . . . . . . . . . . . . . . . . . . . . 202.4 Empirical models used to describe the volume fraction ofstrain-induced  0-martensite. . . . . . . . . . . . . . . . . . . 332.5 Hall-Petch parameters determined at room temperature forvarious fcc materials. . . . . . . . . . . . . . . . . . . . . . . . 444.1 Nominal composition (in wt.%) of the grade used in this study. 724.2 Thermo-mechanical procedure used to generate the  ve con-ditions of grain size studied in this thesis. . . . . . . . . . . . 734.3 Description of the  ve grain size distributions, characterizedby EBSD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.1 Initial values of the lattice parameters used in the Rietveldanalysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875.2 Tensile characteristics of the  ve grain size conditions. . . . . 886.1 Identi cation of the plane/direction matching conditions be-tween  and  0. . . . . . . . . . . . . . . . . . . . . . . . . . . 1176.2 All possible Schmid factors (counted positive) correspondingto the twelve f111g h112i slip systems, in 15 grains. Thef111g planes containing the planar faults are shown in bold.It can be seen that, in 10 grains out of 15, those featuresappeared on the planes with highest Schmid factor. . . . . . . 1186.3 Values retained for the modi ed Olson-Cohen model. . . . . . 1468.1 Input parameters directly determined from tensile experiments.1778.2 Adjustable input parameters, used to model uniaxial tension. 178viiList of Tables8.3 Input parameters directly determined from simple shear ex-periments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1868.4 Adjustable input parameters, used to model simple shear. . . 186viiiList of Figures1.1 Relative performances of three generations of steels. . . . . . 22.1 Relative stability of phases predicted by Thermo-Calc usingthe composition corresponding to the 301LN stainless steelstudied in this work at atmospheric pressure. (a) Evolutionof the Gibbs free energy of fcc ( ), hcp ( ) and bcc ( ) phasesat room temperature, (b) Evolution of the T0 temperaturewith variable nickel content. . . . . . . . . . . . . . . . . . . . 62.2 The Bain correspondence between the fcc unit cell (light gray)and the tetragonally distorted bcc unit cell (black). . . . . . . 102.3 Change in the Ms temperature as a function of the loadingcondition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 Schematic representation of the interrelationships betweenstress-assisted (below M s ) and strain-induced (above M s )martensitic transformations. . . . . . . . . . . . . . . . . . . . 132.5 Plates of  -martensite in grade 304L after a 5% tensile strainat  196 C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.6 Schematic representation of (a) a twin, (b) a thin plate of -martensite. . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.7 TEM Micrograph of an  0 martensite island formed withinan  band. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.8 Evolution of the fraction of  and  0 martensites formed dur-ing room-temperature tensile deformation of grade 301. . . . 182.9 Nucleation of  0 in grade 316 after a 5% tensile strain at 196 C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.10 The distorted bcc unit cell extracted from Figure 2.2, plottedshowing f111gfcc planes (i.e. two Thompson tetrahedra). . . 212.11 Illustration of two di erent behaviours of the transformationkinetics as a function of grain size. . . . . . . . . . . . . . . . 25ixList of Figures2.12 Evolution of martensite volume fraction in 316L stainless steelduring room-temperature deformation and after a  196 Cprestrain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.13 E ect of deformation temperature on the  ! 0 kinetics ofgrade 301LN. . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.14 E ect of strain rate on the  ! 0 kinetics of grade 204M. . . 292.15 (a) Grade 304 deformed at 196 C along 3 di erent paths, ata constant loading rate. (b) Di erence between tension andcompression performed at room temperature on grade 304. . 302.16 Kinetics of two austenitic stainless steels for di erent loadingcombinations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.17 Illustration of the grain size dependence on the Olson-Cohenequation, with  = 8. . . . . . . . . . . . . . . . . . . . . . . . 372.18 Illustration of the temperature dependence on the two Olson-Cohen parameters. . . . . . . . . . . . . . . . . . . . . . . . . 382.19 Illustration of the stress state dependence on the Stringfellowmodel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.20 E ect of the temperature on the yield stress of a 301LN stain-less steel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.21 E ect of the temperature on the tensile curves of a 316L stain-less steel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.22 E ect of the temperature on the tensile curves of a 301LNstainless steel. . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.23 Tensile behaviour of 304L pre-strained at 25 C and furtherdeformed at -196 C. The reloading at -196 C was associatedwith the propagation of a band in which the strain was localized. 472.24 Stress-strain curve of two austenitic stainless steels for di er-ent loading combinations. . . . . . . . . . . . . . . . . . . . . 482.25 Measured elastic strain evolution in  -martensite parallel tothe tensile direction plotted for two individual austenite grains(13 and 18), along with the average X-ray elastic strain. . . . 492.26 Stress level in austenite and  0-martensite phases measuredby neutron di raction in grade 316L. . . . . . . . . . . . . . . 502.27 Stress level in austenite and  0-martensite phases measuredfrom X-ray di raction stress measurements in grade 301LN. . 512.28 Stress level in the austenite phase of grade 301LN, measuredby X-ray Di raction as a function of square root of dislocationdensity of austenite determined by Integral Breadth Method. 522.29 Simulated true stress and work-hardening curves obtained forroom-temperature tension in various austenitic grains. . . . . 55xList of Figures2.30 (a) Evolution of the volume fraction of martensite with strainfor the 301LN stainless steel during tensile testing at 20 C. (b)Evolution of the calculated austenitic and martensitic grainsize during the tensile test. (c) Simulated stress-strain curvesfor the martensitic and austenitic constituents. (d) Experi-mental and modelled stress-strain curves. . . . . . . . . . . . 572.31 Mean chord length of  0-martensite islands in 301LN steel asa function of  0-martensite volume fraction. . . . . . . . . . . 594.1 Schematic of the magnetic induction measurement performedby a Feritscope. . . . . . . . . . . . . . . . . . . . . . . . . . . 664.2 Magnetic signal measured by Feritscope during RT-rollingand cryorolling, for two angular rotations. . . . . . . . . . . . 714.3 Band contrast EBSD map of as-received 301LN, plotted toalso reveal grain boundaries. . . . . . . . . . . . . . . . . . . . 734.4 Band contrast EBSD maps showing grain boundaries illus-trating the microstructure of samples annealed under the con-ditions detailed in Table 4.2 . . . . . . . . . . . . . . . . . . . 754.5 (a) to (e): Histograms of the grain size distributions in termsof number fraction, as a function of the equivalent area diam-eter (EQAD). (f) Superimposition of the  ve grain size distri-butions, represented as a function of the EQAD normalizedby the average of the distribution. . . . . . . . . . . . . . . . 774.6 Histograms of the grain size distributions in terms of areafraction, as a function of the normalized EQAD. . . . . . . . 784.7 Bright  eld TEM image of a sample annealed for 3 minutesat 800a176C resulting in a 0.5 a181m average grain size, illustratingthe presence of chromium nitride precipitates. . . . . . . . . . 794.8 The segregation of nickel as seen with back-scattered electronsimaging in the SEM. . . . . . . . . . . . . . . . . . . . . . . . 815.1 Geometry of the  at tensile test coupons used in this study. . 835.2 Schematic overview of the shear testing apparatus. . . . . . . 845.3 Geometry of the  at shear test coupons used in this study. . . 855.4 Stress-strain curves obtained in uniaxial tension at room tem-perature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885.5 Work-hardening curves obtained under uniaxial tension. . . . 895.6 Stress-strain response of grade 301LN under uniaxial tensionat 80 C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905.7 Comparison of the tensile tests performed at 23 C and at 80 C. 90xiList of Figures5.8 Stress-strain curves obtained under simple shear, at room-temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.9 Comparison of the stress-strain curves from uniaxial tensionand simple shear tests. . . . . . . . . . . . . . . . . . . . . . . 935.10 Comparison of the work-hardening curves from uniaxial ten-sion and simple shear tests. . . . . . . . . . . . . . . . . . . . 935.11 X-ray di raction spectra representing thef0002g andf10 11g peaks for four conditions of strain in the 0.5 a181m condition. Acomparison towards the 28 a181m condition is presented. . . . . 945.12 Evolution of the volume fraction of  0-martensite with straindetermined by Feritscope measurements made after room-temperature uniaxial tension. . . . . . . . . . . . . . . . . . . 955.13 Maximum rate of formation of  0-martensite as a function ofthe initial austenite grain size. . . . . . . . . . . . . . . . . . . 965.14 Evolution of the volume fraction of  0-martensite in simpleshear as measured from X-ray di raction. . . . . . . . . . . . 975.15 Comparison of the volume fraction of  0-martensite in simpleshear, compared to the one measured in uniaxial tension asa function of Von Mises strain. The solid lines are drawn toguide the eyes. . . . . . . . . . . . . . . . . . . . . . . . . . . 985.16 Schematic of the work-hardening behaviour in stainless steels,de ning the three stages of work-hardening at room temper-ature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.17 Flow stress obtained under uniaxial tension at room-temperatureas a function of the volume fraction of  0. . . . . . . . . . . . 1015.18 Average theoretical stress in  0 as a function of applied truestrain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1026.1 Inverse pole  gure, coloured corresponding to crystallographicdirection parallel to sample direction. This colour scheme isused in all following EBSD maps for plotting the orientationof  0-martensite. . . . . . . . . . . . . . . . . . . . . . . . . . 1066.2 Geometry of small tensile coupons used for sequential EBSD. 1076.3 A series of three EBSD maps measured on samples (D=28a181m) deformed to the three indicated levels of strain. . . . . . 1086.4 Orientation maps illustrating two di erent morphologies of 0-martensite. In (a), the  0-martensite (green in colour) ap-pears \blocky", while in (b) it appears in bands within thegrain (here, an annealing twin). . . . . . . . . . . . . . . . . . 110xiiList of Figures6.5 Orientation map showing a few grains of austenite in the 28a181m condition, deformed to 15% strain in uniaxial tension. . . 1116.6 Angle of inclination of f111g traces observed to correspondto low band contrast lines in EBSD maps. These measure-ments come from 25 grains. The inset  gure illustrates therelationship between a f111g plane (coloured in pink) andits trace and de nes the angle of inclination,  . . . . . . . . . 1126.7 (a) Magni ed view of the area underlined in Figure 6.4(b),showing the presence of  -martensite. (b) and (c) show theBurgers orientation relation in this grain between  and the -phase observed. (b) represents the f0001g pole  gure su-perimposed on the f111g pole  gure and (c) represents theh1 210i pole  gure superimposed on the h110i pole  gure. . 1136.8 (a) Bright  eld TEM micrograph of a grain showing a setof planar features (b) Selected area di raction pattern ex-hibiting extra spots characteristic of  -martensite . (c) Dark eld image of the same grain, using theh0 110i re ection un-derlined in (b). (d) Theoretical positions of the re ectionscorresponding to (b). . . . . . . . . . . . . . . . . . . . . . . . 1156.9 Two low magni cation orientation maps used for the Schmidanalysis detailed in Table 6.2. . . . . . . . . . . . . . . . . . . 1166.10 Superimposed (a) f111g and f110g 0 pole  gures and (b)f110g and f111g 0 pole  gures, showing the orientation re-lationship of the four identi ed variants of  0-martensite asobserved in grain 1. Only the two f111g intersecting planesgiving rise to the considered variant of  0 are represented onthe f110g pole  gure. . . . . . . . . . . . . . . . . . . . . . . 1216.11 (a) Observed distribution of Schmid factors forf111g h112i slip on thef111g planes corresponding to the plane matchingcondition in the K-S orientation relationship. (b) Rank of thecorresponding Schmid factors, from highest (1) to lowest (4). 1226.12 (a) Observed distribution of Schmid factors forf111g h110i slip on the intersecting f111g planes de ned in Figure 6.13.(b) Rank of the corresponding Schmid factors, from highest(1) to lowest (4). . . . . . . . . . . . . . . . . . . . . . . . . . 1236.13 Schematic showing the geometry at a  / 0/ interface. . . . . 1256.14 Spacing between observed plates ( or faults bands) when thegrain size is varied. . . . . . . . . . . . . . . . . . . . . . . . . 127xiiiList of Figures6.15 Orientation maps of the D=0.5 a181m grain size condition, de-formed at 18% strain in tension. (b) shows a higher magni -cation view of the highlighted area from (a), illustrating thelack of low index quality bands in the austenite in contrastto the observations in coarse grain size (e.g. Figure 6.5). . . . 1286.16 Nickel segregation corresponding to the region analyzed inFigure 4.8. (a) Back-scattered electron imaging in the SEM,showing the nickel-rich regions in lighter colours. (b) Aus-tenite orientation map of the same area showing the islandswhich remained austenitic. . . . . . . . . . . . . . . . . . . . . 1296.17 (a) Low magni cation bright  eld image of a  ne grainedsample deformed to 5% strain. (b) Selected area di ractionpattern of the region viewed in (a). The lines under the dif-fraction pattern show the expected position of rings for aus-tenite (blue),  0-martensite (red) and  -martensite (green).No clear evidence for di raction from  -martensite could befound. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1306.18 (a) Bright  eld image of a grain oriented close to [110] par-allel to the beam direction, exhibiting stacking faults alongwith a set of  ne twins (determined based on the extra spotsin the accompanying selected area di raction pattern). Thefaults and twins appear to emanate from grain boundaries.(b) Dark  eld image showing one set of twins. . . . . . . . . . 1316.19 Sequential orientation mapping performed for a true strainof (a) 0.15 and (b) 0.2 where several grain boundary nucleiof  0-martensite have been highlighted. Many of these nucleiappear to grow when the strain is increased from 0.15 to 0.2. 1336.20 Fraction of grain boundaries versus boundary disorientationtaken from EBSD maps corresponding to samples with D=28a181m and D=0.5 a181m. . . . . . . . . . . . . . . . . . . . . . . . . 1346.21 Locus of the grain boundary misorientation, represented inthe Frank-Rodrigues space. . . . . . . . . . . . . . . . . . . . 1366.22 Austenite orientation map of a coarse-grained coupon de-formed 41% in uniaxial tension. . . . . . . . . . . . . . . . . . 1376.23 Evolution of the length scale of the microstructure, evaluatedfrom EBSD, in (a) the 0.5 a181m condition, (b) the 2.2 a181m con-dition, (c) the 28 a181m condition. The scale of both phases wasevaluated from the EQAD on di erent EBSD micrographs. . 138xivList of Figures6.24 Evolution of (a) the surfacic rate of  0 nucleation, (b) the rateper grain, for the three conditions of grain size studied. Theerror bars, when they exist, illustrate the variation measuredfrom di erent orientation maps of the same condition. . . . . 1406.25 EBSD inverse pole  gure maps of  0-martensite (colour) over-laid on band contrast maps for austenite illustrating the mi-crostructure of samples deformed in simple shear. . . . . . . . 1416.26 Application of equations 6.8 to 6.10 to reproduce the mea-sured kinetics of formation of  0-martensite, (a) in uniaxialtension, (b) in simple shear. . . . . . . . . . . . . . . . . . . . 1476.27 Variation of the  -parameter as a function of grain size, inpresent empirical model. The validity of this  t is limited to0.5 a181m  D 30 a181m. . . . . . . . . . . . . . . . . . . . . . . 1477.1 Schematic representation of the Villari e ect. . . . . . . . . . 1517.2 Variation of the anhysteretic magnetization curves with truestress, as measured in a Fe-2%Mn steel. . . . . . . . . . . . . 1527.3 Geometry of the tensile coupons used at cryogenic tempera-tures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1557.4 Tensile stress-strain curve of the D=28 a181m test coupon atcryogenic temperature. . . . . . . . . . . . . . . . . . . . . . . 1567.5 (a) Evolution of the Feritscope signal during straining, withmeasurements performed on two samples, under load and un-loaded. The error bars show the range of the signal measuredby the Feritscope during six measurements. (b) Correspond-ing stress-strain data, with the actual measurement pointsindicated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1577.6 (a) Feritscope measurements (FS) obtained in the referencesample, when reloaded elastically at room temperature. (b)Evolution, in the reference, of FS normalized by the the Fer-itscope measurement at zero applied stress (F0S) as a functionof applied stress. The error bars only show the spread in theFS measurement. . . . . . . . . . . . . . . . . . . . . . . . . . 1597.7 (a) Intrinsic stresses measured in the  0-martensite. (b) Sameas (a) but multiplied by the volume fraction of consideredphase. Points are the actual measurements. . . . . . . . . . . 161xvList of Figures7.8 Comparison of the fraction of stresses in the  0-martensiteobtained from Feritscope measurements compared with neu-tron di raction measurements and theoretical stresses ob-tained from Figure 5.18, by extrapolating the behaviour ofthe austenite from 80 C tests. . . . . . . . . . . . . . . . . . 1627.9 Comparison of (a) the  ! 0 transformation kinetics, and(b) overall stress-strain curve in the two grades of 301LN. . . 1637.10 Evolution of the average stresses in austenite and martensite,as a function of the applied strain. . . . . . . . . . . . . . . . 1647.11 Comparison between neutron di raction and Feritscope mea-surements of the stresses borne in the austenite. . . . . . . . . 1657.12 Stress evolution in the di erent condition of grain size. . . . . 1677.13 Stress partitioning ratio as a function of true strain, makingapparent a relation with the three stages of work-hardening. . 1688.1 Schematic representation of the parameters appearing in Equa-tion 8.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1758.2 (a) Simulated stress-strain behaviour of the two single phases and  0. For this model, the scaling stresses of the  0-martensite were obtained from the experimental stress-straincurve of cryorolled material shown in (b). . . . . . . . . . . . 1798.3 (a) Simulated stress-strain curves of the D=28 a181m conditiondeformed in uniaxial tension. (b) Comparison of the simu-lated and experimental work-hardening curves. . . . . . . . . 1808.4 Representation of the three work-hardening (W-H) terms,as de ned in Equation 8.6, obtained from simulation. Thesum of these three terms is in turn compared to the work-hardening measured experimentally. . . . . . . . . . . . . . . 1818.5 In uence of the grain size on the simulated tensile curves. . . 1828.6 Grain size dependence of the three work-hardening (W-H)terms, as de ned in Equation 8.6, obtained from simulation. . 1838.7 Sensitivity of the kinetics on the tensile behaviour. . . . . . . 1848.8 Comparison of the simple shear simulated curves with theexperimental ones. The divergence between model and ex-periment is attributed to the di erent texture evolution . . . 1878.9 Results of the mechanical model applied to the data collectedby Nanga. (a) Simulated stress-strain curves and (b) simu-lated work-hardening curves, for uniaxial tension. . . . . . . . 189xviList of Figures8.10 (a) Orientation map showing the microstructure of 301LNafter cryorolling and annealing at 750 C. It can clearly beseen that the austenite is not fully recrystallized. (b) Stress-strain curve of the condition exhibiting the microstructureshown in (a), during room-temperature uniaxial tension. Thetensile curve shows a long plateau (24% strain) characteristicof strain localization. . . . . . . . . . . . . . . . . . . . . . . . 1918.11 Results of the mechanical model applied to the data collectedby Spencer on grade 316L. (a) Simulated stress-strain curvesand (b) simulated work-hardening curves, for uniaxial tension. 1938.12 Comparison of both approaches applied to simulate the ten-sile curve of the 28 a181m condition. . . . . . . . . . . . . . . . . 195A.1 X-ray di raction patterns illustrating the change in the pro-portion of phases when the strain is increased. . . . . . . . . 224A.2 Calibration curve of the Feritscope towards Rietveld re ne-ment of X-Ray Di raction spectra. . . . . . . . . . . . . . . . 224xviiChapter 1IntroductionCar manufacturers are continuously seeking to decrease vehicle weight andemissions [1], while improving crash performance (energy absorption). Asa consequence, there is demand for high-strength materials that permit re-duced thickness without compromised formability or crashworthiness. Sim-ilar demands are also made in other applications [2]. A range of strengthen-ing methods including grain size reduction [3], solid solution strengthening[4], precipitation hardening [5, 6] and texture optimisation [7] have been ex-plored in this context. Another strengthening approach is the use of compos-ite microstructures where the properties of the parent phase are reinforcedby a harder phase, an approach used in pearlitic and dual-phase (DP) steels[8]. Recent developments have led to the commercialization of multiphaseTRIP (Transformation Induced Plasticity) and TWIP (TWinning InducedPlasticity) steels in which phase transformations and twin formation repre-sent the reinforcing \phase". These materials are particularly complex sincethe microstructure evolves strongly with plastic deformation.According to Figure 1.1, stainless steels displaying the TRIP e ect o eran impressive combination of mechanical and physical properties comparedto high-performance carbon steel grades. The main weakness of the stainlesssteels is, however, in the areas of formability, cost and \experience". Im-provements in these three areas would allow for the expansion of the marketsfor austenitic stainless steels. The lack of \experience" noted above refers,in large part, to the inability to predict with high precision the mechan-ical behaviour of these steels using physically-based models, a gap whichinhibits the development of solutions to improve the formability of thesematerials, e.g.  nite element simulations [9]. Consequently, there is a needfor new physically-based models that predict the mechanical properties and1Chapter 1. Introductiontheir variations with controlled parameters, such as temperature, strain rate,stress state and strain path as well as with microstructure, e.g. grain size,texture and dislocation density.Figure 1.1: Relative performances of three generations of steels. Reproducedfrom Schmitt [2].While there is a relatively large pre-existing literature surrounding themechanical behaviour of austenitic stainless steels, there remain basic ques-tions about the physical processes coupling the strain-induced martensiticphase transformation and plastic deformation. Recent work has questionedwhether this transformation is driven by stress or by plastic strain [10] whileother work has focused on how microstructural parameters such as grain sizecan be used to control mechanical response in these materials [11, 12]. Withthe advent of new experimental techniques, there is an opportunity to exam-ine some of these questions with the aim of developing more physically-basedmodels that couple the microstructure, deformation and phase transforma-tions.In this work, the strain-induced martensitic phase transformations oc-curring in an austenitic stainless steel (grade 301LN) have been studiedexperimentally. Material provided by ArcelorMittal Stainless Steel has beenfurther processed by rolling and annealing to achieve materials with grain2Chapter 1. Introductionsizes as small as D=500 nm. Starting from these materials, mechanicaltesting has been performed at room temperature with a  xed strain rate inboth uniaxial tension and simple shear. These results are meant to comple-ment results obtained on the same grade under uniaxial tension but wherestrain rate and test temperature have been varied [13, 14]. Experimen-tally, the material has been characterized with an eye to linking the mi-crostructural evolution during plastic deformation with the strain-inducedtransformations occurring in this material. The bulk mechanical response ofthe material has been analyzed treating the material as a sort of \dynamiccomposite". A new method for estimating the load transfer between thetwo phases based on the magnetomechanical e ect has been developed andcompared to measurements of lattice strains made in a previous study byneutron di raction.This thesis starts by a review of the literature, aiming at clarifying thee ect of strain-induced phase transformations on the mechanical propertiesof austenitic stainless steels. It continues by describing the methods usedin the initial processing of the as-received material. Following this, themacroscopic mechanical properties and phase fractions are presented. Thisis followed by a description of the microstructural evolution with strain.A magnetic method for estimating the fraction of the macroscopic stresscarried by the  0-martensite phase is next presented and used to developa simple physically based model for the bulk mechanical response. In allcases, this work is speci cally linked to i) the starting austenite grain sizeand ii) the imposed strain path (shear and tension).3Chapter 2Literature ReviewAustenitic stainless steels are relatively mature materials, being widely usedin practical applications. There is, therefore, a large amount of literaturedescribing the behaviour of this class of materials. In the speci c case of theTRIP e ect in austenitic stainless steels, periods of active research into thedetails of the strain-induced martensitic transformations alternated with pe-riods of lower activity. In the past 10 years, there has been growing interestin returning to the basic questions of the mechanisms of the strain-inducedmartensitic transformation induced by increased demand for these materi-als as well as by the availability of new experimental techniques for probingthe material response. This literature review seeks to describe the litera-ture most relevant to the work done in this Ph.D. thesis and, in particular,focuses on new developments in the  eld over the past 10{15 years. Thereader interested in a more detailed overview of the behaviour of austeniticstainless steels is directed to the reviews found in the literature [15{17].2.1 Austenitic Stainless Steels: Stable andMetastable PhasesStainless steels are iron-based alloys containing between 10.5% to 30% chro-mium by weight [18, 19]. The chromium content is responsible for the highresistance to oxidation and corrosion, due to the formation of a passivechromium oxide  lm on the sample surface, in the presence of oxygen. Thetwo most common stainless steel families are ferritic stainless steels andaustenitic stainless steels. Ferritic stainless steels have chromium as the pri-mary alloying addition and the state of the material at room temperature is42.1. Austenitic Stainless Steels: Stable and Metastable Phasesthe stable body centered cubic (bcc) ferrite ( ) phase. Austenitic stainlesssteels are alloyed so as to allow for the retention of the high temperatureface centered cubic (fcc) austenite ( ) phase to room temperature. Nickelis the most common fcc-stabilizing element, although other fcc-stabilizingelements (e.g. manganese, copper) may partially replace nickel in some al-loys [18, 20]. The interstitial elements, namely carbon and nitrogen, alsoincrease the stability of austenite relative to ferrite. The use of carbon tostabilize austenite is avoided as chromium carbide formation depletes grainboundaries of chromium, leading to enhanced intergranular corrosion (sen-sitization) [16]. Instead, nitrogen has been used as an alloying elementproviding austenite stabilization, solid solution strengthening and increasedcorrosion resistance in various austenitic stainless steel grades [21]. An ex-ample of the nitrogen-alloyed steels is the grade AISI 301LN, which has anitrogen content ranging from 0.1 to 0.2 wt %.The retention of austenite at room temperature does not imply that itis the most thermodynamically stable phase at this temperature. Indeed,for all austenitic stainless steels at room temperature and ambient pressure,the  -ferrite is the stable phase. The addition of austenite stabilizers (e.g.nickel) lowers the free energy of austenite relative to ferrite. This is mosteasily seen if one considers the free energy di erence between austenite andferrite as a function of nickel content of the steel, cf. Figure 2.1(a). Thetemperature at which the free energy of austenite and ferrite are equal (theT0 temperature) is important in this context, since the driving force for the ! transformation increases with undercooling below T0. As one can seefrom Figure 2.1(b), increasing nickel additions lower the T0 temperature,thus reducing the driving force for the  !  transformation at a  xedtemperature (e.g. room temperature). Moreover, as T0 decreases, so do thekinetics of di usion required to transform  to  by di usional mechanisms.If T0 is low enough, the combination of lowered driving force for transfor-mation and the slow rate of transformation can \trap" the material in ametastable  state.A third phase, denoted as  , also plays an important role in the defor-mation response of austenitic stainless steels. This phase has a hexagonal52.1. Austenitic Stainless Steels: Stable and Metastable Phasess48s53s49s48s49s53s50s48s45s57s48s45s56s48s45s55s48s45s54s48s45s53s48s45s52s48s71s105s98s115s32s102s114s101s32s101s110s101s114s103s121s32s40s74s46s109s111s108s45s49s41s78s105s99s107s101s108s32s99s111s110s116s101s110s116s32s105s110s32s119s116s37s61537s61541s61543(a)s48s53s49s48s49s53s50s48s53s48s54s48s55s48s56s48s57s48s49s48s84s48s32s116s101s109s112s101s114s97s116s117s114s101s32s40s111s67s41s78s105s99s107s101s108s32s99s111s110s116s101s110s116s32s105s110s32s119s116s37(b)Figure 2.1: Relative stability of phases predicted by Thermo-Calc [22] us-ing the composition corresponding to the 301LN stainless steel studied inthis work (cf. composition in chapter 4) at atmospheric pressure. (a) Evo-lution of the Gibbs free energy of fcc ( ), hcp ( ) and bcc ( ) phases atroom temperature, (b) Evolution of the T0 temperature with variable nickelcontent.close-packed (hcp) crystal structure and is the equilibrium phase only athigh pressure [23{25], although it is often observed to form during the plas-tic deformation of austenitic stainless steels at ambient temperature andpressure. Figure 2.1 (a) illustrates that, at room temperature and at atmo-spheric pressure, the hcp  phase has an energy intermediate between bcc  and fcc  phases.The formation of  , as well as deformation twinning, during the plasticstraining of austenitic stainless steels has been linked to the low stackingfault energy (SFE) of austenitic stainless steels. In this regard, there is aclose association with stacking faults,  and twins 1.Table 2.1 compares typical values of the intrinsic SFE of face centeredcubic materials 2. In comparison to most metals, the intrinsic stacking faultenergy of austenitic stainless steels is low, being between 6 and 60 mJ.m 21 See references [26, 27] for a detailed discussion on partial dislocations and stackingfaults in fcc crystals.2 An extensive survey of the stacking fault energy and its dependence on compositionand temperature for austenitic steels has been carried on recently by Bracke [28].62.1. Austenitic Stainless Steels: Stable and Metastable Phases[29]. The stacking fault energy of austenitic stainless steels is strongly af-fected by the composition [30{32]. For example, carbon is known to stronglyincrease the SFE, while nitrogen and silicon decrease it [32]. Also notable isthe strong temperature dependence of the SFE in austenitic stainless steelscompared to other pure fcc metals [33].Material 301LN 316L 304 Ag Cu Ni AlSFE (mJ.m 2) 14 14 17 22 78 90 166Reference [34] [28] [30] [35] [35] [30] [35]Table 2.1: Experimentally-determined stacking fault energies (SFE) of var-ious fcc materials at room temperature. All measurements were obtainedfrom observation of dissociated triple nodes, except in 301LN where X-raydi raction line broadening was used.The intrinsic SFE has a direct relation to the stability of fcc-austeniterelative to hcp- . A single intrinsic stacking fault results in a two atomiclayer thick crystal having hcp packing [36, 37]. On the basis of this geometricrelationship, it has been proposed that the free energy di erence betweenthe fcc and hcp phases can be related to the intrinsic SFE as [38]:SFE = 2  Gfcc!hcp + 2 fcc=hcp (2.1)where  is the molar surface density of the austenite 3,  Gfcc!hcp isthe Gibbs Free energy di erence in transforming one mole of fcc crystal toone mole of hcp crystal, and  fcc=hcp is the surface energy of the hcp plate.Other, more detailed, models for the SFE have been advanced, for examplethat described by Ferreira [39] for the Fe-Cr-Ni system.Simple models of the stacking fault energy have been used in an attemptto predict the strain-induced deformation response of low-SFE steels. No-table is the work of Allain et al. who have constructed maps allowing for thecorrelation between composition, test temperature and deformation mech-3 The molar surface density can be determined from the lattice parameter a, by: = 4p3 1a2@ where @ is Avogadro’s number.72.1. Austenitic Stainless Steels: Stable and Metastable Phasesanism (e.g.  formation, deformation twinning, only slip) in Fe-Mn basedsteels [38].82.2. Deformation-Induced Martensitic Phase Transformations in Austenitic Stainless Steels2.2 Deformation-Induced Martensitic PhaseTransformations in Austenitic Stainless Steels2.2.1 Ferrous Martensites in Austenitic Stainless SteelsAs noted in section 2.1, austenitic stainless steels retain the fcc crystal struc-ture at room temperature owing to the fact that the kinetics of the di usionalphase transformation from  to  are too slow at temperatures where thedriving force for the transformation is large enough. This leaves the mate-rial in a metastable state with respect to the thermodynamically stable  phase. While di usion may be too slow to allow for the transformation from to  (or to the intermediate  phase), another option for the transitionexists. A martensitic transformation can lead to the transformation from  to  or  without the need for long-range di usion. In contrast to a di usivephase transformation, a martensitic transformation is characterized by itsdisplacive character, the motion of atoms being governed by a homogeneousshearing of atoms at velocities close to the speed of sound [40, 41].Ferrous martensites are most commonly associated with the spontaneousformation of the martensitic phase (e.g. body centered tetragonal  0 mar-tensite from austenite in carbon steels) when the steel is cooled below acertain temperature, known as the martensite start (or Ms) temperature[42]. In most cases, the martensite is formed at velocities close to the speedof sound as soon as the material is cooled below this temperature. Thevolume fraction of martensite increases with undercooling below the Mstemperature until the transformation is complete at the martensite  nish(or Mf) temperature. While this thermal martensite [43] is most common,there are other steels that exhibit a time dependence to the fraction of mar-tensite formed when cooled below the Ms temperature. The kinetics of thisisothermal martensitic transformation are typically linked to the di cultyin the nucleation of martensite plates [44{46]. This can be understood basedon the fact that a large energy barrier must be overcome in order to formferrous martensite. In a plain carbon steel, this barrier is on the order of  1 kJ.mol 1 [47].92.2. Deformation-Induced Martensitic Phase Transformations in Austenitic Stainless SteelsOn the microscopic scale, the martensitic phase transformation from  to  0 can be described on the basis of a uniform and homogeneous strain ofthe crystal lattice. The Bain correspondence between the  and  0 latticesis often used to describe a pathway between these two phases [48]. Thisrelation, schematically shown in Figure 2.2, assumes that one  0 lattice canarise from  , provided it is contracted along one of theh100i directions andexpanded along the two other h100i directions.[001]fccbardbl[001]bcc[100]fccbardbl[110]bcc[010]fccbardbl[¯110]bccBulletCircleBulletCircleBulletCircleBulletCircleBulletCircleBulletCircleBulletCircleBulletCircleBulletCircleBulletCircleBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletCircleBulletCircleBulletCircleBulletCircleFigure 2.2: The Bain correspondence between the fcc unit cell (light gray)and the tetragonally distorted bcc unit cell (black).In the case of pure iron taken at room temperature, the Bain expansion is20% while the Bain contraction is 12%. The magnitude of these distortionsdepends on the composition of the alloy, increasing strongly with the carboncontent. The Bain model, while giving the correct lattice correspondencebetween the  and  lattices, does not predict properly the crystallographicorientation relationship between the two phases [49, 50]. This requires amore detailed understanding of the mechanism of nucleation and growth ofmartensite, which will be described in section 2.2.3.While the  !  0 transformation is the most well-known martensitictransformation in steels, it is also possible to form  from  as a martensitictransformation. As noted in Equation 2.1, a stacking fault has the equivalent102.2. Deformation-Induced Martensitic Phase Transformations in Austenitic Stainless Steelsstacking sequence as a two atomic layer thick plate of hcp crystal. Thus, oneway to obtain a thin plate of  -martensite is by the passage of a Shockleypartial dislocation on every other parallel f111g plane. A small contractionnormal to thef111gplane is also necessary since  -martensite is more densethan austenite [51].In the case of austenitic stainless steels, an Ms temperature below 4K iscommon for the  ! 0 transformation [52]. Therefore, thermal martensiteis not readily obtained even though a signi cant driving force exists. Thereis, however, the possibility to trigger the martensitic transformation by ei-ther: (i) modifying existing nucleation barriers by reducing their activationenergy, or (ii) increasing the number of nucleation sites with low activationbarriers.One of the common ways of modifying existing nucleation sites by low-ering the activation barrier is through the application of an applied stress.The work of Patel and Cohen showed how stress could e ect the Ms temper-ature of thermal martensite [47]. As shown in Figure 2.3, the e ect of stressstate on the transformation can increase or decrease the Ms temperature.s48s53s48s49s48s49s53s48s50s48s50s53s48s45s49s48s48s49s48s50s48s51s48s52s48s32s85s110s105s97s120s105s97s108s32s116s101s110s115s105s111s110s32s85s110s105s97s120s105s97s108s32s99s111s109s112s114s101s115s105s111s110s32s72s121s100s114s111s115s116s97s116s105s99s32s111s109s112s114s101s115s105s111s110s67s104s97s110s103s101s32s105s110s32s77s115s32s40s111s67s41s83s116s114s101s115s32s111s114s32s112s114s101s115s117s114s101s32s40s77s80s97s41Figure 2.3: Change in the Ms temperature as a function of the loadingcondition. Adapted from the work of Patel and Cohen [47].Patel and Cohen argued that this e ect could be explained based on the112.2. Deformation-Induced Martensitic Phase Transformations in Austenitic Stainless Steelsmechanical work done in the transformation of  to  0. In this argument, themechanical work (or interaction energy) done by the phase transformationcorresponds to:Uinter =   +  (2.2)where  is the component of the macroscopic stress acting normal tothe martensite plate, and  is the shear stress acting parallel to the sheardirection of the plate. The strains denoted as  and  are the normal strainand shear strain accompanying the martensitic transformation. In the caseof the steels studied by Patel and Cohen, the values of  and  were takento be 0.04 and 0.20 respectively. The value of the interaction energy basedon this calculation was 0.86 J.mol 1 per MPa of applied (tensile) stress [47].Using a linear temperature-dependence of the free energy di erence betweenaustenite and martensite and under the assumption of a constant activationenergy for martensite nucleation (taken to be 837 J.mol 1), Patel and Cohenobtained a variation of the Ms temperature with tensile stress equalling:dMsd = 0:15 C:MPa 1 (2.3)which corresponds well to the experimental measurements shown in Fig-ure 2.3.In some materials the deformation temperature is close to, but slightlyabove, Ms. In this instance, the imposition of a stress in the elastic regimecan be enough to induce a stress-assisted martensitic phase transforma-tion [53]. This form of transformation is made use of in shape memoryalloys where deformation induces the martensitic transformation which canbe subsequently removed by heating the material to a temperature where  is the stable phase [54{56]. In the case of ferrous alloys, the most commonstress-assisted martensitic transformation occurs in the Fe-Mn-Si systemwhere thin  martensite plates can form under an elastic applied stress [57],consistent with the Patel and Cohen model as de ned by Equation 2.2.If the testing temperature is su ciently above the Ms temperature, thestress required to induce the martensitic transformation can be above the122.2. Deformation-Induced Martensitic Phase Transformations in Austenitic Stainless Steelsyield stress of the alloy. In this case, the martensitic transformation occursduring plastic deformation. Beyond the possible e ect of the applied stresson the activation barrier for martensite nucleation, plastic deformation canalso serve to create nucleation sites not present in the as-annealed material[47, 58, 59]. Martensite formed concurrently with plastic deformation is saidto be strain-induced [43].Stress-assisted nucleationM M Mσss dStrain-induced nucleationYield stress of austeniteσyTemperatureStressσyFigure 2.4: Schematic representation of the interrelationships betweenstress-assisted (below M s ) and strain-induced (above M s ) martensitictransformations. The blue curve indicates the onset temperature for the ! 0 martensitic transformation. In the absence of stress, this onset cor-responds to Ms. When a stress is superimposed, the onset temperature isincreased in agreement with Equation 2.3. Above the yield stress of auste-nite, new nucleation sites are formed by plastic deformation. Plot adaptedfrom the work of Olson and Cohen [60].The dependence of the Ms temperature with stress is schematically il-lustrated in Figure 2.4. This plot also shows how the yield stress of theaustenite changes with temperature and therefore how the dependence ofthe Ms temperature changes with stress. All points lying above the blueline indicate conditions where martensite will form. The M s temperatureis de ned as the corresponding Ms temperature at the yield stress of theaustenite. On the other end of the plot, the Md temperature corresponds tothe temperature above which no strain-induced transformation will occur.132.2. Deformation-Induced Martensitic Phase Transformations in Austenitic Stainless Steels2.2.2 Techniques Used to Measure Martensite Content inAustenitic SteelsTo understand the relationship between deformation and martensitic phasetransformation, it is necessary to have experimental techniques capable ofquantifying the fraction of the microstructure they occupy. Several suchmethods are available, the most important having been recently reviewedby Talonen [61]. Some of the characteristics of these techniques are detailedin Table 2.2.Technique Probed volume Advantages DisadvantagesMagnetic Magnetic saturationmeasurements Bulk High accuracy not always reached,(permeability Edge e ects,or force) Calibration neededNeutron High penetration Need access todi raction Bulk depth ( 20 mm) large facilityFast Small penetrationX-ray di raction Surface layer and relatively depth ( 10 a181m),easy Texture e ectsMetallography / Surface layer Spatial Phase recognition isSEM / EBSD information not straightforwardM ossbauer Thin foil High accuracy sensitive to chemistryTable 2.2: Review of the techniques used to quantify phase fractions of 0-martensite.Phase fractions from X-ray di raction (XRD) and neutron di raction canbe estimated from the ratios of the intensities of Bragg peaks (e.g. Dickson’smethod [62]) or from whole pattern Rietveld  tting [63, 64]. A drawbackof these techniques is that intensities of the di raction peaks are dependentupon crystallographic orientation and that peak overlap can in uence peak tting. Various techniques to reduce these e ects have been proposed [65,66].Measurement of the  0 phase based on magnetic probes have been usedwith good results 4. Measurements of  0 fraction by saturation magnetiza-4 Both austenite and  -martensite being paramagnetic, the ferromagnetic  0-martensiteis the only phase that can be detected by magnetic sensors.142.2. Deformation-Induced Martensitic Phase Transformations in Austenitic Stainless Steelstion [67], magnetometer [68], SQUID [69] and Feritscope [68] or equivalentdevices [70] have been used extensively, while magnetic force measurementsor Satmagan measurements [61] are less common.Finally, direct measurement of the phase fractions by microscopy (opti-cal, SEM, EBSD) have also been attempted (e.g. [71]). These techniques arelimited, however, by the di culty of resolving the  ne scale phases (particu-larly  martensite) and by the poor statistical sampling of the microstructureby these techniques.2.2.3 Mechanisms of Formation of Strain-InducedMartensite in Austenitic Stainless SteelsIn the case of austenitic stainless steels, the martensitic transformationsare typically observed to occur during plastic deformation and thus arestrain-induced according to the de nition given in 2.2.1. It is common inthese steels for both  -martensite and  0-martensite to form during straining.Moreover, it is common for  -martensite to participate in the nucleation of 0-martensite. The result is a complex relationship between microstructure,plastic deformation and martensite fraction.Most recent work supports the view that the strain-induced  ! trans-formation precedes the formation of  0-martensite and that the  -martensiteis important in the nucleation of  0. The strain-induced  -martensite formsin the shape of thin plates, as illustrated in Figure 2.5.As noted in section 2.1, the structure of  -martensite is consistent witha stacking fault lying on every second f111g plane. This results in theBurgers orientation relationship [73]:f111g ==f0001g (2.4)h 1 10i ==h2 1 10i (2.5)which leads to only four distinguishable variants of  -martensite [50, 74, 75].One obvious mechanism for the formation of  -martensite is the co-ordinated motion of a=6h112i Shockley partial dislocations. The similarity152.2. Deformation-Induced Martensitic Phase Transformations in Austenitic Stainless SteelsFigure 2.5: Plates of  -martensite in grade 304L after a 5% tensile strain at 196 C. Reproduced with permission from [72].between the structure of an  -plate and a deformation twin is illustratedin Figure 2.6. While a twin can be described as a stacking fault on everyadjacentf111g plane, a plate of  -martensite only requires a stacking faulton every second plane. This has led to the hypothesis that the formation of -martensite is similar to the mechanism of formation of deformation twins[51]. R emy and Pineau have suggested that there is a continuous transi-tion from deformation twining to  -martensite formation with decreasingstacking fault energy [75, 76].Recently, new experimental work has provided further clues as to thedi erences between the nucleation of twins and  -martensite [77]. The dis-location structure in an Fe-Mn-Al-Si alloy was studied over a range of tem-peratures where the deformed structure contained twins or  -martensite. Inthis experiment, it was found that, under conditions where  -martensite wasformed, extrinsic stacking faults were observed in the material. In contrast,under conditions where twinning was observed, intrinsic stacking faults wereobserved. This contradicts theories that have proposed that intrinsic stack-ing faults are precursors for  -martensite while extrinsic stacking faults areprecursors for twinning (e.g. [78, 79]).162.2. Deformation-Induced Martensitic Phase Transformations in Austenitic Stainless Steels(a) Twin (b)  -martensiteFigure 2.6: Schematic representation of (a) a twin, (b) a thin plate of  -martensite. The atoms belonging to a f100g plane prior to the transforma-tion are underlined to help visualize their motion.It has been experimentally observed in many cases [79, 80] that plates of -martensite are often imperfect and composed of thinner  -layers separatedby retained austenite, which itself may or may not be twinned [28, 38].Such complex structures, as shown in Figure 2.7, may explain why much ofthe early microscopy on austenitic stainless steels referred to the  ne plate-like packets of stacking faults as \shear bands" (e.g. [81]) rather than as -martensite or as deformation twins.Figure 2.7: TEM Micrograph of an  0 martensite island formed within an -band. Reproduced with permission from [28]The formation of  is a very localized process, associated with the pres-172.2. Deformation-Induced Martensitic Phase Transformations in Austenitic Stainless Steelsence of wide stacking faults [45, 78, 82]. Though the formation of -martensiteis still not well understood, some mechanisms have been proposed. Notablemodels include pole mechanisms (e.g. Seeger [83]) and deviation based mech-anisms (e.g. Fujita et al. [84]), the latter mechanism being consistent withthe idea that the nucleation of  requires some overlapping of stacking faults[37]. In all of these cases, the formation of  -martensite is traced to theexistence of extended stacking faults rather than by nucleation at grain ortwin boundaries [85].There has also been a suggestion that the formation of  -martensite mayrespect the Schmid law, in that it occurs at a critical value of the resolvedshear stress on some slip system of the austenite. In a 301 grade, Hedstr omdetermined that the austenitic grains which form  -martensite were thosewith highest Schmidt factor on the f111g [1 21] slip systems [86]. Thesame slip systems were found to verify the Schmid law for the apparition ofmechanical twins [28, 87] in Fe-Mn steels.s48s46s48s46s49s48s46s50s48s46s51s48s46s52s48s46s53s49s48s50s48s51s48s52s48s53s48s54s48s55s48s56s48s61541s61537s39s86s111s108s117s109s101s32s102s114s97s99s116s105s111s110s32s111s102s32s112s104s97s115s101s32s40s37s41 s84s114s117s101s32s83s116s114s97s105s110Figure 2.8: Evolution of the fraction of  and  0 martensites formed duringroom-temperature tensile deformation of grade 301. The volume fractionswere measured by X-ray di raction [88].While  -martensite is often found as an important feature of the mi-crostructure in austenitic stainless steel deformed to low levels of strain,Figure 2.8 illustrates that it never exceeds a small fraction of the total vol-ume of the microstructure [88]. After only a few percent plastic strain, it182.2. Deformation-Induced Martensitic Phase Transformations in Austenitic Stainless Steelsis typically found that the volume fraction of  0-martensite signi cantly ex-ceeds that of  martensite. In these steels, the  0 phase is comparable to thebody centered tetragonal (bct) thermal martensite obtained by quenchingof carbon steels [89{92], though the interstitial content (i.e. carbon plusnitrogen) of austenitic stainless steels tends to be low meaning that thetetragonality of the martensite is nearly zero [17, 93].While the volume fraction of  0 is found to greatly exceed that of  -martensite, the formation of the former is often linked to the existence ofthe latter. In particular, it is very common to observe the nucleation of  0on  -martensite plates as well as at the intersection between  -martensiteplates as illustrated in Figure 2.9.Figure 2.9: Nucleation of  0 in grade 316 after a 5% tensile strain at 196 C.The dark fault bands are associated with  -martensite while  0 martensite ishighlighted at some, but not all, intersections. TEM micrograph reproducedwith permission from [72].Venables was the  rst to describe this mechanism of nucleation in auste-nitic stainless steels [94]. There has been some controversy about whether  appears as an intermediate to  0 [33, 60, 95] or if it is generated to accom-modate plastic strains [95, 96], although the current opinion tends to favourthe concept that  -martensite forms prior to  0 [72].192.2. Deformation-Induced Martensitic Phase Transformations in Austenitic Stainless SteelsBesides nucleation at  intersections,  0 has been observed to nucleateheterogeneously on other deformation-induced defects (cf. Table 2.3). Sim-ilar to the nucleation at  intersections, other researchers have observed thenucleation of  0 at the intersection of deformation twins [82, 97{102]. Othervariations include the proposal for the formation of  0 at the intersectionbetween a plate of  and a deformation twin [103] or an annealing twin [72],or at the intersection of two stacking faults [104]. Twin boundaries have gen-erally been proposed to be less e ective nucleation sites than  -martensite[72, 105].Intersecting features are not always a necessary condition: some re-searchers have observed the nucleation of  0 to take place within a sin-gle  plate [106, 107]. Martensite formation at grain boundaries has beenmuch less reported with the only observations being made in grades dis-playing higher nickel content than traditional commercial stainless steels[33, 108, 109] or in submicron austenitic grains obtained after Equal Chan-nel Angular Pressing (ECAP) [110]. Finally, there has been the proposalthat  0 can form directly from slip in the austenite [95, 111].Type of Nucleation ReferenceIntersection between two  plates [94, 97, 112{115]Intersection between two deformation twins [82, 97{102, 116]Intersection between  and a deformation twin [103]Intersection between  and an annealing twin boundary [72, 117]Nucleation within a single  plate [106, 107]Grain Boundary nucleation from  [33, 108, 109, 118]Direct nucleation from  [95, 96, 111]Table 2.3: Review of the mechanisms of  0 nucleation observed in austeniticstainless steels.A physically-based model capable of explaining the apparent relationshipbetween  -martensite plates and the nucleation of  0-martensite was  rstproposed by Olson and Cohen [60]. Olson and Cohen re ected upon thedouble shear model for  0-martensite from  ,  rst proposed by Bogers and202.2. Deformation-Induced Martensitic Phase Transformations in Austenitic Stainless SteelsBurgers [119]. Bogers and Burgers showed that the homogeneous shearingof the fcc lattice on two di erent f111g planes, illustrated in Figure 2.10,can produce a strain equivalent to the Bain strain.BulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBullet(a)BulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBulletBullet(b)Figure 2.10: The distorted bcc unit cell extracted from Figure 2.2, plottedshowing f111gfcc planes (i.e. two Thompson tetrahedra). If the material ishomogeneously sheared on the f111g planes as shown by the arrows in (a),then one approximately obtains the Bain strain leading to a bcc lattice asin (b).The shears required to give the Bain strain are afcc=12h112i (half thetwinning shear) on one set of f111g planes and afcc=18h112i (one third thetwinning shear) on a second set off111gplanes. Olson and Cohen made thelink between this concept and the presence of  0 at  intersections, notingthat the  rst shear given above corresponds exactly to that caused by an plate, while the second is consistent with the strain-induced formationof a faulted  plate. This model therefore requires that two  plates, oneperfect and the other faulted, must intersect in order to generate strains atthe intersection which, in a continuum sense, give the Bain strain. Whilethe Olson-Cohen model is a continuum model, recent molecular dynamicssimulations have shown that the martensitic transformation can occur at theintersection between the bands, even when the discreetness of the crystallattice is considered [105, 120]. Moreover, the atomistic models indicate212.2. Deformation-Induced Martensitic Phase Transformations in Austenitic Stainless Steelsthat a variety of di erent  and faulted  intersection combinations can leadto conditions resulting in the martensitic transformation. One feature ofthis model which appears at odds with the generally accepted experimentalresults is that the  / 0 orientation relationship obtained from this model(Pitsch) di ers from that observed experimentally (Kurdjumov-Sachs) [120].In the case of strain-induced martensite, there has been little discussionregarding the relative importance of nucleation versus growth. In mostmodels for the evolution of strain-induced martensite nucleation, the kineticsare assumed to be only limited by nucleation. However, experimentally, onecan see that the growth of strain-induced  and  0 is not the same in alldirections [121]. In the work of Spencer [72], careful attention was paid to themorphology of martensite in the early stages of formation under conditionswhere nucleation occurred at the intersection between  -martensite plates.In this condition, the  0-martensite clearly had a preference to grow alongthe prior  plates rather than normal to the plates. This was also observed inmolecular dynamics simulations, where the growth was observed to be nearlyisotropic when the orientation relationship was Pitch but anisotropic withparticularly fast growth along  bands when the  0 had a Kurdjumov-Sachsorientation relationship [105].A  nal point regarding the concept of the potential importance of mar-tensite growth relative to nucleation for the kinetics of strain-induced mar-tensitic transformations relates to the concept of mechanical stabiliza-tion. The concept of mechanical stabilization relates to the fact that dis-locations ahead of a martensite interface will retard the motion of the aus-tenite/martensite interface. In the model of Chatterjee et al. [122], theretarding force from a density of dislocations in austenite was presumedto arise from the stress to move dislocations past one another. When thedensity of dislocations is high enough that the force required to move dis-locations is higher than the driving force for the martensite interface, themartensite will not be able to move further and the remaining austeniteis considered to be stabilized mechanically. Chatterjee et al. applied thisconcept to explain the fact that a maximum of 50{60% strain-induced mar-tensite could be formed in a 316L stainless steel that had undergone severe222.2. Deformation-Induced Martensitic Phase Transformations in Austenitic Stainless Steelsplastic deformation. It was found, in this case, that the e ects of mechanicalstabilization were not important until large strains (  2{3). Conversely, ina crystal-plasticity-based study, it was found that a relatively small level ofprestrain (7.5% strain in the austenitic grain) was su cient to fully suppressthe phase transformation due to austenite stabilization [123].232.3. Factors In uencing the Rate of Strain-Induced Martensitic Transformation in Austenitic Stainless Steels2.3 Factors In uencing the Rate ofStrain-Induced Martensitic Transformationin Austenitic Stainless SteelsGiven the complex nature of the strain-induced martensitic transformationsin austenitic stainless steels, it is not surprising that the rate of transforma-tion is sensitive to a large number of parameters. Some of these parametersare intrinsic to the microstructure of the material (e.g. grain size, disloca-tion density) while other factors are intrinsic to the test conditions (e.g. testtemperature, strain rate, stress state, strain path).One important microstructural feature that impacts on martensitic trans-formations is austenite grain size. The e ect of grain size on the stability ofthermal martensite can be observed as a variation of Ms temperature withgrain size. For both thermal  ! 0 [124] and  ! [125, 126] martensitictransformations, grain size re nement leads to a reduction in the Ms tem-perature. It is possible, in fact, to completely suppress the  0 martensitetransformation via grain size re nement [127, 128].The strain-induced  -martensite obtained in Fe-Mn steels shows similarcharacteristics. Hamada et al. [129] showed that the  ! transformationcould be suppressed by reducing the grain size from 40 a181m to 10 a181m. Theseauthors proposed that the presence of numerous  3 twins in small grainscould impede the motion of the partial dislocations and be an obstacle tothe growth of the  -martensite platelets. Alternatively, they also suggestedthat large austenitic grains may allow for the formation of stacking faultsalong a large number of planes. Thus, the probability of  nding nucleationsites (for  ) formed by the overlapping of faults is higher in larger grains.The literature on metastable austenitic stainless steels provides a muchless complete set of data for the e ect of grain size on transformation ki-netics and mechanical response. The range of grain sizes for which dataexists in the literature ranges principally between 10 and 200 a181m, thougha small number of experiments on materials with smaller grain sizes havebeen recently reported [110, 130]. The trend in the literature is for grain size242.3. Factors In uencing the Rate of Strain-Induced Martensitic Transformation in Austenitic Stainless Steelsre nement to promote the stability of  with respect to the strain-inducedformation of both  [131] and  0 [132, 133] martensites. The reduction of therate of transformation to  0 with decreasing grain size has been observeddirectly by Nohara [134], Leal [135], Gonzales [136], Jeong [137], Varma[138] and P etein [139]. There are, however, cases where exceptions to thisbehaviour have been observed. For example, in grade 304, an increase in therate of transformation to  0 with decreasing grain size has been observed[140, 141]. Examples of these two contradicting behaviours are representedin Figure 2.11.s48s46s48s46s49s48s46s50s48s46s51s48s46s52s50s52s54s56s49s48s49s50s37s32s61537s39s32s109s97s114s116s101s110s115s105s116s101s84s114s117s101s32s83s116s114s97s105s110s53s51s61549s109s49s50s51s61549s49s56s48s61549s109s50s56s53s61549s109(a)s48s46s48s46s49s48s46s50s48s46s51s48s46s52s50s48s52s48s54s48s56s48s49s48s37s32s61537s39s32s109s97s114s116s101s110s115s105s116s101s84s114s117s101s32s115s116s114s97s105s110s56s61549s109s53s56s61549(b)Figure 2.11: Illustration of two di erent behaviours of the transformationkinetics as a function of grain size. Grade 304 deformed in tension (a) atroom temperature [138] and (b) at  50 C [141].There are few mechanisms that have been proposed to explain this trend.In particular, there have been no attempts to separate the grain size depen-dence of the  !  and  !  0 transformation. This is linked to theuncertainty, in many cases, of the mechanism of formation of  0, i.e. is itformed directly from austenite, or does it require  as a precursor?One attempt to rationalize a grain size dependence of the  ! 0 trans-formation mechanism is found in the work of Yang et al. [118]. They ob-served that coarse austenite grains ( 40 a181m) would form  0 at intersectionsof shear bands ( -martensite, deformation twins, stacking faults), whereas insubmicron austenitic grains,  0 would preferentially nucleate at grain bound-252.3. Factors In uencing the Rate of Strain-Induced Martensitic Transformation in Austenitic Stainless Steelsaries. No quantitative attempt to link the rate of transformation to grainsize was attempted in this work, however.Another example of the importance of starting microstructure comesfrom experiments where metastable stainless steels have been subjected topre-strain under one set of test conditions, followed by a second strainingunder di erent conditions. Spencer imposed tensile strain of 10% at 196 Con alloy 316L [72]. This pre-strained material was subsequently allowed toreturn to room temperature and the tensile deformation continued. Whilenormally the rate of formation of  0-martensite at room temperature is lowin this alloy, after pre-straining at  196 C, it was observed that the rate oftransformation at room temperature was extremely high (cf. Figure 2.12).In fact, the rate of transformation at room temperature, just following thetemperature change, was found to be the same as that at 196 C just beforethe test was stopped. Inversely, when pre-straining was conducted  rst atroom temperature followed by testing at 196 C, the subsequent formationof  0 at low temperature resulted in the formation of a L uders-like bandwhich propagated through the material [72] 5. Observation indicated thatthe amount of the  0-martensite within the localization band was higherthan that outside of the band. The conclusion drawn from these results wasthat the rate of  0-martensite transformation is sensitive to the microstruc-ture formed by plastic deformation, and thus to the presence of forest dis-locations and other deformation-induced defects. This phenomenon is notwell-understood, and has only been accounted for in quantitative kineticmodels in a semi-empirical form based on the concept of austenite stabiliza-tion [123].Aside from microstructural e ects on the strain-induced transformation,bulk mechanical testing conditions can also have notable e ects on the trans-formation kinetics. One of the most important parameters, with regards tothe strain-induced martensitic transformation to  0-martensite is the e ectof test temperature. Figure 2.13 illustrates the reduction in the rate of for-mation of  0 with increasing temperature in a 301LN stainless steel. Thisobservation is consistent with a wide range of other alloys, e.g. [134, 142{5 Refer to section 2.5.1 for more details about this behaviour.262.3. Factors In uencing the Rate of Strain-Induced Martensitic Transformation in Austenitic Stainless Steelss48s46s49s48s46s50s48s46s51s48s46s52s48s46s53s48s46s54s48s46s48s48s46s49s48s46s50s48s46s51s77s97s114s116s101s110s115s105s116s101s32s102s111s114s109s101s100s32s117s110s100s101s114s32s109s111s110s111s116s111s110s105s99s32s108s111s97s100s105s110s103s32s97s116s32s50s53s111s67s86s111s108s117s109s101s32s102s114s97s99s116s105s111s110s32s111s102s32s61537s39s32s109s97s114s116s101s110s115s105s116s101s67s117s109s117s108s97s116s105s118s101s32s116s114s117s101s32s83s116s114s97s105s110s77s97s114s116s101s110s115s105s116s101s32s102s111s114s109s101s100s32s97s116s32s50s53s111s67s102s111s108s108s111s119s105s110s103s32s97s32s49s48s37s32s112s114s101s45s115s116s114s97s105s110s32s97s116s32s45s49s57s54s111s67Figure 2.12: Evolution of martensite volume fraction in 316L stainlesssteel during room-temperature deformation and after a  196 C prestrain.Adapted from [72].146]. The decreasing rate of transformation with increasing temperaturecan be understood in relation to the reduction in the driving force for trans-formation ( ! and  ! 0) and the associated rise in the stacking faultenergy of the austenite (linked to the formation of  -martensite), with in-creasing temperature, as described in section 2.1. For instance, in grade304, increasing the temperature from 20 C to 80 C reduces  G ! 0 by300 J:mol 1 and  G ! by 150 J:mol 1 while increasing the stacking faultenergy by 4 mJ:m 2 [33].The strong in uence of temperature on the transformation kinetics canalso be seen in an indirect way if one performs tests at elevated strain rates.While most metals exhibit relatively low strain rate sensitivity, stainlesssteels are an exception. This rate sensitivity is a direct e ect of deformation-induced sample heating, which overwhelms the possible intrinsic e ects ofstrain rate on mechanical response even for relatively moderate increases instrain rate. In this case the sample heating comes from the energy dissipated272.3. Factors In uencing the Rate of Strain-Induced Martensitic Transformation in Austenitic Stainless Steelss48s46s48s48s46s49s48s46s50s48s46s51s48s46s52s48s46s48s48s46s50s48s46s52s48s46s54s48s46s56s49s46s48s45s49s48s111s67s32s45s52s48s111s67s32s50s51s111s67s32s52s48s111s67s32s56s48s111s67s86s111s108s117s109s101s32s102s114s97s99s116s105s111s110s32s111s102s32s61537s39s32s109s97s114s116s101s110s115s105s116s101s84s114s117s101s32s115s116s114s97s105s110Figure 2.13: E ect of deformation temperature on the  ! 0 kinetics ofgrade 301LN. Adapted from [146].by the motion of dislocations and the latent heat of the phase transforma-tion(s). As an example of the sensitivity of austenitic stainless steels tothis e ect, Figure 2.14 shows that increasing the strain rate by one order ofmagnitude (from 1:3 10 3 s 1 to 1:3 10 2 s 1) during the tensile testingof grade 204M causes a 50 C increase of sample temperature at the end ofa uniaxial tensile test. As illustrated, this rise in temperature is enough toreduce the rate of transformation in half [14, 147, 148].While there have been some reports that increasing strain rates promotethe formation of microscopic shear bands (e.g.  -martensite) [68, 97, 149,150], this e ect is best seen at low strains. It is not clear whether this e ectcontinues to be important at larger strains as the e ects of self-heatingdescribed above tend to dominate the material response.As noted above, the in uence of imposed stresses (and stress state) on the282.3. Factors In uencing the Rate of Strain-Induced Martensitic Transformation in Austenitic Stainless Steelss48s46s48s46s49s48s46s50s48s46s51s48s46s52s48s46s53s46s48s46s50s48s46s52s48s46s54s48s46s56s32s49s46s51s120s49s48s45s52s32s115s45s49s32s49s46s51s120s49s48s45s51s32s115s45s49s32s49s46s51s120s49s48s45s50s32s115s45s49s32s49s46s51s120s49s48s45s49s32s115s45s49s61537s39s45s77s97s114s116s101s110s115s105s116s101s32s118s111s108s117s109s101s32s102s114s97s99s116s105s111s110s84s114s117s101s32s83s116s114s97s105s110(a)s48s46s48s46s49s48s46s50s48s46s51s48s46s52s48s46s53s50s48s52s48s54s48s56s48s32s49s46s51s120s49s48s45s49s32s115s45s49s32s49s46s51s120s49s48s45s50s32s115s45s49s32s49s46s51s120s49s48s45s51s32s115s45s49s84s101s109s112s101s114s97s116s117s114s101s32s82s105s115s101s32s40s111s67s41 s84s114s117s101s32s83s116s114s97s105s110(b)Figure 2.14: E ect of strain rate in grade 204M (a) on the  ! 0 kinetics,(b) on the sample temperature. Adapted from [148]formation of thermal martensite are well known and documented. In the caseof strain-induced martensitic transformations, the e ects are more complexowing to the path dependence of plastic deformation. It is therefore oftendi cult to deconvolute the e ects arising from the transformation beingstress-assisted and/or from the plastic strains generating di erent nucleationsites depending on the strain path.Though this topic is complex, there have been a number of studies exam-ining and comparing the mechanical response and transformation kineticsfor di erent modes of mechanical loading. Unfortunately, there are manycon icting results that make a de nitive correlation di cult.As an illustration of the confusion regarding the e ects of stress stateand deformation path, one can compare the e ects of uniaxial tension andcompression found in the literature. Powell [153] and Lebedev [151] showedthat the rate of the  ! 0 transformation was higher in uniaxial tensionthan either in compression or torsion. Contradicting these results are theresults of Iwamoto [152] and Kato [154], who observed that compressionresulted in the formation of more  0 than did tension. Iwamoto rationalizedtheir observations based on di erences in texture evolution between tensionand compression [152].292.3. Factors In uencing the Rate of Strain-Induced Martensitic Transformation in Austenitic Stainless Steelss48s46s48s46s49s48s46s50s48s46s51s48s46s52s48s46s53s48s46s54s50s48s52s48s54s48s56s48s49s48s67s111s109s112s114s101s115s105s111s110s84s111s114s115s105s111s110s37s32s61537s39s32s109s97s114s116s101s110s115s105s116s101s84s114s117s101s32s83s116s114s97s105s110s84s101s110s115s105s111s110(a)s48s46s48s46s50s48s46s52s48s46s54s48s46s56s49s48s50s48s51s48s52s48s53s48 s84s101s110s115s105s111s110s67s111s109s112s114s101s115s105s111s110s37s32s61537s39s32s109s97s114s116s101s110s115s105s116s101s84s114s117s101s32s115s116s114s97s105s110(b)Figure 2.15: (a) Grade 304 deformed at  196 C along 3 di erent paths, ata constant loading rate. The  0 fraction was measured by X-ray di raction[151]. (b) Di erence between tension and compression performed at roomtemperature on grade 304. This di erence is attributed to the deformation-induced anisotropy [152].Hecker found that the use of the Von Mises equivalent strain was appro-priate to compare test data ranging from biaxial tension to uniaxial tensionin terms of  0 formation. It was found that  0 started to form at lowerstrains under biaxial stresses [68], consistent with the observation that theintersection of \shear bands" (e.g.  -martensite, twins, bundles of stackingfaults) started at a lower strain along that path [97]. The observation ofenhanced transformation at crack tips [155] suggests that the  ! 0 trans-formation is accelerated for high values of the stress triaxiality. Experimentsperformed over a wide range of stress states suggest a complex dependence,with an increase in the rate of the  ! 0 martensitic transformation forstress states varying from simple shear to uniaxial tension [146, 151, 156],as illustrated in Figure 2.16.The trend from uniaxial tension to equibiaxial tension is not de nitivehowever, with reports of decreasing [146, 151, 157] and increasing trans-formation rate [156]. Equibiaxial tests are often achieved using cup draw-ing, a complex non-uniform state of deformation [158]. It is possible thatthe use of such complicated, non-monotonic, non-proportional deformation302.3. Factors In uencing the Rate of Strain-Induced Martensitic Transformation in Austenitic Stainless Steelss48s46s48s46s49s48s46s50s48s46s51s48s46s52s48s46s53s48s46s48s46s50s48s46s52s48s46s54s48s46s56s49s46s48s32s85s110s105s97s120s105s97s108s32s116s101s110s115s105s111s110s32s66s97s108s110s99s101s100s32s98s105s97s120s105s97s108s32s116s101s110s115s105s111s110s32s80s108s97s110s101s32s115s116s114s97s105s110s32s83s105s109s112s108s101s32s115s104s101s97s114s86s111s108s117s109s101s32s102s114s97s99s116s105s111s110s32s111s102s32s61537s39s32s109s97s114s116s101s110s115s105s116s101s86s111s110s32s77s105s115s101s115s32s101s113s117s105s118s97s108s101s110s116s32s115s116s114s97s105s110(a)s48s46s48s48s46s53s48s46s49s48s48s46s49s53s48s46s50s48s46s48s46s50s48s46s52s48s46s54s48s46s56s49s46s48s32s107s62s49s32s40s85s110s105s97s120s105s97s108s32s116s101s110s115s105s111s110s41s32s107s61s48s46s53s32s107s61s49s46s52s32s107s61s49s32s40s66s105s97s120s105s97s108s32s116s101s110s115s105s111s110s41s86s111s108s117s109s101s32s102s114s97s99s116s105s111s110s32s111s102s32s61537s39s32s109s97s114s116s101s110s115s105s116s101s86s111s110s32s77s105s115s101s115s32s101s113s117s105s118s97s108s101s110s116s32s115s116s114s97s105s110s8226s84s32s61s32s45s49s48s111s67s61555s32s61s49s53s32s77s80s97s46s115s45s49(b)Figure 2.16: Kinetics of two austenitic stainless steels for di erent loadingcombinations. (a) Grade 301LN Quanti cation of kinetics by saturationmagnetic measurements. Adapted from [146]. (b) 18 10 austenitic stainlesssteel for various ratios of principal stresses (denominated by k). Adaptedfrom Lebedev [151].paths adds extra degrees of complexity that make their direct comparisonagainst monotonic, proportional loading routes (e.g. uniaxial tension, uni-axial compression or shear) impossible. Recent work has started the taskof examining the e ects of controlled non-monotonic deformation paths onthe strain-induced transformation behaviour [158, 159], however, without aproper understanding of the physical mechanisms linking the deformed stateto the rate of martensitic transformation, a physical understanding of theseresults is very di cult.As a  nal comment on the relationship between the imposed deformationpath and strain-induced martensitic transformation, there has been recentlya return to the application of the Patel and Cohen model (described in sec-tion 2.2.3) in an attempt to explain the observation of variant selectionof  0-martensite. It is observed in metastable stainless steels that, out ofthe 24 possible crystallographic variants of  0-martensite that could formwith a Kurdjumov-Sachs orientation relationship, only a small fraction areactually observed. As with the Patel and Cohen model described abovefor thermal martensite and stress-induced martensite, the most favourable312.3. Factors In uencing the Rate of Strain-Induced Martensitic Transformation in Austenitic Stainless Steelscrystallographic variants of  -martensite and  0-martensite could be deter-mined based upon the interaction energies calculated between the shapechange associated with each martensite variant and the imposed stress state[10, 50, 160]. It has been shown, however, that the application of thesemodels requires arbitrary cuto s to be applied so as to limit the numberof predicted variants. In a recent publication [161], the number of thesecuto s has been reduced through simple geometrical arguments consistentwith experimental observations. Further, in this work, it has been shownthat the variant selection can be made without the need to calculate theinteraction energy. Such predictions could be improved further if the exactmechanism(s) of transformation could be better understood.322.4. Modelling of the Kinetics of the Strain-Induced Phase Transformations2.4 Modelling of the Kinetics of theStrain-Induced Phase TransformationsIn the previous section, it has been shown that the strain-induced marten-sitic transformations depend in a complex, and not well-understood way, ona number of microstructural and test parameters. In order to build physi-cal models for the transformation kinetics and subsequently the mechanicalresponse of these materials, one must attempt to capture at least the basicphysical observations consistent with experiments. To date, many of themodels proposed in the literature for the kinetics of the strain-induced mar-tensitic transformations have been empirical in nature. Table 2.4 lists someof the more common empirical models applied in the literature for capturingthe volume fraction of  0 as a continuous function of strain. Such modelsrely on their  tting parameters to empirically capture the e ects of variablessuch as temperature, strain rate, grain size and strain path. Such modelsdo not, however, allow for the complex behaviours illustrated in section 2.3where path changes must be accounted for. It is also important to note thatthe role of the  ! transformation is not explicitly accounted for in thesemodels.Reference Model YearLudwigson-Berger [162] f 0 = 11+ 1k p1969Gerberich [163] f 0 = A 12 1970Guimaraes [164] f 0 = 1 exp( kg z) 1972Sugimoto [165] f 0 = 1 exp( ks ) 1992Pychmintsev [166] f 0 = 1 exp( (ks  hPh) ) 2002Shin [167] f 0 = f 0sat(1 exp(  (   0)n)) 2003Table 2.4: Empirical models used to describe the volume fraction of strain-induced  0-martensite. All parameters are determined from  tting.332.4. Modelling of the Kinetics of the Strain-Induced Phase Transformations2.4.1 Review of the Olson-Cohen Model for TransformationKineticsThe most widely used physically based model is that of Olson and Cohen[81]. The Olson and Cohen (O-C) model is based on the assumption thatthe transformation kinetics are dictated by the rate at which nucleation sitesfor  0 are formed. It is assumed in this model that the nucleation sites areformed via plastic deformation, thus giving the link to the degree of plasticstrain.In the development of the O-C model, the nucleation sites for  0 weredescribed as being \shear band intersections". Neither the nature of thesefault band intersections (i.e. whether they are  plates or some other planarfeature induced by deformation), nor their speci c mechanism of formationare described explicitly by the model. Instead, it is simply assumed that thevolumetric rate of formation of fault bands is constant with strain. Takinginto consideration the evolution of the fraction of austenite then gives therate of shear band formation as:dfsbd =  (1 fsb) (2.6)where fsb is the volume fraction of shear bands and  is their rate offormation, assumed constant with strain. It is expected that this parametershould depend upon stacking fault energy and strain rate, since both in u-ence the formation of planar faults. The (1 fsb) term accounts for the factthat the fraction of non-faulted material is reduced as straining continues.From Equation 2.6, the number of shear bands per unit volume can becalculated assuming a constant volume per shear band ( vsb). The numberof shear band intersections is thus calculated from geometry as,NIv = K(Nsbv )n (2.7)where K is a constant that depends on the austenitic grain size (D).Quantitative stereology predicts that K =  2D216 and n = 2 for shear bandsthat are randomly oriented \thin" plates. Olson and Cohen argued that,342.4. Modelling of the Kinetics of the Strain-Induced Phase Transformationsas shear bands tend to align parallel to each other, they can no longer beconsidered randomly oriented, which would lead to a di erent value for n.They  nally chose n = 4:5 from  tting the experiments of Angel [142].It is assumed in the O-C model that there is a probability, p, that anintersection of shear bands will give rise to the nucleation of  0-martensite.Thus, the number of shear band intersections is related to the number of  0nuclei as,dN 0v = p dNIv (2.8)In this case, the probability p should be related to the chemical drivingforce for the  ! 0 transformation. Olson and Cohen assume a Gaussiandistribution of these probabilities with the chemical driving force (thereforewith temperature). The cumulated distribution is:P = 1p2 Z g 1exp" 12 p  psp 2#dp (2.9)where  p and sp represent the mean and the standard deviation of the distri-bution, determined by best  t.Finally, one obtains the following equation for the volume fraction of 0-martensite (f0 ) as a function of the true plastic strain  :df 0d =  v 0dN 0vd (1 f 0) (2.10)where  v 0 is the critical nucleus size for the nucleated martensite.After integration, this leads to:f 0 = 1 exp(  (1 exp(   ))n) (2.11)with = PK  v 0( vsb)n (2.12)One can directly link the parameter  with the saturated volume fraction352.4. Modelling of the Kinetics of the Strain-Induced Phase Transformationsof  0 (fsat) attained as true strain tends to in nity, = ln(1 fsat) (2.13)The maximum rate of nucleation of  0-martensite, is itself a function ofboth  and  .The importance of  -martensite as a potential site for the nucleation for 0 was demonstrated in section 2.2.3. While the physical basis for the O-Cmodel is far more sound than that of the models presented in Table 2.4, itremains that the basic model for the formation of \shear bands" in the O-Cmodel is itself largely empirical. The hypothesis of constant rate of nucle-ation of shear bands has never been validated. This makes it di cult tolink nucleation of  0 with experimental volume fraction of  -martensite. Ol-son and Cohen themselves pointed out that their choice for the  parameterdoes not match the volume fraction evolution of  -martensite experimentallydetermined by Guntner and Reed [168].2.4.2 In uence of External Parameters on the Olson-CohenModelAlthough the O-C model succeeds in reproducing the sigmoidal shape of themonotonic martensite fraction curve with strain and provides a signi cantphysical basis, its basic formulation requires extension to consider the e ectsresulting from changes in microstructure and testing conditions, as presentedin section 2.3.As an example, consider the in uence of grain size on the transformationkinetics for  0. Assuming  and n not to depend upon grain size, thegrain size dependence predicted by the O-C model comes from the factthat  is proportional to D2. This dependence is shown in Figure 2.17, forsome conventional values of  and n. According to this plot, decreasing theaustenitic grain size tends to decrease the saturated volume fraction of  0.This e ect has been discussed by Iwamoto [169] and is consistent with theargument advanced in section 2.3 that grain re nement makes the austenitemore stable. Unfortunately, however, there is little experimental data in the362.4. Modelling of the Kinetics of the Strain-Induced Phase Transformationss48s46s48s46s49s48s46s50s48s46s51s48s46s52s48s46s53s48s46s48s46s50s48s46s52s48s46s54s48s46s56s49s46s48s32s110s61s50s32s110s61s52s46s53s70s114s97s99s116s105s111s110s32s111s102s32s61537s39s84s114s117s101s32s115s116s114s97s105s110s51s48s32s61549s109s50s32s61549s109s48s46s53s32s61549s109Figure 2.17: Illustration of the grain size dependence on the Olson-Cohenequation, with  = 8. The sensitivity of the grain size dependence increaseswhen n is changed from 2 ("theoretical value") to 4.5 ("Olson and Cohenvalue").literature with which to compare the O-C predicted grain size dependence ofthe transformation rate. However, if one considers typical values of n usedto  t in the limit of large grain sizes (above 10 a181m), then one can see thatthe grain size dependence would be predicted to be very signi cant. Sucha large grain size dependence does not seem to have been reported in theliterature.Yet, the model of Olson and Cohen could be reinterpreted to observe theopposite dependence upon grain size. It is implied by Olson and Cohen thatthe shear bands cross the entire grain, meaning that  vsb = tD2 where t isthe thickness of the shear bands. Substituting this into Equation 2.12 givesa grain size dependence of  / D2D2n. Thus, for n> 1, the O-C model wouldpredict an increasing rate of transformation with decreasing grain size, incontradiction with Figure 2.17 and in contradiction with most experimentspresented in section 2.3.From the perspective of building physical models, one would like to be372.4. Modelling of the Kinetics of the Strain-Induced Phase Transformationsable to directly predict the results for di erent alloy compositions withouthaving to resort to re- tting of parameters for each composition change.Moreover, a model incorporating directly the thermodynamics of the phasescould be valuable for alloy design. Olson and Cohen noted, for example,that substitution of chromium and manganese by nickel could help reducingthe strong temperature-sensitivity of the kinetics of nucleation of  0, basedon qualitative trends they observed in their model  t to experiments. In theO-C model, the chemical composition only appears indirectly through the tting parameters (namely  ,  and n), making the comparison/modellingof the kinetics of two di erent grade of steels di cult. Related to composi-tion is the temperature dependence of the material behaviour, which a ectsboth the driving force for the transformation as well as the mechanisms ofplasticity (via stacking fault energy). Since  is expected to vary inverselywith the stacking fault energy and  is linked to  G ! 0, both  and  should decrease with temperature, as illustrated in Figure 2.18.s45s50s48s45s49s53s48s45s49s48s45s53s48s48s53s48s49s48s48s49s50s61538s84s101s109s112s101s114s97s116s117s114s101s32s40s111s67s41s45s50s48s45s49s53s48s45s49s48s45s53s48s48s53s48s49s48s48s53s49s48s61537Figure 2.18: Illustration of the temperature dependence on the two Olson-Cohen parameters (namely  and  ). Adapted from [81].The O-C model attempts to directly link the rate of transformation tonucleation sites formed due to plasticity (shear band intersections). One382.4. Modelling of the Kinetics of the Strain-Induced Phase Transformationswould expect that such features formed via plasticity should have an e ectas well on the  ow stress and the work-hardening of the austenite. There hasbeen, however, no attempt to make a direct link between these two featuresof the material response. Related to this is the fact that the O-C model isnot well-suited to capture e ects arising from non-monotonic testing. If theO-C model is used in its integrated form (Equation 2.11), then there is noability to deal with behaviours such as that presented in Figure 2.12 wherestraining is continued at a higher temperature after a pre-strain performed ata lower temperature. In this case, a more sophisticated linkage between thestrain-induced nucleation sites, the mechanical behaviour of the austeniteand the transformation kinetics of  ’ would be needed.As with the e ect of composition and temperature, the in uence of stressstate on the rate of transformation was not explicitly outlined in the originalO-C model. However, the observations presented in section 2.3 relating tovariant selection and the stress dependence of the Ms temperature indicatethat stress state should be explicitly incorporated into a model of  0 kinetics.In this direction, the O-C model has been re ned by Stringfellow to includethe e ect of triaxiality 6 [170, 171]. Stringfellow assumes a distribution ofpotencies for \shear band intersections" to form  0, this distribution beinga function of temperature and triaxiality through a parameter g, whichrepresents a net thermodynamic driving force:g = g0 g1T +g2  g3 3 (2.14)where g0, g1 and g2, g3 are dimensionless constants, T is the temperature,and  is the triaxiality. In the case of high triaxiality ratios, as can be foundin crack tips, Stringfellow recommends using the last term in  3, which canbe ignored otherwise.The driving force de ned by Equation 2.14 is the basis for the distribu-tion of the probability that an intersection forms  0-martensite. In parallel6 The triaxiality is de ned here as the ratio of the volumetric and deviatoric stressinvariants:  =  p where p is the hydrostatic pressure and  the Von Mises equivalentstress.392.4. Modelling of the Kinetics of the Strain-Induced Phase Transformationsto Equation 2.9, Stringfellow wrote the cumulative probability as:P = 1p2 Z g 1exp" 12 g0  gsg 2#dg0 (2.15)with  g and sg the mean and the standard deviation of the distribution,determined by best  t. This probability is proportional to the  parameterwhich appears in Equation 2.11.Stringfellow mentions an attempt to model the full stress tensor, insteadof simply the triaxiality, but this resulted in too much transformation athigh levels of stress, an e ect attributed to the di cult propagation of  0when the austenite work-hardens.It is worth comparing the Stringfellow model to the Patel-Cohen modelmentioned in section 2.2.1. While the Stringfellow model predicts that uni-axial compression lowers the rate of transformation (cf. Figure 2.19), thePatel-Cohen model shows that both uniaxial tension and uniaxial compres-sion favour the transformation (cf. Figure 2.3). This di erence comes fromthe fact that the shear component is not taken into account by Stringfellow,while it often dominates in the Patel-Cohen model. This observation ques-tions the relevance of the triaxiality as unique parameter to describe complexstress states. Another scalar variable, known as the Lode parameter, wasproposed to complement the triaxiality [139].Based on the fact that compression would form more shear bands atlow strains, Iwamoto, Tomita and Tsuta re ned the Stringfellow model byadding the triaxiality ratio into the rate of increase of shear bands [152]. Todo this, the  parameter which appears in Equation 2.11 was written as: =  1 + 2T + 3T2  4 (2.16)in which  4 = 0 returns the Stringfellow model.As noted by Serri et al., there is a drastic di erence between the kineticspredicted either by Stringfellow or by Iwamoto [172]. Iwamoto’s model tendsto delay the formation of shear bands in equibiaxial tension, an observationcontrary to the experimental data presented in Figure 2.16.402.4. Modelling of the Kinetics of the Strain-Induced Phase Transformationss48s46s48s48s46s50s48s46s52s48s46s54s48s46s56s48s46s48s48s46s50s48s46s52s48s46s54s48s46s56s49s46s48s112s108s97s110s101s45s115s116s114s97s105s110s32s99s111s109s112s114s101s115s105s111s110s115s105s109s112s108s101s32s99s111s109s112s114s101s115s105s111s110s115s104s101s97s114s115s105s109s112s108s101s32s116s101s110s115s105s111s110s86s111s108s117s109s101s32s102s114s97s99s116s105s111s110s32s111s102s32s61537s39s32s109s97s114s116s101s110s115s105s116s101s69s113s117s105s118s97s108s101s110s116s32s112s108s97s115s116s105s99s32s115s104s101s97s114s32s115s116s114s97s105s110s112s108s97s110s101s45s115s116s114s97s105s110s32s116s101s110s115s105s111s110Figure 2.19: Illustration of the stress state dependence on the Stringfellowmodel. Adapted from [171].The rate of the strain-induced phase transformation might as well havea stress-dependence. Despite some recent work [159], this issue remainsunresolved at present.To summarize, in all models of the  ! 0 transformation kinetics, thein uence of grain size on the kinetics of the  ! 0 phase transformation istreated in the same way as originally proposed in the O-C model, yet thistrend has never been systematically studied experimentally. Many studieshave considered changes of strain path via the triaxiality ratio. This param-eter only accounts for variations of the hydrostatic stress. Despite abundantamount of data relating to changing strain paths, the incorporation of theshear components into a model of the  ! 0 transformation kinetics hasnot advanced signi cantly since the work of Patel and Cohen (1953) [47].412.5. Mechanical Response of Austenitic Stainless Steels2.5 Mechanical Response of Austenitic StainlessSteelsThe strain-induced martensitic phase transformations described above re-sults in a microstructure that evolves strongly during plastic deformation.As the microstructure evolves from being fully austenitic to containing amixture of  0-martensite,  -martensite and austenite, the bulk mechanicalproperties of the material vary as well. This evolution creates a materialwhose properties are not necessarily given by a volume fraction weightedcombination of the properties of the individual phases in bulk form. One canalso incorporate the synergistic e ects arising from one phase on the others(e.g. the accumulation of transformation-induced dislocations in austenitedue to the formation of  0-martensite), as well as other complex features,such as the size and morphology of the martensitic phases. Despite therebeing a signi cant amount of experimental data published in the literature(see for instance the surveys of Powell [153] and Llewellyn [17]) on the bulkmechanical response of metastable austenitic stainless steels, our basic un-derstanding of the physical origins of the mechanical response remains acomplex and challenging topic.2.5.1 Bulk Mechanical ResponseIn most fcc metals, the yield stress is weakly dependent upon temperature,with the main variation coming from the temperature-dependence of theshear modulus ( ) [173]. In contrast, austenitic stainless steel present ahigh-temperature dependence of yield stress, as one can see in Figure 2.20.Despite this strong temperature dependence of the yield strength, thegrain size dependence of yield stress is found to generally follow the Hall-Petch relation [175, 176] for grain sizes ranging from approximately 1 a181mupwards. Some measurements for the Hall-Petch parameters of di erentstainless steels are given in Table 2.5 and compared to the Hall-Petch con-stants for other fcc metals. The important feature of Table 2.5 is that therecan be a large dispersion in the values given for the Hall-Petch slope, this422.5. Mechanical Response of Austenitic Stainless Steelss45s50s48s45s49s48s48s49s48s50s48s50s48s51s48s52s48s53s48s54s48s55s48s56s48s32s69s120s112s101s114s105s109s101s110s116s97s108s32s121s105s101s108s100s32s115s116s114s101s115s101s115s32s65s108s97s105s110s39s115s32s102s105s116s89s105s101s108s100s32s115s116s114s101s115s32s40s77s80s97s41 s84s101s109s112s101s114s97s116s117s114s101s32s40s111s67s41Figure 2.20: E ect of the temperature on the yield stress (determined by0.2% o set method) of a 301LN stainless steel. The experimental datapoints are from Nanga [14],  tted with the equation proposed by Allain foraustenitic Fe-Mn-C steels [174].feature being a re ection of the di culty to reproduce grain size measure-ments.Beyond yield, the evolution of the  ow stress with deformation (the work-hardening) is sensitive to a large variety of parameters including strain rate,test temperature and deformation path. The dependency on these varia-tions are often largely determined by the way in which the test parameterschange the rate of transformation. The uniaxial tension stress-strain curvesof a 316L stainless steel are plotted as a function of test temperature inFigure 2.21(a) [72]. The stress-strain curve at 177 C represents the be-haviour of a fully austenitic material as neither  0 nor  -martensite form atthis temperature. One may also note that, at this temperature, serrationsin the stress-strain curve appear at higher stresses. These serrations are acommon feature in many austenitic steels (stainless as well as non-stainlessalloys [87]) and are associated with dynamic strain-aging. Various explana-tions for the dynamic strain-aging in austenitic steels have been proposed432.5. Mechanical Response of Austenitic Stainless SteelsMaterial Ref. Friction stress Hall-Petch slope 0 (MPa) ky (MPa.a181m1=2)301LN [177] 252 274304 [178] 220 492316L [179] 200 116316L [180] 164 621316L-modi ed [181] 144 580Copper [182] 26 110Nickel [183] 15 238Table 2.5: Hall-Petch parameters determined at room temperature for var-ious fcc materials.(e.g. [184, 185]), though this is still an active area of research.The TRIP e ect manifests itself in this material for temperatures be-low 25 C. The e ect of the transformation to  0 can be recognized fromthe in ection in the stress-strain curve and the consequent increasing rateof work-hardening, at intermediate levels of strain. The e ect of the ap-pearance of strain-induced martensite can be more easily seen in a plot ofwork-hardening rate as presented in Figure 2.21(b).s48s46s48s46s49s48s46s50s48s46s51s48s46s52s48s46s53s48s46s54s53s48s49s48s49s53s48s50s48s48s37s32αs39 s57s37s32αs39s57s48s37s32αs39s49s55s111s67s50s53s111s67s45s54s48s111s67s84s114s117s101s32s115s116s114s101s115s32s40s77s80s97s41 s84s114s117s101s32s115s116s114s97s105s110s45s49s57s54s111s67s56s37s32s39(a)s48s53s48s49s48s49s53s48s50s48s53s48s49s48s48s45s49s57s54s111s67s45s54s48s111s67s50s53s111s67s87s111s114s107s45s104s97s114s100s101s110s105s103s32s114s97s116s101s32s40s77s80s97s41 s84s114s117s101s32s83s116s114s101s115s32s40s77s80s97s41s49s55s111s67s67s111s110s115s105s100s232s114s101s99s114s105s116s101s114s105s111s110(b)Figure 2.21: E ect of the temperature on the tensile curves of a 316L stain-less steel. Adapted from Spencer [72].One of the bene ts of the increasing rate of work-hardening is that it442.5. Mechanical Response of Austenitic Stainless Steelsallows for the onset of strain localization in tension, as determined by theConsid ere criterion,@ @   (2.17)to be suppressed until high strains and stresses. This results in materialsthat have very large combinations of uniform elongation and ultimate tensilestrength.Figure 2.22 illustrates that a similar behaviour is exhibited by grade301LN [186]. In this alloy, however, the formation of  0 during strainingbecomes signi cant at higher temperatures compared to alloy 316L, a con-sequence of the lower stability of austenite in this alloy. For temperaturesbelow -40 C, Figure 2.22 shows the presence of an upper yield point and alower yield point, as well as a long plateau of 10% strain. It was observedthat this corresponds to the formation of a L uders-like band during defor-mation. Only the volume swept by the band in which strain localizationoccurs was found to form  0-martensite. Similar behaviour was obtained inthe condition described in section 2.3 and whose tensile curve is reproducedin Figure 2.23, in which the material was pre-strained at a certain tempera-ture and then was reloaded at a lower temperature [72]. Additionnally, suchdiscontinuous yielding has also been reported in submicron-grained 304 or316 grades, and the magnitude of the L uders strain was found to increasewhen the temperature was lowered from 25 C to -196 C [187]. The explana-tion of this phenomenon is not well understood [187], although recent workhas suggested the possibility that the strong increase of yield stress withdecreasing temperature may lead to the yield stress rising above the levelnecessary to initiate a truly stress-assisted martensitic transformation [10].The role that grain size plays in modifying the work-hardening responseof austenitic stainless steel is also complicated by the fact that grain sizeimpacts both the mechanical response of austenite as well as the kinetics ofmartensitic transformation, as seen from section 2.3. In the case of stableaustenitic stainless steels, several researchers [179, 180, 188, 189] have takenan empirical approach to explaining the grain size dependence through anextended Hall-Petch expression where the Hall-Petch parameters are taken452.5. Mechanical Response of Austenitic Stainless Steelss48s46s48s46s49s48s46s50s48s46s51s48s46s52s48s46s53s48s53s48s49s48s48s49s53s48s50s48s48 s51s37s32αs39s55s53s37s32αs39s57s54s37s32αs39s49s48s37s32αs39s45s52s48°s67s56s48°s67s50s51°s67s84s114s117s101s32s115s116s114s101s115s32s40s77s80s97s41s84s114s117s101s32s83s116s114s97s105s110s45s49s53s48°s67Figure 2.22: E ect of the temperature on the tensile curves of a 301LNstainless steel. Adapted from Nanga [14].to be strain-dependent. A more physical approach to understanding thegrain size dependence of the work-hardening rate of alloy 316L (in whichno martensitic transformation occurred under the testing conditions) wasperformed by Feaugas and Haddou [190]. In this work, the grain size depen-dence of the work-hardening rate was attributed primarily to an increase inthe long-range stresses arising from dislocation pile-ups at grain boundaries.Accordingly, it was argued that the grain size dependence, and thereforekinematic hardening, is higher for lower stacking fault energy alloys. Thisis in qualitative agreement with a recent model proposed for the grain sizedependent work-hardening of fcc metals [191].Varying the strain path has an e ect on the stress-strain response of aus-tenitic stainless steels. The reason for this change in stress-strain behaviourhas been attributed to di erent reasons, e.g. variation of transformationkinetics (cf. section 2.3) and di erent texture evolution [152, 157]. Two ex-amples of strain path dependence of the stress-strain behaviour of austeniticstainless steels are given in Figure 2.24. Very often, the relative position ofthe stress-strain curves is consistent with the measured kinetics, the gradesforming more  0 at a given equivalent strain experiencing higher hardening462.5. Mechanical Response of Austenitic Stainless Steelss48s46s48s48s46s49s48s46s50s48s46s51s48s46s52s48s50s48s52s48s54s48s56s48s49s48s48s49s50s48s49s52s48 s45s49s57s54s111s67s84s114s117s101s32s115s116s114s101s115s115s32s40s77s80s97s41s84s114s117s101s32s115s116s114s97s105s110s50s53s111s67s67s111s110s115s116s97s110s116s32s108s111s97s100Figure 2.23: Tensile behaviour of 304L pre-strained at 25 C and furtherdeformed at -196 C. The reloading at -196 C was associated with the prop-agation of a band in which the strain was localized. Adapted from Spencer[72].[151]. This is evident when comparing Figure 2.16 to Figure 2.24. In Fig-ure 2.24(a), shear and tension initially display similar work-hardening beforediverging for an equivalent true strain of  18%, whereas the di erence be-tween strain path manifested itself much earlier (almost after yielding) inFigure 2.24(b). This inconsistency also appears in the rate of the kinetics,and can be observed when comparing Figure 2.16(a) and Figure 2.16(b).2.5.2 The Intrinsic Mechanical Response of Austenite andMartensiteOne of the most common methods for analyzing the bulk mechanical re-sponse of metastable stainless steels is to consider the bulk response ascoming directly from the weighted response of each of the individual phases.This requires a knowledge of the mechanical response for each of the phasespresent in the microstructure.472.5. Mechanical Response of Austenitic Stainless Steelss48s46s48s46s49s48s46s50s48s46s51s48s46s52s50s48s52s48s54s48s56s48s49s48s49s50s48s49s52s48s84s32s61s32s50s51s111s67s83s105s109s112s108s101s32s115s104s101s97s114s86s111s110s32s77s105s115s101s115s32s101s113s46s32s115s116s114s101s115s32s40s77s80s97s41s86s111s110s32s77s105s115s101s115s32s101s113s117s105s118s97s108s101s110s116s32s115s116s114s97s105s110s85s110s105s97s120s105s97s108s32s84s101s110s115s105s111s110(a)s48s46s48s48s46s53s48s46s49s48s48s46s49s53s48s46s50s48s50s48s52s48s54s48s56s48s49s48s49s50s48s32s107s62s49s32s40s85s110s105s97s120s105s97s108s32s116s101s110s115s105s111s110s41s32s107s61s48s46s53s32s107s49s46s52s32s107s61s49s32s40s66s105s97s120s105s97s108s41s86s111s110s32s77s105s115s101s115s32s101s113s46s32s115s116s114s101s115s32s40s77s80s97s41s86s111s110s32s77s105s115s101s115s32s101s113s117s105s118s97s108s101s110s116s32s115s116s114s97s105s110s8226s84s32s61s32s45s49s48s111s67s61555s32s61s49s53s32s77s80s97s46s115s45s49(b)Figure 2.24: Stress-strain curve of two austenitic stainless steels for di er-ent loading combinations. (a) 301LN adapted from Nanga [146], (b) 18-10austenitic stainless steel for various ratios of principal stresses (denominatedby k). Adapted from Lebedev [151].Although  -martensite is formed in many austenitic stainless steels, itscontribution has largely been ignored in descriptions of the bulk mechanicalresponse of these materials. This is normally justi ed based on the fact that -martensite represents only a small fraction of the microstructure [192]. Arecent study by Hedstr om [86] combined uniaxial tensile experiments withX-ray di raction to monitor the elastic strain in  -martensite during in-situstraining experiments. Over the relatively small range of plastic strainsstudied it was found that the  -martensite had a very low work hardeningrate (Figure 2.25).While there are few systematic studies examining the relationship be-tween the presence of  -martensite and mechanical properties in austeniticstainless steels , there are a large number of studies aimed at correlatingthe bulk mechanical response with the presence of  0-martensite. In orderto understand the relative contributions to the overall work-hardening ratecoming from austenite and 0-martensite, several experiments have been per-formed in an attempt to directly assess (for a given deformation and levelof martensite) the stresses carried by each of the phases. The stress-strainresponse of austenite in the absence of martensite is not directly measurable482.5. Mechanical Response of Austenitic Stainless SteelsFigure 2.25: Measured elastic strain evolution along the tensile directionplotted for two individual austenite grains (13 and 18), along with the av-erage X-ray elastic strain. Grain 13 transformed fully to  -martensite. Re-produced from Hedstr om [86].for conditions where the strain-induced transformations occur, since strain-ing of the austenite a ects the phase transformation. One possibility is toperform mechanical tests at temperatures where no strain-induced marten-sitic transformations are observed (e.g. [193]) and infer the response of theaustenite at lower temperatures. Of course, the strong temperature sensi-tivity of the stacking fault energy (among other properties) for these steelsmakes this extrapolation di cult. Similarly, it is not possible to measuredirectly the bulk mechanical response of the martensitic phase since it isonly formed by straining of the austenite. It is, however, possible to makeindirect measurements that provide estimates for the stress borne by theaustenite and/or martensite phases.In the work performed by Spencer et al., neutron di raction was usedto measure the elastic lattice strains in alloy 316L [72, 194]. In this case,316L test coupons were pre-strained at -196 C so as to obtain a signi -cant fraction of  0-martensite, then tensile tests were continued at roomtemperature under neutron irradiation for di raction experiments. By con-tinuing the testing at room temperature, the fraction of martensite in the492.5. Mechanical Response of Austenitic Stainless Steelsmicrostructure was found to not vary signi cantly beyond the  rst few per-cent strain. In these experiments, the load partitioning in the case of astatic phase fraction could be estimated. As shown in Figure 2.26, it wasobserved that  0-martensite represents a strong reinforcing phase (i.e. hav-ing a higher yield strength than the austenitic matrix) but that the phasesexhibited signi cant plastic co-deformation. Moreover, the apparent rateof work-hardening arising from the  0-martensite was only slightly higherthan the macroscopic work-hardening rate. This implies that the strongincrease in work-hardening rate associated with the TRIP e ect comes notfrom an intrinsically high work-hardening rate of the  0-martensite but in-stead, from the contribution of the evolution of the phase fraction to thework-hardening.Figure 2.26: Stress level in austenite and  0-martensite phases measured byneutron di raction in grade 316L. Reproduced from Spencer [112].Other di raction based studies of the stresses borne by the two phaseshave been carried out using neutron and X-ray di raction under conditionswhere the phase fraction evolves with strain. One example is representedin Figure 2.27, from the work of Talonen [34]. In this case, one observes rst that the di raction-based estimate of the stress borne by the austenitematches the macroscopic stresses over the range of strains where austenite502.5. Mechanical Response of Austenitic Stainless Steelsis the only phase present. Once the fraction of  0 becomes signi cant how-ever, the macroscopic  ow stress is seen to evolve away from the austenite ow curve. While it is impossible to make quantitative comparisons betweenthese results and those from the work of Spencer (Figure 2.26), due to di er-ences in the testing methodology and alloy, one can note that the observedhardening rates for the  0-martensite measured in both cases are similar, asare the hardening rates for the austenite. These results are also very similarto those obtained under similar conditions by Berrahmoune [195] using X-ray di raction, and to the neutron di raction measurements performed byDufour [13].Figure 2.27: Stress level in austenite and  0-martensite phases measuredfrom X-ray di raction stress measurements in grade 301LN. Reproducedfrom Talonen [34].Further analysis of such di raction based data can be used to give deeperinsight into the deformation behaviour of the two phases. Talonen [34]used the results of peak broadening from X-ray di raction measurementsin an attempt to estimate the dislocation density in the austenite and  0-martensite phases in a 301LN alloy. Figure 2.28, from this work, shows thetotal  ow stress as well as the stress in the austenite phase as measured512.5. Mechanical Response of Austenitic Stainless Steelsfrom di raction peak shifting versus the square root of dislocation densityin austenite.Figure 2.28: Stress level in the austenite phase of grade 301LN, measured byX-ray di raction as a function of square root of dislocation density of aus-tenite determined by Integral Breadth Method. Reproduced from Talonen[34].The results show that the Taylor equation relating  ow stress to dislo-cation density: =  0 + T bp (2.18)appears to be well obeyed for the austenite phase. Similar estimations ofthe dislocation density where made for the  0 phase resulting in a high dislo-cation density (  0 6 14 1014 m 2 compared to    1 6 1014 m 2 foraustenite) which did not vary substantially with plastic strain. This disloca-tion density is similar to that found by Narutani [196] and is consistent withthe qualitative TEM observations by Spencer [72] who explained the highdislocation density in the martensite as likely being a consequence of dislo-cations in the prior austenite being incorporated into  0 during the trans-formation. Yet, one must be careful with the interpretation of these results.Beyond the di culties of dislocation density estimates from peak broaden-ing in simple single phase materials, the interpretation of peak broadening inthe  0 phase here are complicated by the fact that the  0 forms progressivelywith strain, meaning that the  0 has a large and continuous distribution ofplastic strains.522.5. Mechanical Response of Austenitic Stainless SteelsThese results give the overall view that both austenite and  0 phasesundergo signi cant plastic co-deformation. However, the  0 phase is seen todeform with a signi cantly higher  ow stress than the austenite, though theapparent work-hardening rate of the two phases is similar.2.5.3 Modelling of the Overall Mechanical Response ofAustenitic Stainless SteelsAs noted above, it is common to interpret the overall mechanical responseof austenitic stainless steels as that of a composite having a dynamicallyevolving phase fraction. According to this approach, the overall response ofthe material may be predicted, provided constitutive laws for the phases canbe identi ed and a scheme for homogenization (i.e. the method to de nethe stress and strain partitioning between phases) can be selected. Amongthe di erent studies that have sought to model the mechanical responseof metastable austenitic stainless steels, the approaches can be generallyseparated into those coming from a mechanics or a materials background.Mechanics-based models for the mechanical response of austenitic stain-less steels are very prevalent in the literature owing to practical application ofthese models in simulations for the forming and in-service mechanical prop-erties of these materials. The general approach, in this case, is to developtensorial expressions for the yield surface and its expansion with strain.Sophisticated homogenization schemes (e.g. tangent [197] or secant [198]self-consistent approaches, Mori-Tanaka [199, 200], or  nite element simula-tion) can be used to obtain the net response based on the constitutive lawsof the individual phases. The constitutive models for the individual phases,as well as the transformation kinetics in these models, however, tend to in-clude many empirical  tting parameters such that the behaviour laws can bemade to coincide with experimental results for a wide range of experimentalobservations.An example of this type of approach is given by the work of Iwamotoand Tsuta’s [169], which aims to describe both the transformation kineticsand the mechanical response of metastable stainless steels as a function of532.5. Mechanical Response of Austenitic Stainless Steelstemperature, strain rates and grain size. In this model, constitutive laws foreach phases are described by: I =  0(I)"_ pslipI_ y#m 0(I) =  y(I) +C1(I)n1 exp  C2(I) pslip(I) oC3(I) y(a) = C4(a) exp  C5(a)T +ky dd0  1=2 y(m) = C4(m) exp  C5(m)T (2.19)where I can be replaced by \a" in case of austenite and \m" in case ofmartensite, where m is the strain rate sensitivity exponent and where C1(I)to C5(I) are material constants. The kinetics of the phase transformation to 0-martensite are described by the Tomita-Iwamoto model [201] presentedin section 2.4.2, including strain rate sensitivity. The homogenization isperformed using a  nite element simulation. To account for any arbitrarystress state, a yield function needs to be de ned. In this particular case, thefollowing has been used: =s3 J2  J3J21=2 (2.20)One can separate the work-hardening into an isotropic component (obey-ing the yield function above) and a kinematic component. In the case ofthe Iwamoto-Tsuta model, the hardening has been assumed to be purelyisotropic (as in many of the mechanics based models) though the presenceof phases having di erent  ow stresses will necessarily result in stress parti-tioning and a kinematic hardening component [158].As an illustration of the  t between experiment and model, the calibratedIwamoto model is shown in Figure 2.29 illustrating its ability to capture theexperimentally observed grain size dependence of the mechanical response.Such a model has the advantage of not making particular assumptionson how the stresses and/or strains are partitioned. However, 24  tting542.5. Mechanical Response of Austenitic Stainless SteelsFigure 2.29: Simulated true stress and work-hardening curves obtained forroom-temperature tension in various austenitic grains. Reproduced from[169].parameters need to be estimated from experiments, with no explicit referenceto the deformation microstructures or many of the other physically-basedobservations described in previous sections of this review.An alternative approach, often less rigorous in terms of mechanics, is touse simple one-dimensional materials-based modelling approaches in whichthe constitutive laws are more generally physical in nature and attempt toexplicitly account for microstructure. These models may use simpler ho-mogenization laws than the mean- eld approaches described above. For in-stance, many authors [58, 112] have used the assumption of uniform strains,the so-called Taylor assumption, which constitutes the upper-boundary ofhomogenization.As noted above, the behaviour of austenite appears to obey the Taylorequation implying the dominance of forest hardening. A popular approachto predicting the evolution of dislocation density is via the Kocks-Meckingand Kocks-Mecking-Estrin equations [202]. In the Kocks-Mecking model,the evolution of the dislocation density is modeled as being associated with552.5. Mechanical Response of Austenitic Stainless Steelsa dislocation storage term and an annihilation term. In the Kocks-Mecking-Estrin model, the Kocks-Mecking equation is expanded to include a secondstorage term associated with the storage of geometrically necessary dislo-cations due to plastic strain gradients. This approach has been successfulin describing the work-hardening in many fcc materials [202] and has beenalso shown to reproduce the work-hardening rate in stable grades of stain-less steels [203] as well as in TWIP steels [204]. In the latter case, defor-mation twins substantially increase the work-hardening rate, and increasethe density of obstacles to planar slip (so-called dynamic Hall-Petch e ect).The presence of these boundaries increases the rate of geometrically neces-sary dislocation storage, while also introducing strong kinematic hardening[205, 206].An example of the Kocks-Mecking-Estrin approach applied to both theaustenitic and martensitic phases is in the work of Bouquerel et al. [207]. Inthis model, the austenitic grain size is considered strain-dependent,d ( ) = dinit (1 f 0)1=3 (2.21)and the dislocation density evolution in austenite is represented by:1Md d =1b 1d ( ) +kp   f (2.22)where k and f are the two Kocks-Mecking parameters representing the rateof accumulation of statistically stored dislocations and the dynamic recov-ery respectively. In this model, it was assumed that the accumulation ofgeometrically necessary dislocations dominated over statistically stored dis-locations.The stress-strain curve of the martensite is described by the Rodriguezand Guttierez relation [208] which is an integrated version of the Kocks-Mecking model:  0 =  0 + M pbs1 exp( Mk2 )k2 (2.23)562.5. Mechanical Response of Austenitic Stainless Steelswhere  is the mean-free path between dislocations within martensite,  isa geometrical constant close to 0.33 and k2 is an adjustable parameter. Thekinetics of the  ! 0 phase transformation is given by the O-C model. Thestress of the composite material is estimated from a Gladman-type mixturelaw, =   (1 f2 0) +  0f2 0 (2.24)with the Taylor assumption (i.e. equal strain) for the strain partitioning.The outputs of the model are represented in Figure 2.30.Figure 2.30: (a) Evolution of the volume fraction of martensite with strainfor the 301LN stainless steel during tensile testing at 20 C. (b) Evolution ofthe calculated austenitic and martensitic grain size during the tensile test.(c) Simulated stress-strain curves for the martensitic and austenitic con-stituents. (d) Experimental and modelled stress-strain curves. Reproducedfrom [207].Such a model has the advantage of describing the stress-strain relationof the two constituents via the same micro-mechanical approach, and it572.5. Mechanical Response of Austenitic Stainless Steelsconsiders the scale reduction in both the austenite and the martensite interms of generating geometrically necessary dislocations. Moveover, unlikeother models which consider the martensite to be a purely elastic phase, themartensite phase in this model is considered to contribute to the mechanicalresponse via plastic co-deformation with the austenite matrix. A criticismof this approach, however, is that it considers that the martensite can bedescribed as a uniform phase, having the same properties throughout. Thisdoes not appear consistent with the fact that the fraction of martensiteis evolving with plastic strain implying that there should be a range ofproperties for the martensite present for any given level of imposed plasticstrains.Other models highlighting the importance of geometrically necessary dis-locations in the strengthening of austenitic stainless steels have also beenproposed (cf. section 2.2.3) [123]. Talonen performed measurements of thechord length distributions of  0 in grade 301LN [34] after di erent levelsof strain. His results, reproduced in Figure 2.31 show that, although the 0 nuclei are initially small, large clusters start appearing very early in thecourse of the deformation. Those measurements can be used to constructa model of dispersion hardening, based on the generation of geometricallynecessary dislocations in the austenite [34, 109].Talonen estimated the density of geometrically necessary dislocationsaccumulated in the austenite phase assuming that the  0-martensite is arigid (non-deforming) phase [34]. The martensite islands were assumed cubicshaped, of volume L3, L being the chord-length presented in Figure 2.31.The density of geometrically necessary dislocations is then given by: G = 4f 0  bL (2.25)with b the Burgers vector of the geometrically necessary dislocations.Talonen found reasonable agreement between the density of geometri-cally necessary dislocations and the dislocation densities estimated from X-ray di raction (Figure 2.28). The fundamental di culty with this approachis that the rate of generation of geometrically necessary dislocations depends582.5. Mechanical Response of Austenitic Stainless SteelsFigure 2.31: Mean chord length of  0-martensite islands in 301LN steel asa function of  0-martensite volume fraction. Reproduced from [34].on the assumption of how much plastic strain is accomplished by the  0 andaustenite phases. The assumption that the  0 phase is a rigid phase is notsupported by the di raction based data shown above (Figure 2.26 and Fig-ure 2.27) nor from in-situ TEM observations where signi cant dislocationactivity in the  0 phase has been observed [72]. Indeed, it has been suggestedthat the partitioning of strain between the austenite and  0 phases might berelated to a percolation threshold, below which most of the imposed strain iscarried by austenite and above which most strain is carried by  0-martensite[34, 109].While describing the mechanical behaviour of austenite appears to bepossible within well-established physically based methodologies (e.g. Kocks-Mecking-Estrin), de ning a constitutive law for the strain-induced 0-martensiteis much more di cult. As noted above, many studies have simply assumedthat the  0-martensite acts as a rigid (or purely elastic), phase though ex-perimental measurements are at odds with this assumption. Further, asdiscussed by Spencer [72], owing to the fact that the  0 phase forms pro-gressively with strain, the  0 formed at one level of strain is formed into avery di erent environment compared to  0 formed at another level of strain.592.5. Mechanical Response of Austenitic Stainless SteelsFor instance, the  rst  0 forms into austenite having a much lower densityof dislocations compared to the  0 formed at larger strain.Spencer has highlighted the composite nature of the material, as wellas the in uence of stress and strain partitioning between the two phases onkinematic hardening [72]. In this work, the role of the transformation strain,i.e. the shape change associated with transforming austenite to  0 (and  )martensite, was noted as:d applied = d(  ! f ) +d   ! 0f 0 + (1 f 0)d  + +f 0d  0 (2.26)where   ! and   ! 0 correspond to the transformation strains associatedwith the transformation from austenite to  and  0-martensite respectively.The terms fi represent the volume fraction of the various phases (i = 0; ; ). Finally, the strain associated with deformation in the austeniteand  -martensite are considered together (  + ). If the rate of transforma-tion is very high, then it was argued that the  rst two terms could dominateleading to a very low work-hardening rate. In this case, it has been arguedthat the transformation strain could explain the observation of L uders likebehaviour described above [72].An equal strain assumption for the plastic response of the austenite and 0 phase was made leading the net work-hardening rate:@ @ = f 0@  0@ + (1 f 0)@  @ +@f 0@ (  0   ) (2.27)One of the important points arising from this equation is that it high-lights the role of the rate of phase transformation df 0=d on the work-hardening rate.In summary, many mechanics based models have been proposed to ex-plain the mechanical response of austenitic stainless steels. These generallyare capable of incorporating the e ects of stress state, but have a weak phys-ical basis when it comes to the constitutive laws of each phase, especially interms of linkage to actual microstructures. Moreover, many of these modelsare based upon purely isotropic models despite experimental measurements602.5. Mechanical Response of Austenitic Stainless Steelsthat show large internal stresses. On the other hand, more physical mod-els have been developed, which capture important microstructural aspectsof the TRIP e ect. One issue is that those are not easy to couple withthe mechanical models mentioned above. What is missing is a way to de-scribe simply the dependence of the mechanical properties of the compositetowards simple variables such as temperature, strain rate, grain size andstrain path. This approach already exists - with degrees of accuracy subjectto discussion - in some kinetics models (e.g. O-C, Stringfellow, Iwamoto-Tsuta) but is almost non-existent in a uni ed theory of work-hardening inthis class of dynamic materials.612.6. Summary of the Literature Review2.6 Summary of the Literature ReviewThis review has highlighted the link between microscopic deformation mech-anisms, strain-induced phase transformations and the macroscopic responseof metastable austenitic stainless steels. The low stability of austenite withrespect to  and  0 martensites means that both phase transformations areoften observed during straining. There is substantial evidence suggestingthat the microscopic mechanisms of plasticity are directly linked to boththe  and  0 transformations. The exact way in which these two phasetransformations are linked to the plastic deformation of the material is,however, still open for debate. At the microscopic scale, the link betweenstarting microstructure and the strain-induced transformations is still poorlyunderstood. For instance, as far as grain size is concerned, the con ictingtrends which have been reported from experiments have generally confusedattempts to provide physically-based models.With respect to the bulk mechanical response, there have been a largenumber of experiments performed to examine the e ect of strain path, stressstate as well as the in uence of microstructure. However, these experimentsare, again, often at odds with one another. This is directly linked to the un-certainty regarding the material behaviour at the microscopic scale. Thesediscrepancies impact on the ability to develop physically based models forthe plastic response of the material, though recent experiments aimed atidentifying the mechanical response of the austenite and martensite individ-ually appear to be promising for developing new understanding.62Chapter 3Scope and ObjectivesIn the previous chapter, the complex mechanical response of austenitic stain-less steels was highlighted. In particular, this review has pointed to severalaspects that remain poorly understood. In some cases there is a general lackof literature, in other areas the existing literature is contradictory.This project has aimed to study both the macroscopic mechanical re-sponse and the microstructural response of a particular industrially suppliedgrade of metastable austenitic stainless steel (AISI 301LN). The deformationconsidered in this work was carried out at low homologous temperature andat low strain rates along monotonic strain paths. Within this study, two pri-mary variables have been investigated. First, the grain size of the materialhas been varied over almost two orders of magnitude. As noted in the litera-ture review, grain size a ects both the strain-induced phase transformationsas well as the intrinsic mechanical response of austenite. To date, there islittle systematic work in the literature that has investigated this relation-ship. The second parameter that has been varied in this study is the modeof deformation. Mechanical testing has been performed in both uniaxialtension as well as in simple shear at various grain sizes in order to help es-tablish the strain path dependence of the mechanical response. Tension andshear have been chosen for this study due to the fact that both strain pathscan be achieved uniformly in a relatively simple experimental route on sheetsamples. Materials tested under these conditions have been characterizedusing electron microscopy, X-ray di raction and magnetic measurements todeduce the strain-induced evolution of the microstructure.This project  ts within a larger framework organized by the stainlesssteel manufacturer ArcelorMittal, whose aim is to develop models for thephase transformations and mechanical properties of these stainless steels.63Chapter 3. Scope and ObjectivesRelated to the project carried out here, a second study [14] at  Ecole desMines de Paris has studied the same material but has focused on thein uence of temperature and strain rate on the mechanical response of thesame grade.Based on the results of these experiments, and with reference to theliterature, the main objective of this thesis has been to advance the state-of-the-art in physically based modelling of the mechanical response of thesematerials. In particular, this requires the incorporation of grain size andstrain path e ects on (i) the evolution of the deformation microstructures,(ii) the nucleation of  -martensite and  0-martensite, (iii) the kinetics of thephase transformations and (iiii) the stresses carried by individual phases.The modelling work detailed in chapter 8 has relied on previous formulations[204, 206], to which signi cant extensions were made to account for theexperimental results obtained as a part of this work.64Chapter 4Processing andCharacterization of 301LNSheet to Develop Grain Sizesin the Micrometer toNanometer Range4.1 IntroductionAs a  rst step in the experimental program of this thesis, it was necessaryto modify the microstructure of the as-received stainless steel sheet so asto obtain a range of grain sizes. These materials would then serve as thestarting materials for experiments on the link between microstructure andmechanical properties in this alloy. There have been a number of recentstudies describing routes for generating materials having a sub-micron grainsize in austenitic stainless steels [181, 209, 210] by  0-martensite rever-sion. While the goal of this work is not to study the mechanisms for thisstrong grain size re nement, we have used a route similar to that in otherstudies. The reader with interest in the mechanisms of grain size re nementvia martensite reversion is directed to recent literature [211{215].This chapter begins by introducing the experimental procedures used toi) generate the microstructures for further testing, and ii) characterize thesemicrostructures. Following this, the material received from ArcelorMittalStainless Steel is described in terms of its chemistry and microstructure.654.2. Experimental Methodology: Materials CharacterizationThis is followed by a description of the thermo-mechanical processing routesused to achieve di erent grain sizes. The microstructures of the as-processedmaterials are described with a particular emphasis on the grain size distri-bution. Finally, the presence of secondary phases and solute segregation inthe processed microstructures are examined.4.2 Experimental Methodology: MaterialsCharacterization4.2.1 Quanti cation of  0-Martensite Content viaFeritscope MeasurementsIn this work, a Feritscope MP30E (Fischer) was used to determine the vol-ume fraction of  0-martensite. The Feritscope is a commercially availabledevice that has been developed for the non-destructive measurement of theferrite content in austenitic and duplex steels in the range of 0.1 to 80%  -Fe[216]. This probe has been used extensively in the past for measuring thefraction of  0-martensite in austenitic stainless steels (see e.g. the reviewin reference [61]). The principle of measurement used by the Feritscope isillustrated schematically in Figure 4.1.Soft iron coreLow frequencyalternating magnetic fieldExciter currentMeasured voltageProbed materialPickup coilFigure 4.1: Schematic of the magnetic induction measurement performedby a Feritscope.664.2. Experimental Methodology: Materials CharacterizationA low frequency alternating magnetic  eld is generated by a coil sur-rounding the soft iron core of the probe. A second ‘pickup’ coil has aninduced current due to the  eld generated by the  rst coil. When the probeis brought into contact with a ferromagnetic sample, the magnetic  eld ismodi ed resulting in a change in induced current in the ‘pickup’ coil. Theresulting change in voltage across the pickup coil is proportional to the fer-romagnetic content of the sample. In the case of austenitic stainless steels,both the austenite and  -martensite are paramagnetic, meaning that onlyferromagnetic  0 martensite contributes to the resulting signal. To obtaina quantitative measure of the volume fraction of  0-martensite, the volt-age change needs to be calibrated against standards of known  0 martensitecontents. Because the magnetic permeability depends upon chemical com-position (especially carbon content, e.g. [217, 218]), it is important to pre-pare calibration standards from the speci c material under study. In thiswork, a set of calibration standards were prepared from the 301LN alloy,by pre-deforming samples then quantifying the  0 martensite content by X-ray di raction. This calibration method, including the preparation of thecalibration standards, is detailed in Appendix A.Correction curves describing the e ect of the specimen thickness andproximity to the sample edge are provided by the manufacturer [216]. Thesecorrections are not necessary for samples thicker than 0.6 mm and as long asthe probe is placed no closer than 5 mm from the edge of the sample. Thegeometry of all the samples used in this study was chosen with these dimen-sions in mind, so that these e ects would be negligible. Feritscope readingswere made with the probe in contact with the specimen, perpendicular to itssurface. Measurements were made with samples in the unloaded state so asto avoid the in uence of stress on the magnetic susceptibility (the so-calledVillari e ect, or magnetomechanical e ect [219, 220]). A detailed evaluationof the magnetomechanical e ect will be described in chapter 7.674.2. Experimental Methodology: Materials Characterization4.2.2 Materials Characterization by Electron MicroscopySamples for scanning electron microscopy (SEM) and back-scattered elec-trons (BSE) observation were  rst prepared by mechanical polishing withSiC emery paper to 1200 grit followed by polishing using 6 a181m and 1 a181mdiamond paste. Electropolishing was next performed using a solution of 90vol.% acetic acid and 10 vol.% perchloric acid at room temperature. Thissolution was found to give good results when electropolishing was performedat room temperature with a current density of  50 mA/cm2. Water cool-ing was used to ensure that the temperature of the electropolishing solutiondid not exceed 25 C. After electropolishing, the specimen was  rst cleanedwith distilled water to remove residues of acetic/perchloric acid, then withdenatured alcohol, before being dried by a  ow of compressed air.To ensure that the mechanical preparation did not induce deformation-induced martensite on the sample surface, back-scattered electron contrastin the SEM was used to ensure the absence of  0-martensite on the polishedsurface of as-recrystallized (fully austenitic) samples. It was found thatshort polishing times (2 minutes) were su cient to remove the deformedlayer induced by mechanical polishing in small grain size samples, whilelonger electropolishing times (6 minutes) were required in coarse grain sizedsamples.Two scanning electron microscopes were used in this study, dependingon the spatial resolution required. The  rst one was a tungsten- lamentHitachi S-570. Higher resolution imaging was achieved with a JEOL JSM-7000F SEM equipped with a Field Emission Gun (FEG). Imaging by back-scattered and forescattered electron detectors was used as a technique forillustrating the microstructure. For quantitative measurements of micro-texture, electron back-scattered di raction (EBSD) systems were attachedto each of the SEMs used. In the lower resolution Hitachi microscope, anHKL Channel5 Flamenco acquisition system was used providing a spatialresolution of  1 a181m, whereas the JEOL FEG SEM was equipped withan Oxford INCA Crystal acquisition system. For data collected from bothmicroscopes and EBSD systems, data post-processing was conducted using684.2. Experimental Methodology: Materials CharacterizationHKL Channel5 software (Oxford Instruments).The EBSD acquisition allows for indexing of di erent phases based ontheir crystal structure. While in later chapters multiple phases will beindexed, here only austenite was considered during acquisition. The as-measured EBSD maps were cleaned by  rst performing a wild spike re-moval. This process changes isolated pixels that are misoriented by morethan 6{7a176 from their eight neighbours to being non-indexed. Following this,noise reduction was applied to remove non-indexed points. This cleaningconsists of attributing to a non-indexed pixel an orientation calculated fromthe mean orientation of the neighbouring points. Noise reduction was per-formed until no non-indexed points remained. Great care was exercised toensure that no artifacts were created during this cleaning procedure.In order to visualize the results, band contrast maps were used. Bandcontrast is a measure of the quality of the Kikuchi pattern recorded by theEBSD acquisition system at each point of measurement [221]. This bandcontrast is sensitive to defects, e.g. grain boundaries which tend to givelow band contrast [106], as well as to crystallographic orientation of thegrain. Also plotted, are the position of grain boundaries. These boundariesare calculated based on a misorientation of more than 2a176 between adjacentEBSD points.Energy dispersive spectroscopy (EDS) and wavelength dispersive spec-troscopy (WDS) were performed in order to estimate the heterogeneity inchemical composition. The chemical analysis was only qualitative. It wascarried out in the FEG-SEM described above, using a Si (Li) detector with asuper atmospheric supporting thin window (SATW) (Oxford Instruments).EDS maps, with dimension 512 430, were acquired with the software INCAEnergy. The acquisition was run for 170 minutes with a step size of  0.1a181m.In order to investigate the role of other phases in the recrystallized mi-crostructures, thin foils for transmission electron microscopy were preparedby mechanical polishing using 600 and 1200 grit emery paper to a thicknessof 100 a181m. Circular samples were subsequently punched from the sheet andjet-polished using a Struers Tenupol 2. The polishing solution used was the694.3. Experimental Methodology: Materials Processingsame as that described above, i.e. a solution of 10 vol.% percholoric acidand 90 vol.% acetic acid. The thinning was performed at room tempera-ture, under an imposed voltage of 9 V corresponding to a current of 0.1 A.The thin foils were subsequently examined in a Hitachi H-800 TransmissionElectron Microscope (TEM) operating at 200 kV.4.3 Experimental Methodology: MaterialsProcessing4.3.1 Cold Rolling of As-Received SheetIn section 2.3, it was shown that temperature and strain rate have an im-portant e ect on the rate of martensite formation. Therefore, a preliminarystudy was done to determine the relationship between rolling conditions(temperature and strain rate) and  0-martensite volume fraction, the latterbeing monitored by the Feritscope. This preliminary study is importantsince the deformation-induced martensite formed during rolling a ects thegrain size after annealing [178, 181, 209, 222, 223]. Conventional rolling wasperformed on a laboratory mill (Stanat, model TA-215) equipped with 105mm diameter rolls. Two di erent schedules were followed for rolling thematerials. Room temperature rolling (\RT-rolling") was performed usingkerosene as a lubricant. Rolling was also performed with no lubrication onmaterial that was cooled to  196 C in a liquid nitrogen bath prior to eachpass. This processing route will be denominated as \cryorolling" in the fol-lowing. For most experiments, samples having initial dimension of  150mm  50 mm  2.1 mm were used. Rolling was always performed parallelto the industrial rolling direction of the sheet.Figure 4.2 shows that higher amounts of  0-martensite can be obtainedby cryorolling. Also, the  0-martensite content formed during RT-rolling waslowered with the angular speed of the rolls (an e ect due to higher sampleheating at high velocities). This e ect was not observed during cryorolling.In order to achieve the widest range of grain sizes upon annealing, bothRT-rolling and cryorolling were used with an angular speed of 51.4 rpm to704.3. Experimental Methodology: Materials Processinga total reduction of 62% (from 2.1 mm to 0.8 mm thickness). This wasaccomplished in 15 passes for the RT-rolled material and in 30 passes forthe cryorolled material. In both cases the largest reduction per pass possiblewas attempted.s48s49s48s50s48s51s48s52s48s53s48s54s48s55s48s48s49s48s50s48s51s48s52s48s53s48s54s48s55s48s56s48s70s101s114s114s105s116s101s115s99s111s112s101s32s114s101s97s100s105s110s103s32s40s37s41s82s97s116s101s32s111s102s32s114s101s100s117s99s116s105s111s110s32s40s37s41s82s84s32s82s111s108s108s105s110s103s53s49s46s52s32s114s112s109s82s84s32s82s111s108s108s105s110s103s49s55s46s50s32s114s112s109s67s114s121s111s114s111s108s108s105s110s103s53s49s46s52s32s114s112s109s67s114s121s111s114s111s108s108s105s110s103s49s55s46s50s32s114s112s109Figure 4.2: Magnetic signal measured by Feritscope during RT-rolling andcryorolling, for two angular rotations. The Feritscope signal is proportionalto the volume fraction of  0-martensite, as described in Appendix A.4.3.2 Post-Rolling Annealing TreatmentsRolled materials were isothermally annealed in a tube furnace under  owingargon. Two annealing times were chosen, namely 3 minutes and 30 minutes,and the annealing temperature was varied between 800 C and 1050 C. Af-ter the furnace temperature was stabilized at the desired temperature, thesample was manually inserted. The initial heating rate of the sample wasmeasured to be 17 C:s 1 and the target temperature was reached in thesample after 70 seconds. After heat treatment, the samples were cooled in714.4. As-Received Materialair. A very thin chromium oxide layer formed on the sample surface andwas removed by polishing.4.4 As-Received MaterialThe work presented here was focused on an AISI 301LN austenitic stainlesssteel provided by the ArcelorMittal Stainless Steel Research Centre in Isber-gues France. The nominal composition of this grade is given in Table 4.1.Low carbon and high nitrogen contents are also characteristics of this grade(hence the \LN" denomination).Element C N Cr Ni Mn Si Cu Mo Cowt.% 0.022 0.107 17.33 6.62 1.77 0.53 0.24 0.21 0.14Table 4.1: Nominal composition (in wt.%) of the grade used in this study.The material was received as cold rolled and recrystallized sheets 2.1mm thick. The state of surface was 2D, meaning that there has been noskin-pass operation after annealing and pickling. An EBSD map illustratingthe microstructure of the as-received steel is presented in Figure 4.3. Thematerial had a relatively uniform and equiaxed grain size D = 10 a181m. Asmall amount of  -ferrite (less than 1 vol.%) was found in the as-receivedmaterial, similar to what had been previously reported for this grade [181].4.5 Generation of Materials With Varying GrainSizesThe  rst step in this thesis was to generate a range of grain sizes, the goalbeing to span as wide a range of sizes possible (from sub-micron to tens ofmicrons). After a series of preliminary annealing experiments, a set of  veannealing conditions were decided upon to generate materials having grainsizes ranging from 0.5 a181m to 28 a181m. These annealing conditions are givenin Table 4.2.724.5. Generation of Materials With Varying Grain SizesRDND50 μmFigure 4.3: Band contrast EBSD map of as-received 301LN, plotted to alsoreveal grain boundaries (misorientation > 2a176, black lines). The map wasacquired on the plane containing the rolling direction (RD) and normaldirection (ND) of the sheet.The grain sizes resulting from these  ve di erent thermo-mechanical pro-cessing routes were characterized by electron backscatter di raction (EBSD).Although the as-measured maps were of high quality (more than 75% pointsindexed for the low-resolution EBSD system and more than 90% indexed forthe high-resolution EBSD system), some cleaning of the data was requiredas described above. Representative EBSD band contrast maps includinggrain boundaries (black lines) from each of the conditions given in Table 4.2Rolling % Annealing AnnealingCondition procedure Reduction temperature timeA Cryorolling 62% - 30 passes 800 C 3 minutesB Cryorolling 62% - 30 passes 850 C 3 minutesC Cryorolling 62% - 30 passes 950 C 3 minutesD Cryorolling 62% - 30 passes 1050 C 3 minutesE RT rolling 62% - 15 passes 1050 C 30 minutesTable 4.2: Thermo-mechanical procedure used to generate the  ve condi-tions of grain size studied in this thesis.734.5. Generation of Materials With Varying Grain Sizesare shown in Figure 4.4.For each condition, the EBSD data was analyzed to determine the grainsize. Grains were reconstructed from the EBSD data by grouping togethertouching measurement points having a misorientation of less than 2a176. Thesize of each of these grains was quanti ed as its equivalent area diameter(EQAD), i.e. the diameter of a circle having an area equal to that of themeasured grains. Grains in contact with the edge of the measurement framewere not considered for grain size estimation. In this grain reconstruction,any grains smaller than 3 pixels were discounted. The removal of these clus-ters was checked visually to ensure that they did not represent a realisticgrain. The mean grain size, as well as other characteristics of the grain sizedistribution, measured in this way for these  ve conditions, is given in Ta-ble 4.3. To be consistent with industrial practice, two di erent assumptionswere made regarding the inclusion of annealing twin boundaries in the grainsize measurement. In a  rst measurement, any grains separated by anneal-ing twins were combined so as to neglect the annealing twin boundary whencalculating grain size (a in Table 4.3). In the second method, annealingtwins were considered as being equivalent to grain boundaries and thus usedto de ne the grain size (b in Table 4.3). For the purposes of this thesis, thelatter de nition of grain size is used.The measured EQAD grain size distributions are illustrated in Fig-ure 4.5. The distributions are observed to be well represented by log-normaldistributions (straight lines in Figure 4.5 (a) to (e)) and appear to be self-similar when the grain size is normalized by the average grain size (Fig-ure 4.5(f)).While the number average grain size distributions presented above ap-pear to give self-consistent results, Figure 4.4 (a) and to a lesser extent (b)give the impression of having a wide size distribution. Similar observationshave been previously made on nanocrystalline austenitic stainless steels andhave been linked to the nucleation of new grains in regions containing  0-martensite versus austenite [11, 139]. It is known that the number sizedistribution as plotted in Figure 4.5 tends to suppress the importance of asmall number of large grains within a microstructure [224]. As an alternative744.5. Generation of Materials With Varying Grain Sizes(a) Condition A (b) Condition B(c) Condition C (d) Condition D(e) Condition E (f)Figure 4.4: Band contrast EBSD maps including grain boundaries (misorien-tation > 2a176, black lines). The conditions A { E correspond to the annealingconditions in Table 4.2. The average grain sizes from these samples are givenin Table 4.3.754.5. Generation of Materials With Varying Grain SizesCondition A B C D EStep size 0.06 a181m 0.12 a181m 0.5 a181m 1.5 a181m 2.5 a181mAverage grain size (a) 0.70 a181m 1.4 a181m 2.4 a181m 19 a181m 43 a181mASTM number (a) 17.8 15.6 13.6 11.2 6.0Average grain size (b) 0.52 a181m 0.91 a181m 2.2 a181m 14 a181m 28 a181mASTM number (b) 18.6 14.7 13.7 9.0 7.7Standard deviation ofthe distribution (b) 0.4 a181m 0.7 a181m 1.6 a181m 8 a181m 18 a181mNumber of grains ana-lyzed (b) 2 000 11 000 2 500 3 000 1 500Proportion of twinboundaries 20% 30% 15% 30% 38%Table 4.3: Description of the  ve grain size distributions, characterized byEBSD. The average grain sizes and ASTM numbers are given both (a)without and (b) with twin boundaries. The ASTM number was calculatedas ASTM = 3:322 log(Na) 2:95, Na being the number of grains permm2. The proportion of twin boundaries as de ned in this table is the ratioof twin boundaries over total amount of boundary (grain boundaries andtwin boundaries).to Figure 4.5, Figure 4.6 presents the distribution under the form of an areafraction. These histograms are normalized by the mean of the distributionto enable comparison of the widths, and clearly indicate that the widestdistributions appear in both 0.5 a181m and 0.9 a181m conditions of grain size. Itis also evident that the distributions, while wide, are not bimodal.764.5. Generation of Materials With Varying Grain Sizes012300. Equivalent area diameter (µµµµm)N u m b e r  f r a c t i o n(a) D0 = 0.5 a181m024600. Equivalent area diameter (µµµµm)N u m b e r  f r a c t i o n(b) D0 = 0.9 a181m02468101200. Equivalent area diameter (µµµµm)N u m b e r  f r a c t i o n(c) D0 = 2.2 a181m0102030405000.050.10.15 Equivalent area diameter (µµµµm)N u m b e r  f r a c t i o n(d) D0 = 14 a181m05010015000.050.10.15 Equivalent area diameter (µµµµm)N u m b e r  f r a c t i o n(e) D0 = 28 a181m0 1 2 3 4 5 6 7 800. EQAD, D / D0Probability density  (f) All conditionsFigure 4.5: (a) to (e): Histograms of the grain size distributions in terms ofnumber fraction, as a function of the equivalent area diameter (EQAD). (f)Superimposition of the  ve grain size distributions, represented as a functionof the EQAD (D) normalized by the average of the distribution (D0). Thedots correspond to the classes represented by the histograms in (a) to (e).774.5. Generation of Materials With Varying Grain Sizes0 1 2 3 4 5 6 7 800. EQADArea fraction(a) D0 = 0.5 a181m0 1 2 3 4 5 6 7 800. EQADArea fraction(b) D0 = 0.9 a181m0 1 2 3 4 5 6 7 800. EQADArea fraction(c) D0 = 2.2 a181m0 1 2 3 4 5 6 7 800. EQADArea fraction(d) D0 = 14 a181m0 1 2 3 4 5 6 7 800. EQADArea fraction(e) D0 = 28 a181mFigure 4.6: (a) to (e): Histograms of the grain size distributions in terms ofarea fraction, as a function of the normalized EQAD (D=D0).784.6. Presence of Other Phases in the Recrystallized Microstructures4.6 Presence of Other Phases in theRecrystallized MicrostructuresWhile to this point, this thesis has focused on three phases, austenite,  0-martensite and  -martensite, other phases may exist in 301LN, in the formof precipitates. Transmission electron microscopy analysis was made on thematerials processed by the thermo-mechanical route described above. It wasfound (Figure 4.7) that nanometric second phase precipitates existed in thematerial annealed at 800 C, whereas these precipitates were not found in anyof the other processed materials. This observation is consistent with that ofRajasekhara et al. [177] who, also in a 301LN stainless steel, identi ed theseprecipitates formed at low annealing temperatures as Cr2N precipitates. 0.5 μmFigure 4.7: Bright  eld TEM image of a sample annealed for 3 minutes at800a176C resulting in a D=0.5 a181m average grain size, illustrating the presence ofchromium nitride precipitates. Such precipitates were not found in samplesannealed at higher temperatures.In normal practice, nitrides are not formed in the 300 series austeniticstainless steels, owing to the high solubility of nitrogen [21, 225]. However,800a176C is much lower than the temperatures normally used for the processingof these grades. In the work of Rajasekhara et al. [177], it was estimated thatthe amount of nitrogen contained in nitrides was 0.01 wt.% after annealingat 800a176C.794.7. Solute Segregation4.7 Solute SegregationGiven the importance of alloying elements like nickel and nitrogen in deter-mining the stability of austenite, and therefore the mechanical response ofthe material studied, it was attempted to quantify the chemical homogeneityof the material studied. In this case, materials having been processed viaroute A above (800a176C, 3 minutes) were subjected to energy dispersive spec-troscopy (EDS) and wavelength dispersive spectroscopy (WDS) analyses inthe SEM. These analyses were performed on the plane containing the rollingand normal directions of the sheet. Given the low annealing temperatureand the short annealing time, it is believed that this sample would repre-sent the segregation pattern exhibited by the as-received material. EDSanalysis of alloying elements Cr, Mn, Si did not reveal any speci c patternof segregation on the scale of the sample thickness. Higher sensitivity tochemical heterogeneities was achieved using WDS on detection of chromiumand nitrogen, and con rmed the absence of heterogeneities. However, EDSanalysis of nickel did indicate segregation as bands parallel to the rollingdirection as illustrated in Figure 4.8. The intensity ratio characteristic ofthe Ni K 1 electronic transition could vary by 25% from within the seg-regation bands to oustide the bands. Once related to the nominal nickelcontent, this intensity ratio enables rough quanti cation of the nickel withinthe segregation. The level of nickel could thus vary by  1.3 wt.% aroundits average value (6.62 wt.%). It was found, that this segregation patterncould be seen in backscatter electron (BSE) images in the SEM. Figure 4.8shows that the regions of segregation appear brighter than the surroundings.It is,  nally, worth noting that the spacing of the segregation bands is onthe order of 0.5 a181m. This spacing is similar to the average grain size in the nest-grained material, indicating that the e ect of segregation may lead tomore heterogeneous behaviour in these microstructures. In larger grain sizematerials, all grains should be expected to contain regions of high and lownickel content.804.8. SummaryFigure 4.8: The segregation of nickel as seen with back-scattered electronsimaging in the SEM. The correspondence between the lighter coloured bandsand regions of nickel enrichment can be seen when compared to the EDSline scan.4.8 SummaryIn this chapter, the preliminary processing of the as-received material hasbeen described. It has been found that materials having mean grain sizesbetween 0.5 a181m and 28 a181m could be formed by a combination of conventionalrolling and annealing. These results are in good agreement with previouswork on the recrystallization behaviour of the same alloy [181]. It has beenshown that, while the grain size distribution appears self-similar, and con-sistent with a log-normal distribution, there are deviations at the smallestgrain sizes associated with the presence of some larger grains in the mi-crostructure. Also, it has been shown that in the material with smallestgrain size, chromium nitride precipitates are present and that segregationon the scale of the grain size exists. These factors will be returned to whenthe deformation microstructures of this material are discussed.81Chapter 5MacroscopicCharacterization of theMechanical Properties andPhase Fraction5.1 IntroductionIn this chapter, the macroscopic mechanical behaviour of the alloy (grade301LN) is described. In particular, the behaviour in uniaxial tension andsimple shear will be compared for samples having the grain size range de-scribed in the previous chapter. These mechanical properties are, in turn,related to the evolution of the average content of  and  0-martensites mea-sured using the Feritscope described in the previous chapter.5.2 Experimental Methods5.2.1 Uniaxial Tensile TestingTensile coupons were machined by electro-discharge machining, prior to an-nealing. The geometry of the tensile coupons (Figure 5.1) is di erent fromthe ASTM 7 standard [226]. It was found that, for the standard geometry,the high rate of work-hardening of the material resulted in plasticity spread-ing from the gauge section into the head of the sample. The sample wastherefore re-designed to ensure that at maximum force, the stress in the grip7 the American Society for Testing and Materials.825.2. Experimental Methodssection (head) was lower than the yield stress of the material in the largestgrain size condition.100 mm18 mm20 mm6 mm20 mm 28 mmR=6 mm0.8 mmRDTDFigure 5.1: Geometry of the  at tensile test coupons used in this study.Room-temperature tensile tests were conducted in displacement controlusing a computer-controlled servohydraulic load frame. The tensile axis wasalways parallel to the rolling direction. Strains were measured with an axialextensometer of 12.5 mm gage length. The rate of data acquisition was 2points per second.As noted in section 2.3, the strain rate sensitivity of austenitic stain-less steels can be high, particularly when the rate of phase transformationresults in large heating of the sample. Therefore care was taken in the se-lection of strain rate such that the temperature rise in the sample was notsigni cant. After di erent trials, it was found that a cross-head speed of0:04 mm:s 1 was adequate for this purpose. Considering the geometry ofthe coupons, this corresponds to a nominal strain rate of 1:4 10 3 s 1.The temperature rise in the specimen was monitored on test samples havinga K-type thermocouple welded in the center of the gauge section. It was ob-served that the temperature increased with strain, but that the temperaturechange never exceeded 10 C. Tests were also performed at lower strain rates(down to 4 10 5 s 1) and it was found that the stress-strain response ofthe material was not signi cantly di erent from that of the sample testedat 1:4 10 3 s 1.Uniaxial tension was also performed at 80 C. These tests were conductedin a small (5 kN capacity) screw-driven Instron load frame where the samplecould be fully immersed in a constant temperature bath. For these tests, astirred bath of water heated with a hot plate was used and the temperaturewas monitored with a thermometer. The geometry of the samples was the835.2. Experimental Methodssame as those for room temperature testing (Figure 5.1) as was the imposeddisplacement rate. Because classical extensometers could not be used atthis elevated temperature, displacement of cross-head was used to calculatestrain in the sample, a correction being made for the machine sti ness. Thesti ness correction was performed so as to ensure consistency of the elasticmodulus of the samples.5.2.2 Testing in Simple ShearSimple shear was performed at the 3S-R laboratory, Institut NationalPolytechnique de Grenoble, France, the setup being illustrated in Fig-ure 5.2. A detailed description of this testing setup can be found elsewhere(cf. [227{230]).9354 786121 sample2  xed grip3 movable grip4 upper frame part5 lower frame part6 LVDT7 load frame8 frame9 computerFigure 5.2: Schematic overview of the shear testing apparatus.The test coupons for shear were rectangular (25 mm  18 mm  0.8mm) with a 3 mm-large gauge area, as illustrated in Figure 5.3. The dis-placement of one side of the coupon relative to the other was monitored by845.2. Experimental Methodsa Linear Variable Di erential Transformer (LVDT), and used to calculateshear strains.3 mm25 mm18 mm0.8 mmRDTD(a) Before deformation0.8 mmh effθDeformed areaRDTD(b) After deformationFigure 5.3: Geometry of the  at shear test coupons used in this study. Theshear angle is de ned by  . While the distance out of the grips is set tobe 3 mm, the deformation actually occurs on a larger width represented byhe .It is known that this testing procedure results in shear strains that arenot fully homogeneous across the width of the gauge area. Thus, a correctionto the width of the deformed region of the sample is conventionally used[229]. This correction uses the shear angle measured from  ducial linesinscribed on the sample ( in Figure 5.3 (b)), after unloading of the coupon.In this case the  ducial lines were made using a  ne point marker. The  nalshear strain ( ) was calculated from  according to: = 12 tan (5.1)This correction was found to reproduce the shear strains measured on othersamples using digital image correlation in previous work [229] as well asin samples of austenitic stainless steel observed in this thesis. The sheartests were performed with the direction of the shear parallel to the rollingdirection, at a cross-head speed of 3 10 3 mm.s 1, which after correction855.2. Experimental Methodscorresponds to _  2:9 10 4 s 1. In order to compare to uniaxial tension,it can be useful to express this strain rate using the Von Mises equivalentstrain. This equivalent strain rate is _  3:3 10 4 s 1, which is 4 timessmaller than the one imposed in tension.5.2.3 Phase Quanti cation by X-Ray Di ractionIn order to complement the Feritscope measurements presented in section4.2.1 for the quanti cation of  0-martensite, and to estimate the amount of -martensite, X-ray di raction spectra were analysed by means of Rietvieldtre nement (see [63, 64]).For tensile samples, a Bruker D8 advance di ractomer equipped withthe Cu K radiation was used. The acquisition of the di raction spectra isdocumented in reference [139].For shear samples, a PANalytical X’Pert PRO goniometer was used witha monochromatic Co K radiation. The di ractometer operated under avoltage of 45 kV and an intensity of 35 mA. A 1 mm beam size was usedwith a step of 0.033a176/s. Samples were carefully positioned so that onlythe uniformly strained portion of the sample was analysed. In order toreduce the e ect of crystallographic texture, the coupons were placed on arotating sample holder and scans were summed up for di erent radial angles . Twelve scans were performed for  varying from 0a176 to 55a176 (by stepsof 5a176) and the sum of these scans was considered. The Rietveld analysiswas performed with X’Pert software. The corresponding space groups wereimposed for the three phases:  -austenite,  0-martensite and  -martensite.Both the volume fractions and the lattice parameters (Table 5.1) were usedas  tting parameters in the re nement. It was ensured that, at the end of there nement, the converged values of the lattice parameters were reasonable.865.3. Mechanical Properties of 301LN in Uniaxial TensionLattice parameter   0  a 3.61  A 2.86  A 2.54  Ac - - 4.16  ATable 5.1: Initial values of the lattice parameters used in the Rietveld anal-ysis. Those were taken from the X-ray di raction work of P etein [139].5.3 Mechanical Properties of 301LN in UniaxialTensionThe tensile response of 301LN at room temperature is shown in Figure 5.4.While this  gure shows only one curve per condition, a minimum of twotensile tests were performed up to failure to con rm reproducibility, andmany other tests, performed to variable levels of strain, justify the con -dence in these measurements. The stress-strain curves in Figure 5.4 havebeen stopped after the onset of necking, the onset of necking correspond-ing well to the Consid ere criterion. This is despite the fact that the phasetransformation causes volume to not be conserved during the test.Several of the characteristics of the material in tension are given in Ta-ble 5.2. The variation of 0.2% o set yield stress with grain size (includingtwin boundaries) obeys a Hall-Petch relation ( =  0 +kyD 1=2) with a fric-tion stress of  0=230 MPa and a Hall-Petch slope of ky=312 MPa.a181m 1=2.Discontinuous yielding was observed for grain sizes in the range of 2.2 a181mto 28 a181m. This is consistent with previous observations on the yielding be-haviour for similar grain sizes in grades 304 and 316 [187]. The discontinuousyielding in these steels has previously been attributed to a low initial num-ber of mobile dislocations per grain, leading to a high yield stress associatedwith the nucleation of dislocations from grain boundaries [187].875.3. Mechanical Properties of 301LN in Uniaxial Tensions48s46s48s48s46s49s48s46s50s48s46s51s48s46s52s48s46s53s48s53s48s49s48s48s49s53s48s8226s84s32s61s32s50s53s111s67s61541s32s61s32s49s46s52s120s49s48s45s51s32s115s45s49s32s48s46s53s32s61549s109s32s48s46s57s32s61549s109s32s50s46s50s32s61549s109s32s49s52s32s61549s109s32s50s56s32s61549s109s84s114s117s101s32s115s116s114s101s115s115s32s40s77s80s97s41 s84s114s117s101s32s115s116s114s97s105s110Figure 5.4: Stress-strain curves obtained in uniaxial tension at room tem-perature.Average Yield stress True stress True straingrain size at necking at neckingD=0.5 a181m 676 MPa 1460 MPa 0.403D=0.9 a181m 593 MPa 1393 MPa 0.413D=2.2 a181m 409 MPa 1412 MPa 0.437D=14 a181m 334 MPa 1411 MPa 0.435D=28 a181m 280 MPa 1339 MPa 0.404Table 5.2: Tensile characteristics of the  ve grain size conditions.In contrast to the yield strength, the ultimate tensile strength and uni-form elongation were not found to vary strongly for grain sizes between 0.5a181m to 28 a181m. The fact that, unlike yield stress, the tensile strength hasa weak grain size dependence suggests that the hardening behaviour of the885.3. Mechanical Properties of 301LN in Uniaxial Tensions48s53s48s49s48s48s49s53s48s48s49s48s48s50s48s48s51s48s48s52s48s48s53s48s48s8226s87s111s114s107s45s104s97s114s100s101s110s105s110s103s32s114s97s116s101s32s40s77s80s97s41s84s114s117s101s32s115s116s114s101s115s115s32s40s77s80s97s41s32s48s46s53s32s61549s109s32s48s46s57s32s61549s109s32s50s46s50s32s61549s109s32s49s52s32s61549s109s32s50s56s32s61549s109s67s111s110s115s105s100s232s114s101s99s114s105s116s101s114s105s111s110 s84s32s61s32s50s53s111s67s61541s32s61s32s49s46s52s120s49s48s45s51s32s115s45s49Figure 5.5: Work-hardening curves obtained under uniaxial tension. Theonset of plastic instability (Consid ere criterion) is represented by the dashedline.material is strongly grain size dependent, with the larger grain size havingthe higher hardening rate. This is indeed observed in the work-hardeningcurves, presented in Figure 5.5, these having been obtained by numericaldi erentiation of the stress-strain curves. In all but the D=0.5 a181m condi-tion, an increase in austenitic grain size corresponds to a higher maximumrate of work-hardening. The maximum rate of work-hardening are in goodagreement with those measured by Talonen also on a 301LN alloy undersimilar conditions of temperature and strain rate [150].The work of Nanga [146, 186] on a similar steel grade showed that verylittle  0-martensite was formed in tension at a temperature of 80 C. There-fore, in order to collect information on the behaviour of austenite with littleor no formation of  0-martensite, tensile tests were carried out at 80 C (Fig-ure 5.6).895.3. Mechanical Properties of 301LN in Uniaxial Tensions48s46s48s46s49s48s46s50s48s46s51s48s46s52s48s46s53s48s53s48s49s48s48 s32s78s101s99s107s105s110s103s32s111s98s115s101s114s118s101s100s8226s84s32s61s32s56s48s111s67s61541s32s61s32s49s46s52s120s49s48s45s51s32s115s45s49s49s52s61549s109s32s45s32s49s48s37s32s61537s39s50s46s32s61549s109s32s45s32s52s37s32s61537s39s48s46s57s32s61549s109s32s45s32s50s37s32s61537s39s48s46s53s32s61549s109s32s45s32s54s37s32s61537s39s84s114s117s101s32s115s116s114s101s115s32s40s77s80s97s41s84s114s117s101s32s115s116s114s97s105s110s50s56s61549s109s32s45s32s49s48s37s32s61537s39Figure 5.6: Stress-strain response of grade 301LN under uniaxial tensionat 80 C. The tensile tests were not conducted to failure. The maximumfraction of  0 formed during uniform deformation of the sample is indicatedas determined from Feritscope measurements.s48s46s48s48s46s49s48s46s50s48s46s51s48s46s52s48s46s53s48s53s48s49s48s48s49s53s48s32s50s51s111s67s32s56s48s111s67s84s114s117s101s32s115s116s114s101s115s115s32s40s77s80s97s41 s84s114s117s101s32s115s116s114s97s105s110s49s52s61549s109s50s46s50s32s61549s109s48s46s57s32s61549s109s48s46s53s61549s109s50s56s61549s109s8226s61541s32s61s32s49s46s52s120s49s48s45s51s32s115s45s49Figure 5.7: Comparison of the tensile tests performed at 23 C (from Fig-ure 5.4) and at 80 C (from Figure 5.6). In the latter, the yield stress wascorrected to be at the same level as that measured at 23 C.905.4. Mechanical Properties of 301LN in Simple ShearIn order to compare these elevated temperature tests with those per-formed at room temperature, it is necessary to correct for the temperature-dependence of the yield stress. The comparison is shown in Figure 5.7, forwhich the yield stresses obtained at 80 C were adjusted to match the yieldstresses obtained at 23 C.In the case of tensile tests performed at elevated temperature, the frac-tion of martensite formed outside of the neck was measured at the end of thetest using a Feritscope. It can be seen that while some  0-martensite wasformed during testing, the fraction formed is low. This is consistent withthe mechanical response of these samples in that they appear qualitativelysimilar to the response obtained in stable austenitic grades.An important feature of Figure 5.7 is that the grain size dependenceof the work-hardening of austenite is low, i.e. all samples show similarhardening.5.4 Mechanical Properties of 301LN in SimpleShearThe stress-strain response of 301LN in simple shear is presented in Figure 5.8for three conditions of grain size. It was impossible, in this case, to observethe presence of discontinuous yielding. This may be due to the di erentcharacteristics of the load frames used in tension and shear testing. In thesetests, the stress-strain curves plotted only up to the point where the strainsremained relatively uniform within the gauge section.In order to compare these deformation curves to those obtained in uni-axial tension, Von Mises equivalent strains and stresses were calculated ac-cording to: = 2p3 (5.2) =p3 (5.3)These are plotted alongside the tensile results in Figure 5.9.915.4. Mechanical Properties of 301LN in Simple Shears48s46s48s48s46s50s48s46s52s48s46s54s48s50s48s52s48s54s48s56s48s8226s50s56s32s61549s109s50s46s50s32s61549s109s84s32s61s32s50s53s111s67s61543s32s61s32s50s46s57s120s49s48s45s52s32s115s45s49s83s104s101s97s114s32s115s116s114s101s115s115s32s40s77s80s97s41 s83s104s101s97s114s32s115s116s114s97s105s110s48s46s53s32s61549s109Figure 5.8: Stress-strain curves obtained under simple shear, at room-temperature.Figures 5.9 and 5.10 show that, while at small to intermediate levels ofstrain the stress-strain responses are nearly equivalent in these two strainpaths, at higher strains the work-hardening rates are higher in uniaxialtension than in simple shear. The divergence between the curves for a givengrain size are observed at true strains between 0.2 to 0.25.925.4. Mechanical Properties of 301LN in Simple Shears48s46s48s48s46s50s48s46s52s48s46s54s48s46s56s48s52s48s56s48s49s50s48s49s54s48 s32s85s110s105s97s120s105s97s108s32s116s101s110s115s105s111s110s32s61541s32s61s32s49s46s52s32s120s32s49s48s45s51s32s115s45s49s32s83s105s109s112s108s101s32s115s104s101s97s114s32s61541s32s61s32s51s46s51s32s120s32s49s48s45s52s32s115s45s49s32s84s32s61s32s50s53s111s67s69s113s117s105s118s97s108s101s110s116s32s115s116s114s101s115s115s32s40s77s80s97s41s69s113s117s105s118s97s108s101s110s116s32s115s116s114s97s105s110s50s56s32s61549s109s50s46s50s32s61549s109s48s46s53s32s61549s109s8226s8226Figure 5.9: Comparison of the stress-strain curves from uniaxial tension andsimple shear tests.s48s53s48s49s48s48s49s53s48s48s49s48s48s50s48s48s51s48s48s52s48s48s53s48s48 s8226s32s85s110s105s97s120s105s97s108s32s116s101s110s115s105s111s110s32s32s61541s32s61s32s49s46s52s32s120s32s49s48s45s51s32s115s45s49s32s83s105s109s112s108s101s32s115s104s101s97s114s32s61541s32s61s32s51s46s51s32s120s32s49s48s45s52s32s115s45s49s84s32s61s32s50s53s111s67s87s111s114s107s45s104s97s114s100s101s110s105s110s103s32s114s97s116s101s32s40s77s80s97s41s69s113s117s105s118s97s108s101s110s116s32s115s116s114s101s115s32s40s77s80s97s41s50s56s32s61549s109s50s46s50s32s61549s109s48s46s53s32s61549s109s8226Figure 5.10: Comparison of the work-hardening curves from uniaxial tensionand simple shear tests.935.5. Quanti cation of the Volume Fractions of Strain-Induced Martensitic Phases5.5 Quanti cation of the Volume Fractions ofStrain-Induced Martensitic Phases5.5.1 Quanti cation of  martensiteFigure 5.11 shows the di raction spectrum of the 0.5 a181m grain size conditionat various levels of strain, compared to the 28 a181m grain size condition. Theseresults are qualitatively similar to the results obtained in previous studiesparticularly with respect to the grain size dependence of  -martensite forma-tion with re nement of grain size [139]. No evidence of  -martensite could befound for the smallest grain sized materials in this study. In coarse grainedsamples, the maximum volume fraction of  -martensite also remained small,with a maximum of 2:4% 0:6% observed for a true strain of 0.1.s52s53s52s54s52s55s52s56s52s57s53s48s48s49s48s50s48s51s48s52s48s123s48s48s50s125s61541s123s49s48s49s125s61541 s48s46s53s32s61549s109s32s45s32s61541s32s61s32s52s46s37s48s46s53s32s61549s109s32s45s32s61541s32s61s32s57s46s52s37s48s46s53s32s61549s109s32s45s32s61541s32s61s32s49s52s46s51s37s48s46s53s32s61549s109s32s45s32s61541s32s61s32s50s51s46s55s37s73s110s116s101s110s115s105s116s121s32s40s99s112s115s41 s50s61553s32s40s111s41 s50s56s32s61549s109s32s45s32s61541s32s61s32s53s37Figure 5.11: X-ray di raction spectra representing thef0002g andf10 11g peaks for four conditions of strain in the 0.5 a181m condition. A comparisontowards the 28 a181m condition is presented.5.5.2 Quanti cation of  0 martensiteThe evolution of  0-martensite was monitored by Feritscope, according tothe protocol described in section 4.2.1. For room-temperature uniaxial ten-945.5. Quanti cation of the Volume Fractions of Strain-Induced Martensitic Phasession, the tests were interrupted in increments of 0.03 true strain. Six mea-surements at di erent locations within the gauge length were performedin the unloaded state. The as-measured Feritscope numbers were then con-verted to volume fraction  0-martensite using the calibration in Appendix A.The average of these measurements appears in Figure 5.12, with the errorbars representing the maximum and minimum of the Feritscope readingswithin the same sample. Each test was repeated twice and the variationin Feritscope readings were found to be smaller between two tests thanbetween measurements within the same sample. The deviation betweenmeasurements increased with strain, being less than 0.1% at low strains andfrom 1% to 3% at high strains.s48s46s48s48s46s49s48s46s50s48s46s51s48s46s52s48s46s53s48s46s48s48s46s50s48s46s52s48s46s54s48s46s56s49s46s48s32s50s56s32s61549s109s32s49s52s32s61549s109s32s50s46s50s32s61549s109s32s48s46s57s32s61549s109s32s48s46s53s32s61549s109s86s111s108s117s109s101s32s102s114s97s99s116s105s111s110s32s111s102s32s61537s39s84s114s117s101s32s115s116s114s97s105s110s8226s84s32s61s32s50s53s111s67s61541s32s61s32s49s46s52s120s49s48s45s51s32s115s45s49Figure 5.12: Evolution of the volume fraction of  0-martensite with straindetermined by Feritscope measurements made after room-temperature uni-axial tension. The error bars represent the spread in Feritscope measurementon a single sample. The solid lines correspond to sigmoidal  ts drawn toguide the eyes.955.5. Quanti cation of the Volume Fractions of Strain-Induced Martensitic PhasesThe experimental measurements displayed in Figure 5.12 show a globaltrend suggesting a slow down of the kinetics with re nement of the auste-nitic grain sizes, with the exception of the D=0.5 a181m condition, for whichthe kinetics is accelerated. This non-monotonic e ect can also be seen inFigure 5.13 which shows the value of the maximum rate of formation of  0 asa function of grain size. Such non-monotonic trend cannot be described bythe usual Olson-Cohen model [81] detailed in section 2.4. Indeed, it has beenshown in section 2.4.2 that grain size re nement could be interpreted to slowdown ( /D2) or accelerate ( /D2(1 n)) the formation of  0-martensite,but always monotonically. Moreover, as mentioned in Figure 2.17, the origi-nal grain size dependence postulated by Olson and Cohen (i.e.  /D2) is fartoo strong to capture the experimental saturation fractions of  0-martensitepresented in Figure 5.12. This non-monotonic grain size e ect will be revis-ited in section 6.6, in relation with observed mechanisms of nucleation forboth  -martensite and  0-martensite.s49 s49s48s51s46s48s51s46s50s51s46s52s51s46s54s51s46s56s52s46s48 s50s56s61549s109s49s52s61549s109s50s46s50s32s61549s109s48s46s57s32s61549s109s77s97s120s105s109s117s109s32s114s97s116s101s32s111s102s32s61537s39s32s102s111s114s109s97s116s105s111s110s71s114s97s105s110s32s115s105s122s101s32s40s61549s109s41s48s46s53s32s61549s109Figure 5.13: Maximum rate of formation of  0-martensite as a function ofthe initial austenite grain size.Considering the samples deformed in shear, due to the small dimensions965.5. Quanti cation of the Volume Fractions of Strain-Induced Martensitic Phasesof the gauge section of shear coupons, the Feritscope was not suitable formeasurements in these samples. Instead, X-ray di raction was performedwith the experimental protocol described in section 5.2.3. Each measure-ment was performed on a di erent coupon. These measurements appear inFigure 5.14. Because the correction applied from the shear angle measuredpost-mortem could vary signi cantly from one sample to another, it was notpossible to reproduce the same shear strain for di erent condition of grainsize. One can observe larger scatter in the volume fractions under shearwhen compared to tension, a scatter of the order of the variation of volumefractions with grain size. Therefore, from these results, it is not possibleto de nitively comment on the grain size dependence of the transformationkinetics in simple shear.s48s46s48s48s46s50s48s46s52s48s46s54s48s46s56s48s50s48s52s48s54s48s56s48s49s48s84s32s61s32s50s53s111s67s32s61541s32s61s32s51s46s51s32s120s32s49s48s45s52s32s115s45s49s32s50s56s61472s61549s109s32s50s46s50s32s61472s61549s109s32s48s46s53s61472s61549s109s86s111s108s117s109s101s32s102s114s97s99s116s105s111s110s32s111s102s32s61537s39 s69s113s117s105s118s97s108s101s110s116s32s115s116s114s97s105s110s8226Figure 5.14: Evolution of the volume fraction of  0-martensite in simpleshear as measured from X-ray di raction.Figure 5.15 compares the  0 volume fractions measured in simple shearand in uniaxial tension as a function of Von Mises equivalent strain. Theseplots suggest a dependence of the fraction of  0 with the strain path. Similar975.5. Quanti cation of the Volume Fractions of Strain-Induced Martensitic Phasesto the stress-strain curves presented above, it is observed that the fractionof  0-martensite formed in tension and shear are similar over the  rst   0.2{0.3. Beyond this level of strain, the fraction of  0-martensite in simpleshear are systematically lower than those in uniaxial tension. This e ect onthe kinetics is the one observed in Figure 2.15 (a) [151] but not in Figure 2.19[170].s48s46s48s46s49s48s46s50s48s46s51s48s46s52s48s46s53s46s48s46s50s48s46s52s48s46s54s48s46s56s49s46s48s32s84s101s110s115s105s111s110s32s45s32s48s46s53s32s61549s109s32s79s45s67s32s102s105s116s32s116s111s32s116s101s110s115s105s111s110s32s100s97s116s32s83s104s101s97s114s32s45s32s48s46s53s32s61549s109s32s79s45s67s32s102s105s116s32s116s111s32s115s104s101s97s114s32s100s97s116s86s111s108s117s109s101s32s102s114s97s99s116s105s111s110s32s111s102s32s61537s39 s69s113s117s105s118s97s108s101s110s116s32s115s116s114s97s105s110s84s32s61s32s50s53s111s67s48s46s48s46s49s48s46s50s48s46s51s48s46s52s48s46s53s46s48s46s50s48s46s52s48s46s54s48s46s56s49s46s48s32s84s101s110s115s105s111s110s32s45s32s50s46s32s61549s109s32s79s45s67s32s102s105s116s32s116s111s32s116s101s110s115s105s111s110s32s100s97s116s32s83s104s101s97s114s32s45s32s50s46s32s61549s109s32s79s45s67s32s102s105s116s32s116s111s32s115s104s101s97s114s32s100s97s116s86s111s108s117s109s101s32s102s114s97s99s116s105s111s110s32s111s102s32s61537s39 s69s113s117s105s118s97s108s101s110s116s32s115s116s114s97s105s110s84s32s61s32s50s53s111s67s48s46s48s46s49s48s46s50s48s46s51s48s46s52s48s46s53s46s48s46s50s48s46s52s48s46s54s48s46s56s49s46s48s32s84s101s110s115s105s111s110s32s45s32s50s56s32s61549s109s32s79s45s67s32s102s105s116s32s116s111s32s116s101s110s115s105s111s110s32s100s97s116s32s83s104s101s97s114s32s45s32s50s56s32s61549s109s32s79s45s67s32s102s105s116s32s116s111s32s115s104s101s97s114s32s100s97s116s86s111s108s117s109s101s32s102s114s97s99s116s105s111s110s32s111s102s32s61537s39 s69s113s117s105s118s97s108s101s110s116s32s115s116s114s97s105s110s84s32s61s32s50s53s111s67Figure 5.15: Comparison of the volume fraction of  0-martensite in simpleshear, compared to the one measured in uniaxial tension as a function ofVon Mises strain. The solid lines are drawn to guide the eyes.985.6. Relationship between Mechanical Response and Volume Fraction of Phases5.6 Relationship between Mechanical Responseand Volume Fraction of PhasesConsistent with previous studies, this work shows only a very small fractionof  -martensite formation during tensile testing. This is not particularlysurprising given the discussion in section 2.2.3 regarding the tendency forthe  0-martensite transformation to be preceded by  -martensite. The otherimportant point is illustrated in Figure 5.11 which shows that the grain sizestrongly a ects the formation of  -martensite. This trend is consistent withthe reported increase in the Ms temperature for the  ! phase transfor-mation when the grain size is increased, in various austenitic stainless steels[125, 126, 131]. This trend will be further discussed in chapter 6.While it is di cult to make de nitive correlations between mechanicalresponse and  -martensite fraction owing to the low volume fraction of thisphase, relationships between the formation of  0-martensite and mechanicalresponse can be made. To assist in describing this relationship below, thework-hardening response of the alloy is divided into three stages, those arerepresented in Figure 5.16. Note that the three stages de ned here are notmeant to have any relationship to the stages of work-hardening typicallyused to describe the behaviour of single crystals.Work-hardening rateTrue stress I II IIIBAFigure 5.16: Schematic of the work-hardening behaviour in stainless steels,de ning the three stages of work-hardening at room temperature.The work-hardening within stage I is seen to be similar to the work-995.6. Relationship between Mechanical Response and Volume Fraction of Phaseshardening of stable austenitic grades that do not undergo phase transforma-tion. Figure 5.7 showed that the work-hardening behaviour observed duringthis stage was very close to that observed at 80 C where little transforma-tion occurred. Some authors have commented on a rapid decrease in stage Iwork-hardening and have related this to the rapid formation of  -martensite[141, 168], or to the nucleation of  0 at shear bands [111, 231]. The endof stage I is marked by a minimum of work-hardening (noted \A" in Fig-ure 5.16).Stage II work-hardening is the stage which accounts for the in ectionin the stress-strain curve, typical to materials exhibiting pronounced TRIPe ect. The end of this stage II is marked by a maximum of work-hardening(noted \B" in Figure 5.16). With the exception of the D=0.5 a181m condition,the most intense work-hardening peaks were found in the conditions of grainsize displaying the highest rates of  0 formation. This suggests a stronglink between the rate of formation of  0-martensite and the rate of work-hardening.Some authors have considered  0-martensite as an ideally plastic phase.Under such an assumption, once formed, the  0-martensite would alwayscarry the same stress (  0). Assuming the volume fraction of  being smallenough to be neglected, the true stress of this dynamic composite materialwould then be given by a simple rule of mixtures (consequence of mechanicalequilibrium) between only  and  0: =   (1 f 0) +  0f 0 (5.4)which can be rewritten as: =   +f 0(  0   ) (5.5)Equation 5.5 would result in a linear dependence of the true stress ofthe composite as a function of the volume fraction of  0. Such a plot ispresented in Figure 5.17. It can be seen that the  ow stress does scale withthe fraction of  0-martensite but that the relationship is not linear. Similar1005.6. Relationship between Mechanical Response and Volume Fraction of Phasesfeature was observed in simple shear. This observation suggests that, whilethe rate of transformation plays a very important role in the macroscopicwork-hardening response of the material, one cannot reasonably assume aperfect plastic response for the  0-martensite. This point will be discussedin more detail in chapter 7 and chapter 8.s48s46s48s48s46s50s48s46s52s48s46s54s48s46s56s49s46s48s48s53s48s49s48s48s49s53s48 s32s48s46s53s32s61549s109s32s48s46s57s32s61549s109s32s50s46s50s32s61549s109s32s49s52s32s61549s109s32s50s56s32s61549s109s84s114s117s101s32s115s116s114s101s115s115s32s40s77s80s97s41 s86s111s108s117s109s101s32s102s114s97s99s116s105s111s110s32s111s102s32s61537s39s8226s84s32s61s32s50s53s111s67s61541s32s61s32s49s46s52s120s49s48s45s51s32s115s45s49Figure 5.17: Flow stress obtained under uniaxial tension at room-temperature as a function of the volume fraction of  0.One can use similar reasoning in an attempt to estimate the  ow stressin the  0 phase, recalculated from the macroscopic mechanical equilibrium: theo 0 =   (1 f 0)  f 0(5.6)In this case, the behaviour of the austenite is estimated from the stress-strain response measured at 80 C, during which less than 10% of  0 wasformed.The variation of this theoretical stress in  0-martensite is plotted againstapplied true strain in Figure 5.18. It is important to note that the x-axisof this graph represents the macroscopic strain which is di erent from the1015.7. Summarystrain carried by the  0-martensite. The di erence between these strainsis related to the way in which strains must be partitioned through the mi-crostructure.s48s46s48s48s46s49s48s46s50s48s46s51s48s46s52s48s46s53s54s48s56s48s49s48s48s49s50s48s49s52s48 s32s48s46s53s32s61549s109s32s48s46s57s32s61549s109s32s50s46s50s32s61549s109s32s49s52s32s61549s109s32s50s56s32s61549s109s84s104s101s111s114s101s116s105s99s97s108s32s115s116s114s101s115s115s32s105s110s32s61537s39s32s40s77s80s97s41s65s112s108s105s101s100s32s116s114s117s101s32s115s116s114s97s105s110Figure 5.18: Average theoretical stress in  0 as a function of applied truestrain.In this plot, the theoretical stresses below true strains of  0.2 are sub-ject to large scatter. This is because f 0 appears in the denominator inEquation 5.6, and a small scatter in the volume fraction of  0-martensiteresults in a large scatter of the theoretical stress. For this reason, the cal-culated values corresponding to an  0 volume fraction below 30% have notbeen plotted. Despite this limitation, this plot shows that the (theoreti-cal) mechanical behaviour of  0 is nearly independent of the austenite grainsize. This is an important point that will be revisited when a model for themechanical behaviour of the material is proposed in chapter 8.5.7 SummaryIn this chapter, the mechanical response of grade 301LN has been describedfor tension and shear tests performed at room temperature. It has been1025.7. Summaryfound that the rate of formation of  0 is important for determining themacroscopic work-hardening response of the material. The reduction of therate of  0-martensite with decreasing grain size explains (at least partially)the reduction in macroscopic hardening. This is re ected in the relativelyweak variation of the ultimate tensile strength with grain size comparedto the strong grain size dependence of yield stress with grain size. Alsoobserved here is the fact that testing in simple shear results in a lower rateof transformation compared to tension, particularly at higher strains. Whilethe response in the two strain paths (when compared on the basis of VonMises equivalent stress and strain) are nearly equivalent for   0.2{0.3,above this level of strain the rate of  !  0 transformation slows morequickly in shear than it does in tension, leading to a lower rate of work-hardening in shear.103Chapter 6Characterization of theDeformed Microstructures6.1 IntroductionIn the last chapter, the mechanical properties and phase transformationfrom austenite to  0-martensite were explored from a macroscopic point ofview. The close link between the transition to  0-martensite and the work-hardening rate of the material was illustrated and discussed in relation toprevious work. In this chapter, the details of the autenite to  -martensiteand (particularly)  0-martensite transformation are examined at the micro-scopic scale in order to provide an understanding of the relationship betweenplasticity in austenite and the rate of formation of the martensitic phases.Also, the apparent grain size dependence of the transformation kinetics areexplored. The main tool used for the microstructural observations is electronbackscatter di raction (EBSD). The EBSD technique provides su cient res-olution to allow for identi cation of all three phases (although not all withthe same ease) while providing a more statistically reliable overview of themicrostructure compared to transmission electron microscopy. This chapterbegins by focusing on the transformations and microstructure evolution dur-ing straining for the material with the largest (28 a181m) grain size. Followingthis, observations on the two  nest grain sizes are presented (0.5 a181m and2.2 a181m) and compared against the behaviour of the coarse grained material.Finally, a comparison will be made between the microstructures of samplesdeformed in tension and in shear.1046.2. Experimental Techniques and Representation Convention6.2 Experimental Techniques and RepresentationConventionHigh-resolution EBSD maps were measured using the FEG-SEM describedin chapter 4. The experimental procedure (sample preparation, data treat-ment) is the same as previously described in section 4.2.2, the one di erencebeing that EBSD patterns were indexed considering fcc austenite, bcc  0-martensite and hcp  -martensite as possible phases. The EBSD maps pre-sented in this chapter have not been cleaned so that unindexed points wouldstill be visible. This avoids the possible creation of false information in thedeformed microstructure. The HKL wild spike extrapolation procedure [232]was used to remove single pixels corresponding to wrongly indexed points 8.To represent the microstructures resulting from EBSD maps, a conven-tion has been adopted in this chapter to display austenite using band con-trast maps. While this does not provide explicit information on orientations,it does allow one to identify features such as grain boundaries and planar,deformation-induced, bands. The  0-martensite has been shown superim-posed on the austenite band contrast maps as orientation maps with colourscorresponding to the Inverse Pole Figure (IPF) code shown in Figure 6.1.The colour on the maps corresponds to the crystallographic direction alignedparallel to the rolling direction (RD) of the sample, also parallel to the ten-sile direction or the shear direction. Finally,  -martensite, when present, hasbeen plotted as being red. As will be seen, very little amount of  -martensitehas been indexed and therefore its presence will be explicitly noted in anyof the maps presented.Trace analysis was performed on EBSD maps using a script writtenin MATLAB. This script reads in EBSD data and allows one to comparecrystallographic traces identi ed manually by the user on the map withcrystallographic planes and directions. In particular, this script has beenused in an attempt to identifyf111g planes associated with planar features8 As mentioned in section 4.2.2, wrongly indexed points (or wild spike) are de nedwithin the HKL software to be single pixels that have a misorientation higher than 6{7a176with all surrounding points.1056.2. Experimental Techniques and Representation Convention111101001Figure 6.1: Inverse pole  gure, coloured corresponding to crystallographicdirection parallel to sample direction. This colour scheme is used in all fol-lowing EBSD maps for plotting the orientation of  0-martensite. In thischapter, the  0-martensite maps have been coloured using the rolling direc-tion (RD) as the sample direction in this scheme.observed on the maps. A criterion of less than 2a176 angular di erence betweenexperimental and theoretical traces was used to determine which f111g plane coincided to the observed trace. This allowed, in all cases, for anunambiguous determination of the plane.The Schmid factor of a particular slip system was analysed using theinformation from trace analysis. The systems considered are f111g h211i for the apparition of  -martensite and f111g h110i for the apparition of 0. The Schmid factor (m) was calculated as:m = ( n)k k  b (6.1)where  is the macroscopic stress tensor, n the slip plane normal and bthe slip direction on the slip plane. For each f111g plane, there are threepossible f110g or f112g slip directions, and consequently three possibleSchmid factors. Since the slip direction cannot be unambiguously deter-mined, the Schmid factor reported in this chapter is the highest of the three(in absolute value) for the considered f111g plane.To make a more speci c link between tensile deformation and microstruc-ture evolution, a series of sequential EBSD experiments were performed. Inthese experiments, sub-size tensile samples (small enough to  t inside the1066.3. Microstructure Evolution in Uniaxial Tension: Large Grain Size LimitSEM chamber) were prepared (Figure 6.2) and pre-strained by 15% in ten-sion. The gauge section of this sample was then electropolished and threeareas for each sample were then de ned and marked with microhardnessindents. A high resolution EBSD scan was then made in each of these areas,prior to the material being strained in tension a further 5%. After straining,this area was again measured by EBSD without any further preparation.This allowed for exactly the same area to be observed after the two levels ofstrain. Due to surface roughness, it was found that the results deterioratedsigni cantly between the  rst and second measurement. Attempting a thirdmeasurement on the same area without re-preparation resulted in very lowindexing rates (e.g. 30{70% misindexed phase).72 mm16 mm14 mm6 mm14 mm 30 mmR=5 mm0.8 mmRDTDFigure 6.2: Geometry of small tensile coupons used for sequential EBSD.Finally, TEM observations were made on selected samples after di erentlevels of deformation following the sample preparation procedure de ned insection Microstructure Evolution in UniaxialTension: Large Grain Size LimitThis section starts with an overview of the behaviour of coarse grainedsamples deformed in uniaxial tension. This condition is most similar to themajority of data in the literature and therefore allows for comparisons withexisting data.1076.3. Microstructure Evolution in Uniaxial Tension: Large Grain Size Limit6.3.1 General Overview of Microstructure Evolution as aFunction of StrainAs expected based on the results from the last chapter, a strong evolution ofthe microstructure of the large grain sized samples (D=28 a181m) was observedas a function of tensile strain. Figure 6.3 illustrates the evolution from theaustenite being the majority phase (5% strain) to  0-martensite being themajority phase (41% strain).20 μm RDND(a)  = 5%20 μm (b)  = 10%20 μm (c)  = 41%Figure 6.3: A series of three EBSD maps measured on samples (D=28 a181m)deformed to the three indicated levels of strain. The qualitative evolutionfrom an austenitic microstructure to a martensitic microstructure is clearlyevident.1086.3. Microstructure Evolution in Uniaxial Tension: Large Grain Size LimitAt a qualitative level, one can see that, in each map, at least a smallamount of  0-martensite is present in each grain. The  0-martensite in thesemaps presents a Kurdjumov-Sachs (K-S) orientation relationship with thesurrounding austenite, consistent with the majority of observations in theliterature [74, 110, 233]. At 10% and 41% strain, one can observe that mostgrains have more than one variant of  0-martensite present (as indicated bythe di erent colours of martensite), though many fewer than the 24 possiblefrom crystallographic considerations. At 10% and 41% strain, one can seedark straight features in the band contrast maps for the austenite phase. Thepresence of  -martensite is not readily apparent from maps at this resolution,however, evidence for its existence will be presented below.The  0-martensite observed in the EBSD maps appears in di erent mor-phologies. In some locations, the  0 martensite appears as \blocky" plates(as in Figure 6.4(a)), though the presence of straight, sharp facets on thesefeatures suggests the in uence of crystallography. In other areas, the mar-tensite appears much narrower, appearing in parallel bands across the grains(as in Figure 6.4(b)).1096.3. Microstructure Evolution in Uniaxial Tension: Large Grain Size LimitRDND D=28 μmε=15%5 μm(a)RDND 5 μmD=28 μmε=15%(b)Figure 6.4: Orientation maps illustrating two di erent morphologies of  0-martensite. In (a), the  0-martensite (green in colour) appears \blocky",while in (b) it appears in bands within the grain (here, an annealing twin).1106.3. Microstructure Evolution in Uniaxial Tension: Large Grain Size Limit6.3.2 Relationship between Martensite Morphology andCrystallographyThe appearance of the  0-martensite observed in these maps suggests a linkto the underlying crystallography of the transformation. For example, it wasfound that the thin, parallel  0-martensite plates were usually associatedwith bands of low band contrast observed in the austenite matrix. The mappresented on Figure 6.4(b), taken from a sample strained 15% in tension,illustrates the presence of thin, parallel  0 plates within, and parallel to,these lines of low band contrast.20 μm (111)χ=47°γ(111)χ=83°γ(111)χ=65°γ(111)χ=79°γ(111)χ=89°γ(111)χ=72°γ(111)χ=32°γ(111)χ=77°γRDNDD=28 μmε=15%Figure 6.5: Orientation map showing a few grains of austenite in the 28 a181mcondition, deformed to 15% strain in uniaxial tension. Some planar featuresare underlined, together with their corresponding plane and their inclinationangle (denoted  ) towards TD.1116.3. Microstructure Evolution in Uniaxial Tension: Large Grain Size LimitFigure 6.5 shows the same sample as shown in Figure 6.4(b) but at alower magni cation. By performing trace analysis within this map it wasfound that the low band contrast features are consistent with the trace off111g planes (see thef111g traces in Figure 6.5). Moreover, it was foundthat the vast majority of these f111g are highly inclined with respect tothe surface of the EBSD sample, as illustrated in Figure 6.6.0 10 20 30 40 50 60 70 80 90012345678Inclination angle (χ) in degreesPlanar features frequencyRDNDTD // observation           directionθχPlane normal(111)γFigure 6.6: Angle of inclination of f111g traces observed to correspond tolow band contrast lines in EBSD maps. These measurements come from 25grains. The inset  gure illustrates the relationship between a f111g plane(coloured in pink) and its trace and de nes the angle of inclination,  .The above results show that the observed lines of low band contrastcorrespond to the trace made between one f111g plane and the polishedsurface of the sample, these traces being particularly visible when the associ-atedf111g planes are highly inclined. If one imagines the low band-contrast1126.3. Microstructure Evolution in Uniaxial Tension: Large Grain Size Limitregions in these maps to be thin plates, then the unambiguous observationof the plates, regardless of their internal structure, is easier when they areinclined edge-on (  90a176) than when they are parallel to the surface of thesample. In the case where the plates are parallel to the surface of the sample,two factors will tend to degrade the measurement. First, if the plates arevery thin, then it is likely that the signal from the sample will be dominatedby that of the austenite matrix. Second, given that the signal for EBSDarises only from within the  rst few nanometers of the sample surface [234]and that the volume fraction of plates is low, then the probability of  ndinga plate within the measurement volume will decrease rapidly with angle ofinclination.When the plates are \edge on" and a su ciently small probe is used, amore distinct pattern may emerge, though the surrounding matrix may stilldominate the signal. This would lead to low band contrast, consistent withwhat is observed here. This is consistent with what has been reported inthe case of materials containing  ne deformation twins [235{237].0.5 μm α'α'εγ(a)RDND{111}    pole{0001}   polege(b)<110>    pole<1210>   polegeRDND(c)Figure 6.7: (a) Magni ed view of the area underlined in Figure 6.4(b), show-ing the presence of  -martensite. (b) and (c) show the Burgers orientationrelation in this grain between  and the  -phase observed. (b) representsthef0001g pole  gure superimposed on thef111g pole  gure and (c) rep-resents the h1 210i pole  gure superimposed on the h110i pole  gure.To investigate the cause of the low band contrast along the traces of thef111g planes, a closer examination of the measured data has been made.1136.3. Microstructure Evolution in Uniaxial Tension: Large Grain Size LimitFocusing on Figure 6.4(b), several small regions indexed as  -martensite(0.3{0.5% in area fraction) were found within the low band contrast lines.Figure 6.7 shows one region, taken from the highlighted box in Figure 6.4(b),of pixels indexed as  -martensite. Con dence in these measured fragmentswas provided by the fact that the orientation relationship between the  -martensite points and the surrounding austenite coincides with the Burgersorientation relationship, as expected from the literature [50, 73{75]. More-over, trace analysis shows that the  -martensite’s (0001) plane is parallel tothe f111g whose trace coincides with the line of low band contrast. Thisis consistent with the view that the observed fragment of  -martensite isactually in the form of a very thin plate whose habit plane is parallel to thef111g plane whose trace is observed [80].To con rm the hypothesis of  -martensite in the form of thin plates in thedeformed microstructure of the coarse grained samples, TEM observationswere performed on samples at low levels of strain ( 5% tensile strain).Figure 6.8 shows a region containing planar faults that gave rise to extraspots in the selected area di raction pattern, consistent with  -martensite.6.3.3 Schmid Analysis of f111g Planes Associated withTrace AnalysisAs noted in the literature review (section 2.2.3), previous studies have sug-gested that  -martensite obeys the Schmid law [86]. This interpretationwould be consistent with a mechanism of  -martensite formation associatedwith the passing of partial dislocations on every other f111g h112i slipsystem (cf. section 2.2.3) leading to the shear plane (f111g ) being parallelto thef0001g plane of the  -martensite. Although the EBSD results abovedo not provide evidence that  -martensite exists along each of the low bandcontrast features in the maps, several examples of the correlation betweenthe latter and  -martensite were observed. Here, it is hypothesized that thelow band contrast features are a result of non-indexed  -martensite plates,much like  ne deformation twins in other steels [145, 233].Based on the discussion above, one would expect the majority of low1146.3. Microstructure Evolution in Uniaxial Tension: Large Grain Size Limit0.2 μmD=28 μmε=5%(a) (b)0.2 μm(c)000200012110000100020112011101100111011201120111011001110112022111200200111111022111011(d)Figure 6.8: (a) Bright  eld TEM micrograph of a grain showing a set ofplanar features (b) Selected area di raction pattern exhibiting extra spotscharacteristic of  -martensite . (c) Dark  eld image of the same grain, usingthe h0 110i re ection underlined in (b). (d) Theoretical positions of there ections corresponding to (b).band contrast traces in the EBSD maps to correspond tof111g h112i slipsystems having high Schmid factors. Calculation of Schmid factors in thiscase is not unambiguous since theh112i direction involved in the formationof  -martensite cannot be fully determined 9.The 15 grains shown in Figure 6.9, whose orientations are given in Ta-ble 6.1, were examined to identify the f111g planes associated with thelow band contrast features. In parallel, the Schmid factors for each f111g 9 Shear on the three h112i directions of a given f111g plane all give the same variantof  -martensite [50].1156.3. Microstructure Evolution in Uniaxial Tension: Large Grain Size Limit20 μm RDND2314568D=28 μmε=15%720 μm RDNDD=28 μmε=10%9101211151413Figure 6.9: Two low magni cation orientation maps used for the Schmidanalysis detailed in Table 6.2.h112i system in the 15 grains were calculated, the results being tabulatedin Table 6.2. In this same table, those f111g planes observed to be asso-ciated with low band contrast bands are highlighted in bold. The resultsshow that 10 grains out of the 15 grains analyzed had planar faults on thef111g plane with the highest Schmid factor.As noted in the literature review (section 2.3), the Patel-Cohen calcu-lation of interaction energy has been recently adopted [10, 50, 160] in anattempt to predict variant selection in strain-induced martensitic transfor-mations. Humbert et al. [50] have applied this methodology to predict theexpected variants of  -martensite preceding the formation  0-martensite.One can readily show that, in the case of the formation of  -martensite intension, the interaction energy predicted from the Patel-Cohen model is pro-portional to the Schmid factor for the f111g h112i system for the  ! transformations (for details, see Appendix B). Thus, the results above arefully consistent with the interaction energy hypotheses in that the selec-tion of  -martensite on systems with high Schmid factor is equivalent to theformation of  -martensite having the high interaction energies.1166.3. Microstructure Evolution in Uniaxial Tension: Large Grain Size LimitEuler angles Euler angles Plane Direction Interaction RankingGrain of  of  0 matching matching Energy 1   2  1   2 Condition Condition (J m 3)1 54 38 76 162 19 66 (111) ==(101) 0 [0 11] ==[ 111] 0 0.018 340 34 40 (111) ==(011) 0 [0 11] ==[1 11] 0 -0.031 -219 40 51 (111) ==(101) 0 [10 1] ==[111] 0 -0.034 -230 4 88 (1 11) ==(011) 0 [110] ==[11 1] 0 -0.031 -2 139 45 7 209 48 48 (1 11) ==( 101) 0 [011] ==[111] 0 0.038 2275 51 41 (1 11) ==(1 10) 0 [011] ==[111] 0 0.026 4156 41 42 (111) ==(101) 0 [ 101] ==[ 111] 0 -0.023 -3 103 34 18 110 24 55 (111) ==(101) 0 [0 11] ==[11 1] 0 -0.032 -355 24 36 (111) ==(011) 0 [1 10] ==[11 1] 0 0.01 4121 30 46 (111) ==(101) 0 [10 1] ==[ 111] 0 0.038 14 63 35 63 195 15 30 (1 11) ==( 101) 0 [110] ==[1 11] 0 0.009 461 26 23 (111) ==(011) 0 [0 11] ==[1 11] 0 -0.034 -5 324 24 9 30 50 46 ( 111) ==(110) 0 [0 11] ==[ 111] 0 -0.018 -325 17 56 (111) ==(101) 0 [0 11] ==[11 1] 0 0.036 1183 29 57 (111) ==(011) 0 [10 1] ==[1 11] 0 -0.006 -9 133 34 50 152 33 77 (1 11) ==(0 11) 0 [10 1] ==[ 111] 0 -0.019 -133 45 9 (11 1) ==(0 11) 0 [101] ==[111] 0 -0.014 -31 28 52 ( 111) ==(0 11) 0 [101] ==[111] 0 0.028 2218 32 59 ( 111) ==(110) 0 [01 1] ==[ 111] 0 0.52 1242 21 37 (1 11) ==( 101) 0 [011] ==[111] 0 0.051 210 73 14 51 24 21 58 ( 111) ==( 101) 0 [01 1] ==[1 11] 0 -0.038 -11 283 34 34 3 36 63 ( 111) ==(110) 0 [01 1] ==[ 111] 0 -0.037 -277 42 86 (11 1) ==( 101) 0 [101] ==[1 11] 0 -0.011 -12 82 41 56 85 29 8 (111) ==(011) 0 [10 1] ==[11 1] 0 -0.039 -13 304 37 19 305 29 60 (111) ==(101) 0 [0 11] ==[11 1] 0 0.046 1148 54 34 (111) ==(011) 0 [0 11] ==[11 1] 0 -0.0077 -152 10 78 ( 111) ==(0 11) 0 [101] ==[111] 0 0.036 214 291 44 21 1 44 70 ( 111) ==(110) 0 [110] ==[1 11] 0 0.035 1238 5 66 ( 111) ==( 101) 0 [110] ==[111] 0 -0.035 -15 300 41 15 301 27 59 (111) ==(101) 0 [0 11] ==[11 1] 0 0.043 1147 44 24 ( 111) ==(101) 0 [0 11] ==[11 1] 0 -0.036 -Table 6.1: Identi cation of the plane/direction matching conditions between and  0. 15 grains of austenite (each containing 0{5  0 variants) wereanalyzed. The Euler angles were determined in Bunge’s convention withx1//RD, x2//ND and x3//TD. The interaction energy used to rank thedi erent variants is calculated based on Humbert’s methodology [50], asdetailed in Appendix B.1176.3.MicrostructureEvolutioninUniaxialTension:LargeGrainSizeLimit(111) (  111) (1  11) (  1  11) [  211] [1  21] [11  2]  [211] [  1  21] [  11  2]  [21  1] [121] [1  1  2]  [2  11] [  121] [  1  1  2]1 77 a176 0.12 0.24 0.37 14 a176 0.07 0.15 0.08 56 a176 0.38 0.25 0.13 79 a176 0.10 0.18 0.082 73 a176 0.35 0.05 0.40 67 a176 0.34 0.39 0.05 4 a176 0.07 0.03 0.03 71 a176 0.36 0.17 0.193 73 a176 0.11 0.22 0.33 45 a176 0.07 0.16 0.09 32 a176 0.40 0.25 0.15 81 a176 0.13 0.17 0.034 85 a176 0.10 0.24 0.34 21 a176 0.06 0.13 0.07 50 a176 0.41 0.26 0.14 78 a176 0.14 0.20 0.065 89 a176 0.39 0.03 0.43 57 a176 0.33 0.34 0.00 21 a176 0.13 0.08 0.06 66 a176 0.41 0.13 0.276 87 a176 0.09 0.24 0.33 24 a176 0.05 0.11 0.06 48 a176 0.42 0.28 0.15 79 a176 0.18 0.22 0.047 73 a176 0.06 0.01 0.05 14 a176 0.22 0.41 0.19 58 a176 0.07 0.01 0.08 81 a176 0.25 0.50 0.238 82 a176 0.43 0.10 0.34 81 a176 0.45 0.35 0.10 21 a176 0.33 0.16 0.17 50 a176 0.49 0.26 0.239 73 a176 0.21 0.07 0.14 70 a176 0.43 0 .36 0.08 39 a176 0.31 0.04 0.27 43 a176 0.43 0.43 0.0110 90 a176 0.10 0.32 0.42 23 a176 0.01 0.02 0.01 50 a176 0.36 0.29 0.07 74 a176 0.11 0.30 0.1911 86 a176 0.19 0.15 0.34 43 a176 0.19 0.34 0.15 28 a176 0.25 0.13 0.13 86 a176 0.08 0.04 0.0412 84 a176 0.13 0.12 0.25 31 a176 0.15 0.35 0.19 40 a176 0.34 0.14 0.19 89 a176 0.03 0.04 0.0713 81 a176 0.25 0.11 0.36 55 a176 0.27 0.40 0.13 16 a176 0.12 0.05 0.06 77 a176 0.22 0.14 0.0814 75 a176 0.13 0.10 0.23 50 a176 0.17 0.39 0.21 22 a176 0.31 0.12 0.19 88 a176 0.00 0.10 0.1015 74 a176 0.19 0.14 0.33 54 a176 0.20 0.36 0.16 18 a176 0.24 0.12 0.13 84 a176 0.09 0.06 0.02T able 6.2: All p ossible Sc hmid factors (coun ted p ositiv e) corresp onding to the t w elv e f 111 g  h 112 i  slip systems,in 15 grains. The f 111 g  planes con taining the planar faults are sho wn in b old. It can b e seen that, in 10 grainsout of 15, those features app eared on the planes with highest Sc hmid factor.1186.3. Microstructure Evolution in Uniaxial Tension: Large Grain Size Limit6.3.4 Formation of  0-Martensite and Variant SelectionAs discussed in section 2.2.1, various hypotheses for the mechanism of forma-tion of  0-martensite have been presented in the literature. In this section,the formation of  0-martensite is explored by examining the morphology,crystallography and spatial distribution of this phase. The crystallographyis of particular interest since, as noted above, many fewer than the maxi-mum number of 24 possible crystallographic variants are observed to formin a given grain. The particular selection of variants should relate to themechanism of transformation.In Figure 6.7, a local region from the grain shown in Figure 6.4(b) washighlighted where austenite,  0-martensite and -martensite all co-exist. Theorientation relationship observed for the three phases was identi ed to bevery close 10 to:f111g ==f0002g ==f011g 0 (6.2)h110i ==h2 110i ==h111i 0 (6.3)corresponding to the Burgers orientation relationship between austenite and -martensite and to the Kurdjumov-Sachs (K-S) orientation relationship be-tween  and  0-martensite. These orientation relationships have been com-monly observed before (e.g. [111, 238]) and have been used as evidence for aco-ordinated transformation from austenite to  -martensite to  0-martensite[50, 239].To help explore the crystallographic relationship between phases, a de-tailed analysis of several austenite grains has been made. The grain shownin Figure 6.4(b) and redrawn in Figure 6.10 illustrates one example of an an-alyzed grain where four  0-martensite variants are observed along with one(indexed)  -martensite variant. Three of the  0-variants (numbered #1{3)are seen to co-exist on the samef111gplane as the  -martensite. These threevariants (green, pink and yellow) can be distinguished based on the colourcode of Figure 6.1 showing the crystallographic direction aligned with RD.10 There is always a small amount of spread in orientation relationship associated withthe fact that the phases are plastically deformed.1196.3. Microstructure Evolution in Uniaxial Tension: Large Grain Size LimitA fourth  0-martensite variant (labeled as #4) appears on a second f111gplane parallel to a low band contrast trace. Although variant #1 and #4share the same colour in this map (h110iclose to RD), they are not equiva-lent in orientation.In Figure 6.10, the crystallographic orientation relationship between thethree phases are represented in the form of pole  gures. The top two pole gures reproduce the superimposed austenite and  -martensite pole  gurespreviously shown in Figure 6.7. In the subsequent pole  gures, the close-packed planes (f111g superimposed with f011g 0) and the close-packeddirections of austenite and  0-martensite (h110i superimposed withh111i 0)are presented for each of the four  0-martensite variants. The plane anddirection parallelisms de ned by these pole  gures allow one to distinguishdi erent K-S variants.Consider  rst the plane matching condition f111g // f011g 0 repre-sented in the left most column of pole  gures in Figure 6.10. One can seethat  0-martensite variants #1{3 all have a f011g 0 plane parallel to thesame (111) plane. This (111) plane is parallel to the f0001g plane andboth have their trace parallel to one set of low band contrast features in themap. Consistent with the discussion in the previous section, this f111g //f011g 0 // f0001g plane has the second highest Schmid factor for f111g h112i slip.Variant #4, forms with the plane matching condition corresponding tothe (1 11) plane (the plane having the highest Schmid factor for f111g h112i slip) rather than the (111) plane. The trace of the (1 11) planeis parallel to the second set of low band contrast features observed in thisgrain.1206.3. Microstructure Evolution in Uniaxial Tension: Large Grain Size Limit#2#3#1#4RDND5 μmεD=28 μmε=15%Var. (a) f111g Pole Figure (b) f110g Pole Figure RDNDRDND(111) ==(0001) h110i ==h2 1 10i #1RDNDRD(111)(−111)  ND(111) ==(101) 0 [0 11] ==[ 111] 0#2RDNDRD(111)(−111)ND(111) ==(101) 0 [0 11] ==[1 11] 0#3RDNDRD(111)(1−11)ND(111) ==(101) 0 [10 1] ==[111] 0#4RDNDRD(−111)(1−11)ND(1 11) ==(011) 0 [110] ==[11 1] 0Figure 6.10: Superimposed (a) f111g and f110g 0 pole  gures and (b)f110g andf111g 0 pole  gures, showing the orientation relationship of thefour identi ed variants of  0-martensite as observed in grain 1. Only thetwof111g intersecting planes giving rise to the considered variant of  0 arerepresented on the f110g pole  gure.1216.3. Microstructure Evolution in Uniaxial Tension: Large Grain Size LimitTo provide a more statistical analysis, the 32  0-variants observed in the fteen grains shown in Figure 6.9 have been analyzed. The orientations ofthe austenite grains and the  0-martensite variants are given in Table 6.1.Figure 6.11(a) shows the maximum Schmid factors for f111g h112i 0 slipconsidering the speci c f111g planes satisfying the plane matching condi-tion in the K-S orientation relationship. Based on the plane matching con-dition f111g // f0001g // f011g 0 and the above discussion concerningthe tendency for  -martensite to form onf111g planes having high Schmidfactors, it should be expected that thef111g //f011g 0 planes should alsohave a high Schmid factor. One can indeed see a statistical preference forthe formation of  0-martensite on planes having a high Schmid factor inFigure 6.11(b), with 78% of observed planes having either the highest orsecond highest Schmid factor within the grain. Schmid factorNumber1234051015Schmid factor RankNumberFigure 6.11: (a) Observed distribution of Schmid factors for f111g h112i slip on the f111g planes corresponding to the plane matching conditionin the K-S orientation relationship. (b) Rank of the corresponding Schmidfactors, from highest (1) to lowest (4).These results support the view that  0-martensite forms on pre-existing -martensite plates, which have formed by the glide of f111g h112i dislo-cations on slip systems having high Schmid factors. Correspondingly, theplane matching condition between austenite and  0-martensite should bepredictable based on a prediction of most active austenite slip systems. Thisview corresponds very well with the previous TEM observations of Suzuki etal. [111] who observed a similar tendency for the plane matching conditionin the K-S orientation relationship to correspond to f111g planes havinghigh Schmid factors. Suzuki et al. argued that the presence of  0-martensite1226.3. Microstructure Evolution in Uniaxial Tension: Large Grain Size Limit00. Schmid factorNumber1234051015Schmid factor RankNumberFigure 6.12: (a) Observed distribution of Schmid factors for f111g h110i slip on the intersecting f111g planes de ned in Figure 6.13. (b) Rank ofthe corresponding Schmid factors, from highest (1) to lowest (4).on planes with low Schmid factor increased with increasing fraction of  0-martensite, suggesting that the complex stress state imposed by the alreadyformed  0-martensite would alter the local stress state from the macroscopicstress state assumed in the Schmid factor calculations.The plane matching condition discussed above is necessary but not suf- cient to determine the crystallographic variant of  0-martensite formed.A second condition that can be used to unambiguously identify di erentvariants is that of direction matching (i.e. the parallelism between close-packed directions in austenite and  0-martensite, h110i ==h111i 0). Forexample, the  0-martensite variants (#1{3) in Figure 6.10 share the sameplane matching condition but each can be uniquely identi ed as they havedi erent direction matching conditions.As noted above, the Patel-Cohen interaction energy method for variantselection has been recently applied to the prediction of the  ! ,  ! 0and  !  0 transformations [10, 50, 160]. Here, this method is used toexamine the speci c  0-martensite variants in Table 6.1 assuming that thetransformation proceeds from  !  !  0 and that the plane matchingcondition in the K-S orientation relationship can be predicted based on theSchmid factor argument presented above. Thus, the Patel-Cohen interac-tion energy (based on the methodology presented by Humbert et al. [50],cf. Appendix B for a detailed description of the methodology) has beencalculated for all possible variants of  0-martensite in a grain. Only thosevariants with high, positive values of the interaction energy would be ex-1236.3. Microstructure Evolution in Uniaxial Tension: Large Grain Size Limitpected to form [50]. The results of the interaction energy calculations foreach of the observed variants are shown in Table 6.1. Out of 32 observed  0-martensite variants, 17 are predicted to have negative interaction energies,meaning that they should be energetically unfavourable and those would notbe predicted to form.The limitations of the interaction energy approach for predicting speci c 0-martensite variants was discussed previously by Suzuki et al. [111] whopointed out that only one of two variants having anti-parallel common di-rections (e.g. variants #1 and #2 in Figure 6.10) will tend to be favouredby the interaction energy. Despite this, Suzuki et al. observed several caseswhere both variants were observed experimentally consistent with the obser-vations presented here where anti-parallel variants were observed in grains1, 9 and 14 when only one of the two variants is predicted to be energeticallyfavourable.In the recent work of Malet et al. [161], an alternative geometrical ar-gument was made in order to predict observed variants. In this work, the 0-martensite was observed to form largely at the intersection between  -martensite plates. In this case, the common plane condition of the K-Sorientation relationship was satis ed with one of the two  -martensite plateswhile the common direction condition was de ned by the line of intersec-tion between the two  -martensite plates, ah110i direction. In the work ofSuzuki et al. [111], similar observations were made, though in that case the 0-martensite was triggered by the intersection of dislocations on one glideplane with a fault band ( -martensite) on a second f111g plane. Again,the common direction condition of the  0-martensite was found predomi-nantly to be that de ned by the line of intersection between the twof111g planes. These observations are common with other mechanisms proposedfor  0-martensite nucleation that involve intersecting f111g planes, for ex-ample the Olson-Cohen model for nucleation at intersecting  -martensiteplates [37].In contrast to the work of Malet et al. [161], there is no evidence in thematerials studied here for nucleation at  -martensite intersections. Instead,a more likely situation would be that observed by Suzuki et al. [111] where1246.3. Microstructure Evolution in Uniaxial Tension: Large Grain Size Limitslip on af111g plane intersected an  -martensite plate (parallel to a secondf111g plane) and that this event triggers the nucleation of  0-martensite inthe  -martensite plate. The geometry implied by this process is illustratedin Figure 6.13. The intersecting plane is the f111g plane that shares theK-S common direction with the K-S common plane.Figure 6.13: Schematic showing the geometry at a  / 0/ interface. Inparticular, this shows how the direction parallelism and the plane parallelismfrom the K-S orientation relationship de ne a second austenite plane, theintersecting plane. The transformation strain needed to change  -martensite(blue box) to  0-martensite (red parallelepiped) is dominated by a shearstrain which is illustrated on the box at the point of intersection betweenthe habit plane of the  -martensite (light blue plane) and the intersectingplane (light red plane).The above considerations of slip on the intersecting plane leading tothe nucleation of  0-martensite on an intersected  -martensite plate wouldsuggest the possibility that the direction matching condition in the K-Sorientation relationship could be predicted by looking for candidate inter-secting planes with high slip activity. In order to check this hypothesis,the \intersecting" planes for the 32 variants of  0-martensite in Figure 6.91256.3. Microstructure Evolution in Uniaxial Tension: Large Grain Size Limithave been identi ed and their Schmid factors calculated 11, the results beingpresented in Figure 6.12. Compared to the Schmid factors for the f111g plane satisfying the plane matching condition, the intersecting planes ap-pear to correlate less strongly to a high Schmid factor. Only 62% of theintersecting planes were found to have the  rst or second highest Schmidfactor compared to 78% for the plane matching condition. One of the possi-ble reasons for this is that the intersecting plane must be di erent from thef111g plane satisfying the plane matching condition. This means that it ismore likely that the intersecting plane will necessarily be one of the f111g planes having the lower Schmid factor.Returning to the speci c grain shown in Figure 6.10, one  nds that vari-ant #3 can be associated with thef111g planes having the  rst and secondhighest Schmid factors (speci cally (111) as the plane matching conditionand (1 11) as the intersecting plane). This variant, however, is observed tobe one of the minority variants within the grain. Variants #1, #2 and #4,on the other hand, all share ( 111) as intersecting plane. This plane hasthe lowest maximum Schmid factor amongst the four f111g planes in thisgrain.To summarize the above results, it appears possible to predict the planematching condition of the K-S orientation relationship based on the obser-vation that the variants which form tend to have a high Schmid factor forthe f111g plane which is parallel to the corresponding f110g 0 plane inthe  0-martensite. The prediction of the direction matching condition doesnot seem to reliably correlate with either a condition based on interactionenergy, nor with a condition based on a maximum Schmid factor on an in-tersecting plane. As pointed out by Suzuki et al. [111], there are a largenumber of possible dislocation reactions that one could consider leading toa speci c  0-martensite variant. Not all of these necessitate slip on the in-tersecting plane as de ned above. More detailed work is needed to identifythe speci c mechanisms leading to the selection of the direction matchingcondition in the formation of  0-martensite.11 The Schmid factors used here are for f111g h110i slip. The results are qualitativelythe same if f111g h112i slip is assumed.1266.4. The E ect of Grain Size on the Strain-Induced Formation of Martensite6.4 The E ect of Grain Size on theStrain-Induced Formation of MartensiteIn section 5.5.2, it was shown that the macroscopic kinetics of  0-martensiteformation exhibited a non-monotonic grain size dependence over the rangeof grain sizes prepared in this study. In this section, the microstructuresassociated with these di erent grain sizes will be compared in order to ex-amine possible reasons for this behaviour. In particular this section willfocus on contrasting the largest grain size (D=28 a181m) and smallest grainsize (D=0.5 a181m) samples.2 μm(a) D = 0.5 a181m {  = 0.202 μm(b) D = 2.2 a181m {  = 0.152 μm(c) D = 28 a181m {  = 0.15Figure 6.14: Spacing between observed plates ( or faults bands) when thegrain size is varied.1276.4. The E ect of Grain Size on the Strain-Induced Formation of MartensiteFigure 6.14 illustrates the microstructures of samples having grain sizesof D=0.5 a181m, 2.2 a181m and 28 a181m, all viewed at (approximately) the samemagni cation, after similar levels of strain. Signi cant changes can be seenin the morphology and spatial distribution of  0-martensite within thesemaps. Most clearly, one can see that, by D=0.5 a181m, the grain size is of asimilar order of magnitude as the  0-martensite spacing in the case of theD=28 a181m grain size sample. Thus, whereas in the case of the coarse grainedmaterial several 0-martensite variants were observed in each austenite grain,in the case of the D=0.5 a181m samples, each grain typically contains only oneor two  0-martensite variant.5 μm RDNDD=0.5 μmε=18%(a)2 μm(b)Figure 6.15: Orientation maps of the D=0.5 a181m grain size condition, de-formed at 18% strain in tension. (b) shows a higher magni cation viewof the highlighted area from (a), illustrating the lack of low index qualitybands in the austenite in contrast to the observations in coarse grain size(e.g. Figure 6.5).Figure 6.15 examines in more detail the microstructure of a D=0.5 a181msample, revealing several other features of note. First, from the lower mag-ni cation image of Figure 6.15(a), one has the impression that  0-martensitehas preferentially formed within a band of austenite grains that are parallelto the prior rolling direction. Indeed this observation was made in severalcases and can be linked to the observation of nickel segregation presented in1286.4. The E ect of Grain Size on the Strain-Induced Formation of Martensitesection 4.7.Figure 6.16 shows a correspondence between nickel segregation (appear-ing lighter in Figure 6.16(a)) and the areas that have remained austeniticaccording to the EBSD map in Figure 6.16(b). As was previously noted, thee ect of this segregation is more apparent for the  ne grain sized materialsowing to the similar scale of the segregation bands and the grain size.(a)20 μm RDTDD=0.5 μmε=19%(b)Figure 6.16: Nickel segregation corresponding to the region analyzed inFigure 4.8. (a) Back-scattered electron imaging in the SEM, showing thenickel-rich regions in lighter colours. (b) Austenite orientation map of thesame area showing the islands which remained austenitic.Returning to Figure 6.15 there are several other features that di er,compared to the coarse grained materials. Comparing Figure 6.15 to Fig-ure 6.14(c), one notices that the low band contrast traces observed in thecoarse grained samples are not apparent in the band contrast maps fromthe  ne grained samples. Given the discussion in section 6.3, this wouldsuggest a lack of  -martensite in the case of the D=0.5 a181m samples. This isconsistent with the macroscopic X-ray di raction results from section 5.11,which also showed no evidence for  -martensite in samples with D=0.5 a181m.TEM samples prepared from a  ne-grained sample deformed 5% in ten-sion were also observed. Figure 6.17 shows a low magni cation view of themicrostructure along with a selected area di raction pattern which showsrings characteristic of austenite and  0-martensite, but not of  -martensite.1296.4. The E ect of Grain Size on the Strain-Induced Formation of MartensiteA higher magni cation view of an area from a di erent region in Figure 6.18shows no  -martensite, although extended faults and  ne twins are observed.1 μmD=0.5 μmε=5%(a)(b)Figure 6.17: (a) Low magni cation bright  eld image of a  ne grained sampledeformed to 5% strain. (b) Selected area di raction pattern of the regionviewed in (a). The lines under the di raction pattern show the expectedposition of rings for austenite (blue),  0-martensite (red) and  -martensite(green). No clear evidence for di raction from  -martensite could be found.1306.4. The E ect of Grain Size on the Strain-Induced Formation of Martensite0.5 μmD=0.5 μmε=5%(a)0.5 μm(b)Figure 6.18: (a) Bright  eld image of a grain oriented close to [110] parallelto the beam direction, exhibiting stacking faults along with a set of  netwins (determined based on the extra spots in the accompanying selectedarea di raction pattern). The faults and twins appear to emanate fromgrain boundaries. (b) Dark  eld image showing one set of twins. 1316.4. The E ect of Grain Size on the Strain-Induced Formation of MartensiteIt was argued in section 6.3 that the formation of  0-martensite occurs bythe sequence  ! ! 0. Based on the results presented here for the case ofD=0.5 a181m, this sequence cannot occur due to the absence of  -martensite.In this case, an alternative mechanism leading to the direct transformationfrom austenite to  0-martensite needs to be envisaged.In an attempt to identify more clearly the formation and propagation of 0-martensite in the D=0.5 a181m condition, a series of sequential deformationexperiments were performed. The results of this sequential mapping is shownin Figure 6.19 where the same area is viewed after tensile strains of 0.15 and0.2. In Figure 6.19(a), a number of  0-nuclei are highlighted, all existingat austenite grain boundaries. In this case, it appears that the nucleationof  0-martensite is triggered by events at austenite grain boundaries. Inthe map of the same region following re-straining (Figure 6.19(b)), one ob-serves an increase in  0-martensite fraction. In some areas, this has occurredwith a sudden growth of martensite nearly completely  lling prior austenitegrains, while in other instances the small martensite nuclei observed in Fig-ure 6.19(a) appear to propagate outwards and along grain boundaries.Compiling together the data collected from several maps similar to thoseshown in Figure 6.19, one can examine the distribution of disorientationangles between austenite and  0-martensite both in the case of the coarsegrained (D=28 a181m) and  ne grained (D=0.5 a181m) samples, as shown inFigure 6.20. Here, one can see that both samples show a peak correspondingto the K-S orientation relationship. However, only 26% of the total  0= boundary length does not show the K-S orientation relationship in the caseof the coarse grained samples, while 60% of the boundary length in the ne grained material was non-K-S. These non-K-S misorientations are dueto K-S oriented  0-martensite which is situated at a grain boundary, thedisorientation angle now being characterisitic of the relationship between 0-martensite and the neighbouring austenite grain.1326.4. The E ect of Grain Size on the Strain-Induced Formation of Martensite2 μm RDTDD=0.5 μmε=15%(a)2 μm RDTDD=0.5 μmε=20%(b)Figure 6.19: Sequential orientation mapping performed for a true strain of(a) 0.15 and (b) 0.2 where several grain boundary nuclei of  0-martensitehave been highlighted. Many of these nuclei appear to grow when the strainis increased from 0.15 to 0.2.1336.4. The E ect of Grain Size on the Strain-Induced Formation of MartensiteD=28 m : 26%D=0.5 m : 61%{K-S a' in contact with grain boundariesDeviation from K-S at g/a' grain boundaries (°)Fraction of pixels on a g/a'  boundaryFigure 6.20: Fraction of grain boundaries versus boundary disorientationtaken from EBSD maps corresponding to samples with D=28 a181m and D=0.5a181m. The fraction of boundary length that has a disorientation of greaterthan 6 from the ideal K-S orientation is 26% in the case of coarse grainedsamples while it is 61% in the case of  ne grained samples.The di erence in formation mechanism for  0-martensite in  ne grainedand coarse grained samples appears to also in uence the morphology of the 0-martensite plates. In coarse grained samples, where nucleation is linkedto planar  -martensite plates, the  0-martensite appears to have a planarmorphology. In the  ne grained material, where nucleation appears to belinked to grain boundaries, the  0-martensite appears much less crystallo-graphic and more irregular in shape. No planar  0-martensite was found inthe  ne grained austenite.Given the apparent importance of grain boundaries as sites for the nu-cleation of  0-martensite in  ne grained samples, a question that arises iswhether grain boundary character is important. In particular the role ofannealing twin boundaries is of particular importance given the crystallo-graphic similarity between twin and  -martensite boundaries (cf. section2.2.3). It was reported in the work of Spencer that annealing twin bound-1346.4. The E ect of Grain Size on the Strain-Induced Formation of Martensitearies were one location for  0-martensite nucleation [72]. In order to examinethe possible in uence of grain boundary character on preferred nucleation of 0-martensite, EBSD data from deformed  ne grained samples were analyzedwith speci c reference to orientation relationship of the  0-martensite withits surrounding austenite matrix. In Figure 6.19, only 3 of 30  0-martensitenuclei appear to be in contact with austenite annealing twin boundaries. Toprovide better statistics, the disorientation distribution from several mapshas been examined to look for evidence of particular  0-martensite / auste-nite twin boundary relationships. The K-S orientation relationship can bedescribed in terms of a disorientation angle ( = 42:8 ) about a commoncrystallographic direction (n = [0:97 0:17 0:17]). Twin boundaries in aus-tenite also present a speci c crystallographic orientation described by theaxis/angle combination of  = 60 and n = [1 1 1]. A plate of  0-martensitenucleated on an annealing twin boundary will have a K-S orientation rela-tionship with one of the austenite grains, and a second speci c orientationrelationship with the adjacent twin. This speci c relationship can be de-scribed by the axis/angle combination of  = 42:8 and n = [0:82 0:47 0:32].This relationship has the same disorientation angle as the ideal K-S orien-tation relationship and therefore will not appear as a distinct peak in Fig-ure 6.20. The K-S/Twin relationship does, however, have a di erent axiscompared to the ideal K-S relationship. Therefore, plotting the disorienta-tion data in Frank-Rodrigues space should allow for di erentiation betweenthe exact K-S relationship and K-S/Twin related  0-martensite and auste-nite. Figure 6.21 shows Frank-Rodrigues space12 for the same data as usedin Figure 6.20, where the data has been plotted showing only the surfacecorresponding to an intensity of 10 times the mean intensity. The ideal K-Sorientation relationship is clearly seen at r = [0:38 0:07 0:07]. However, nopeak in intensity is observed corresponding to the location associated withthe K-S/Twin relationship, its ideal location (r = [0:32 0:18 0:13]) beingindicated by the blue circle. These results seem to suggest that there is12 Frank-Rodrigues space represents orientations or misorientations as points at theend of a vector that is de ned as r = tan ( =2)n where n and  are the axis and angledescribing the orientation relationship.1356.4. The E ect of Grain Size on the Strain-Induced Formation of Martensiteno particular (statistical) signi cance of annealing twin boundaries on thenucleation of  0-martensite. In fact, beyond the peak in Figure 6.21 corre-sponding to the K-S orientation relationship, no other strong peaks in theFrank-Rodrigues space could be identi ed suggesting no special relationshipbetween particular boundaries and  0-martensite nucleation.K-STwin + K-SNoiseD=0.5 μmε=20%Figure 6.21: Locus of the grain boundary misorientation, represented in theFrank-Rodrigues space. No particular cluster of orientations can be found atthe twinning relation. The intensity observed close to r = [0:41 0:41 0:17] aresmall misorientations due to misindexing. Note that, due to the symmetry,r = [0:41 0:41 0:17] is a location equivalent to r = [0 0 0] [221].The results above do not prove direct nucleation on grain boundariesbut do seem to indicate that nucleation occurs either at or close to grainboundaries in the case of  ne grained austenite. Indeed, even in the case ofthe coarse grained samples, careful observation reveals many  0-martensiteplates that interact with grain boundaries (e.g. Figure 6.5). Recent work1366.4. The E ect of Grain Size on the Strain-Induced Formation of Martensiteby other authors has also suggested di erences between the behaviour in ne grained and coarse grained samples. In the work of Yang et al., apossible change of the nucleation mechanism for  0-martensite in a 304Lgrade deformed by ECAP was suggested [110]. Di erent morphologies of  0martensite have also been described for austenite grain sizes below 1 a181m [110,240]. In grade 301LN, while the intersections of so-called \shear bands" wereobserved to be at the origin of laths of  0-martensite in the coarse-grainedmaterial, the formation of microtwins (occasionally associated with shortstacking faults) was observed to prevail in the submicron grain condition[240].From EBSD data such as that shown in Figure 6.22, one can estimatethe characteristic scale of both austenite and  0-martensite based on theaverage equivalent area diameter (EQAD) of the phases. The evolution ofthis quantity is plotted as a function of the true strain in Figure 6.23.20 μm RDNDD=28 μmε=41%Figure 6.22: Austenite orientation map of a coarse-grained coupon deformed41% in uniaxial tension.1376.4. The E ect of Grain Size on the Strain-Induced Formation of Martensites48s46s48s46s49s48s46s50s48s46s51s48s46s52s48s46s48s46s49s48s46s50s48s46s51s48s46s52s48s46s53s83s99s97s108s101s32s108s101s110s103s116s104s32s40s61549s109s41s32 s84s114s117s101s32s115s116s114s97s105s110s61537s39s61543(a) D = 0.5 a181ms48s46s48s46s49s48s46s50s48s46s51s48s46s52s48s46s48s46s53s49s46s48s49s46s53s50s46s48s50s46s53s61537s39s83s99s97s108s101s32s108s101s110s103s116s104s32s40s61549s109s41s32 s84s114s117s101s32s115s116s114s97s105s110s61543(b) D = 2.2 a181ms48s46s48s46s49s48s46s50s48s46s51s48s46s52s53s49s48s49s53s50s48s50s53s51s48s83s99s97s108s101s32s108s101s110s103s116s104s32s40s61549s109s41s32s84s114s117s101s32s115s116s114s97s105s110s61537s39s61543(c) D = 28 a181mFigure 6.23: Evolution of the length scale of the microstructure, evaluatedfrom EBSD, in (a) the 0.5 a181m condition, (b) the 2.2 a181m condition, (c) the28 a181m condition. The scale of both phases was evaluated from the EQADon di erent EBSD micrographs.In Figure 6.23(a), one can see that the EQAD of the austenite and  0-martensite in the material having D = 0.5 a181m vary only slightly duringtesting. This is consistent with the observations made in, for example Fig-ure 6.19, which showed that individual austenite grains are consumed by nomore than 3 or 4 (and in many cases only 1)  0-martensite nuclei. It is alsointeresting to note that in Figure 6.19 an apparent growth of pre-existing 0-martensite variants occurred. The results in Figure 6.23(a) show that,1386.4. The E ect of Grain Size on the Strain-Induced Formation of Martensitewhile some apparent increase in the average size of  0-martensite nuclei oc-curs during straining, the change in average size with strain is relativelysmall.In Figure 6.23(b) and (c), a slightly di erent situation can be seen. Here,a substantial re nement of the austenite is observed with straining, againconsistent with many  0-martensite nuclei forming in each grain. In the caseof D=28 a181m, the size of the remaining austenite islands after a strain of 0.4is on the order of 1 a181m. In both cases of D=2.2 a181m and D=28 a181m, the scaleof both  0-martensite and  -martensite converge by the a strain of 0.4. Thesize of  0-martensite islands, however, is relatively constant in both the casesof D=2.2 a181m and D=28 a181m, with the average  0-martensite size increasingfrom 0.3 to 0.7 to 7.2 for grain sizes of D=0.5, 2.2 and 28 a181m respectively.One does, however, note a decrease in the average size of  0-martensite nucleibetween strains of 0.2 and 0.4 for a starting austenite grain size of D=28a181m. Interestingly, at strains below 0.2, the average austenite size is largerthan that  0-martensite size. As the strain is increased, however, the size ofthe austenite continues to decrease. Given that new  0-martensite can onlyform within remaining austenite islands, the average size of  0-martensiteislands must also decrease.The above results would indicate that the transformation kinetics aredominated by the rate of nucleation of  0-martensite, which forms withan approximately strain independent (but austenite grain size dependent)size. This latter point can be understood on the basis that the size of 0-martensite formed will be controlled by microstructural features withinthe material. As the  0-martensite cannot cross austenite grain bound-aries (without destroying the preferred K-S orientation relationship) thelargest possible size of  0-martensite should be the starting austenite grainsize. However, other features such as  -martensite, stacking faults, other  0-martensite variants will also act as obstacles to  0-martensite. In the caseof the  ne grain sizes (D=0.5 a181m and D=2.2 a181m), it is more likely thatan  0-martensite variant can traverse the entire grain before being stopped.In contrast, one would expect more obstacles per grain in the coarse grainlimit and therefore more nuclei per grain for a given volume fraction of1396.4. The E ect of Grain Size on the Strain-Induced Formation of Martensite 0-martensite.s48s46s48s48s46s49s48s46s50s48s46s51s48s46s52s49s69s45s51s48s46s48s49s48s46s49s49s49s48s50s46s50s32s61549s109s48s46s53s32s61549s109s78s117s109s98s101s114s32s111s102s32s61537s39s32s105s115s108s97s110s100s115s32s112s101s114s32s117s110s105s116s32s97s114s101s97s32s40s61549s109s45s50s41s32s84s114s117s101s32s115s116s114s97s105s110s50s56s32s61549s109(a)s48s46s48s48s46s49s48s46s50s48s46s51s48s46s52s48s46s48s49s48s46s49s49s49s48s49s48 s32s50s56s32s61549s109s32s50s46s50s32s61549s109s32s48s46s53s32s61549s109s78s117s109s98s101s114s32s111s102s32s61537s39s32s105s115s108s97s110s100s115s32s112s101s114s32s103s114s97s105s110s84s114s117s101s32s115s116s114s97s105s110(b)Figure 6.24: Evolution of (a) the surfacic rate of  0 nucleation, (b) the rateper grain, for the three conditions of grain size studied. The error bars,when they exist, illustrate the variation measured from di erent orientationmaps of the same condition.1406.5. Microstructure Evolution in Simple ShearOne can investigate the grain size and strain dependence of the numberof  0-nuclei by further interrogating the measured EBSD maps. The numberof unique  0-martensite islands observed per unit observation area is plottedin Figure 6.24(a). This  gure shows a much higher nuclei density in the  negrained materials compared to the coarse grained materials, however, if thedata is plotted as the number of nuclei per grain then one sees that thevalues for grain sizes of D=0.5 a181m and 2.2 a181m are very similar while D=28a181m is higher as would be expected from the description above.6.5 Microstructure Evolution in Simple ShearD=0.5 a181m D=28 a181mRDTD 2 μmD=0.5 μmε=0.12RDTD 10 μmD=28 μmε=14%RDTD 2 μmD=0.5 μmε=25%RDTD 10 μmD=28 μmε=30%Figure 6.25: EBSD inverse pole  gure maps of  0-martensite (colour) over-laid on band contrast maps for austenite illustrating the microstructure ofsamples deformed in simple shear.1416.6. The Link Between Macroscopic Transformation Kinetics and MicrostructureThe macroscopic transformation kinetics presented in section 5.5.2 showedthat the rate of formation of  0-martensite in tension and shear are very sim-ilar when compared on the basis of Von Mises equivalent strain. Particularly,the kinetics are nearly identical up to strains of 0.2{0.3. Above these lev-els of strain, the rate of transformation in shear tended to be lower thanin tension. Following from the results presented above, the microstructureof samples deformed in simple shear can be compared directly with thosedeformed in tension. Consistent with the results presented in section 5.5.2,few di erences in the morphology or crystallography of  0-martensite couldbe found in samples deformed in shear versus tension (Figure 6.25).As in tension,  -martensite was indexed lying along traces of low bandcontrast leading to  0-martensite with similar morphologies, sizes and crys-tallographic orientation relationship as observed in tension. Figure 6.25shows the variation of the microstructure sheared to di erent levels of strain.Though the macroscopic kinetics of  0-martensite transformation appeardi erent in tension compared to shear, no signi cant microstructural di er-ences could be found to correlate with this macroscopic behaviour.6.6 The Link Between MacroscopicTransformation Kinetics and MicrostructureThe results of this section illustrate the complexity of the austenite to  0-martensite transformation in these grades of steel. In the limit of large grainsize, it was observed that the transformation primarily takes place by nu-cleation from  -martensite plates, though some grain boundary nucleationmay occur. While the formation of  -martensite plates appears to dependon the magnitude of the applied stress, no good correlation between the ap-plied stress and the formed  0-martensite could be found. As noted before,this may be due to the fact that one must consider the local state of stressrather than the global state of stress when applying criteria such as thoseapplied above. These results, do however, raise questions about the applica-tion of interaction energy based methods for the prediction of  0-martensite1426.6. The Link Between Macroscopic Transformation Kinetics and Microstructureformation under the conditions considered here.The strongest trend observed here is the correlation between the grainsize, macroscopic transformation kinetics and the role of  -martensite. Itwas argued here that re ning the starting austenite grain size decreasesthe fraction of  -martensite formed and therefore the rate of  0-martensiteformed. Such a decrease in  -martensite formation with decreasing grainsize can be argued in a simple way. The O-C model presented in section 2.4starts from a simple model for the kinetics of \shear band" formation. Inthe material studied here, these shear bands can be identi ed with plates of -martensite. The parameter  in Equation 2.6 can be re-written explicitlyin terms of the volume of an  -martensite plate and the number rate offormation of  -martensite plates,11 f df d =  v dN vd (6.4)where N v is the number of  -martensite plates per unit volume. If theepsilon plates are assumed to be circular plates of constant thickness t, thenone can re-write Equation 6.4 as,11 f df d = t 2 dN vd (6.5)where the size of the  -martensite plates will be determined by the aus-tenite grain size (i.e.   = D). For larger grains, the plates may not be ableto cross an entire austenite grain due to features such as other  -martensiteplates formed in another part of a grain,  0-martensite and dislocation cellwalls, as seen in, e.g. Figure 6.5 and Figure 6.14.Assuming that dN v=d is independent of austenite grain size, Equa-tion 6.5 would predict a rapid decrease in the rate of formation of -martensitewith decreasing grain size. One might, however, argue that grain boundarynucleation of  -martensite may become dominant with a decrease in grainsize. In this case, the grain size dependence of the number density of nucleiwould depend on the grain boundary surface area to volume ratio,1436.6. The Link Between Macroscopic Transformation Kinetics and MicrostructuredN vd /1D (6.6)meaning that overall,11 f df d /D (6.7)While the simple explanation given above qualitatively describes thedecrease in the rate of formation of  -martensite, and therefore a decreasein the rate of formation of  0-martensite, with decreasing austenite grainsize, it does not explain the observed increase in the rate of  0-martensiteformation for D < 1 a181m. It was argued above that this change in  0-martensite transformation rate was due to a change in dominant mechanismof transformation from being dominated by nucleation on -martensite platesto being dominated by nucleation on austenite grain boundaries. In thiscase, one might expect that the rate of grain boundary nucleation of  0-martensite should vary with the grain boundaries’ surface area to volumeratio (1/D).Such a grain size dependence qualitatively predicts the trends observedin Figure 5.12, with the minimum rate of transformation occurring for D=1a181m. Such a model, however, tends to strongly over predict the magnitude ofthe grain size e ect on the macroscopic kinetics. This points out a numberof important questions.As noted above, in the limit of large D,  0-martensite appears to nu-cleate from pre-existing  -martensite plates. However, it was also shown(Figure 6.24) that there was a strong re nement of the size of the austenitephase with strain and the formation of  0-martensite. Based on the abovearguments, this should lead to a decreasing rate of formation of  -martensitein the remaining austenite. Indeed, it may lead to the complete suppressionof  -martensite formation in the  nest austenite regions.Similarly, in the case of the  nest grain sized material, it was observedthat nucleation early in the deformation occurred on grain boundaries.These nuclei, however, were often observed to be smaller than the auste-nite grain size (Figure 6.19). Such a situation could lead to a rapid and1446.6. The Link Between Macroscopic Transformation Kinetics and Microstructurecomplete consumption of all grain boundary nucleation sites as f 0 tends to1.These two observations point to the fact that another mechanism forthe formation of  0-martensite that does not require  -martensite or aus-tenite grain boundaries must exist. In Figure 6.19, it was observed that 0-martensite appeared to \grow" between two sequential images. Similar,though less conclusive, observations were made on sequential deformationexperiments made on coarse grained samples. Such an observation may pointto a further mechanism for the formation of  0-martensite either by a truegrowth of pre-existing nuclei (much as in the case of isothermal martensite[46]) or by a form of nucleation just ahead of pre-existing  0-martensite is-lands. Owing to the di culty of resolving  ne details of the microstructuresof these samples at high strain, no detailed observations on this mechanismcould be made.Finally, it is interesting to note that there were no perceptible changesin the morphology or geometry of samples tested in simple shear comparedto tension, regardless of grain size. The fact that the interaction energywas not capable of predicting the variants of  0-martensite formed in ten-sion supports the idea that the macroscopic stress state is less importantthan what has been recently suggested on the basis of the Patel and Cohenmodel [241]. Indeed, it is perhaps not surprising that this model, originallydeveloped to describe the e ects of an elastic (i.e. nearly uniform) stresson the transformation of thermal martensite does not adequately capturethe e ects of stress on the transformation kinetics in a plastically deform-ing material where large stress concentrations may occur at, for example,dislocation pileups or grain boundaries.Based on the results presented above, the original O-C model (includ-ing grain size dependence) is not capable of predicting the results obtainedhere. However, it remains valuable as an empirical model for describing thepresented data. In this way, some of the ideas presented above have beenused to modify the O-C model so that the observed grain size dependence ofdf 0=d can be captured. First, the fraction of shear bands (fsb) describedin the O-C model is replaced here with the volume fraction of  -martensite1456.6. The Link Between Macroscopic Transformation Kinetics and Microstructure(f ). Considering the comments made above, it is assumed that the rate offormation of  -martensite (df =d ) can be written as:11 f df d =  (6.8)where  is the rate of the  ! transformation, assumed to be inde-pendent of strain. An empirical  t was adopted so that  has the grain sizedependence described above, i.e. = pD +qDr (6.9)where p, q and r have been taken to  t the experimental data displayedin Figure 5.13. Following Olson and Cohen [81], the rate of  0-martensite istaken as:11 f 0df 0d =  (f )n (6.10)where  is taken to be grain size independent, but temperature andstress state dependent. The integrated form of this equation is the same asthe O-C equation presented in Equation 2.11, with the di erence that the -parameter is now dependent on grain size according to Equation 6.9. Forconvenience, n is  xed for all conditions as it was found to be possible to  tall  0 formation kinetics considering n = 5. The best  t values of p, q and rare given in Table 6.3, considering  and n to be independent of grain size.Deformation mode p q r  nUniaxial Tension 0.42 3.41 0.08 4.52 5Simple Shear 0.42 3.41 0.08 3.40 5Table 6.3: Values retained for the modi ed Olson-Cohen model.Using those values, the obtained  ts to the kinetics are presented inFigure 6.26, while the non-monotonic grain-size dependence of  is plottedin Figure 6.27. This is viewed as a purely empirical model that will beused in chapter 8 when a model for the macroscopic mechanical behaviour1466.6. The Link Between Macroscopic Transformation Kinetics and Microstructureis presented.s48s46s48s46s49s48s46s50s48s46s51s48s46s52s48s46s53s46s48s46s50s48s46s52s48s46s54s48s46s56s49s46s48s32s50s56s32s61549s109s32s49s52s32s61549s32s50s46s32s61549s109s32s48s46s57s32s61549s32s48s46s53s32s61549s109s86s111s108s117s109s101s32s102s114s97s99s116s105s111s110s32s111s102s32s61537s39 s84s114s117s101s32s115s116s114s97s105s110s8226s84s32s61s32s50s53s111s67s61541s32s61s32s49s46s52s120s49s48s45s51s32s115s45s49(a)0 .0 0 .2 0 .4 0 .6 0 .80 .00 .20 .40 .60 .81 .0  0 .5  m m 2 .2  m m 2 8  m mVolume fraction of a’ E q u iv a le n t  s t r a in T  =  2 5 o C(b)Figure 6.26: Application of equations 6.8 to 6.10 to reproduce the measuredkinetics of formation of  0-martensite, (a) in uniaxial tension, (b) in simpleshear.s48s46s49s49s49s48s49s48s48s50s52s54s56s61537s71s114s97s105s110s32s115s105s122s101s32s40s61549s109s41Figure 6.27: Variation of the  -parameter as a function of grain size, inpresent empirical model. The validity of this  t is limited to 0.5 a181m D 30 a181m.1476.7. Summary6.7 SummaryIn this section, the detailed microstructural evolution occurring during thestraining of samples having di erent grain sizes in tension and shear havebeen summarized. The key  ndings are that the transformation is nucleatedat  -martensite (coarse grain size) and grain boundaries (small grain size). Ithas been argued that a third mechanism must also exist in order to explainthe continued formation of  0-martensite in the absence of  -martensite oraustenite grain boundaries. The grain size dependence of the transformationkinetics can be qualitatively understood in terms of the grain size depen-dence of the formation of  -martensite as well as the increasing density ofgrain boundary nuclei with decreasing grain size. This picture would leadto the non-monotonic transformation kinetics observed here, for which anempirical model is proposed. Owing to the failure of criteria based on themacroscopic stress to predict the formation of individual  0-martensite nu-clei, further work is needed to help provide insight into the mechanisms thatlead to nucleation of  0-martensite during straining.148Chapter 7A Novel Method ofEstimating the Stresses in 0-Martensite7.1 IntroductionIn chapter 5, the bulk mechanical properties of 301LN stainless steel mea-sured in tension and shear were presented. To develop a physically baseddescription for these results, one must understand the contributions comingfrom the individual phases [242]. The complexity of this for the presentmaterial comes from the fact that three phases can co-exist (austenite,  -martensite,  0-martensite) and that the fraction of these phases is continu-ously evolving with strain. In the previous two chapters it has been shownthat  -martensite exists only as a minority phase (if it is formed at all).Owing to this observation, and to simplify the following discussion, the me-chanical behaviour of 301LN will be attributed to the austenite and the  0-martensite phases only, the possible e ect of  -martensite being neglected.This approach is the one that has been generally adopted in the descriptionof metastable austenitic stainless steels in the literature [34, 207].Understanding the contributions to the mechanical response arising fromthe austenite and  0-martensite phases requires some estimation of thestresses carried by these two phases in situ. Measurements of the in situpartitioning of stresses between  and  0 during co-deformation of austeni-tic stainless steels have previously been made using di raction-based tech-niques (e.g. neutron, X-ray) [13, 194, 195, 243, 244]. While powerful, these1497.2. Magnetostriction and the Magnetomechanical E ecttechniques have limitations, not the least of which is the need for access tomajor facilities in the case of experiments requiring a neutron or synchrotronsource.This chapter presents an alternative method for the estimation of thestress partitioning in metastable austenitic stainless steels by means of themagnetomechanical e ect. It will be shown that this relatively simple tech-nique gives results that are in excellent agreement with the results of neutrondi raction and X-ray di raction measurements on the same grade of stain-less steel. Moreover, the results of these measurements give important in-formation required for the interpretation of the overall mechanical responsein terms of the behaviour of the individual phases.7.2 Magnetostriction and theMagnetomechanical E ectMagnetostriction describes the shape change of a ferromagnetic materialduring the process of magnetization [245]. Under an applied magnetizing eld H, favourably oriented magnetic domains will tend to grow by meansof magnetic domain wall migration and/or domain rotation. This occurs soas to reduce the magnetic energy at the expense of increased elastic strainenergy in the crystal. In the case of linear magnetostriction, sometimesreferred to as the Joule e ect, a ferromagnetic sample magnetized in a uni-form  eld will undergo a strain denoted as  . Normally, it is the value of measured at magnetic saturation ( s) that is reported in the literature.The magnetostriction strain, is actually a tensor,  s being the componentof the magnetostriction strain observed parallel to the direction of magneti-zation. Linear magnetostriction is volume conserving so that the principalcomponents of the magnetostriction strain obey  1 +  2 +  3 = 0. Magne-tostriction is a function of imposed magnetic  eld, the magnetic propertiesof the material being magnetized and also the crystallographic directionparallel to the axis of magnetization. In the case of pure iron, the magne-tostriction coe cients at saturation parallel to theh100iandh111idirections1507.2. Magnetostriction and the Magnetomechanical E ectare  h100i = 15 10 6 and  h111i = 21 10 6, i.e. magnetization parallelto h100i leads to an extension of the material while magnetization paralleltoh111ileads to contraction parallel to this direction. An untextured poly-crystal behaves isotropically, the magnetostriction coe cient of iron in thiscase being equal to  iso = 7 10 6.From an engineering point of view, magnetostriction is a well knownphenomenon that is central to many technologies [246]. For instance, mag-netostriction is used in the production of high frequency actuators as wellas in magnetic sensors. The most well-known engineering consequence ofmagnetostriction is that it is the source of the \hum" coming from electricaltransformers.Randomly oriented domains(a)Resulting FieldResulting Fieldσ(b)Figure 7.1: Schematic representation of the Villari e ect. (a) In the absenceof a mechanical stress, the magnetic domains tend to display randomly-oriented magnetization and the overall magnetic  eld is zero. (b) Oncesubmitted to a mechanical stress, those favourably oriented start expandingto the detriment of others, thus creating a magnetic  eld.In this study, it is the inverse e ect, often called the magnetomechanicale ect, that is of interest. While magnetostriction describes the straining ofa ferromagnetic sample due to an imposed magnetic  eld, the magnetome-1517.2. Magnetostriction and the Magnetomechanical E ectchanical e ect describes the change in magnetization induced by an appliedstress. Other names are used in conjunction with the magnetomechanicale ect. Piezomagnetism is sometimes used to describe this e ect, while theterm Villari e ect is commonly used to describe the changes in the magneticsusceptibility at low magnetization due to an imposed stress or strain.An imposed elastic strain (stress) couples to magnetic domains in a fer-romagnetic material causing those domains that are favourably oriented togrow by means of domain wall migration and (to a lesser extent) domainrotation. This occurs to reduce the total combined magnetic and elasticenergy of the system. Figure 7.2 shows the magnetization of an initiallydemagnetized sample of a low carbon steel under di erent levels of appliedstress [247].s48s53s49s48s49s53s48s46s48s48s46s53s49s46s48s49s46s53 s61555s32s61s32s50s48s32s77s80s97s61555s32s61s32s49s48s32s80s97s61555s32s61s32s45s50s48s32s77s80s97s61555s32s61s32s48s32s77s80s97s65s112s108s105s101s100s32s109s97s103s110s101s116s105s99s32s102s105s101s108s100s32s66s32s40s84s101s115s108s97s41s77s97s103s110s101s116s105s122s105s110s103s32s102s105s101s108s100s32s72s32s40s107s65s46s109s45s49s41s61555s32s61s32s45s49s48s32s80s97Figure 7.2: Variation of the anhysteretic magnetization curves with truestress, as measured in a Fe-2%Mn steel. Adapted from [247].The e ect is clearly complex as the magnetomechanical e ect does notonly change the saturation magnetization but also the apparent magneticsusceptibility. The origins of this e ect are complex and di cult to describeanalytically since it arises from electronic phenomena at the atomistic scale.1527.2. Magnetostriction and the Magnetomechanical E ectJiles et al. have developed a methodology based on thermodynamic argu-ments giving the result that the e ect of an imposed stress can be taken asbeing equivalent to an extra \e ective" applied magnetic  eld [248]. Thise ective  eld can be written as,H = 32   0 d dM  (7.1)where M = B= 0 H is the magnetization,  is the applied stress and 0 is the permeability. The calculation of d =dM requires a knowledgeof  as a function of both M and  . Analytical models describing thisfunctional dependence are not generally available. Jiles et al. have taken anempirical approach to describing  as a function of M and  [248], theseresults showing good agreement with the anhysteritic behaviour shown inFigure 7.2.Although less well understood compared to magnetostriction, the mag-netomechanical e ect has been used practically in a number of sensor tech-nologies, including the estimation of stresses within engineering components(see e.g. [246]).As described in Appendix A and section 4.2.1, a magnetic sensor (Fer-itscope MP30) has been used to estimate the volume fraction of ferromag-netic  0-martensite in this thesis. Other authors using the same approachhave noted that the measurements of  0-martensite made in this way shouldbe carefully analyzed to avoid the magnetomechanical e ect arising fromstresses in the  0-martensite phase [34, 249{251]. In a limited number ofcases, however, the application of the magnetomechanical e ect to estimat-ing stresses in ferromagnetic materials has been explored. Kaleta et al.explored the possibility of using the magnetomechanical e ect as a way ofassessing fatigue life for samples where the cyclic strains are very small(stresses close to the fatigue limit) [252]. It was shown that the magneticmeasurements could be well correlated to the fatigue life. Moreover, it wasshown that a measurement of the magnetizing  eld strength (H) correlatedwell with the stress carried by a sample of pure nickel.The recent work of Post et al. [70] is notable in that it marks a  rst1537.3. Experimental Techniquesattempt at using the variation in magnetic induction with applied stress asan estimator for the stresses in strain-induced  0-martensite. In this case, asensor was attached to a tensile sample and measurements of induced volt-age di erence between two sensing coils was used to estimate the fraction of 0-martensite. It was found, however, that the same sample had a di erentmagnetic response if the induction was performed under load or on unload-ing. A systematic decrease in induction was observed with higher strain andvolume fraction of  0-martensite. Post et al. [70] used the signal measuredunder load compared to samples under no load as a way of correcting theinduction measurement so as to be able to have a measure of the fractionof  0-martensite during continuous loading of the sample. In this work, itwas implied that the e ect of the stress on the magnetic induction was dueto the hydrostatic component of the macroscopically imposed stress state,though in reality the stress state in the  0-martensite will not be the sameas that imposed macroscopically. As discussed above the magneto-elasticcoupling arising from the magnetomechanical e ect in strong ferromagneticmaterials (such as  0-martensite) is not purely dilational. Moreover, thestress-state within the individual  0-martensite nuclei should vary stronglyfrom the macroscopic state of stress (particularly at low volume fractionsof  0-martensite). Despite these issues, the results of Post et al. are veryencouraging as one could argue that the measured variation in magneticinduction with stress in  0-martensite, if appropriately calibrated, could beused as a scalar estimate of the in situ stresses carried by  0-martensiteduring the deformation of a metastable austenitic stainless steel. In thischapter, an attempt to make this correlation is presented.7.3 Experimental TechniquesIn these experiments, the variation of Feritscope signal (directly related tothe magnetic induction measurements made by Post et al. [70]) have beenmade during the tensile deformation of 301LN. Unless otherwise speci ed,the geometry of the tensile coupons is the one which appears in Figure 5.1,and the deformation setup is the one de ned section 5.2.1, both for room-1547.3. Experimental Techniquestemperature and 80 C tensile loadings. The Feritscope measurements weremade according to the procedure described in section 4.2.1. Six measure-ments were made in the gage length of the sample for each condition mea-sured in order to assess the uniformity of the measurement.As noted above, the magnetomechanical e ect is very complex and, atpresent, only semi-empirical relations exist to describe it. Moreover, the Fer-itscope used in these experiments does not produce a simple axial magnetic eld of uniform strength [216]. Indeed, the Feritscope induces a spatiallynon-uniform  eld in the sample to be measured. Given all of these complex-ities, a simpli ed empirical procedure has been used in an attempt to cali-brate the e ect of applied stress on the Feritscope response. This involvedthe production of a sample containing approximately 100% martensite as areference sample from which to calibrate the stress sensitivity of the results.The reference sample was produced by performing a tensile test to thepoint of necking in a bath of liquid nitrogen at -196 C). In this case the rateof work-hardening in the sample was very large (Figure 7.4) and, therefore,to avoid the spreading of plasticity into the grip section the width of thegrip area had to be increased compared to the sample geometry describedin Figure 5.1. This new sample geometry is given in Figure 7.3.R=15 mm6 mm0.8 mm158 mm50 mm 28 mm 50 mm36 mmRDTDFigure 7.3: Geometry of the tensile coupons used at cryogenic temperatures.Tensile samples for low temperature testing were prepared from rolledsheet and annealed following the annealing procedure required to produce a28 a181m average grain size (condition E in Table 4.2). The tensile curve fromthis test (Figure 7.4) exhibits an upper and lower yield stress, as well as aplateau indicating strain localization. As mentioned in section 2.5.1, thisfeature is common for low temperature testing of this grade of austeniticstainless steels [10].1557.4. Estimation of Stress Carried by  0-Martensite via the Magnetomechanical E ects48s46s48s48s46s49s48s46s50s48s46s51s48s53s48s49s48s48s49s53s48s50s48s48s50s53s48s84s114s117s101s115s32s115s116s114s101s115s115s32s40s77s80s97s41s84s114s117s101s32s115s116s114s97s105s110s8226s84s32s61s32s45s49s57s54s111s67s61541s32s61s32s49s46s52s120s49s48s45s51s32s115s45s49Figure 7.4: Tensile stress-strain curve of the D=28 a181m test coupon at cryo-genic temperature.The volume fraction of  0-martensite in this reference sample was mea-sured using the Feritscope and found to be equal to 91%. This comparesto a maximum of 81%  0-martensite measured at the onset of necking in asample tested at room temperature.7.4 Estimation of Stress Carried by  0-Martensitevia the Magnetomechanical E ectThe magnitude of the magnetomechanical e ect on Feritscope measurementsis illustrated in Figure 7.5(a), where the results of Feritscope measurementsare plotted for two tensile loading experiments 13 made on samples havingan austenitic grain size D=28 a181m. The results for other conditions of grainsize will be discussed in section 7.5. In the  rst experiment, the test wasinterrupted and the sample periodically unloaded (to 0 MPa stress) and a13 The experimental setup for shear testing did not allow for in situ measurement usingthe Feritscope during testing. Thus, only results for uniaxial tension are shown here.1567.4. Estimation of Stress Carried by  0-Martensite via the Magnetomechanical E ects48s46s48s46s49s48s46s50s48s46s51s48s46s52s49s48s50s48s51s48s52s48s53s48s54s48s85s110s108s111s97s100s101s100s32s40s70s83s48s41s32s85s110s100s101s114s32s108s111s97s100s32s40s70s83s41s70s101s114s105s116s115s99s111s112s101s32s40s97s114s98s46s32s117s110s105s116s115s41 s84s114s117s101s32s115s116s114s97s105s110s68s61s50s56s61549s109s84s61s50s51s111s67s48s46s48s46s49s48s46s50s48s46s51s48s46s52s50s48s52s48s54s48s56s48s49s48s49s50s48s49s52s48 s68s61s50s56s61549s109s84s61s50s51s111s67s84s114s117s101s32s115s116s114s101s115s32s40s77s80s97s41 s84s114s117s101s32s115s116s114s97s105s110s85s110s108s111s97s100s101s100s110s100s101s114s32s108s111s97s100Figure 7.5: (a) Evolution of the Feritscope signal during straining, withmeasurements performed on two samples, under load and unloaded. Theerror bars show the range of the signal measured by the Feritscope duringsix measurements. (b) Corresponding stress-strain data, with the actualmeasurement points indicated.Feritscope measurement made. In the second experiment, a sample of thesame material was tested in the same way, being interrupted at the samelevels of strain but instead of unloading, the sample was held under load ata  xed actuator position and Feritscope measurements were made. For themeasurements under load, the tensile tests had to be interrupted with thetensile machine in displacement control. During these holds the load wasobserved to drop by 10{12% (relative to the stress at the point where thetest was stopped) because of stress relaxation. The results presented beloware all plotted with respect to the stress measured at the point where thetest was stopped.While, overall, the two curves in Figure 7.5 are similar, both exhibitinga sigmoidal shape characteristic of the transformation kinetics, they beginto diverge above a true strain of approximately 0.2. These results are qual-itatively very similar to the results presented by Post et al.[70] and indicatean increasing importance of the magnetomechanical e ect with increasingmacroscopic stress and transformed fraction. Here, the results obtained inthe macroscopically unloaded state (F0S) are used to estimate the volumefraction of  0-martensite as described in section 5.5.2. Even when the two-1577.4. Estimation of Stress Carried by  0-Martensite via the Magnetomechanical E ectphase austenite/ 0-martensite mixture is macroscopically in the unloadedstate, the individual phases will contain residual stresses. While this couldlead to errors in the estimated fraction of  0-martensite due to the magne-tomechanical e ect even in the macroscopically unloaded state, the resultsshown below will indicate that the stress e ect on Feritscope measurement isnegligible for stresses below 600 MPa. A  nal point on these measurementsis that it has been indicated that plastic deformation can also in uence themagnetic response of a ferromagnetic material [70]. This can be due tomacroscopic texture changes (magnetic anisotropy) as well as through thee ect of dislocations on the domain structure. While it is not envisionedthat the texture of the  0-martensite changes drastically with deformationin this material [238], it is very di cult to separate the remaining e ects ofplastic deformation from the e ects of stress (elastic deformation) over therange of strains applied to the material. Thus, in this work we consider allchanges in Feritscope signal to arise from either changing volume fractionof  0-martensite or to intrinsic stresses in pre-existing  0-martensite.In order to interpret the results in Figure 7.5(a) in terms of the stresscarried by the  0-martensite phase, it is necessary to use the reference sam-ple de ned in section 7.3 which contains nearly 100%  0-martensite. Thispre-deformed sample was incrementally re-loaded elastically at room tem-perature. At predetermined levels of elastic strain, Feritscope measurementswere made while the sample was held under load. These measurements areshown in Figure 7.6(a), where the raw Feritscope measurement is givenwithout the correction factor described in Appendix A. As can be seen,the Feritscope reading is not very sensitive to stress for low stresses but thesignal decreases rapidly with increasing stress. In order to normalize thisresult with respect to the fraction of  0-martensite in the sample, one canplot the ratio of the Feritscope measurement under load to the Feritscopemeasurement under no load, i.e. Fs/F0s .In a fully martensitic sample, Figure 7.6(a) should be su cient to use asa calibration of stress born by the  0-martensite phase as a function of Fs/F0sfor any value of pre-strain and therefore any value of F0s . In the referencesample considered here, the microstructure is not fully martensitic but (as1587.4. Estimation of Stress Carried by  0-Martensite via the Magnetomechanical E ects48s53s48s49s48s49s53s48s52s48s52s53s53s48s53s82s101s102s101s114s101s110s99s101s32s57s49s37s32s61537s39s84s32s61s32s50s51s111s67s70s101s114s105s116s115s99s111s112s101s32s114s101s97s100s105s110s103s32s40s70s83s41s32 s84s114s117s101s32s115s116s114s101s115s32s40s77s80s97s41s70s83s48(a)s48s53s48s49s48s49s53s48s48s46s54s48s46s55s48s46s56s48s46s57s49s46s48s32s77s101s97s115s117s114s101s32s112s111s105s110s116s115s32s80s111s108s121s110s111s109s105s97s108s32s102s105s116s58s97s46s61555s50s32s43s32s98s46s61555s32s43s32s99s32s32s97s32s61s32s45s49s46s52s69s45s55s32s77s80s97s45s50s32s32s98s32s61s32s51s46s56s54s45s53s32s97s45s49s32s32s99s32s32s49s46s48s50s70s83s32s47s70s83s48s84s114s117s101s32s115s116s114s101s115s32s105s110s32s61537s39s32s40s77s80s97s41s82s101s102s114s101s110s99s101s32s57s49s37s32s61537s39s84s32s61s32s50s51s111s67(b)Figure 7.6: (a) Feritscope measurements (FS) obtained in the referencesample, when reloaded elastically at room temperature. (b) Evolution, inthe reference, of FS normalized by the the Feritscope measurement at zeroapplied stress (F0S) as a function of applied stress. The error bars only showthe spread in the FS measurement.noted above) was measured to contain  9% retained austenite. As a  rstorder approximation, it is assumed that the fraction of the stresses borne bythe austenite ((1 f 0)  ) are small relative to the fraction of the stressesborne by the  0-martensite. The relationship between the macroscopic stress( ref) and the stress carried by the martensite (  0) can then be given bythe condition for stress equilibrium as, ref = f 0  0 + (1 f 0)   f 0  0 (7.2)and therefore,  0  ref=f 0 (7.3)The resulting relationship between the Feritscope measurement and thestress   0 is illustrated in Figure 7.6(b). To be able to use this result asa calibration curve for other samples, an empirical polynomial  t has beenmade to the experimental data, as shown in Figure 7.6(b).Returning to the data shown in Figure 7.5 it is now possible to estimatethe level of stress carried by the  0-martensite using the calibration given1597.4. Estimation of Stress Carried by  0-Martensite via the Magnetomechanical E ectin Figure 7.6(b). As one can see, the sensitivity of this technique is poor atstresses   0 < 600 MPa, a result that could re ect the di culty of movingdomain walls below a critical threshold stress [220]. However, for data abovethis level of stress, the e ect of stress on Fs=F0s is signi cant and easilydetectable. By taking the data from the two curves given in Figure 7.5(a) togenerate Fs=F0s and using the empirical  t to the data in Figure 7.6(b), onecan obtain an estimate for   0 as shown in Figure 7.7(a). The error bars on  0 have been estimated based on the the uncertainty in f 0. Though regularmeasurements of Fs and F0s were made as a function of strain, it was foundthat reliable estimates of   0 could only be found for measurements abovestrains of 0.2. Below this level of strain the uncertainty in the Feritscopemeasurement (as re ected by the variation coming from the 6 measurementson a single sample) overwhelmed the ratio Fs=F0s , which is seen in Figure 7.5to be very small for strains below 0.2.As will be discussed further in chapter 8, the critical parameter relatingto the stress borne by the  0-martensite phase is f 0  0. Plotting f 0  0rather than   0 has the added advantage that it reduces the signi cance ofthe errors noted above when f 0 is low. In order to compare f 0  0 withthe macroscopic  ow stress, both are plotted in Figure 7.7(b). The shapeof f 0  0 appears sigmoidal re ecting the evolution of f 0 with strain. Boththe magnitude and shape of f 0  0 are, however, in uenced by   0. This isconsistent with the discussion in section 5.6, where it was shown that themacroscopic hardening rate could not be explained by df 0=d alone and thatsome intrinsic hardening of the  0-martensite phase must also be present..7.4.1 Comparison of Results with Other Estimates forStresses in MartensiteThe results presented in Figure 7.7 can be compared to various other experi-mental methods for estimating the value of f 0  0. Here, such a comparisoncan be made against estimates arising from an assumed behaviour for aus-tenite and, more reliably, against results arising from neutron di raction1607.4. Estimation of Stress Carried by  0-Martensite via the Magnetomechanical E ects48s46s48s46s49s48s46s50s48s46s51s48s46s52s48s46s53s50s48s52s48s54s48s56s48s49s48s49s50s48s49s52s48s68s32s61s32s50s56s61549s109s84s32s61s32s50s51s111s67s73s110s116s114s105s110s115s105s99s32s116s114s117s101s32s115s116s114s101s115s32s40s77s80s97s41 s65s112s108s105s101s100s32s116s114s117s101s32s115s116s114s97s105s110s126s61472s61549s47s49s48s61555s116s111s116s61555s61537s39(a)s48s46s48s46s49s48s46s50s48s46s51s48s46s52s48s46s53s50s48s52s48s54s48s56s48s49s48s49s50s48s49s52s48 s61555s116s111s116s102s61537s39s32s61555s61537s39s73s110s116s114s105s110s115s105s99s32s116s114s117s101s32s115s116s114s101s115s32s40s77s80s97s41 s65s112s108s105s101s100s32s116s114s117s101s32s115s116s114s97s105s110s68s32s61s32s50s56s61549s109s84s32s61s32s50s51s111s67(b)Figure 7.7: (a) Intrinsic stresses measured in the  0-martensite. (b) Sameas (a) but multiplied by the volume fraction of considered phase. Points arethe actual measurements.experiments.In chapter 5, macroscopic tensile curves for 301LN were measured at80 C. These results indicated that, after correcting the yield strength fortemperature, the hardening rates of the samples tested at 80 C was nearlyidentical to the behaviour measured at room temperatures for strains up toapproximately 0.15 (i.e. up to the point where df 0=d becomes signi cant).If it is hypothesized that this behaviour can be extrapolated to explain thebehaviour of austenite over the full range of strains investigated at roomtemperature, then an estimate for f 0  0 based on equilibrium of stressescan be obtained as:f 0  0 =  tot (1 f 0)  (7.4)The magnitude of f 0  0 estimated in this way based on the 80 C tensiledata from Figure 5.6 is compared to the Feritscope estimate of f 0  0 inFigure 7.8.While the above analysis requires strong assumptions (both in terms ofthe estimates in   as well as in the Feritscope estimated   0), the resultsshow remarkably good agreement. This is particularly true considering that1617.4. Estimation of Stress Carried by  0-Martensite via the Magnetomechanical E ects48s46s48s48s46s49s48s46s50s48s46s51s48s46s52s48s46s53s48s53s48s49s48s48 s32s80s114s101s115s101s110s116s32s115s116s117s100s121s32s102s114s111s109s32s97s117s115s116s101s110s105s116s101s32s97s116s32s56s48s111s67s32s78s101s117s116s114s111s110s32s100s105s102s114s97s99s116s105s111s110s32s100s97s116s97s102s61537s39s32s61555s61537s39s32s40s77s80s97s41 s65s112s108s105s101s100s32s116s114s117s101s32s115s116s114s97s105s110Figure 7.8: Comparison of the fraction of stresses in the  0-martensite ob-tained from Feritscope measurements compared with neutron di ractionmeasurements [13] and theoretical stresses obtained from Figure 5.18, byextrapolating the behaviour of the austenite from 80 C tests.the behaviour of the austenite has had to be extrapolated well beyond whereit can be independently measured at room temperature and to strains wherethe remaining volume fraction of austenite is low.A second check of the Feritscope estimated f 0  0 can be made in com-parison with neutron di raction measurements. An independent study, byDufour [13] at the Universit e Catholique de Louvain was undertakenusing in situ time of  ight neutron di raction measurements during tensiletesting of the same steel as studied here. These experiments were performedto estimate the stresses in both austenite and  0-martensite by means ofchanges in the lattice parameters of the austenite and  0-martensite phases.Rietveldt whole-pattern analysis of neutron di raction spectra provided theevolution of the lattice parameters for di erent crystallographic planes ofeach phases. The determination of a strain-free lattice parameter for the 0-martensite was carried on by di erent methods, including the X-ray dif-fraction measurement of internal stresses by the sin2 method. Knowing1627.4. Estimation of Stress Carried by  0-Martensite via the Magnetomechanical E ectthe change in unit cell parameters as a function of macroscopic strain allowsfor an average elastic strain to be estimated (using the stress-free latticeparameter) for both austenite and  0-martensite. Based on the Rietveldanalysis, an average behaviour of the phase was used to estimate the av-erage stresses carried by each phase. The details of the calculation of theintrinsic stresses can be found in reference [13].While the same grade of steel (provided by ArcelorMittal) as analyzedby Dufour was used in this study, the material used by Dufour had beengiven a small skin pass as a  nal step in the processing before  nal coiling.It was found that the macroscopic tensile response of the material studiedby Dufour matched very well to the stress-strain response of the materialstudied here if their data was o set by a 6% strain. This is illustrated inFigure 7.9. Moreover, if the work-hardening rates of the two materials arecompared on the basis of stress rather than strain, one  nds that they arenearly identical, as one would expect. Thus, in the following comparisonsDufour’s data has been corrected by adding 6% strain corresponding to theshift shown in Figure 7.9.s48s46s48s46s49s48s46s50s48s46s51s48s46s52s48s46s53s46s48s46s50s48s46s52s48s46s54s48s46s56s49s46s48s32s68s117s102s111s117s114s39s115s32s116s117s100s121s32s80s114s101s115s101s110s116s32s119s111s114s107s86s111s108s117s109s101s32s102s114s97s99s116s105s111s110s32s111s102s32s61537s39s32s109s97s114s116s101s110s115s105s116s101s84s114s117s101s32s115s116s114s97s105s110(a)s48s46s48s46s49s48s46s50s48s46s51s48s46s52s48s46s53s53s48s49s48s49s53s48s32s68s117s102s111s117s114s39s115s32s116s117s100s121s32s80s114s101s115s101s110s116s32s119s111s114s107s84s114s117s101s32s115s116s114s101s115s32s40s77s80s97s41 s84s114s117s101s32s115s116s114s97s105s110(b)Figure 7.9: Comparison of (a) the  ! 0 transformation kinetics, and (b)overall stress-strain curve in the two grades of 301LN. The data originatingfrom Dufour’s work [13] was shifted 6% so that the  ! 0 transformationkinetics would match present study.The neutron di raction results from Dufour are reproduced in Figure 7.10,1637.4. Estimation of Stress Carried by  0-Martensite via the Magnetomechanical E ectshowing only the average response for the austenite and  0-martensite curvesalong with the total macroscopic stress. It is notable that the  0-martensiteappears under compression at small strains, though the low  0-martensitevolume fractions at these levels of strain makes the measurements very sen-sitive to any sources of error.s48s46s48s48s46s49s48s46s50s48s46s51s48s46s52s48s53s48s49s48s48s49s53s48 s109s97s99s114s111s115s99s111s112s105s99s97s117s115s116s101s110s105s116s101s84s114s117s101s32s115s116s114s101s115s32s40s77s80s97s41s65s112s108s105s101s100s32s116s114s117s101s32s83s116s114s97s105s110s61537s39s32s109s97s114s116s101s110s115s105s116s101s32s56s48s111s67s32s116s101s115s116Figure 7.10: Evolution of the average stresses in austenite and martensite,as a function of the applied strain. Adapted from Dufour [13].Figure 7.8 shows the value of f 0  0 taken from Dufour’s data comparedwith the estimate of f 0  0 from the Feritscope estimates produced here.Again, the agreement between the two methods is excellent. The diver-gence between the data at low strains and  0-martensite fractions could beattributable to measurement errors from either the Feritscope or neutron dif-fraction measurements as the uncertainty in both techniques grows rapidlywith decreasing volume fraction of  0-martensite. In the neutron di rac-tion measurements, the measurement of strain, and the resulting calculatedstresses, are made independently for the austenite and martensite phases.While the comparisons made above focused on   0 one can also comparethe Feritscope and neutron di raction measurements on the basis of thebehaviour of austenite. In Figure 7.11, the neutron di raction estimates of1647.4. Estimation of Stress Carried by  0-Martensite via the Magnetomechanical E ect  and (1 f 0)  are compared against these same parameters calculatedfrom the Feritscope estimates of   0. In the case of the Feritscope data,  and (1 f 0)  have been calculated on the basis of stress equilibriumEquation 5.4. It can be seen that, again, the agreement between the twomethods is very good consistent with the above discussion. These results alsopoint to the fact that the neutron di raction estimated   is quite similarto the (temperature corrected) stress-strain curve measured at 80 C. Thisis shown in Figure 7.10.s48s46s48s48s46s49s48s46s50s48s46s51s48s46s52s48s46s53s48s53s48s49s48s48s49s53s48 s61555s61543s32s78s101s117s116s114s111s110s32s100s105s102s114s97s99s116s105s111s110s32s100s97s116s97s32s80s114s101s115s101s110s116s32s115s116s117s100s121 s40s49s45s102s61537s39s41s32s61555s61543s73s110s116s114s105s110s115s105s99s32s116s114s117s101s32s115s116s114s101s115s32s40s77s80s97s41s65s112s108s105s101s100s32s116s114s117s101s32s115s116s114s97s105s110Figure 7.11: Comparison between neutron di raction [13] and Feritscopemeasurements of the stresses borne in the austenite.The good agreement between the estimated stresses borne by 0-martensiticas measured from neutron di raction, estimated from extrapolation of thebehaviour of austenite and from the new magnetic method proposed heresuggest that the magnetomechanical e ect can provide a route for assessingthe mechanical response of an embedded ferromagnetic phase in a complexmaterial such as the one studied here. The robust nature of these resultshas been checked as well against the data of stresses in  0-martensite mea-sured via in situ X-ray di raction in the work of Talonen [34]. This dataset collected on the same grade (301LN, though manufactured by a di erent1657.5. Measurement of Stresses in Samples of Di erent Grains Sizes and the Impact on Overall Mechanical Responsecompany) shows very similar response to that found by Dufour. Moreover,these results also suggest that the estimated behaviour of austenite comingfrom the extrapolation of data measured at 80 C is close to being correct,at least over the range of strains where (1 f 0)  is signi cant comparedto f 0  0.While di raction based estimates of the stresses are certainly more ro-bust and provide deeper understanding of the deformation behaviour, sincethey discriminate based on elastic strains along selected crystallographic di-rections, the relatively inexpensive and simple magnetomechanical methodproposed here has been shown to provide useful and complimentary esti-mates of the load partitioning in a material containing a ferrite-austenitemixture.7.5 Measurement of Stresses in Samples ofDi erent Grains Sizes and the Impact onOverall Mechanical ResponseThe procedure described above for the D=28 a181m sample has been equallyapplied to the other four conditions of grain size highlighted in chapter 4.The estimated variation of f 0  0 with strain in these samples is shown inFigure 7.12One can see in Figure 7.12 that the load borne by the  0-martensite is notstrongly depending on the starting austenite grain size, as was previouslysuggested from the macroscopic data from Figure 5.18. The fact that thebehaviour of the  0-martensite is nearly independent of austenite grain size(aside from the e ect on transformation kinetics) is not surprising giventhe results presented in Figure 6.23 where it was shown that the size andmorphology of the  0-martensite nuclei, while not exactly the same, weresimilar for all of the di erent grain sizes.Re ecting upon the results given above for f 0  0 in relation to the over-all mechanical response of the samples, one can make some general state-ments regarding the relative importance of austenite and  0-martensite on1667.5. Measurement of Stresses in Samples of Di erent Grains Sizes and the Impact on Overall Mechanical Response0 .0 0 .1 0 .2 0 .3 0 .4 0 .505 0 01 0 0 01 5 0 0Intrinsic true stress (MPa)T r u e  s t r a in 0 .5  m m 0 .9  m m 2 .2  m m  1 4  m m  2 8  m mFigure 7.12: Stress evolution in the di erent condition of grain size. Thesymbols are the real data points calculated from Feritscope measurements,while the lines are the result from the applied  t.the work-hardening response. First, it would appear that the contributionfrom  0-martensite to the work-hardening response is largely determined bythe rate of transformation df 0=d , but that there is a non-negligible hard-ening rate attributable to the  0-martensite. Assuming the  0-martensite tobe a perfectly elasto-plastic material is therefore not appropriate in this case(see section 2.5.2). Careful examination of the   0 curve shown in Figure 7.7raises some questions, however, given that the initial rate of hardening pre-dicted for the  0-martensite is actually high and sustained at   =10. The-oretically, one expects the highest rate of work-hardening in cubic metalsto be of the order of  =20, a rate which drops rapidly with strain due todynamic recovery [202, 253]. This particularity will be discussed in moredetail in the following chapter, where it will be argued that this is a conse-quence of the \dynamic composite" behaviour of the material. Examining1677.5. Measurement of Stresses in Samples of Di erent Grains Sizes and the Impact on Overall Mechanical Responsethe behaviour of   0 further, it is interesting to also note that it is predictedthat the stresses in the  0-martensite increase from 800 MPa at a strain of0.2 to 1500 MPa at a strain of 0.4. This would suggest that the  ow stressof  0-martensite, at least at small strains, is not much higher than that ofthe austenite (which can reach  1000 MPa at a strain of 0.4). Thus, formuch of the stress-strain curve the austenite contributes signi cantly to theoverall work-hardening rate.s48s46s48s48s46s49s48s46s50s48s46s51s48s46s52s48s46s53s48s46s48s48s46s50s48s46s52s48s46s54s48s46s56s49s46s48 s32s80s114s101s115s101s110s116s32s119s111s114s107s32s68s117s102s111s117s114s32s84s97s108s111s110s101s110s83s116s114s101s115s115s32s112s97s114s116s105s116s105s111s110s105s110s103s32s114s97s116s105s111s84s114s117s101s32s115s116s114s97s105s110s48s46s48s48s46s49s48s46s50s48s46s51s48s46s52s48s46s53s48s49s48s48s50s48s48s51s48s48s52s48s48s53s48s48 s73s73s73s87s111s114s107s45s104s97s114s100s101s110s105s110s103s32s40s77s80s97s41s84s114s117s101s32s115s116s114s97s105s110s73Figure 7.13: Stress partitioning ratio as a function of true strain, makingapparent a relation with the three stages of work-hardening. Those are inturn compared to the work of Dufour [13] and Talonen [34].1687.6. SummaryIn an attempt to quantify the relative contributions of the  0-martensiteand austenite, it is interesting to examine the ratio (r) between the stressescarried by these two phases,r = f 0  0 tot(7.5)= 1 (1 f 0)   tot(7.6)This quantity is plotted for the sample with D=28 a181m in Figure 7.13.It can be deduced from this plot that the austenite dominates the tensilestress-strain response below 0.20 true strain, while above 0.35 true strain,80% of the total stress is borne by  0-martensite. This point is importantas it suggests that the possible e ects arising from austenite scale re ne-ment on its mechanical response with increasing f 0 will not be dominant,and can perhaps be ignored. This could explain why the extrapolation ofthe austenite  ow stress measured at 80 C provides a reasonable descrip-tion of the austenite behaviour without accounting for any strain dependentscale re nement (as considered by Talonen to be dominant [34]). For strainsbetween 0.2 and 0.35, the contributions from both phases in uence the hard-ening behaviour to a similar amount. These three domains can be linked tothe three work-hardening stages described in section SummaryIn this chapter, a novel way of using the magnetoelastic e ect to estimatethe stresses borne  0-martensite has been presented. While the use of thise ect has been previously employed in sensors and an e ect similar to theone presented here was observed by Post et al. [70], it is believed that thiswork represents the  rst attempt at making quantitative predictions of stresspartitioning based on this e ect.Although the analysis presented here is very simple, neglecting the po-tential for multiaxial states of stress or for residual stresses and lacking a1697.6. Summaryprecise description of the underlying physics of the magnetomechanical ef-fect, it has been shown that the calbration and analysis procedure usedhere compared very well with measurements on the same steel made us-ing di raction by Talonen [34] and Dufour [13]. In agreement with thesedi raction experiments, the measure re ects the variation in importance ofthe austenite and  0-martensite behaviour as a function of strain. More-over, the results presented here con rm that the mechanical behaviour ofthe austenite is similar to the mechanical response of austenite measured at80 C.The results of this work also help to provide important details needed forthe construction of a physically based mechanical model for the tensile re-sponse of this alloy. It shows, for example, that the contribution of austeniteand  0-martensite to the total  ow stress of the material can be separatedinto three regimes, the  rst dominated by austenite, the last dominated by 0-martensite and the intermediate range of strains being controlled by bothphases in similar proportions. The behaviour of  0-martensite, while largelycontrolled by df 0=d , itself has an intrinsic hardening rate. This hardeningrate is surprisingly high and sustained up to relatively large strains whereit begins to saturate. In the next chapter a description for this behaviour ispresented which attempts to capture the fact that the measured response of 0-martensite re ects a range of behaviours resulting from the progressivenature of the formation of  0-martensite.170Chapter 8Modelling of the MechanicalResponse of 301LN8.1 IntroductionIt has been shown in chapters 5 and 7 that the mechanical behaviour hasstrong contributions from both austenite and  0-martensite. Based on theresults presented in chapter 7, it is possible to develop a description forthe bulk mechanical response based on the measured mechanical responseof the individual phases. In section 2.5.3, it was noted that two di erentapproaches to the modelling of the mechanical response of austenitic stain-less steels can be found in the literature. The mechanics-based models, suchas the one previously highlighted by Iwamoto and Tsuta [169], tend to becomplex, including many empirical parameters allowing for the inclusion ofstrain path and stress state e ects. On the other hand, material-based mod-els such as those developed by Olson [58], Spencer [112], Bouquerel [207] andTalonen [34], tend to use simpler one-dimensional descriptions of mechanicalresponse while focusing more heavily on capturing physical aspects of themicrostructural contributions to strength. In the present work, an attempthas been made to develop a description more aligned with the latter groupof models.1718.2. Review of Mechanical Response and Previous Microstructural Based Models8.2 Review of Mechanical Response andPrevious Microstructural Based ModelsIn chapter 7, it was shown that the behaviour of austenite back-extrapolatedfrom both di raction and from the magnetic method proposed in this thesiswas consistent with the behaviour of austenite measured at 80 C when theyield strength was corrected for temperature. In particular, this appearsvery consistent when one views the early stages of deformation prior to theformation of signi cant fractions of  0-martensite. In this case, the work-hardening rates of the curves measured at room temperature and at 80 C arevery similar. Previous models (e.g. [207] and [34]) have highlighted the factthat plastic incompatibility between the austenite and  0-martensite alongwith the scale re nement of the austenite (cf. Figure 6.23) should leadto extra hardening of the austenite due to the formation of geometricallynecessary dislocations. This would seem incompatible with the assertionmade above that the behaviour of austenite measured at 80 C (where little 0-martensite forms) appears to be the same as that at room temperature.Figure 7.13 however shows that the relative importance of austenite to thework-hardening and  ow stress drops rapidly with strain such that at a strainof  0.3, the austenite contributes only 20% to the overall  ow stress. Thisis clearly seen if one examines the estimated value of (1 f 0)  shownin Figure 7.11. The sensitivity of the overall behaviour of the materialto the description of the austenite at higher levels of strain (where extrastrengthening due to geometrically necessary dislocations will be important)is expected to be low, in this case. This may help to explain why it ispossible to extrapolate the behaviour of the austenite measured at 80 C tolarge strains without needing to account for extra hardening.In the case of  0-martensite, various proposals have been made for its be-haviour in the literature. Bouquerel [207] treated it has a monolithic phaseand described it as obeying a Voce type hardening law. Others, e.g. Talo-nen [34], have treated the  0-martensite as a rigid, non-plastically deformingphase. Under this assumption, the mechanical response of the material hasbeen attributed to the hardening arising from the dislocation content in1728.2. Review of Mechanical Response and Previous Microstructural Based Modelsaustenite. Neither of these cases appears to capture the response of the 0-martensite found here. Examination of Figure 7.7 reveals the behaviourof  0-martensite to be complex. The behaviour of  0-martensite shows anextremely large rate of apparent hardening between the strains of 0.2 and0.3. Between these strains, the stress carried by  0-martensite appears torise linearly with a slope of   =10 where  is the shear modulus of themartensite. In cubic metals, the maximum rate of hardening due to dislo-cations is generally found to be of the order of  =20, this occurring at theonset of general yield in a well annealed material [202, 253]. One must becareful as the behaviour of  0-martensite in Figure 7.7 is plotted against themacroscopic strain. In reality the stains carried by  0-martensite and auste-nite could be very di erent. The apparent hardening rate of  0-martensitein Figure 7.7 could be written as:d  0d =d  0d  0d  0d (8.1)where  is the macroscopic strain and   0 is the strain carried by  0. Inorder for d  0=d  0 to be smaller than d  0=d , d  0=d needs to be largerthan 1, meaning that the strain in  0-martensite would have to be higherthan the macroscopic strain. If  0-martensite is considered to be the hardestof the two phases, then it would be expected to have a lower strain comparedto the macroscopic strain. Thus, non-uniform strain partitioning wouldappear not able to explain this behaviour. The sustained high apparenthardening rate of the  0-martensite suggests that another mechanism mustbe accounted for.An important aspect of the mechanical response of austenitic stainlesssteels is that they behave as a \dynamic composite" in that the microstruc-ture is gradually converting from austenite to  0-martensite. The behaviourshown in Figure 7.7 can therefore be misleading, as it has to be viewed asan average behaviour of  0-martensite formed at various levels of strain.This is a point which has not been adequately explored in previous mod-els for the mechanical response of dynamically transforming materials (e.g.TRIP, TWIP steels) and will form the basis of the model developed below.1738.3. A Dynamic Composite Model for 301LN Stainless Steel8.3 A Dynamic Composite Model for 301LNStainless SteelThe approach taken here, motivated by the results presented in chapters 5and 7, is to describe the behaviour of the material not as a mixture betweentwo monolithic materials, but as an n-phase composite composed of auste-nite and a continuous distribution of  0-martensites, the behaviour of themartensite depending on the strain at which it has been formed. In devel-oping the basic components of this model, the data for the coarse grained(D=28 a181m) sample deformed in tension will be treated  rst. Both the bulkmechanical response presented in chapter 5 as well as the behaviour of theindividual phases chapter 7 are used to  t the model. A vital component tothe model is a good description of the kinetics of the  ! 0 transformation.For simplicity, the basic O-C equation (Equation 2.11) has been  t to theexperimental data as shown in Figure 5.12. The deformation of the materialin this model is considered based on an equivalent stress and strain basisand is therefore one-dimensional. In the model, strain is assumed uniformthrough all phases. More sophisticated homogenization schemes (such as the -model [254] or the iso-work approach [255]) could be applied, however, thischoice does not change the physical nature of the model presented.8.3.1 Behaviour of AusteniteThe work-hardening response of the austenite has simply been  t to thisexperimental data using a Voce law:  =   0 +  s 1 exp(   0  s ) (8.2)in which   0,   0 and   s respectively stand for the yield stress, initialhardening rate and scaling stress of the austenite. The physical meaning ofthese hardening parameters is shown schematically in Figure 8.1.A Voce law has been selected as the parameters in this model (i.e.   0and   s) can be attributed to dislocation based hardening mechanisms in fcc1748.3. A Dynamic Composite Model for 301LN Stainless Steelmaterials [202, 253] through a Kocks-Mecking approach. The temperaturedependence of the yield strength of this alloy can also be described based onthe recent model presented by Allain et al. [174], as shown in Figure 2.20.In the present case, for simplicity, the yield strength of the tensile curvemeasured at 80 C has simply been adjusted to  t the yield strength of thesamples tested at room temperature. The form of the Voce law used does notinclude a term accounting for geometrically necessary dislocations. Such aterm could be included but, as discussed above, the sensitivity of the resultsto the inclusion of this term are considered small.True stressTrue strains0s  ss+0(a)Work-hardening rateTrue stresq0s0s  ss+0(b)Figure 8.1: Schematic representation of the parameters appearing in Equa-tion Behaviour of  0-MartensiteThe model developed here is applied incrementally such that the fraction ofthe total stress carried by austenite, (1 f 0)  , and  0-martensite f 0  0are computed at  xed increments of the strain (d ). The increment of  0-martensite formed in this strain increment (df 0) is also calculated for eachincrement of strain. Each df 0 increment of  0-martensite formed is treatedas a separate phase in the calculation.Critical to the explanation of the behaviour observed in Figure 7.7 is theassumption that each increment of  0-martensite formed is an elasto-plasticelement and that the  0-martensite, when formed, is in compression by anamount denoted as   000. As previously discussed in sections 2.2.1 and 6.3.4,1758.3. A Dynamic Composite Model for 301LN Stainless Steelthe formation of  0-martensite from austenite involves a shear strain, as wellas an expansion along one direction, and contraction along the two perpen-dicular directions. Overall, the transformation occurs with an increase involume from austenite to  0-martensite. The net result of this process is thatthe formation of  0-martensite should lead to a local unloading of the ma-terial while also giving rise to an extra amount of \transformation strain".Various detailed micromechanical models have been previously built to ex-amine the combination of these two e ects (see e.g. [256, 257]). In simplerone-dimensional models, this complex situation is often considered eitherin terms of an extra transformation strain added to the macroscopic straindue to the transformation (e.g. [112]) at  xed stress, or the formed phase isconsidered to be under compression at  xed (uniform) strain (see e.g. [257],for a discussion of these limits). In the present case, the latter descriptionhas been selected as it matches better to the results shown in Figure 7.7 andis consistent with the assumption of uniform strains. This behaviour is con-sistent with the neutron di raction measurements presented in Figure 7.10,which suggest that the  0-martensite is under compression at small strains[13].Once formed, each incremental element of  0-martensite is assumed toload elastically in tension until it reaches the stress   00. Above this level ofstress it is assumed to deform plastically, where each  0-martensite elementis assumed to obey a Voce law. Therefore, the constitutive behaviour formartensite can be described by:  0 =(   000 +E 0 for   0 <  00  00 +  0s 1 exp(   00  0s ) for   0 >  00 (8.3)withE 0 being the Young modulus of 0-martensite, estimated to be 200 GPa(e.g. [13]). Based on this constitutive model for each  0-martensite element,the high initial hardening rate in Figure 7.7 is explained to be due to theinjection, at each step of deformation, of new  0-martensite elements whichare initially in compression and which have to load elastically to their yieldstress. Thus, the high hardening rate can be interpreted as a sort of extended1768.3. A Dynamic Composite Model for 301LN Stainless Steelelasto-plastic transition.At each step of the calculation, the material will consist of n elements of 0-martensite, each carrying a di erent level of stress. The total contributionto the stress from the  0-martensite is the sum of the contributions of then di erent islands of  0 which were formed at each time step weighted bytheir individual volume fractions: tot 0 =nXi=1dfn i 0   i 0 (8.4)in which the exponent refers to the step when  0 was formed.Finally, macroscopic equilibrium imposes that: = (1 f 0)  + tot 0 (8.5)8.3.3 Choice of the ParametersAs already noted, some of the parameters used in the model can be derivedfrom experimental measurements. This is the case for the yield stress andwork-hardening rate of the austenite as well as the kinetics of the  ! 0phase transformation, extracted from Figure 5.12 or Figure 5.15. The grainsize, and yield strength of austenite derived from experiments are given inTable 8.1 as are the parameters for the O-C model for the  ! 0 kinetics.Yield stress of Uniaxial TensionGrain size Austenite O-C parametersD   0   n0.5 a181m 670 MPa 4.06 4.52 50.9 a181m 610 MPa 3.85 4.52 52.2 a181m 440 MPa 3.83 4.52 514 a181m 370 MPa 4.28 4.52 528 a181m 280 MPa 4.52 4.52 5Table 8.1: Input parameters directly determined from tensile experiments.The yield stresses were determined from Figure 5.4, while the kinetics weretaken from Table 6.3.1778.3. A Dynamic Composite Model for 301LN Stainless SteelThe parameters for the Voce law used to describe austenite have been  tto the behaviour of 301LN measured at 80 C and are shown in Table 8.2. Itshould be noted that   0    =30 and   00    0=20, where   and   0 arethe shear moduli of austenite and  0-martensite. These values are consis-tent with the theory of work-hardening of polycrystals [202, 253]. The onlytemperature-dependence of this parameter comes from the temperature-dependence of the shear modulus. In the range of temperature consideredhere (e.g. from  196 C to 150 C),  0 can be considered independent oftemperature. On the other hand, the scaling stress   s is expected to betemperature-dependent.The four parameters corresponding to the martensitic mechanical be-haviour have been determined by  tting to the bulk mechanical response aswell as the evolution of f 0  0 given in Figure 7.12. It was found that thetwo hardening parameters for  0-martensite (  00 and   0s) had a relativelysmall impact on the overall stress-strain response of the material. On theother hand, the values of   000 and   00 were found to strongly in uencethe predicted behaviour. The selected values of all those parameters arepresented in Table 8.2.Austenite ( ) Martensite ( 0)  0   s   000   00   00   0s2500 MPa 1000 MPa -2200 MPa 1050 MPa 3500 MPa 900 MPaTable 8.2: Adjustable input parameters, used to model uniaxial tension.The respective behaviours of each phase using the parameters from Ta-ble 8.2 are represented in Figure 8.2. It can be seen from this plot that thechoice of the hardening parameters for  0-martensite is compatible with theyield stress measured at room temperature on a sample of 301LN which hadbeen previously cryorolled. Because cryorolling results in samples containingnearly 100%  0-martensite (cf. Figure 4.2), the yield stress of such a samplede nes an upper limit for the  ow stress of  0, upon speci ed strain route.This argument is consistent with the low work-hardening, as observed oncryorolled specimens experimentally deformed in shear. Figure 8.2 suggests1788.3. A Dynamic Composite Model for 301LN Stainless Steelthat this maximum  ow stress is di erent from uniaxial tension to simpleshear. This last point will be revisited in section 8.3.6.s48s46s48s46s50s48s46s52s48s46s54s48s46s56s49s46s48s49s46s50s45s50s48s48s45s49s48s48s48s49s48s48s50s48s48s32s85s110s105s97s120s105s97s108s32s116s101s110s115s105s111s110s32s83s105s109s112s108s101s32s115s104s101s97s114s61555s61543s86s111s110s32s77s105s115s101s115s32s115s116s114s101s115s32s40s77s80s97s41 s86s111s110s32s77s105s115s101s115s32s115s116s114s97s105s110s61555s61537s39s48s46s48s48s46s50s48s46s52s48s46s54s45s50s48s48s45s49s48s48s48s49s48s48s50s48s48s32s85s110s105s97s120s105s97s108s32s84s101s110s115s105s111s110s32s79s110s115s101s116s32s111s102s32s110s101s99s107s105s110s103s32s105s110s32s116s101s110s115s105s111s110s32s83s105s109s112s108s101s32s83s104s101s97s114s86s111s110s32s77s105s115s101s115s32s115s116s114s101s115s32s40s77s80s97s41s86s111s110s32s77s105s115s101s115s32s115s116s114s97s105s110Figure 8.2: (a) Simulated stress-strain behaviour of the two single phases and  0. For this model, the scaling stresses of the  0-martensite wereobtained from the experimental stress-strain curve of cryorolled materialshown in (b).8.3.4 Discussion of Model Results for D=28 a181m inUniaxial TensionThe results of the model  t to the sample having D=28 a181m is shown inFigure 8.3. One can see that the model reasonably predicts both the macro-scopic stress-strain response as well as the Feritscope measured variation off 0  0.The various contributions to the macroscopic work-hardening behaviourcan be identi ed more easily if one examines the rate of work-hardeningbased on the derivative of Equation 5.5,d d = (1 f 0)d  d | {z }Term 1+ df 0d  (  0   )| {z }Term 2+f 0 d  0d | {z }Term 3(8.6)1798.3. A Dynamic Composite Model for 301LN Stainless Steels48s46s48s46s49s48s46s50s48s46s51s48s46s52s48s46s53s53s48s49s48s49s53s48s32s61555s32s109s111s100s101s108s32s61555s32s101s120s112s32s61555s61543s32s109s111s100s101s108s32s61555s61543s32s101s120s112s32s61555s61537s39s116s111s116s32s109s111s100s101s108s32s102s61537s39s61555s61537s39s32s101s120s112s84s114s117s101s32s115s116s114s101s115s32s40s77s80s97s41 s84s114s117s101s32s115s116s114s97s105s110(a)s48s53s48s49s48s49s53s48s49s48s50s48s51s48s52s48s53s48s87s111s114s107s45s104s97s114s100s101s110s105s103s32s114s97s116s101s32s40s77s80s97s41 s84s114s117s101s32s115s116s114s101s115s32s40s77s80s97s41s32s77s111s100s101s108s32s69s120s112s101s114s105s109s101s110s116 s67s111s110s115s105s100s232s114s101s32s108s105s110s101(b)Figure 8.3: (a) Simulated stress-strain curves of the D=28 a181m conditiondeformed in uniaxial tension. Those are compared to the experimentalstress-strain curves obtained from Figure 5.4 for the  / 0 aggregate andfrom Figure 5.7 for the austenite. The simulated stress in  0-martensite(Equation 8.4) is compared to the experimental stresses determined fromFeritscope measurements (Figure 7.8). (b) Comparison of the simulatedand experimental work-hardening curves.The three terms highlighted in the above equation describe the work-hardening of the austenite (Term 1), the rate of the  ! 0 phase transfor-mation and the mechanical contrast between austenite and martensite (Term2), and the apparent work-hardening arising from the net behaviour of the 0-martensite (Term 3). These three components of the work-hardening rateare compared in Figure 8.4.For strains between 0 to 0.10, work-hardening is dominated by the me-chanical behaviour of the austenite (Term 1). This corresponds to the stage Ias identi ed in chapters 5 and 7. The in ection characteristic of the TRIP ef-fect is the dominant feature of stage II. It can be seen that this correspondsto the transition during which Terms 2 (related to the rate of the phasetransformation) and 3 (related to the apparent work-hardening of  0) startbecoming more important than the work-hardening of the austenite. In thelast work-hardening stage (stage III), Term 1 becomes negligible and Term3 has the largest contribution. A consequence of this observation is that the1808.3. A Dynamic Composite Model for 301LN Stainless Steels48s46s48s46s49s48s46s50s48s46s51s48s46s52s48s46s53s45s49s48s48s48s49s48s48s50s48s48s51s48s48s52s48s48 s73s73s73s87s111s114s107s45s104s97s114s100s101s110s105s110s103s32s114s97s116s101s32s40s77s80s97s41s84s114s117s101s32s115s116s114s97s105s110s32s69s120s112s46s32s87s45s72s32s83s105s109s46s32s45s72s32s83s105s109s46s32s84s101s114s109s32s49s32s83s105s109s46s32s84s101s114s109s32s50s32s83s105s109s46s32s84s101s114s109s32s51s73Figure 8.4: Representation of the three work-hardening (W-H) terms, asde ned in Equation 8.6, obtained from simulation. The sum of these threeterms is in turn compared to the work-hardening measured experimentally.kinetics of the  ! 0 phase transformation only dominates in stage II, ane ect already highlighted in the discussion on stress partitioning evolution(Figure 7.13). A  nal point on this plot is that Term 2 becomes negativein the early stages of deformation due to the fact that the stress carriedby austenite is initially higher than that carried by  0-martensite. This canhave an important consequence when the austenite has a high yield strength(i.e.  ne grain sized samples) as will be discussed in sections 8.3.5 and Application of Model to the Grain Size Dependenceof Mechanical ResponseAs shown by the tensile tests carried out at elevated temperature (Fig-ure 5.6), the grain size apparently has little impact on the work-hardeningrate of austenite. This is also con rmed by the early stages of deformationat room temperature where, as discussed above, the work-hardening rate isdominated by the behaviour of austenite.1818.3. A Dynamic Composite Model for 301LN Stainless Steels48s46s48s46s49s48s46s50s48s46s51s48s46s52s48s46s53s53s48s49s48s49s53s48s32s61555s32s109s111s100s101s108s32s61555s32s101s120s112s32s61555s61543s32s109s111s100s101s108s32s61555s61543s32s101s120s112s32s61555s61537s39s116s111s116s32s109s111s100s101s108s32s102s61537s39s61555s61537s39s32s101s120s112s84s114s117s101s32s115s116s114s101s115s32s40s77s80s97s41 s84s114s117s101s32s115s116s114s97s105s110(a) 14 a181ms48s46s48s46s49s48s46s50s48s46s51s48s46s52s48s46s53s53s48s49s48s49s53s48s32s61555s32s109s111s100s101s108s32s61555s32s101s120s112s32s61555s61543s32s109s111s100s101s108s32s61555s61543s32s101s120s112s32s61555s61537s39s116s111s116s32s109s111s100s101s108s32s102s61537s39s61555s61537s39s32s101s120s112s84s114s117s101s32s115s116s114s101s115s32s40s77s80s97s41 s84s114s117s101s32s115s116s114s97s105s110(b) 2.2 a181ms48s46s48s46s49s48s46s50s48s46s51s48s46s52s48s46s53s53s48s49s48s49s53s48s32s61555s32s109s111s100s101s108s32s61555s32s101s120s112s32s61555s61543s32s109s111s100s101s108s32s61555s61543s32s101s120s112s32s61555s61537s39s116s111s116s32s109s111s100s101s108s32s102s61537s39s61555s61537s39s32s101s120s112s84s114s117s101s32s115s116s114s101s115s32s40s77s80s97s41 s84s114s117s101s32s115s116s114s97s105s110(c) 0.9 a181ms48s46s48s46s49s48s46s50s48s46s51s48s46s52s48s46s53s53s48s49s48s49s53s48s32s61555s32s109s111s100s101s108s32s61555s32s101s120s112s32s61555s61543s32s109s111s100s101s108s32s61555s61543s32s101s120s112s32s61555s61537s39s116s111s116s32s109s111s100s101s108s32s102s61537s39s61555s61537s39s32s101s120s112s84s114s117s101s32s115s116s114s101s115s32s40s77s80s97s41 s84s114s117s101s32s115s116s114s97s105s110(d) 0.5 a181mFigure 8.5: In uence of the grain size on the simulated tensile curves. Thoseare compared to the experimental stress-strain curves obtained from Fig-ure 5.4 for the  / 0 aggregate and from Figure 5.7 for the austenite. Thesimulated stress in  0-martensite (Equation 8.4) is compared to the experi-mental stresses determined from Feritscope measurements (Figure 7.12).This e ect is seen in Figure 5.5 where the hardening rates for the initialportion (Stage I) of the curves collapse on top of one another if the yieldstrength is subtracted from the stress axis of the plot. According to theresults shown in Figure 7.12, the mechanical behaviour of the  0-martensitehas been taken to be independent of the austenite grain size. Based onthese two observations, it has been assumed that the hardening parametersfor each phase in Table 8.2 are independent of grain size, at least in the1828.3. A Dynamic Composite Model for 301LN Stainless Steelrange of grain size considered here (0.5 a181m < D < 50 a181m). Consequently,the austenite grain size is considered to only a ect the mechanical propertiesthrough the kinetics of the  ! 0 transformation and the yield stress ofaustenite, all parameters are easily obtained from experiment.The results of the predicted mechanical response as a function of grainsize are shown in Figure 8.5, the individual contributions to the work-hardening rate being shown in Figure 8.6.s48s46s48s46s49s48s46s50s48s46s51s48s46s52s48s46s53s45s49s48s49s48s50s48s51s48s52s48s87s111s114s107s45s104s97s114s100s101s110s105s103s32s114s97s116s101s32s40s77s80s97s41s84s114s117s101s32s115s116s114s97s105s110s32s69s120s112s46s32s87s45s72s32s83s105s109s46s32s45s32s105s46s32s84s101s114s109s32s49s32s83s105s109s46s32s101s114s32s50s32s105s46s32s84s101s114s109s32s51s73s73s73(a) 14 a181ms48s46s48s46s49s48s46s50s48s46s51s48s46s52s48s46s53s45s49s48s49s48s50s48s51s48s52s48s87s111s114s107s45s104s97s114s100s101s110s105s103s32s114s97s116s101s32s40s77s80s97s41 s84s114s117s101s32s115s116s114s97s105s110s32s69s120s112s46s32s87s45s72s32s83s105s109s46s32s45s32s105s46s32s84s101s114s109s32s49s32s83s105s109s46s32s101s114s32s50s32s105s46s32s84s101s114s109s32s51s73s73s73(b) 2.2 a181ms48s46s48s46s49s48s46s50s48s46s51s48s46s52s48s46s53s45s49s48s49s48s50s48s51s48s52s48s87s111s114s107s45s104s97s114s100s101s110s105s103s32s114s97s116s101s32s40s77s80s97s41 s84s114s117s101s32s115s116s114s97s105s110s32s69s120s112s46s32s87s45s72s32s83s105s109s46s32s45s32s105s46s32s84s101s114s109s32s49s32s83s105s109s46s32s101s114s32s50s32s105s46s32s84s101s114s109s32s51s73s73s73(c) 0.9 a181ms48s46s48s46s49s48s46s50s48s46s51s48s46s52s48s46s53s45s49s48s49s48s50s48s51s48s52s48s87s111s114s107s45s104s97s114s100s101s110s105s103s32s114s97s116s101s32s40s77s80s97s41 s84s114s117s101s32s115s116s114s97s105s110s32s69s120s112s46s32s87s45s72s32s83s105s109s46s32s45s32s105s46s32s84s101s114s109s32s49s32s83s105s109s46s32s101s114s32s50s32s105s46s32s84s101s114s109s32s51s73s73s73(d) 0.5 a181mFigure 8.6: Grain size dependence of the three work-hardening (W-H) terms,as de ned in Equation 8.6, obtained from simulation.It can be seen that changing the austenite grain size mainly e ects thecontribution of Term 2, which includes the di erence in stress carried byaustenite and  0-martensite. The increase of peak work-hardening (peakB, according to the nomenclature used in Figure 5.16) with grain size isprimarily due to the di erence between the  ow stress of austenite and  0-1838.3. A Dynamic Composite Model for 301LN Stainless Steelmartensite.An important e ect that comes from Term 2 is its dependence on (  0   ).As one can see in Figure 8.6, this term becomes increasingly negative in theearly stages of deformation as grain size is decreased. This is because of theincreasing yield strength of austenite relative to the  0-martensite (whoseproperties have been assumed independent of austenite grain size). The endresult is that, as the grain size of austenite is re ned, the in uence of theformation of  0-martensite can actually be to soften the material. This isobserved, for example, in the case of the D=0.5 a181m condition where the  owstress of austenite lies above that of the macroscopic material.Another important point concerning Term 2 is the importance of the ! 0 kinetics. A weak change in transformation kinetics results in a largedi erence in the predicted stress-strain curve. One example to illustrate thispoint is given in Figure 8.7 in which the transformation kinetics of the 28 a181mcondition have been computed assuming the transformation kinetics of theD=2.2 a181m condition. It is clear that even small errors in the prediction of thetransformation kinetics can have large e ects on the predicted mechanicalresponse. In the case considered here, the use of the kinetics for the D=2.2a181m condition leads to a reduction of 150 MPa in the true stress at necking.s48s46s48s46s49s48s46s50s48s46s51s48s46s52s48s46s53s46s48s46s50s48s46s52s48s46s54s48s46s56s49s46s48 s50s46s32s61549s109s86s111s108s117s109s101s32s102s114s97s99s116s105s111s110s32s111s102s32s61537s39s45s109s97s114s116s101s110s115s105s116s101s84s114s117s101s32s115s116s114s97s105s110s50s56s32s61549s109(a)s48s46s48s46s49s48s46s50s48s46s51s48s46s52s48s46s53s53s48s49s48s49s53s48s84s114s117s101s32s115s116s114s101s115s32s40s77s80s97s41 s84s114s117s101s32s115s116s114s97s105s110s50s46s32s61549s109s50s56s32s61549s109(b)Figure 8.7: Sensitivity of the kinetics on the tensile behaviour. Comparisonof (a) kinetics and (b) simulated tensile curves, using in one case the kineticsof the 28 a181m condition, and in the other case, the kinetics of the 2.2 a181mcondition.1848.3. A Dynamic Composite Model for 301LN Stainless Steel8.3.6 Application of Model to the Mechanical Response inShearIt was shown in chapter 5 that the mechanical response in shear was ini-tially very similar to the mechanical response in tension (compared on thebasis of Von Mises equivalent stress and strain) but that, as the strain andfraction of  0-martensite increased, the  ow stress for the two strain pathsincreasingly deviated from one another. It was also shown that the  ! 0transformation kinetics were similar for the two strain paths over the ini-tial stages of deformation but that the rate of transformation slowed morerapidly in the case of the shear tests than in the case of tension.In an attempt to adapt the model developed for tension to the case ofshear, some modi cations are necessary. First, the O-C parameters need tobe changed to describe the kinetics of the  ! 0 transformation observedexperimentally (cf. Figure 5.15). Second, the hardening parameters arealso required to be adapted to the variation of strain path. A  rst exampleof this can be found in the work of Allain focusing on the deformationmechanisms of Fe-Mn steels, in the absence of  ! 0 phase transformation[87]. In this work, discrepancies were observed between the mechanicalproperties of austenite measured along uniaxial tension and pure shear, thesebeing explained by the di erence in texture evolution and the fact that theequivalent strain (according to Von Mises) does not take into account therotation of principal axes (this point will be revisited in the end of thissection). A second example, in terms of the mechanical properties of  0only, was presented in Figure 8.2. The mechanical behaviours of cryorolledmaterials, plotted in terms of Von Mises equivalents, suggest that the scalingstresses of  0-martensite need to be smaller in shear than in tension. Thesescaling stress can be calculated using the fact that the maximum  ow stressof a cryorolled specimen along a speci c strain path is equal to   00 +  0s.The actual values deduced from Figure 8.2 show that the scaling stress of 0 is decreased from 900 to 700 MPa.Since the change from uniaxial tension to simple shear tends to lowerthe  ow stress of both austenite and  0-martensite, the overall e ect on the1858.3. A Dynamic Composite Model for 301LN Stainless Steel ow stress of austenitic stainless steels is also a decrease in  ow stress, atrend consistent with the experiments reported in references [146, 186].In order to account for these changes, the scaling stress of austenite   0was lowered from 1000 to 800 MPa, while that of  0  00 was also decreasedfrom 900 to 700 MPa, according to Figure 8.2. All other parameters, partic-ularly the initial hardenings, were left unchanged. The full list of parametersis detailed in Table 8.4. The model results in Figure 8.8.Simple ShearGrain size Austenite O-C parametersD   0   n0.5 a181m 670 MPa 4.06 3.4 52.2 a181m 440 MPa 3.83 3.4 528 a181m 280 MPa 4.52 3.4 5Table 8.3: Input parameters directly determined from simple shear exper-iments. The yield stresses were the same as those determined in uniaxialtension, while the kinetics were taken from Table 6.3.Austenite ( ) Martensite ( 0)  0   s   000   00   00   0s2500 MPa 800 MPa -2200 MPa 1050 MPa 3500 MPa 700 MPaTable 8.4: Adjustable input parameters, used to model simple shear. Whencompared to tension, only the scaling stresses were changed in agreementwith the experimental data from Figure 8.2.1868.4. Application of Model to Literature Datas48s46s48s46s49s48s46s50s48s46s51s48s46s52s48s46s53s48s46s54s48s46s55s53s48s49s48s49s53s48s32s77s111s100s101s108s32s69s120s112s101s114s105s109s101s110s116s86s111s110s32s77s105s115s101s115s32s101s113s46s32s115s116s114s101s115s32s40s77s80s97s41 s86s111s110s32s77s105s115s101s115s32s101s113s46s32s115s116s114s97s105s110s48s46s53s61549s109s50s46s61549s109s50s56s61549s109(a)s53s48s49s48s49s53s48s48s49s48s50s48s51s48s52s48s53s48s48s46s53s61549s109s50s46s61549s109s32s77s111s100s101s108s32s69s120s112s101s114s105s109s101s110s116s87s111s114s107s45s104s97s114s100s101s110s105s103s32s114s97s116s101s32s40s77s80s97s41s86s111s110s32s77s105s115s101s115s32s101s113s46s32s115s116s114s101s115s32s40s77s80s97s41s50s56s61549s109(b)Figure 8.8: Comparison of the simple shear simulated curves with the exper-imental ones. The divergence between model and experiment is attributedto the di erent texture evolutionIt can be noticed that the shear behaviour above 40% true strain di erswhen one compares experiment and model predictions. This feature could bea re ection of the di culty to capture strains using the Von Mises equivalentstrain. Indeed, the equivalent strain was already reported to overestimatethe true strains in simple shear [258, 259]. The underlying reason is thatthe Von Mises strain does not account for the incremental rotation of theprincipal axes. This e ect would help explain the results observed here,since an overestimation of the strains results in an overestimation of both 0-martensite fraction and equivalent individual stresses. All these e ectstend to an overestimation of the overall  ow stress of the composite material,as observed in Figure Application of Model to Literature DataIf the model presented here is to be of general applicability, it should bepossible to extend it to explain the mechanical behaviour at di erent tem-peratures, and for di erent alloys. As was mentioned earlier in this thesis,the work carried out here was performed in parallel with a second studyfocused on the e ects of temperature and strain rate on the same alloy1878.4. Application of Model to Literature Data[146, 186]. The e ects of strain rate, as discussed in section 2.3, mainlycome from self-heating of the sample. These non-isothermal conditions aredi cult to model as they require a model for the temperature as a functionof strain (time) in the sample. It is, however, possible to compare the modeldeveloped here with the data collected under isothermal conditions.It was assumed, in section 8.3.3, that temperature does not signi cantlyin uence the work-hardening rate of austenite in the range 20 C to 80 C.In reality, the work-hardening response of austenite should be a functionof temperature primarily through the   s term (related to dynamic recov-ery) and to a much smaller extent through   0 (through the temperaturedependence of the shear modulus). Similarly, the  ow stress and hardeningrate of  0-martensite should also be a function of temperature. As a simplecheck on the robustness of the model developed here, a simple comparisonbetween the data collected by Nanga [146, 186] has been made assumingthat the work-hardening behaviours of austenite and  0-martensite weretemperature independent. Consequently, the temperature dependence wasconsidered only in relation to the yield strength of austenite and the rateof  !  0 transformation. The rate of  !  0 transformation has beencaptured using a  t to the O-C model, as for the results presented above.The results of this simulation are shown in Figure 8.9.1888.4. Application of Model to Literature Datas48s46s48s48s46s49s48s46s50s48s46s51s48s46s52s48s46s53s48s53s48s49s48s48s49s53s48 s56s48°s67s52s48°s67s50s51°s67s45s52s48°s67s45s49s48°s67s32s69s120s112s101s114s105s109s101s110s116s32s77s111s100s101s108s84s114s117s101s32s115s116s114s101s115s32s40s77s80s97s41s84s114s117s101s32s115s116s114s97s105s110s8226s71s114s97s100s101s32s51s48s49s76s78s61541s32s61s32s53s120s49s48s45s52s32s115s45s49s71s114s97s105s110s32s115s105s122s101s32s61s32s56s32s61549s109s32(a)s53s48s49s48s48s49s53s48s48s49s48s48s50s48s48s51s48s48s52s48s48s53s48s48s87s111s114s107s45s104s97s114s100s101s110s105s110s103s32s114s97s116s101s32s40s77s80s97s41s84s114s117s101s32s115s116s114s101s115s32s40s77s80s97s41s67s111s110s115s105s100s101s114s101s56s48°s67s52s48°s67s50s51°s67s45s52s48°s67s45s49s48°s67(b)T   0 O-C values( C) (MPa)   n 100 620 4.3 57.7 4.5 40 600 4.6 14.4 4.523 440 3.2 12.7 4.540 400 3.9 0.9 4.580 360 3.0 0.1 4.5Figure 8.9: Results of the mechanical model applied to the data collected byNanga [146, 186]. (a) Simulated stress-strain curves and (b) simulated work-hardening curves, for uniaxial tension. For the two lowest temperatures, thesimulated work-hardening curves start laying below the Consid ere line atthe exact moment when strain localization is observed.Despite the fact that the temperature dependence of the  ow stress of1898.4. Application of Model to Literature Dataaustenite and  0-martensite have been excluded in Figure 8.9, the modelpredictions capture the behaviour well. Indeed, the model has also beensuccessfully applied to the data from the work of Talonen where measure-ments were made over a similar temperature range on a similar grade of301LN stainless steel [34].Interestingly, the model predicts the observed localization (similar to thepropagation of a L uders band) at  40 C and  100 C from Nanga’s data.This comes from the increasing di erence in  ow stress between austeniteand  0-martensite and therefore an increasing negative contribution fromTerm 2 in Equation 8.6. At low temperature, this negative contribution tothe work-hardening rate results in the Consid ere criterion (Equation 2.17)being satis ed early in the stress-strain curve (Figure 8.9(b)). Yet, as Term2 rises and becomes positive with increasing fraction of  0-martensite, theinstability is ended and uniform straining is continued again.This e ect of strain localization has similarly been observed during thepresent study for a sample deformed in liquid nitrogen (cf. Figure 7.4).Aside from tension, it is also possible to induce L udering if the grain size issu ciently reduced or if the austenite is su ciently work-hardened withoutthe formation of  0-martensite. Indeed, the tensile sample having D=0.5 a181m(  0=670 MPa) nearly reaches this condition as illustrated by Figure 8.5.A second example of a high yield stress resulting in strain localization isdetailed below.A sample was produced with a very  ne grain size (high yield strength)by cryorolling 301LN followed by annealing at 750 C during 30 minutes.This treatment generated a partially recrystallized microstructure ( 80%recrystallized), resulting in a yield stress of 970 MPa. The starting mi-crostructure of this material was free of  0-martensite in the as-annealedstate. As expected, the tensile curve of this material presented sharp dis-continuous yielding followed by a long plateau ( 24% of strain) of strainlocalization (Figure 8.10(b)).1908.4. Application of Model to Literature Data10 μm RDTD(a)s48s46 s48s46s49s48s46s50s48s46s51s53s48s49s48s49s53s48s84s114s117s101s32s83s116s114s101s115s32s40s77s80s97s41 s84s114s117s101s32s83s116s114s97s105s110(b)Figure 8.10: (a) Orientation map showing the microstructure of 301LN aftercryorolling and annealing at 750 C. It can clearly be seen that the austeniteis not fully recrystallized. (b) Stress-strain curve of the condition exhibitingthe microstructure shown in (a), during room-temperature uniaxial tension.The tensile curve shows a long plateau (24% strain) characteristic of strainlocalization.The observation of strain localization has previously been discussed inthe literature. In the work of Spencer [72], the formation of apparent L udersbands at low temperature was attributed to a a high rate of transformation,resulting in a large contribution to the strain (at constant stress) from thetransformation strain. This explanation is similar to the one given here,since, as discussed above, the assumption of an extra transformation strain(at constant stress) and an unloading e ect (at constant strain) are upperand lower bounds on the true expected behaviour of the material. However,the advantage of the current proposal is that it shows that the rate of trans-formation does not need to be particularly high in order to induce strain lo-calization. A key factor that must be considered is the relative di erence inthe  ow stress contribution coming from austenite and  0-martensite. Thus,the experimental observations made by Spencer [72] of very large L udersplateaux following a change of testing condition from room temperature (no 0-martensite formation but work-hardening of austenite) to  196 C (highrate of  0-martensite formation) should be considered in terms of both therate of  !  0 transformation as well as the di erence in  ow stress of1918.4. Application of Model to Literature Dataaustenite and  0-martensite, as suggested by Term 2 in Equation 8.6.It has also been recently suggested [10] that the formation of strainlocalization bands in metastable austenitic stainless steels is an indication ofa transition from a strain-induced to stress-assisted transformation. Whilethe work here does not preclude the possibility for such a transition, itdoes show that such a transition is not necessary to explain the observedbehaviour.So far, the proposed model has been applied only to a 301LN grade ofstainless steel. This grade has the advantage, compared to other grades, thatit is relatively unstable and therefore forms a high fraction of  0-martensiteby testing at room temperature. Similar e ects, however, can be seen inother grades tested at low temperatures.Spencer et al. performed tests on a 316L grade of stainless steel at lowtemperatures [72]. Since no  0-martensite is formed in 316L at room temper-ature, the stress-strain curve at 25 C is a good estimate for the mechanicalbehaviour of the austenite. Comparing the results of tensile testing at roomtemperature on 316L with that on 301LN tested at 80 C revealed that thebehaviours were quite similar. The Voce parameters required to  t the be-haviour of austenite in 316L at this temperature, are given in Figure 8.11.As in other cases, the experimental  ! 0 transformation kinetics reportedby Spencer have been  t to an O-C model for the various test temperatures,and the yield strength of the austenite has been matched to the experimentalcurves. The behaviour of the austenite and  0-martensite have been takento be the same as that used for the 301LN alloy described above, except forthe yield stress of the  0-martensite, which was increased from 1050 MPato 1550 MPa to better reproduce the experimental stress-strain curve inFigure 8.11.1928.4. Application of Model to Literature Datas48s46s48s48s46s49s48s46s50s48s46s51s48s46s52s48s46s53s48s53s48s49s48s48s49s53s48s50s48s48 s49s55s111s67s50s53s111s67s45s54s48s111s67s32s77s111s100s101s108s32s69s120s112s101s114s105s109s101s110s116s84s114s117s101s32s115s116s114s101s115s32s40s77s80s97s41s84s114s117s101s32s115s116s114s97s105s110s45s49s57s54s111s67s71s114s97s100s101s32s51s49s54s76(a)s48s53s48s49s48s48s49s53s48s50s48s48s48s50s48s48s52s48s48s54s48s48s56s48s48s49s48s48s49s50s48s48s87s111s114s107s45s104s97s114s100s101s110s105s110s103s32s114s97s116s101s32s40s77s80s97s41s84s114s117s101s32s115s116s114s101s115s32s40s77s80s97s41s32s77s111s100s101s108s32s69s120s112s101s114s105s109s101s110s116(b)T   0 O-C values( C) (MPa)   n 196 550 4.5 9.6 4.5 60 420 3.9 5.8 4.525 260 0.4 65 4.5177 160 0.0 0.0 4.5  s 1000 MPa  0 2500 MPa  0s 900 MPa  00 3500 MPa  000 2200 MPa  00 1550 MPaFigure 8.11: Results of the mechanical model applied to the data collectedby Spencer on grade 316L [72]. (a) Simulated stress-strain curves and (b)simulated work-hardening curves, for uniaxial tension.Once again, the model provides a good qualitative prediction for the1938.5. De ning an Average  0-Martensite Behaviour in the Dynamic Composite Modelexperimental tensile response. In the case of testing at  196 C, one seesthat the maximum stress at necking is not properly captured. This can becorrected if the value of   0s is made to be a function of temperature suchthat the  0-martensite strength increases with decreasing temperature asone would expect.8.5 De ning an Average  0-Martensite Behaviourin the Dynamic Composite ModelIn the model presented above, it is necessary to treat each new incrementof  0-martensite formed in a given increment of applied strain as a separatephase with its own stress and strain. Each of these  0-martensite elementsmust be tracked individually in terms of the strain that they have undergoneand therefore the stress that they carry. It would be much easier if theaverage behaviour of the  0-martensite could be de ned directly based onthe current volume fraction of  0-martensite. One way of doing this in such adynamic composite has been recently proposed by Bouaziz et al. [260]. Thekey point is that, while each of the pre-existing df 0 should see the sameincrement of imposed strain (d ), the newly formed  0-martensite is formedwith no strain. Now, consider an average increment of strain hd i appliedto all elements of  0-martensite equally. This will also lead to an averagestrain, h i in the  0-martensite. Equating these two approaches gives:f 0 (h i) +df 0 (0) = (f 0 +df 0) [h i+hd i] (8.7)This can be simpli ed to:hd i= f 0 df 0f 0[h i+d ] h i (8.8)Neglecting the second order terms, one can write:hd i= d  df 0f 0h i (8.9)This allows one to calculate, for a given applied strain increment d , an1948.6. Summaryaverage strain increment for  0-martensite that accounts for the formationof df 0 at each step of the calculation having zero strain. In this way, one cantrack only the average strain in the  0-martensiteh irather than individualstrains for each of the df 0 formed. This has signi cant advantage in termsof the simplicity of calculating the incremental material response.The results of applying this method to the model developed above isillustrated in Figure 8.12 and compared to the full approach described above.The results are nearly identical as expected. While only shown here for thecase of a single grain size, this simpli ed average approach works equally wellfor all of the other conditions of grain size, temperature and compositiondescribed above.s48s46s48s48s46s49s48s46s50s48s46s51s48s46s52s48s46s53s48s53s48s49s48s48s49s53s48s84s114s117s101s32s115s116s114s101s115s115s32s40s77s80s97s41s84s114s117s101s32s115s116s114s97s105s110s32s69s120s112s101s114s105s109s101s110s116s32s77s111s100s101s108s32s65s118s101s114s97s103s101s32s109s111s100s101s108Figure 8.12: Comparison of both approaches applied to simulate the tensilecurve of the 28 a181m condition.8.6 SummaryThe novelty of the approach described in this chapter resides in the de-scription of the mechanical behaviour of the  0-martensite via a n-phase1958.6. Summarycomposite model, stressing the di erence in work-hardening between freshlyformed and more ancient  0-martensite. One of the assets of this model isits ability to describe, with minimal number of adjustable parameters, thegrain size dependence and the temperature dependence of the mechanicalproperties of grade 301LN. In particular, the model succeeds in explainingthe strain localization occurring at low temperatures or in a partially recrys-tallized sample. The extension to other austenitic grades seems possible aswell, as shown by the comparison of the simulation with Spencer’s results.Complex strain paths may be more challenging to model, given the uncer-tainty of describing a deformation by its Von Mises equivalents. In this case,incorporating the model within a more rigorous mechanical framework (e.g.crystal plasticity calculations) would avoid this problem.196Chapter 9Conclusion9.1 Summary and Key ResultsThe aim of this thesis, as de ned in chapter 2, was to implement a physically-based model for the mechanical response of 301LN stainless steel able ofcapturing the e ects of grain size, strain path and deformation tempera-ture. Beyond this, the work presented here has helped to advance our basicunderstanding of the relationship between the mechanical response and mi-crostructure in these complex materials. Here is how these objectives wereaccomplished in present work.A particularly important outcome of this work is the observation of thee ect of austenite grain size on the strain-induced martensitic phase trans-formations. As highlighted in the literature review (chapter 2), until now,there had been no single study with systematic observations on the trendsrelated to the rate of transformation with grain size re nement, particularlywith grain sizes in the nanometric scale. The results of this work shows thatthe e ect of starting austenite grain size on the transformation kinetics isnot large (much smaller than that predicted from the classical O-C model)but that this change in the kinetics does have a large e ect on the pre-dicted mechanical response (see e.g. Figure 8.7). Also interesting was thefact that the grain size dependence of the transformation kinetics were notmonotonic,  rst decreasing with grain size then increasing with grain size,the change in trend occurring for D 1 a181m. It was found that this tran-sition occurred at around the same grain size range where  -martensite wasfound to be suppressed. Thus, it has been hypothesized that the decreasingrate of  !  0 transformation corresponds to a decreasing proportion of -martensite to serve as nucleation sites for the  0-martensite. The increase1979.1. Summary and Key Resultsin the rate of transformation with further grain size re nement (below D 1 a181m) was suggested to be linked to a change in nucleation mechanism.In particular, for su ciently  ne grains, it was observed that the transfor-mation may nucleate preferentially at grain boundaries (in the absence of -martensite).The e ect of strain path on the mechanical response and transformationkinetics were also monitored by comparing the results of tests in uniaxialtension and simple shear. As with the e ect of grain size, the experimentaldata available in the literature (cf. chapter 2) does not present a consistentpicture with respect to the e ect of stress state and strain path. Consistentwith other work [151], it was found that the e ect of shear is to reduce theapparent rate of formation of  0-martensite, when tension and shear testswere compared on the basis of Von Mises equivalent strain. It has been ar-gued here that the use of the Von Mises equivalent strain (and similarly theVon Mises equivalent stress) is not appropriate for making direct compar-isons between the data as they ignore the e ects of plastic anisotropy. It hasbeen suggested here that the di erence between shear and tension, both interms of the transformation kinetics and mechanical response, could possi-bly be resolved if one were to account for texture and its evolution (e.g. viacrystal plasticity simulations). This goes against traditional models whichhave argued for the predominance of triaxiality [171]. Detailed observationson the crystallography of the  ! ! 0 transformation, however, also leadto questions about the importance of the macroscopic stress state on thetransformation kinetics. In the case of tension, no clear correlation could befound between the interaction energy for the  ! 0 transformation and thespeci c variants selected during this transformation based on the imposedstress. Attempts to correlate to slip activity were similarly unsuccessful. Onthe other hand, it was shown that the interaction energy and Schmid factors(which are shown to be proportional to one another) were well-correlated tothe observed variants of  -martensite.The estimation of the stresses borne in the  0-martensite during ten-sile loading by the technique based on the magnetomechanical e ect pre-sented in chapter 7, compared well with measurements performed using1989.1. Summary and Key Resultswell-established di raction techniques [13, 34]. A clear advantage for thistechnique is its relative simplicity both in terms of equipment required andthe processing of data. With this technique, it was demonstrated that thehardening of the austenite, when the alloy was deformed at room tempera-ture, was well approximated by deformation at 80 C, despite the fact thatsigni cant scale re nement occurs within the austenite due to the formationof  0-martensite. It was suggested that this scale re nement does not make alarge contribution to the macroscopic  ow stress because of the diminishingfraction of austenite with strain. In this sense, it was argued that it is mostrelevant to discuss the contributions of the individual phases as f 0  0 and(1 f )  rather than as   0 or   . These quantities were found to be ofprime importance in validating the model proposed in chapter 8.The unique feature of the model presented in chapter 8 is in its in-corporation of a \dynamic composite e ect", where consideration is givento \fresh"  0-martensite being formed at each increment of strain. To ourknowledge, this approach has not been previously adopted, most microstruc-turally based models considering monolithic behaviour for the  0-martensitephase. The other key feature is the requirement that  0-martensite formswith a compressive stress. This e ect, arising from the transformation strainassociated with the phase transformation, has previously been accounted forby means of an extra (positive) transformation strain associated with eachnew fraction of  0-martensite formed. This \dynamic composite" e ect con-tributes to the hardening coming from the elastic loading of the freshlyformed  0-martensite. This phenomenon helps to explain the observed highwork-hardening behaviour of the  0-martensite. The input parameters ofthis model have been held to a minimum, with most being identi able fromexperiments, the only parameters requiring adjustment being those de ningthe constitutive law of  0-martensite, i.e.   000,   00 and   00, these havingrelatively low sensitivity to the overall response.The model is capable of predicting grain size and temperature depen-dence of the  ow stress as well as the response in shear. Notably, it wassuccessful in predicting the apparition of tensile instabilities. Previously,such instabilities have been attributed to very high rates of  0-martensite1999.1. Summary and Key Resultstransformation and large transformation strains [72]. Here, it is proposedthat, under certain circumstances, the  0-martensite can lead to a softeningas its contribution to the  ow stress is lower than that of the austenite.With weight reduction and improved crashworthiness in mind, grain sizestrengthening processing routes have been examined with growing interestby steel industries and manufacturers. Such thermo-mechanical routes havebeen shown to be easily applicable to austenitic stainless steels in whichmartensite reversion enables substantial grain size re nement, e.g. [213].On a macroscopic level, the gain brought by grain size re nement is mostlyre ected on the yield strength of the material, the results of present studyshowing that its ultimate tensile strength and uniform elongation would besimilar for all grain sizes. More speci cally, in grade 301LN, grain re nementbelow  0.5 a181m will not be bene cial in forming operations, due to thestrain localization which appears when the yield strength of austenite istoo high. Speci cally, in this grade, localization occurred when the yieldstress of the austenite was raised above  700 MPa, regardless of whetherthe strengthening was due to temperature reduction or grain size re nement.The model for the development of such tensile instability, as proposed in thisstudy, should therefore constitute a valuable tool when forming operationsare envisaged.The modelling presented here also demonstrated the importance of therate of transformation on the mechanical properties, with a weak change inthe kinetics of  0-martensite formation translating to a large di erence intensile stress at necking (of the order of 150 MPa in Figure 8.7). Hence,the advantage of monitoring the volume fraction of  0-martensite in situduring forming operations. The Feritscope used in present study could bean appropriate tool to perform such measurements. However, as shown inFigure 7.5, the Feritscope measurements require correction due to the factthat the  0-martensite is not free of stresses. This has been raised previ-ously as a point of concern when using magnetic techniques for measure-ment of  0-martensite fraction [34]. It was shown, here, that for uniaxialstresses lower than  600 MPa (a value to adapt to the considered defor-mation route), corrections due to stress can be neglected. Another possible2009.2. Future Workuse of the Feritscope method presented in chapter 7 is in the evaluation ofresidual stresses. Evaluating the usefulness of this technique for assessingresidual stresses in deep drawn parts could be very useful, particularly whenconsidered in relation to problems such as delayed cracking [261].Finally, it was noted in chapter 1 that one of the key limitations to theimplementation of austenitic stainless steel sheet in formed parts comes froma lack of \knowledge" about its behaviour. This thesis provides signi cantnew insights both into the behaviour of this speci c alloy, as well as in thecoupling between phase transformations and mechanical properties moregenerally in austenitic stainless steels and therefore helps push towards abetter \knowledge" of behaviour for these steels.9.2 Future WorkThis thesis showed the di culty to model the grain size dependence of the ki-netics of  ! 0 transformation (the equations proposed in section 6.6 tend-ing to over predict the kinetics, although they capture the non-monotonice ect). The mechanisms of formation of  0, notably the reduction in  -martensite with scale re nement and the other mechanisms that may notinvolve nucleation on -martensite or grain boundaries demand deeper analy-sis before a complete model of the kinetics of  0 formation can be advanced.One of the questions that needs to be resolved clearly is the in uence ofstress (rather than strain) on the transformation kinetics. It has been re-cently proposed [262] that experiments should be performed where the ma-terial is pre-strained at a temperature where no transformation takes placefollowed by deformation at a lower temperature where transformation canoccur. In this way, the e ect of hardening of the austenite (increasing itsyield stress) by plastic deformation can a ect the subsequent transformationkinetics. Finally, with regards to the transformation kinetics, it was shownhere that the formation of  -martensite could be explained based on themacroscopic stress, but that the speci c variants of  0-martensite could notbased upon the concept of interaction energy nor based upon slip systemactivity. This is an area that needs much work as a good physical expla-2019.2. Future Worknation for the transformation kinetics requires a clear understanding of thedominant mechanisms of nucleation of  0-martensite. Further detailed mi-croscopy is needed to identify these mechanisms and to provide statistics forevaluating the di erent hypotheses that exist in the literature.With regard to the mechanical model derived in chapter 8, an impor-tant requirement of the model is that the  0-martensite forms in compres-sion. This part of the model could be validated, for instance, by X-raymicrobeam di raction experiments [263]. Another possible route to assessthe importance of the mechanical contrast between the phases would be bymeasurement of the Bauschinger e ect. Such experiments on 301LN havebeen performed and will be analyzed to compare with the predictions of themodel developed here.A natural next step to re ne the model developed here would be toincorporate it within the framework of crystal plasticity so that the variationof crystallographic textures along di erent strain paths can be considered.As noted already in this thesis, the important di erences in work-hardeningrates of austenite between tension and shear cannot be captured by a simpleVon Mises yield surface. Incorporation into such a model would also allow forarbitrary stress states and strain paths without the need for an assumptionabout the yield surface. Ideally, this model could then be used to simulatethe kinetics of phase transformation and the deformation curve along planestrain deformation, a common deformation route in industrial practice.Finally, as seen in chapter 8, very few parameters need to be varied in themechanical model developed here to account for the mechanical behaviourof other grades of materials presenting the TRIP e ect. The knowledge ofthe evolution of those parameters with some thermodynamics variables (e.g.stacking fault energy and Gibbs energies of the phases) would be an impor-tant contribution in terms of alloy design to tailor the chemical compositionof a material to match a predetermined range of speci cations. This re-quires that more experiments be performed on alloys of known chemistry toevaluate the e ects of solute on the overall mechanical response.202Bibliography[1] \Ultralight steel auto body, ulsab  nal report," tech. rep., American Iron and SteelInstitute, Washington, D.C., March 1998.[2] J. H. Schmitt, \New trends in austenitic stainless steel  at products for structuralapplications," in 4th Stainless Steel Science and Market Congress, 2002.[3] R. Armstrong Canadian Metallurgical Quarterly, vol. 13, p. 187, 1974.[4] N. Ohkubo, \E ect of alloying elements on the mechanical properties of the stableaustenitic stainless steel.," ISIJ International, vol. 34, no. 9, pp. 764{772, 1994.[5] F. Danoix and P. Auger, \Atom probe studies of the Fe-Cr system and stainlesssteels aged at intermediate temperature: A review," Materials Characterization,vol. 44, no. 1-2, pp. 177{201, 2000.[6] C. N. Hsiao, C. S. Chiou, and J. R. Yang, \Aging reactions in a 17-4 PH stainlesssteel," Materials Chemistry and Physics, vol. 74, no. 2, pp. 134{142, 2002.[7] C. L. Xie and E. Nakamachi, \Design of texture for improved formability of high-strength steel," Materials Science and Engineering A, vol. 340, no. 1-2, pp. 130{138,2003.[8] M. Sarwar and R. Priestner, \In uence of ferrite-martensite microstructural mor-phology on tensile properties of dual-phase steel," Journal of Materials Science,vol. 31, no. 8, pp. 2091{2095, 1996.[9] H. Takuda, K. Mori, T. Masachika, E. Yamazaki, and Y. Watanabe, \Finite elementanalysis of the formability of an austenitic stainless steel sheet in warm deep draw-ing," Journal of Materials Processing Technology, vol. 143{144, pp. 242{248, 2003.Proceedings of the International Conference on the Advanced Materials ProcessingTechnology, 2001.[10] E. PerdahcIoglu, H. Geijselaers, and M. Groen, \In uence of plastic strain ondeformation-induced martensitic transformations," Scripta Materialia, vol. 58,no. 11, pp. 947{950, 2008.[11] S. Rajasekhara, Development of Nano/Sub-micron Grain Structures in MetastableAustenitic Stainless Steels. PhD thesis, The University of Texas, Austin, U.S.A.,2007.[12] C.-S. Yoo, Y.-M. Park, Y.-S. Jung, and Y.-K. Lee, \E ect of grain size ontransformation-induced plasticity in an ultra ne-grained metastable austeniticsteel," Scripta Materialia, vol. 59, no. 1, pp. 71 { 74, 2008.203Bibliography[13] P. Dufour, \D etermination des propri et es microm ecaniques d’un acier TRIP pardi raction de neutrons et par corr elation directe d’images digitales," Master’s thesis,Universit e Catholique de Louvain, Belgium, 2007.[14] S. Nanga, Comportement et Transformations Martensitiques de deux aciers inoxyd-ables aust enitiques: e ets de la temperature, de la vitesse et du chargement. PhDthesis,  Ecole Nationale Sup erieure des Mines de Paris, France, 2008.[15] P. Lacombe, B. Baroux, and G. Beranger, eds., Stainless Steels. Les editions dephysique - Les Ulis, France, 1993.[16] A. J. Sedriks, Corrosion of Stainless Steels, ch. 2: Composition, Structure andMechanical Properties, pp. 18{21. second edition ed., 1996.[17] D. T. Llewellyn, \Work-hardening e ects in austenitic stainless steels," Journal ofMaterials Science & Technology, vol. 13, no. 5, pp. 389{400, 1997.[18] ASM Handbook, ch. Properties and Selection: Irons, Steels, and High-PerformanceAlloys - Section Wrought Stainless steels. ASM International, 2002.[19] \British-Adopted European Standard BS EN 10088-1:2005 - Stainless steels. List ofstainless steels."[20] J. Charles, \The new 200-series: an alternative answer to Ni surcharge? Risks oropportunities?," La Revue de M etallurgie-CIT, pp. 308{317, 2007.[21] M. Byrnes, M. Grujicic, and W. Owen, \Nitrogen strengthening of a stable austeniticstainless steel," Acta Metallurgica, vol. 35, no. 7, pp. 1853{1862, 1987.[22] Thermo-Calc, database TCFE 2000, S-version, 2000.[23] D. Bancroft, E. L. Peterson, and S. Minshall, \Polymorphism of iron at high pres-sure," Journal of Applied Physics, vol. 27, no. 3, pp. 291{298, 1956.[24] T. Takahashi and W. Bassett, \High-pressure polymorph of iron," Science, vol. 145,no. 3631, pp. 483{486, 1964.[25] L. Dougherty, G. Gray, E. Cerreta, R. McCabe, R. Field, and J. Bingert, \Raretwin linked to high-pressure phase transition in iron," Scripta Materialia, vol. 60,no. 9, pp. 772{775, 2009.[26] J. Friedel, Dislocations. Pergamon Press, Oxford, 1964.[27] D. Hull and D. J. Bacon, Introduction to Dislocations. Butterworth-Heinemann,2001.[28] L. Bracke, Deformation Behaviour of Austenitic Fe-Mn Alloys by Twinning andMartensitic Transformation. PhD thesis, Ghent University, Netherlands, 2007.[29] S. Tavares, J. Pardal, M. G. da Silva, H. Abreu, and M. da Silva, \Deformationinduced martensitic transformation in a 201 modi ed austenitic stainless steel,"Materials Characterization, vol. 60, no. 8, pp. 907{911, 2009.204Bibliography[30] L. R emy, Maclage et transformation martensitique cfc ! hc induite par d eformationplastique dans les alliages aust enitiques  a basse  energie de d efaut d’empilement dessyst emes Co-Ni et Fe-Mn-Cr-C. PhD thesis, Universit e de Paris-Sud, Orsay, France,1975.[31] R. Schramm and R. Reed, \Stacking fault energies of seven commercial austeniticstainless steels," Metallurgical and Materials Transactions A, vol. 6, no. 7, pp. 1345{1351, 1975.[32] F. B. Pickering, \Physical metallurgical development of stainless steels," in Proc.Conf. Stainless Steels, Gothenburg (E. G. Dunlop, ed.), 1984.[33] F. Lecroisey and A. Pineau, \Martensitic transformations induced by plastic de-formation in the Fe-Ni-Cr-C system," Metallurgical and Materials Transactions B,vol. 3, no. 2, pp. 391{400, 1972.[34] J. Talonen, E ect of strain-induced  0 martensite transformation on mechanicalproperties of metastable austenitic stainless steels. PhD thesis, Aalto UniversitySchool of Science and Technology (TKK), Finland, 2007.[35] L. E. Murr, Interfacial Phenomena in Metals and Alloys. Addison-Wesley Pub. Co.,1975.[36] J. P. Hirth, \Thermodynamics of stacking faults," Metallurgical Transactions B,vol. 1, no. 9, pp. 2367{2374, 1970.[37] G. B. Olson and M. Cohen, \A general mechanism of martensitic nucleation. part I:General concepts and fcc-hcp transformation," Metallurgical and Materials Trans-actions A, vol. 7, no. 12, pp. 1897{1904, 1976.[38] S. Allain, J. P. Chateau, O. Bouaziz, S. Migot, and N. Guelton, \Correlationsbetween the calculated stacking fault energy and the plasticity mechanisms in Fe-Mn-C alloys," Materials Science and Engineering A, vol. 387-389, pp. 158{162, 2004.[39] P. J. Ferreira and P. M ullner, \A thermodynamic model for the stacking-fault en-ergy," Acta Materialia, vol. 46, no. 13, pp. 4479{4484, 1998.[40] R. Bunshah and R. Mehl, \Rate of propagation of martensite," Transactions of theAIME, vol. 197, pp. 1251{1258, 1953.[41] D. A. Porter and K. E. Easterling, Phase Transformations in Metals and Alloys.London: Chapman & Hall, 2nd edition ed., 1993.[42] R. Grange and H. Stewart, \The temperature range of martensite formation," Trans-actions of the AIME, vol. 167, pp. 467{497, 1946.[43] P. Maxwell, A. Goldberg, and J. Shyne, \Stress-assisted and strain-induced marten-sites in Fe-Ni-C alloys," Metallurgical and Materials Transactions B, vol. 5, no. 6,pp. 1305{1318, 1974.[44] I. Y. Georgieva and I. I. Nikitina, \Isothermal and athermal martensitic transfor-mations," Metal Science and Heat Treatment, vol. 14, no. 5, pp. 452{458, 1972.205Bibliography[45] Z. Nishiyama, Martensitic Transformation. Academic Press, London, 1978.[46] A. Borgenstam and M. Hillert, \Nucleation of isothermal martensite," Acta Mate-rialia, vol. 48, no. 11, pp. 2777{2785, 2000.[47] J. Patel and M. Cohen, \Criterion for the action of applied stress in the martensitictransformation," Acta Metallurgica, vol. 1, no. 5, pp. 531{538, 1953.[48] E. Bain, \The nature of martensite," Transactions of the AIME, vol. 70, p. 2546,1924.[49] J. J. Jonas, Y. He, and S. Godet, \The possible role of partial dislocations in fa-cilitating transformations of the Nishiyama-Wassermann type," Scripta Materialia,vol. 52, no. 3, pp. 175{179, 2005.[50] M. Humbert, B. Petit, B. Bolle, and N. Gey, \Analysis of the  !  !  0 vari-ant selection induced by 10% plastic deformation in 304 stainless steel at -60 C,"Materials Science and Engineering A, vol. 454-455, pp. 508{17, 2007.[51] H. Bhadeshia, Worked examples in the Geometry of Crystals, ch. 3, pp. 26{27. TheInstitute of Materials, London, 2001.[52] P. Mangonon and G. Thomas, \The martensite phases in 304 stainless steel," Met-allurgical and Materials Transactions B, vol. 1, no. 6, pp. 1577{1586, 1970.[53] M. Andersson, R. Stalmans, and J.  Agren, \Uni ed thermodynamic analysis of thestress-assisted  !  martensitic transformation in Fe-Mn-Si alloys," Acta Materi-alia, vol. 46, no. 11, pp. 3883{3891, 1998.[54] H. Funakubo, Shape Memory Alloys. Gordon and Breach Science Publishers, 1987.[55] K. Otsuka and C. Wayman, Shape Memory Materials. Cambridge University Press,1998.[56] H. Yu, \A new model for the volume fraction of martensitic transformations," Met-allurgical and Materials Transactions A, vol. 28, no. 12, pp. 2499{2506, 1997.[57] K. Tsuzaki, Y. Natsume, and T. Maki, \Transformation reversibility in Fe-Mn-Sishape memory alloy," Journal de Physique IV, vol. 5, pp. 409{414, 1995. Interna-tional Conference on Martensitic Transformations (ICOMAT).[58] G. B. Olson and M. Azrin, \Transformation behavior of TRIP steels," Metallurgicaland Materials Transactions A, vol. 9, no. 5, pp. 713{721, 1978.[59] G. F. Bolling and R. H. Richman, \The in uence of stress on Martensite-starttemperatures in Fe-Ni-C alloys," Scripta Metallurgica, vol. 4, no. 7, pp. 539{543,1970.[60] G. B. Olson and M. Cohen, \A mechanism for the strain-induced nucleation ofmartensitic transformations," Journal of the Less Common Metals, vol. 28, no. 1,pp. 107{118, 1972.206Bibliography[61] J. Talonen, P. Aspegren, and H. H anninen, \Comparison of di erent methods formeasuring strain induced  0 martensite content in austenitic steels," Materials Sci-ence And Technology, vol. 20, no. 12, pp. 1506{1512, 2004.[62] M. Dickson Journal of Applied Crystallography, vol. 2, p. 176, 1969.[63] H. M. Rietveld, \A pro le re nement method for nuclear and magnetic structures,"Journal of Applied Crystallography, vol. 2, no. 2, pp. 65{71, 1969.[64] A. Albinati and B. T. M. Willis, \The Rietveld method in neutron and X-ray powderdi raction," Journal of Applied Crystallography, vol. 15, no. 4, pp. 361{374, 1982.[65] R. D. Arnell J. Iron Steel Inst., vol. 206, pp. 1035{1036, 1968.[66] R. W. Cheary and Y. Ma-sorrell J. Mater. Sci., vol. 35, pp. 1105{1113, 2000.[67] M. Radu, J. Valy, A. Gourgues, F. L. Strat, and A. Pineau, \Continuous magneticmethod for quantitative monitoring of martensitic transformation in steels contain-ing metastable austenite," Scripta Materialia, vol. 52, no. 6, pp. 525{530, 2005.[68] S. Hecker, M. Stout, K. Staudhammer, and J. Smith, \E ects of strain state andstrain rate on deformation-induced transformation in 304 stainless steel. Part I: Mag-netic measurements and mechanical behavior," Metallurgical and Materials Trans-actions A, vol. 13, no. 4, pp. 619{626, 1982.[69] L. Zhao, N. H. van Dijk, E. Br uck, J. Sietsma, and S. van der Zwaag, \Magneticand X-ray di raction measurements for the determination of retained austenite inTRIP steels," Materials Science and Engineering A, vol. 313, no. 1-2, pp. 145{152,2001.[70] J. Post, H. Nolles, K. Datta, and H. Geijselaers, \Experimental determination of theconstitutive behaviour of a metastable austenitic stainless steel," Materials Scienceand Engineering A, vol. 498, no. 1-2, pp. 179{190, 2008.[71] A. K. De, D. C. Murdock, M. C. Mataya, J. G. Speer, and D. K. Matlock, \Quan-titative measurement of deformation-induced martensite in 304 stainless steel byX-ray di raction," Scripta Materialia, vol. 50, pp. 1445{1449, 2004.[72] K. Spencer, The Work-Hardening of Austenitic Stainless Steel, applied to the Fab-rication of High-Strength Conductors. PhD thesis, McMaster University, Canada,2004.[73] W. G. Burgers Physica, vol. 1, p. 561, 1934.[74] G. Stone and G. Thomas, \Deformation induced alpha’ and epsilon martensites inFe-Ni-Cr single crystals," Metallurgical and Materials Transactions B, vol. 5, no. 9,pp. 2095{2102, 1974.[75] T.-H. Lee, E. Shin, C.-S. Oh, H.-Y. Ha, and S.-J. Kim, \Correlation between stack-ing fault energy and deformation microstructure in high-interstitial-alloyed austeni-tic steels," Acta Materialia, vol. 58, no. 8, pp. 3173{3186, 2010.207Bibliography[76] L. R emy and A. Pineau, \Twinning and strain-induced fcc ! hcp transformationin the Fe-Mn-Cr-C system," Materials Science and Engineering B, vol. 28, no. 1,pp. 99{107, 1977.[77] H. Idrissi, L. Ryelandt, M. V eron, D. Schryvers, and P. Jacques, \Is there a rela-tionship between the stacking fault character and the activated mode of plasticityof Fe-Mn-based austenitic steels?," Scripta Materialia, vol. 60, no. 11, pp. 941{944,2009.[78] J. Brooks, M. Loretto, and R. Smallman, \Direct observations of martensite nucleiin stainless steel," Acta Metallurgica, vol. 27, no. 12, pp. 1839{1847, 1979.[79] J. Putaux and J. Chevalier, \HREM study of self-accomodated thermal  martensitein an Fe-Mn-Si-Cr-Ni shape memory alloy," Acta Materialia, vol. 44, pp. 1701{1716,1996.[80] K. Verbeken, N. Van Caenegem, and D. Raabe, \Identi cation of epsilon martensitein a Fe-based shape memory alloy by means of EBSD," Micron, vol. 40, no. 1,pp. 151{156, 2009.[81] G. B. Olson and M. Cohen, \Kinetics of strain-induced martensitic nucleation,"Metallurgical and Materials Transactions A, vol. 6, no. 4, pp. 791{795, 1975.[82] J. W. Brooks, M. H. Loretto, and R. E. Smallman, \In-situ observations of the for-mation of martensite in stainless steel," Acta Metallurgica, vol. 27, no. 12, pp. 1829{1838, 1979.[83] Seeger Zeitschrift fur Metallkunde, vol. 47, p. 653, 1956.[84] H. Fujita and S. Ueda, \Stacking faults and f.c.c. ( ) ! h.c.p. ( ) transformationin 18/8-type stainless steel," Acta Metallurgica, vol. 20, no. 5, pp. 759{767, 1972.[85] B. Jiang, X. Qi, S. Yang, W. Zhou, and T. Y. Hsu, \E ect of stacking fault prob-ability on  !  martensitic transformation and shape memory e ect in Fe-Mn-Sibased alloys," Acta Materialia, vol. 46, no. 2, pp. 501{510, 1998.[86] P. Hedstr om, U. Lienert, J. Almer, and M. Od en, \Elastic strain evolution and -martensite formation in individual austenite grains during in situ loading of ametastable stainless steel," Materials Letters, vol. 62, no. 2, pp. 338{340, 2008.[87] S. Allain, Caract erisation et mod elisation thermom ecaniques multi- echelles desm ecanismes de d eformation et d’ ecrouissage d’aciers aust enitiques  a haute teneuren mangan ese. Application  a l’e et TWIP. PhD thesis, INP Lorraine, Metz, France,2004.[88] G. Blanc, R. Tricot, and R. Castro, \Transformations martensitiques dans les aciersinoxydables aust enitiques Fe-Cr-Ni. Relation entre les param etres de la phase  etles m ecanismes de la transformation.," Revue de la M etallurgie, vol. 70, no. 7-8,pp. 527{541, 1973.[89] W. L. Fink and E. D. Campbell Transactions of the American Society for SteelTreating, vol. 9, p. 717, 1926.208Bibliography[90] N. Sljakov, G. Kurdjumov, and N. Goufdob Z. Phys., vol. 45, p. 384, 1927.[91] C. S. Roberts, \E ect of carbon on the volume fractions and lattice parameters ofretained austenite and martensite," Transactions of the AIME, vol. 197, pp. 203{204, 1953.[92] G. Krauss, \Deformation and fracture in martensitic carbon steels tempered at lowtemperatures," Metallurgical and Materials Transactions B, vol. 32, no. 2, pp. 205{221, 2001.[93] L. Xiao, Z. Fan, and Z. Jinxiu, \Lattice parameter variation with carbon content ofmartensite (I)," Physical Review, vol. 52, no. 14, pp. 9970{9978, 1995.[94] J. A. Venables, \Martensite transformation in stainless steel," Philosophical Maga-zine, vol. 7, no. 73, pp. 35{44, 1962.[95] J. Breedis and L. Kaufman, \The formation of hcp and bcc phases in austenitic ironalloys," Metallurgical and Materials Transactions B, vol. 2, no. 9, pp. 2359{2371,1971.[96] J. Dash and H. M. Otte, \The martensite transformation in stainless steel," ActaMetallurgica, vol. 11, no. 10, pp. 1169{1178, 1963.[97] L. E. Murr, K. Staudhammer, and S. Hecker, \E ects of strain state and strain rateon deformation-induced transformation in 304 stainless steel. Part II: Microstruc-tural study," Metallurgical and Materials Transactions A, vol. 13, pp. 627{635, 1982.[98] J.-Y. Choi and W. Jin, \Strain-induced martensite formation and its e ect on strain-hardening behavior in the cold-drawn 304 austenitic stainless steels," Scripta Ma-terialia, vol. 36, no. 1, pp. 99{104, 1997.[99] H. Fujita and T. Katayama, \In-situ observation of strain-induced  ! ! 0 and !  0 martensitic transformations in Fe-Cr-Ni alloys," Material Transactions ofthe Japanese Institute of Metals, vol. 33, no. 3, pp. 243{252, 1992.[100] V. Tsakiris and D. V. Edmonds, \Martensite and deformation twinning in austeniticsteels," Materials Science and Engineering A, vol. 273-275, pp. 430{436, 1999.[101] T. Inamura, K. Takashima, and Y. Higo, \Crystallography of nanometre-sized  0-martensite formed at intersections of mechanical  twins in an austenitic stainlesssteel," Philosophical Magazine, vol. 83, no. 8, pp. 935{954, 2003.[102] N. Nakada, H. Ito, Y. Matsuoka, T. Tsuchiyama, and S. Takaki, \Deformation-induced martensitic transformation behavior in cold-rolled and cold-drawn type 316stainless steels," Acta Materialia, vol. 58, no. 3, pp. 895{903, 2010.[103] P. Mangonon and G. Thomas, \Structure and properties of thermal-mechanicallytreated 304 stainless steel," Metallurgical and Materials Transactions B, vol. 1, no. 6,pp. 1587{1594, 1970.[104] X. Wang, \Private communication." 2010.209Bibliography[105] C. Sinclair, \A molecular dynamics study of deformation induced phase transfor-mations at fault band intersections," in 15th International Conference on Strengthin Materials (ICSMA), 2009.[106] N. Gey, B. Petit, and M. Humbert, \Electron backscattered di raction study of  / 0-martensite martensitic variants induced by plastic deformation in 304 stainlesssteel," Metallurgical And Materials Transactions A, vol. 36, no. 12, pp. 3291{3299,2005.[107] Y.-K. Lee and C.-S. Choi, \E ects of thermal cycling on the kinetics of the  ! martensitic transformation in an Fe-17%Mn alloy," Metallurgical and MaterialsTransactions A, vol. 31, no. 11, pp. 2735{2738, 2000.[108] G. B. Olson and M. Cohen, \A general mechanism of martensitic nucleation. partII: fcc!bcc and other martensitic transformations," Metallurgical and MaterialsTransactions A, vol. 7, no. 11, pp. 1905{1914, 1976.[109] J. Guimaraes and S. F. De Oliveira, \Work-hardening and martensitic transforma-tion in Fe-27% Ni-0.23%C at 263 K," Scripta Metallurgica, vol. 13, no. 7, pp. 537{542, 1979.[110] G. Yang, C. Huang, S. Wu, and Z. Zhang, \Strain-induced martensitic transforma-tion in 304L austenitic stainless steel under ECAP deformation," Acta MetallurgicaSinica, vol. 45, no. 8, pp. 906{911, 2009.[111] T. Suzuki, H. Kojima, K. Suzuki, T. Hashimoto, and M. Ichihara, \An experimen-tal study of the martensite nucleation and growth in 18/8 stainless steel," ActaMetallurgica, vol. 25, no. 10, pp. 1151{1162, 1977.[112] K. Spencer, J. D. Embury, K. T. Conlon, M. V eron, and Y. Br echet, \Strengtheningvia the formation of strain-induced martensite in stainless steels," Materials Scienceand Engineering A, vol. 387-89, pp. 873{881, 2004.[113] W.-S. Lee and C.-F. Lin, \The morphologies and characteristics of impact-inducedmartensite in 304L stainless steel," Scripta Materialia, vol. 43, no. 8, pp. 777 { 782,2000.[114] L. Bracke, L. Kestens, and J. Penning, \Transformation mechanism of  0 martensitein an austenitic Fe-Mn-C-N alloy," Scripta Materialia, vol. 57, no. 5, pp. 385{388,2007.[115] A. Das, S. Sivaprasad, M. Ghosh, P. Chakraborti, and S. Tarafder, \Morphologiesand characteristics of deformation induced martensite during tensile deformationof 304 LN stainless steel," Materials Science and Engineering A, vol. 486, no. 1-2,pp. 283{286, 2008.[116] C. Huang, G. Yang, Y. Gao, S. Wu, and S. Li, \Investigation on the nucleationmechanism of deformation-induced martensite in an austenitic stainless steel undersevere plastic deformation," Journal of Materials Research, vol. 22, no. 3, pp. 724{729, 2007.210Bibliography[117] K. Spencer, M. V eron, K. Yu-Zhang, and J. D. Embury, \The strain induced mar-tensite transformation in austenitic stainless steels. part 1: In uence of temperatureand strain history," Materials Science And Technology, vol. 25, no. 1, pp. 7{17, 2009.[118] H.-S. Yang and H. Bhadeshia, \Austenite grain size and the martensite-start tem-perature," Scripta Materialia, vol. 60, no. 7, pp. 493{495, 2009.[119] A. Bogers and W. Burgers, \Partial dislocations on the f110g planes in the bcclattice and the transition of the fcc into the bcc lattice," Acta Metallurgica, vol. 12,no. 2, pp. 255{261, 1964.[120] C. Sinclair and R. Hoagland, \A molecular dynamics study of the fcc ! bcc trans-formation at fault intersections," Acta Materialia, vol. 56, no. 16, pp. 4160{4171,2008.[121] C. Hayzelden, K. Chattopadhyay, J. Barry, and B. Cantor, \Transmission electronmicroscopy observations of the f.c.c.-to-h.c.p. martensite transformation in Co-Nialloys," Philosophical Magazine A, vol. 63, no. 3, pp. 461{470, 1991.[122] S. C. H.-S. W. J. R. Y. . H. K. D. H. Bhadeshia, \Mechanical stabilisation ofaustenite," Materials Science and Technology, vol. 22, pp. 641{644, 2006.[123] V. Kouznetsova and M. Geers, \Modeling the interaction between plasticity and theaustenite-martensite transformation," International Journal for Multiscale Compu-tational Engineering, vol. 5, no. 2, pp. 129{140, 2007.[124] M. Umemoto and W. Owen, \E ects of austenitizing temperature and austenitegrain size on the formation of athermal martensite in an iron-nickel and an iron-nickel-carbon alloy," Metallurgical and Materials Transactions B, vol. 5, no. 9,pp. 2041{2046, 1974.[125] B. H. Jiang, L. Sun, R. Li, and T. Y. Hsu, \In uence of austenite grain size on ! martensitic transformation temperature in Fe-Mn-Si-Cr alloys," Scripta Met-allurgica et Materialia, vol. 33, no. 1, pp. 63{68, 1995.[126] T. Durlu, \E ect of austenite grain size on  martensite formation in an Fe-Mn-Moalloy," Journal of Materials Science Letters, vol. 16, no. 4, pp. 320{321, 1997.[127] Y. Inokuti and B. Cantor, \Splat-quenched Fe{Ni alloys," Scripta Metallurgica,vol. 10, no. 7, pp. 655{659, 1976.[128] Y. Inokuti and B. Cantor, \Overview 15: The microstructure and kinetics of marten-site transformations in splat-quenched Fe and Fe-Ni alloys { II," Acta Metallurgica,vol. 30, no. 2, pp. 343{356, 1982.[129] A. Hamada, P. Sahu, S. Ghosh Chowdhury, L. Karjalainen, and T. Levoska,J.And Oittinen, \Kinetics of the  !  martensitic transformation in  ne-grained Fe-26Mn-0.14C austenitic steel," Metallurgical And Materials TransactionsA, vol. 39, pp. 462{465, 2008.211Bibliography[130] K. Tao, H. Choo, H. Li, B. Clausen, J.-E. Jin, and Y.-K. Lee, \Transformation-induced plasticity in an ultra ne-grained steel: An in situ neutron di raction study,"Applied Physics Letters, vol. 90, no. 10, p. 101911, 2007.[131] J.-H. Jun and C.-S. Choi, \Variation of stacking fault energy with austenite grainsize and its e ect on the Ms temperature of  !  martensitic transformation inFe-Mn alloy," Materials Science and Engineering A, vol. 257, no. 2, pp. 353{356,1998.[132] T. Maki, Y. Tomota, and I. Tamura, \E ect of grain size on the transformation-induced plasticity in metastable austenitic Fe-Ni-C alloy," Journal of the JapaneseInstitute of Metals, vol. 38, pp. 871{876, 1974.[133] E. Jimenez-Melero, N. van Dijk, L. Zhao, J. Sietsma, S. O erman, J. Wright, andS. van der Zwaag, \Martensitic transformation of individual grains in low-alloyedTRIP steels," Scripta Materialia, vol. 56, no. 5, pp. 421{424, 2007.[134] K. Nohara, Y. Ono, and N. Ohashi, \Composition and grain-size dependencies ofstrain-induced martensitic transformation in metastable austenitic stainless steels,"The Iron and Steel Institute of Japan (ISIJ), vol. 63, pp. 212{222, 1977.[135] R. H. Leal and J. Guimaraes, \Microstructure evolution during mechanically in-duced martensitic transformation in Fe-33%Ni-0.1%C," Materials Science and En-gineering, vol. 48, no. 2, pp. 249{254, 1981.[136] J. Gonzales, R. Aranda, and M. Jonap a, \The in uence of grain size on the kineticsof strain induced martensite in type 304 stainless steel," in Applications of stainlesssteel ’92, Stockholm, Sweden, pp. 1009{1016, 1992.[137] W. Jeong, D. Matlock, and G. Krauss, \E ects of tensile-testing temperature on de-formation and transformation behavior of retained austenite in a 0.14C-1.2Si-1.5Mnsteel with ferrite-bainite-austenite structure," Materials Science and Engineering A,vol. 165, no. 1, pp. 9{18, 1993.[138] S. K. Varma, J. Kalyanam, L. E. Murr, and V. Shrinivas, \E ect of grain size ondeformation-induced martensite formation in 304 stainless steel and 316 stainlesssteel during room temperature tensile testing," Journal of Materials Science Letters,vol. 13, no. 2, pp. 107{111, 1994.[139] A. P etein, On the Interaction between strain-induced Phase Transformations andMechanical Properties in Mn-Si-Al Steels And Ni-Cr Austenitic Stainless Steels.PhD thesis, Universit e Catholique de Louvain, Belgium, 2007.[140] V. Shrinivas, S. K. Varma, and L. E. Murr, \Deformation-induced martensitic char-acteristics in 304 and 316 stainless steels during room-temperature rolling," Metal-lurgical and Materials Transactions A, vol. 26, pp. 661{671, 1995.[141] A. De, J. Speer, D. Matlock, D. Murdock, M. Mataya, and R. Comstock,\Deformation-induced phase transformation and strain-hardening in type 304 aus-tenitic stainless steel," Metallurgical and Materials Transactions A, vol. 37, no. 6,pp. 1875{1886, 2006.212Bibliography[142] T. Angel, \Formation of martensite in austenitic stainless steels - e ects of defor-mation, temperature, and composition," Journal of Iron & Steel Institute, vol. 177,pp. 165{174, 1954.[143] K. Mumtaz, S. Takahashi, J. Echigoya, L. Zhang, Y. Kamada, and M. Sato, \Tem-perature dependence of martensitic transformation in austenitic stainless steel,"Journal of Materials Science Letters, vol. 22, no. 6, pp. 423{427, 2003.[144] R. Kubler,  Etude du comportement des aciers  a e et TRIP : approches mi-crom ecaniques et ph enom enologiques. Application  a la mise en forme. PhD thesis,Universit e Paul Verlaine de Metz, France, 2004.[145] J. Talonen, H. H anninen, P. Nenonen, and G. Pape, \Formation of shear bandsand strain-induced martensite during plastic deformation of metastable austeniticstainless steels," Acta Materialia, vol. 55, no. 18, pp. 610{6118, 2007.[146] S. Nanga, A. Pineau, B. Tanguy, L. Naz e, and P.-O. Santacreu, \Plasticity andstrain-induced martensitic transformation in two austenitic stainless steels," in In-ternational Conference on Martensitic Transformations (ICOMAT), 2008.[147] V. Talyan, R. H. Wagoner, and J. K. Lee, \Formability of stainless steel," Metal-lurgical And Materials Transactions A, vol. 29, no. 8, pp. 2161{2172, 1998.[148] H. N. Han, C. G. Lee, C.-S. Oh, T.-H. Lee, and S.-J. Kim, \A model for deforma-tion behavior and mechanically induced martensitic transformation of metastableaustenitic steel," Acta Materialia, vol. 52, no. 17, pp. 5203{5214, 2004.[149] S. G. S. Raman and K. A. Padmanabhan, \Tensile deformation-induced martensi-tic transformation in AISI 304LN austenitic stainless steel," Journal of MaterialsScience Letters, vol. 13, pp. 389{392, 1994.[150] J. Talonen, P. Nenonen, G. Pape, and H. H anninen, \E ect of strain rate on thestrain-induced  ! 0 martensite transformation and mechanical properties of aus-tenitic stainless steels," Metallurgical and Materials Transactions A, vol. 36, no. 2,pp. 421{432, 2005.[151] A. A. Lebedev and V. V. Kosarchuk, \In uence of phase transformations on the me-chanical properties of austenitic stainless steels," International Journal of Plasticity,vol. 16, no. 7-8, pp. 749{767, 2000.[152] T. Iwamoto, T. Tsuta, and Y. Tomita, \Investigation on deformation mode depen-dence of strain-induced martensitic transformation in TRIP steels and modellingof transformation kinetics," International Journal of Mechanical Sciences, vol. 40,no. 2-3, pp. 173{182, 1998.[153] G. W. Powell, E. R. Marshall, and W. A. Backofen, \Strain-hardening of austeniticstainless steel," ASM Transactions, vol. 50, pp. 478{497, 1958.[154] M. Kato and T. Mori, \Orientation of martensite formed in Fe-23Ni-5Cr crystalsunder uniaxial stress along [001]," Acta Metallurgica, vol. 25, no. 8, pp. 951{956,1977.213Bibliography[155] P. Jacques, Q. Furn emont, T. Pardoen, and F. Delannay, \On the role of martensi-tic transformation on damage and cracking resistance in TRIP-assisted multiphasesteels," Acta Materialia, vol. 49, no. 1, pp. 139{152, 2001.[156] P. Jacques, Q. Furn emont, F. Lani, T. Pardoen, and F. Delannay, \Multiscale me-chanics of TRIP-assisted multiphase steels. Part I: Characterization and mechanicaltesting," Acta Materialia, vol. 55, no. 11, pp. 3681{3693, 2007.[157] DeMania, \The in uence of martensitic transformation on the formability of 304Lstainless steel sheet," Master’s thesis, Massachusetts Institute of Technology, 1995.[158] D. Mohr and J. Jacquemin, \Large deformation of anisotropic austenitic stainlesssteel sheets at room temperature: Multi-axial experiments and phenomenologicalmodeling," Journal of the Mechanics and Physics of Solids, vol. 56, no. 10, pp. 2935{2956, 2008.[159] E. PerdahcIoglu, H. Geijselaers, and J. Hu etink, \In uence of stress state and strainpath on deformation-induced martensitic transformations," Materials Science andEngineering A, vol. 481{482, pp. 727{731, 2008.[160] S. Kundu and H. Bhadeshia, \Transformation texture in deformed stainless steel,"Scripta Materialia, vol. 55, no. 9, pp. 779{781, 2006.[161] L. Malet, C. Sinclair, P. Jacques, and S. Godet, \Grain scale analysis of variantselection during the  !  !  0 phase transformation in austenitic steels," inPTM, 2010.[162] D. C. Ludwigson and J. A. Berger, \Plastic behaviour of metastable austeniticstainless steels," Journal of Iron & Steel Institute, vol. 207, pp. 63{69, 1969.[163] W. W. Gerberich, G. Thomas, E. R. Parker, and V. F. Zackay vol. 3, pp. 849{899,ASM, Metals Park, 1970.[164] J. R. C. Guimaraes, \The deformation-induced martensitic reaction in polycrys-talline Fe-30.7Ni-0.06C," Scripta Metallurgica, vol. 6, p. 795, 1972.[165] K. Sugimoto, M. Kobayashi, and S. Hashimoto, \Ductility and strain-induced trans-formation in a high-strength transformation-induced plasticity-aided dual-phasesteel," Metallurgical Transactions A, vol. 23, no. 11, pp. 3085{3091, 1992.[166] I. Pychmintsev, R. Savrai, B. De Cooman, and O. Moriau, \High strain rate be-haviour of TRIP-aided automotive steels," in Proceedings of the International Con-ference on TRIP-aided high strength ferrous alloys, GRIPS, Aachen:Mainz, pp. 299{302, 2002.[167] H. C. Shin, T. K. Ha, W. J. Park, and Y. W. Chang, \Deformation-induced marten-sitic transformation under various deformation modes," Key Engineering Materials,vol. 223, no. 236, pp. 667{672, 2003.[168] C. Guntner and R. Reed, \The e ect of experimental variables including the mar-tensitic transformation on the low-temperature mechanical properties of austeniticstainless stees," ASM Transactions, vol. 55, pp. 399{419, 1962.214Bibliography[169] T. Iwamoto and T. Tsuta, \Computational simulation of the dependence of theaustenitic grain size on the deformation behavior of TRIP steels," InternationalJournal of Plasticity, vol. 16, no. 7-8, pp. 791{804, 2000.[170] R. G. Stringfellow, Mechanics of Strain-induced Transformation Toughening inMetastable Austenitic Stainless Steels. PhD thesis, Massachusetts Institute of Tech-nology, U.S.A., 1990.[171] R. G. Stringfellow, D. M. Parks, and G. B. Olson, \A constitutive model fortransformation plasticity accompanying strain-induced martensitic transformationsin metastable austenitic steels," Acta Metallurgica et Materialia, vol. 40, no. 7,pp. 1703{1716, 1992.[172] J. Serri, M. Martiny, and G. Ferron, \A numerical analysis of the formability ofunstable austenitic steels," Journal of Materials Processing Technology, vol. 164-165, pp. 1241{1247, 2005. AMPT/AMME05 Part 2.[173] U. F. Kocks, \Laws for work-hardening and low-temperature creep," Journal ofEngineering Materials and Technology - Transactions of the ASME, pp. 76{85, 1976.[174] S. Allain, O. Bouaziz, and J. Chateau, \Thermally activated dislocation dynamicsin austenitic FeMnC steels at low homologous temperature," Scripta Materialia,vol. 62, no. 7, pp. 500{503, 2010.[175] E. Hall, \The deformation and ageing of mild steel," Proceedings of the PhysicalSociety B, vol. 64, pp. 747{753, 1951.[176] N. Petch, \Ductile fracture of polycrystalline -iron," Philosophical Magazine, vol. 1,pp. 186{190, 1956.[177] S. Rajasekhara, P. J. Ferreira, L. P. Karjalainen, and A. Kyrolainen, \Hall-Petchbehaviour in ultra- ne grained AISI 301LN stainless steel," Metallurgical and Ma-terials Transactions A, vol. 38, no. 6, pp. 1202{1210, 2007.[178] A. Di Schino, M. Barteri, and J. M. Kenny, \E ects of grain size on the properties ofa low nickel austenitic stainless steel," Journal of Materials Science, vol. 38, no. 23,pp. 4725{4733, 2003.[179] B. Kashyap and K. Tangri, \On the hall-petch relationship and substructural evo-lution in type 316L stainless steel," Acta Metallurgica et Materialia, vol. 43, no. 11,pp. 3971{3981, 1995.[180] S. M. G. Singh, K.K.; Sangal, \Hallpetch behaviour of 316L austenitic stainless steelat room temperature," Materials Science and Technology, vol. 18, no. 2, pp. 165{172, 2002.[181] S. Brochet, Compr ehension du r^ole de la microstructure d’aciers inoxydablesaust enitiques  a grains  ns sur le comportement en fatigue. PhD thesis, Universit edes Sciences et Technologies de Lille, France, 2007.[182] M. A. Meyers and C. K. Kumar, Mechanical Metallurgy: Principles and applications,ch. 14: Grain size Strengthening, pp. 494{514. Prentice-Hall Inc., 1984.215Bibliography[183] A. W. Thompson, \E ect of grain size on work hardening in nickel," Acta Metal-lurgica, vol. 25, pp. 83{86, 1977.[184] T. Lebedkina, M. Lebyodkin, J.-P. Chateau, A. Jacques, and S. Allain, \On themechanism of unstable plastic  ow in an austenitic Fe-Mn-C TWIP steel," MaterialsScience and Engineering, A, vol. 519, no. 1-2, pp. 147{154, 2009.[185] K. Renard, S. Ryelandt, and P. Jacques, \Characterisation of the Portevin-LeCh^atelier e ect a ecting an austenitic TWIP steel based on digital image corre-lation," Materials Science and Engineering, A, vol. 527, no. 12, pp. 2969{2977,2010.[186] S. Nanga, A. Pineau, B. Tanguy, and P.-O. Santacreu, \Strain-induced martensitictransformations in two austenitic stainless steels: macro-micro behaviour," in 17thEuropean Conference on Fracture (ECF17), 2008.[187] R. A. Varin, B. Mazurek, and D. Himbeault, \Discontinuous yielding in ultra ne-grained austenitic stainless steels," Materials science and Engineering, vol. 94,pp. 109{119, 1987.[188] K. Singh, \Strain hardening behaviour of 316L austenitic stainless steel," MaterialsScience and Technology, vol. 20, no. 9, pp. 1134{1142, 2004.[189] B. P. Kashyap and K. Tangri, \Hall-Petch relationship and substructural evolu-tion in boron containing type 316L stainless steel," Acta Materialia, vol. 45, no. 6,pp. 2383{2395, 1997.[190] X. Feaugas and H. Haddou, \Grain-size e ects on tensile behavior of nickel and AISI316L stainless steel," Metallurgical and Materials Transactions A, vol. 34, no. 10,pp. 2329{2340, 2003.[191] C. W. Sinclair, H. Proudhon, and J. D. Mithieux, \Work-hardening in a  ne grainedaustenitic stainless steel," Materials Science Forum, vol. 539-543, pp. 4714{4719,2007.[192] D. Rousseau, G. Blanc, R. Tricot, and A. Gueussier, \Structure stability underdeformation and at low temperatures for austenitic stainless steels with Cr-Ni,"Mem. Sci. Revue de la M etallurgie, vol. 67, no. 5, p. 315, 1970.[193] P.-O. Santacreu, J.-C. Glez, G. Chinouilh, and T. Frohlich, \Behaviour model ofaustenitic stainless steels for automotive structural parts," Steel Research Interna-tional, vol. 77, pp. 686{691, 2006.[194] K. Spencer, K. T. Conlon, Y. Br echet, and J. D. Embury, \The strain inducedmartensite transformation in austenitic stainless steels. part 2: E ect of internalstresses on mechanical response," Materials Science And Technology, vol. 25, no. 1,pp. 18{28, 2009.[195] M. R. Berrahmoune, Martensitic transformation and delayed cracking phenomenonin the 301LN unstable austenitic steel. PhD thesis, ENSAM, Metz, France, 2006.216Bibliography[196] T. Narutani, \E ect of deformation-induced martenic transformation on the plas-tic behavior of metastable austenitic stainless steel," Material Transactions of theJapanese Institute of Metals, vol. 30, no. 1, pp. 33{45, 1989.[197] A. Molinari, \Extensions of the self-consistent tangent model," Modelling and Sim-ulation in Materials Science & Engineering, vol. 7, no. 5, pp. 683{697, 1999.[198] L. Bardella, \An extension of the secant method for the homogenization of thenonlinear behavior of composite materials," International Journal of EngineeringScience, vol. 41, no. 7, pp. 741{768, 2003.[199] Y. Benveniste, \A new approach to the application of Mori-Tanaka’s theory incomposite materials," Mechanics of Materials, vol. 6, no. 2, pp. 147{157, 1987.[200] C. Garion, B. Skoczen, and S. Sgobba, \Constitutive modelling and identi cation ofparameters of the plastic strain-induced martensitic transformation in 316L stainlesssteel at cryogenic temperatures," International Journal of Plasticity, vol. 22, no. 7,pp. 1234{1264, 2006.[201] Y. Tomita and T. Iwamoto, \Constitutive modelling of trip steel and its applicationto the improvement of mechanical properties," International Journal of MechanicalSciences, vol. 37, no. 12, pp. 1295{1305, 1995.[202] U. F. Kocks and H. Mecking, \Physics and phenomenology of strain-hardening: thefcc case," Progress in Materials Science, vol. 48, pp. 171{273, 2003.[203] N. Tsuchida, Y. Tomota, H. Moriya, O. Umezawa, and K. Nagai, \Application ofthe Kocks-Mecking model to tensile deformation of an austenitic 25Cr-19Ni steel,"Acta Materialia, vol. 49, no. 15, pp. 3029{3038, 2001.[204] S. Allain, J. P. Chateau, and O. Bouaziz, \A physical model of the twinning-inducedplasticity e ect in a high manganese austenitic steel," Materials Science and Engi-neering A, vol. 387-389, pp. 143{147, 2004.[205] O. Bouaziz and N. Guelton, \Modelling of TWIP e ect on work-hardening," Mate-rials Science and Engineering A, vol. 319-321, pp. 246{249, 2001.[206] O. Bouaziz, S. Allain, and C. Scott, \E ect of grain and twin boundaries on thehardening mechanisms of twinning-induced plasticity steels," Scripta Materialia,vol. 58, no. 6, pp. 484{487, 2008.[207] J. Bouquerel, K. Verbeken, and B. C. de Cooman, \Microstructure-based modelfor the static mechanical behaviour of multiphase steels," Acta Materialia, vol. 54,no. 6, pp. 1443{1456, 2006.[208] R. Rodriguez and I. Gutierrez, \Uni ed formulation to predict the tensile curvesof steels with di erent microstructures," Materials Science Forum, vol. 426-432,pp. 4525{4530, 2003.[209] A. Di Schino, M. Barteri, and J. M. Kenny, \Development of ultra  ne grain struc-ture by martensitic reversion in stainless steel," Journal of Materials Science Letters,vol. 21, no. 9, pp. 751{753, 2002.217Bibliography[210] H. W. Zhang, Z. K. Hei, G. Liu, J. Lu, and K. Lu, \Formation of nanostructuredsurface layer on AISI 304 stainless steel by means of surface mechanical attritiontreatment," Acta Materialia, vol. 51, no. 7, pp. 1871{1881, 2003.[211] K. Tomimura, S. Takaki, and Y. Tokunaga, \Reversion mechanism from deforma-tion induced martensite to austenite in metastable austenitic stainless steels," ISIJInternational, vol. 31, pp. 1431{1437, 1991.[212] A. F. Padilha, R. L. Plaut, and P. R. Rios, \Annealing of cold-worked austeniticstainless steels," ISIJ International, vol. 43, no. 2, pp. 135{143, 2003.[213] A. Poulon-Quintin, S. Brochet, J. B. Vogt, J. C. Glez, and J. D. Mithieux, \Finegrained austenitic stainless steels: The role of strain induced  0 martensite and thereversion mechanism limitations," ISIJ International, vol. 49, no. 2, pp. 293{301,2009.[214] R. Misra, S. Nayak, P. Venkatasurya, V. Ramuni, M. Somani, and L. Karjalainen,\Nanograined/ultra ne-grained structure and tensile deformation behavior of shearphase reversion-induced 301 austenitic stainless steel," Metallurgical and MaterialsTransactions A, vol. 41, no. 8, pp. 2162{2174, 2010.[215] F. Forouzan, A. Naja zadeh, A. Kermanpur, A. Hedayati, and R. Surkialiabad,\Production of nano/submicron grained AISI 304L stainless steel through the mar-tensite reversion process," Materials Science and Engineering A, vol. 527, no. 27-28,pp. 7334{7339, 2010.[216] F. T. Inc., Feritscope MP30E - Operator’s Manual, 2006.[217] R. Bozorth, Ferromagnetism, ch. 2: Factors a ecting Magnetic Quality, pp. 14{19.Van Nostrand, 1951.[218] K. M. Olsen and R. C. Sto ers, \E ect of carbon content on the magnetic propertiesof iron-30 % cobalt-15 % chromium alloys," Journal Of Applied Physics, vol. 42,no. 4, pp. 1792{1793, 1971.[219] E. du Tr emolet de Lacheisserie, Magnetostriction, Theory and Applications of Mag-netoelasticity, ch. 3: Magnetoelasticity of Soft Ferromagnets: The Physical E ects,pp. 198{211. CRC Press, 1993.[220] S. Chikazumi, Physics of Magnetism, ch. 8: Magnetostriction, pp. 161{185. JohnWiley & Sons, Inc., 1964.[221] V. Randle and O. Engler, Texture Analysis: Macrotexture, Microtexture & Orien-tation Mapping, ch. 6: The Kikuchi Di raction Pattern, pp. 148{151. Gordon andBreach Science Publishers, 2000.[222] S. Takaki, S. Tanimoto, and Y. Tokunaga, \Grain re ning of austenitic stainlesssteels by  0-reversion to  -reversion," Transactions of the Iron and Steel Instituteof Japan, vol. 25, no. 9, pp. B223{B223, 1985.218Bibliography[223] R. Ueji, N. Tsuji, Y. Minamino, and Y. Koizumi, \Ultragrain re nement of plain lowcarbon steel by cold-rolling and annealing of martensite," Acta Materialia, vol. 50,no. 16, pp. 4177{4189, 2002.[224] B. Raeisinia, Modelling the E ect of Grain Size Distribution on the MechanicalResponse of Metals. PhD thesis, The University of British Columbia, Vancouver,Canada, 2008.[225] A. Ramirez, J. Lippold, and S. Brandi, \The relationship between chromium nitrideand secondary austenite precipitation in duplex stainless steels," Metallurgical andMaterials Transactions A, vol. 34, no. 8, pp. 1575{1597, 2003.[226] ASM Handbook Online, ch. 8: Mechanical Testing and Evaluation. ASM Interna-tional, 2000.[227] C. G’Sell, S. Boni, and S. Shrivastava, \Application of the plane simple shear test fordetermination of the plastic behaviour of solid polymers at large strains," Journalof Materials Science, vol. 18, pp. 903{918, 1983.[228] E. Rauch and C. G’Sell, \Flow localization induced by a change in strain path inmild steel," Materials Science and Engineering A, vol. 111, pp. 71{80, 1989.[229] P.-Y. Manach,  Etude du comportement thermom ecanique d’alliages  a m emoire deforme Ni-Ti. PhD thesis, Institut National Polytechnique de Grenoble, France,1993.[230] P. Y. Manach and N. Couty, \Elastoviscohysteresis constitutive law in convectedcoordinate frames: application to  nite deformation shear tests," ComputationalMechanics, vol. 28, no. 1, pp. 17{25, 2002.[231] I. Tamura, T. Maki, and H. Hato Trans. ISIJ, vol. 10, p. 163, 1970.[232] O. Instruments, \HKL - Channel 5 User manual," 2007.[233] K. Datta, R. Delhez, P. Bronsveld, J. Beyer, H. Geijselaers, and J. Post, \A low-temperature study to examine the role of  -martensite during strain-induced trans-formations in metastable austenitic stainless steels," Acta Materialia, vol. 57, no. 11,pp. 3321{3326, 2009.[234] J. Goldstein, D. E. Newbury, D. C. Joy, and C. E. Lyman, Scanning Electron Mi-croscopy and X-ray Microanalysis, vol. 1, ch. 3: Electron Beam { Specimen Inter-actions, pp. 75{86. Springer; 3rd edition, 2003.[235] D. Barbier, N. Gey, N. Bozzolo, S. Allain, and M. Humbert, \EBSD for analysingthe twinning microstructure in  ne-grained TWIP steels and its in uence on workhardening," Journal of Microscopy, vol. 235, pp. 67{78, 2009.[236] X. Liang, J. McDermid, O. Bouaziz, X. Wang, J. Embury, and H. Zurob, \Mi-crostructural evolution and strain hardening of Fe-24Mn and Fe-30Mn alloys duringtensile deformation," Acta Materialia, vol. 57, no. 13, pp. 3978{3988, 2009.219Bibliography[237] I. Gutierrez-Urrutia, S. Zae erer, and D. Raabe, \Electron channeling contrastimaging of twins and dislocations in twinning-induced plasticity steels under con-trolled di raction conditions in a scanning electron microscope," Scripta Materialia,vol. 61, no. 7, pp. 737{740, 2009.[238] B. Petit,  Etude du comportement m ecanique et des modi cations de texture et demicrostructure induites par la transformation de phase  ! 0 sous contrainte d’unacier AISI 304. Aspects exp erimentaux et mod elisations. PhD thesis, Universit ePaul Verlaine de Metz, France, 2006.[239] C. Cayron, F. Barcelo, and Y. de Carlan, \The mechanisms of the fcc-bcc martensitictransformation revealed by pole  gures," Acta Materialia, vol. 58, no. 4, pp. 1395{1402, 2010.[240] R. Misra, Z. Zhang, Z. Jia, M. Somani, and L. Karjalainen, \Probing de-formation processes in near-defect free volume in high strength-high ductilitynanograined/ultra ne-grained metastable austenitic stainless steels," Scripta Ma-terialia, vol. 63, no. 11, pp. 1057{1060, 2010.[241] H. Geijselaers and E. PerdahcIoglu, \Mechanically induced martensitic transfor-mation as a stress-driven process," Scripta Materialia, vol. 60, no. 1, pp. 29 { 31,2009.[242] J. D. Embury and C. W. Sinclair, \The mechanical properties of  ne-scale two-phasematerials," Materials Science and Engineering A, vol. 319-321, pp. 37{45, 2001.[243] K. Tao, J. J. Wall, H. Li, D. W. Brown, S. C. Vogel, and H. Choo, \In situ neu-tron di raction study of grain-orientation-dependent phase transformation in 304Lstainless steel at a cryogenic temperature," Journal of Applied Physics, vol. 100,no. 12, p. 123515, 2006.[244] P. Hedstr om, L. E. Lindgren, J. Almer, U. Lienert, J. Bernier, M. Terner, andM. Od en, \Load partitioning and strain-induced martensite formation during ten-sile loading of a metastable austenitic stainless steel," Metallurgical And MaterialsTransactions A, vol. 40, no. 5, pp. 1039{1048, 2009.[245] R. Bozorth, Ferromagnetism, ch. 13: Stress and magnetostriction, pp. 595{712. VanNostrand, 1951.[246] B. Westermo and L. Thompson, \Magnetic strain measurement methodology forstructural damage assessment and monitoring," in Advances in Instrumentation,Proceedings, vol. 47, pp. 1295{1303, 1992.[247] D. Jiles and D. Atherton, \Theory of the magnetisation process in ferromagnetsand its application to the magnetomechanical e ect," Journal of Physics D, vol. 17,pp. 1265{1281, 1984.[248] D. C. Jiles, \Theory of the magnetomechanical e ect," Journal of Physics D, vol. 28,pp. 1537{1547, 1995.220Bibliography[249] M. Smaga, F. Walther, and D. Ei er, \Deformation-induced martensitic transforma-tion in metastable austenitic steels," Materials Science and Engineering A, vol. 483-484, pp. 394{397, 2008. 14th International Conference on the Strength of Materials(ICSMA).[250] A. M. Beese, D. Mohr, and P.-O. Santacreu, \Isotropic phase transformation inanisotropic stainless steel 301LN sheets," in European Symposium on MartensiticTransformations (Esomat), 2009.[251] A. M. Beese, \Quanti cation of phase transformation in stainless steel 301LNsheets," Master’s thesis, Massachusetts Institute of Technology, 2009.[252] J. Kaleta and J. Zebracki, \Application of the Villari e ect in a fatigue examinationof nickel," Fatigue & Fracture Of Engineering Materials & Structures, vol. 19, no. 12,pp. 1435{1443, 1996.[253] H. Mecking and U. F. Kocks, \Kinetics of  ow and strain-hardening," Acta Metal-lurgica, vol. 29, pp. 1865{1875, 1981.[254] G. Cailletaud and K. Sai, \A polycrystalline model for the description of ratchet-ting: E ect of intergranular and intragranular hardening," Materials Science andEngineering A, vol. 480, no. 1-2, pp. 24{39, 2008.[255] O. Bouaziz and P. Buessler, \Iso-work increment assumption for heterogeneous ma-terial behavior modelling," Advanced Engineering Materials, vol. 6, no. 1-2, pp. 79{83, 2004.[256] L. Delannay, P. Jacques, and T. Pardoen, \Modelling of the plastic  ow of TRIP-aided multiphase steel based on an incremental mean- eld approach," InternationalJournal of Solids and Structures, vol. 45, no. 6, pp. 1825{1843, 2008.[257] L. Mazzoni-Leduc, T. Pardoen, and T. Massart, \Strain gradient plasticity analysisof transformation induced plasticity in multiphase steels," International Journal ofSolids and Structures, vol. 45, no. 20, pp. 5397{5418, 2008.[258] A. Rolett, Strain Hardening at Large Strains in Aluminum Alloys. PhD thesis,Drexel University, Philadelphia, U.S.A., 1987.[259] S. Shrivastava, J. Jonas, and G. Canova, \Equivalent strain in large deformationtorsion testing : Theoretical and practical considerations," Journal of the Mechanicsand Physics of Solids, vol. 30, no. 1-2, pp. 75{90, 1982.[260] O. Bouaziz, C. W. Sinclair, and M. Goun e, \An improved approach for the descrip-tion of the behaviour of dynamic composites," 2009.[261] M. Berrahmoune, S. Berveiller, K. Inal, and E. Patoor, \Delayed cracking in 301LNaustenitic steel after deep drawing: Martensitic transformation and residual stressanalysis," Material Science and Engineering A, vol. 438-440, pp. 262 { 266, 2006.[262] A. Das, P. C. Chakraborti, S. Tarafder, and H. K. D. H. Bhadeshia, \Analysisof deformation induced martensitic transformation in stainless steels," MaterialsScience and Technology, vol. 27, pp. 366{370, 2011.221[263] D. Stephan and K. Richter, \Fast local stress measurement by the x-ray microbeamdi raction technique using an annular proportional counter," Crystal Research andTechnology, vol. 16, pp. 57{61, 1981.[264] A. M. Beese and D. Mohr, \Identi cation of the direction-dependency of the mar-tensitic transformation in stainless steel using in situ magnetic permeability mea-surements," 2010.[265] S. Kundu and H. Bhadeshia, \Crystallographic texture and intervening transforma-tions," Scripta Materialia, vol. 57, no. 9, pp. 869{872, 2007.222Appendix ACalibration of the Feritscopein Grade 301LNThis appendix de nes the calibration necessary to adapt the Feritscope mea-surements to the volume fractions of  0-martensite actually present in grade301LN after straining. The present calibration was performed towards vol-ume fractions measured by Rietveld  tting of X-ray di raction spectra, amethod already used in the past for phase quanti cation in austenitic stain-less steels [139].This calibration was performed on only one condition of grain size,namely the D=28 a181m condition. Given that the size of  0-martensite varieslittle when the austenitic grain size is varied, it was assumed that the calibra-tion curve could be applied to all grain size conditions of 301LN presentedin this work.Rectangular strips ( 100 mm  20 mm  0.8 mm) were strained inuniaxial tension, the tensile direction corresponding to the prior rolling di-rection. The X-ray di raction spectra were acquired using the Bruker D8di ractomer presented in section 5.2.3, with no texture correction. Thosespectra are presented in Figure A.1.Six Feritscope measurements were performed on each samples, accordingto the procedure described in section 4.2.1. An average Feritscope reading(denoted as FS0 14) and a dispersion were calculated. The correlation be-tween Feritscope measurements and Rietveld calculation of  0-martensiteappear in Figure A.2.14 The notation emphasizes the fact that the measurement is performed under no appliedstress.223Appendix A. Calibration of the Feritscope in Grade 301LNs52s48s53s48s54s48s55s48s56s48s57s48s49s48 s48s49s48s48s50s48s48s51s48s48s52s48s48s49s48s49s53s50s48s50s53s51s48s51s53s52s48s52s51s32s84s114s117s101s32s115s116s114s97s105s110s32s40s37s41s50s61553s32s40s111s41s73s110s116s101s110s115s105s116s121s32s40s99s111s117s110s116s115s41s40s50s50s41γs40s49s48s41αs39s40s49s49s41γs40s50s48s41γs40s50s48s41αs39s40s50s48s41γs40s50s49s41αs39s40s51s49s41γs40s50s48s41αs39Figure A.1: X-ray di raction patterns illustrating the change in the propor-tion of phases when the strain is increased.s48s49s48s50s48s51s48s52s48s53s48s54s48s48s50s48s52s48s54s48s56s48s49s48s77s97s114s116s101s110s115s105s116s101s32s99s111s110s116s101s110s116s44s32s102s61537s39s32s40s37s41s70s101s114s105s116s115s99s111s112s101s32s114s101s97s100s105s110s103s44s32s70s83s48s32s40s37s41s102s61537s39s32s61s32s49s46s54s56s32s70s83s48Figure A.2: Calibration curve of the Feritscope towards Rietveld re nementof X-Ray Di raction spectra.224Appendix A. Calibration of the Feritscope in Grade 301LNFigure A.2 shows that the relation between the Feritscope readings andthe actual volume fraction (f 0) can be described by the linear equation:f 0 = 1:68 FS0 (A.1)which was already veri ed in grade 301LN [61, 250, 264].The calibration was performed for Feritscope readings ranging from 0 to60% only. The Feritscope may su er from saturation of the magnetic signalfor higher volume fractions [72], consequently, such high volume fractionsneed to be treated with caution.225Appendix BThe Patel-Cohen Model forVariant SelectionHere we summarize the methodology for computing the interaction energyas outlined by Humbert et al. [50]. A more detailed description can be foundin this reference.The interaction energy as de ned by Patel and Cohen [47] can be writtenas,W =   (B.1)where  is the macroscopically imposed stress tensor and  is the trans-formation strain associated with the transformation between austenite and -martensite or austenite and  0-martensite. As suggested by Kundu andBhadeshia [265], the factor of 0.5 introduced by Humbert et al. has beendropped. In the case of a uniaxial tensile test,  is written as, =0B@ 11 0 00 0 00 0 01CA (B.2)The deformation gradient associated with the austenite to  -martensitetransformation can be written as [50]D ! =0B@1 0 1=p20 1 00 0 11CA (B.3)where the x direction is parallel to a h112i (shearing direction) and z226Appendix B. The Patel-Cohen Model for Variant Selectionis parallel to a h111i .In the case of the austenite to  0-martensite transformation the defor-mation gradient can be written as,D ! 0 =0BB@a 0p32a  a 06a 00 a 0p83a 00 0 a 0p2c 1CCA (B.4)where x is parallel to the common close packed directions in the austeniteand  0-martensite and z is perpendicular to the close packed planes of theaustenite and  0-martensite. Also, a = 0.254 nm, c = 0.415 nm, anda 0 = 0.287 nm correspond to the lattice parameters of  -martensite and 0-martensite respectively, the values being taken from Humbert [50].To convert the above deformation gradients to strain tensors one canwrite  = 1=2 D +DT where the superscript T denotes transpose.In order to calculate the interaction energy, one must re-write both theimposed stress and the transformation strain in the same coordinate frame.In the case of Humbert et al., this was chosen to be the macroscopic frame ofreference associated with the tensile test. One can, however, calculate thiswithin the frame of reference that the transformation strains are written in.In the speci c case of the  -martensite transformation occurring during atensile test, this has obvious advantages.The transformation strain associated with the austenite to  -martensitetransformation is a pure shear (i.e.  13 =  31 =  2p2  1 and all other ij = 0) when calculated in the frame of reference coinciding with the x = b= h112i , z = n = h111i coordinate system de ned above. A macroscopictensile stress  can be resolved onto this slip system as,  = ( n) b givingthe interaction energy as,W ! = ( n) b 13 (B.5)Given that the transformation strain is  xed at that the applied tensilestress ( 11) is a constant, then this can be re-written as,227Appendix B. The Patel-Cohen Model for Variant SelectionW ! = m 11 13 (B.6)where m is the Schmidt factor for thef111g h112i slip system in ques-tion. Thus, the Patel-Cohen interaction energy is directly proportional tothe Schmid factor for the austenite to  -martensite transformation.228


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