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Evaluation of in-situ laser ultrasonics for abnormal grain growth in a plain carbon steel Tran, Brian 2018

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EVALUATION OF IN-SITU LASER ULTRASONICS FOR ABNORMAL GRAIN GROWTH IN A PLAIN CARBON STEEL  by  Brian Tran  B.S., University of Washington, 2015  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Materials Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)  February 2018  © Brian Tran, 2018 ii  Abstract  The control of grain size distribution is an important factor governing the homogeneity of mechanical properties for steels and other alloys used in a wide range of applications. In the present work, the evolution of austenite grain size is in-situ monitored in an A36 plain carbon steel during isothermal holding between 900°C and 1150°C using laser ultrasonics. In this technique, the attenuation of laser generated ultrasound is related to a representative mean grain size in the material. In A36 steel, the coarsening and dissolution of AlN precipitates at 1000°C leads to a stage of abnormal austenite grain growth with the development of a bimodal grain size distribution. This stage corresponds to a period of high grain growth rate when measured by laser ultrasonics, suggesting that the technique is sensitive to the progression of abnormal grain growth stages. For abnormal grain growth scenarios, the laser ultrasonic measurements are compared with ex-situ metallographic measurements of large grain area fraction on quenched samples.   Experimental correlation between laser ultrasonic grain size measurement and area fraction of large grains is validated with a finite element analysis of ultrasound wave propagation in an anisotropic polycrystalline aggregate of controlled grain size distribution. The numerical analysis provides important insight into the scattering of ultrasound waves in a material of heterogeneous grain size. The range of applicability for the laser ultrasonic technique is evaluated using grain size calculations from simulated attenuation for selected grain size distributions.    iii  Lay Summary  Polycrystalline materials such as alloys and ceramics are composed of many grains, where each grain has a crystallographic orientation different from its neighbors. The strength of polycrystalline materials is generally higher when they are composed of fine grains. Precipitates promote the retention of fine grains but also enable the development of abnormal grain growth during heat treatment. Abnormal grain growth is defined by the rapid growth of a few large grains, which results in a heterogeneous grain structure and jeopardizes consistent mechanical properties.  Laser ultrasonics uses optically generated ultrasound waves to rapidly evaluate materials in real-time. It is a non-contact technique that is compatible for evaluating high temperature processes such as austenite grain growth in steels. This thesis studies the laser ultrasonic response during abnormal grain growth in a plain carbon steel. Findings are supported by finite element simulations of ultrasound wave propagation through various grain structures.  iv  Preface  The work presented in this thesis was conducted by the author at the University of British Columbia within the Department of Materials Engineering. This included all experimental design, sample preparation, heat treatments, in-situ observation, metallography, experimental analysis, and finite element simulations. Small strip samples described in Chapter 4 were designed by Thomas Garcin and fabricated by the Materials Engineering machine shop. Laser ultrasonic analysis was performed with the CTOME software developed by Thomas Garcin.  v  Table of Contents  Abstract .......................................................................................................................................... ii Lay Summary ............................................................................................................................... iii Preface ........................................................................................................................................... iv Table of Contents ...........................................................................................................................v List of Tables .............................................................................................................................. viii List of Figures .................................................................................................................................x List of Symbols ........................................................................................................................... xvi List of Abbreviations ...................................................................................................................xx Acknowledgements .................................................................................................................... xxi Chapter 1: Introduction ................................................................................................................1 Chapter 2: Literature Review .......................................................................................................4 2.1 Grain growth ................................................................................................................... 4 2.1.1 Theory of grain growth ............................................................................................... 4 2.1.1 Particle pinning ........................................................................................................... 9 2.1.2 Austenite grain growth in aluminum-killed steel...................................................... 15 2.2 Prior austenite grain size measurement ......................................................................... 20 2.2.1 Summary of techniques............................................................................................. 20 2.2.2 Laser ultrasonics ....................................................................................................... 22 2.3 Ultrasonic scattering theory .......................................................................................... 27 2.3.1 Rayleigh-stochastic scattering regime transition ...................................................... 27 2.3.2 Consideration of grain size distribution .................................................................... 32 vi  Chapter 3: Scope and Objectives................................................................................................37 Chapter 4: Experimental Methodology .....................................................................................38 4.1 Material and sample geometry ...................................................................................... 38 4.2 In-situ microstructural characterization ........................................................................ 39 4.2.1 Heat treatment ........................................................................................................... 39 4.2.2 In-situ laser ultrasonic measurement ........................................................................ 40 4.2.3 Interpretation of ultrasound pulse waveforms .......................................................... 41 4.3 Ex-situ microstructural characterization ....................................................................... 44 4.3.1 Specimen preparation................................................................................................ 44 4.3.2 Characterization of microstructure ........................................................................... 44 4.4 Finite element simulation of ultrasound pulse propagation .......................................... 45 4.4.1 Generation of sub-domain......................................................................................... 45 4.4.2 Generation of Abaqus simulation domain ................................................................ 51 4.4.3 Simulation of ultrasound pulse propagation ............................................................. 54 Chapter 5: Austenite Grain Growth in A36 Steel .....................................................................57 5.1 Evolution of austenite grain size during isothermal heat treatment .............................. 57 5.2 Characterization of interrupted heat treatment microstructures .................................... 60 5.2.1 Laser ultrasonic measurements ................................................................................. 60 5.2.2 Ex-situ metallography ............................................................................................... 61 5.3 Interpretation of laser ultrasonic measurements ................................................................. 66 Chapter 6: Finite Element Simulation of Ultrasound Pulse Propagation in Austenite .........70 6.1 Unimodal grain size distributions ................................................................................. 70 6.1.1 Generation of sub-domain......................................................................................... 70 vii  6.1.2 Effect of mesh template size ..................................................................................... 72 6.1.3 Narrow grain size distributions ................................................................................. 74 6.1.4 Wide grain size distributions .................................................................................... 76 6.2 Bimodal grain size distributions ................................................................................... 78 6.2.1 Sensitivity of grain cluster spatial configuration ...................................................... 78 6.2.2 Sensitivity of grain cluster position with respect to ultrasound pulse ...................... 81 6.2.3 Variations with bimodal area ratio and mean grain sizes ......................................... 88 6.2.4 Area fraction rule of mixtures model ........................................................................ 91 6.3 Application to LUMet ................................................................................................... 93 6.4.1 Grain size calculation from FE simulated attenuation spectra.................................. 93 6.4.2 Comparison of attenuation from domains with similar calculated grain size ........... 95 6.4.3 Grain size calculation uncertainty ............................................................................. 99 Chapter 7: Conclusion ...............................................................................................................102 7.1 Summary ..................................................................................................................... 102 7.2 Future work ................................................................................................................. 104 Bibliography ...............................................................................................................................105  viii  List of Tables  Table 2.1 Composition in weight percent of DQSK steel used by Cheng et al. [42] ................... 15 Table 2.2 Composition in weight percent of the steels used by Militzer et al. [48] ..................... 17 Table 4.1 Composition in weight percent of the as-received A36 steel ....................................... 38 Table 4.2 Summary of parameters used in MICRESS® generation of narrow grain size distribution sub-domains ............................................................................................ 46 Table 4.3 Summary of parameters used in MICRESS® generation of wide unimodal grain size distribution sub-domains. Columns “Lseed = R” and “Number of grains” list 12 values corresponding to 12 grain types. ..................................................................... 47 Table 4.4 Summary of parameters used in MICRESS® generation of bimodal grain size distribution sub-domains. Seeds of each grain type were randomly dispersed throughout the entire sub-domain. ............................................................................. 49 Table 4.5 Summary of parameters used in MICRESS® generation of bimodal grain size distribution sub-domains with ordered configurations. Seeds of each grain type were randomly dispersed throughout rectangular regions of the sub-domain defined by two points (x1, y1) x (x2, y2). ............................................................................................. 50 Table 4.6 Material properties of austenite at 1000°C. Elastic constants use Hill averaging [106]. .................................................................................................................................... 53 Table 5.1 Mean values for tests of laser ultrasonic grain size measurements between 500 seconds and 600 seconds of isothermal holding ...................................................................... 58 Table 5.2 Data of large cluster area fraction in etched A36 steel following interrupted 1000°C isothermal holding...................................................................................................... 64 ix  Table 6.1 Data of effective large grain area fraction in simulation domains and from calculations using correlated laser ultrasonic grain size ................................................................ 86 Table 6.2 Grain size and uncertainty from calculation using the FE simulated attenuation spectra in Figure 6.17 ............................................................................................................. 98 x  List of Figures  Figure 2.1 Idealized two dimensional model for grain growth [14] ............................................... 5 Figure 2.2 Schematic drawing of tetrakaidecahedron [16] ............................................................. 6 Figure 2.3 Optical micrographs of etched AISI 304L austenitic stainless steel a) as-received b) after annealing at 850°C for 3 hours [27] .................................................................. 12 Figure 2.4 Grain growth evolution for 15 independent Monte Carlo Potts simulations with particle volume fraction of 10%. Grain diameter uses arbitrary units and time uses Monte Carlo timestep units [31]. ............................................................................... 13 Figure 2.5 a) Micrographs showing networks of AlN particles in DQSK steel samples held for 24 hours at 700°C and annealed for a) 30 minutes at 1000°C, imaged using STEM b) 10 minutes at 1100°C, imaged using TEM. c) Measured and modelled evolution of mean AlN particle diameter during 1000°C annealing [42]. ..................................... 16 Figure 2.6 Metallographic measurement of austenite grain size during isothermal holding after heating at 5°C/s in a) DQSK steel and b) A36 steel [48] ........................................... 18 Figure 2.7 Evaluation of the prior austenite grain structure during abnormal grain growth in an A36 steel at 1000°C [52] ............................................................................................ 19 Figure 2.8 Calibration of b  parameter from experimental mean austenite grain size (EQAD) measurements [4] ....................................................................................................... 24 Figure 2.9 Evolution of mean grain size (EQAD) during soaking of INCONEL 718 at 1050°C a) measured with LUMet and EBSD metallography, and b) modeled with a parabolic growth equation for coarse grains and Zener pinning for small grains [5] ................ 26 xi  Figure 2.10 Measured attenuation (points) and model (lines) for longitudinal waves in aluminum rods with indicated mean grain diameters [91]. Indicated scattering range denoted the frequencies in which comparisons were made to a previous study [77]. ................... 28 Figure 2.11 Measured longitudinal wave attenuation in an equiaxed iron-30% nickel alloy, modeled with a curve of monotonically decreasing slope connecting Rayleigh and stochastic scattering regimes. Frequency uses units of megacycles and FB  is the boundary frequency between scattering regimes defined at λ = 2πD [85]. .............. 29 Figure 2.12 Theoretical model of attenuation for longitudinal ultrasound waves in polycrystalline steel at 1000°C [6] a) for various mean grain size b) normalized with respect to mean grain size, pink dotted line indicates the transition between Rayleigh and stochastic scattering regimes defined by λ = 2πD ..................................................................... 31 Figure 2.13 Calculation of ultrasonic attenuation using modified Roney’s theory [83,95] and theoretical grain size distributions (solid) and comparison to Rayleigh scattering law (dashed). (∆K/K)2 is average elastic mismatch parameter [96]. .............................. 33 Figure 2.14 Calculated attenuation (points) and linear regressions (straight lines) for single power-law grain size distributions, (∆K/K)2 =  π2/2, K = 1, D1 = 0.02 μm, D2 = 320 μm, ξ = 0.5, 1.6, 2.4, 3.6 [90] .............................................................................. 35 Figure 4.1 a) Microstructure of as-received A36 steel b) schematic of small strip sample and LUMet experimental setup......................................................................................... 39 Figure 4.2 Schematic thermal histories for all heat treatments completed in this investigation     .................................................................................................................................... 40 Figure 4.3 Schematics depicting laser ultrasonic measurements of a) reference signal waveform at 900°C and b) current signal waveform. Red dotted lines indicate the first echo xii  measured after ultrasound pulse generation. Experimental data examples of c) frequency content of individual echoes and d) calculated attenuation spectrum with fitted Equation 2.14. Pink dotted lines indicate the frequency bandwidth used for fitting Equation 2.14. Amplitude uses arbitrary units. ............................................... 43 Figure 4.4 Schematic of mesh template used for generation of simulation domains ................... 52 Figure 4.5 Double Ricker wavelet applied to simulation domain surface in FE a) time dependence b) frequency dependence and c) an example at time 200 ns of the Gaussian normalization of applied amplitude across the generation surface, where the Gaussian peak at position 0 mm is centered on the mid-length of the simulation domain. Amplitude uses arbitrary units. .................................................................... 55 Figure 5.1 Laser ultrasonic measurement of austenite grain growth in A36 steel during isothermal holding tests at the indicated temperatures a) 950°C, 1050°C, and 1100°C b) 1000°C. Red points are ex-situ metallographic data [48] and pink dotted lines indicate the onset of isothermal holding. ................................................................... 59 Figure 5.2 Laser ultrasonic measurement of austenite grain growth in A36 steel during tests of interrupted 1000°C isothermal holding. Red points are ex-situ metallographic data [48]. ............................................................................................................................ 60 Figure 5.3 Micrograph of etched A36 steel following 80 seconds of 1000°C isothermal holding and helium gas quenching (Test 3) ............................................................................ 61 Figure 5.4 Micrographs of etched A36 steel following interrupted 1000°C isothermal holding, through-thickness cross-section at the LUMet generation laser spot center. a) Test 2, b) Test 3, and c) Test 4 ............................................................................................... 62 xiii  Figure 5.5 Tracing of large clusters (black) in etched A36 steel following interrupted 1000°C isothermal holding and schematic illustration of region probed by laser ultrasonics   .................................................................................................................................... 64 Figure 5.6 Comparison between large cluster fractions measured by metallography and by a normalization approximation from laser ultrasonic grain size measurements. Green line depicts perfect agreement. ................................................................................... 65 Figure 5.7 Thermodynamic stability of AlN precipitates in A36 steel as obtained from Thermocalc v4.0......................................................................................................... 66 Figure 6.1 a) A sub-domain of narrow grain size distribution and mean grain size 89 μm, and b) cumulative grain size plot of narrow size distribution sub-domains ......................... 71 Figure 6.2 Distributions of misorientation obtained in the sub-domain with 41 μm mean grain size compared with the ideal Mackenzie distribution ................................................ 72 Figure 6.3 FE simulated attenuation spectra from narrow grain size distribution domains with varying mesh template sizes a) 131 μm mean grain size b) 240 μm mean grain size  .................................................................................................................................... 73 Figure 6.4 FE simulated attenuation spectra from narrow grain size distribution domains of mean grain size a) 10 µm, 41 µm, 56 µm and 89 µm, b) 131 µm and 240 µm. Green lines represent best-fit α = a0 + bF3. ................................................................................ 74 Figure 6.5 A sub-domain of log-normal grain size distribution and mean grain size 82 μm ....... 76 Figure 6.6 Wide grain size distributions in terms of number fraction and area fraction a) log-normal b) normal, and c) corresponding FE simulated attenuation spectra ............... 77 Figure 6.7 a) Dual layered 41 µm and 131 µm bimodal grain size distribution sub-domain and b) corresponding FE simulated attenuation spectrum with constituent narrow grain size xiv  distribution attenuation spectra overlaid. Green line represents best-fit α = a0 + bF3 up to 15 MHz. ............................................................................................................ 79 Figure 6.8 a) 41 µm and 131 µm bimodal grain size distribution sub-domains of various grain cluster configuration and b) corresponding FE simulated attenuation spectra .......... 80 Figure 6.9 a) 41 µm and 131 µm bimodal grain size distribution domains of various large grain cluster position (vgroup,   Y is group velocity of a pressure wave in the vertical (Y) direction) and b) corresponding FE simulated attenuation spectra ............................ 82 Figure 6.10 a) 10 µm and 131 µm bimodal grain size distribution domains of various large grain cluster position and b) corresponding FE simulated attenuation spectra ................... 83 Figure 6.11 Large grain cluster from the 41 µm and 131 µm bimodal grain size distribution “Cluster at center” domain in Figure 6.9a) ................................................................ 85 Figure 6.12 Comparison between effective large grain area fraction in simulation domains and by a normalization approximation from laser ultrasonic grain size calculations. Green line depicts perfect agreement. ................................................................................... 87 Figure 6.13 FE simulated attenuation spectra from 41 µm and 131 µm bimodal grain size distribution domains of various area ratio .................................................................. 88 Figure 6.14 FE simulated attenuation spectra from bimodal grain size distribution domains of mean grain size populations a) 10 µm and 131 µm and b) 41 µm and 240 µm. Green lines represent best-fit α = a0 + bF3. ....................................................................... 89 Figure 6.15 Area fraction rule of mixtures modeling for FE simulated attenuation spectra from bimodal grain size distribution domains of mean grain size populations a) 10 µm and 131 µm (b), c)) 41 µm and 131 µm with varying area ratio, d) 41 µm and 240 µm    .................................................................................................................................... 92 xv  Figure 6.16 Comparison between grain size calculated directly from FE domain grain structures and from application of Section 4.2.3 methodology to FE simulated attenuation spectra. Featured grain size distributions are a) unimodal with FE domain grain size in AWGD, b) bimodal with FE domain grain size in EQAD and AWGD. Green line depicts perfect agreement. .......................................................................................... 94 Figure 6.17 Comparison between FE simulated attenuation spectra from domains with similar LUMet grain size of approximately a) 41 μm, and b) 56 μm .................................... 96 Figure 6.18 Plot of normalized grain size uncertainty and grain size from calculation on FE simulated attenuation spectra ..................................................................................... 99  xvi  List of Symbols  𝑎0  Absorption contribution to attenuation 𝐴  Area 𝐴𝑖  Area of a given grain 𝐴𝑗  Total area of a given grain family 𝐴𝑡𝑜𝑡𝑎𝑙  Total sampling area ?̅?  Mean area ?̃?  Amplitude ?̃?𝑜  Initial amplitude 𝑏  Scattering parameter for attenuation 𝑏1  Fitting constant for the attenuation model of Mason and McSkimin 𝑏2  Fitting constant for the attenuation model of Mason and McSkimin 𝑐𝐺  Temperature dependent grain growth parameter 𝑐𝑁  Constant for the grain size distribution expression of Nicoletti  𝐶𝐼  Solute concentration in the matrix at a precipitate interface 𝐶𝑀  Average solute concentration in the matrix 𝐶𝑃  Solute concentration in precipitates 𝑑  Mean grain diameter of an individual grain 𝐷  Grain size 𝐷1  Smallest grain size in a distribution 𝐷2  Largest grain size in a distribution xvii  𝐷𝑐𝑟  Critical grain size for abnormal grain growth 𝐷𝑓𝑖𝑛𝑎𝑙   Final grain size 𝐷𝑖𝑛𝑖𝑡𝑖𝑎𝑙  Initial grain size  𝐷𝐿𝑈𝑀𝑒𝑡  Grain size calculated using LUMet methodology 𝐷𝑚  Median grain size ?̅?  Mean grain size ?̅?0  Initial mean grain size ?̌?𝑟  Bulk matrix diffusivity of the rate-limiting species 𝐸  Total energy 𝑓𝐴  Area fraction of large grains 𝑓𝐹𝐸   Effective area fraction of large grains beneath the simulated ultrasound pulse 𝑓𝐿  Line fraction of large grains 𝑓𝐿𝑈𝑀𝑒𝑡  Area fraction of large clusters calculated from 𝐷𝐿𝑈𝑀𝑒𝑡 𝑓𝑍  Total volume fraction of second-phase particles 𝐹  Frequency 𝐹𝐵  Boundary frequency between Rayleigh and stochastic scattering regimes 𝐹𝑝  Peak frequency in Ricker wavelet 𝑘 Dimensionless variable indicating the constituent concentration gradient near the  particle/matrix interface 𝐾  Temperature dependent material constant for scattering attenuation 𝐾𝑔  Temperature dependent material constant for geometric scattering attenuation 𝐾𝐺  Temperature dependent constant for normal grain growth xviii  𝐾𝑟  Temperature dependent material constant for Rayleigh scattering attenuation 𝐾𝑠  Temperature dependent material constant for stochastic scattering attenuation 𝐿  Mesh template length 𝐿𝑠𝑒𝑒𝑑  Minimum distance between grain seeds 𝑀  Mobility 𝑀0  Mobility pre-exponential factor 𝑛  Scattering exponent for attenuation 𝑁𝑗  Total number of grains in a grain family  𝑁𝑡𝑜𝑡𝑎𝑙   Total number of grains in a domain 𝑁(𝐷)  Function to describe grain size distributions 𝑃𝑛𝑒𝑡  Net driving pressure for curved grain boundary motion 𝑃𝐺   Driving pressure for curved grain boundary motion 𝑃𝑍  Retaining pressure exerted by second-phase particles (Zener pressure) 𝑞(𝐷)  Function to describe log-normal grain size distributions 𝑄  Activation energy 𝑟  Radius of second-phase particles 𝑟0  Initial radius of second-phase particles ?̅?  Mean radius of second-phase particles 𝑅   Radius of a growing grain 𝑅𝑎𝑑   Radius of a shrinking grain adjacent to a growing grain of radius 𝑅 𝑅𝑔𝑎𝑠   Universal gas constant 𝑡  Time xix  𝑇  Absolute temperature 𝑣𝑏  Velocity of a boundary 𝑣𝑔𝑟𝑜𝑢𝑝,   𝑌 Group velocity of a pressure wave in the vertical (Y) direction 𝑣𝑤  Velocity of a wave 𝑉  Molar volume 𝑥  Position in X (horizontal) direction y  Position in Y (vertical) direction z  Distance travelled by a wave 𝛼  Attenuation 𝛼𝑔  Geometric scattering attenuation 𝛼𝑟  Rayleigh scattering attenuation 𝛼𝑠  Stochastic scattering attenuation 𝛽  Dimensionless constant for Zener pinning model 𝜃  Specimen thickness 𝛾𝑔𝑏  Grain boundary energy per unit grain boundary area 𝛾𝑝 Interfacial energy between a grain and a second-phase particle per unit interface area 𝜀  Constant in the amplitude expression of a Ricker wavelet 𝜆  Wavelength 𝜉  Grain size distribution exponent 𝜌∗  Radius of curvature 𝜎  Standard deviation xx  List of Abbreviations  2-D   two dimensional 3-D   three dimensional AlN   aluminum nitride AWGD area weighted grain diameter EBSD   electron backscatter diffraction EQAD  equivalent area diameter EDX   energy dispersive X-ray analysis FE   finite element LUMet  laser ultrasonics for metallurgy STEM  scanning transmission electron microscopy TEM   transmission electron microscopy   xxi  Acknowledgements  First and foremost, I would like to thank Dr. Matthias Militzer and Dr. Thomas Garcin for supervising this work. None of this would be possible if not for your enduring guidance, expertise, and support.  To my colleagues in particular, Isaac, Alyssa, Debanga, Madhumanti, Mojtaba, Mahsa, Raina, Nicolas, and Mariana, thank you for all your helpful insights and for fostering a fun environment outside of research. Thanks also to Jacob Kabel for lending a firm, helpful hand with the cutting machines in the EM lab.  To my friends and family, thank you so much for your love, caring, and patience throughout the past couple years. Thinking of you will always inspire me to become better as a scholar and as a person.  Lastly, I would like to thank the Natural Sciences and Engineering Research Council of Canada for the provision of funding for this study.     1  Chapter 1: Introduction  Grain size is an important microstructural parameter that affects the strength and toughness of a polycrystalline material [1,2]. Electrical and magnetic properties are also strongly influenced by the crystallographic orientation of grains, i.e. those properties can be controlled by the size distribution of grains with favorable crystallographic orientations [3]. Grain growth is the process by which grain size increases. It is driven by the reduction of grain boundary energy per unit volume by means of reducing the total grain boundary surface area. Normal grain growth is defined by self-similar growth of grains, resulting in the preservation of a unimodal grain size distribution. Abnormal grain growth is defined by the rapid growth of a few large grains into a matrix of otherwise unchanged small grains, resulting in a bimodal grain size distribution until all the small grains are consumed.   Particle pinning is a mechanism that inhibits the migration of grain boundaries, thus promoting grain refinement and higher strength of polycrystalline materials. During heat treatment, particle dissolution enables the development of abnormal grain growth. Plain carbon steels killed with aluminum commonly precipitate aluminum nitride particles, and exhibit abnormal austenite grain growth at specific heat treatment temperatures. In steels, austenite grain size distribution must be carefully controlled during heat treatment to provide consistent mechanical properties upon cooling to ambient conditions. In some instances however, abnormal grain growth may be used to form large grains of favorable crystallographic orientation. For example in electrical steels, the direction of easy magnetization is <100>. Large grains with the {110}<100> or {001}<100> 2  crystallographic orientations in the sheet plane are preferred such that the direction of easy magnetization is in the sheet plane [3].   The process of revealing prior austenite grain size is challenging and time-intensive, especially in low carbon steels due to the difficulty of obtaining a fully martensitic microstructure for chemical etching. Laser ultrasonics is a technique that generates and detects ultrasound waves by optical means, enabling non-contact rapid in-situ evaluation of materials at high temperatures. Ultrasonic attenuation is correlated to mean austenite grain size using an existing empirical calibration for various plain carbon steels [4]. In a previous study, laser ultrasonic measurements were used to detect a distinct rapid grain growth stage correlating to abnormal grain growth in a nickel-based superalloy [5].  The main contributor to ultrasonic attenuation is grain scattering, and three scattering regimes have been classified based on the relative values of grain size, 𝐷, and ultrasound wavelength, 𝜆. The Rayleigh scattering regime is defined when 𝐷 ≫ 𝜆 and the stochastic regime is defined when 𝐷 ≅ 𝜆 . The scattering behavior during the transition between Rayleigh and stochastic regimes is especially relevant for ultrasonic evaluation of polycrystalline materials. Theoretical models of scattering commonly assume that a grain size distribution can be represented by a single representative value of mean grain size [6,7]. Recent studies have expanded beyond this assumption by deriving an expression for ultrasonic attenuation in a log-normal grain size distribution [7]. During abnormal grain growth, a bimodal distribution emerges for which the modeling of attenuation is complicated.  3  This thesis investigates to what extent abnormal grain growth can be measured using laser ultrasonics. The laser ultrasonic response is studied during abnormal grain growth in a plain carbon steel. A finite element modeling framework is used to simulate ultrasound wave propagation in heterogeneous grain structures of various grain size distributions. Attenuation spectra from finite element simulations are compared in terms of representative grain size to determine the degree to which ultrasonic attenuation responses can be discerned for various grain size distributions.    4  Chapter 2: Literature Review  2.1 Grain growth  2.1.1 Theory of grain growth  Grain growth is the process by which the mean grain size increases in a polycrystalline material. It is driven by the reduction of grain boundary energy such that grain boundaries migrate towards their center of curvature [8]. The driving pressure for curved grain boundary motion, 𝑃𝐺 , is inversely proportional to the radius of curvature, 𝜌∗, i.e. [9]   𝑷𝑮 = 𝟐𝜸𝒈𝒃 𝝆∗⁄   (2.1) where 𝛾𝑔𝑏 is the grain boundary energy per unit grain boundary area. The velocity of boundary motion. 𝑣𝑏, is such that   𝒗𝒃 = 𝑴𝑷𝒏𝒆𝒕  (2.2) where 𝑀 is the boundary mobility and 𝑃𝑛𝑒𝑡 is the net driving pressure for boundary motion, i.e. the sum of curvature driving pressure and retaining pressures such as those originating from second-phase particle pinning and solute drag [10,11]. Mobility is frequently described by an Arrhenius relationship, i.e.   𝑴 = 𝑴𝟎𝒆𝒙𝒑(−𝑸 𝑹𝒈𝒂𝒔𝑻)⁄   (2.3) where 𝑀0 is the pre-exponential mobility factor, 𝑄 is the activation energy, 𝑅𝑔𝑎𝑠 is the universal gas constant, and 𝑇 is the absolute temperature.   5  Grain shape must satisfy two requirements at equilibrium: space-filling and minimization of surface tension [12]. If all grain boundaries possess equivalent 𝛾𝑔𝑏 then 120° is the equilibrium angle at which grain boundaries meet [12–14]. In two dimensions (2-D), this condition corresponds to a hexagonal equilibrium grain shape. For grains with more than six neighbors to possess internal angles of 120°, their grain boundaries must arc inwards. This concept is schematically illustrated in Figure 2.1. Grain boundaries that arc inwards are conducive for a grain to grow outward into neighboring grains.   Figure 2.1 Idealized two dimensional model for grain growth [14]  In three dimensions (3-D), the edges of a grain must meet at 109.5° for grain boundaries to meet at the 120° equilibrium angle [12]. No regular polyhedron with planar faces possesses exactly 109.5° between its edges. Figure 2.2 shows a tetrakaidecahedron composed of 6 four-sided faces and 8 six-sided faces. This grain shape is space-filling and attains 109.5° between edges when double curvatures are introduced to the hexagonal faces [15]. The requisite presence of curvature in the equilibrium 3-D grain shape suggests the inevitability of grain growth in 3-D. 6   Figure 2.2 Schematic drawing of tetrakaidecahedron [16]  Two types of grain growth can be discriminated by the way large grains emerge from an initial microstructure. Normal grain growth is defined by the self-similar growth of grains, i.e. grains grow and shrink such that a scaling log-normal size distribution is retained [17]. Burke and Turnbull [18] derived a grain growth law from Equation 2.1 using the following set of assumptions: • The only forces which act on any grain boundary are due to surface curvature, i.e. 𝑃𝑛𝑒𝑡 = 𝑃𝐺 . • Radius of curvature 𝜌∗ is proportional to the mean grain diameter of an individual grain, 𝑑. • Grain boundary mobility 𝑀  is independent of 𝑑  and grain boundary velocity 𝑣𝑏  is proportional to 𝜕(𝑑)/𝜕(𝑡), i.e. the time rate of change of 𝑑. Then from Equation 2.2, curvature driven pressure 𝑃𝐺  is proportional to 𝜕(𝑑)/𝜕(𝑡). • 𝑑 is equal to the mean grain size ?̅?. • 𝛾𝑔𝑏 is independent of grain size and time and is equal for all grain boundaries.  7  Incorporating these assumptions into Equation 2.1 results in  𝝏(?̅?) 𝝏(𝒕)⁄ = 𝒄𝑮(𝟐𝜸𝒈𝒃 ?̅?⁄ ) Here 𝑐𝐺  is a temperature dependent grain growth parameter and 𝑡  is time. Rearranging the expression with regard to the differential terms 𝜕(?̅?) and 𝜕(𝑡), and taking the integral both sides results in      ∫ ?̅? 𝝏(?̅?)?̅?𝟎=  ∫ 𝒄𝑮(𝟐𝜸𝒈𝒃) 𝝏(𝒕)𝒕𝟎  Evaluating the definite integral results in an expression for normal grain growth kinetics which is described by a parabolic relationship with respect to time, i.e. [18–20]     [𝟏𝟐?̅?𝟐]?̅?𝟎=  [𝒄𝑮(𝟐𝜸𝒈𝒃)𝒕]𝒕𝟎  ?̅?𝟐 − ?̅?𝟎𝟐 = 𝟒𝒄𝑮𝜸𝒈𝒃𝒕 (2.4) where ?̅?0  is the initial mean grain size. The first time derivative of the mean grain size in Equation 2.4 is proportional to 𝑡−1/2, meaning that the normal grain growth rate decreases with time. This is due to the radius of curvature increasing with grain size such that the driving force for grain boundary migration decreases as normal grain growth progresses (Equation 2.1).  Abnormal grain growth is defined by the rapid growth of relatively few large grains in an otherwise unchanged initial microstructure. A characteristic feature of abnormal grain growth is that growth rate increases with grain size [21]. Hillert introduced an expression to describe abnormal grain growth such that growth rate increases with grain size above a critical grain size 𝐷𝑐𝑟, i.e.   𝝏(𝑫) 𝝏(𝒕)⁄ = 𝑴𝜸𝒈𝒃(𝟏 𝑫𝒄𝒓⁄ − 𝟏 𝑫⁄ ) (2.5) 8  Another explanation for the growth of large grains at the expense of small grains originates from the theory of Gladman [16]. Considering how a large grain possesses less grain boundary area per unit volume than a small grain, the rate of total energy change, 𝑑𝐸, is expressed in terms of the radii of a growing tetrakaidecahedral grain and its corresponding adjacent shrinking tetrakaidecahedral grain, 𝑅 and 𝑅𝑎𝑑, respectively, i.e.   𝝏(𝑬) 𝝏(𝑹)⁄ = (𝟐 𝑹⁄ − 𝟑 𝟐𝑹𝒂𝒅⁄ )𝜸𝒈𝒃 (2.6) Here, it is assumed that grains can be expressed in terms of an equivalent sphere having the same volume. According to Gladman’s theory, net reduction of total energy, i.e. 𝛿(𝐸)/𝛿(𝑅)  < 0, is obtained when a growing grain’s radius is a factor of 4/3 larger than that of a neighboring grain, i.e. 𝑅 >43𝑅𝑎𝑑. This energetic advantage favors the growth of large grains at the expense of small grains [16]. The extent to which the growth of large grains is energetically favored depends on the relative values of 𝑅  and 𝑅𝑎𝑑 . Polycrystalline grain sizes are typically of log-normal distribution [22] such that many grains possess the 4/3 size advantage factor necessary for growth according to Equation 2.6. If 𝑅𝑎𝑑 is fixed, grains possessing a 4/3 size advantage factor will continuously grow according to Equation 2.6. The increasing difference between 𝑅 and 𝑅𝑎𝑑 results in an increasing energetic favorability for growth of large grains, i.e. 𝛿2(𝐸)/𝛿(𝑅)2  < 0. This concept was considered by Gladman to be the basis for abnormal grain growth.   However, a further investigation of the Gladman theory [23] concluded that the theory “used a ‘mean field’ approach in which the stored grain boundary energy is uniformly distributed around the growing grain, i.e. 𝑅 ≫ 𝑅𝑎𝑑 .” This assumption of 𝑅 ≫ 𝑅𝑎𝑑  conflicted with the theory’s conclusion that the minimum size advantage for a grain to grow was 𝑅 >43𝑅𝑎𝑑 . When the 9  assumption 𝑅 ≫ 𝑅𝑎𝑑 is applied to Equation 2.6, the resulting grain growth kinetics agree with those of Hillert’s abnormal grain growth expression in Equation 2.5.  It is generally accepted that abnormal grain growth is promoted by anisotropy of grain boundary mobility, i.e. abnormal grains emerge from the migration of highly mobile grain boundaries [24–27]. Anisotropy of grain boundary mobility, and consequently abnormal grain growth, can be caused by grain boundary structure, i.e. texture and grain boundary faceting, [27,28], and dissolution of second-phase particles [5,14,16,27,29–33]. In the case of second-phase particles, the theoretical work of Rios [30] concluded that the microstructural characteristics that lead to abnormal grain growth susceptibility consist of a narrow grain size distribution, thermodynamically unstable pinning particles, and a sufficiently high grain boundary mobility, i.e. at high temperatures.  2.1.1 Particle pinning   An important effect of second-phase particles is in grain size control, particularly the achievement of grain refinement in alloys and increased yield strength according to the Hall-Petch relationship [1,2]. The mechanism behind this type of grain refinement is the obstruction of grain boundary migration by second-phase particles. The retaining pressure exerted by second-phase particles is also called Zener pressure, 𝑃𝑍, and is defined by [34,35]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    𝑷𝒁 = 𝜷𝜸𝑔𝑏𝒇𝒁 ?̅?⁄   (2.7) This pressure depends on the total volume fraction of the second-phase particles, 𝑓𝑍, and the mean particle radius, ?̅? . The dimensionless constant 𝛽  depends on features such as particle 10  distribution, morphology, and grain geometry [36]. The classical derivation of 𝑃𝑍 used 𝛽 = 3/2 [34], whereas a modified derivation used 𝛽 = 12  [10,37]. The relative values of 𝑃𝐺  and 𝑃𝑍 determine the extent to which grain boundary pinning affects grain growth. When 𝑃𝑍 is larger than 𝑃𝐺 , no grain growth occurs. When 𝑃𝑍 is lower than 𝑃𝐺 , grain boundaries can unpin from particles and grain growth occurs.   Reduction of 𝑃𝑍 in Equation 2.7 can arise from the enlargement of particle radius. Driven by total interfacial energy reduction, the process of particle coarsening enlarges the particle radius via the diffusion of atoms from smaller particles to larger particles [38]. Coarsening of particles commonly obey a diffusion-controlled growth equation [39,40], i.e.    𝒓𝟑 − 𝒓𝟎𝟑 = 𝟖𝜸𝒑?̌?𝒓𝑪𝑴𝑽𝟐𝒕 𝟗𝑹𝒈𝒂𝒔𝑻⁄  (2.8) where 𝛾𝑝 is the interfacial energy between a grain and a particle, 𝑟 is the particle radius, 𝑟0 is the initial particle radius, ?̌?𝑟  is the bulk diffusivity of the particle’s rate-limiting species in the matrix, 𝐶𝑀 is the average solute concentration in the matrix, and 𝑉 is molar volume.  Elimination of 𝑃𝑍 arises from particle dissolution. For spherical second-phase particles uniformly distributed in a finite matrix, the dissolution rate can be approximated as [41,42]  𝝏(𝒓)/𝝏(𝒕) = −𝒌(?̌?𝒓 𝒓⁄ + √?̌?𝒓/𝝅𝒕 )  (2.9) where k is a dimensionless variable indicating the constituent concentration gradient near the  particle/matrix interface, i.e.  𝒌 =   (𝑪𝑰 − 𝑪𝑴) (𝑪𝑷 − 𝑪𝑰)⁄  (2.10) 11  where 𝐶𝐼  is the solute concentration in the matrix at the interface, and 𝐶𝑃  is the solute concentration in precipitates. For an MX-type precipitate where M is a substitutional alloy element in the matrix (i.e. Nb, V, and Al in Fe), and X is an interstitial element in the matrix (i.e. C and N in Fe), diffusivity of the substitutional element is usually several orders of magnitude lower than that of the interstitial element. Thus it can be assumed that the kinetics of particle dissolution and coarsening are “conrolled by the lower diffusion rate of the substitutional alloying element, and the concentration gradient of the interstitial element in the matrix can be ignored” [43].  In both cases of particle dissolution and coarsening, unpinning of grain boundaries is a local behavior dependent on the position of dissolved particles and large coarsening particles. Particle dissolution results in a sudden increase in grain boundary mobility that can enable rapid grain growth and the emergence of abnormal grains. In comparison, particle coarsening occurs at a relatively slow rate proportional to 𝑡−1/3, and thus is less favorable for abnormal grain growth emergence than particle dissolution [17].  Since particle dissolution and coarsening are thermally activated diffusion processes, abnormal grain growth is typically observed during high temperature heat treatment of alloys containing second-phase particles [5,27,32]. Above a certain temperature however, particle dissolution rate becomes high enough for many grains to unpin within a short amount of time [30]. The resulting grain growth, although temporarily abnormal, behaves more like normal grain growth [14]. Thus, the temperature range at which abnormal grain growth is observed can be very specific.  12  For example, Shirdel et al. [27] observed abnormal grain growth of an AISI 304L austenitic stainless steel at temperatures between 850°C and 900°C, corresponding to homologous temperatures between 0.65 and 0.7. Figure 2.3 displays optical micrographs of etched austenite grains in the as-received material and a sample exhibiting abnormal growth during an 850°C annealing test. Using energy dispersive X-ray (EDX) analysis, it was determined that abnormal grain growth coincided with partial dissolution of Ti-rich carbide particles. Once all small grains were consumed by abnormal grain growth, normal grain growth of the abnormally grown large grains commenced. The completion of abnormal grain growth corresponded with stagnation of abnormal grain size measured using a linear intercept method.   Figure 2.3 Optical micrographs of etched AISI 304L austenitic stainless steel a) as-received b) after annealing at 850°C for 3 hours [27]  Owing to its localized nature, abnormal grain growth can also occur despite stable particle populations [44]. In these rare instances, Rios [45] theorized that locally higher 𝑃𝐺  or lower 𝑃𝑍 arise from probabilistic differences in grains’ local environments. To investigate this topic 13  further, Holm et al. [31] simulated grain growth from equiaxed, nontextured, three dimensional grain structures using a Monte Carlo Potts model. Uniform, isotropic properties were assigned to grains and pinning particles were randomly deposited at grain boundaries. Simulations thus isolated the possibility of grain boundary curvature driven grain growth in a stable particle pinned system. Figure 2.4 shows the grain growth evolution for 15 independent simulation trials from an initial structure of grain diameter 10 a.u. and particle fraction 10%.   Figure 2.4 Grain growth evolution for 15 independent Monte Carlo Potts simulations with particle volume fraction of 10%. Grain diameter uses arbitrary units and time uses Monte Carlo timestep units [31].  After long simulated times, most systems’ grain growth was hindered by the stable particle population except for three cases which exhibit an abrupt increase in mean grain diameter. Analysis of these three simulation trials identified the initiating event for abnormal grain growth being one grain boundary that thermally fluctuated off its pinning particles. Once unpinned, that 14  grain rapidly grew into the entire system based on its size advantage. The different times at which rapid grain growth was observed for the three trials indicated that the process of abnormal grain growth in stable particle populations is probabilistic.   Another consideration to particle pinning theory is the positioning of particles with respect to grain boundaries. The classical Zener derivation implicitly assumed that particles are numerous enough for the net pinning pressure to be independent of grain size [21,34]. As grain size decreases however, the density of triple junctions and quadruple points increases. Particles located at triple junctions and quadruple points respectively contact 1.5 and 2 times more grain boundary area than particles located at grain boundaries [21]. From consideration of these concepts, Bréchet and Militzer [21] determined that pinning force is enhanced when grain size and particle spacing are of similar values. This concept was then used to rationalize abnormal austenite grain growth in a low-carbon steel with a stable TiN precipitate configuration.     15  2.1.2 Austenite grain growth in aluminum-killed steel  During liquid processing, steel can be killed with aluminum to remove dissolved oxygen and prevent the formation of blowholes upon solidification [46]. The added aluminum also precipitates with dissolved nitrogen as aluminum nitride (AlN) during subsequent batch-annealing [46]. It is well established that precipitated AlN is the main cause of austenite grain boundary pinning in Al-killed steels [47]. Cheng et al. [42] studied the characteristics of AlN particles in a low-carbon drawing quality special Al-killed (DQSK) steel that was subject to hot rolling, coiling, and annealing. Table 2.1 lists the chemical composition of the DQSK steel.   Table 2.1 Composition in weight percent of DQSK steel used by Cheng et al. [42] C Mn S Si Cr Al N 0.037 0.30 0.008 0.009 0.033 0.040 0.0052  Transmission electron microscopy and EDX revealed fine 10 nm diameter AlN particles located along ferrite grain boundaries in as-received coil samples. Isothermal annealing between 650°C and 750°C resulted in rapid growth of AlN particles followed by slow coarsening to mean particle diameters of roughly 50 nm. Subsequent rapid heating and isothermal holding at 1000°C produced the scanning transmission electron microscopy (STEM) micrograph and particle size distributions of Figure 2.5. 16   Figure 2.5 a) Micrographs showing networks of AlN particles in DQSK steel samples held for 24 hours at 700°C and annealed for a) 30 minutes at 1000°C, imaged using STEM b) 10 minutes at 1100°C, imaged using TEM. c) Measured and modelled evolution of mean AlN particle diameter during 1000°C annealing [42].  Figure 2.5a) and Figure 2.5b) display networks of AlN particles in the micrographs of DQSK steel samples held for 30 minutes at 1000°C and 10 minutes at 1100°C. The networks were believed to represent prior ferrite grain boundaries. The evolution of mean AlN particle diameter shown in Figure 2.5c) showed that mean particle size increased during 1000°C isothermal holding, which was attributed to particle coarsening. It was also suggested that AlN particles 17  underwent partial dissolution upon reaching high temperatures due to the high solubility of AlN in austenite compared to in ferrite [42,47]. TEM revealed that AlN particles fully dissolved after 30 minutes of isothermal holding at 1150°C.  In a similar study, Militzer et al. [48] investigated the austenite grain growth evolution in Al-killed steels during isothermal holding. Table 2.2 lists the chemical composition of the DQSK steel and A36 steel used by Militzer et al. [48].  Table 2.2 Composition in weight percent of the steels used by Militzer et al. [48] Steel C Mn P S Si Cu Ni Cr Al N DQSK 0.038 0.30 0.010 0.008 0.009 0.015 0.025 0.033 0.040 0.0052 A36 0.17 0.74 0.009 0.008 0.012 0.016 0.010 0.019 0.040 0.0047  Tubular samples were heated at 5°C/s from room temperature to various isothermal holding temperatures and quenched. Helium gas quenching was used for samples containing austenite grain size above 35 µm and water quenching was used for samples containing austenite grain size below 35 µm. Prior austenite grain boundaries were revealed by chemical etching. Figure 2.6a) displays measured mean austenite grain size in DQSK steel, with time at 0 seconds being defined at the onset of isothermal holding. Very little growth was observed during holding at 1000°C and 1050°C, as well as in the first 60 seconds of 1100°C isothermal holding. A stage of rapid grain growth was observed after 60 seconds of 1100°C isothermal holding. Metallographic observations during this rapid grain growth stage identified it as abnormal grain growth. Rapid 18  grain growth occurred during heating up to 1150°C such that the austenite grain size measurement at the beginning of isothermal holding was approximately 140 μm.  Figure 2.6 Metallographic measurement of austenite grain size during isothermal holding after heating at 5°C/s in a) DQSK steel and b) A36 steel [48]  A36 is an ASTM classification for Al-killed low-carbon structural steel commonly used in civil infrastructure such as bridges and buildings [49–51]. Grain growth behavior of A36 during isothermal holding in Figure 2.6b) exhibited similar trends to that of DQSK steel in Figure 2.6a). Very little growth was observed during 950°C and the first 100 seconds of 1000°C isothermal holding. A stage of rapid grain growth was observed after 100 seconds of 1000°C isothermal holding, which was identified as abnormal grain growth. Rapid grain growth occurred during heating up to 1100°C and 1150°C such that the austenite grain size measurements at the beginning of isothermal holding were respectively 90 μm and 100 μm.  19  Giumelli et al. [32,52] conducted further tests on A36 steel using the same methodology of Militzer et al. [48]. To illustrate abnormal grain growth, Figure 2.7 displays micrographs of etched A36 steel at 10 seconds, 120 seconds, and 600 seconds of 1000°C isothermal holding.   Figure 2.7 Evaluation of the prior austenite grain structure during abnormal grain growth in an A36 steel at 1000°C [52]  The prior austenite microstructure evolves from a fine grain distribution at 10 seconds (Figure 2.7a)) to a combination of fine and coarse grains at 120 seconds (Figure 2.7b)) and a coarse grain distribution at 600 seconds (Figure 2.7c)). Abnormal grain growth was attributed to the influence of AlN precipitates. Grain growth is initially limited due to AlN particle pinning. At longer times of 1000°C holding, gradual dissolution of AlN particles occurs and grain boundaries become 20  unpinned. The result of this gradual unpinning is abnormal grain growth and the bimodal grain size distribution observed in Figure 2.7b). Once large abnormal grains consume all the small grains, the grain size distribution of coarse grains returns to a log-normal distribution.  2.2 Prior austenite grain size measurement  2.2.1 Summary of techniques  Austenite decomposes to various forms of ferrite, cementite, pearlite, bainite, or martensite upon cooling [53]. These phase transformations depend on cooling rate, alloy composition, and austenite grain size, which can make it difficult to reveal the prior austenite grain size [54]. Techniques generally consist of preparing polished sample surfaces using conventional metallographic methods and an etching method such as chemical, thermal, or oxidative etching.   Chemical etching of prior austenite grain boundaries typically requires a fully martensitic microstructure, which is difficult to achieve for low carbon steels [55]. A successful chemical etchant is a solution of 100 mL saturated aqueous picric acid, six drops of concentrated hydrochloric acid, and 2 mL sodium alkylsulfonate as a wetting agent [55–57]. Nital (solution of nitric acid in ethanol) chemically attacks ferrite and can also reveal prior austenite grain boundaries if they are decorated with ferrite upon cooling [54]. Tempering at temperatures suitable for segregating phosphorous to grain boundaries can also aid in chemical etching [56].  21  Thermal etching is conducted by heating polished samples in a furnace with a vacuumed or inert atmosphere. At the intersection of a grain boundary and the sample surface, a visible groove forms due to surface tension and evaporation effects [58,59]. Thermal etching combined with  hot stage optical microscopy [59] or laser scanning confocal microscopy [60,61], can provide in-situ measurement of austenite grain size. A drawback of this in-situ technique originates from the hindrance of grain boundary migration by thermal grooves. Traces of old grooves complicate the identification of austenite grains during in-situ observation [59,61]. Groove traces can be eliminated upon quenching by the strain associated with martensitic transformation [59].  Oxidative etching is conducted by heating polished samples in a furnace with an oxidizing atmosphere [62]. Austenite grain boundaries accumulate oxides or experience decarburization that is observable upon cooling. Immersion into molten glass can enhance the oxidative contrast created at austenite grain boundaries by increased ionic transport [63]. An alternative to etching is electron backscatter diffraction (EBSD), in which prior austenite grains are identified based on crystallographic orientation relationships with transformed phases [64–68]. However, EBSD requires careful sample preparation and complex image analysis compared to etching techniques.  Once prior austenite grain boundaries are revealed, the grain size can be determined using various techniques such as linear intercept or grain count per unit area correlation [13,69] to a standard grain size number [70]. Alternatively, grain boundaries can be traced and quantitatively analyzed using an image processing program [71]. Grain size can be expressed as an equivalent area diameter (EQAD) of the mean grain area ?̅?, i.e.  𝑬𝑸𝑨𝑫 = √𝟒?̅? 𝝅⁄   (2.11) 22  An alternative expression of grain size incorporates a factor for the relative area of each grain [72]. Area weighted grain diameter (AWGD) can be calculated as follows  𝐀𝐖𝐆𝐃 = ∑ (√𝟒𝑨𝒊 𝝅⁄ ∗ 𝑨𝒊 𝑨𝒕𝒐𝒕𝒂𝒍⁄ )𝒊   (2.12) For the area of a given grain, 𝐴𝑖, the equivalent area diameter √4𝐴𝑖 𝜋⁄  is calculated and scaled by the fraction of area it occupies in the total sampling area 𝐴𝑖 𝐴𝑡𝑜𝑡𝑎𝑙⁄ . Such an expression is useful to describe wide grain size distributions in which a few grains occupy a large volume of the sample [73,74].  2.2.2 Laser ultrasonics  Ultrasound technology is widely established in its industrial application to non-destructive thickness gauging, flaw detection, and to a limited capacity microstructural feature characterization [75,76]. Conventional ultrasound technology uses piezoelectric transducers that require either physical contact or a coupling fluid medium to propagate an ultrasound wave through samples [77,78]. Laser ultrasonics is a development of ultrasound technology that generates and detects ultrasound waves by optical means, which enables non-contact rapid in-situ evaluation of materials. The technique is especially useful for monitoring high temperature, high speed operations such as industrial metal processing [75].  Among the capabilities of laser ultrasonics is the measurement of austenite grain size via correlation to ultrasonic attenuation. The method is a valuable alternative to other austenite grain 23  size measurement techniques which are time and labor intensive. As an ultrasound wave travels through a material, its amplitude, ?̃?, decays according to [79]  ?̃? = ?̃?𝒐𝒆𝒙𝒑(−𝜶𝒛)  (2.13) where the initial unattenuated amplitude is ?̃?𝑜, attenuation is 𝛼, and distance travelled by the wave is 𝑧. Attenuation is caused by absorption, diffraction, and scattering. Absorption is the conversion of sound energy to other forms of energy such as heat, primarily by inelastic damping and internal friction [80,81]. Diffraction can be understood as the spreading of an ultrasound pulse depending on sample geometry. If sample geometry, sample material, and distance traveled by the ultrasound pulse are preserved during attenuation calculation, then the contribution of diffraction to calculated attenuation is nullified [82].   Scattering changes the direction of a sound wave by reflection at internal inhomogeneities which constitute an elastic discontinuity, such as voids, inclusions, and grains [5,83]. If a material is without voids and inclusions, then the scattering contribution to attenuation can be correlated to grain size. Assuming attenuation is calculated such that diffraction can be neglected, ultrasonic attenuation can be described as the sum of absorption, 𝑎0, and scattering, 𝑏𝐹𝑛  𝜶(𝒇, 𝑫) = 𝒂𝟎 + 𝒃𝑭𝒏  (2.14) Here, frequency is 𝐹, and the scattering exponent 𝑛 is commonly approximated with a value of 3 for the frequency range available in laser ultrasonics for metallurgy (LUMet) (2 – 20 MHz) [82,84,85]. Other assumptions include that absorption is frequency-independent and that grains are equiaxed. The scattering parameter, 𝑏, depends on grain size, 𝐷, according to   𝒃 = 𝑲𝑫𝒏−𝟏  (2.15) 24  where 𝐾 is a material constant that depends on temperature. The terms 𝑎0  and 𝑏 are used as fitting parameters to fit Equation 2.14 with experimental ultrasonic attenuation measurements. The best-fit 𝑏  value can then be correlated to a representative grain size based on existing calibrations. For example, Kruger et al. [4] constructed an empirical calibration for austenite grain growth in a range of plain carbon steels. Extensive laser ultrasonic experiments coupled with interrupted heat treatment and quantitative metallographic analysis was used to accumulate data for 𝑏 and mean austenite grain sizes. The resulting calibration curve is shown in Figure 2.8. Grain size was calculated as the EQAD of austenite grain areas (Equation 2.11).   Figure 2.8 Calibration of 𝒃 parameter from experimental mean austenite grain size (EQAD) measurements [4]  This empirical calibration for austenite grain size has been validated for laser ultrasonic studies on various steels  [37,84,86–88]. Additional calibrations have been constructed for grain size in cobalt-based and nickel-based superalloys [5,89]. This laser ultrasonic technique has also shown the capability of detecting discrete stages of grain growth. Garcin et al. [5] measured the grain 25  size evolution in a nickel-based superalloy Inconel 718 using laser ultrasonics and EBSD metallography. Samples were isothermally held at 1050°C, above the super-solvus temperature for δ-phase precipitates in Inconel 718. Measurements of the LUMet grain size during isothermal holding coincided with the dissolution of δ-phase precipitates.   Figure 2.9a) shows the repeatability of LUMet grain size measurements for various soaking times at 1050°C and their agreement with EBSD metallographic measurements. A distinct stage of rapid grain growth in between two stages of limited grain growth was identifiable from the LUMet grain size evolution. Figure 2.9b) shows that the initial stage of limited grain growth can be modeled by Zener type pinning and the final stage of limited grain growth can be modeled by the ideal parabolic law, i.e. negligible pinning (Equation 2.4). EBSD maps of microstructures during the rapid grain growth stage revealed a heterogeneous grain structure attributed to abnormal grain growth. Similar grain growth stages have been identified for laser ultrasonic measurements in A36 steel during 1000°C and 1050°C isothermal holding [84]. 26   Figure 2.9 Evolution of mean grain size (EQAD) during soaking of INCONEL 718 at 1050°C a) measured with LUMet and EBSD metallography, and b) modeled with a parabolic growth equation for coarse grains and Zener pinning for small grains [5]   27  2.3 Ultrasonic scattering theory  2.3.1 Rayleigh-stochastic scattering regime transition  Wavelength is inversely proportional to frequency by  𝝀 = 𝒗𝒘 𝑭⁄  (2.16) where 𝑣𝑤 is the velocity of a wave. Ultrasonic scattering is described by a general expression [90]  𝜶(𝑫,  𝝀) = 𝑲𝑫𝒏−𝟏(𝟏 𝝀⁄ )𝒏  (2.17) The values for scattering exponent, 𝑛, and material constant, 𝐾, depend on the relative values of grain diameter, 𝐷, and ultrasound wavelength, 𝜆. Three distinct scattering regimes for ultrasound waves have been studied and classified: the Rayleigh, stochastic, and geometric regimes [7].  𝑹𝒂𝒚𝒍𝒆𝒊𝒈𝒉: 𝜶𝒓(𝑫,  𝝀) = 𝑲𝑟𝑫𝟑(𝟏 𝝀⁄ )𝟒, 𝑫 ≪ 𝝀𝑺𝒕𝒐𝒄𝒉𝒂𝒔𝒕𝒊𝒄: 𝜶𝒔(𝑫,  𝝀) = 𝑲𝒔𝑫(𝟏𝝀⁄ )𝟐, 𝑫 ≅ 𝝀𝑮𝒆𝒐𝒎𝒆𝒕𝒓𝒊𝒄: 𝜶𝒈(𝑫,  𝝀) = 𝑲𝒈𝑫−𝟏,  𝑫 ≫ 𝝀   Note how the scattering regimes are rather loosely defined in terms of relative 𝐷 and 𝜆 values. The ultrasonic evaluation of metals tends to involve wavelengths that are larger than grain size [80], so the transition from Rayleigh to stochastic scattering regimes is of particular interest.  An early study of ultrasonic scattering in metals was conducted by Mason and McSkimin [91] on aluminum rods immersed in a water bath. A piezoelectric transducer was used to generate and detect an ultrasound pulse that travels through the water bath and aluminum rod. Corrections were made for the energy loss of wave propagation through water. The procedure was repeated 28  for various ultrasound frequencies and calculated attenuation points are shown in Figure 2.10. Attenuation can be described by  𝜶 = 𝒃𝟏𝑭 + 𝒃𝟐𝑭𝟒  (2.18) in which 𝑏1and 𝑏2 are fitting constants. The terms 𝑏1𝐹 and 𝑏2𝐹4 in Equation 2.18 respectively represent the contributions of hysteresis damping and scattering to the total ultrasonic attenuation.  Figure 2.10 Measured attenuation (points) and model (lines) for longitudinal waves in aluminum rods with indicated mean grain diameters [91]. Indicated scattering range denoted the frequencies in which comparisons were made to a previous study [77].  Attenuation was higher for the aluminum rod of 230 μm grain size for low frequencies up to 18 MHz. At higher frequencies up to 35 MHz, the attenuation spectrum flattened and became less than that of the 130 μm grain size rod. Rayleigh and stochastic scattering expressions were calculated to be valid for the conditions 𝐷 < 𝜆/3 and 𝐷 > 3𝜆, respectively. At the time, no adequate theory existed to describe ultrasound attenuation between those two limits. 29  Figure 2.11 shows attenuation measurements by Papadakis [85] in an equiaxed iron-30% nickel alloy. The measurements were plotted on a double logarithmic plot such that the scattering exponent, 𝑛, was represented by the slope and the Rayleigh and stochastic scattering laws were visualized by lines of slope 4 and 2, respectively. Here, 𝐹𝐵 is the boundary frequency between Rayleigh and stochastic scattering regimes and was defined at 𝜆 = 2𝜋?̅?, where the mean grain size ?̅? was 56 μm for the tested samples.   Figure 2.11 Measured longitudinal wave attenuation in an equiaxed iron-30% nickel alloy, modeled with a curve of monotonically decreasing slope connecting Rayleigh and stochastic scattering regimes. Frequency uses units of megacycles and 𝑭𝑩 is the boundary frequency between scattering regimes defined at 𝝀 = 𝟐𝝅?̅? [85].  30  The three attenuation measurements near 𝐹𝐵  appeared to have a slightly less frequency dependence of attenuation than the Rayleigh scattering law. It was postulated that since there were some relatively large grains in the sample, the large grains scattered by the stochastic process and lowered the overall value of scattering exponent in the sample. A simple model consisting of a monotonically decreasing slope connecting the Rayleigh and stochastic regions agreed well with the attenuation measurements in Figure 2.11. The monotonically decreasing slope indicates that a scattering exponent between 2 and 4 is necessary to describe attenuation measurements near 𝐹𝐵 , and suggests that the scattering exponent value of 3 commonly approximated in LUMet is applicable to attenuation measurements near 𝐹𝐵.  A unified theory of ultrasonic scattering was developed by Stanke and Kino [6] that covered any ratio of wavelength and grain size, including ratios corresponding to transitions between scattering regimes. The theoretical model provided a self-consistent method for determining attenuation in polycrystalline media [92,93] and began with the following assumptions: • Equiaxed, single-crystal grains each of single phase with no voids or inclusions • Weak elastic anisotropy of grains (elastic constant mismatch is much less than the average elastic modulus) • Preferred crystallographic orientation must be absent (random texture) The model was developed using the concept of mean free path length, where the free path was defined as a line segment passing through one grain with endpoints on grain boundaries. To a first approximation, the mean free path length can be assumed to be equivalent to a representative mean grain size [69].   31  Figure 2.12a) shows the theoretical model plotted for various mean grain sizes of polycrystalline austenite at 1000°C, with select grain sizes labeled on the plot. Attenuation increases with the mean grain size and above 150 μm an inflection point signals the transition from Rayleigh to stochastic scattering regimes. The plot in Figure 2.12b) depicts this transition more clearly by normalizing both axes with respect to the mean grain size and plotting on double logarithmic scales. The scattering exponent, 𝑛, is again represented by the slope in Figure 2.12b).  Figure 2.12 Theoretical model of attenuation for longitudinal ultrasound waves in polycrystalline steel at 1000°C [6] a) for various mean grain size b) normalized with respect to mean grain size, pink dotted line indicates the transition between Rayleigh and stochastic scattering regimes defined by 𝝀 = 𝟐𝝅?̅?   The transition from Rayleigh to stochastic scattering regime is characterized by a hump in the normalized attenuation plot in Figure 2.12b). This hump, “where the frequency dependence of attenuation is less than second order” [6], meets with the stochastic scattering regime at 32  approximately 2𝜋?̅?(1  𝜆⁄ ) = 10. The horizontal portion of the normalized attenuation plot in Figure 2.12b) corresponds to the geometric scattering regime in which attenuation is independent of frequency.  This decreased frequency dependence in the stochastic regime compared to the Rayleigh regime is attributed to the different mode conversions of scattering in each regime. Sound waves travel predominantly in longitudinal and shear modes, where the motion of atoms respectively is parallel and perpendicular to the wave propagation direction [80]. Theoretical analysis of scattering by individual particles concluded that energy is scattered mostly as shear waves in the Rayleigh regime and as longitudinal waves in the stochastic regime [6]. Attenuation in a polycrystalline austenitic stainless steel has been shown to be higher for shear waves than longitudinal waves [94]. Stanke and Kino postulated that the attenuation hump represented a “transition from a mechanism controlled by scattering into shear waves (Rayleigh regime) to one by longitudinal waves (stochastic regime)”.  2.3.2 Consideration of grain size distribution   A common assumption in theoretical models of ultrasonic scattering is that the grain size distribution can be represented by a single value of mean grain size. However, the largest grain size in a distribution can be significantly larger than the mean size. In such cases a grain size distribution can span multiple scattering regimes for a given ultrasound wavelength. Using a theoretical model that incorporated a function describing grain size distribution, 𝑞(𝐷), [95], 33  Smith [96] investigated how the frequency dependence of ultrasonic attenuation was affected by grain size distribution. Log-normal grain size distributions were defined by   𝒒(𝑫) =𝟏𝑫√𝟐𝝅𝝈𝟐𝐞𝐱𝐩 (−(𝐥𝐨𝐠(𝑫/𝑫𝒎))𝟐𝟐𝝈𝟐)  (2.19) where 𝐷𝑚and 𝜎 are respectively the median grain size and standard deviation of the log-normal distribution. Figure 2.13 shows attenuation calculated from the theoretical model using various grain size distributions.   Figure 2.13 Calculation of ultrasonic attenuation using modified Roney’s theory [83,95] and theoretical grain size distributions (solid) and comparison to Rayleigh scattering law (dashed). (∆𝑲/𝑲)𝟐̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅  is average elastic mismatch parameter [96]. 34  The distributions labelled as ?̅? = 10 μm, ?̅? = 50 μm, and {?̅? = 20 μm, 𝐷𝑚 = 20 μm, 𝜎 = 0.0} were uniform, i.e. consisting of only one grain size. At low frequencies below 10 MHz, the calculated attenuation followed the Rayleigh scattering law for all cases. The grain size distribution with {?̅? = 20 μm, 𝐷𝑚 = 12.1 μm and 𝜎 = 1.0} was significantly different from the two other distributions with identical mean grain size of 20 μm. At low frequencies that distribution’s attenuation was dominated by its few large grains of size up to 80 μm, resulting in a total attenuation that was greater than that of the ?̅? = 50 μm distribution. At higher frequencies, the large grains contributed less attenuation as they approached the stochastic scattering regime and subsequently attenuation fell below that of the 50 μm distribution. Above 15 MHz, the 50 μm distribution also had an attenuation of smaller frequency dependence than the Rayleigh scattering law.   Nicoletti [7] introduced a function 𝑁(𝐷)  to describe grain size distribution such that the attenuation resulting from a grain size distribution was expressed as     𝜶(𝝀) = ∫ 𝑵(𝑫)𝜶𝝀,𝑫(𝝀, 𝑫)𝒅𝑫∞𝟎  (2.20) where 𝑁(𝐷)𝑑𝐷 is the number of grains from size 𝐷 to 𝐷 + 𝑑𝐷 and 𝛼𝜆,𝐷(𝜆, 𝐷) is the scattering attenuation due to a single grain size, i.e. from Equation 2.17. A log-normal grain size distribution of large variance was assumed, and was approximated using an inverse power-law  𝑵(𝑫) = 𝒄𝑵𝑫−𝝃, 𝑫𝟐 ≥ 𝑫 ≥ 𝑫𝟏   (2.21) where 𝑐𝑁 is a constant, 𝜉 is the grain size distribution exponent, and 𝐷2 and 𝐷1 are respectively the largest and smallest grain size in the distribution. Substituting Equations 2.17 and 2.21 into Equation 2.20, the attenuation resulting from a grain size distribution was re-formulated as 35   𝜶(𝝀) = 𝒄𝑵𝝀−𝝃 ∫ 𝒑−𝝃𝜶𝒑(𝒑)𝒅𝒑𝑫𝟐/𝝀𝑫𝟏/𝝀   (2.22) where 𝑝 = 𝐷 𝜆⁄  and 𝛼𝑝(𝑝) = 𝜆𝛼𝜆,𝐷(𝜆, 𝐷) such that attenuation can be expressed solely in terms of 𝑝. Consider for example the Rayleigh scattering regime, for which 𝛼𝜆,𝐷(𝜆, 𝐷) = 𝐾𝑟𝐷3(1 𝜆⁄ )4 and 𝛼𝑝(𝑝) = 𝜆𝛼𝜆,𝐷(𝜆, 𝐷) = 𝐾𝑟𝐷3(1 𝜆⁄ )3 = 𝐾𝑟𝑝3 . Evaluating the limit as 𝑝 → 0  in Equation 2.22 results in the convergence of 𝛼(𝜆) ∝  𝜆−4 , correctly corresponding to the Rayleigh scattering regime. Similarly evaluating the limit as 𝑝 → ∞  in Equation 2.22 results in the convergence of 𝛼(𝜆) ∝ 𝜆0, correctly corresponding to the geometric scattering regime.  Further derivation by Nicoletti equated the grain size distribution exponent, 𝜉, and the scattering exponent, 𝑛. This result is shown in Figure 2.14, in which calculated attenuation is plotted on double-logarithmic axes such that the slope represents 𝑛.   Figure 2.14 Calculated attenuation (points) and linear regressions (straight lines) for single power-law grain size distributions, (∆𝑲/𝑲)𝟐̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ =  𝝅𝟐/𝟐, K = 1, D1 = 0.02 μm, D2 = 320 μm, 𝝃 = 0.5, 1.6, 2.4, 3.6 [90]  36  When 𝜆 ≫ 𝐷2, linear regressions produce 𝑛 = 4 for all grain size distributions, which correctly corresponds to the Rayleigh scattering regime. For 𝜆 < 𝐷2, linear regressions produced 𝑛 values that matched the 𝜉 from each grain size distribution. Additional evidence of this equivalence exists as experimental studies on pure nickel [97]. Grain size distributions were measured and followed an inverse power-law relationship with 𝜉  ranging from 2.22 to 2.45. Ultrasonic attenuation measurements also followed an inverse power-law relationship with 𝑛 ranging from 2.33 to 2.40 for wavelengths between the smallest and largest grain size. The correlation between 𝜉 and 𝑛 established Equation 2.22 as a straightforward relationship linking log-normal grain size distribution to ultrasonic scattering attenuation.   The correlation from Nicoletti relied on the assumption of a log-normal grain size distribution and the approximation of log-normal distributions by an inverse power-law (Equation 2.21). The choice of an inverse power-law was important due to its similarity to the inverse power-law relationship between attenuation and frequency (Equation 2.17), which enabled the derivation of a relatively simple attenuation expression in Equation 2.22. In cases where the grain size distribution is not log-normal, i.e. during abnormal grain growth when a bimodal distribution emerges, the description of grain size distribution and the subsequent modeling of attenuation are complicated. 37   Chapter 3: Scope and Objectives  The aim of this study is to evaluate whether abnormal grain growth can be measured in the laser ultrasonics for metallurgy (LUMet) technique. The LUMet response during abnormal grain growth using an A36 plain carbon steel is examined. A finite element (FE) modeling framework is used to simulate the propagation of an ultrasound wave in heterogeneous grain structures. Using simulated attenuation, the LUMet technique’s range of applicability is evaluated in terms of maximum grain size and discernment between grain size distributions.  The sub-tasks necessary to achieve the objective of this study are as follows: 1. Correlate laser ultrasonic measurement of grain size with ex-situ metallographic measurements. 2. Simulate the ultrasonic attenuation spectra for narrow, log-normal, and bimodal grain size distributions. 3. Determine the representative grain size parameter best suitable to build a correlation between metallographic and laser ultrasonic grain size.  38   Chapter 4: Experimental Methodology  4.1 Material and sample geometry  The material used for this study was an A36 plain carbon steel supplied by United States Steel Gary Works in the form of a transfer bar extracted before finish rolling in the hot strip mill [32,48,84,86]. The steel chemistry is listed in Table 4.1 in weight percent of each alloying element. Figure 4.1a) shows the microstructure of the as-received A36 steel revealed using 0.5% Nital etching. The microstructure is composed of grain boundary ferrite surrounding bainite islands. Small strip samples (Figure 4.1b)) of 60 x 10 x 3 mm were machined from the as-received material with the strip longitudinal direction parallel to the transfer bar transverse direction. The specimen thickness of 3 mm optimizes the ultrasonic signal by being thin enough to avoid excessive attenuation and thick enough to avoid overlap between consecutive echoes.    Table 4.1 Composition in weight percent of the as-received A36 steel C Mn P S Si Cu Ni Cr Al N 0.17 0.74 0.009 0.008 0.012 0.016 0.010 0.019 0.040 0.0047  39    Figure 4.1 a) Microstructure of as-received A36 steel b) schematic of small strip sample and LUMet experimental setup  To understand the different grain growth behaviors observed for isothermal holding temperatures between 950°C and 1100°C, CALPHAD software Thermocalc v4.0 [98] was used to model the thermodynamic stability of AlN precipitates in the exact composition of A36 steel listed in Table 4.1. The TCFE7 database was used with the equilibrium condition of Gibbs free energy minimization and the following phases selected: bcc ferrite, fcc austenite, cementite, and AlN.   4.2 In-situ microstructural characterization  4.2.1 Heat treatment   Experiments were conducted in a Gleeble 3500 thermomechanical simulator (Dynamic System Inc. Poestenkill, NY) equipped with a Laser Ultrasonics for Metallurgy (LUMet) sensor. Low friction force jaws were used to hold the specimen to ensure zero force on the specimen during 40   the entire duration of the treatment. Prior to the tests, the sample chamber was evacuated down to 0.05 Pa. Sample temperature was measured and controlled using a pair of K-type thermocouples spot-welded along the mid-length of the small strip longitudinal face (Figure 4.1b)). Samples were heated using alternating current and the principle of resistive heating.   Figure 4.2 charts the different heat treatments conducted during this study. Samples were heated at 5°C/s up to 950°C, 1000°C, 1050°C and 1100°C and held for 10 minutes prior to rapid quenching using high pressure helium gas. Additional samples were quenched from 1000°C after 80 seconds and 140 seconds to observe the microstructure at different stages. The cooling rate measured at the surface was on the order of 150°C/s.  Figure 4.2 Schematic thermal histories for all heat treatments completed in this investigation      4.2.2 In-situ laser ultrasonic measurement  Laser ultrasonic measurements were conducted using a LUMet sensor attached to the rear door of the Gleeble chamber. The LUMet sensor is composed of a generation and a detection laser. 41   The generation laser is used to generate ultrasonic pressure waves by ablating thin layers of material from the specimen surface. The laser has a frequency-doubled Q-switched Nd:YAG laser cavity giving a radiation with a wavelength of 532 nm; the pulse has a duration of approximately 6 ns and maximum energy of 72 mJ [82]. The repetition rate was set to 1 pulse per second. The generated pressure pulse propagates with a direction normal to the incident specimen surface and experiences amplitude decrease as it interacts with the material and its microstructure. The volume travelled by the pressure pulse can be approximated as a cylinder, i.e. the 2 mm diameter laser spot multiplied by the 3 mm specimen thickness. Ultrasonic measurements in this technique are representative of the average properties of the material over this cylindrical volume.  To detect generated ultrasound pulses, the detection system illuminates the specimen surface at the generation laser spot. The detection laser has a frequency-stabilized Nd:YAG laser cavity operating at an infrared wavelength of 1064 nm with a pulse duration of 90 μs. The repetition rate of the detection laser was 50 pulses per second. The infrared detection laser light reflected on the specimen surface is demodulated inside a photo-refractive crystal using an active interferometer approach.   4.2.3 Interpretation of ultrasound pulse waveforms  Figure 4.3a) and Figure 4.3b) show typical laser ultrasound waveforms measured in this material. The large amplitude echoes correspond to the generated ultrasound pulse returning periodically to the generation surface. Smaller oscillations are measured in between echoes, and correspond 42   to shear waves. The first echo measured after generation was selected for all analysis, as indicated by the red dotted lines in Figure 4.3a) and Figure 4.3b). For each test, the reference waveform was defined as the waveform measured at 900°C. The use of a reference waveform in this manner negates the diffraction contribution to attenuation and isolates scattering attenuation. This method of calculating attenuation is called the single echo technique [82,99].   A fast Fourier transform was used to calculate the frequency content of individual echoes (Figure 4.3c)). Attenuation was calculated in the frequency domain by the ratio of the amplitude of a current echo to a reference echo such that [4]  𝜶(𝑭) =  𝟐𝟎𝟐𝜽𝒍𝒐𝒈𝟏𝟎(?̃?𝒄𝒖𝒓𝒓𝒆𝒏𝒕(𝑭)  ?̃?𝒓𝒆𝒇𝒆𝒓𝒆𝒏𝒄𝒆(𝑭)⁄ )  (4.1) The units for attenuation in Equation 4.1 is dB/mm, 𝜃 is the specimen thickness in units mm, and 𝐹 is frequency in units MHz.  Resulting attenuation spectra were fit with Equation 2.14, and the best-fit values of 𝑏  were correlated to grain size values using the empirical calibration from Figure 2.8. An example of this process is shown in Figure 4.3d). The minimum and maximum frequencies of analysis were respectively 2 MHz and 15 MHz. The methodology for ultrasound waveform analysis was conducted using the in-house software CTOME (Computational Tools for Metallurgy) v2.29 [82]. CTOME functionality included weighting of signal to noise ratio and fit quality into the calculation of grain size. The criteria for a waveform analysis to be accepted by the CTOME software were a signal to noise ratio of more than 1 and a standard fit error of less than 15%. 43   Details regarding the assumptions and methodology of the CTOME software can be found in its official documentation [82].  Figure 4.3 Schematics depicting laser ultrasonic measurements of a) reference signal waveform at 900°C and b) current signal waveform. Red dotted lines indicate the first echo measured after ultrasound pulse generation. Experimental data examples of c) frequency content of individual echoes and d) calculated attenuation spectrum with fitted Equation 2.14. Pink dotted lines indicate the frequency bandwidth used for fitting Equation 2.14. Amplitude uses arbitrary units.  44   4.3 Ex-situ microstructural characterization  4.3.1 Specimen preparation  All specimens were cut through-thickness using a Struers Accutom-2 water-cooled diamond cut-off machine at the position of laser ultrasonic measurement, i.e. along the mid-length of the generation laser spot. The cut sections were mounted in phenolic and prepared for optical microscopy by surface grinding using successive 320, 400, 600 and 1200 grit silicon carbide grinding papers. The specimen surface was rinsed with water after each grinding step. The specimen surfaces were then mechanically polished using 6 µm and 1 µm diamond suspensions. Between each mechanical polishing step, specimens were immediately rinsed with denatured ethanol and dried with compressed air.  4.3.2 Characterization of microstructure  Microstructure characterization was conducted exclusively on microstructures quenched during abnormal grain growth. Etching of specimens was done using 0.5% Nital (solution of 0.5% nitric acid in ethanol) for 15 to 20 minutes. Micrographs were taken using a Nikon MA200 optical microscope between 5x and 50x magnification. Micrographs were printed out and large grain clusters were delineated by hand-tracing onto transparent film, which was subsequently scanned into an image file. ImageJ v1.50b software was used to threshold the traced outlines and measure the area of each outlined region.   45   4.4 Finite element simulation of ultrasound pulse propagation  4.4.1 Generation of sub-domain  MICRESS® is a commercial software that employs the phase-field method to simulate microstructure evolution. The Voronoi tessellation algorithm in MICRESS® v6.200 was used to generate sub-domains composed of grains with various size distribution [92,100,101]. Generated sub-domains were limited to a finite size of at maximum 2000 μm by 2000 μm, with boundary conditions being periodic. Sub-domains were extracted at time step 0 seconds so it was only necessary to consider the Voronoi tessellation algorithm of MICRESS®.   The tessellation algorithm randomly positioned seeds within the sub-domain in accordance with a defined value of minimum distance between seeds. Cells were allocated to each seed in accordance with a defined value of grain radius. Overlapping cells were then allocated to whichever seed they were spatially nearest to. In this way interfaces were defined by the cells that are equidistant from two seeds, resulting in straight line interfaces. Interfaces were defined to consist of four cells across their thickness.  Cell dimensions were chosen such that grains possessed at least eight cells across their smallest dimension. For domains containing grains of size 35 μm or less, cell dimensions were 1 μm by 1 μm. For all other domains, cell dimensions were 4 μm by 4 μm.  A sub-domain’s grain size distribution was controlled by defining the minimum distance between seeds, 𝐿𝑠𝑒𝑒𝑑 , and grain radius, 𝑅 . Sub-domains of narrow unimodal grain size 46   distribution were obtained by setting identical values for 𝐿𝑠𝑒𝑒𝑑 and 𝑅, and setting the maximum possible number of grains that fit into the finite sub-domain dimensions. Parameters for generation of narrow grain size distribution sub-domains are summarized in Table 4.2.   Table 4.2 Summary of parameters used in MICRESS® generation of narrow grain size distribution sub-domains Sub-domain dimensions (μm) Cell dimension (μm) EQAD (μm) 𝐿𝑠𝑒𝑒𝑑 = 𝑅 (μm) Number of grains 500 x 500 1 10 7.5 3027 2000 x 2000 4 41 30 3027 2000 x 2000 4 56 41 1602 2000 x 2000 4 89 65 641 2000 x 2000 4 131 100 296 2000 x 2000 4 240 192 88    47   Wide unimodal grain size distributions were generated by defining twelve types of grains within the subdomain. Each of the twelve grain types were of a narrow size distribution, but of such sizes that their distributions overlapped to create a unimodal distribution. The number of grains for each grain type was adjusted to produce the desired size distribution. Parameters for generation of wide grain size distribution sub-domains are summarized in Table 4.5. The first and second rows in the “Number of grains” column respectively produced log-normal and Gaussian grain size distributions.  Table 4.3 Summary of parameters used in MICRESS® generation of wide unimodal grain size distribution sub-domains. Columns “𝑳𝒔𝒆𝒆𝒅 = 𝑹” and “Number of grains” list 12 values corresponding to 12 grain types. Sub-domain dimensions (μm) Cell dimension (μm) EQAD (μm) 𝐿𝑠𝑒𝑒𝑑 = 𝑅 (μm) Number of grains 2000 x 2000 4 82 102, 94, 86, 78, 70, 62, 54, 46, 38, 30, 22, 18 15, 24, 30, 36, 54, 66,  84, 144, 105, 90, 78, 35 6, 14, 15, 16, 90, 108, 120, 140, 105, 70, 48, 35     48   Bimodal size distributions were generated by defining two types of grains within the sub-domain, each of narrow size distribution, but of such sizes such that their distributions did not overlap. The Voronoi tessellation algorithm seeded grain types in ascending order, i.e. all grains of grain type 1 are seeded first, followed by all grains of grain type 2. Area ratio between the bimodal distribution’s two grain size populations was controlled by the number of grains defined for each grain type.   Parameters for generation of bimodal grain size distribution sub-domains are summarized in Table 4.4. The cases in Table 4.4 randomly dispersed seeds throughout the entire sub-domain. Ordered configurations of grain clusters, consisting of layering and cluster positioning, was achieved by defining rectangular regions of the sub-domain for each grain type to randomly disperse seeds within. Rotation and translation of sub-domains was accomplished using the software CTOME. Table 4.5 summarizes the parameters for generation of ordered configuration bimodal grain size distribution sub-domains. 49   Table 4.4 Summary of parameters used in MICRESS® generation of bimodal grain size distribution sub-domains. Seeds of each grain type were randomly dispersed throughout the entire sub-domain. Sub-domain dimensions (μm) Cell dimension (μm) Grain type 1 Grain type 2 EQAD (μm) 𝐿𝑠𝑒𝑒𝑑 = 𝑅 (μm) Number of grains EQAD (μm) 𝐿𝑠𝑒𝑒𝑑 = 𝑅 (μm) Number of grains 1000 x 1000 1 131 75 39 10 7.3 5659 2000 x 2000 4 131 85 230 41 28 683 2000 x 2000 4 131 81 157 41 29 1455 2000 x 2000 4 131 81 70 41 29 2381 2000 x 2000 4 240 140 46 41 29  1455    50   Table 4.5 Summary of parameters used in MICRESS® generation of bimodal grain size distribution sub-domains with ordered configurations. Seeds of each grain type were randomly dispersed throughout rectangular regions of the sub-domain defined by two points (x1, y1) x (x2, y2). Sub-domain dimensions (μm) Cell dimension (μm) Grain type 1 Grain type 2 EQAD (μm) 𝐿𝑠𝑒𝑒𝑑 = 𝑅 (μm) Number of grains Seeding region (μm) EQAD (μm) 𝐿𝑠𝑒𝑒𝑑 = 𝑅 (μm) Number of grains Seeding region (μm) 1000 x 500 1 131 75 5 (75, 75) x (250, 425) 10 7.3 5352 (0, 0) x (1000, 500) 2000 x 2000 4 131 100 157 (0, 0) x (2000, 1000) 41 30 1455 (0, 1000) x (2000, 2000) 2000 x 2000 4 131 82 45 (0, 1050) x (2000, 1490) 41 30 2539 (0, 0) x (2000, 2000) 51   Once a sub-domain was generated, the MICRESS® program assigned each grain a random crystallographic orientation. Note in Table 4.2 that the number of grains is a finite value limited by the sub-domain dimensions and grain size. To achieve better statistics, five replications of each sub-domain were created. Each sub-domain replication was assigned a different set of random crystallographic orientations. The different sets of random crystallographic orientations were obtained from the MICRESS® program by regenerating each sub-domain with different randomization integers. No other changes are made to sub-domains besides the assignment of different random crystallographic orientation sets. The distribution of misorientation assigned by MICRESS® was calculated by importing grain crystallographic orientation files (.ang) into the EDAX-TSL software OIM Analysis v7.2 [102].  4.4.2 Generation of Abaqus simulation domain  Simulation domains were created by periodic repetition of a MICRESS®-generated sub-domain until it met a desired mesh template size. Figure 4.4 schematically illustrates the mesh template. The default mesh template dimensions were {𝐿, 𝜃} = {10 mm, 2 mm}, where 𝐿 is length and 𝜃 is thickness. A larger mesh template of dimensions {𝐿, 𝜃} = {40 mm, 8 mm} was additionally chosen for sub-domains containing grains sizes larger than 100 μm.  52    Figure 4.4 Schematic of mesh template used for generation of simulation domains  The mesh consisted of four-node bi-linear quadrilateral plane strain (CPE4R) elements. The simulation domain is sub-divided into a centered region with a fine mesh size of either 4 μm or 1 μm and a graded coarse mesh on each side. Manual mesh refinement gradually transitions the coarse mesh into the fine mesh. On the domain edges, coarse mesh size was on the order of 50 times larger than fine mesh size. The “Mechanical-ENCASTRE” boundary condition (𝑈1  = 𝑈2  =  𝑈3  =  𝑈𝑅1  =  𝑈𝑅2  =  𝑈𝑅3 = 0) was applied to left and right vertical edges of the simulation domain to fix the model in space. 𝑈  and 𝑈𝑅  are respectively displacement components and rotational displacement components, and the subscripts 1, 2, 3 correspond to the X, Y, and Z dimensions in Abaqus. The model assumed linear elasticity for large displacements.  For each set of elements contained in one grain, effective elastic constants were calculated using stiffness tensor rotation [103] on elastic constants of austenite at 1000°C. Table 4.6 shows material properties of austenite at 1000°C. The stiffness tensor rotation accounted for the 53   dependency of elastic constants on crystallographic orientation. The effective elastic constants were assigned to each grain’s set of elements.  To simulate reference waveforms for attenuation analysis, elastically isotropic domains were used. Isotropic elasticity was imposed by conducting the aforementioned process of stiffness tensor rotation on the elastic constants of 1000°C isotropic polycrystalline austenite [104,105]. The elastic constants of isotropic polycrystalline austenite in Table 4.6 represent the Hill-averaged elastic constants in a polycrystalline sample of infinitely small austenite grain size, i.e. a sample in which all possible crystallographic orientations exist. The software CTOME was used to conduct the steps of sub-domain mirroring, simulation domain meshing, stiffness tensor rotation [103], and elastic property assignment to element sets.  Table 4.6 Material properties of austenite at 1000°C. Elastic constants use Hill averaging [106].  Density (g/cm3) C11  (GPa) C12  (GPa) C44  (GPa) Single crystal 7.603 154.0 122.0 77.0  Isotropic polycrystal 7.603 188.3 104.9 41.7     54   4.4.3 Simulation of ultrasound pulse propagation  Abaqus CAE v6.13 was used to conduct explicit dynamic 2-D planar finite element modeling simulations of ultrasound propagation through anisotropic polycrystalline grain structures [72,107–109]. The simulations consist of an ultrasound pulse generation step followed by a propagation step. During the first step, an initial displacement in the form of a double Ricker wavelet (Figure 4.5a)) was imposed on every node located along a 2 mm length region of the simulation domain surface (Figure 4.4). The displacement amplitude for a double Ricker wavelet is given by  ?̃?(𝒕) =  ∑  (𝟏 − 𝟐𝝅𝟐(𝑭𝒑)𝒊𝟐(𝒕 − 𝜺𝒊)𝟐)𝒆−𝝅𝟐(𝑭𝒑)𝒊𝟐(𝒕−𝜺𝒊)𝟐𝟐𝒊=𝟏   (4.2) where 𝐹𝑝 is the peak frequency in units MHz, 𝑡 is time in units seconds, and 𝜀 is a constant. Figure 4.5b) displays the frequency content of the wavelet in Figure 4.5a), as calculated using a fast Fourier transform. The frequency bandwidth exhibited in Figure 4.5b) is similar to the frequency bandwidth obtained with LUMet (Figure 4.3c)). The double Ricker wavelet amplitude was distributed across the 2 mm generation surface using a Gaussian function given by  ?̃?(𝒙) = ?̃?(𝒕)𝒆−(𝒙𝟐(𝟎.𝟓)𝟐⁄)   (4.3) The Gaussian distribution of amplitude reduces the stress discontinuity between nodes in the generation region and their neighboring nodes outside the generation region. Equation 4.3 is plotted in Figure 4.5c) at time 200 ns.  55       Figure 4.5 Double Ricker wavelet applied to simulation domain surface in FE a) time dependence b) frequency dependence and c) an example at time 200 ns of the Gaussian normalization of applied amplitude across the generation surface, where the Gaussian peak at position 0 mm is centered on the mid-length of the simulation domain. Amplitude uses arbitrary units.  Lasers generally emit beams with a Gaussian spatial profile of intensity. Recall however in the LUMet system that the ultrasound pulse is generated by laser ablation. The spatial profile of the generated displacement pulse may not follow the spatial profile of the incident laser pulse due to the nonlinear nature of the vaporization process [110]. Therefore it is unknown the extent to which the generated ultrasound pulse in the LUMet system is simulated by the Gaussian spatial distribution of Equation 4.3.  To ensure adequate discretization of the simulated ultrasound pulse, mesh dimensions were confirmed to describe the smallest longitudinal wavelength by at least 20 mesh elements. The minimum group velocity for longitudinal wave propagation through austenite is approximately 4000 m/s and the maximum frequency generated in the initial impulse is 30 MHz, so the minimum mesh dimension is calculated as 4000 / (20 * 30) = 6.7 μm. Both cell dimensions used 56   in Section 4.4.1 (4 μm or 1 μm) satisfy this requirement, and thus Abaqus mesh dimensions were defined to be identical to MICRESS® cell dimensions.  During the second simulation step, the generation surface nodes are released and ultrasound pulse propagation is simulated for a duration of 2000 ns or 3500 ns respectively for {10 mm, 2 mm} and {40 mm, 8 mm} mesh templates. For both simulation steps a time increment was chosen to ensure adequate sampling of the ultrasound pulse propagation. The time increment was defined such that the time for a node to move the distance of one mesh dimension takes at least four time steps. For example, a mesh dimension of 1 μm requires a maximum time step of (1 / 5000) / 4 = 0.05 ns, where 5000 m/s is the approximate speed of ultrasound wave propagation through austenite. The time increment was chosen as either 0.05 ns or 0.2 ns, respectively for mesh dimensions of 1 μm and 4 μm. Sensitivity analysis of the selected mesh sizes has been completed previously [82], whereas sensitivity analysis of the selected time increments has not been attempted.  Displacement data in the vertical (Y) direction was extracted every 8 ns and averaged along the nodes of the 2 mm generation surface for {10 mm, 2 mm} mesh templates. Displacement data was averaged along the nodes opposite of the 2 mm generation surface (i.e. the centered 2 mm length region along the bottom surface of the simulation domain) for the {40 mm, 8 mm} mesh template. Calculation of attenuation from FE simulated displacement data followed the methodology of Section 4.2.3. Reference displacement data was selected as the FE simulation results for an elastically isotropic domain of matching mesh template size. 57   Chapter 5: Austenite Grain Growth in A36 Steel  5.1 Evolution of austenite grain size during isothermal heat treatment  Figure 5.1 displays the laser ultrasonic austenite grain size measured during isothermal holding at 950°C, 1050°C, and 1100°C. Time at 0 seconds was defined at the onset of isothermal holding. Consequently, times prior to 0 seconds were during 5°C/s heating. All results in Figure 5.1a) exhibit limited grain growth from the onset of isothermal holding. Rapid grain growth is observed during the 5°C/s heating up to 1050°C and 1100°C holding temperatures. Good repeatability of measured austenite grain evolution is observed amongst multiple tests for 1050°C, and 1100°C isothermal holding. The two tests of 950°C isothermal holding diverge slightly after 200 seconds.  Figure 5.1b) displays the evolution of laser ultrasonic austenite grain size during 1000°C isothermal holding, which exhibits a distinct stage of rapid grain growth between stages of limited grain growth. Clear variability exists amongst the multiple tests for the onset of rapid grain growth, which ranges from 0 to 100 seconds. The laser ultrasonic grain size during isothermal holding also varies significantly amongst tests of 1000°C isothermal holding. Table 5.1 reports mean values for each test of laser ultrasonic grain size measurements between 500 seconds and 600 seconds of isothermal holding, i.e. the time at which a limiting grain size appears to have been established. The mean grain size values for 950°C and 1000°C tests both span a relatively wide range that surpasses the 15% error of the LUMet technique [82].    58   Table 5.1 Mean values for tests of laser ultrasonic grain size measurements between 500 seconds and 600 seconds of isothermal holding Thold (°C) ?̅?500𝑠−600𝑠 (μm) 950 15, 29 1000 95, 96, 106, 110, 112, 137, 153 1050 95, 95, 100, 104 1100 124, 126  Ex-situ metallographic data from Militzer et al. [48] was overlaid onto the laser ultrasonic measurements in Figure 5.1. The laser ultrasonic measurements agree reasonably well with the ex-situ metallographic data. Due to the variability amongst the 1000°C tests, only one test in Figure 5.1b) agrees with the two ex-situ metallographic points between 50 seconds and 150 seconds. 59    Figure 5.1 Laser ultrasonic measurement of austenite grain growth in A36 steel during isothermal holding tests at the indicated temperatures a) 950°C, 1050°C, and 1100°C b) 1000°C. Red points are ex-situ metallographic data [48] and pink dotted lines indicate the onset of isothermal holding. 60   5.2 Characterization of interrupted heat treatment microstructures  5.2.1 Laser ultrasonic measurements  Additional laser ultrasonic tests were done to observe the microstructure at different stages of grain growth during 1000°C isothermal holding. Figure 5.2 displays the results of these additional tests. Helium gas quenching was initiated at times when rapid grain growth was expected based on the laser ultrasonic measurements shown in Figure 5.1. Laser ultrasonic grain size measurements ranged between 40 µm and 104 µm at the time of quenching, which indicated the four tests were quenched at various points of rapid grain growth. Variability amongst tests again lead to only one test in Figure 5.2 agreeing with the ex-situ metallographic point at 120 seconds.  Figure 5.2 Laser ultrasonic measurement of austenite grain growth in A36 steel during tests of interrupted 1000°C isothermal holding. Red points are ex-situ metallographic data [48]. 61   5.2.2 Ex-situ metallography  To characterize the microstructure probed by LUMet, the quenched specimens from the tests in Figure 5.2 were cut through-thickness at the generation laser spot center. Figure 5.3 shows the etched microstructure of A36 steel following interrupted 1000°C isothermal holding and reveals a non-homogeneous microstructure with large clusters of martensite standing out against a fine matrix of bainite. The large martensitic clusters can be attributed to clusters of large austenite grains that underwent abnormal grain growth. The fine bainitic matrix can be attributed to small austenite grains that did not undergo abnormal grain growth.  Figure 5.3 Micrograph of etched A36 steel following 80 seconds of 1000°C isothermal holding and helium gas quenching (Test 3)  Figure 5.4 shows the etched microstructures at a lower magnification such that the entire through-thickness cross-section is shown. Two types of variability are apparent in Figure 5.4, i.e. variability amongst tests and variability within each specimen. Consider the two micrographs shown in Figure 5.4a) and Figure 5.4b). The area fraction of abnormal grains is clearly larger in 62   Figure 5.4a) than in Figure 5.4b), despite the two tests being of identical heat treatment. The micrograph shown in Figure 5.4c) corresponds to a test that was isothermally held for 60 seconds longer than the other tests. Nevertheless, it produced an area fraction of abnormal grains that was less than the other tests.   Figure 5.4 Micrographs of etched A36 steel following interrupted 1000°C isothermal holding, through-thickness cross-section at the LUMet generation laser spot center. a) Test 2, b) Test 3, and c) Test 4  63   The area fraction of abnormal grains is also highly variable across each specimen’s cross-section. Take for example the micrograph in Figure 5.4c), in which the area fraction of abnormal grains is near zero for most of the cross-section. However, the rightmost 2 mm width area of the micrograph is completely occupied by abnormal grains. Recall that in laser ultrasonics, the ultrasound pulse travels through a finite volume of 2 mm width. The heterogeneous distribution of abnormal grains exhibited in Figure 5.4 is on a scale wider than the laser spot width, such that the finite sampling volume can lead to variability of laser ultrasonic grain size measurement. That is, laser ultrasonic grain size is a local measurement that might not be representative of the entire specimen volume. Altogether, the variability of emerging abnormal grains evidenced in Figure 5.4 can explain the variability of laser ultrasonic grain size measurements during 1000°C isothermal holding observed in Figure 5.1b) and Figure 5.2.  Table 5.2 presents the results of outlining large clusters (Figure 5.5) and calculating their area fraction with respect to 2 mm width regions of the photographed specimen cross-section. The column “Total” denotes the area fraction of large clusters in the whole cross-section. 𝐷𝐿𝑈𝑀𝑒𝑡 is the measured laser ultrasonic grain size at quenching. The highlighted “4 – 6 mm” region in Table 5.2 corresponds to the region travelled by an ultrasound pulse during laser ultrasonic measurements. This region is also indicated by the dashed lines in Figure 5.5.    64    Figure 5.5 Tracing of large clusters (black) in etched A36 steel following interrupted 1000°C isothermal holding and schematic illustration of region probed by laser ultrasonics   Table 5.2 Data of large cluster area fraction in etched A36 steel following interrupted 1000°C isothermal holding Test # 𝐷𝐿𝑈𝑀𝑒𝑡 (μm) Area fraction of large clusters per region 0 – 2 mm 2 – 4 mm 4 – 6 mm 6 – 8 mm 8 – 10 mm Total 1 104 1 1 1 1 1 1 2 79 1 0.87 0.62 0.49 0.41 0.68 3 43 0.67 0.46 0.25 0.05 0.05 0.28 4 40 0.09 0.01 0.08 0.16 0.98 0.27  Note in Table 5.2, there is a correlation between 𝐷𝐿𝑈𝑀𝑒𝑡 and area fraction of large clusters in the region travelled by the ultrasound pulse. This correlation can be clarified by using a 65   normalization approximation to calculate the area fraction of large clusters corresponding to each 𝐷𝐿𝑈𝑀𝑒𝑡, i.e.  𝒇𝑳𝑼𝑴𝒆𝒕 = (𝑫𝑳𝑼𝑴𝒆𝒕 −  𝑫𝒊𝒏𝒊𝒕𝒊𝒂𝒍)  (𝑫𝒇𝒊𝒏𝒂𝒍 −  𝑫𝒊𝒏𝒊𝒕𝒊𝒂𝒍)⁄   (5.1) Here, 𝐷𝑖𝑛𝑖𝑡𝑖𝑎𝑙 is the initial grain size obtained from the isothermal holding data of Section 5.1, i.e. 15 μm. 𝐷𝑓𝑖𝑛𝑎𝑙 is the final grain size obtained from the isothermal holding data of Section 5.1. 𝐷𝑓𝑖𝑛𝑎𝑙 for a given isothermal holding temperature can be defined as the ?̅?500𝑠−600𝑠 values from Table 5.1. Defining 𝐷𝑓𝑖𝑛𝑎𝑙 as the average of the seven ?̅?500𝑠−600𝑠 values in Table 5.1 for 1000°C isothermal holding, Figure 5.6 displays good correlation between the calculated large cluster fraction, 𝑓𝐿𝑈𝑀𝑒𝑡, and the metallographically measured values from Table 5.2, 𝑓𝑚𝑒𝑡𝑎𝑙𝑙𝑜. The error bars in Figure 5.6 correspond to the standard deviation of 𝑓𝐿𝑈𝑀𝑒𝑡 calculations using the seven values of ?̅?500𝑠−600𝑠 for 𝐷𝑓𝑖𝑛𝑎𝑙.  Figure 5.6 Comparison between large cluster fractions measured by metallography and by a normalization approximation from laser ultrasonic grain size measurements. Green line depicts perfect agreement. 66   5.3 Interpretation of laser ultrasonic measurements  Figure 5.7 shows the thermodynamic stability of AlN precipitates in austenite for the exact composition of A36 steel used in this study. Per the Thermocalc v4.0 TCFE7 database thermodynamic model, AlN precipitates begin to dissolve significantly at 840°C and dissolve completely at 1098°C. The equilibrium fractions of AlN at 950°C, 1000°C, 1050°C, and 1100°C are respectively 86%, 70%, 42%, and 0% of the equilibrium fraction at 700°C.   Figure 5.7 Thermodynamic stability of AlN precipitates in A36 steel austenite as obtained from Thermocalc v4.0  Considering how abnormal grain growth can be caused by second-phase particle dissolution [30], abnormal grain growth can be expected for all temperatures above 840°C in A36 steel. However, the development of abnormal grain growth depends on the kinetics of second-phase particle dissolution. In a previous study using the same A36 steel and heat treatments, Militzer et al. [48] observed abnormal grain growth during 950°C and 1000°C isothermal holding. 67   Abnormal grain growth was also observed during 5°C/s heating up to 1050°C and 1100°C. In the present study, a distinct stage of abnormal grain growth measurements was observed during 1000°C isothermal holding but not observed during 950°C, 1050°C and 1100°C isothermal holding.   These observations can be explained by considering the interplay of grain growth and particle dissolution with respect to time and temperature. The rates of grain growth (Equation 2.4) and particle dissolution (Equation 2.9) with respect to time both increase as temperature increases. A heating rate of 5°C/s and isothermal holding at 1000°C evidently produce the proper combination of grain growth and particle dissolution kinetics for abnormal grain growth to develop during isothermal holding.   Further evidence for this explanation originates from Cheng et al. [42], who characterized particle size distributions of AlN precipitates in a low carbon steel after 1000°C isothermal holding. Their results suggested that AlN precipitates partially dissolve during the early stages of 1000°C holding, accompanied by coarsening of undissolved precipitates. Such changes to precipitate size distribution enable abnormal grain growth [17,30,35]. This rationale is corroborated by previous experimental observations of abnormal grain growth in A36 steel during 1000°C isothermal holding [32,48,84,86]. In future work, a systematic study of the AlN precipitate distribution during abnormal grain growth at 1000°C isothermal holding can be conducted by TEM analysis.  At 950°C, the rate of particle dissolution is lower than at 1000°C such that pinned grain growth occurs until a sufficiently long time of isothermal holding produces enough particle dissolution 68   for abnormal grain growth. Above 1000°C, the rates of particle dissolution and grain growth are higher such that abnormal grain growth develops rapidly, i.e. during non-isothermal heating to higher holding temperatures. These explanations are consistent with the experimental observations of abnormal grain growth in A36 steel from Militzer et al. [48]. Future work involves the investigation of abnormal grain growth development during longer isothermal holding times at 950°C. The existing heating rate of 5°C/s can also be increased such that abnormal grain growth develops during isothermal holding at temperatures above 1000°C instead of during non-isothermal heating.  The variability of abnormal grain area fraction exhibited by the micrographs in Figure 5.4 can be attributed to the probabilistic, localized character of abnormal grain growth [31,35]. Abnormal grain growth depends on microstructural features, such as second-phase particle distribution, that may vary from specimen to specimen and locally within each specimen. Heterogeneous distribution of emerging abnormal grains can also explain the variability amongst laser ultrasonic grain size measurements during 1000°C isothermal holding tests from Table 5.1. The size of an abnormal grain depends on the amount of surrounding small matrix grains that it can grow into [27,35]. A locally sparse population of abnormal grains allows each abnormal grain to grow to a larger size than in a locally dense population.   This interpretation can also explain why some of the limiting grain size measurements for 1000°C holding in Table 5.1 are higher than the limiting grain size measurements at 1050°C and 1100°C holding. In a dense population of abnormal grains, which can be expected during heating above 1000°C, physical impingement amongst abnormal grains restrains the limiting grain size. 69   A dense population of abnormal grains also eliminates the variability of abnormal grain size associated with locally sparse populations of abnormal grains, which explains the consistent limiting grain size measurements at 1050°C and 1100°C holding. However, the size of abnormal grains depends on the temperature-dependent mobility of grain boundaries. It therefore is understandable how the limiting grain size at 1100°C holding is larger than the limiting grain size at 1050°C holding despite similarly dense populations of abnormal grains. Another future work is to reveal prior austenite grain boundaries such that austenite grain size can be measured and used to validate laser ultrasonic measurements.   70   Chapter 6: Finite Element Simulation of Ultrasound Pulse Propagation in Austenite  6.1 Unimodal grain size distributions  6.1.1 Generation of sub-domain  Figure 6.1a) displays a sub-domain generated by Voronoi tessellation in MICRESS® in the case of a narrow grain size distribution with mean grain size 89 µm. Each grain is shown as a different color. Figure 6.1b) shows the cumulative grain size distribution with respect to reduced grain size, i.e. the ratio of grain size to the mean of the size distribution, for sub-domains with mean grain size ranging from 10 µm to 240 µm. All the cases show the same narrowness irrespective of the mean grain size in the sub-domain. The distribution for the sub-domain with a mean grain size of 240 µm is not as smooth as the other cases due to the much lower number of grains in that sub-domain. The distributions are symmetrically distributed around a mean value, i.e. a Gaussian distribution with a standard deviation that is 18% of the mean value. 71    Figure 6.1 a) A sub-domain of narrow grain size distribution and mean grain size 89 μm, and b) cumulative grain size plot of narrow size distribution sub-domains  The distribution of misorientation for the sub-domain with 41 µm mean grain size is shown in Figure 6.2 in comparison to the ideal Mackenzie distribution calculated for a perfectly random set of misorientation. The excellent agreement between the measured distribution and the Mackenzie distribution demonstrates the randomness of texture generated by MICRESS®.    72    Figure 6.2 Distributions of misorientation obtained in the sub-domain with 41 μm mean grain size compared with the ideal Mackenzie distribution  6.1.2 Effect of mesh template size  Figure 6.3a) and Figure 6.3b) compare the attenuation in domains when using either a mesh template of thickness 2 mm or 8 mm. The domains consisted of narrow grain size distributions with a mean grain size of 131 µm and 241 µm. The points in Figure 6.3 correspond to the average attenuation from the five different simulations where new sets of crystallographic orientations are allocated to the grains. The error bar corresponds to the standard deviation resulting from the five different simulations. For both grain sizes, the standard deviation is smaller when simulation is conducted in the domain of 8 mm thickness, illustrating the influence of grain size statistics and crystallographic orientation set in the resulting attenuation.  73   The ratio of grain size to sample thickness influences the resulting through-thickness texture. When grain size is large, the number of grains present across the sample thickness can be small, which results in a stronger effect of some crystallographic orientations in the scattering of the wave [82]. For example, the 240 µm mean grain size domain had only ten grains spanning the thickness of the 10 mm by 2 mm mesh template. In view of this observation, future simulations involving grain size of 131 µm and 240 µm should be conducted in the domain of 8 mm thickness. Further sensitivity analysis can confirm if simulations involving grain size less than 130 µm can be conducted in the domain of 2 mm thickness.  Figure 6.3 FE simulated attenuation spectra from narrow grain size distribution domains with varying mesh template sizes a) 131 μm mean grain size b) 240 μm mean grain size    74   6.1.3 Narrow grain size distributions  Figure 6.4 shows the attenuation calculated for narrow grain size distributions of various grain sizes. Green lines represent the best-fit 𝛼 = 𝑎0 + 𝑏𝐹3 for attenuation up to 15 MHz. For the attenuation of the 240 µm grain structure, the green line represents the best-fit 𝛼 = 𝑎0 + 𝑏𝐹3 for attenuation up to 12 MHz.   Figure 6.4 FE simulated attenuation spectra from narrow grain size distribution domains of mean grain size a) 10 µm, 41 µm, 56 µm and 89 µm, b) 131 µm and 240 µm. Green lines represent best-fit 𝜶 = 𝒂𝟎 + 𝒃𝑭𝟑.  In Figure 6.4a), the attenuation in mean grain sizes below 90 µm appear to obey the power-law expression 𝛼 = 𝑎0 + 𝑏𝐹3 up through 15 MHz. In contrast, the attenuation at 131 µm and 240 µm mean grain sizes (Figure 6.4b)) appear to transition towards a smaller frequency dependence 75   above a certain frequency. In the approximation of a single scattering regime, the attenuation for these large grain sizes should only be used in a limited range of frequency, i.e. up to 15 MHz and 12 MHz, respectively. The frequencies of 15 MHz and 12 MHz equate to wavelengths of approximately 330 µm and 400 µm for ultrasound wave propagation through austenite. These wavelength values are comparable to the respective mean grain sizes of 131 µm and 240 µm in the grain structures, which satisfies the condition of the stochastic scattering regime (Equation 2.17). Thus, the transition to smaller frequency dependence of attenuation in Figure 6.4b) can be linked to the transition from the Rayleigh to the stochastic scattering regime.  76   6.1.4 Wide grain size distributions  At this stage, it is interesting to consider a grain structure of wide unimodal size distribution, such as the scaling log-normal distribution that is usually encountered during normal grain growth [22]. Figure 6.5 displays a sub-domain of log-normal grain size distribution with mean grain size 82 µm and a Dmax/EQAD of 2.15.   Figure 6.5 A sub-domain of log-normal grain size distribution and mean grain size 82 μm  The log-normal grain size distribution is compared to a Gaussian grain size distribution to evaluate if a variation in the area fraction of larger grains can be detected using attenuation. The two grain size distributions are shown in Figure 6.6 in terms of number and area fraction distribution. For both distributions, the maximum grain size (176 μm) and the mean grain size (82 μm) are identical and only the area fraction of grains with size between 80 μm and 176 μm is different. The Gaussian grain size distribution has a standard deviation that is 83% of its mean value. 77    Figure 6.6c) shows the attenuation corresponding to these two grain size distributions. The attenuation is nearly identical within the degree of uncertainty caused by the variation in random crystallographic orientations from simulation to simulation. This suggests that the differences in those two grain size distributions are not large enough to be measurable using ultrasonic measurements.   Figure 6.6 Wide grain size distributions in terms of number fraction and area fraction a) log-normal b) normal, and c) corresponding FE simulated attenuation spectra      78   6.2 Bimodal grain size distributions  6.2.1 Sensitivity of grain cluster spatial configuration  In the case of a bimodal grain size distribution, the spatial distribution of the two families of grains may influence the resulting attenuation measured on the surface of the sample. A systematic study is conducted by generating several configurations of a bimodal grain size distribution.  Figure 6.7a) shows the first case composed of two horizontal layers of grains with the top and bottom layers respectively consisting of 41 µm and 131 µm mean grain size. The area fractions for the fine and coarse grain layer are 0.48 and 0.52 respectively.   Figure 6.7b) shows the attenuation calculated in this bimodal grain size distribution. For sake of comparison, the attenuation calculated from each constituent layer are overlaid in Figure 6.7b). The attenuation calculated in the bimodal grain size distribution lies within the bounds of its constituent attenuation spectra. The bimodal grain size distribution attenuation also exhibits a smaller frequency dependence above 15 MHz, which is similar to the attenuation of the narrow 131 µm mean grain size distribution.  79    Figure 6.7 a) Dual layered 41 µm and 131 µm bimodal grain size distribution sub-domain and b) corresponding FE simulated attenuation spectrum with constituent narrow grain size distribution attenuation spectra overlaid. Green line represents best-fit 𝜶 = 𝒂𝟎 + 𝒃𝑭𝟑 up to 15 MHz.  A total of five configurations are shown in Figure 6.8a) where the spatial configuration of the two grain size families is modified so the sensitivity of grain cluster configuration is systematically investigated. The first configuration in Figure 6.8a) interchanges the spatial configuration of fine and coarse grain layers from the Figure 6.7a) sub-domain. The other three configurations are randomly dispersed, vertically layered, and horizontally layered large grains amongst a matrix of small grains. The corresponding attenuation spectra are shown in Figure 6.8b). The attenuation calculated in these configurations is not measurably different, indicating that the spatial distribution of grain cluster does not play an important role in these cases. 80    Figure 6.8 a) 41 µm and 131 µm bimodal grain size distribution sub-domains of various grain cluster configuration and b) corresponding FE simulated attenuation spectra 81   6.2.2 Sensitivity of grain cluster position with respect to ultrasound pulse  In view of the results shown in Figure 6.8b), the spatial distribution of the large grains does not influence the attenuation to a measurable extent. An additional series of simulations is conducted to explore specifically the influence of the scale of these large grain clusters with respect to the size of the ultrasound pulse propagating in the sample. Figure 6.9a) displays a set of configurations for which the bimodal grain structure consists of 41 µm and 131 µm mean grain size families vertically aligned in the sample. The area fraction covered with small grains is set to 85%, resulting in a 0.5 mm width cluster of large grains separated by a 1.5 mm width cluster of small grains. The position of the cluster of large grains is then shifted horizontally by 0.5 mm increments with respect to the position of the ultrasound pulse to examine the influence on the resulting attenuation. The color scale in Figure 6.9a) represents the velocity for an ultrasound pressure wave propagating in the vertical (Y) direction, i.e. each grain is shown as a different color.    Figure 6.9b) shows the attenuation calculated in these different configurations. The attenuation increases as the vertical cluster of large grains moves towards the center of the ultrasound pulse. This trend is explained by considering the Gaussian distribution of ultrasound pulse amplitude across the generation surface (Figure 4.5c)), in which the amplitude is maximum at the pulse center and the full width at half of the maximum amplitude is 1.16 mm. Grains at the outer edge of the 2 mm pulse generation surface therefore have a lesser influence on the resulting attenuation than the grains located near the center of the ultrasound pulse. The results shown in Figure 6.9b) are consistent with these considerations where a larger attenuation is calculated 82   when the large grain cluster is located where amplitude is higher, and a smaller attenuation is calculated when it is located where amplitude is smaller.   Figure 6.9 a) 41 µm and 131 µm bimodal grain size distribution domains of various large grain cluster position (𝒗𝒈𝒓𝒐𝒖𝒑,   𝒀 is group velocity of a pressure wave in the vertical (Y) direction) and b) corresponding FE simulated attenuation spectra  This effect is investigated further by changing the ratio between small and large grain size in the domain. The coarse grain family is kept at 131 µm mean grain size and the fine grain size family is changed to a narrow size distribution of mean size 10 µm. Figure 6.10a) displays two such configurations with different positions of the large grain cluster respective to the pulse location. 83   Figure 6.10b) displays the resulting attenuation in these cases. The attenuation is again influenced by the spatial location of the large grain cluster. This result is important for the application of ultrasonic attenuation techniques to microstructures possessing a non-random distribution of grains, for example banding or segregation during casting. The severity of this effect depends on the difference between the large and small grain size, i.e. how large the scattering difference is between the two grain size families. Comparing the attenuation results in Figure 6.9b) and Figure 6.10b), the effect of large grain cluster positioning is more pronounced for wider differences between small and large grain size, i.e. when the small grain size contributes less attenuation.  Figure 6.10 a) 10 µm and 131 µm bimodal grain size distribution domains of various large grain cluster position and b) corresponding FE simulated attenuation spectra  84   To further illustrate the effect of these large grains clusters as well as the impact of their position relative to the propagating ultrasound pulse, the methodology for the calculation of grain size in LUMet (detailed in Section 4.2.3) is applied to the attenuation shown in Figure 6.9 and Figure 6.10. Resulting 𝐷𝐿𝑈𝑀𝑒𝑡 are reported in Table 6.1. The normalization approximation in Equation 5.1 is used again to calculate the fraction of large grains corresponding to each 𝐷𝐿𝑈𝑀𝑒𝑡. 𝐷𝑓𝑖𝑛𝑎𝑙 and 𝐷𝑖𝑛𝑖𝑡𝑖𝑎𝑙 are the LUMet grain sizes calculated from unimodal domains of the large and small grain families, respectively. The 𝐷𝐿𝑈𝑀𝑒𝑡  calculations show how the presence of large grain clusters with interspacing on the same order of magnitude as the width of the ultrasound pulse can lead to variation of the LUMet grain size.  To account for the spatial distribution of ultrasound wave amplitude, an expression for effective area fraction of large grains beneath the simulated ultrasound pulse, 𝑓𝐹𝐸 , is introduced by   𝒇𝑭𝑬 = ∫ 𝒇𝑳(𝒙)?̃?(𝒙)𝒅𝒙𝟏−𝟏   (6.1) Here, 𝑥 is horizontal position in the X direction with units mm, and 𝑥 = 0 is defined at the center of the ultrasound pulse. ?̃?(𝑥) is the Gaussian spatially distributed amplitude from Equation 4.3. 𝑓𝐿(𝑥) is length fraction of large grains, i.e. the length of large grains intercepted by a vertical line at 𝑥 per length 𝜃, where 𝜃 is thickness of the domain in the Y direction. In essence, 𝑓𝐿(𝑥) is weighted by ?̃?(𝑥) in Equation 6.1 for horizontal positions −1 𝑚𝑚 ≤ 𝑥 ≤ 1 𝑚𝑚  beneath the simulated ultrasound pulse. To a first approximation, Equation 6.1 can be calculated using the area fraction of large grains instead of length fraction, i.e.   𝒇𝑭𝑬 = ∑ (𝒇𝑨(𝒙𝒊+𝟏, 𝒙𝒊) (∫ ?̃?(𝒙)𝒅𝒙𝒙𝒊+𝟏𝒙𝒊 ∫ ?̃?(𝒙)𝒅𝒙𝟏−𝟏 ⁄ ))𝒊  , −𝟏 ≤ 𝒙𝒊 < 𝒙𝒊+𝟏 ≤ 𝟏  (6.2) 85   where 𝑓𝐴(𝑥𝑖+1, 𝑥𝑖) is the area fraction of large grains within the area {(𝑥𝑖+1 − 𝑥𝑖), 𝜃}. Take for example the “Cluster at center” domain in Figure 6.9a), in which 𝑥𝑖 and 𝑥𝑖+1 can be respectively defined as -0.25 mm and 0.25 mm for the large grain cluster. 𝑓𝐴 in this region is calculated as 𝑓𝐴(0.25, −0.25) = 0.60 and weighted by the value of  (∫ ?̃?(𝑥)𝑑𝑥0.25−0.25 ∫ ?̃?(𝑥)𝑑𝑥1−1 ⁄ ) = 0.40  Note that 𝑓𝐴(0.25, −0.25) does not equal 1 due to presence of small grains amongst the large grain cluster. This observation is shown in Figure 6.11 by displaying only the large grain cluster from Figure 6.9a). Pink dotted lines indicate the region from which 𝑓𝐴(0.25, −0.25)  is calculated.  Figure 6.11 Large grain cluster from the 41 µm and 131 µm bimodal grain size distribution “Cluster at center” domain in Figure 6.9a)  86   The remainder of the domain is comprised entirely of small grains, i.e.   𝑓𝐴(𝑥𝑖+1, 𝑥𝑖) = 0  {   𝑥𝑖 ∪ 𝑥𝑖+1 < −0.25𝑥𝑖 ∪ 𝑥𝑖+1 > 0.25  The calculation for 𝑓𝐹𝐸  thus simplifies to  𝑓𝐹𝐸 = 0 + 𝑓𝐴(0.25, −0.25) (∫ ?̃?(𝑥)𝑑𝑥0.25−0.25 ∫ ?̃?(𝑥)𝑑𝑥1−1 ⁄ ) =  0.24  Equation 6.2 was used to calculate 𝑓𝐹𝐸  for all domains in Section 6.2.2 and the values are presented in Table 6.1.  Table 6.1 Data of effective large grain area fraction in simulation domains and from calculations using correlated laser ultrasonic grain size  Grain size families Large grain cluster position 𝑓𝐹𝐸    𝐷𝐿𝑈𝑀𝑒𝑡 (μm) 𝑓𝐿𝑈𝑀𝑒𝑡  41 μm, 131 μm Center 0.24 55 0.15 Off-center 0.15 50 0.09 Edges 0.06 39 0 10 μm, 131 μm Center 0.23 38 0.25 Edges 0.06 11 0.04    87   Figure 6.12 displays a comparison between the effective large grain area fraction, 𝑓𝐹𝐸 , and the calculated values of 𝑓𝐿𝑈𝑀𝑒𝑡 from Table 6.1. This result corroborates the experimental result from Figure 5.6 in which grain size calculated from LUMet methodology can be correlated to the area fraction of large grains beneath the ultrasound pulse.   Figure 6.12 Comparison between effective large grain area fraction in simulation domains and by a normalization approximation from laser ultrasonic grain size calculations. Green line depicts perfect agreement.    88   6.2.3 Variations with bimodal area ratio and mean grain sizes  Figure 6.13 shows the attenuation calculated for bimodal grain size distributions in which the family of 131 µm mean grain size occupies successively 0, 23, 48, 77 and 100% of the domain area. The remainder of the domain area is occupied by a unimodal 41 µm mean grain size family. The spatial distribution of large grains is randomly dispersed as shown in Figure 6.8. The results are not surprising in that the attenuation for a bimodal structure increases with the area fraction of large grains. This suggests that a rule of mixtures weighted by the area fraction of large grains can provide an appropriate description of the attenuation from a bimodal grain structure.  Figure 6.13 FE simulated attenuation spectra from 41 µm and 131 µm bimodal grain size distribution domains of various area ratio 89   Two additional sets of simulations are conducted to further investigate the characteristics of attenuation in bimodal grain structures. It is especially important to evaluate bimodal grain size distributions in which the two grain families are further apart in terms of grain size. Such grain size distributions can represent cases where one family of grains is strictly in the Rayleigh scattering regime and the other approaches the transition between the Rayleigh and the stochastic scattering regimes. Figure 6.14 shows the attenuation calculated for a bimodal grain size distribution where the mean grain sizes are 10 µm and 131 µm (Figure 6.14a)) and a second distribution where the mean grain sizes are 41 µm and 240 µm (Figure 6.14b)). In both these distributions the total area fraction of large grains is set to 52%. For sake of comparison, the attenuation simulated for relevant narrow grain size distributions are inserted in Figure 6.14.   Figure 6.14 FE simulated attenuation spectra from bimodal grain size distribution domains of mean grain size populations a) 10 µm and 131 µm and b) 41 µm and 240 µm. Green lines represent best-fit 𝜶 = 𝒂𝟎 + 𝒃𝑭𝟑. 90   Here again, the attenuation calculated in bimodal grain structures lies in between the attenuation calculated for both unimodal grain structures. An exception is at approximately 15 MHz in Figure 6.14 once the narrow 41 µm and narrow 240 µm attenuation spectra overlap. At that moment, the bimodal grain structure attenuation is larger than either unimodal attenuation by approximately 0.2 dB/mm.   The trend of attenuation in bimodal grain structures presenting a non-singular frequency dependence continues for the results in Figure 6.14. The attenuation in Figure 6.14a) displays a smaller frequency dependence above approximately 15 MHz, which is similar to the attenuation of the narrow 131 µm mean grain size distribution. The attenuation in Figure 6.14b) displays a smaller frequency dependence above approximately 17 MHz. This transition occurs at a higher frequency than for the attenuation of the narrow 240 µm mean grain size distribution. This observation suggests a similar result to that of Section 6.2.2, i.e. the effect of large grains on attenuation is more pronounced when the small grain size population contributes less attenuation.   91   6.2.4 Area fraction rule of mixtures model  These sets of simulation are now suitable to test the assumption that was proposed by Nicoletti et al. [97] concerning the calculation of attenuation accounting for the presence of a distribution in grain size (Equation 2.20). In essence, Equation 2.20 modeled total attenuation as the sum of individual grain attenuation. The formalism can be adjusted by incorporating grain area weighting and is expressed as   𝜶(𝑭) = ∫ 𝑨(𝑫) 𝜶(𝑭, 𝑫) 𝒅𝑫∞𝟎  (6.2) Here, 𝐴(𝐷)𝑑𝐷 is defined as the area of grains of size between 𝐷 and 𝐷 + 𝑑𝐷. For the simulated bimodal grain size distributions, each family of grains possesses a narrow grain size distribution that does not overlap that of the other grain family. The model can thus be simplified as the sum of relative attenuation contribution from each grain family. This reformulation is expressed by   𝜶(𝑭) = ∑  (𝑨𝒋 𝑨𝒕𝒐𝒕𝒂𝒍⁄ ) ∗ (𝜶(𝑭, 𝑫))𝒋𝒋   (6.3) where 𝐴𝑗  is the total area occupied by each grain family, 𝑗 , in a domain, 𝐴𝑡𝑜𝑡𝑎𝑙  is the total domain area, and 𝛼𝑗  is the attenuation calculated for a domain consisting entirely of a grain family. 𝐴𝑗/𝐴𝑡𝑜𝑡𝑎𝑙 is in other terms the total area fraction of each grain family.   Figure 6.15 shows the attenuation calculated from Equation 6.3 in comparison with the attenuation calculated from FE simulations in cases of bimodal grain size distribution. The predicted attenuation is in excellent agreement with those of the FE simulation throughout the available frequency range. Area fraction therefore seems to be an appropriate weighting parameter for this rule of mixture in a 2-D simulation.  92     Figure 6.15 Area fraction rule of mixtures modeling for FE simulated attenuation spectra from bimodal grain size distribution domains of mean grain size populations a) 10 µm and 131 µm (b), c)) 41 µm and 131 µm with varying area ratio, d) 41 µm and 240 µm  93   6.3 Application to LUMet   6.4.1 Grain size calculation from FE simulated attenuation spectra  The grain size for all FE simulated attenuation spectra is calculated using the established methodology and applying the empirical calibration for austenite grain size. As this methodology is used in LUMet, grain size calculated in this manner is labelled as “𝐷𝐿𝑈𝑀𝑒𝑡”. Figure 6.16 compares 𝐷𝐿𝑈𝑀𝑒𝑡 calculated from FE simulated attenuation spectra to mean grain size calculated directly from grain area analysis of FE simulation domain grain structures, 𝐷𝐹𝐸 . Error bars are the standard deviation of calculated 𝐷𝐿𝑈𝑀𝑒𝑡  from each domain’s five simulated attenuation spectra using different crystallographic orientations. Figure 6.16a) displays the grain size comparison for unimodal grain size distributions and Figure 6.16b) that for bimodal grain size distribution. The AWGD (Equation 2.12) is chosen to calculate 𝐷𝐹𝐸  to better observe the differences between the Gaussian and log-normal distributions with identical EQAD described in Figure 6.6. In the case of narrow grain size distributions, AWGD was a factor of 1.01 larger than EQAD. For the wide Gaussian and log-normal grain size distributions, AWGD was a factor of 1.2 larger than EQAD. 94    Figure 6.16 Comparison between grain size calculated directly from FE domain grain structures and from application of Section 4.2.3 methodology to FE simulated attenuation spectra. Featured grain size distributions are a) unimodal with FE domain grain size in AWGD, b) bimodal with FE domain grain size in EQAD and AWGD. Green line depicts perfect agreement.   The grain sizes 𝐷𝐿𝑈𝑀𝑒𝑡 and 𝐷𝐹𝐸  agree very well up to 150 μm. Above 150 μm, 𝐷𝐿𝑈𝑀𝑒𝑡 is much lower than 𝐷𝐹𝐸 , which suggests that the LUMet empirical calibration is valid only up to about 150 µm mean grain size. This upper limit of mean grain size corresponds to a smaller frequency dependence of attenuation emerging below 15 MHz, as was observed in the unimodal grain structure attenuation plots in Figure 6.4.   In the case of bimodal grain size distribution, EQAD (black symbols) as well as AWGD (red symbols) are used to evaluate 𝐷𝐹𝐸  in Figure 6.16b). The agreement is better when correlating the calibration grain size 𝐷𝐿𝑈𝑀𝑒𝑡 with the AWGD rather than with the EQAD. The red point at 150 95   μm in 𝐷𝐹𝐸  corresponds to a bimodal structure containing 52% area as 240 μm grains, which does not agree well with the calibration as suggested from the unimodal grain size distribution study in Figure 6.16a). In summary, both the EQAD and the AWGD provide an appropriate estimate of the true mean grain size. The AWGD seems to be a better representative grain size for the cases of bimodal grain size distribution, i.e. in case of severe abnormal grain growth.   The excellent agreement between 𝐷𝐿𝑈𝑀𝑒𝑡  and 𝐷𝐹𝐸  for both unimodal and bimodal grain structures indicates that the LUMet methodology is applicable to various grain size distributions. Further, the excellent agreement constitutes a quantitative validation of the FE modeling framework with respect to LUMet methodology. Recall that the empirical calibration for austenite grain growth was constructed using extensive experimental work involving interrupted heat treatment and quantitative metallographic analysis. The work presented here shows that FE simulation can substitute costly experiments and provide rapidly a sense of the complicated scattering behavior in complex grain structures. The FE simulation is an efficient alternative for the development of future calibrations between ultrasonic attenuation and mean grain size.  6.4.2 Comparison of attenuation from domains with similar calculated grain size  Throughout Chapter 6, it was shown that attenuation in bimodal grain structures displayed non-singular frequency dependence that distinguished them from the attenuation of unimodal grain structures.  Now that grain size is calculated for all simulated attenuation spectra (Figure 6.16), it is of interest to compare the attenuation for domains with similar calculated grain size but different grain size distribution. In this way, the influence of a grain size distribution in the 96   calculations of attenuation and grain size can be visualized. Figure 6.17a) and Figure 6.17b) show the attenuation for domains with calculated grain size of approximately 41 μm and 56 μm, respectively.  Figure 6.17 Comparison between FE simulated attenuation spectra from domains with similar LUMet grain size of approximately a) 41 μm, and b) 56 μm  In both Figure 6.17a) and Figure 6.17b), the attenuation from unimodal grain structures and bimodal grain structures follow similar single-frequency dependencies up to approximately 15 MHz. Above 15 MHz, the attenuation in the bimodal grain structures display a smaller frequency dependence and diverge from the attenuation of the unimodal grain structures. As was observed in Figure 6.4, 15 MHz is when the attenuation in a unimodal 131 μm grain structure transitions towards the stochastic scattering regime. Both bimodal grain structures in Figure 6.17 have a large grain population of mean size 131 μm. This coincidence suggests the possibility of 97   correlating an attenuation spectrum inflection point to a domain’s maximum grain size, in accordance with the hypotheses of Smith and Nicoletti. [7,96,97].  Recall however that the maximum frequency used for grain size calculation was 15 MHz. The similarity of singular frequency dependence below 15 MHz produces similar calculated values of grain size (𝐷𝐿𝑈𝑀𝑒𝑡 ) as shown in Table 6.2. Calculation of 𝐷𝐿𝑈𝑀𝑒𝑡  is given by the software CTOME with an uncertainty expressed in µm. This uncertainty is estimated by the weighted average of the standard fit error for the nonlinear regression applied to an attenuation spectrum. In other words, the grain size uncertainty is an indicator of how well the equation 𝛼 = 𝑎0 + 𝑏𝐹3 fits an attenuation spectrum. Details of how uncertainty is quantified are clarified further in the CTOME documentation [82].  Table 6.2 shows the uncertainty, in µm and normalized as a fraction of 𝐷𝐿𝑈𝑀𝑒𝑡, for the grain size calculations using the FE simulated attenuation spectra in Figure 6.17. Recall that each domain had five different simulations where five different sets of random crystallographic orientations are allocated to the grains. The reported grain size uncertainty is the mean of five grain size uncertainties from each domain`s five simulated attenuation spectra.       98   Table 6.2 Grain size and uncertainty from calculation using the FE simulated attenuation spectra in Figure 6.17 Domain 𝐷𝐿𝑈𝑀𝑒𝑡 (μm) 𝐷𝐿𝑈𝑀𝑒𝑡 uncertainty (μm) / 𝐷𝐿𝑈𝑀𝑒𝑡 (μm/μm) Narrow 41 μm 41 4.3 0.11 Bimodal (10 μm. 131 μm) Area Ratio 85:15 38 7.9 0.21 Narrow 56 μm 53 7.9 0.15 Bimodal (41 μm. 131 μm)  Area Ratio 77:23 58 9.7 0.17  The uncertainty values for the bimodal grain structure attenuation spectra in Figure 6.17 are slightly higher than the uncertainty for unimodal grain structures of similar 𝐷𝐿𝑈𝑀𝑒𝑡. The higher uncertainties are due to slight deviations from singular frequency dependence in the attenuation from bimodal grain structures below 15 MHz. For example, the bimodal (10 μm, 131 μm) domain in Figure 6.17a) displays smaller frequency dependence above 10 MHz. This smaller frequency dependence appears as a slight attenuation dip of 0.3 dB/mm compared to the attenuation in the unimodal 41 μm grain structure in Figure 6.17a).  99   6.4.3 Grain size calculation uncertainty  With the large number of simulated attenuation spectra presented in this study, it is interesting to extend the grain size uncertainty analysis in Table 6.2 to a broader consideration of domains. Figure 6.18 plots normalized grain size uncertainty and 𝐷𝐿𝑈𝑀𝑒𝑡 for all the grain structures in this study.   Figure 6.18 Plot of normalized grain size uncertainty and grain size from calculation on FE simulated attenuation spectra 100   Uncertainty is relatively stable with values near 0.2 for 𝐷𝐿𝑈𝑀𝑒𝑡 up to 60 μm. Those domains have at least 75% of their area comprised of grains smaller than 60 μm, which means a large contribution to their attenuation is well described by power-law frequency dependence (Figure 6.4). The two domains with at least 85% of their area comprised of 10 μm grains display a relatively large uncertainty compared to other domains with 𝐷𝐿𝑈𝑀𝑒𝑡 less than 60 μm. Figure 6.4 and Figure 6.10b) showed that these domains had very low attenuation values of less than 1 dB/mm. When attenuation values are so low, any slight deviation disproportionately increases the nonlinear regression fit error and uncertainty.   Above 𝐷𝐿𝑈𝑀𝑒𝑡 of 60 μm, uncertainty increases as grain size becomes of a comparable value to ultrasound wavelength within the 2 – 15 MHz bandwidth of analysis. As was described in Section Error! Reference source not found., a smaller frequency dependence emerges below 15 MHz for mean grain sizes above 130 μm that is not described well by the existing calibration. Consequently, the uncertainty for bimodal grain structures containing 131 μm and 240 μm mean grain size families is higher than the uncertainty for unimodal grain structures of similar 𝐷𝐿𝑈𝑀𝑒𝑡. The domains containing a grain size family of 240 μm display much higher uncertainty than other grain structures with 𝐷𝐿𝑈𝑀𝑒𝑡 of approximately 90 μm. This is due to the smaller frequency dependence of attenuation emerging well below 15 MHz for structures containing 240 μm grains, as was shown in Figure 6.4.   Continuing the investigation of mesh template size from Section 6.1.2, the uncertainty for unimodal 131 µm mean grain size structures is presented in Figure 6.18 for domains of 2 mm and 8 mm thickness. The 𝐷𝐿𝑈𝑀𝑒𝑡 from the domains of 2 mm and 8 mm thickness are respectively 101   89 μm and 135 μm. Considering the known mean grain size of 131 μm in this unimodal grain structure, the mesh template of 8 mm thickness produces a more accurate result for 𝐷𝐿𝑈𝑀𝑒𝑡 . Additionally, the uncertainty for the 8 mm thickness domain is less than the uncertainty for the 2 mm thickness domain. These results reinforce the conclusion that a mesh template of at least 8 mm thickness is necessary for FE simulations using unimodal grain structures with mean grain size larger than 130 μm.   Excluding the outlier points corresponding to the domains containing a 240 μm grain size family and the domain of unimodal 131 μm grain size with 2 mm mesh template thickness, the uncertainty from bimodal grain structures in Figure 6.18 is slightly higher than the uncertainty from unimodal grain structures. This result confirms the trend in Table 6.2, and introduces a quantitative distinction between the attenuation in a unimodal and bimodal grain structure of similar 𝐷𝐿𝑈𝑀𝑒𝑡. However, the observation in Figure 6.17 that slightly increased uncertainty arises from slight deviations from singular frequency dependence suggests that this distinction is of a small magnitude that is difficult to measure using the existing LUMet methodology.   If a higher maximum frequency is used for analysis, i.e. 20 MHz instead of 15 MHz, then the pronounced smaller frequency dependence in bimodal grain structures is better accounted for. Such an analysis would increase the grain size uncertainty for bimodal grain structures, but also alter the calculated 𝐷𝐿𝑈𝑀𝑒𝑡  and compromise the agreement between 𝐷𝐹𝐸  and 𝐷𝐿𝑈𝑀𝑒𝑡  in Figure 6.16. Future work involves improving methodology to either identify or account for the pronounced smaller frequency dependence of attenuation in bimodal grain structures. Such a development may involve incorporating a more complex expression for scattering attenuation.  102   Chapter 7: Conclusion  7.1 Summary  This thesis contributes understanding to the ultrasonic attenuation response measured from various grain size distributions, such as bimodal distributions that are a product of abnormal grain growth. The evolution of laser ultrasonic austenite grain size measurements is studied for isothermal heat treatments conducted on A36 steel. A rapid grain growth stage is observed between stages of limited grain growth during 1000°C isothermal holding. At the position of laser ultrasonic measurement, optical microscopy on etched cross-sections reveals heterogeneous microstructures corresponding to abnormal grain growth during the rapid grain growth stage. The heterogeneous microstructures exhibit variability on a scale larger than the generation laser spot size and variability amongst tests. This variability is a reminder that the laser ultrasonic method probes a finite sampling volume that might not be representative of the entire sample in abnormal grain growth scenarios. Metallographic measurement of large cluster fraction correlates well to a normalization approximation from laser ultrasonic grain size measurements.   Attenuation spectra are simulated for domains of narrow, log-normal, and bimodal grain size distributions using a finite element modeling framework. A transition of attenuation frequency dependence is observed when grain size approaches ultrasound pulse wavelength, suggesting a transition from Rayleigh scattering to stochastic scattering. Attenuation spectra are sensitive to the positioning of large grain clusters with respect to the simulation’s spatially distributed ultrasound pulse amplitude but relatively insensitive to the positioning of grains within a domain. 103   Although lasers generally emit beams with a Gaussian spatial profile of intensity, the extent to which the experimental amplitude spatial distribution is represented by the FE simulation is unknown due to the nonlinear nature of the laser ablation process [110]. Nevertheless the FE simulation result represents an additional consideration of uncertainty for the LUMet methodology if large grain clusters are present and ultrasound pulse amplitude is spatially distributed. Attenuation in bimodal grain structures is bounded by the attenuation of its constituent grain families. An area fraction rule of mixtures model is used to describe the simulated attenuation spectra of bimodal grain size distribution domains.  Mean grain size is calculated from simulated attenuation spectra using the fitting and empirical calibration methodology employed in LUMet. These mean grain sizes correlate well with mean grain sizes calculated from the simulation domain. The finite element modeling framework is therefore validated as a supplementary option for development of future calibrations between ultrasonic attenuation and mean grain size. AWGD and EQAD are both capable measures of mean grain size for correlation between attenuation spectra and grain size, and AWGD is slightly better at accounting for log-normal or bimodal grain size distributions containing large grains. Comparison between unimodal and bimodal grain structures of similar 𝐷𝐿𝑈𝑀𝑒𝑡   reveal that attenuation in bimodal grain structures display slight deviations from singular frequency dependence. However, these deviations are of a small magnitude that is difficult to measure using the existing LUMet methodology.     104   7.2 Future work  Although experimental results correlated a rapid grain growth stage of laser ultrasonic measurements to abnormal grain growth, further work is necessary to understand the emergence of abnormal grain growth during isothermal heating of A36 steel. Higher heating rates and longer holding times could be employed to investigate the possibility of abnormal grain growth at isothermal holding temperatures above and below 1000°C, respectively.  An incomplete aspect of the experimental results was the measurement of large grain clusters instead of austenite grain size. Prior austenite grain boundaries must be revealed in order to measure the austenite grain size and validate laser ultrasonic measurements. Water could be used instead of helium gas to obtain quenched microstructures that are more conducive for prior austenite grain etching.   Regarding finite element simulation results, simulated attenuation spectra for unimodal mean grain size domains between 140 μm and 200 μm are necessary to clarify the upper grain size limit for the existing empirical calibration. Existing modeling of attenuation can be refined by accounting for the smaller frequency dependence exhibited from bimodal grain structures possessing large grains. 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