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Laser ultrasonic investigations of recrystallization and grain growth in cubic metals Keyvani, Mahsa 2018

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 Laser Ultrasonic Investigations of Recrystallization and Grain Growth in Cubic Metals  By   Mahsa Keyvani   B.Sc., University of Tehran, 2008 M.Sc., University of Tehran, 2010  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF   DOCTOR OF PHILOSOPHY  in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES  (Materials Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)  April 2018  © Mahsa Keyvani, 2018  ii  Abstract This study evaluates the applicability of laser ultrasonics for metallurgy (LUMet) as a non-destructive tool in measurement of grain size evolution and quantifying recrystallization in FCC polycrystalline metallic materials. A systematic investigation has been conducted to correlate the ultrasonic attenuation parameter with the effective average grain size determined by metallography in two cobalt-based superalloys and pure copper.  To correlate the ultrasonic attenuation parameter with the metallographic grain size, a series of thermo-mechanical treatments were carefully designed to generate samples with different average grain sizes. Equivalent area diameter (EQAD) and area weighted grain diameter (AWGD) including twin boundaries were selected as the measures of average grain size. The developed correlation for cobalt superalloys was used to monitor the recrystallization kinetics of cold-rolled and hot deformed specimens in real time through the refinement of the mean grain size as well as the grain growth kinetics. Furthermore, it was found that when a substantial tail exists in the microstructure, the AWGD-based correlation provides a better estimation of the average grain size than the EQAD-based correlation since the former changes according to the changes in grain size distribution. Moreover, the finite element modelling of wave propagation revealed that twin boundaries have similar scattering behavior as other high-angle grain boundaries. This suggests that the LUMet technique cannot be used to extract the fraction of twins.  A versatile method was then introduced to harmonize all the existing empirical equations to evaluate the grain size change in FCC metals. It was observed that the amount of grain scattering is controlled by the single crystal elastic constants which should be known apriori. The harmonized equation can be used to measure the grain size evolution in other metals without the need to develop a new calibration or at least reduces the number of experiments and labor-intensive ex-situ characterizations required for the design of a new calibration.   iii  Lay summary Metals and alloys are widely used in many industrial applications. On the microscopic scale, metallic structures consist of small crystallites or grains. The grain structure governs the performance of engineering products and is tuned by high-temperature processing treatments. Laser Ultrasonics for Metallurgy (LUMet) is a new technique that is capable of tracking microstructure evolution in-situ during thermo-mechanical processing. In this technique, microstructure features such as grain size are correlated with ultrasound properties. In the present work, application of the LUMet system is extended to measure grain size evolution in cobalt superalloys and pure copper during processing. This sensor significantly reduces the necessity of ex-situ and labor-intensive microstructure characterization and thus provides a tool to expedite optimization of processing routes. Based on the results obtained in this study, a generalized method is introduced to measure grain size in a wide range of metals and alloys using laser ultrasonics.            iv  Preface The present work was conducted in the Department of Materials Engineering at the University of British Columbia with financial support received from the Natural Sciences and Engineering Research Council (NSERC) of Canada. The majority of experiments and simulations were conducted, analyzed, and completed by the author. My supervisors, Dr. Matthias Militzer and Dr. Thomas Garcin, were involved and assisted me to complete the present thesis. Analysis of the ultrasonic waveforms was carried out using the CTOME software designed and developed by Dr. Garcin. Additionally, Dr. Garcin developed the finite element modeling of wave propagation and provided guidance in generating the results presented in Figure 8.7.  A part of chapter 5 has been published in a refereed journal paper: M. Keyvani, T. Garcin, D. Fabrègue, M. Militzer, K. Yamanaka, and A. Chiba, “Continuous Measurements of Recrystallization and Grain Growth in Cobalt Superalloys”, Metall. Mater. Trans. A, pp. 2363–2374, 2017. Dr. Damien Fabrègue was involved in the discussions held to write the journal paper. Dr. Akihiko Chiba, Dr. Damien Fabrègue, and Dr. Kenta Yamanaka collaborated to provide the cobalt superalloy specimens. Dr. Kenta Yamanaka machined the strip specimens used to conduct the tests reported in the paper. All the co-authors read and edited the manuscript.    v  Table of contents Abstract .......................................................................................................................................... ⅱ Lay summary ................................................................................................................................ ⅲ Preface........................................................................................................................................... ⅳ Table of contents  ........................................................................................................................... ⅴ List of tables ................................................................................................................................... ⅹ List of figures ................................................................................................................................ ⅺ List of symbols  ........................................................................................................................... ⅺⅹ List of abbreviations  ................................................................................................................. ⅹⅹⅴ Acknowledgments .................................................................................................................... ⅹⅹⅴⅰ Chapter 1     Introduction  .............................................................................................................. 1 Chapter 2     Literature review  ...................................................................................................... 4      2.1 Introduction ......................................................................................................................... 4      2.2 Propagation of ultrasound waves in polycrystalline materials ........................................... 5           2.2.1 Ultrasonic velocity ................................................................................................... 5           2.2.2 Ultrasonic attenuation .............................................................................................. 8      2.3 Application of contact ultrasonics for material characterization ...................................... 11      2.4 Laser generation and detection of ultrasound pulse .......................................................... 17 vi       2.5 Laser ultrasonic characterization of metallurgical phenomena in metals ......................... 19           2.5.1 Grain size and grain growth studies ....................................................................... 19           2.5.2 Recovery ................................................................................................................ 25           2.5.3 Recrystallization .................................................................................................... 29           2.5.4 Phase transformation .............................................................................................. 32      2.6 Summary ........................................................................................................................... 35 Chapter 3     Scopes and objectives .............................................................................................. 36 Chapter 4     Methodology ........................................................................................................... 37      4.1 Materials ........................................................................................................................... 37      4.2 Processing set up ............................................................................................................... 38           4.2.1 Gleeble and Laser Ultrasonics for Metallurgy (LUMet) system ........................... 38           4.2.2 Instron tensile testing machine and DIC camera ................................................... 40      4.3 Processing routes .............................................................................................................. 42           4.3.1 Thermal treatments of cobalt superalloys .............................................................. 42           4.3.2 Thermo-mechanical treatments of cobalt superalloys ........................................... 44           4.3.3 Thermal treatments on copper ............................................................................... 48           4.3.4 Strain annealing treatments on copper ................................................................... 49      4.4 Microstructure characterization ........................................................................................ 50           4.4.1 Sample preparation ................................................................................................ 50 vii            4.4.2 Scanning Electron Microscopy (SEM) .................................................................. 51           4.4.3 Electron Back Scatter Diffraction (EBSD) mapping ............................................. 51           4.4.4 Quantification of metallographic grain size ........................................................... 52           4.4.5 X-ray diffraction (XRD) ........................................................................................ 54           4.4.6 Uncertainties in microstructure measurements ...................................................... 54      4.5 Ultrasonic waveform analysis ........................................................................................... 55      4.6 Finite element simulation of wave propagation ................................................................ 57 Chapter 5     Kinetics of recrystallizaion and grain growth in cold rolled cobalt superaloys ...... 63      5.1 Introduction ....................................................................................................................... 63      5.2 Microstructure evolution during annealing ....................................................................... 63      5.3 Laser ultrasonic grain growth measurements ................................................................... 69           5.3.1 Laser ultrasonic grain size ..................................................................................... 69           5.3.2 Recrystallization during continuous heating .......................................................... 72           5.3.3 Isothermal grain growth ......................................................................................... 74      5.4 Laser ultrasonic grain size measurement precision .......................................................... 75      5.5 Grain growth modelling .................................................................................................... 76      5.6 Summary ........................................................................................................................... 79 Chapter 6     Recrystallization and grain growth in hot-deformed L605 cobalt superalloy ......... 80      6.1 Introduction ....................................................................................................................... 80 viii       6.2 High temperature flow behavior ....................................................................................... 80      6.3 Laser ultrasonic grain size after hot deformation ............................................................. 82      6.4 Microstructure evolution after hot deformation ................................................................ 83      6.5 Calculation of softening fraction ...................................................................................... 89      6.6 Relation between laser ultrasonic grain size and static recrystallization behavior ........... 92      6.7 Summary ........................................................................................................................... 93 Chapter 7     Laser ultrasonic response in pure copper ................................................................ 94      7.1 Introduction ....................................................................................................................... 94      7.2 Laser ultrasonic response in a homogeneous microstructure ........................................... 94      7.3 Development of a graded microstructure .......................................................................... 97      7.4 Laser ultrasonic response in a graded microstructure ..................................................... 103      7.5 Summary ......................................................................................................................... 106 Chapter 8     Discussion ............................................................................................................. 108      8.1 Harmonization of grain size measurements with laser ultrasonics ................................. 108      8.2 Numerical investigation of grain boundary type effects on ultrasonic attenuation ........ 121 Chapter 9     Conclusions and future work................................................................................. 129      9.1 Conclusions ..................................................................................................................... 129      9.2 Suggestions for future work ............................................................................................ 132 Bibliography .............................................................................................................................. 134 ix  Appendix 1 ................................................................................................................................. 151                 x  List of tables   Table 4.1 Chemical composition of cobalt superalloys being used in this research (wt. %)  38 Table 4.2 Detailed heat treatment conditions (WQ: water quenched; HQ: Helium quenched) 44 Table 4.3 Detailed thermo-mechanical treatments employed for compression tests on L605 47 Table 4.4 Material properties of copper at room temperature [53] 60 Table 5.1 Metallographic grain size data for L605 and CCM samples rapidly cooled from isothermal heat treatments 67 Table 5.2 Numerical values of the fit parameters used in the grain growth model 78 Table 6.1  Metallographic grain size data with twins for as-received and deformed specimens helium quenched from isothermal holding after deformation 88 Table 6.2 Maximum true stress, yield stress values and calculated softening fraction from double-hit experiments 91 Table 7.1 Metallographic grain size parameters for sheet samples annealed at various temperature for one hour 95 Table 7.2 Microstructure parameters measured from EBSD maps shown in Figure 7.6 101 Table 8.1 Single crystal elastic constants of FCC cobalt, 𝛾-iron, and copper and compliance constants of nickel superalloys reported in literature 110 Table 8.2 𝜀𝑙𝑅(𝑇) and 𝑣𝑙0(𝑇) for nickel, cobalt, and copper at 27°C and for 𝛾-iron at 1155°C  112 Table 8.3 Microstructure parameters of the L1050_300s and L1100_1000s samples measured from EBSD maps 125 Table 8.4 Scaled microstructure parameters used for the FE simulations 126    xi  List of figures    Figure 2.1 Schematics of wave scattering; velocity of ultrasound changes at boundaries.   9 Figure 2.2 Effective grain size determined from attenuation measurement versus metallographic ferrite grain size in A36 and A242 steel plates [69]. 13 Figure 2.3 a) Attenuation at 40 MHz versus temperature. b) Exponent n (red line) versus temperature. % recrystallization determined from the JMAK equation (solid black curve) and from optical metallography (black dotted curve) are inserted [73]. 15 Figure 2.4 Ultrasonic velocity versus annealing time measured at different frequencies at room temperature [10]. 16 Figure 2.5 Schematics of wave generation by a) thermo-elastic, and b) ablative mechanisms [6]. 18 Figure 2.6 Comparison of austenite grain size measured by LUMet and metallography (large solid square symbols) for A36 steel austenitized at 1100°C [2]. 20 Figure 2.7 𝑏 parameter determined by the single-echo technique as a function of metallographic grain size for seamless tubes [61]. 22 Figure 2.8 Comparison between grain sizes measured by metallography (points) with those obtained from LUMet experiments (lines) for X80 linepipe steel [36]. 23 Figure 2.9 a) Variation of 𝑏 with time during annealing of INCONEL 718 at 1050 ⁰C [84]. b) Large grains fraction with respect to annealing time calculated by applying a lever rule to laser ultrasonic grain size data and fitted grain growth models [37]. 24 Figure 2.10 a) Correlation between LUS parameter and metallographic grain size in 316 stainless steel. b) Continuous measurement of grain growth during isothermal holding at 1000°C and 1050°C using the established correlation [3]. 25 Figure 2.11 Normalized frequency as a function of grain size in low carbon steels (dashed lines are experimental values and solid lines are those from theoretical 26 xii  calculations). Bold line connects the predicted grain size at each frequency. The specimen grain size was 7 μm [31]. Figure 2.12 a) Fractional velocity change and b) ultrasonic attenuation at 8 MHz measured after deformation at 550°C, 730°C, and 800°C to a total true strain of 0.15 and strain rate of 0.1 s-1. c) Evolution of dislocation density after deformation at 550°C calculated from ultrasonic velocity and attenuation and stress values using equations (2.19), (2.20), and (2.21) [25]. 28 Figure 2.13 Variation of velocity during recrystallization of IF steel [96]. 30 Figure 2.14 Recrystallization kinetics of cold worked AA5754 Aluminum alloy measured by laser ultrasonics, metallography and mechanical testing [28]. 31 Figure 2.15 Effect of strain on grain size evolution after hot deformation of a model steel [33]. 32 Figure 2.16 Evolution of ultrasonic velocity during continuous cooling and heating of 1020 steel at a rate of 5°C/s in the range 500°C to 1000°C [13]. 33 Figure 2.17 a) Temperature dependence of velocity for different phases in steel. b) Fraction austenite measured by laser ultrasonics and dilatometry during continuous cooling in a low alloy steel [14]. 34 Figure 4.1 a) Schematic representation of principles of the LUMet technique [37]. b) Optical set up used in the two-wave mixing interferometer [107]. 40 Figure 4.2 Instron tensile testing machine and the DIC camera configuration. The camera was set up to cover 30 mm of the sample gauge length. 41 Figure 4.3 a) Schematic of sheet specimens. b) The set up used to perform thermal treatments. 43 Figure 4.4 Schematics of thermal treatments conduced on L605 and CCM alloys. 43 Figure 4.5 a) Polishing kit used for faceting L605 cylindrical specimens. b) Cylindrical specimen was placed inside the polishing kit. c) Faceted samples. 45 Figure 4.6 a) Schematic of faceted cylindrical specimen prepared for hot compression tests. b) Experimental configuration image during the test. 46 Figure 4.7 Schematic illustration of thermo-mechanical treatments carried out on the L605 alloy in the Gleeble system. 47 xiii  Figure 4.8 Heat treatments conducted on 99.998% copper in a tubular furnace. Laser ultrasonic measurements were carried out in the Gleeble system at room temperature. 48 Figure 4.9 Geometry of tapered tensile specimens. Dimensions are in mm. Tapered samples scanned with LUMet system along the gauge before and after strain annealing treatment. 49 Figure 4.10 Strain annealing treatment applied on the tapered copper specimens to generate a graded microstructure. 50 Figure 4.11 BSE micrograph with marked grain boundaries of CCM alloy annealed at 1100°C for 600 s a) excluding twin boundaries and b) including twin boundaries. 54 Figure 4.12 Ultrasonic waveform generated from a cobalt superalloy at room temperature. Here, the first echo is selected for calculation of velocity and attenuation. 56 Figure 4.13 Attenuation spectrum determined by single-echo technique for a sheet sample of cobalt superalloy during isothermal holding at 1100°C. 57 Figure 4.14 Schematic of the mesh applied to the FE template in the CTOME software. 59 Figure 4.15 Finite element template used for wave propagation simulation. The microstructure is generated including twin boundaries. Colors represents longitudinal velocity in Y direction.  60 Figure 4.16 Ricker wavelet used to impose a displacement on top boundary of simulation box in a) time domain, and b) frequency domain. 61 Figure 5.1 Inverse Pole Figure (IPF) map of a) as-received cold rolled L605 alloy, b) as-received cold-rolled CCM alloy , c) L605 sample heat treated for 5 s at 1100°C, and d) the associated Kernel average misorientation (KAM) distributions. 64 Figure 5.2 BSE micrographs of L605 samples held at a) 1150 °C for 200 s, b) 1150 °C for 1000 s, c) 1200 °C for 100 s, d) 1200 °C for 1000 s, e) 1200 °C for 3 h, and f) 1200 °C for 18 h.  65 xiv  Figure 5.3 BSE micrographs of CCM samples held at a) 1100 °C for 600 s, b) 1150 °C for 200 s, c) 1150 °C for 600 s, d) 1150°C for 1000 s, e) 1200 °C for 100 s, and f) 1200 °C for 1000 s. 66 Figure 5.4 a) Correlation between different grain size parameters extracted from metallographic analysis and b) Cumulative area fraction with respect to the reduced grain size (D/EQAD) for the selected L605 samples shown in Table 5.1. 68 Figure 5.5 a) Grain size (EQAD) correlation with the parameter 𝑏 measured at room temperature. Three samples were used as reference, L605 held at 1100 °C for 5s (circles), 600 s (squares), and at 1200 °C for 100 s (triangles). Vertical error bars are 5.6% of the averaged value. b) Temperature contribution to the ultrasonic grain size parameters for two CCM samples. The solid circle is the reference point at room temperature. 71 Figure 5.6 a) Evolution of the grain size (EQAD) measured in-situ for the CCM cobalt superalloy at 1150°C by LUMet (lines) and by metallography (open squares).b) Signal to noise ratio evaluated for test 3.  72 Figure 5.7 Evolution of grain size (EQAD) during heating-isothermal holding tests at a rate of 50°C/s for L605 (open squares) and CCM (open triangles) superalloys.  73 Figure 5.8 In-situ measurement of grain size during isothermal annealing using the established calibration in a) L605 and b) CCM superalloys. 75 Figure 5.9 Normalized attenuation parameter 𝑏 with respect to the relative count measured for 60 waveforms at the same position at room temperature for L605 samples with different average grain sizes. 76 Figure 5.10 Grain growth model applied to the grain size evolution up to 1000 s measured during isothermal annealing in a) L604 and b) CCM superalloys. 78 Figure 6.1 a) Diametric true stress versus diametric true strain after friction correction for L605 deformed at elevated temperatures and b) work hardening rate versus true strain calculated from the friction-compensated stress-strain curves. 81 xv  Figure 6.2 Evolution of the mean grain size after deformation at a) 1000°C, b) 1050°C, and c) 1100°C measured by laser ultrasonics (lines). Open circles represent the mean grain size defined by metallography. The time scale refers to the time after deformation. 83 Figure 6.3 IPF maps of a) as-received L605, b) sample deformed and helium-quenched 1 second after deformation at 1000 °C, and c) 1100 °C. GOS maps of d) as-received specimen, e) sample deformed at 1000 °C, and f) 1100 °C. 84 Figure 6.4 IPF maps of specimens deformed and isothermally held at 1000 °C for a) 15 min, and b) 30 min. c) and d) GOS maps of specimens shown in Figure 6.4.a and Figure 6.4.b. 85 Figure 6.5 IPF maps of specimens deformed and isothermally held at a) 1050 °C for 300 s, and b) 1100 °C for 73 s. c) and d) GOS maps of specimens shown in Figure 6.5.a (c) and Figure 6.5.b (d). 86 Figure 6.6 Back Scattered Electron (BSE) micrographs of specimens deformed and isothermally held for 30 min at a) 1050°C, and b) 1100°C. 87 Figure 6.7 Correlation between mean grain sizes (EQAD) measured from BSE micrographs and EBSD maps. 88 Figure 6.8 GOS distributions of the selected L605 specimens. 89 Figure 6.9 Friction-corrected true stress true strain curves of interrupted compression tests on L605 samples at a) 1000°C, and b) at 1050°C. Second hits were conducted after 15 and 30 min holding at 1000°C and after 300 s of holding at 1050°C. 91 Figure 7.1 Back-scattered contrast micrographs of sheet specimens annealed at a) 360°C, b) 400°C, c) 500°C, d) 600°C, and e) 700°C for one hour. 95 Figure 7.2 Cumulative volume fraction versus reduced grain size (D/EQAD) in samples with a homogeneous microstructure. 96 Figure 7.3 Square root of attenuation parameter 𝑏 measured at room temperature versus square root of the relative grain size change (EQAD) in Cu samples with a homogeneous microstructure. 97 xvi  Figure 7.4 Inverse Pole Figure (IPF) map in RD-TD plane of the a) as-received and b) annealed conditions at 400°C for one hour. c) True stress-true strain curves of the respective conditions. 98 Figure 7.5 Engineering plastic strain measured with the DIC technique along the sample gauge length at the end of the tensile test.  99 Figure 7.6 PF maps of strain annealed specimen at positions of a) 0 mm, b) 6 mm, c) 12 mm, d) 16 mm, e) 30 mm, and f) 40 mm. 100 Figure 7.7 GOS maps of strain-annealed specimen at positions of a) 0 mm, b) 6 mm, c) 12 mm, d) 16 mm, e) 30 mm, and f) 40 mm. 101 Figure 7.8 Partitioned grain size distribution of strain-annealed specimen at positions of a) 6 mm, b) 12 mm, c) 16 mm, d) 30 mm, and e) 40 mm. 103 Figure 7.9 a) Square root of 𝑏 parameter measured along the gauge of tapered samples in the as-received, annealed, deformed (tensile test 1 and 2), and strain annealed conditions. Red solid lines are the two linear fits used to apply the lever rule method. b) Evolution of the fraction recrystallized obtained from LUMet parameter 𝑏 (open circles) and metallography (closed circles). 104 Figure 7.10 Correlation between the relative change in grain size and ultrasonic parameter 𝑏 measured at room temperature on sheet specimens with a homogeneous microstructure and on the strain annealed specimen with a graded microstructure using a) EQAD and b) AWGD as the measure of mean grain size. 106 Figure 8.1 Temperature dependence of scaling parameter in the Rayleigh regime for cobalt, nickel, 𝛾-iron, and copper. The data points were calculated using Equations (8.1) and (8.5), and the single crystal elastic constants reported in the respective metals. 111 Figure 8.2 a) Attenuation spectrum in the studied metals in the Rayleigh regime and b) Attenuation spectrum scaled by 𝜉𝑅(13).  112 Figure 8.3 Correlation between square root of laser ultrasonic parameter 𝑏 and square root of the relative change in square of grain size for a) steels in austenite 115 xvii  phase developed at 900, 1050, and 1300°C [158], b) nickel at 1050°C [37], c) cobalt at 900, 1050, and 1300°C [38], and d) copper at 27°C. Figure 8.4 Correlation between AWGD and EQAD values measured in a) Inconel, b) cobalt superalloys, and c) copper.  116 Figure 8.5 Square root of 𝑏 measured with laser ultrasonics versus square root of the relative change in grain size in terms of a) EQAD and b) AWGD in austenite [158], nickel [37], cobalt superalloys [38], and copper.  117 Figure 8.6 Square root of scaled attenuation parameter 𝑏 with respect to the relative change in square of grain size (AWGD) in the studied metals.  118 Figure 8.7 45° ODF intensity projections of a) Inconel, b) cobalt, and c) pure copper references.  119 Figure 8.8 Grain size evolution (AWGD) in a) L605 and b) CCM superalloys during isothermal holding at 1200°C. Grain size evolution in L605 after hot deformation at c) 1000°C, d) 1050°C, and e) 1100°C using the cobalt calibration and the harmonized Equation (8.10). 120 Figure 8.9 a) Microstructure with hexagonal grains generated by centroidal Voronoi tessellation. b) misorientation distribution of grain boundaries for microstructures with 0% (top) and 14.7% (bottom) twin boundaries. c) FE-simulated attenuation spectra of samples with different percent twin boundaries ranging from 0 to 25% [157]. 123 Figure 8.10 Grain map of the L1050_300s sample a) with and b) without twin boundaries. Grain boundary misorientation distributions c) including and d) excluding twin boundaries. The MacKenzie distribution is also included for comparison.  124 Figure 8.11 Grain map of the L1100_1000s sample a) with and b) without twin boundaries. Grain boundary misorientation distributions c) including and d) excluding twin boundaries The MacKenzie distribution is also included for comparison. 125 Figure 8.12 Voigt averaged longitudinal velocity in the y direction for the L1050_300s sample a) with and b) without twins. Position of the mirror plane was changed for each FE simulation. 126 xviii  Figure 8.13 Comparison between the FE simulated frequency dependence of attenuation for templates with and without twin boundaries for the a) L1050_300s and b) L1100_1000s samples. 127 xix  List of symbols   Symbol Description ?̅? arithmetic average of all the grain areas in a microstructure  𝐴 radius of the laser generation source 𝑎 Frequency-independent attenuation parameter 𝐴𝑡 total area of grains in a micrograph 𝐴(𝑡) amplitude of the Ricker signal  𝐴(𝑥) peak amplitude of the Ricker signal 𝐴𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 cross correlation amplitude 𝐴𝑀 and 𝐴𝑅 amplitude spectra of the same echo in the tested and reference waveforms  𝐴𝑛(𝑓, 𝑇) and 𝐴𝑚(𝑓, 𝑇) amplitude spectra of the 𝑛𝑡ℎ and 𝑚𝑡ℎ echoes at temperature 𝑇 in a waveform 𝑏 frequency-dependent attenuation parameter   𝐵 damping constant  𝑏𝑇𝑟𝑒𝑠𝑐𝑎 barreling factor determined under the Tresca criterion L longitudinal wave 𝑐 invariant anisotropy factor 𝐶 dislocation line tension  𝑐110  Hill-averaged 𝑐11 in an isotropic polycrystal xx  𝑐11, 𝑐12, 𝑐44 independent components of elastic stiffness tensor in a single crystal with cubic symmetry  〈𝑐11〉𝑉𝑜𝑖𝑔𝑡, 〈𝑐44〉𝑉𝑜𝑖𝑔𝑡 Voigt-averaged elastic constants in an isotropic polycrystal with cubic symmetry  𝑐𝑖𝑗𝑘𝑙(𝑚) elastic stiffness tensor in the 𝑚𝑡ℎ grain  〈𝑐𝑖𝑗𝑘𝑙〉𝑉𝑜𝑖𝑔𝑡 Voigt-averaged elastic stiffness tensor in an isotropic polycrystal 𝐷 grain size  𝐷0 average grain size of reference sample 𝐷𝑑 domain size  𝑑0 initial diameter of a cylindrical sample  𝑑𝑖 instantaneous diameter of a cylindrical sample measured by dilatometry Dmax maximum grain diameter 𝐸𝑠𝑎𝑡 Young’s modulus at saturation 𝑓 ultrasonic frequency 𝑓0𝑟 relaxation frequency 𝐹𝐹𝑇 fast Fourier transformation 𝐹𝑖  applied force by load cell of Gleeble system 𝐹𝑟𝑒𝑥 Recrystallized fraction 𝐹𝑠 softening fraction 𝐺 shear modulus ℎ thickness of a sheet specimen  xxi  ℎ0 initial height of a cylindrical sample before compression ℎ𝑓𝑖𝑛𝑎𝑙  final height of a cylindrical sample after compression  𝐼𝐹𝐹𝑇 inverse fast Fourier transformation 𝐼𝑠 saturation magnetization 𝑘 wave vector 𝑙 ultrasonic wavelength 𝐿 average pinning point separation 𝑚 friction coefficient  𝑀 Taylor factor 𝑚𝑖 mesh size  𝑛 scattering exponent 𝑁 number of grains 𝑃 compressive flow stress in absence of friction  𝑅0 initial radius of a cylindrical sample before compression  𝑅𝑒 electrical resistivity 𝑅𝑡ℎ𝑒𝑜𝑟𝑦 theoretical final radius of a cylindrical sample after compression  S shear wave 𝑠11, 𝑠12, 𝑠44 independent components of compliance tensor in a single crystal with cubic symmetry 〈𝑠11〉𝑅𝑒𝑢𝑠𝑠, 〈𝑠44〉𝑅𝑒𝑢𝑠𝑠 Reuss-averaged elastic constants in an isotropic polycrystal with cubic symmetry 𝑆𝐹 Fresnel parameter xxii  𝑠𝑖𝑗𝑘𝑙(𝑚) compliance tensor in the 𝑚𝑡ℎ grain 〈𝑠𝑖𝑗𝑘𝑙〉𝑅𝑒𝑢𝑠𝑠 Reuss-averaged compliance tensor in a random polycrystal 𝑇 temperature 𝑡𝑃𝐷 time after deformation 𝑡 isothermal holding time  𝑣𝑑(0) velocity in absence of dislocations 𝑣𝑑(ф) velocity for a dislocation density of ф 𝑉𝑔 grain volume 〈𝑣𝑙〉𝑖 longitudinal wave velocity in an isotropic polycrystal estimated using the Voigt or the Reuss averaging methods 𝑣𝑙0 Hill-averaged longitudinal wave velocity in an isotropic medium 〈𝑣𝑠〉𝑖  shear wave velocity in an isotropic polycrystal estimated using the Voigt or the Reuss averaging methods 𝑣𝑠0 Hill-averaged shear wave velocity in an isotropic medium 𝑣𝑇𝐸  velocity determined by the two-echo method 𝑋 recrystallized fraction 𝑧 wave propagation distance 𝛼 ultrasonic attenuation 𝛼𝐷 diffraction contribution to attenuation 𝛼𝑑𝑖𝑠𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛 attenuation induced by dislocations 𝛼𝐼𝐹 internal friction contribution to attenuation 𝛼𝑙𝑅 attenuation of a longitudinal wave in the Rayleigh regime xxiii  𝛼𝑙𝑆 attenuation of a longitudinal wave in the Stochastic regime 𝛼𝑚𝑒 magnetomechanical damping contribution to attenuation 𝛼𝑠𝑐 grain scattering induced attenuation Б Burgers vector Γ(𝑇) temperature contribution to attenuation function   ∆ℎ change in a cylindrical sample height after compression ∆𝑅 difference between the maximum radius and radius at a cylindrical sample surface after compression  𝛥𝑡 finite element simulation time step 𝜀𝑖 diametric true strain 𝜀𝑖𝑗(𝑚) strain in the 𝑚𝑡ℎ grain in the Reuss method 𝜀𝑘𝑙 applied strain 𝜀𝑙𝑅 relative inhomogeneity in the Rayleigh regime 𝜀𝑙𝑆 relative inhomogeneity in the Stochastic regime 𝜆 and µ Lamé moduli 𝜇𝑖 initial permeability 〈µ〉𝑅𝑒𝑢𝑠𝑠 and 〈𝜆〉𝑅𝑒𝑢𝑠𝑠 Reuss-averaged Lamé moduli 𝜆𝑠 magnetostriction constant 〈µ〉𝑉𝑜𝑖𝑔𝑡 and 〈𝜆〉𝑉𝑜𝑖𝑔𝑡 Voigt-averaged Lamé moduli ᴧ single crystal elastic anisotropy 𝜉𝑅 relative inhomogeneity factor in the Rayleigh regime 𝜌 density xxiv  𝜎 compressive flow stress corrected for friction   𝜎𝑓 relaxation stress 𝜎𝑖 diametric true stress 𝜎𝑚𝑎𝑥 maximum stress in the first hit of a double-hit test 𝜎𝑦 yield stress 𝜎𝑦1 yield stress in the first hit of a double-hit test 𝜎𝑦2 yield stress in the second hit of a double-hit test 𝜏 time delay between two successive echoes in a waveform 𝜏𝑖𝑗(𝑚) stress in the 𝑚𝑡ℎ grain in the Voigt method 𝜏𝑘𝑙 applied stress  ф dislocation density          xxv  List of abbreviations   AISI American iron and steel institute  AWGD area weighted grain diameter  BCC body-centered cubic  CCD charge-coupled device  CI confidence index DIC digital image correlation  EBSD electron backscatter diffraction EDM electro-discharge machine  EQAD equivalent area diameter  FCC face-centered cubic FEM finite element modelling  GOS grain orientation spread IPF inverse pole figure KAM kernel average misorientation  LUMet laser ultrasonics for metallurgy  MTEX microstructure and texture analysis  ODC orientation distribution coefficient SEM scanning electron microscopy   xxvi  Acknowledgments  First and foremost, I wish to thank my advisors, Professor Matthias Militzer and Dr. Thomas Garcin for their constant support, encouragement, and advice during my PhD studies. I would like to specially thank Professor Militzer for giving me a chance to work under his supervision in one of the most resourceful and dynamic research groups in the world. Many thanks go to Dr. Garcin for guiding and teaching me with tremendous patience throughout this research. He has mentored me to become an independent researcher and to never give up resolving challenges. Without their consistent help, I would never be able to go the distance. I would like to express gratitude to Professor Warren Poole for giving me guidance and support for the copper research project. I would like to thank Professor Damien Fabrègue for providing great ideas and advices for the cobalt superalloys research project. His enthusiasm for science and devotion to support our collaboration inspired me. I would like to thank Dr. Kenta Yamanaka for providing me with the cobalt superalloy test samples. I would also like to thank the staff of the Materials Engineering department for all their help and support, especially Ross Mcleod, David Torok, Carl Ng, Michelle Tierney, Mary Jansepar, Fiona Webster, Norma Donald, and Jacob Cable. Thank you Ross Macleod for keep reminding me that I have got what it takes to finish my PhD. Thank you Michelle Tierney for your support to resolve my administrative matters. I would like to thank Dr. Hamid Azizi Alizamini, Dr. Ghazal Nayyeri, Mojtaba Mansuri, Arthur Depres, and Jingqi Chen for their assistance on my research work. I would also thank my office mates in AMPEL 349 for creating a friendly atmosphere during the past couple of years.  I am very thankful to all my dear friends who supported me during this amazing journey. Special thanks go to my best friend ever, Dr. Ghazal Nayyeri, for her endless care and support from the minute I came to Canada. She always believed in me and inspired me to be a better person and taught me how to power through difficult times. I would also thank my other best friend Dr. Nastaran Arianpoo who constantly made me feel special and encouraged me particularly when I was writing my thesis. They are a part of my family. I cherish all the amazing times we spent together and all the happy and sad memories we share over the years. I would like to thank my dear friend Dr. Shima Karimi for her kindness and helps. Thanks to my friends who did not forget me after I migrated to Canada. Special thanks to Niloofar Eshghi and Dr. Azadeh Mobasher for xxvii  being my amazing best friends over the past 20 years always kind and supportive. Also, I would like to thank my dear friend Dr. Maryam Akhlaghi who has been a great inspiration for me. I wish to thank my amazing mother, Soheila, and my wonderful father, Ardeshir, who devoted and dedicated their lives to support me and my sisters. Many thanks to them for always encouraging me to pursue my dreams and for always believing in me. They showed me how to be patient, persistent, and positive in life. Without their tremendous kindness and every day support, this journey would not be possible. Soheila and Aredeshir, you are all my reasons. I would like to thank my uncle, Mohamad Khoshsiar, and my aunt, Setareh Khoshsiar, who taught me how to be strong and how to smile when there are least reasons to do so. I would like to thank my lovely sister, Mahboobeh, and my awesome brother-in-law, Javad Soleimani, for all their love and support. Thanks Mahboobeh for being a role model for me and always backing me up. I would like to thank my lively and energetic sister, Farangis, for always cheering me up and for always being there for me. Thanks for keep reminding me that life is a dance and we should dance gracefully as long as we can. Last but not least, I would like to thank my adorable niece, Saba, for coming into our lives. Watching you growing up every day learning new things and experiencing the world, has been filled my heart with everlasting joy.  1  Chapter 1 Introduction  Polycrystalline metals and alloys are widely used in a multiplicity of engineering applications including the construction, energy and transportation sectors. Microstructural features such as size, morphology, and size distribution of grains and crystallographic texture govern physical and mechanical properties. The concept of correlating microstructural features with material properties goes back to the late 19th century when the metallographic techniques were used for the first time in modern steel making research facilities [1]. The forming processes such as rolling and forging were improved and it was found that microstructure and consequently properties can be optimized through suitable thermo-mechanical treatments. Later on, physically based models have been developed to describe the evolution of microstructure with respect to process conditions by conducting thermo-mechanical simulations coupled with dilatometry measurements and labor intensive ex-situ quantitative metallographic techniques. To expedite process development aiming to obtain the desired physical and mechanical properties, rapid in-situ knowledge on microstructure is needed. Further, a number of microstructural parameters, such as the austenite grain size in low and ultra-low carbon steels, are challenging to measure using ex-situ methods. In-situ sensors capable of monitoring the evolution of microstructure can therefore bring more insight into the development of such process models. Among these sensors, ultrasonics has been of particular interest in non-destructive evaluation (NDE) of polycrystalline materials. Since an ultrasonic wave can propagate deeply into polycrystalline metals, ultrasonics has been a tremendously valuable technique to evaluate the change in microstructural features. For more than two decades, lasers have been used to generate and detect the ultrasonic wave required for the NDE examinations. The developed laser ultrasonic technology has now reached sufficient maturity to be used as a new tool for metallurgists. The Laser Ultrasonics for Metallurgy (LUMet) system was developed as an attachment to the Gleeble thermo-mechanical simulator. The LUMet technology is dedicated to the in-situ monitoring of microstructure evolution during thermo-mechanical treatments. The system is especially oriented for research in metallurgy and is capable of providing useful data in a wide range of materials and diverse sample geometries [2]–[4].  2  Ultrasonic waves are extensively used for flaw detection, thickness measurements and microstructure characterization of various materials. Traditionally, piezoelectric transducers are used for the generation and detection of ultrasound. A direct contact with the sample is often a limitation to application at high temperature. In the laser ultrasonic technology, lasers are used to remotely generate and detect a broadband ultrasonic pulse in the material. First, a short-pulse, high energy laser generates a broadband longitudinal ultrasound pulse by ablating a thin layer of the material (typically 10nm per pulse). This pulse travels through the sample, interacts with the microstructure and eventually comes back at the generation surface after reflection on the opposite surface. The detection laser is used to detect the small surface displacement caused by the successive arrival of the ultrasonic echoes. The surface displacement modifies the phase of the light collected after reflection on the sample surface which is directly demodulated by an active interferometer to extract a signal proportional to the surface displacement [4]–[6]. The signal also called as waveform shows a succession of echoes which corresponds to the bouncing of a single ultrasonic pulse between the parallel surfaces of the sample [7]. The waveform is used to extract the ultrasonic velocity and attenuation of the longitudinal ultrasonic pulse.  Correlating ultrasonic properties, velocity and attenuation, to material properties is of high significance for the non-destructive evaluation of microstructural evolution in metals and alloys during thermo-mechanical processing. The ultrasound velocity is evaluated by the ratio of propagation distance to the time delay between two successive echoes and is correlated to the bulk elastic properties of the material. The ultrasonic attenuation corresponds to the decrease in amplitude of the pulse per unit length. In elastically anisotropic polycrystalline materials, attenuation is primarily affected by grain scattering. Grain boundary scattering corresponds to the dispersion of the wave in different directions at grain boundaries. In weakly anisotropic materials like aluminum or magnesium, the contribution of grain scattering to attenuation has a smaller effect compared to other dispersive contributions [6].  The principle aim of the present thesis is to further extend the LUMet potential for in-situ grain size measurements. In particular, it is sought to extend grain size measurements to elastically anisotropic materials other than austenitic steels following cold and hot deformation processes. A particular series of thermo-mechanical scenarios are designed to generate samples with different 3  average grain sizes. A versatile method is then introduced to harmonize all the existing empirical equations to evaluate the grain size change in weakly textured FCC metals. In addition, it is proposed to extend the classical understanding of ultrasonic attenuation in polycrystalline cubic metals to more complex issues such as effects of grain boundary type using the finite element modelling (FEM) approach. FE modelling of ultrasonic wave propagation in polycrystalline metals was initiated in early 1990s [8]. The FE simulation approach provides a more rigorous way to describe the complex characteristics of ultrasonic wave scattering. The knowledge obtained through this thesis is valuable to evaluate the applicability of laser ultrasonics as a non-destructive tool in measurement of grain size evolution and quantifying recrystallization in FCC polycrystalline metallic materials.     4  Chapter 2  Literature review  2.1 Introduction Ultrasonics is one of the most widely used non-destructive evaluation (NDE) techniques for detection of flaws, characterization of microstructure and mechanical properties of materials, quality control of processing treatments, and evaluation of damage in industrial parts during service [9]. In ultrasonics, velocity and attenuation of ultrasound waves are used to correlate with material properties. An ultrasonic wave is a mechanical wave that can only travel in elastic media. The wavelength of ultrasound depends on the elastic properties and atomic vibrations of its propagation medium, which allows for nondestructive evaluation of materials [4]. Ultrasonic velocity is related to density and bulk elastic constants of the probed material. It can be used to determine elastic constants and to evaluate any metallurgical phenomenon that affects elastic constants and density, such as recrystallization [10]–[12], phase transformations [13], [14], and precipitation [15], [16]. Ultrasonic attenuation is the decrease in amplitude of the wave per unit length. In anisotropic polycrystalline materials, attenuation is primarily affected by grain scattering. The grain scattering is caused by the reflection and transmission of the wave at grain boundaries and varies with the incidence angle, the elastic mismatch between grains, and the size of the grain relative to the ultrasonic wavelength. The principle method to measure ultrasonic properties, velocity and attenuation, is to generate an ultrasonic pulse propagating through the sample and to detect echoes afterwards. The required time for the ultrasound to travel into a given sample is called delay. The time delay between two echoes is related to sample thickness and can be estimated accurately to determine velocity. Attenuation can be determined by measuring the relative decrease of amplitude of two successive echoes in a waveform [9].   The ultrasound wave can be generated and detected by piezoelectric, magnetic, and electromagnetic transducers and lasers. Piezoelectric transducers produce ultrasound by converting electrical energy to mechanical energy. The transducer has to be in contact with the probed specimen through an adhesive bonding or a fluid layer. These sensors are therefore not 5  suitable for high-temperature characterization [9]. Electromagnetic acoustic transducers (EMAT) generate and receive ultrasound by imposing a high-intensity magnetic field in the specimen without any physical contact. The non-contact nature of EMAT allows fast operation at high temperatures [17]. This transducer however needs to be placed near the probing material which is not feasible in most industrial environments. Further, industrial products and processes become increasingly more complex which substantially limits the application of contact and near-contact ultrasonic transducers [18]. More recently, lasers have been used to generate and detect ultrasonic pulses (laser ultrasonics). These lasers can be a meter-distant away from the tested material. Laser ultrasonics was initially developed to inspect composite laminates made of different materials with complex geometries [19]. The technique was used to detect and characterize internal and surface- breaking cracks, porosity, impact damage, corrosion damage and delamination in different aeronautical parts [20]–[24]. The system is capable of providing useful data in a wide range of materials and diverse sample geometries such as sheets, tubes, cylinders, and bars [4]. Here, at least one longitudinal echo can be measured if sheet specimens with thickness of 1 to 15 mm and cylindrical specimens with a diameter of 10 to 12 mm and a ratio of height to diameter in the range of 1 to 1.5 are used. Also, surface roughness plays an important role in detecting the ultrasonic echoes which is recommended to be smaller than 100 μm. Furthermore, laser ultrasonics offers real-time monitoring of microstructure evolution caused by different metallurgical phenomena such as recovery [25], recrystallization [26]–[29], grain growth [30]–[38], and phase transformation [39]–[41] during thermomechanical treatments.  2.2 Propagation of ultrasound waves in polycrystalline materials    2.2.1 Ultrasonic velocity   Based on the wave propagation mode, four types of ultrasonic velocities are considered, longitudinal (L), shear (S), Lamb, and surface. L and S wave velocities are of particular interest for non-destructive characterization as they are dependent on the stiffness tensor and density but independent of the given sample dimension [9]. Metals are generally composed of a multitude of grains having a particular crystallographic orientation. A weakly textured material in which sizes of grains are small compared to the dimensions of the sample, is assumed isotropic on the 6  macroscopic scale [42]. In such a polycrystalline metal, two methods have been often used to calculate the average elastic constants and velocities from single crystal elastic constants, i.e. Voigt and Reuss [43], [44]. In the Voigt method [43], it is assumed that strain 𝜀𝑘𝑙 is homogeneous through the material.  Stress in the 𝑚th grain 𝜏𝑖𝑗(𝑚) is expressed as [45]:  𝜏𝑖𝑗(𝑚)= 𝑐𝑖𝑗𝑘𝑙(𝑚)𝜀𝑘𝑙 (2.1) where 𝑐𝑖𝑗𝑘𝑙(𝑚) is the elastic stiffness tensor in the 𝑚th constituent grain. The average stress 〈𝜏𝑖𝑗〉 is defined by spatial averaging of elastic stiffness constants over all possible grain orientations in the polycrystal [46]:  〈𝜏𝑖𝑗〉 = (∑ 𝑐𝑖𝑗𝑘𝑙(𝑚)𝑁𝑚=1𝑁) 𝜀𝑘𝑙 = 〈𝑐𝑖𝑗𝑘𝑙〉𝑉𝑜𝑖𝑔𝑡𝜀𝑘𝑙 (2.2) where 𝑁 is the number of grains and 〈𝑐𝑖𝑗𝑘𝑙〉𝑉𝑜𝑖𝑔𝑡 is Voigt-averaged elastic stiffness tensor. In the Reuss method [44], stress 𝜏𝑘𝑙 is considered uniform throughout the aggregate. Strain in the 𝑚th grain 𝜀𝑖𝑗(𝑚) is then expressed as [45]: 𝜀𝑖𝑗(𝑚)= 𝑠𝑖𝑗𝑘𝑙(𝑚)𝜏𝑘𝑙 (2.3) where 𝑠𝑖𝑗𝑘𝑙(𝑚) is the compliance tensor in the 𝑚th constituent grain. The average strain 〈𝜀𝑖𝑗〉 is determined by spatial averaging of the compliance constants over all grain orientations as [45]: 〈𝜀𝑖𝑗〉 = (∑ 𝑠𝑖𝑗𝑘𝑙(𝑚)𝑁𝑚=1𝑁) 𝜏𝑘𝑙 = 〈𝑠𝑖𝑗𝑘𝑙〉𝑅𝑒𝑢𝑠𝑠𝜏𝑘𝑙 (2.4) Where 〈𝑠𝑖𝑗𝑘𝑙〉𝑅𝑒𝑢𝑠𝑠 is the Reuss-averaged elastic compliance tensor.  In a single crystal with cubic symmetry, the stiffness tensor is expressed by three independent elastic constants 𝑐11, 𝑐12, and 𝑐44 [47], [48]. The Voigt matrix formalism is often used to transform the fourth-rank elasticity tensor to a 6×6 symmetric second-rank stiffness tensor. Using the second-rank stiffness tensor notation, 7  Voigt and Reuss averages of elastic constants in a weakly textured polycrystal with cubic symmetry are defined as [49], [50]:  〈𝑐11〉𝑉𝑜𝑖𝑔𝑡 = 𝑐11 −25𝑐 〈𝑐44〉𝑉𝑜𝑖𝑔𝑡 = 𝑐44 +𝑐5 〈𝑠11〉𝑅𝑒𝑢𝑠𝑠 =2(𝑠11 + 𝑠12 −𝑠5)(𝑠11 + 2𝑠12)(𝑠44 +4𝑠5 ) 〈𝑠44〉𝑅𝑒𝑢𝑠𝑠 = (𝑠44 +4𝑠5)−1 (2.5) where 𝑐 = 𝑐11 − 𝑐12 − 2𝑐44 and 𝑠 = 𝑠11 − 𝑠12 −𝑠442. The bracket sign refers to the average quantity in the polycrystal. Quantities used on the right side are the constants of the single crystal. Further, Voigt and Reuss averaged elastic constants can be expressed in terms of Lamé moduli 𝜆 and µ: 〈µ〉𝑉𝑜𝑖𝑔𝑡 = 〈𝑐44〉𝑉𝑜𝑖𝑔𝑡     and    〈𝜆〉𝑉𝑜𝑖𝑔𝑡 = 〈𝑐11〉𝑉𝑜𝑖𝑔𝑡 − 2〈𝑐44〉𝑉𝑜𝑖𝑔𝑡   〈µ〉𝑅𝑒𝑢𝑠𝑠 = 〈𝑠44〉𝑅𝑒𝑢𝑠𝑠    and    〈𝜆〉𝑅𝑒𝑢𝑠𝑠 = 〈𝑠11〉𝑅𝑒𝑢𝑠𝑠 − 2〈𝑠44〉𝑅𝑒𝑢𝑠𝑠 (2.6) The Voigt method overestimates and the Reuss method underestimates the elastic constants of a polycrystal. Further, longitudinal and shear wave velocities in a weakly textured polycrystal can be estimated as [42]: 〈𝑣𝑙〉𝑖 = √〈𝜆 + 2µ〉𝑖𝜌 〈𝑣𝑠〉𝑖 = √〈µ〉𝑖𝜌 (2.7) where 𝑖 refers to the averaging method, Voigt or Reuss and 𝜌 is density.  8  In practice, the velocity of the longitudinal wave can be calculated as the ratio of the propagation distance to the time delay between two successive echoes [7]. Different methods have been used for precise measurement of the time delay [51]. Among these techniques, numerical cross-correlation is one of the most accurate methods to determine the delay between two echoes [52]. It was reported that the experimentally-determined velocity lies between the upper bound (Voigt) and lower bound (Reuss) of the averaging models. An arithmetic average, the so-called Hill average, is thus often used to calculate the wave velocities [53].  2.2.2 Ultrasonic attenuation   Ultrasonic attenuation α corresponds to the loss of energy per unit length of propagation distance. Mainly three phenomena contribute to the attenuation in metals: diffraction (αD), internal friction (αIF) and grain scattering (αSC). The diffraction contribution is generally difficult to quantify and varies with the wavelength, sample dimension and size of the generation spot. However, the magnitude of this contribution can be qualitatively estimated by the evaluation of the Fresnel parameter S𝐹 such as [7]:  S𝐹 =𝑙𝑧𝐴2 (2.8) where 𝑙 is the ultrasonic wavelength, 𝑧 the propagation distance and 𝐴 the radius of the generation source. In the near field region (S𝐹<0.1), the front of the ultrasonic pulse remains approximately planar and the diffraction has no or weak influence on the signal. In the far field region (S𝐹>10), the front of the wave becomes spherical and its magnitude decreases inversely with the propagation distance. Finally, when the Fresnel parameter ranges between 0.1 and 10, complex phenomena occur and make the analysis more challenging.   Internal friction corresponds to the absorption of some of the pulse energy into anelastic phenomena, such as magneto-mechanical damping and motions of dislocations and/or interstitial atoms [54]–[57]. Measurement of internal friction has been used for instance to monitor processes such as recovery during which the dislocation structure evolves [25]. Stanke and Kino [58] stated that in materials with highly anisotropic elastic properties, the influence of absorption on velocity 9  and attenuation can be assumed as a second-order effect compared to that of texture, grain size and phase changes. Grain boundary scattering is caused by the difference of elastic properties between grains with respect to an incoming wave propagating in a given direction of the sample as illustrated in Figure 2.1. The amount of scattering is proportional to the single crystal elastic anisotropy of the studied material. In cubic materials, single crystal elastic anisotropy ᴧ is characterized by: ᴧ =2𝑐44𝑐11 − 𝑐12 (2.9) Furthermore, the scattering behavior, 𝛼𝑠𝑐, depends on the ratio between ultrasonic wavelength and mean grain size in the microstructure as [2]: 𝛼𝑠𝑐(𝑓, 𝑇) = Γ(𝑇)𝐷𝑛−1𝑓𝑛 (2.10) where the function Γ(𝑇) accounts for the evolution of elastic anisotropy with temperature, 𝐷 is grain size and 𝑓 is the frequency which is related to the wavelength. The parameter 𝑛 is an exponent between 0 and 4 that varies with the scattering regime, i.e. with the ratio of the ultrasonic wavelength to the average grain size in the material. Equation (2.10) is validated empirically for weakly textured polycrystalline materials with polygonal grain structures and narrow grain size distribution.        Figure 2.1 Schematics of wave scattering; velocity of ultrasound changes at boundaries. 10  Three scattering regimes can be defined. In the Rayleigh regime, the average grain size is small compared to the wavelength (𝑙>> 𝐷) and 𝑛 is 4. In the stochastic regime the grain size is comparable to the wavelength (𝑙≈𝐷) and 𝑛 is 2. In the geometric regime, grains are larger compared to the wavelength (𝑙<<𝐷) and n is 0. In practice, the scattering exponent 𝑛 generally has values ranging from 2 to 4 [30], [41]. The reason can be explained as follows. Laser-generated ultrasound has a broad range of frequencies from a fraction of MHz to few hundreds of MHz. In metals, the grain size generally ranges from 1 to 500 µm which is smaller than or comparable to the wavelength generated with the LUMet system. Note that other microstructural features such as precipitates, inclusions and defects can also induce scattering of the ultrasonic pulse. Due to the relatively small size of these features compared to the size of the grains, their effect on ultrasonic attenuation is generally small [59]–[63].   Two methods have been established to calculate ultrasonic attenuation in practice, referred to as two-echo method and single-echo technique, respectively [34], [42], [63]. In the two-echo method, ultrasonic attenuation is calculated from the ultrasonic waveform by: 𝛼(𝑓, 𝑇) =202(𝑚 − 𝑛)ℎ𝑙𝑜𝑔𝐴𝑛(𝑓, 𝑇)𝐴𝑚(𝑓, 𝑇) (2.11)  where 𝐴𝑛(𝑓, 𝑇) and 𝐴𝑚(𝑓, 𝑇) are the amplitude spectrum of the 𝑛th and 𝑚th echoes at temperature T in each waveform and ℎ is the sample thickness. Attenuation values are generally expressed in nepers/mm or dB/mm. Ultrasonic attenuation evaluated with this method is related to both scattering and diffraction contributions. The two-echo method has been used in contact, near contact, and laser ultrasonic techniques [9], [13], [39], [64]. The single-echo technique was developed by NRC at Boucherville in collaboration with the Timken Company to measure the austenite grain size in steels using laser ultrasonic attenuation [59], [60]. This method consists of comparing the amplitude spectrum of an echo s in the frequency domain extracted from a waveform M with an echo s measured in another waveform R acquired in a reference sample of the same geometry and a sufficiently small grain size such that attenuation due to grain scattering is negligible. Under these conditions, the ultrasonic attenuation is expressed by:  11  𝛼(𝑓, 𝐷𝑖, 𝑇) =202𝑠. ℎ𝑙𝑜𝑔 (𝐴𝑅(𝑓, 𝐷0, 𝑇0 )𝐴𝑀(𝑓, 𝐷𝑖 , 𝑇)) (2.12) where f is the ultrasonic frequency, ℎ is the sample thickness, 𝐴𝑀 is the echo amplitude spectrum for the measured sample with an unknown average grain size 𝐷𝑖 and 𝐴𝑅 is the echo amplitude spectrum for the sample used as reference with average grain size 𝐷0. Both echoes are measured when the ultrasonic pulse has travelled the same propagation distance. In this approach, the small variations associated with thermal expansion and/or phase transformation are neglected. Further, it is considered that the diffraction is not measurably dependent on temperature and is not affected by the characteristics of the microstructure. Diffraction is therefore identical for both amplitudes, 𝐴𝑀 and 𝐴𝑅, and is cancelled out in Equation (2.12). The ultrasonic attenuation calculated in this approach is therefore only related to the relative change in grain scattering between the measured sample and the reference sample which is the main advantage of the single-echo technique over the two-echo method [37].  2.3 Application of contact ultrasonics for material characterization   Several studies have been conducted to determine microstructural, physical, and mechanical properties of metals and alloys using contact and near-contact ultrasonics.   Measurement of elastic constants Longitudinal and shear ultrasonic velocities were used to measure elastic constants, 𝑐11, 𝑐12, and 𝑐44, of iron single crystals in the temperature range 4.2 K to 1160 K [65], [66]. Later on, Haldipur et al. [67] determined the single crystal elastic constants of nickel superalloys by evaluating three independent ultrasonic properties, i.e. longitudinal velocity, shear velocity, and longitudinal attenuation at 7.5 MHz using piezoelectric transducers. Velocities and attenuation were measured from specimens containing randomly-oriented polygonal grains with known density and mean grain size. Single crystal elastic constants were determined by applying the Hill averaging method in an iterative way.  12  Estimation of average grain size  Willems and Goebbels [68] used ultrasonic backscattered signals to determine the mean grain size in steels at room temperature adopting the Rayleigh regime. Back scattered signals are the signals that appear between ultrasonic echoes. These signals were measured using a piezoelectric transducer in a similar manner to that of ultrasonic echoes. To measure grain size from back scattering measurements, hundreds of rectified signals were spatially averaged and digitized for each specimen. Clearly, grain size measurement with the backscattering method requires a sophisticated signal processing procedure and it is only suitable for a limited grain size range, i.e. from 10 μm up to 100μm.  Ultrasonic attenuation has been used to assess grain size in polycrystalline materials. Takafuji and Sekiguchi [69] developed a first method to obtain grain size from attenuation in steels at room temperature. In this study, the two-echo method was used to determine ultrasonic attenuation α in steel samples with known grain sizes. The obtained attenuation contained grain scattering, reflection losses at sample surfaces, and diffraction contributions. All the sources of attenuation except grain scattering contribution were determined experimentally. Accordingly, a correction function was established to separate the grain scattering contribution from the total measured attenuation. The scattering-induced attenuation was then used to establish an empirical relationship between α, grain diameter (EQAD), and frequency at each scattering regime (Equation (2.10)). Finally, the established correlation was used to determine the average grain diameter in samples with unknown grain sizes referred to as effective grain size in the range of about 10 to 50 μm. Figure 2.2 compares the effective grain sizes (effective ASTM No.) in two steel plates with metallographically determined grain sizes (metallographic ASTM No.). ASTM numbers determined by these two methods are in good agreement. It was pointed out that the effective ASTM No. can be determined if attenuation values are equal to or above 0.5 dB/cm, in sheet samples with thickness equal to or below 15 cm, with an accuracy of ±0.5 of the metallographic ASTM No. Later, Palanichamy et al. [70] proposed another method to estimate grain size from ultrasonic velocity in AISI 316 austenitic steel at room temperature. Here, a similar approach to that of 13  Takafuji and Sekiguchi [69] was used to determine mean grain size from velocity measurements. It was observed that longitudinal and shear wave velocities decrease with increase in grain size from 60 µm to 170 µm, the former decreased by 0.5% and the latter by 1.23%. The accuracy of grain size measurement in this method was estimated to be 20% which was smaller than the one obtained by attenuation measurements, i.e. 35%, on the same steel specimens [71]. Haldipur et al. [72] estimated the average grain size of nickel superalloy billets by combined measurements of ultrasonic velocity, attenuation, and backscattered signal. It was observed that attenuation and back scattered noise capacity vary with depth, being highest where the average grain diameter is the largest. Correlations were established between attenuation, noise level, and average grain size excluding twin boundaries. However, experimental points scattered significantly from the fitted model maybe due to the way in which micrographs were selected for metallographic characterizations and the twin boundaries were identified in the micrographs. Further, ultrasonic velocity was constant throughout the billets. All ultrasonic parameters were found to be independent of the wave propagation direction into the samples, suggesting the presence of an equiaxed microstructure with a weak crystallographic texture.   Figure 2.2 Effective grain size determined from attenuation measurements versus metallographic ferrite grain size in A36 and A242 steel plates [69]. 14  Measurement of recrystallization Many researchers used ultrasonics to study recrystallization behavior after deformation in different materials. Generazio [73] developed a method to determine the onset, progress, and completion of recrystallization after cold rolling in pure nickel with ultrasonic attenuation measurements using the two-echo method. The attenuation spectrum was measured ex-situ at room temperature after different annealing treatments. Figure 2.3.a shows the attenuation values at 40 MHz as a function of annealing temperature. Below 800 K, attenuation remains constant, from 800 K to 975 K, it increases with increase in temperature, and above 975 K, it increases with the highest rate within the studied treatments. The attenuation spectrum was fitt with a power-law equation in the form 𝛼(𝑓, 𝑇) = 𝑏 × 𝑓𝑛. The calculated frequency-dependent exponent 𝑛 is shown in Figure 2.3.b as a function of annealing temperature. Here, the recrystallized fraction determined from the JMAK equation [74] and from optical metallography measurements are also included. Recrystallization kinetics were evaluated from both attenuation and 𝑛 value in the following way. At the onset of recrystallization, small strain-free grains (Rayleigh scatterers) nucleate causing a large value of exponent 𝑛 (green shaded area in Figure 2.3.a). As the annealing temperature increases, both nucleation and growth rate of recrystallized grains enhances. The microstructure therefore consists of a mixture of small (Rayleigh scatterers) and large (Stochastic scatterers) grains, which causes attenuation to increase and the exponent 𝑛 to decrease (blue shaded area in Figure 2.3.a). Recrystallization ends when exponent 𝑛 remains constant, while the attenuation still increases with a slow rate (red shaded area in Figure 2.3.a). Above 975 K (cross-hatched area in Figure 2.3.a), the 𝑛 value is constant but attenuation increases with a high rate, this indicates a pure grain growth scenario, i.e. the microstructure only consists of large grains.  15   Figure 2.3.a) Attenuation at 40 MHz versus temperature. b) Exponent n (red line) versus annealing temperature. % recrystallized determined from the JMAK equation (solid black curve) and from metallography (dotted black curve) are inserted [73]. Vasudevan and Palanichamy [10] monitored the onset, progress, and completion of recrystallization in a cold-worked Ti-microalloyed stainless steel with ultrasonic velocity measurements using contact longitudinal wave probes. Figure 2.4 shows the evolution of velocity with annealing time measured at different frequencies at room temperature. Three distinct regimes are observed at each frequency. Below 10 hours, velocity measured at 2 MHz slightly increases while it remains constant at higher frequencies. This increase is attributed to the occurrence of recovery during which the lattice distortions are reduced. At higher frequencies, the velocity increase is negligible which may be due the enhanced interaction of ultrasonic waves with TiC precipitates. However, the precipitates were not visible in the optical micrographs as their size was in the range of 15 to 30 nm. This stage is followed by a sharp decrease in velocity which is due to the overall textural change occurring during the progress of recrystallization in this material. Beyond 100 hours of annealing, the velocity saturates with annealing time indicating the completion of recrystallization. It was stated that measurement of ultrasonic velocity is a more accurate method than measurement of mechanical properties for quantification of a recrystallization coupled with precipitation. However, no evidence was provided regarding the presence of TiC precipitates in this steel. Further, precipitates with the size range of 15 to 30 nm have negligible effects on the scattering of ultrasonic waves.  16   Figure 2.4 Ultrasonic velocity versus annealing time measured at different frequencies at room temperature [10]. Pandey [11] utilized ultrasonic longitudinal and shear wave velocities and attenuation measured by contact longitudinal wave and delay line probes to describe the microstructural changes caused by recrystallization in cold-rolled interstitial-free (IF) steel sheets. Here, the longitudinal velocity increased during the course of recrystallization. It should be mentioned that while the abovementioned investigations brought insight into the evolution of microstructure during recrystallization, they were only conducted ex-situ after annealing treatments and most of the assumptions had not yet been validated.  Evaluation of precipitation   Rosen [16] and Kumaran et al. [15] established correlations between ultrasonic properties, velocity and attenuation, and hardness to evaluate the precipitation behavior of age-hardened aluminum alloys. Rosen used a contact Matec ultrasound generator and detector at room temperature and Kumaran et al. used X-cut and Y-cut transducers from room temperature to 623 K by applying a proper couplant for high-temperature measurements. They observed that the changes in velocity and attenuation are related to the type, size, and distribution of precipitates and the composition of the matrix. The highest velocity measured (lowest hardness) is associated with 17  the formation of large incoherent θ, S, and S՛ precipitates which have substantially higher elastic moduli and higher velocities than the base aluminum. The highest hardness is achieved by formation of the GP zones and the coherent θ՛, S, and S՛ phases which have slightly higher velocities than the matrix. In agreement with the velocity measurements, the lowest attenuation corresponds to formation of the GP zones and coherent precipitates. These low-temperature phases are too small to contribute to the scattering of ultrasound. The highest attenuation obtained after prolonged aging treatments during which incoherent large precipitates form in the microstructure. It should be pointed out that interpreting the changes in velocity and attenuation regarding precipitation during and/or after an aging treatment is only sensible if an extensive knowledge of the precipitation behavior of the studied material exists apriori. Furthermore, size of precipitates in metals and alloys are often too small to be detected by the ultrasonic waves.     Measurement of mechanical properties Many researchers have used ultrasonic parameters, particularly attenuation, to predict fatigue damage in metals and alloys. Schenck et al. [75] evaluated attenuation using the two-echo method in a carbon steel during rotational bending fatigue. It was observed that attenuation slightly increases in the beginning, reaches a long steady-state period, and then substantially increases just before failure. Similar behavior was reported by Joshi and Green [76] during cyclic fatigue of steel and aluminum alloys. The gradual increase in longitudinal wave attenuation was attributed to the enhanced dislocation density. The attenuation peak prior to fracture was caused by the propagated microcracks which are substantial scatterers of ultrasound energy. Kenawy et al. [77] evaluated the microstructure and mechanical properties of cast iron using ultrasonic properties. Ultrasonic velocity increases with increase in nodularity of cast iron, while ultrasonic attenuation decreases. Empirical correlations were found between velocity and mechanical properties, i.e. hardness, yield strength, and ultimate tensile strength, in this material. Ultrasonics have also been used to estimate creep life in metals and alloys [78], [79]. At 60% of creep life, attenuation shows a peak value associated with the enhanced density and rearrangement of mobile dislocations.   18  2.4 Laser generation and detection of ultrasound pulse  Lasers have been utilized to generate ultrasonic pulses in solids since 1963 [80]. Two dominant mechanisms are involved in generation of an ultrasound pulse using a pulsed laser source, i.e. thermo-elastic and ablative [4]. Figure 2.5 schematically shows the generation of an ultrasonic pulse via different mechanisms. In the thermo-elastic mechanism (Figure 2.5.a), the optical power of the laser is low so that the material exposed to the laser does neither melt nor vaporize. The laser light is absorbed by the material in a certain depth, d, depending on heat diffusion and causes a localized increase in temperature and thermal expansion. The localized thermal expansion generates a strain which is the source of the ultrasound pulse in the material. Surface and plane waves can be generated efficiently through the thermo-elastic mechanism when the source is emitted on the surface in a small circle [6]. At higher energy densities, a very thin layer of the exposed material melts and eventually evaporates in addition to the thermo-elastic expansion. The vaporized material containing ions and electrons forms a plasma. The plasma cloud then expands away from the point of generation causing a recoil effect normal to the surface [81]. This momentum transfer generates an epicentral longitudinal wave that travels through the material. The ablative mechanism of generating ultrasound is not entirely non-destructive as a thin layer of the material surface is vaporized with each laser impingement. After generation of a substantial number of pulses, a crater mark with the same diameter as the laser spot may be seen on the surface (Figure 2.5.b) [6].  Figure 2.5 Schematics of wave generation by a) thermo-elastic, and b) ablative mechanisms [6]. 19  To detect the ultrasound, a second laser continuously or in long-duration pulses illuminates the surface. Upon hitting, the laser light is scattered and/or reflected and then collected using an interferometric-based device [82]. In the interferometric detection techniques, the phase or frequency modulation in the laser light generated by the ultrasonically-induced surface motion is converted to an intensity-modulated signal. The intensity-modulated laser light is then detected by an optical detector. The optical interferometers mostly used in laser ultrasonics are reference beam interferometers and self-referential interferometers [4]. In the reference beam interferometers, the scattered signal from the sample surface (signal beam) interferes with a planar signal (pump beam) at a photodetector. These interferometers best perform with mirror-polished surfaces. If the surface is not perfectly polished, the signal beam will have a speckled pattern. In such a scenario, mixing a planar beam with a speckled beam significantly reduces the performance of the interferometer. To address this problem, self-referential interferometers were introduced. In this class of interferometers, the speckled signal beam is interfered with a beam of a matching wave front, named as reference beam. The wave front-matched reference beam can be either collected from the sample or reconstructed holographically in a photorefractive crystal [5], [6].    2.5 Laser ultrasonic characterization of metallurgical phenomena in metals 2.5.1 Grain size and grain growth studies The measurement of austenite grain size in steel is one of the success stories of the laser ultrasonic technology in view of the challenges faced for the ex-situ characterization of austenite microstructures in low carbon steels. A first method to determine average grain size from ultrasonic attenuation measurements was developed by Dubois and co-workers for a series of C-Mn steels with carbon contents in the range of 0.03 to 0.72 wt% [2], [41]. In this method, a correlation was developed between the ultrasonic attenuation at a frequency of 15 MHz obtained using the two-echo method and the metallographic grain size. The temperature dependence of the attenuation was measured separately such as: 𝛼15(𝑇) = 𝛼15(1100) − 𝐴𝑐 × (𝑇 − 1100) (2.13) 20  where 𝐴𝑐 = −2.26 × 10−3(db/mm)/°C and 𝛼15(1100) is the attenuation at 1100°C. Figure 2.6 illustrates the use of this method for the determination of the grain size in an A36 steel (0.17wt%C-0.74wt%Mn). The laser ultrasonic grain size was compared with the metallographic grain size and excellent agreement was observed. Here, the mean equivalent area diameter (EQAD) was used as the metallographic grain size and calculated by √4?̅? 𝜋⁄  where ?̅? is the arithmetic average over all the grain areas. As an additional validation, this technique was used to measure grain growth occurring during thermal cycles for other FCC materials such as pure Nickel [83].  Figure 2.6 Comparison of austenite grain size measured by LUMet and metallography (large solid square symbols) for A36 steel austenitized at 1100°C [2]. Krakauer et al. [30] proposed another approach for the construction of a correlation between attenuation and grain size in an AISI 304 stainless steel over a frequency range of 20 to100 MHz. They stated that since the value of the scattering exponent 𝑛 depends on both the grain size and frequency, attenuation data should be normalized with respect to the wave vector 𝑘: 𝛼𝑠𝑐(𝑓, 𝑇)𝑘= 𝛤(𝑇)(𝑘𝐷)𝑛−1       (2.14) where 𝑘 =2𝜋𝑓𝑣. A log-log plot of 𝛼𝑠𝑐/𝑘 versus 𝑘𝐷 was used as a calibration curve for grain sizes in the range 10 to 67 μm. However, their experimental data showed substantial scattering from the 21  master curve. It was also observed that when the microstructure varies from a homogeneous polygonal grain structure, these calibrations are not providing relevant results.  Jeskey and co-workers [59], [60] developed the single-echo technique to calculate the attenuation spectrum in steels (Equation (2.12)). When the reference material has a very small average grain size, the attenuation measured in this method can be correlated with the absolute value of the grain size [33]. The attenuation spectrum is fitted using the following equation: 𝛼(𝑓, 𝑇) = 𝑎 + 𝑏𝑓𝑛 (2.15) This expression contains a frequency-dependent grain size contribution 𝑏 and a frequency-independent parameter 𝑎 which accounts for frequency-independent contributions such as those causing internal friction as well as external factors associated with the variation of the laser source intensity and/or the reflectivity of the sample surface. 𝑛 is taken to be 3. By substitution with Equation (2.10), the grain diameter can be related to the 𝑏 parameter using the following expression:     b =  Γ(T)𝐷𝑛−1 (2.16) The calibration curve shown in  Figure 2.7 was generated for tubes of various steel grades with a wide range of austenite grain size (20 μm to 300 μm), thickness (up to 30 mm) and temperature (900 to 1250 °C). It can be seen that the fit applied to the data points is non-linear, suggesting that the scattering exponent is actually slightly less than 3 in the studied steels.  22   Figure 2.7 𝑏 parameter determined by the single-echo technique as a function of metallographic grain size for seamless tubes [61]. This technique was used to investigate the evolution of austenite grain size in a model steel (C 0.05, Mn 1.88, Nb 0.048, Mo 0.49, Al 0.05 wt%) and X80 linepipe steel [33]–[36]. Figure 2.8 shows that the average grain size values determined by metallography and laser-ultrasonics agree well except for short holding times at 1150 ⁰C where the LUMet grain size is markedly higher than that obtained from metallography. At this temperature, dissolution of Nb-rich precipitates promotes abnormal grain growth. During abnormal grain growth, a few grains grow and generate a heterogeneous microstructure consisting of fine grains and a few large grains [35], [37], [84]. In such microstructures, however, characterization by only a single average metallographic grain size is not relevant.  23   Figure 2.8 Comparison between grain sizes measured by metallography (points) with those obtained from LUMet experiments (lines) for X80 linepipe steel [36]. Recently, Garcin et al [37], [84] studied the grain growth behavior of Inconel 718 during dissolution of the δ phase with the LUMet system. They observed three regimes in variation of parameter 𝑏 with respect to the holding time at 1050⁰C, indicative of negligible grain growth in the first 30 s, followed by abnormal growth of precipitate-free grains and subsequently normal growth of large grains generated during the first 230 s of annealing (Figure 2.9.a). An empirical correlation was established between parameter 𝑏 and the average grain size measured by metallography. Evolution of grain size in the first 30 s of holding was described by the Zener pinning grain growth model in which the presence of precipitates decreases the growth rate [85]. In the third regime of grain growth, i.e. after 230s of holding, all the precipitates were in solution and the grain growth rate can be expressed by the parabolic grain growth model. They stated that by applying a lever rule to the laser ultrasonically measured grain size using the two grain growth models applied on the first and third stages of grain growth, the onset and completion of abnormal grain growth can be determined. Using this methodology, the fraction of large grains, indicative of the degree of growth heterogeneity, was calculated from laser ultrasonic and metallographic measurements as shown in Figure 2.9.b. Here, EQAD including twin boundaries were selected as the measure of average grain size [37]. In some studies, twins were removed from the constituent grains such as [86], while in others for instance [87], twinned grains were considered independent 24  from the parent grains. In these investigations, no reasons were provided for either including or excluding the twin boundaries in the statistics.   Figure 2.9 a) Variation of 𝑏 with time during annealing of INCONEL 718 at 1050 ⁰C [84]. b) Large grains fraction with respect to annealing time calculated by applying a lever rule to laser ultrasonic grain size data and fitted grain growth models [37]. Further, Kruger et al. [88] measured the backscattered noise signals to evaluate grain size in a low-carbon steel (0.06 wt% C). A backscattered noise corresponds to the signal measured prior to the arrival of the first compressive echo. In this approach, such signals are converted to the frequency domain and fitted by a Gaussian function. Then the frequency of the maximum amplitude from each fitted spectrum is taken and plotted against the grain size. It was stated that a correlation can be considered between the peak frequency and the grain size even though the percent error in predicting the average grain size can be as high as 48%. In another study, Hutchinson et al. [3] used the amplitude spectrum to monitor grain growth in 316 stainless steel. First, a series of specimens with different grain size values were generated. Then, the “LUS parameter” which is the ratio of intensity in the high-frequency part to that in the low-frequency part of the amplitude spectrum was calculated (Figure 2.10.a). It was stated that ASTM number and LUS parameter have a linear relationship. The obtained empirical equation was used to evaluate grain growth during isothermal annealing (Figure 2.10.b). As can be seen, 25  each test was repeated twice to show the repeatability of laser ultrasonic measurements at elevated temperatures. The grain growth kinetics was also described by parabolic relationships (solid black lines). The fit quality was acceptable particularly in the early stages of grain growth.   Figure 2.10    a) Correlation between LUS parameter and metallographic grain size in 316 stainless steel. b) Continuous measurement of grain growth during isothermal holding at 1000°C and 1050°C using the established correlation [3]. Sundin et al. [31] proposed another method to acquire grain size in commercial low carbon steels from attenuation of longitudinal waves. Figure 2.11 illustrates their procedure. A normalized experimental attenuation (αD) at each frequency is plotted with respect to the grain size obtained from metallography. Then, the attenuation values were calculated using the unified scattering theory and plotted in the same graph as the experimental values. At each frequency, theoretical and experimental attenuation has the same value at a particular grain size which can be considered as the laser ultrasonic grain size. In this approach, the measured grain size was a function of frequency. To address this problem, a specific frequency range (27.5 to 37.5 MHz) was selected within which the laser ultrasonic grain size was almost independent of frequency. It was explained that the grain size values measured in this way were in good agreement with those obtained by conventional metallographic techniques. However, this technique may not be applicable for a broad range of grain sizes as the suitable frequency range may not be included in the available bandwidth in a particular laser ultrasonic system. 26   Figure 2.11 Normalized frequency as a function of grain size in low carbon steels (dashed lines are experimental values and solid lines are those from theoretical calculations). Bold line connects the predicted grain size at each frequency. The specimen grain size was 7 µm [31]. 2.5.2 Recovery  Smith et al. [25] used laser-ultrasonic and stress relaxation techniques to monitor recovery through dislocation density variations in an ultra-low carbon steel following warm deformation. The principle consists of first measuring simultaneously velocity and attenuation. Figure 2.12.a shows fractional velocity change measured following deformation at 550°C, 730°C, and 800°C to a total true strain of 0.15 at a strain rate of 0.1 s-1. Velocity decreases with time for all the annealing temperatures. Further, the initial velocity decreases with increase in temperature. It was pointed out that ultrasonic velocity is a function of texture [28], dislocation structure [89], magnetomechanical damping [90], and degree of grain scattering [91]. During the progress of recovery, grain size and texture remain constant. It was also assumed that interaction between ultrasound and magnetic domain walls is negligible as compared to interaction between ultrasound and dislocations. The fractional change in velocity during recovery was therefore attributed only to dislocation damping. Figure 2.12.b shows ultrasonic attenuation at 8 MHz measured following warm deformation at 550°C, 730°C, and 800°C using the two echo method. Attenuation is approximately constant with time for all the annealing temperatures, being higher at higher temperatures. Ultrasonic attenuation is caused by grain scattering [58], diffraction [92], 27  magnetomechanical damping [93], and dislocation damping [89]. The grain scattering contribution was calculated using Equation (2.10) in the Rayleigh regime. The diffraction contribution is related to the sample geometry and could not be measured due to the geometry of the deformed cylinder. The magnetomechanical damping contribution 𝛼𝑚𝑒 was estimated as [94]: 𝛼𝑚𝑒 = (0.45𝑓0.115)(𝜇𝑖𝜆𝑠2𝐸𝑠𝑎𝑡𝐼𝑠2)(𝑓/𝑓0𝑟1 + 𝑓2/𝑓0𝑟2 )  (2.17)  where 𝜇𝑖 is the initial permeability, 𝜆𝑠 the magnetostriction constant, 𝐸𝑠𝑎𝑡 the Young’s modulus at saturation, 𝐼𝑠 the saturation magnetization, and 𝑓0𝑟 the relaxation frequency given by: 𝑓0𝑟 =𝜋𝑅𝑒24𝜇𝑖𝐷𝑑2  (2.18) where 𝑅𝑒 is the electrical resistivity and 𝐷𝑑 is the domain size. By subtracting grain scattering contribution calculated by Equation (2.10) and magnetomechanical damping contribution calculated by Equations (2.17) and (2.18) from the total measured attenuation, Smith et al. [25] obtained the attenuation that was only caused by dislocation damping. The authors however neglected the diffraction contribution effect.    Afterwards, both experimentally measured ultrasonic parameters were correlated to dislocation density ф and the average pinning point separation 𝐿 by approximating the vibrating string theory proposed by Granato and Lüke to calculate the dislocation density as [89]: 𝑣𝑑(ф) − 𝑣𝑑(0)𝑣𝑑(0)=4𝐺Б2ф𝐿2𝜋4𝐶  (2.19) 𝛼𝑑𝑖𝑠𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛 =16𝐺𝐵Б2𝑓2ф𝐿40.115𝜋4𝐶2 (2.20) where 𝑣𝑑(ф) is the velocity for a dislocation density of ф, 𝑣𝑑(0) the velocity in absence of dislocations, 𝐺 the shear modulus, Б the Burgers vector, 𝐿 the average pinning point separation, and 𝐶 the dislocation line tension. 𝛼𝑑𝑖𝑠𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛 is attenuation due to presence of dislocations, 𝐵  the 28  damping constant, and 𝑓 the frequency. The parameters 𝐺, Б, and 𝐶 were obtained from previous investigations. The value of 𝐵 on the other hand, depends on damping caused by phonons, electrons, and impurities. The damping constant was estimated empirically as a function of carbon content and temperature by applying a fit to the existing experimental data in literature. Finally, Equations (2.19) and (2.20) were solved simultaneously to obtain the dislocation density.   Figure 2.12.c exhibits the dislocation density calculated using equations (2.19) and (2.20). Similar to fractional velocity change, the dislocation density decreases with annealing time [25].   Figure 2.12 a) Fractional velocity change and b) ultrasonic attenuation at 8 MHz measured after deformation at 550°C, 730°C, and 800°C to a total true strain of 0.15 and strain rate of 0.1 s-1. c) Evolution of dislocation density after deformation at 550°C calculated from ultrasonic velocity and attenuation and stress values using equations (2.19), (2.20), and (2.21) [25]. Stress measured during the stress relaxation test was also used to calculate the dislocation density by [95]: 𝜎𝑓 − 𝜎𝑦 = 𝑀𝛼1𝐺Б√ф  (2.21) 29  where 𝜎𝑓 is the relaxation stress measured experimentally, 𝜎𝑦 the yield stress, 𝑀 the Taylor factor, and 𝛼1 is a constant. The evolution of dislocation density following deformation at 550°C from laser ultrasonics and stress relaxation methods is compared in Figure 2.12.c. The dislocation densities calculated from laser-ultrasonic measurements are two orders of magnitude lower than those calculated from stress relaxation experiments. This difference is, in part, due to the fact that only mobile dislocations are sensed by ultrasonics, whereas the evolution of the overall dislocation density is considered in a stress relaxation test [25].  It is worth noting that the studied material’s constants Б, 𝐺, 𝐶, 𝜇𝑖, 𝜆𝑠, 𝐸𝑠𝑎𝑡, 𝐼𝑠, 𝑅𝑒, 𝐵, and 𝐷𝑑 are needed apriori to monitor recovery with laser ultrasonics. The damping constant 𝐵 is difficult to measure and a number of assumptions have to be made to estimate it. Also, mean grain size and texture coefficients are required to determine the grain scattering and texture contributions to velocity. Moreover, it is essential to know the magnetomechanical damping, diffraction, and grain scattering effects to separate dislocation damping from the total measured attenuation. Magnetomechanical damping and grain scattering contributions can be obtained using the existing theories, although a few assumptions have to be considered. The diffraction contribution is often complicated and impossible to measure.  2.5.3 Recrystallization Many researchers used laser ultrasonics to evaluate recrystallization after cold and hot deformation in different materials. Kruger et al. [96] studied static recrystallization of cold-rolled low carbon and IF steel sheets in the ferrite region. Figure 2.13 depicts variation of velocity during isothermal annealing at temperatures above 600°C after cold deformation. At each temperature, velocity changes are observed during recrystallization as the overall texture of the material evolves [3]. Similar results were found in dual-phase steel sheets [26], [35], [97]. It has been stated that if velocity increases monotonically during the treatment, the recrystallized fraction 𝑋 can be described by the fractional changes in velocity:  𝑋 =𝑣 − 𝑣𝐶𝑜𝑙𝑑𝑤𝑜𝑟𝑘𝑒𝑑𝑣𝑅𝑒𝑐𝑟𝑦𝑠𝑡𝑎𝑙𝑙𝑖𝑧𝑒𝑑 − 𝑣𝐶𝑜𝑙𝑑𝑤𝑜𝑟𝑘𝑒𝑑 (2.22) 30  i.e. 𝑋 is determined by applying a lever rule to velocity vs. time curves [28]. Evolution of velocity during recrystallization is sometimes non-monotonic. The non-monotonic changes in velocity are thought to be due to the selection of texture components during the recrystallization process [98]. In such a scenario, a modified lever rule is used to determine the fraction recrystallized [26], [35].   Figure 2.13 Variation of velocity during recrystallization of IF steel [96]. Hutchison et al. [3] successfully monitored the progress of recrystallization in cold-rolled 316-stainless steel sheets. During recrystallization, the overall texture change results in a significant decrease in velocity and the recrystallized fraction was obtained by applying a lever rule on velocity vs. temperature curves. Static recrystallization in cold-rolled Aluminum alloys was also studied using laser ultrasonics [28], [29]. Figure 2.14 shows that the onset and progress of recrystallization measured by laser ultrasonics is in good agreement with those acquired from the ex-situ methods, yield stress and metallography. However, the final stages of recrystallization, particularly the end of recrystallization, measured by laser ultrasonics is inconsistent with that obtained from the conventional methods. In addition, Militzer et al. [26], [27] investigated the microstructural evolution associated with intercritical annealing of a cold-rolled dual-phase steel (C 0.10, Mn 1.86, Si 0.16, Cr 0.34 wt %). Recrystallized fraction was quantified using the modified lever rule method from ultrasonic velocity only between 35 and 100% recrystallized where texture changes happened. Further, variation of velocity was used to determine the completion of 31  recrystallization and austenite formation in scenarios where recrystallization finished before austenite formation.  Figure 2.14 Recrystallization kinetics of cold worked AA5754 Aluminum alloy measured by laser ultrasonics, metallography and mechanical testing [28]. Austenite recrystallization in C-Mn steels after hot compression was first characterized by Smith et al. [99] and further pursued by other research groups. Sarkar et al. [33] studied recrystallization of a low carbon steel (C 0.05, Mn 1.88, Nb 0.048, Mo 0.49, Al 0.05 wt.%) in the austenite region. In uniaxial hot deformation experiments, cylindrical samples were used. Here, attenuation rather than velocity was used to identify start and finish time of recrystallization. Figure 2.15 illustrates grain size evolution after deformation at 1000⁰C for various strain levels.  32   Figure 2.15 Effect of strain on grain size evolution after hot deformation of a model steel [33]. Three distinct regimes were observed. Initially, the grain size remains approximately constant. During the occurrence of recrystallization, a significant decrease in grain size was measured and was followed by a stage of grain growth after completion of recrystallization. The end of the first step matches with the time required for 5% recrystallization predicted by an independently, based on double hit tests, developed softening model [100]. Further, the minimum grain size at the end of the second stage corresponds to the time of 95% recrystallization and is, therefore, a measure of the recrystallized grain size. Another interesting point from Figure 2.15 is that the degree of deformation which relates to the degree of elongation of the austenite grains affects the laser-ultrasonically measured grain size prior to recrystallization. This indicates that it may be possible to distinguish with the LUMet system between microstructures having similar grain size values but different grain aspect ratios.   2.5.4 Phase transformation Temperature, crystallographic texture, and composition of constituent phases affect ultrasonic velocity, as the above mentioned factors affect the elastic properties and density of the probed material. Dubois and coworkers [13] extensively studied the phase transformations in carbon steels. Figure 2.16 shows continuous measurement of velocity during heating and cooling cycles at a rate of 5°C/s for a 1020 steel as an example. It can be observed that upon heating, velocity 33  decreases with an increasing rate up to 750°C and with a linear rate from 750°C to 1000°C. A cusp-shaped feature was detected around 750°C associated with the ferromagnetic to paramagnetic transition. During cooling of steels with carbon concentration higher than 0.06 wt.%, a hysteresis in the temperature dependence of velocity was observed indicative of austenite decomposition. Further, velocity measured in steels with strong texture was used to calculate the orientation distribution coefficients (ODC) in the initial ferrite, austenite and ferrite formed from austenite. The measured ODC in ferrite phases was then used to calculate longitudinal velocity with respect to temperature and found to be in good agreement with the experimentally-determined values.   Figure 2.16 Evolution of ultrasonic velocity during continuous cooling and heating of 1020 steel at a rate of 5°C/s in the range 500°C to 1000°C [13]. In another study Kruger et al. [14] quantitatively monitored austenite decomposition in low alloy steels based on ultrasonic velocity differences between the parent and product phases. The proposed method consists of applying a lever rule to the plot of velocity versus temperature assuming that only two phases co-exist during the transformation progress (Figure 2.17.a). It can be seen that this method is precise to measure the austenite-ferrite transformation if velocity of the two phases are different, e.g. below the Curie point in this scenario. If the two phases have similar velocities, which can be seen above the Curie point in this steel, the lever rule cannot be used. It should be mentioned that the velocities of ferrite and austenite depend strongly on their respective textures and can differ from one another in either direction of the Curie transformation point. 34  Figure 2.17.b compares the austenite fraction determined from velocity and dilatometry measurements as a result of austenite decomposition during cooling. As can be seen, similar transformation kinetics are obtained from both methods, though the early and late stages of decomposition are quantitatively different. Ultrasonic attenuation was also used to monitor phase transformations in steels indirectly through grain size variations. The sudden increase or decrease in attenuation at single frequency during heating or cooling is associated with the nucleation and growth of the product phase having a different grain size compare to the parent phase [41].   Figure 2.17 a) Temperature dependence of velocity for different phases in steel. b) Fraction austenite measured by laser ultrasonics and dilatometry during continuous cooling in a low alloy steel [14]. Zamiri et al [40] monitored the α to β phase transformation in Ti-6Al-4V titanium alloy by analyzing the temperature dependence of longitudinal and shear wave velocities, from which the transformation start temperature was detected. More recently, Shinbine et al. [39] quantitatively assessed the α to β phase transformation in pure titanium by evaluating longitudinal velocity measured during cyclic heating and cooling treatments. It was observed that in the α region, the velocity decreased linearly with increase in temperature. In the β region, velocity values varied from sample to sample indicative of textural effects of sampling very few large β grains by the ultrasound. A model was introduced to calculate the velocity profiles by the knowledge of temperature dependence of elastic constants and density and the Hill-averaged stiffness tensor for polycrystalline aggregates.  Temperature dependence of density in single-phase regions were 35  calculated based on the CALPHAD method [101]. Temperature dependence of elastic constants were obtained from previous studies [102]–[104]. The averaged stiffness tensor was calculated from elastic constants and orientation distribution function constructed from the experimentally measured texture in this material. The β phase fraction during the course of transformation was obtained from the modelled velocity assuming a linear contribution of the constituent phases [39].     2.6 Summary   Ultrasonics has been extensively used for flaw detection, microstructure characterization and assessment of mechanical properties in metals and alloys. Laser ultrasonics has been used to monitor various annealing phenomena during thermo-mechanical treatments of metals and alloys, such as recovery, recrystallization, grain growth, and phase transformations, in real time. Most of the laser ultrasonic studies were conducted for steels in the austenite region and they are generally restricted to measurements of normal grain growth where the grain size distribution remains self-similar. Systematic grain growth studies for other FCC metals such as copper and cobalt superalloys are therefore needed. In case of recrystallization, static recrystallization after cold and hot deformation was mostly investigated in steels. Though, characterization of static recrystallization and subsequent grain growth after deformation in other systems such as cobalt superalloys where precipitation may occur concurrently with recrystallization have yet to be explored in more detail. Further, to the best knowledge of the author, there has been no attempt to harmonize the existing empirical correlations developed to measure grain size based on the classical unified scattering theory. In addition, an in-depth knowledge on the ultrasonic wave interactions with various types of grain boundaries present in the microstructure, e.g. random high-angle boundaries, special boundaries, and the resultant scattering behavior is still lacking.       36   Chapter 3  Scopes and Objectives The goal of the present research is first to provide experimental data on the laser ultrasonic response of materials with FCC crystallographic structures other than steels in the austenite phase during thermo-mechanical processing such as copper and cobalt superalloys. The second goal is then to obtain a thorough understanding of the effects of microstructural features such as grain size and grain boundary type on ultrasonic attenuation using a finite element modelling approach. Results of these experimental and simulation studies are designed to further evaluate the applicability of laser ultrasonics as a non-destructive tool in measurements of mean grain size and quantifying recrystallization in FCC polycrystalline metallic materials.  The detailed objectives of the present study are: I. Develop laser ultrasonic calibration curves to measure grain size in single-phase FCC materials other than austenite grain sizes in steels, including copper and cobalt-based superalloys. The alloys are selected in a way that their microstructures can be straightforwardly measured with standard metallographic techniques.  II. Application of the developed calibration curves to continuous measurements of grain size evolution in cold-rolled and hot deformed cobalt superalloys to determine in-situ recrystallization and grain growth kinetics. III. Explore the applicability of the LUMet system to make measurements in materials with carefully selected graded microstructures for high throughput microstructure investigations.  IV. Harmonize existing empirical grain size calibrations to one single equation based on scattering theory.   V. Quantify the effect of grain boundary type, i.e. regular high angle grain boundaries and twin boundaries, on laser ultrasonic grain size.   37  Chapter 4 Methodology 4.1 Materials The materials chosen for this study must fulfill a number of criteria in order to facilitate the interpretation of measurements. Materials have to be sufficiently elastically anisotropic such that grain scattering can be considered as the major contribution to ultrasonic attenuation. Also, for easy ex-situ evaluation, a single-phase material is preferred. Further, the microstructures should consist of randomly-oriented polygonal grains. Accordingly, two cobalt-based superalloys and two pure copper samples, i.e. 99.998 pct high-purity copper and 99.9 pct commercially pure copper, were selected.  The chemical compositions of the cobalt superalloys used for this study are shown in Table 4.1. L605 and CCM alloys were initially 23% cold rolled at Tohoku University and supplied to UBC in the form of sheet specimens with dimensions of 60×10×3 mm. Isothermal heat treatments were then conducted on these specimens. Further, high-temperature deformation experiments were conducted using cylindrical samples of the L605 alloy with a diameter of 10 mm and a height of 15 mm. The procedure used to prepare cylindrical specimens at the Carpenter Technology Co. was as follows [86]. Ingots of L605 alloy with a diameter of 150 mm and weight of 27 kg were prepared by induction melting. The ingots were homogenized at 1230°C for 6 hours followed by hot forging, hot rolling, and hot swaging at 1200°C. The 14 mm rods obtained from high-temperature deformation steps were cold swaged to 10.92 mm. The area reduction of the final cold swaging was 42 pct. Subsequently the samples were annealed for 1 hour at 1150°C to homogenize the microstructure and dissolve the existing precipitates. Different types of carbides may form in cobalt superalloys due to aging treatment or during cooling from processing temperatures. The most frequently formed carbide in alloys with more than 5 pct chromium is M23C6 where M stands for metallic alloying elements. M6C type carbide is another common precipitate observed in alloys with high chromium and carbon content as well as molybdenum or tungsten concentrations higher 38  than 6-8 pct [105]. Finally, the cylindrical specimens with a diameter of 10 mm and a height of 15 mm were machined from the solutionized rods.   Table 4.1 Chemical composition of cobalt superalloys being used in this research (wt. %) Alloy Cr W Ni Mo Mn Fe Si C N Co L605 20.3 14.6 9.95 - 1.46 2.1 <0.1 0.054 0.0021 bal. CCM 27.8 - <0.01 6.03 0.55 <0.1 0.56 0.05 0.1367 bal. In addition, laser ultrasonic studies were also conducted on oxygen-free high-purity 99.998% copper. Sheet specimens of 35×10×2 mm were extracted from samples that were 93% cold-rolled at 77 K in a previous study conducted by Sinclair and Poole [106]. The machined specimens were then annealed at different temperatures to generate microstructures with various mean grain sizes. Tapered tensile specimens with a gauge length of 40 mm, thickness of 2 mm, and a width varying from 6 mm to 4 mm were machined from 99.9% pure copper blocks using an electrical discharge wire cutting machine (EDM). The tapered specimens were then strained and annealed to generate a graded microstructure.  4.2 Processing set up 4.2.1 Gleeble and Laser Ultrasonics for Metallurgy (LUMet) system A Gleeble 3500 thermo-mechanical simulator (Dynamic System Inc. Poestenkill, NY) equipped with a LUMet sensor was used for the heat treatment and uniaxial compression tests. Prior to the tests, the sample chamber was evacuated down to a pressure of 0.05 Pa to minimize surface oxidation. To conduct thermal treatments, a set of low-force jaws was used while high-force jaws were employed to run the thermo-mechanical treatments. The temperature of specimens was measured and controlled with a pair of K-type (NiCr-Ni) thermocouples spot welded at the centre length of the samples. The LUMet sensor is attached to the back door of the Gleeble chamber. The principles of the LUMet technique is shown schematically in Figure 4.1.a. In this sensor, a frequency-doubled Q-39  switched Nd:YAG with a wavelength of 532 nm generates a broad band ultrasonic pulse by ablation of a very thin layer of the surface. The available bandwidth of the ultrasound wave is 2 to 30 MHz. The generated pulse travels through the sample, bounces at the back wall, and returns to the generation surface. The arrival of the pulse generates a small surface displacement that is recorded with the detection laser, a frequency-stabilized Nd:YAG laser with a wavelength of 1064 nm. The detection laser light reflected on the samples surface is modulated by the surface displacement caused by the arrival of the ultrasound wave and demodulated inside an interferometer. Two-wave mixing interferometry is used to detect ultrasonic waves in the present work [107]. Figure 4.1.b shows the optical set up used in this method. The signal beam is mixed with the planar pump beam emitted from the detection laser inside the photorefractive crystal and a refraction grating is generated. This refraction index changes the wave front of the pump beam to best match with that of the signal beam. The interfered beams are then directed to a retardation plate and a polarizing beam splitter which make the necessary phase shifts and polarizations to the components of the mixed signals. Intensity of these signals are eventually measured by two photodiodes. The response is shown as a plot of echo amplitude in volts with respect to time. Both generation and detection laser beams are collinearly aligned at the centre length of the sample on the opposite side of the thermocouple junction. Also, the laser beam is mounted on a translation stage linked to the Gleeble console and moves during deformation tests to remain at the centre of the specimen. The angle between the laser beam and the surface of sample is 20° which is measured with a digital angle meter and controlled with a pair of stainless steel anvils. The pulse generation rate can be as high as 50 Hz and can be adjusted during the experiments. Furthermore, the number of pulses per second is chosen in a way that the resulting surface damage does not affect the ultrasonic attenuation. The maximum number of pulses varies with temperature, grain size, material type, and laser spot diameter. Generally, up to 1000 pulses can be generated prior to substantial surface damage. The characteristics of the ultrasound pulse are representative of the average properties of the material over a volume created by the surface of the laser spot (2 mm) multiplied by the sample thickness. 40   Figure 4.1 a) Schematic representation of principles of the LUMet technique [37]. b) Optical set up used in the two-wave mixing interferometer [107]. 4.2.2 Instron tensile testing machine and DIC camera The work hardening of copper in this study is characterized using an Instron tensile testing machine. An extensometer with a gauge length of 12.5 mm is attached to regular specimens to measure the displacement. Tensile tests were carried out with a load cell of 5 kN. A digital image correlation (DIC) camera La Vision is employed for an accurate measurement of strain gradients on the tapered sample surface during the tests. Figure 4.2 shows the experimental set up used for this study. The camera is mounted on an aluminum rail positioned as close as possible to the sample gauge covering at least 30 mm of the gauge. To have an optimum amount of light for imaging, two LED lights are mounted on the aluminum rail close to the sample. Images were captured with a frequency of 3 Hz. Acquiring and post processing of images are carried out using the La Vision’s 41  Davis8 software. Calculation of strain using a DIC system is explained in detail elsewhere [108], [109]. For this purpose, two images are quantitatively compared, before and after deformation. Each image contains a random pattern referred to as speckle pattern. Before deformation, the calculation area is defined by plotting a rectangle in the reference image. The software uses a cross-correlation method to track the position of each pixel in each image and calculate the displacement field over the gauge surface.  Before measuring strain with the DIC method, the CCD camera has been calibrated using a standard calibration target consisting of an array of points with a diameter of 1 mm and a centre-to-centre distance of 5 mm. The centre-to-centre distance between points in the calibration target was similar to that in the speckle pattern applied on the gauge surface. A series of images were then captured from the target at different orientations and used to track the position of each pixel in the speckle pattern during deformation.   Figure 4.2 Instron testing machine and the DIC camera configuration. The camera was set up to cover 30 mm of the sample gauge length.  42  4.3 Processing routes  4.3.1 Thermal treatments of cobalt superalloys      A series of thermal treatments were conducted on sheet samples of L605 and CCM cobalt superalloys. Figure 4.3.a exhibits the schematics of specimen geometry and measurement configuration, and Figure 4.3.b is an image of the experimental set up in the Gleeble chamber. Samples were heated at 50 °C/s to a specific temperature between 1100 °C and 1200 °C where they were held for a time ranging from 5 s to 1000 s prior to cooling to room temperature. The cooling rate measured during helium quench (HQ) was 150 °C/s. In addition, two heat treatments with larger holding times, i.e. 3 and 18 hours, were conducted in a dedicated vertical furnace. Specimens and a thermocouple were placed in the central position of the vertical furnace under normal atmospheric condition. These two samples were then water quenched (WQ) at the end of the holding time. Figure 4.4 and Table 4.2 summarize the heat treatment conditions. The label refers to the name of the alloy (“L” for L605 and “C” for CCM, respectively), temperature, and time of holding at high temperature. For all heat treatments conducted in the Gleeble 3500 equipped with LUMet, the ultrasonic attenuation was measured during the heating and soaking steps in the plane perpendicular to the wave propagation direction (Figure 4.3.a). Following each cycle, a series of waveforms were acquired with LUMet. The temperature dependence of the ultrasonic attenuation was measured on the CCM alloy during continuous cooling at 5 °C/s from 1200 °C.   43   Figure 4.3 a) Schematic of sheet specimens. b) The set up used to perform thermal treatments.   Figure 4.4 Schematics of thermal treatments conduced on L605 and CCM alloys.    44  Table 4.2 Detailed heat treatment conditions (WQ: water quenched; HQ: Helium quenched) Alloy Label TH (°C) tH Cooling L605 L1100_5s 1100 5 s HQ L1100_100s 100 s L1100_600s 600 s L1150_200s 1150 200 s L1150_1000s 1000 s L1200_100s 1200 100 s HQ L1200_1000s 1000 s L1200_3h 3 h WQ L1200_18h 18 h CCM C1100_600s 1100 600 s HQ C1150_200s 1150 200 s C1150_600s 600 s C1150_1000s 1000 s C1200_100s 1200 100 s C1200_1000 1000 s C1200_300s_5 300 s 5 °C/s C1200_1000s_5 1000 s 4.3.2 Thermo-mechanical treatments of cobalt superalloys       Recrystallization and grain growth behavior of the L605 cobalt superalloy after hot deformation was studied by a series of high-temperature uniaxial compression tests on cylindrical samples using the Gleeble 3500 system equipped with the LUMet system. The surfaces of the samples were faceted by an in-house designed polishing kit to enhance the LUMet signal quality. The kit 45  assembly is shown in Figure 4.5a. Samples were placed in the kit (Figure 4.5.b) and polished mechanically on a 1200 µm sand paper. Following this procedure, the same geometry was obtained for all the specimens (Figure 4.5.c).  Figure 4.5 a) Polishing kit used for faceting L605 cylindrical specimens. b) Cylindrical specimen placed inside the polishing kit. c) Faceted samples. The faceted samples were placed between a fixed stainless steel jaw and a moving piston. A compression force of 300 N was applied to maintain the sample in position and regulated to be less than 1 kN. Carbon sheets with a nickel paste were employed to limit the friction between sample and compression anvils. A tantalum foil was placed on the carbon sheet in contact with the sample to reduce the thermal gradient along the sample length.  Figure 4.6.a represents the schematics of the sample assembly, and Figure 4.6.b is a photo taken during the hot deformation experiment. During deformation, the piston moved and the subsequent diametric displacement was monitored by a dilatometer positioned in the centre plane of the sample. The true stress was calculated as 𝜎𝑖 =𝐹𝑖𝜋𝑑𝑖2/4 where 𝐹𝑖 is the applied force measured by the load cell and 𝑑𝑖 is the instantaneous sample diameter measured by the dilatometer during deformation. The diametric true strain is given by  𝜀𝑖 = 2ln (𝑑0𝑑𝑖) where 𝑑0 is the initial sample diameter. The obtained stress-strain curves were corrected for friction using the equations explained in the Appendix 1. The test matrix is summarized in Table 4.3 and shown schematically in Figure 4.7. The sample labels for single-hit tests indicate the name of the alloy, the deformation temperature, the following 46  isothermal holding time, and the test number. The labels for double-hit tests show the deformation condition (double-hit), the name of the alloy, the deformation temperature and the isothermal holding time between the first and the second hit. Prior to deforming, samples were heated at a rate of 5°C/s to the deformation temperature in the range of 1000°C to 1100°C. The samples were then held at the deformation temperature for 30 s to minimize thermal gradients. Subsequently, the samples were deformed with a constant stroke rate of 0.1 s-1 and controlled stroke displacement to have a true strain of 0.2. The deformed samples were then held isothermally at the deformation temperature from 1 s to 30 minutes (𝑡𝑃𝐷). To preserve the high-temperature microstructure, samples were rapidly cooled to room temperature by helium quenching. A cooling rate as high as 150°C/s was obtained on the sample surface.    Figure 4.6 a) Schematic of faceted cylindrical specimen prepared for hot compression tests. b) Experimental configuration image during the test.     47  Table 4.3 Detailed thermo-mechanical treatments employed for compression tests on L605 Test  Label T (°C) Diametric true strain stroke rate (s-1) 𝑡𝑃𝐷 cooling Single-hit L1000_1s_1 1000 0.2 0.1 1 s HQ L1000_15min_1 15 min L1000_30min_1 30 min L1000_30min_2 L1050_15min_1 1050 15 min L1050_30min_1 30 min L1050_30min_2 L1100_1s_1 1100 1 s L1100_15min_1 15 min L1100_15min_2 L1100_30min_1 30 min L1100_30min_2 Double-hit DH_L1000_15min 1000 0.4 15 min DH_L1000_30min 30 min DH_L1050_300s 1050 300 s  Figure 4.7 Schematic illustration of thermo-mechanical treatments carried out on the L605 alloy in the Gleeble system. 48  To further characterize the softening behavior of the L605 alloy, double-hit experiments were conducted at selected deformation conditions. These conditions are summarized in Table 4.3. In the double-hit tests, specimens were heated to the deformation temperature with the same heating rate as in the single-hit compression tests. After the first hit, samples were unloaded and held at the deformation temperature for a specific period of time. Then, the second hit was applied to the specimens followed by fast cooling to room temperature with helium gas (cooling rate 150°C/s). The ultrasonic attenuation was measured during the entire thermo-mechanical treatments. The in-situ measurement of grain size after deformation was carried out using the calibration developed in the present study to relate the frequency dependence of ultrasonic attenuation with the metallographically-measured mean grain size.   4.3.3 Thermal treatments on copper        Strip specimens machined from the cold-rolled copper plate were isothermally annealed at different temperatures in the range 360°C to 700°C for one hour in a tubular furnace under argon atmosphere (Figure 4.8). Following the heat treatments, specimens were water quenched. Ultrasonic attenuation was then obtained from the average of 60 waveforms acquired at room temperature on annealed samples.   Figure 4.8 Heat treatments conducted on copper in a tubular furnace. LUMet measurements were carried out in the Gleeble system at room temperature. 49  4.3.4 Strain annealing treatments on copper         Strain annealing treatment was applied to the tapered specimens of pure copper. Figure 4.9 shows the geometry of tapered specimens used for this study. To apply the required strain, annealed samples were uniaxially deformed in tension up to the necking point at room temperature. Figure 4.10 illustrates the strain annealing cycle carried out on tapered specimens. Tensile tests were conducted using a 5 kN Instron tensile testing machine at a nominal strain rate of 2×10-3 s-1. The strain gradient was measured with a DIC camera from position 20 to 40 mm with respect to the zero position of the sample (see Figure 4.9). The tapered geometry was used to achieve a strain gradient in the range of 0 to 50% along the gauge length. Due to the strain gradient, a graded microstructure can be produced after the annealing step. Following the tensile tests, samples were annealed first at 225°C for 14 hours and then at 325°C for 6 hours in a box furnace. The furnace was programmed to conduct the second part of the heat treatment after the first part ends. To minimize the oxidation, the furnace was purged with argon gas during the entire annealing treatment. The strain annealing cycle used here was adopted from the investigation on the optimization of grain boundary character distribution in copper by King et al. [110]. They stated that a two-step thermal treatment in a strain annealing procedure can significantly enhance the fraction of special boundaries formed in a copper microstructure.  Figure 4.9 Geometry of tapered tensile specimens. Dimensions are in mm. Tapered samples scanned with LUMet system along the gauge before and after strain annealing treatment. 50  LUMet experiments were conducted at room temperature before the tensile test, after the tensile test, and after the annealing cycle. The centre line of the sample starting at position zero was scanned by moving the laser position manually with a step size of 2 mm. At each position, 60 waveforms were obtained to determine the frequency dependence of attenuation. Afterwards, the specimens were cut along the centre line to characterize the selected areas, i.e. at the positions of 0, 6, 12, 16, 30, and 40 mm, by conventional metallographic techniques.   Figure 4.10 Strain annealing treatment applied on the tapered copper specimens to generate a graded microstructure. 4.4 Microstructure characterization  4.4.1 Sample preparation   Cobalt superalloy samples were prepared for scanning electron microscopy (SEM) and electron backscatter diffraction (EBSD) analysis in the following way. Specimens were cut with a precision cutting saw at the thermocouple position, i.e. in the plane perpendicular to the rolling direction (RD). Samples were then hot-mounted and mechanically polished to a final roughness of 1 µm prior to a last step of vibratory polishing for 15 hours in a 0.05 µm colloidal suspension. The sample surface was cleaned in an ultrasonic bath of denatured ethanol for 5 minutes. Similarly, the 51  microstructure of copper specimens was examined in the plane perpendicular to the rolling direction. To this aim, samples were mechanically ground and polished with 1 µm alumina suspension followed by electropolishing in 60% phosphoric acid aqueous solution [111].    4.4.2 Scanning Electron Microscopy (SEM)  The microstructure of specimens was analyzed by back-scattered electron (BSE) contrast imaging using a Zeiss Sigma field emission SEM. An accelerating voltage of 10 kV and 20 kV was applied to reveal the grains in copper and cobalt samples, respectively. SEM micrographs of cold deformed and partially recrystallized specimens, on the other hand, had poor BSE contrast due to the enhanced misorientation spread within grains. For such microstructures, the grain boundaries could therefore not be distinguished and EBSD mapping was used instead to reveal the grain structure.  4.4.3 Electron Back Scatter Diffraction (EBSD) mapping  EBSD mapping was carried out using a DigiView IV EBSD camera attached to the Zeiss Sigma field emission SEM. To conduct an EBSD scan, the specimen was tilted 70° and focused using 60 µm aperture. A step size of 0.3 µm was used to scan cobalt specimens while varying step sizes from 0.3 µm to 2 µm were selected to probe the copper specimens. Analysis of the EBSD measurements was performed with the TSL Orientation Imaging Microscopy (OIM) 6 Data Analysis software. Here, points with confidence index less than 0.1 were omitted. The confidence index (CI) standardization and grain dilation methods were used to clean the data set [112]. In the grain CI standardization, if a low-confidence index pixel is surrounded by high-confidence index pixels within a recognized grain, it is assumed that the orientation of that specific pixel is defined correctly. The confidence index of all pixels within that particular grain is therefore changed according to the pixel in the grain with a maximum confidence index. Clearly, a number of pixels would appear on the map that do not fit into any grains after the CI standardization step. The dilation process modifies the orientation of such pixels. If most neighboring points of an unindexed pixel belong to one identified grain, a matching orientation with that grain is allocated to the pixel. If an unindexed pixel is surrounded by grains with different orientations, a random orientation is 52  assigned to the pixel. The dilation process can iterate until all non-indexed pixels are associated with a grain. Grains contained at least 5 pixels in multiple rows and grain boundaries were defined with a misorientation angle larger than 15°.  To quantify the state of the microstructure, the kernel average misorientation (KAM) and grain orientation spread (GOS) were calculated for each map. The former was determined by averaging the misorientation between the centre point of the kernel and the points at the kernel perimeter using 3rd nearest neighbors with a square grid. The latter was calculated as the average misorientation deviation of all individual pixels within a grain from the average orientation of the grain. The KAM and GOS values are associated with the extent of the local misorientation [113]. Moreover, twin length fraction was defined from Inverse Pole Figure (IPF) maps by taking the ratio of twin boundaries length to the total boundary length defined in the map. In an FCC material, twin boundaries are defined as a 60° misorientation between two grains about the <111> axis. Twin boundaries were identified with a tolerance angle of 3°. Misorientation distribution was calculated to define the fraction of boundaries present in microstructures. Aspect ratio is defined as the ratio of minor axis length to major axis length of the ellipse fitted to a grain. Average aspect ratio of each specimen is calculated by averaging aspect ratios of all grains in the map. Further, EBSD grain files consisting of coordinates (x and y) and orientation (Euler angles) of each pixel were obtained using the TSL OIM data collection software. The MTEX texture analysis toolbox was then used to create a discretized orientation distribution assuming orthotropic sample symmetry with a step size of 5°.  4.4.4 Quantification of metallographic grain size  The area of grains of fully-annealed specimens was measured from the BSE micrographs by identifying the position of the grain boundaries for at least 300 grains per sample to provide sufficient statistics to determine the average grain size. The latter was determined by adding continuously more grains in the calculated average such that the relative accuracy of the mean values was at least 10%. Twin boundaries were identified from BSE micrographs based on their different geometry from regular grain boundaries, i.e. planar versus curved. For the sake of comparison, the mean grain size was defined first by including all types of boundaries (i.e. with 53  twins) and was then also estimated by excluding the twin boundaries from the map (i.e. without twins) [114]. For the measurements, the positions of grain boundaries were marked over the BSE micrographs. Figure 4.11 shows BSE micrographs with marked boundaries excluding (Figure 4.11.a) and including (Figure 4.11.b) twin boundaries. For each specimen, at least 5 BSE micrographs were included in the analysis. The area of grains was then measured with the software ImageJ. Furthermore, the microstructures of deformed and partially recrystallized specimens were identified from EBSD scans since the BSE contrast was substantially reduced in the SEM images by the applied deformation. Here, the grain areas were directly obtained from the TSL OIM following the procedure explained in section 4.4.3. Grain boundaries were defined with a misorientation angle larger than 15°. The grain size D was then quantified as the equivalent area diameter (EQAD) by calculating √4?̅? 𝜋⁄  where ?̅? is the arithmetic average over the grain areas obtained from the BSE micrographs and EBSD maps. The grain size D was also quantified as the area weighted grain diameter (AWGD) defined as:   𝐴𝑊𝐺𝐷 =1𝐴𝑡∑ 𝐷𝑖 × 𝐴𝑖𝑛𝑖=1 (4.1) where 𝑛 is number of grains, 𝐷𝑖  and 𝐴𝑖 are diameter and area of a given grain, and 𝐴𝑡 is the total area of all grains within a map. The maximum grain diameter (DMAX) was defined by calculating the average of the 1% largest grain diameters measured on the distribution of grain areas. In this description, the ratio of DMAX over EQAD represents an indication of the width of the grain size distribution. Finally, the ratio of the grain size over the EQAD (D/EQAD) was used as a representation of the reduced grain size for comparison of cumulative area distributions. 54   Figure 4.11 BSE micrograph with marked grain boundaries of CCM alloy annealed at 1100°C for 600 s a) excluding twin boundaries and b) including twin boundaries. 4.4.5 X-ray diffraction (XRD) XRD scanning was conducted using a Rigaku Multi-Flex diffractometer on cobalt specimens before and after deformation in the temperature range 1000°C to 1100°C to investigate the presence of precipitates. Cu-Kα radiation was used as the x-ray source with a voltage of 50 kV and a current of 30 mA. Noise reduction was achieved using a graphite monochromator. Continuous scanning was carried out in the angle range 20°-50° with a step size (2θ) of 0.02° and step time of 4 s. The parameters were selected in a way that small fractions of precipitates can be detected. The XRD data was analyzed with the Match3 software based on the previous investigations on the precipitation behavior in L605 cobalt superalloy [86], [115].   4.4.6 Uncertainties in microstructure measurements  When applied to the measurement of grain size, the laser ultrasonics technology requires the construction of a calibration based on the metallographically measured grain size. Many factors influence the precision in the determination of the average grain size by metallography as summarized in the ASTM standard E112-12 which include the selected magnification, the ability to properly delineate grain boundaries, the number of grains considered, the total sample area scanned and number of fields of view permitted for the selected magnification as well as the 55  representativeness of the specimens selected and the area chosen for the measurement [116]. Following all guidelines for optimum estimation of the average grain size, and according to the various sources of errors in the procedure, the relative accuracy obtained in this study for the metallographic grain size value was determined to be 15%, i.e. 10% from statistics and 5% from the uncertainty in delineation of grain boundaries. The relative accuracy from statistics was determined as follows. The average grain sizes in a series of specimens were calculated by considering more than 1000 grains in the statistics. Then, the average grain sizes in the same specimens were calculated from 300 grains. In each specimen, the average grain size determined from 300 grains was within ±5% of the one measured from more than 1000 grains. On the other hand, the method to estimate the accuracy for the metallographic grain size, twin fraction, Dmax/EQAD, and fraction recrystallized in tapered copper specimens was different from cobalt superalloys and sheet copper specimens. In the case of tapered samples, large EBSD maps were acquired (1.2 mm by 1.4 mm). The grain size parameters were then estimated in cropped sections of these maps which areas were sufficient to include at least 300 grains. Then, this operation was repeated 10 times by gradually increasing the cropped area up to the total area of the map. The relative accuracy was defined as 𝑚𝑎𝑥−𝑚𝑖𝑛𝑎𝑣𝑒𝑟𝑎𝑔𝑒, where 𝑚𝑖𝑛, 𝑚𝑎𝑥, and 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 are respectively the smallest, the largest, and the average of grain size parameter of the cropped areas. Out of the 6 cases studied, the relative accuracy was obtained to be 15%, i.e. this is the same number as the one obtained for the more complicated conventional measurements.  4.5 Ultrasonic waveform analysis  Analysis of ultrasonic waveforms acquired from laser ultrasonics experiments were carried out with the CTOME software [117].  Figure 4.12.b shows a typical waveform obtained in a CCM cobalt superalloy at room temperature. The oscillations observed in the first 0.5 µs are caused by the ultrasound generation. The signal shows a succession of echoes which corresponds to the bouncing of a single ultrasonic pulse between parallel surfaces of the sample [118]. The waveform was used to extract the signal to noise ratio, ultrasonic velocity and attenuation of the longitudinal ultrasonic pulse. The signal to noise ratio, is calculated by the ratio of the amplitude of the selected echo to the noise amplitude measured prior to generation. The maximum frequency resolved is 56  further defined by the frequency for which the amplitude of the selected echo is lower than the noise amplitude. The delay between echoes is calculated using the cross-correlation function. First, the cross correlation amplitude, 𝐴𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛, between two successive echoes in one waveform is calculated as [119]: 𝐴𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 = 𝐼𝐹𝐹𝑇(𝐹𝐹𝑇(𝑓) × 𝐹𝐹𝑇(𝑔)) (4.2) Where 𝐹𝐹𝑇 and 𝐼𝐹𝐹𝑇 are the fast Fourier and inverse fast Fourier transformations of the echoes, respectively. The overbar on 𝐹𝐹𝑇 represents the complex conjugate. The maximum of the cross-correlation amplitude is then chosen as the time where two echoes are aligned. Using this method, the velocity was calculated by:   𝑣𝑇𝐸 =2(𝑚−𝑛)ℎ𝜏  (4.3) where 𝑚 and 𝑛 are two successive echo numbers and 𝜏 is the time delay. Sample thickness ℎ was measured before each test and corrected for temperature based on the coefficient of thermal expansion in the studied system, i.e. cobalt or copper. It was reported that the accuracy in measurement of velocity is 1%, while the precision of velocity measurements is about 0.01% [39].   Figure 4.12 Ultrasonic waveform generated from a cobalt superalloy at room temperature. Here, the first echo is selected for calculation of velocity and attenuation.  57  Ultrasonic attenuation for a given travelled distance corresponds to a gradual decay of the amplitude of the echoes as the pulse energy is scattered in the sample by interacting with its microstructure (Figure 4.12). In this study, the single-echo technique was used to calculate attenuation from ultrasonic waveforms (Equation 2.12) [59]. As explained in the literature review, in metals, the mean grain size is usually smaller or equal to the wavelength. In these regimes of scattering, i.e. stochastic and/or Rayleigh, the attenuation varies with the square and fourth power of the frequency, respectively. To simplify the approach, the attenuation spectrum is correlated with the third power of the frequency (𝑛 = 3) for all grain size ranges considered as proposed in the literature (Equation 2.15) [60]. Grain size is therefore measured by computing the frequency-dependent term 𝑏 from the attenuation spectrum. Figure 4.13 shows a typical attenuation spectrum together with the power-law fit. To calculate attenuation spectra in sheet samples of cobalt superalloys, waveforms acquired at room temperature were used as the reference while for cylindrical samples a particular waveform after deformation at high temperature was considered as the reference. In the case of sheet and tapered samples of pure copper, the waveform acquired at room temperature from a sheet sample annealed at 360°C for 1 hour was used as the reference waveform. The second echo was used for calculation of attenuation.   Figure 4.13 Attenuation spectrum determined by single-echo technique for a sheet sample of cobalt superalloy during isothermal holding at 1100°C.  58  4.6 Finite element simulation of wave propagation  A 2D planar dynamic explicit model was used to simulate ultrasound propagation in the anisotropic polycrystals with FCC crystallographic structures using Abaqus CAE v6.13. The simulation toolbox was developed by Garcin [120]. Here, a 2D simulation was appropriate due to LUMet configuration symmetry, i.e. along the 60 mm sample length. Also, it can be assumed that the displacement propagates in the selected 2D plane since the nature of the wave propagation is predominantly longitudinal. The purpose of the FE modelling was to gain a better understanding on the effect of grain boundary type on ultrasonic attenuation. The grain structure for the simulations was taken from the microstructures of single-phase equiaxed grains with random texture measured from EBSD maps on cobalt specimens. It should be mentioned that the cobalt EBSD maps were only used to obtain the crystallographic orientations associated with an experimental microstructure within which grains have FCC structure and are randomly oriented. EBSD maps of another alloy with a random texture and an FCC structure can also be used for this purpose. The dimensions of the maps were roughly 1mm by 1 mm. The simulation domain was created by the repetitive mirroring of EBSD maps into a large mesh template. The EBSD map was translated along and/or rotated about the mirror plane.  Figure 4.14 shows the mesh template schematically. The default mesh template dimensions were 10 mm by 2 mm. The mirroring plane in the central region was shifted to different locations while the displacement was imposed at the same location. A sensitivity analysis was conducted to obtain the optimum sample thickness that would minimize the calculation time while conserving large enough number of grains through thickness, i.e. a random texture. This analysis was conducted for a mean grain size of 100, 150, 200, and 300 μm in samples with 2 mm and 8 mm thicknesses. Each case was repeated 5 times by changing the set of crystallographic orientations. It was observed that the attenuation became sensitive to the texture, i.e. sets of orientation through the thickness, in the sample of 2 mm thickness for grain size larger than 150 μm. It was then decided to set the thickness to 2 mm for structures in which the grain size is smaller than 150 μm.  59   Figure 4.14 Schematic of the mesh applied to the FE template in the CTOME software. The mesh consisted of quadrilateral four-node bi-linear plane strain (CPE4R) elements. The simulation domain was subdivided into a central region with a fine mesh size (𝑚𝑖) of 4 μm and a graded coarse mesh on each side. Manual mesh refinement was used to gradually transition from the fine mesh to a coarse mesh. On the domain edges, coarse mesh size was on the order of 50 times larger than fine mesh size. The boundary conditions “Mechanical ENCASTRE” were applied to left and right vertical edges of the simulation domain to fix the model in space, i.e. no rotations about x or y axis and no vertical and horizontal displacements. A sensitivity analysis was conducted to obtain the optimum mesh size that would minimize the calculation time while conserving a sufficient number of elements per grain. For the sensitivity analysis, the attenuation spectrum was calculated for a wave propagating in a sample with a mean grain size of 30, 20 and 10 μm using a mesh size of 4μm and 1 μm. The results were not sensitive to the mesh size for the grain size of 30 μm and became measurably different at lower grain size. Therefore, the mesh size was set to 1 μm in structures where the minimum grain size is lower than 30 μm. Simulations were conducted in structures which grain sizes ranges from 32 to 50 μm suggesting that 2 mm-thick template meshed with 4 μm mesh size can be selected.  The TSL OIM software was used to extract the crystallographic orientations assigned to elements of the FE mesh. A step of average orientation per grain was applied prior to generation of the grain file. Twins were removed in some cases by defining twin boundaries as 60° rotation about <111> direction with tolerance angle of ±3°, i.e. grain boundaries with 3° deviation from the exact FCC 60  twin misorientation were also considered as twin boundaries. with CTOME software was used to allocate material properties to each of the elements of the FE template (Figure 4.15). The elements contained in each grain were assigned the properties of single crystal of copper at room temperature (Table 4.4). Material properties were defined as density and single crystal elastic stiffness tensor rotated by the associated Euler angle at each position [42]. The coarse meshed domain in Figure 4.15 were defined considering isotropic assumption for elastic properties. The isotropic areas had one stiffness tensor averaged over all orientations in the microstructure using the Hill averaging [121].   Figure 4.15 Finite element template used for wave propagation simulation. The microstructure is generated including twin boundaries. Colors represents longitudinal velocity in Y direction. Table 4.4 Material properties of copper at room temperature [53]. Condition  Density (g/cm3) 𝑐11 (GPa) 𝑐12 (GPa) 𝑐44 (GPa) Single crystal  8.938 170.0 123.0 75.0 Polycrystalline (isotropic)  8.938 200.7 105.3 47.6 61  The first step of simulation consisted of generation of a pulse for 0.5 μs. During this step, displacement of surface nodes over 2mm length located at the centre of the template (Figure 4.15) was imposed as the sum of the two Ricker wavelets. The amplitude of each Ricker wavelet 𝐴(𝑡) is a function of time and is defined as [122]: 𝐴(𝑡) = ∑(1 − 2𝜋2𝑓𝑖2(𝑡 − 𝑏𝑖)2)𝑒−𝜋2𝑓𝑖2(𝑡−𝑏𝑖)22𝑖=1 (4.4)   where 𝑓𝑖 is the peak frequency in Hz, 𝑡 the time in s, and 𝑏𝑖 a scalar. The frequency spectrum of the Ricker wavelets can be obtained by applying an FFT transformation to Equation (4.4), i.e. by converting the wavelet amplitude in the time domain to the one in the frequency domain. Figure 4.16.a and b demonstrate the Ricker wave amplitude in time and frequency domain, respectively. In order to prevent the effect of discontinuities of the nodes at the end of the generation line, a Gaussian distribution was chosen to define the peak amplitudes of the Ricker wavelet along the 2 mm line as:  𝐴(𝑥) = 𝐴(𝑡)𝑒−(𝑥2(0.5)2) (4.5)   The frequency bandwidth used in Equation (4.4) was chosen to be in the same range of frequency as for the laser ultrasonic pulse acquired with LUMet system.   Figure 4.16 Ricker wavelet used to impose a displacement on top boundary of simulation box in a) time domain, and b) frequency domain. 62  During the second step of the simulation, the surface nodes were released and the pulse was propagating freely for a period of 2 μs allowing enough time for the pulse to propagate at least once through the sample. For such a pulse propagating into copper, the smallest wavelength can be defined as the 𝑣𝑓𝑚𝑎𝑥=470030(𝑚.𝑠−1𝑀𝐻𝑧) = 156 μm. For appropriate resolution of this wavelength in the FE template, i.e. for adequate discretization of the simulated ultrasound wave, the mesh size must be at least 20 times smaller than the wavelength, i.e. 15620= 8 μm. This criterion is required to ensure that the simulation calculation converges. Therefore, 4 μm mesh size was suitable for the simulation. Furthermore, to ensure of the convergence of simulation, the time step must be defined such that the maximum displacement of a node during one-time step is lower than one mesh size. In practice, the time step was defined to be 4 times smaller, i.e. 𝛥𝑡 =𝑚𝑖4×𝑣=  4 4×4700(𝜇𝑚𝑚.𝑠−1) =0.2 ns. Following the application of the imposed pulse, vertical displacements (Y direction) of the nodes located along the 2 mm line at the bottom boundary were averaged after the wave traveled into the sample, i.e. after 2.5 μs. The total simulation time was adjusted in a way that at least 3 echoes can be acquired from the resulting waveforms. Attenuation spectrum was calculated from FE simulated displacement data following the methodology explained in section 4.5. The displacement data obtained by travelling the FE pulse in an elastically isotropic domain with the same mesh template size as others was used as the reference. The isotropic media was obtained by assigning one stiffness tensor to all the grains using the properties of polycrystalline copper at room temperature (Table 4.4).       63   Chapter 5  Kinetics of recrystallization and grain growth in cold rolled cobalt superalloys 5.1 Introduction  This section describes LUMet measurements of recrystallization and grain growth in L605 and CCM cobalt-based superalloys. To this aim, a calibration was developed. The evolution of grain size was examined continuously during rapid heating from the cold-rolled state and during isothermal holding treatments. Quantitative evaluation of the relation between ultrasonic attenuation and average grain size in these alloys provides a critical step toward the application of this technology for the investigation of microstructure evolution after hot-forging processes. 5.2 Microstructure evolution during annealing  Inverse Pole figure (IPF) maps of the as-received microstructure for the L605 and CCM alloys are shown in Figure 5.1.a and Figure 5.1.b in the plane perpendicular to the rolling direction (RD). The microstructures consist of grains with no or little apparent elongation in this plane. Average aspect ratios of as-received L605 and CCM samples are 0.3 and 0.4, respectively. The EQAD including twin boundaries was determined to be 30 and 15 µm for the L605 and CCM samples, respectively. The IPF map presented in Figure 5.1.c shows the microstructure of the L605 sample held for 5 seconds at 1100°C. The mean grain size of 4 µm for this case is much smaller than that of the as-received samples and depicts only polygonal grains with small or no internal misorientation. Here, the average aspect ratio is 0.4. The distribution of the KAM values for these three cases is presented in Figure 5.1.d. As much as 88% of the KAM values in the L605 and 50% in the CCM as-cold-rolled samples are larger than 1°, this indicates a high degree of internal deformation in these alloys [123]. On the other hand, 95% of the measured KAM values are lower than 1° for the sample heat treated 5 seconds at 1100°C. Both the dramatic evolution in the internal misorientation and the reduction of mean grain size indicates that recrystallization had occurred during the heating stage or at the beginning of the holding time [124].  64   Figure 5.1 Inverse Pole Figure (IPF) map of a) as-received cold rolled L605 alloy, b) as-received cold-rolled CCM alloy , c) L605 sample heat treated for 5 s at 1100°C, and d) the associated Kernel average misorientation (KAM) distributions. The evolution of grain size during holding at high temperatures is illustrated in Figure 5.2 and Figure 5.3. Figure 5.2.a to Figure 5.2.f show selected BSE micrographs of the L605 samples, and Figure 5.3.a to Figure 5.3.f depict the micrographs of the CCM samples. In both cases, a polygonal structure with a large amount of annealing twins is observed. The grain size increases with holding time and temperature, as summarized in Table 5.1.  65   Figure 5.2 BSE micrographs of L605 samples held at a) 1150 °C for 200 s, b) 1150 °C for 1000 s, c) 1200 °C for 100 s, d) 1200 °C for 1000 s, e) 1200 °C for 3 h, and f) 1200 °C for 18 h. 66   Figure 5.3 BSE micrographs of CCM samples held at a) 1100 °C for 600 s, b) 1150 °C for 200 s, c) 1150 °C for 600 s, d) 1150°C for 1000 s, e) 1200 °C for 100 s, and f) 1200 °C for 1000 s.  67  Table 5.1 Metallographic grain size data for L605 and CCM samples rapidly cooled from isothermal heat treatments Sample EQAD with twins (µm) EQAD without twins (µm) Dmax with twins (µm) Dmax/EQAD with twins AWGD with twins (µm) L1100_5s 4 7 12 3.0 5 L1100_100s 4 6 12 3.0 6 L1100_600s 7 10 24 3.4 12 L1150_200s 19 43 73 3.8 32 L1150_1000s 32 85 126 3.9 47 L1200_100s 24 41 76 3.2 37 L1200_1000s 43 78 115 2.7 63 L1200_3h 60 101 191 3.2 99 L1200_18h 64 100 227 3.5 103 C1100_600s 13 24 40 3.1 20 C1150_200s 16 41 66 4.1 28 C1150_600s 25 46 70 2.8 36 C1150_1000s 28 48 107 3.8 38 C1200_100s 20 38 66 3.3 32 C1200_300s_5 24 52 85 3.5 37 C1200_1000s 35 71 134 3.8 55 In this analysis, four grain size parameters were examined, i.e. (i) the equivalent area diameter with twins, (ii) the area weighted grain diameter with twins, (iii) the equivalent area diameter without twins and (ⅳ) the maximum grain diameter 𝐷𝑚𝑎𝑥 with twins. The heat treatment cycles selected produced microstructures with mean grain sizes, i.e. EQAD with twins, ranging from 4 to 64 µm for L605 and from 13 to 35 µm for CCM. The ratio of 𝐷𝑚𝑎𝑥/𝐸𝑄𝐴𝐷 reported in Table 5.1 ranges from 2.7 to 4.1 indicating that none of the tested conditions led to the development of a substantial tail in the grain size distribution. The obtained range here is in agreement with the range reported in a nickel-based superalloy with a lognormal grain size distribution. It was stated that the ratio of 𝐷𝑚𝑎𝑥/𝐸𝑄𝐴𝐷 in this superalloy increased to values as high as 6.7 when a tail appears in the grain size distribution [37]. Figure 5.4.a shows the correlation between the mean EQAD 68  with twins and the mean EQAD without twins. Open symbols are used for L605 and closed symbols for CCM. Relative accuracy of grain size measurements with metallography is 15 pct. Accordingly, each error bar shown in Figure 5.4.a was calculated as 15 pct of the measured grain size value. These two metallographic grain size values are correlated with a constant ratio of 0.54 ± 0.02 suggesting that the relative fraction of twin boundaries in the microstructure does not evolve to a measureable extent. EQAD and AWGD with twins are also linearly correlated such as 𝐴𝑊𝐺𝐷 = (1.58 ± 0.02) × 𝐸𝑄𝐴𝐷. Figure 5.4.b shows the evolution of the cumulative area fraction with respect to the reduced EQAD for selected samples. The fact that all cases overlap in this representation provides confirmation of the self-similarity of the grain size distribution in these heat treatment conditions [3, 4]. One single grain size parameter is therefore sufficient to describe the evolution of the average grain size in the present study. Further, the construction of a correlation between a metallographic grain size and the ultrasonic attenuation parameter is equivalent whether or not the grain size is determined by including twin boundaries.   Figure 5.4 a) Correlation between different grain size parameters extracted from metallographic analysis and b) Cumulative area fraction with respect to the reduced grain size (D/EQAD) for the selected L605 samples shown in Table 5.1. 69  5.3 Laser ultrasonic grain growth measurements 5.3.1 Laser ultrasonic grain size  As discussed, the ultrasonic attenuation spectrum measured from a LUMet experiment is fitted with a power-law relationship (Equation (2.15)). The frequency dependence of the attenuation parameter 𝑏 accounts for grain size and temperature contribution. Applied to Equation (2.12) and Equation (2.16), 𝑏 measured in a sample with a grain size 𝐷𝑖 using a reference sample of grain size 𝐷0 at a temperature 𝑇 can then be expressed as: √|1000𝑏| =  Γ(𝑇) (√|𝐷𝑖2 − 𝐷02|)1−𝜀 (5.1) where 𝜀 is a fitting parameter and the function 𝛤(𝑇) represents the temperature dependence of 𝑏. As it was shown in Figure 2.8, correlation between square root of the 𝑏 parameter and the steel grain size is non-linear, i.e. exponent 𝑛 is slightly less than 3 [61]. Here, the parameter 𝜀 is introduced to compensate for the possible non-linearity of the experimental data points. Equation ((5.1) is analogous to that recently constructed for an Inconel 718 superalloy [37]. Figure 5.5.a shows the correlation between the metallographic grain sizes (EQAD with twins) and the ultrasonic parameter 𝑏 measured at room temperature. The values of 𝑏 correspond to the arithmetic mean of a series of 60 attenuation measurements conducted on each sample at room temperature. Each vertical error bar was calculated as 5.6% of the averaged 𝑏 value. Horizontal error bars are removed for the sake of clarity. Selected samples of both alloys are included in Figure 5.5.a, i.e. open symbols correspond to L605 and full symbols to CCM. The evaluation of 𝑏 requires the selection of a reference amplitude spectrum 𝐴𝑅(𝑓, 𝐷0, 𝑇0 ). Figure 5.5.a shows three different choices for the reference state, i.e. samples held at 1100 °C for 5 seconds (circles), 600 seconds (squares) and at 1200 °C for 100 seconds (triangle). The relative change in grain size is not affected by the choice of the reference state as long as its amplitude spectrum has relatively small attenuation [125]. For small 𝐷0 (lower than 5 µm) the laser ultrasonic grain size measurement becomes an absolute measurement. Within the scatter of the data, the grain size of both alloys can be described with one common correlation. The optimum value for the parameter 𝜀 was determined from the least square method to be 𝜀 = 0.23 ± 0.06 when expressing 𝐷𝑖 and 𝐷0 in µm and 𝑏 in 70  dBmm-1MHz-3. The standard error expressed for these two parameters was evaluated with the algorithms available in the software Gnuplot using the variance-co-variance matrix in a similar way to the method used for determining the standard errors in a linear least square fitting problem [126]. To find a correlation for 𝛤(𝑇) the evolution of the parameter 𝑏 was measured during cooling from 1200°C to 200°C in two different samples in which the mean grain size was known and remained constant during cooling. To this aim, two CCM samples were continuously cooled at 5°C/s from 1200ºC after a 300 s and a 1000 s hold, respectively to generate two different mean grain sizes. The variation of 𝛤(𝑇) for both samples is shown in Figure 5.5.b. The temperature dependence in the range 300-1200°C can be expressed by an empirical relationship in the form 𝛤(𝑇)  = 𝛿 + 𝜅𝑇1.5. The 𝛤(𝑇) empirical equation was fitted on each data set by the least square method. The obtained fitting parameters for C120-300-5 and C120-1000-5 specimens respectively are 𝛿 = 0.025 ± 0.006 and 𝜅 = (2.561 ± 0.008) × 10−5°𝐶−3/2, and 𝛿 = 0.023 ± 0.006 and 𝜅 = (2.346 ± 0.008) × 10−5°𝐶−3/2. Selection of the fitting parameter sets does not affect the measured laser-ultrasonic grain size within the accuracy of the measurement. The optimum values for 𝛿 and 𝜅 are therefore found by averaging the upper bound and lower bound values, i.e. 𝛿 = 0.024 ± 0.006 and 𝜅 = (2.453 ± 0.008) × 10−5°𝐶−3/2, and shown in Figure 5.5.b.     71   Figure 5.5 a) Grain size (EQAD) correlation with the parameter 𝑏 measured at room temperature (20°C). Three samples were used as reference, L605 held at 1100 °C for 5s (circles), 600 s (squares), and at 1200 °C for 100 s (triangles). Vertical error bars are 5.6% of the averaged value. b) Temperature contribution to the ultrasonic grain size parameters for two CCM samples. The solid circle is the reference point at room temperature. The attenuation due to ultrasound scattering in metals is a function of the temperature through the temperature dependence of the elastic constants and the elastic anisotropy ratio both of which are usually not strongly dependent on small variations in alloy composition. The value found for the function 𝛤(𝑇) can therefore, in a first approximation, be applied to both alloy compositions, CCM and L605. The obtained 𝛤(𝑇) function depends on how grain size is defined, i.e. EQAD or AWGD. If AWGD with twins are used as the metallographic mean grain size, the ratio of AWGD to EQAD scales the obtained fit parameters, i.e.  𝛿 = 0.015 ±  0.001 and 𝜅 = (3.925 ± 0.008) ×10−5°𝐶−3/2. Evolution of parameter 𝑏 was measured during the thermal treatments and converted to grain size (referred to as laser ultrasonic grain size) using Equation (5.1).  In order to test the repeatability of the method for grain size measurements at high temperature, the calibration established in Figure 5.5.a and Figure 5.5.b was applied to monitor grain growth in three separate tests conducted at a temperature of 1150 °C for times up to 200, 600, and 1000 72  seconds on the CCM alloy (Figure 5.6.a). The metallographic grain size measured on quenched samples are shown in this figure with open square symbols. The three LUMet measurements show acceptable repeatability and are in excellent agreement with the metallographic measurements.   Figure 5.6 a) Evolution of the grain size (EQAD) measured in-situ for the CCM at 1150°C by LUMet (lines) and by metallography (open squares).b) Signal to noise evaluated for test 3. It should be mentioned that the scatter in measured grain sizes increases with time due to surface damage. Also, there is an apparent slight decrease in grain size after 600 s of isothermal holding (test 3). Amplitude of the second echo (signal) was divided by the amplitude of the noise received prior to the generation of ultrasound (noise) from each waveform acquired during test 3 and plotted in  Figure 5.6.b. The signal to noise ratio constantly decreases with time and after 600 s, it becomes very low. The observed decrease in grain size after 600 s and the increased scatter in the measured grain size is therefore related to the poor signal quality caused by substantial surface damage. 5.3.2 Recrystallization during continuous heating  The evolution of the laser ultrasonic grain size (EQAD) measured during heating from the as-cold-rolled state at a rate of 50°C/s followed by isothermal holding at 1200°C for 100 seconds is plotted in Figure 5.7 for the L605 and CCM alloys. Two sets of measurements are shown for each 73  alloy in order to validate the repeatability of the measurements. Metallographic grain sizes measured at room temperature and after 100 seconds at the holding temperature are shown in the figure for comparison. The agreement between the two techniques is quite reasonable in the cold-rolled sample at room temperature.   Figure 5.7 Evolution of grain size (EQAD) during heating-isothermal holding tests at a rate of 50°C/s for L605 (open squares) and CCM (open triangles) superalloys.  The subsequent fluctuations of the signal between room temperature and 300°C remain unclear at this stage. It is believed that a more sophisticated description of the temperature contribution is necessary in this range. Above 300°C, the laser ultrasonic grain size remains approximately constant up to a temperature of 900°C. A sharp drop down to a laser ultrasonic grain size of about 7 µm is then recorded in the range of 900°C to 1100°C for the CCM alloy and between 1000°C and 1200°C for L605. The evolution of the mean grain size here correlates well with the occurrence of recrystallization of the cold-rolled microstructure.  The in-situ and quantitative evaluation of such a brief recrystallization period is a strength of the laser-ultrasonic system. This result is consistent with the previous investigations on the static recrystallization behavior of the L605 superalloy after cold deformation using ex-situ EBSD mapping [128]. Further, the incubation time is longer in L605 than in the CCM alloy. The amount 74  and type of deformation, deformation temperature, strain rate, annealing temperature after deformation, heating rate, the deformation texture, solutes, and initial grain size may affect the kinetics of recrystallization [128]. The alloys studied were 23% cold rolled with the same strain rate at room temperature followed by heating at 50 °C/s to the same holding temperatures. Further, both alloys have similar deformation textures, but the amount and type of alloying elements as well as the initial grain size are different in L605 and CCM alloys, i.e. 20Cr, 15Ni, 10W in L605 with 30 µm mean grain size and 28Cr, 6Mo in CCM with 15µm mean grain size. It was found that recrystallization in a material with a fine-grained microstructure occurs more readily and in a shorter time span than in a material with a coarse-grained microstructure [129]. In agreement with previous studies, the longer incubation time of the L605 alloy as compared to the CCM alloy can be attributed to its chemical composition and a larger initial mean grain size. During further heating, the grain size increases and substantial grain growth occurs during holding at the soaking temperature of 1200°C. The metallographic grain sizes measured on samples quenched after 100 seconds of holding are in agreement with the laser ultrasonic grain size in these conditions. The laser-ultrasonic recrystallized grain size is consistent with the metallographically-measured grain size for L605 at 1100°C, c.f. Figure 5.1.c.   5.3.3 Isothermal grain growth  Figure 5.8.a and Figure 5.8.b show the evolution of the laser ultrasonic grain size (EQAD) measured continuously during isothermal holding at 1100°C for 600 seconds, 1150°C and 1200°C for 1000 seconds in L605 and CCM alloys, respectively. Due to surface damage, scattering in the measured grain size values increases during isothermal holding at all temperatures. At 1150°C for both superalloys, the extent of scatter after 600 seconds of holding became so high that the acquired signals were no longer acceptable. At this particular temperature, therefore, the test was repeated to obtain waveforms for only the last 400 seconds of holding. Metallographic grain sizes measured on samples quenched at various times during holding are shown in Figure 5.8 for comparison. For all cases, the agreement between metallographic and laser ultrasonic grain size is in the range of the accuracy estimated for the technique. Further, growth rates measured in L605 are larger than those in CCM at 1150°C and 1200°C. The growth rate in L605 at 1100°C is however smaller than that in CCM.  75   Figure 5.8 In-situ measurement of grain size (EQAD) during isothermal annealing using the established calibration in a) L605 and b) CCM superalloys. 5.4 Laser ultrasonic grain size measurement precision  As discussed, measurement of grain size with the LUMet system requires developing a calibration based on grain sizes measured by metallographic techniques. The constructed calibration is shown in Figure 5.5.a and Figure 5.5.b. According to the calibration (Equation ((5.1)), the accuracy in the measurement of grain size by metallography, the measurement of  𝑏  at room temperature, and elevated temperatures contribute to the total accuracy of the laser ultrasonic grain size measurement. The metallographic grain sizes were measured with a relative accuracy of 15%. Figure 5.9 shows the estimated precision in the measurement of the ultrasonic parameter 𝑏 for four selected cases. This figure was plotted by computing the distribution of measured values over 60 waveforms acquired at the same position in a sample at room temperature. Abscises for the data shown in this figure are normalized with respect to the arithmetic average of each dataset. A Gaussian distribution is applied by setting the mean value to 1 and adjusting the relative standard deviation for each case using a least square fitting method. The standard deviations resulting from the regressions range from 0.020 to 0.028 indicating that the uncertainty in the measurement of the attenuation parameter 𝑏, at room temperature, is at least 0.056 (5.6%) with a confidence 76  interval of 95%. Further, 𝑏 depends on temperature and the uncertainty associated with measurements at high temperatures was determined from the relative square error for all data shown in Figure 5.5.b, i.e. (𝛤𝑚𝑜𝑑𝑒𝑙 − 𝛤𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑)2/𝛤𝑚𝑜𝑑𝑒𝑙2 . The overall uncertainty resulting from this analysis amounted to 10%. Considering that these error sources are not correlated, the total relative error can be estimated by the square root of the sum of each contribution squared. The total relative error amounted to 26%, i.e. laser ultrasonics grain size was measured with a relative precision of ±13%.   Figure 5.9 Normalized attenuation parameter 𝑏 with respect to the relative count measured for 60 waveforms at the same position at room temperature for L605 samples with different grain sizes.  5.5 Grain growth modelling Heat treatment conditions up to 1000 s shown in Figure 5.10 are relevant for industrial processing. The observed evolution of the mean grain size, 𝐷, for these isothermal holding times can in a first approximation be expressed by the general relationship for normal grain growth such that: 𝐷𝑚 − 𝐷𝑖𝑛𝑖𝑡𝑚 = 𝛷(𝑇)𝑡                   (5.2) 77  where 𝐷𝑖𝑛𝑖𝑡 is the initial mean grain size, 𝑚 is the growth rate exponent, and 𝛷(𝑇) is a temperature dependent rate parameter that is described by:  𝛷(𝑇) = 𝜆1𝑒𝑥𝑝 (−Q𝑅𝑇) (5.3)  where 𝜆1 is a constant, Q is an activation energy, 𝑅 is the universal gas constant (8.314 ×10−3 kJ. mol−1), and 𝑇 is the temperature in Kelvin. Here, 𝐷𝑖𝑛𝑖𝑡 is the laser ultrasonic grain size measured at the beginning of holding at high temperatures. A least square method was applied to the measured laser ultrasonic grain size evolution to find the optimum values for the model parameters 𝜆1 and Q in Equations (5.2) and (5.3), as reported in Table 5.2. A common growth rate exponent, i.e. 𝑚 = 3.5, is used for all temperatures and both alloys. This value is consistent with the grain growth rate exponent expected for grain growth under solute drag conditions, as for example quantified by Shahandeh et al. using phase field simulations [130]. The obtained activation energies are higher than that for the self-diffusion in cobalt (440 kJ.mol-1) [131]. The result of the proposed model is shown with solid lines in Figure 5.10.a and b. The agreement with the measured data is acceptable for most of the measured data considering the degree of uncertainty of grain size measurements for the laser ultrasonic technique. The model overestimates the experimental measurements at the end of the holding time for the temperature of 1200°C, i.e. for 3 and 18 hours, indicating that the approach adopted in Equation (5.2) may no longer be valid in these conditions.  78   Figure 5.10 Grain growth model applied to the grain size evolution up to 1000 s measured during isothermal annealing in a) L604 and b) CCM superalloys. Table 5.2     Numerical values of the fit parameters used in the grain growth model Alloy Temperature (°C) 𝐷𝑖𝑛𝑖𝑡(µm) 𝑚 𝜆1(s−1) 𝑄 (kJ. mol−1) L605 1100 4 3.5 223 × 1035 ± 394 × 1032  960 ± 50 1150 10 1200 15 CCM 1100 5 383 × 1020 ± 250 × 1018 560 ± 80 1150 11 1200 14 The proposed grain growth model is applicable to heat treatment times up to 1000 s in the temperature range 1000°C to 1200°C. For longer holding times grain growth may be affected by the formation of precipitates. The small change in the metallographic grain size measured in L605 79  on samples quenched from 1200°C after 3 and 18 hours of holding (Table 5.1) indicates that a limiting grain size is approached which is expected as a result of a pinning pressure on boundary migration due to precipitates. This observation is consistent with the precipitation behavior in both alloys reported in the literature. Formation of different precipitates such as M23C6 and M6C-M12C, where M is a metallic element, was observed in L605 subjected to isothermal heat treatments of 10 hours at temperatures ranging from 600°C to 1350°C [11, 12]. Further, the σ phase precipitates above 900°C in CCM [133], [134].  5.6 Summary For the first time, the LUMet technology was used for the investigation of recrystallization and grain growth in cobalt superalloys. The evolution of the mean grain size was monitored in-situ during continuous heating and isothermal holding between 1000°C to 1200°C. The following important observations were made:  1) The grain size distribution remains self-similar during the grain growth scenarios investigated here in the temperature range of 1100°C to 1200°C for holding times up to 18 hours.  2) The relative fraction of annealing twin boundaries does not evolve to a measurable extent in this range of heat treatment conditions.  3) The recrystallization kinetics of the cold-rolled microstructure is monitored in real time during continuous heating through the refinement of the mean grain size.  4) The recrystallization process is completed at a lower temperature for the CCM alloy as compared to the L605 alloy.  5) The grain growth rate is substantially higher for the L605 alloy as compared to the CCM alloy.  6) A power law grain growth model can be used to describe the grain growth kinetics in the temperature range of 1100°C to 1200°C for holding times up to 1000 s.    80  Chapter 6  Recrystallization and grain growth in hot-deformed L605 cobalt superalloy 6.1 Introduction   In this section, microstructure evolution of L605 cobalt superalloy after hot deformation is being examined in-situ by laser-ultrasonics using the established correlation between the frequency-dependent attenuation parameter and mean grain size introduced in chapter 5. To this aim, samples of L605 alloy were compressed uniaxially in the temperature range of 1000°C to 1100°C and held isothermally at the deformation temperature for various times from 1 second to 30 minutes. The ultrasound attenuation was measured during the holding step to measure mean grain size. To evaluate ex-situ the state of microstructure, the holding treatment following the deformation step was interrupted at selected times. The evolution of the mean grain size measured by laser-ultrasonics was validated by conventional metallographic techniques, i.e. SEM and EBSD. Internal grain orientation spread was also measured from EBSD maps. Subsequently, double-hit compression experiments were conducted on the selected samples to calculate the softening fraction. This study verifies that measurement of grain size by laser-ultrasonics after conducting a series of deformation experiments can provide insight to identify the state of the ongoing softening processes in real time in cobalt superalloys.  6.2 High temperature flow behavior Figure 6.1.a shows the friction-corrected true stress true strain curves acquired at temperatures in the range of 1000°C to 1100°C at a constant stroke rate of 0.1 s-1. The deformation temperatures, strain rate, and total strain were selected from a previous recrystallization study on the L605 alloy [134]. Two tests were conducted for each condition to show the repeatability of the deformation tests. The total engineering strain was imposed to acquire a total true strain of 0.2. However, the amount of true strain decreases with decrease in temperature which is due to higher friction coefficient at lower temperatures. The flow stress decreases with increase in temperature. Figure 6.1.b shows the work hardening rate with respect to true strain calculated from the compressive flow curves shown in Figure 6.1.a. Strain hardening varies with deformation temperature. At 81  1100°C, the work hardening rate first decreases rapidly (the recovery stage) and then with a lower rate (the recrystallization stage) with true strain which is the characteristic work hardening behavior of metals plastically deforming by dislocation slip [135]. At 1050°C and 1000°C on the other hand, a different work hardening behavior is observed. This type of work hardening has been observed in metals with low stacking fault energy (SFE) such as copper and nickel superalloys, that plastically deform by both dislocation slip and mechanical twinning [136], [137]. Here, three distinct strain hardening regimes can be identified. First, the work hardening rate rapidly decreases with strain up to a strain of about 0.02. This initial decrease can be attributed to dynamic recovery occurring in FCC metals. The dynamic recovery occurs more readily at 1100°C as compared to 1000°C and 1050°C. The recovery regime is followed by a plateau (at 1000°C) or a small rise in the hardening rate (at 1050°C). This regime is referred to as primary twinning in which deformation dominantly takes place by mechanical twining. In the third regime, the primary twining rate linearly decreases with strain with a lower rate as compared to the first regime. At this stage, deformation can no longer proceed by twining as the density of mechanical twins saturates in the microstructure and deformation is again controlled by dislocation slip [138].   Figure 6.1 a) Diametric true stress versus diametric true strain after friction correction for L605 deformed at elevated temperatures and b) work hardening rate versus true strain calculated from the friction-compensated stress-strain curves. 82  6.3 Laser ultrasonic grain size after hot deformation  The frequency dependence of attenuation measured after a deformation experiment can be presented as laser ultrasonic grain size using the developed calibration explained in chapter 5. To calculate the attenuation spectra, the first echo of the 213th waveform from the sample deformed at 1050°C and held for 30 minutes was used as the reference signal. This particular waveform number shows the minimum frequency dependence value during the entire holding time. It should be noted that any other reference signal acquired during the isothermal holdings following deformation could be used to obtain the 𝑏 values. The difference between the final diameter of the samples deformed at 1000, 1050, and 1100°C was 0.1 mm. The wave propagation distance was 20 mm since the first echo was used to calculate the attenuation spectra. The 0.1 mm difference between diameter of the deformed samples has a negligible effect on the diffraction contribution in a 20 mm propagation distance.  Figure 6.2.a through Figure 6.2.c show the evolution of the grain size (EQAD) measured by LUMet continuously during isothermal holding following deformation at 1000°C, 1050°C, and 1100°C for 30 min, respectively. The metallographic grain sizes measured on quenched specimens are included in the graphs with open circles for comparison. Each experiment was conducted twice to test the repeatability of the method for grain size measurement. Laser ultrasonic measurements show good repeatability and are in reasonable agreement with the ex-situ metallographic measurements. Grain size evolution at each temperature can be categorized into three separate regimes. In the first regime, the  grain size remains constant for a few seconds depending on the deformation temperature from 1 to 30 s at 1000°C (Figure 6.2.a), 1 to 10 s at 1050°C (Figure 6.2.b), and 1 to 2 s at 1100°C (Figure 6.2.c). The second regime shows a measurable decrease in grain size. At 1000°C, small grain refinement was observed, the mean grain size decreases from 32 µm to 20 µm. At 1050°C on the other hand, significant grain refinement was obtained from 32 µm to about 12 µm. Similarly, at 1100°C the grains become much finer as compared to the initial grain size, i.e. 32 µm to about 15 µm. In the final regime, the grain size increases and the extent of the increase depends on temperature. At 1000°C, the grain size change is negligible. The apparent increase in grain size observed in the last 600 seconds of isothermal holding was due to a reduced signal to noise ratio. At 1050°C, there is at least a tendency for grain growth to occur, i.e. the grain size increases from 12 µm to 14 µm. This change in grain 83  size is however within the accuracy of measurement. At 1100°C on the other hand, grain growth occurs, i.e. the grain size increases from 15 µm to 28 µm.   Figure 6.2 Evolution of the mean grain size after deformation at a) 1000°C, b) 1050°C, and c) 1100°C measured by laser ultrasonics (lines). Open circles represent the mean grain size defined by metallography. The time scale refers to the time after deformation. 6.4 Microstructure evolution after hot deformation   Inverse pole figure (IPF) and grain orientation spread (GOS) maps of the as-received L605 are shown in Figure 6.3.a. and Figure 6.3.d in the plane normal to the compression axis. The microstructure consists of polygonal grains and annealing twins with an EQAD of 27 µm including twin boundaries. As much as 98% of grains have a GOS value smaller than 1°, i.e. the internal misorientation of the grains is negligible. IPF and GOS maps of a specimen deformed to a total strain of 0.2 at 1000°C are depicted in Figure 6.3.b and Figure 6.3.e. These maps represent the evolution in the microstructure generated by deformation. The polygonal grains have no apparent change in their aspect ratio compared to the as-received condition. The EQAD including twin boundaries is 28 µm similar to the mean grain size in the as-received material. The GOS map on the other hand shows that 94% of grains have a GOS larger than 1° indicating a high internal misorientation due to deformation. The constant mean grain size and the observed change in internal misorientation after deformation at 1000°C indicates that recrystallization did not occur during deformation at this temperature. IPF and GOS maps of a specimen deformed to a total strain of 0.2 at 1100°C are depicted in Figure 6.3.c and Figure 6.3.f. As much as 78% of the grains have 84  GOS values larger than 1° indicating a typical microstructure of a deformed sample. In the GOS map on the other hand a few small grains with an average size of 5 µm and negligible orientation spread exist at some of the original grain boundaries. In agreement with the previous investigation on hot deformation behavior of L605, this map shows that either dynamic recrystallization has started or static recrystallization has occurred at grain boundaries during 1 s holding time before quenching. Even though recrystallization has started, it can be seen that it is in its early stages as the area fraction of these small grains (recrystallized grains) is below 5% [134]. Further, the presence of these recrystallized grains did not significantly affect the grain size measured by LUMet (Figure 6.2.c) and measured by metallography (Table 6.1).   Figure 6.3 IPF maps of a) as-received L605, b) sample deformed and helium-quenched 1 second after deformation at 1000 °C, and c) 1100 °C. GOS maps of d) as-received specimen, e) sample deformed at 1000 °C, and f) 1100 °C. Figure 6.4.a and Figure 6.4.c show the IPF and GOS maps for a sample deformed at 1000 °C and held isothermally for 15 min. Two distinct groups of grains can be observed, one with GOS values above 1° (22%) and the other with GOS values below 1° (78%). The mean grain size is 25 µm, slightly smaller than the mean grain size right after deformation, i.e. 28 µm. These results 85  indicate that static recrystallization has progressed during the 15 min holding after deformation resulting in a heterogeneous partially recrystallized microstructure. IPF and GOS maps for a sample deformed at 1000°C and held isothermally for 30 min are shown in Figure 6.4.b and d. Similar to Figure 6.4.a and c, the microstructure is partially recrystallized. As much as 83% of grains have GOS values below 1° suggesting little progress in static recrystallization as compared to the sample held for 15 min at 1000°C.   Figure 6.4 IPF maps of specimens deformed and isothermally held at 1000 °C for a) 15 min, and b) 30 min. c) and d) GOS maps of specimens shown in Figure 6.4.a and Figure 6.4.b. Figure 6.5. a and Figure 6.5.b show the IPF maps of specimens deformed and held isothermally at 1050°C for 300 s and at 1100 °C for 73 s, respectively. At these specific holding times, the minimum grain size was reached after deformation at 1050°C (Figure 6.2.b) and 1100°C (Figure 6.2.c). As can be seen, both microstructures consist of small polygonal grains with an average size 86  of 12 µm (Figure 6.5.a) and 15 µm (Figure 6.5.b) with negligible internal misorientation (Figure 6.5.c and d). 99% of GOS values are below 1° indicating full recrystallization for these isothermal holding times after deformation.   Figure 6.5 IPF maps of specimens deformed and isothermally held at a) 1050 °C for 300 s, and b) 1100 °C for 73 s. c) and d) GOS maps of specimens shown in Figure 6.5.a (c) and Figure 6.5.b (d). Further holding at 1050°C up to 30 min results in limited growth of recrystallized grains (Figure 6.6.a and Table 6.1). Prolonged isothermal annealing following deformation at 1100°C up to 30 min is more favorable for grain growth where an average size as large as 28 µm was obtained (Figure 6.6.b and Table 6.1). 87   Figure 6.6 Back Scattered Electron (BSE) micrographs of specimens deformed and isothermally held for 30 min at a) 1050°C, and b) 1100°C. Grain size parameters, i.e. EQAD, AWGD, maximum grain diameter (Dmax), and Dmax/EQAD ratio were quantified including twins from both SEM micrographs and EBSD scans and are summarized in Table 6.1. Microstructure parameters of deformed and partially recrystallized specimens were only identified from the EBSD scans. The EQAD ranges from 12 µm in a fully-recrystallized specimen (L1050_300s) to 29 µm in the as-received hot rolled material. The ratio of Dmax/EQAD varies between 2.9 and 5.6. Partially recrystallized specimens (L1000_15min and L1000_30min) have longer tails in their size distribution as compared to the deformed or fully recrystallized cases. Mean grain size of recrystallized grains and non-recrystallized grains in L1000_15min specimen is 22 µm and 33 µm, respectively. For L1000_30 min specimen the recrystallized and non-recrystallized mean grain sizes are 20 µm and 29 µm, respectively. Figure 6.7 exhibits the correlation between grain size parameters EQAD and Dmax measured from EBSD scans and BSE images. Open squares are used for EQAD and closed squares for Dmax. Microstructure parameters determined using SEM and EBSD techniques are linearly correlated suggests that either value can be used as the single grain size parameter representing the microstructure. Here, consistent with the established calibration between metallographic grain size and the ultrasonic attenuation parameter detailed in section 5, the EQAD including twins obtained from SEM micrographs is used to compare with the laser ultrasonic grain size values.   88  Table 6.1 Metallographic grain size data with twins for as-received and deformed specimens helium quenched from isothermal holding after deformation Sample Method EQAD (µm) Dmax (µm) Dmax/EQAD  AWGD (µm) As-received SEM 29 84 2.9 46  EBSD 27 91 3.4 45 L1000_1s 28 75 2.7 39 L1000_15min 25 125 5.0 43 L1000_30min 22 123 5.6 40 L1050_300s SEM 12 56 4.6 27 EBSD 12 52 4.3 26 L1050_30min SEM 14 56 4.0 25 L1100_1s EBSD 28 82 3.0 46 L1100_73s SEM 12 59 4.9 22 EBSD 15 52 3.5 22 L1100_15min SEM 21 75 3.6 36 L1100_30min 28 72 2.6 38  Figure 6.7 Correlation between mean grain sizes (EQAD) measured from BSE micrographs and EBSD maps. 89  6.5 Calculation of softening fraction   Cumulative number fractions of GOS values in selected specimens measured from EBSD maps are shown in Figure 6.8. Except for the as-received, L1050_300s, and L1100_73s samples, it can be seen that grains are distributed into two populations. The common approach to quantify the fraction recrystallized from an EBSD map is first to define the orientation spread within each grain of the scanned area and then to define a threshold value below which grains are considered as recrystallized. The way in which the threshold value is selected depends on the stage of recrystallization and the type of material tested [139-141]. Here, GOS values of 99% of grains in a fully recrystallized specimen such as L1100_73s or L1050_300s were below 1°. A sensitivity analysis was conducted to select a threshold value for orientation spread. When the threshold value was changed from 0.5 to 1° (50% increase), the fraction recrystallized increased only by 2%. Based on the sensitivity analysis, a constant threshold value of 1° was selected for this study. Recrystallized fraction for specimens deformed at 1000°C and held isothermally for 15 min and 30 min is 0.78 and 0.83, respectively. Analysis of GOS values also shows that recrystallization is stagnated in these scenarios.   Figure 6.8 GOS distributions of the selected L605 specimens. 90  Double-hit compression tests were conducted for selected thermo-mechanical scenarios to quantify the softening behavior of the material. Figure 6.9.a and Figure 6.9.b show the friction-corrected true stress true strain curves measured from double hit tests at 1000°C and 1050°C, respectively. The time intervals between the first and second hits, i.e. 15 min and 30 min at 1000°C and 300 s at 1050°C, were chosen based on laser ultrasonic and metallographic characterizations (Table 6.1 and Figure 6.2). The 15 min and 30 min isothermal holding after deformation at 1000°C lead to partially recrystallized microstructures, whereas a fully recrystallized microstructure was obtained by deformation at 1050°C and isothermal holding for 300 s.  The percent softening 𝐹𝑠 is calculated from interrupted compressions tests by [142]:  𝐹𝑠 =𝜎𝑚𝑎𝑥 − 𝜎𝑦2𝜎𝑚𝑎𝑥 − 𝜎𝑦1× 100 (6.1) where 𝜎𝑚𝑎𝑥 is the maximum stress in the first hit, 𝜎𝑦1 and 𝜎𝑦2 are the yield stresses (0.2% offset) of the first and second hit, respectively. The flow stress values used to calculate 𝐹𝑠 are summarized in Table 6.2. The softening percent for the double-hit tests at 1000°C is 63% while the fraction is 96% for the test at 1050°C. These results indicate that deformation and isothermal holding up to 30 min at 1000°C leads to a partial and stagnated softening while 300 s holding at 1050°C after deformation is sufficient to reach full softening. Kwon and DeArdo [136] stated that static softening measured from a mechanical test is correlated to the state of recrystallization quantified by metallographic techniques. It was reported that 20% of softening is associated with recovery in medium and high-stacking fault energy materials such as copper and HSLA steels [143], [144]. L605 superalloy has a medium stacking fault energy in the temperature range 1000°C to 1100°C, i.e. 162 mJ/m2 to 173 mJ/m2 [86]. Here, a sensitivity analysis was conducted to investigate the effect of fraction attributed to recovery, i.e. 15% and 25%, on the fraction recrystallized defined from the double-hit tests (Table 6.2). 𝐹𝑅𝑒𝑥 values obtained based on these assumptions are 14% less than 𝐹𝑅𝑒𝑥 measured from GOS maps, i.e. 78% and 83%. The possible explanation could be the way in which a threshold value for orientation was selected, i.e. a constant value as opposed to a varying value at different stages of recrystallization [141].  91   Figure 6.9 Friction-corrected true stress true strain curves of interrupted compression tests on L605 samples at a) 1000°C, and b) at 1050°C. Second hits were conducted after 15 and 30 min holding at 1000°C and after 300 s of holding at 1050°C. Table 6.2 Maximum true stress, yield stress values and calculated softening fraction from double-hit experiments Sample  L1000_15min L1000_30min L1050_300s 𝑚𝑎𝑥 (MPa) 377 375 295 𝜎𝑦1 (MPa) 155 155 155 𝜎𝑦2 (MPa) 227 220 160 𝐹𝑠 68 70 96 𝐹𝑅𝑒𝑥  (recovery: 15%) 60 62 95 𝐹𝑅𝑒𝑥  (recovery: 25%) 55 57 95 XRD scans were conducted on the as-received and selected heat treated specimens to find the possible reasons behind the stagnated recrystallization after deformation at 1000°C. The as-received material consists of a single phase cobalt matrix. Deformation and isothermal holding in the temperature range of 1000°C to 1100°C lead to formation of M6C carbides in the specimens. 92  Favre also reported that in the L605 alloy M6C carbides may precipitate in the temperature range of 800°C to 1100°C [86]. However, the amounts of precipitates formed at different temperatures were not measurably different. Further, the precipitates were not visible in the SEM micrographs. Additional investigations are therefore required to further analyze the fraction of carbides precipitated in the studied deformation conditions.   6.6 Relation between laser ultrasonic grain size and static recrystallization behavior Laser ultrasonic grain size evolution after deformation shows three regimes (Figure 6.2).  The first regime where the grain size remains unchanged, corresponds to recovery prior to the onset of recrystallization. The decrease observed in grain size in the second regime is associated with the occurrence of static recrystallization. It was reported that recrystallized grain size is a function of initial grain size prior to deformation, amount of applied strain, and strain rate [1]. Here, the initial grain size, amount of strain, and strain rate were constant. The only varying parameter was deformation temperature which was demonstrated to have no effects on the recrystallized grain size. Therefore, similar minimum grain sizes (recrystallized grain sizes) were expected to be observed for all the deformation conditions. On the contrary, at 1000°C the minimum grain size is 20 µm, larger than the one at 1050°C (12 µm) and at 1100 °C (15 µm). Consistent with the metallographic observations and softening fractions calculated from double-hit tests (cf. Figure 6.4, Figure 6.5, and Table 6.2), these results indicate partial recrystallization at 1000°C while full recrystallization occurred at 1050°C and 1100°C. The larger minimum grain size obtained at 1100°C as compared to 1050°C can be due to the concurrent occurrence of recrystallization and grain growth leading to a slightly larger mean grain size at the end of recrystallization. Extended isothermal holding at 1000°C after deformation did not cause further grain refinement which indicated that no further progress was achieved in recrystallization. Possible reasons for the stagnated recrystallization could be the insufficient grain boundary mobility (temperature) and presence of the precipitates prior to deformation. At 1100°C, the grain size increases in the third regime due to the occurrence of grain growth following recrystallization. The limited grain growth observed at 1050°C as compared to 1100°C is due to the reduced grain boundary mobility at the lower temperatures [33].  93  The total relative error in grain size measurement is similar to the one estimated for grain size evaluations in sheet specimen, i.e. 26 pct. Additionally, the laser ultrasonically measured grain size 1 s after deformation was systematically higher than the metallographic grain size. This increase could be due to the change in the grains shape right after deformation which may cause an effect on ultrasonic attenuation measurements. It must be also mentioned that evolution of ultrasonic velocity can be utilized to quantify fraction recrystallized if texture of the studied material evolves during recrystallization [28]. Evolution of velocity after deformation was measured in the studied superalloy. The observed changes in the relative velocity following deformation were within the measurement precision, i.e. 1 pct., indicating that texture does not significantly evolve with the progress of recrystallization for the studied conditions.       6.7 Summary  The LUMet technology was used for the investigation of recrystallization and grain growth following hot deformation in an L605 cobalt superalloy. In this study, frequency dependence of attenuation was measured and converted to mean grain size based on the established grain size calibration. The constructed calibration was used without any corrections for the cylindrical shape of specimens or the change in grain shape due to deformation, this indicates the versatility of the laser ultrasonic technique for real time measurements of microstructural evolution in metals and alloys at high temperatures. Evolution of the mean grain size was monitored in-situ during isothermal holding between 1000°C to 1100°C after deformation. It was demonstrated that grain size evolution can be used to indicate the progress of static recrystallization. Analysis of the recrystallized grain size as obtained by LUMet at different deformation temperatures revealed that recrystallization was partial after deformation at 1000°C while full recrystallization occurred after deformation at 1050°C and a small overlap of recrystallization and grain growth was observed after deformation at 1100°C. Further, evolution of relative velocity after deformation was analyzed. It was observed that velocity does not vary significantly during recrystallization in the investigated superalloy. This may indicate that recrystallization had occurred without any major textural changes. This conclusion needs further validation with proper texture measurements.   94  Chapter 7  Laser ultrasonic response in pure copper 7.1 Introduction  In this section laser ultrasonic measurement of grain size is extended to another FCC material, i.e. pure copper. Two different approaches were examined. Similar to the measurements conducted on austenite, nickel, and cobalt superalloys, the first approach consisted of generating microstructures with various grain sizes using conventional cold rolling and annealing treatments, i.e. homogeneous microstructures. The second approach on the other hand consisted of developing a graded microstructure within a single specimen using a specially-designed thermo-mechanical treatment on a tapered sample. Ultrasonic attenuation was measured at room temperature on both types of microstructures and correlated with the relevant mean grain sizes measured by metallographic techniques, i.e. SEM and EBSD.  7.2 Laser ultrasonic response in a homogeneous microstructure  Cold-rolled sheet specimens of 99.99% pure copper were annealed at different temperatures in the range of 360°C to 700°C for one hour to generate microstructures with different average grain sizes. Figure 7.1 depicts back-scattered contrast SEM micrographs of the annealed samples. Polygonal structures with a broad range of grain size, i.e. in terms of EQAD from 3 µm to 115 µm, are obtained through recrystallization and grain growth (see Table 7.1). AWGD values are also included in this table which are linearly correlated with the EQAD values. Here, the fit parameters are similar to those obtained for cobalt superalloys, i.e. 𝐴𝑊𝐺𝐷 = (1.59 ± 0.01) × 𝐸𝑄𝐴𝐷. The Dmax/EQAD ratio reported in Table 7.1 ranges from 2.9 to 3.7, this indicates that no substantial tail is developed in the grain size distributions. Further, grain size distributions in terms of cumulative volume fraction with respect to reduced grain size (D/EQAD) are plotted in Figure 7.2.  The fact that all 5 cases overlap confirms the self-similarity of the grain size distribution obtained in these annealing conditions.  95   Figure 7.1 Back-scattered contrast micrographs of sheet specimens annealed at a) 360°C, b) 400°C, c) 500°C, d) 600°C, and e) 700°C for one hour. Table 7.1 Metallographic grain size parameters for sheet samples annealed at various temperature for one hour. Annealing temperature (°C) EQAD (µm) AWGD (µm) AWGD/EQAD Dmax/EQAD 360 3.6 5.5 1.5 3.2 400 13 23 1.8 3.5 500 24 38 1.6 2.9 600 31 50 1.6 3.7 700 115 183 1.6 3.0 96   Figure 7.2 Cumulative volume fraction versus reduced grain size (D/EQAD) in samples with  a homogeneous microstructure. The copper specimen with 3.0 µm EQAD is a suitable reference sample due to its fine grain size and narrow grain size distribution. LUMet experiments were carried out at room temperature (20°C) on these samples (dimensions were 2×10×35 mm). The 𝑏 values are extracted from the measured attenuation spectrum and plotted in Figure 7.3 with respect to the square root of relative change in grain size, i.e. √𝐷2 − 𝐷02 . The vertical error bars inserted in the figure were obtained with the same method as the one used for cobalt superalloys, i.e. the uncertainty in the measurement of parameter 𝑏 at room temperature is 5.6%. The horizontal error bars were obtained by considering relative accuracy of 15% for metallographic grain size measurements. Similar to the results obtained in cobalt superalloys, the attenuation parameter 𝑏 increases with increase in average grain size. Here, the measured 𝑏 value in each specimen can provide an absolute measurement of grain size since the selected reference grain size 𝐷0 is smaller than 5 µm. The attenuation parameter 𝑏 and average grain size obtained in the homogeneous samples and those acquired in the graded specimens which are introduced in the following section, are correlated using the empirical equation (5.1) and shown in Figure 7.10 (section 7.4).  97   Figure 7.3 Square root of attenuation parameter 𝑏 measured at room temperature versus square root of the relative grain size change (EQAD) in Cu samples with a homogeneous microstructure. 7.3 Development of a graded microstructure  The inverse pole figure (IPF) map of the as-received 99.9% pure copper obtained in the RD-TD plane is shown in Figure 7.4.a. The microstructure consists of grains elongated with respect to the rolling direction with an average aspect ratio of 0.4. The EQAD including twin boundaries is determined to be 22 µm. As much as 62% of GOS values are larger than 1° indicating a high level of internal deformation in the as-received state. A uniaxial tensile test at a strain rate of 0.1 s-1 was conducted using a standard tensile specimen at room temperature. The obtained true stress-true strain curve is shown in Figure 7.4.c. The total fracture strain in the as-received state is as small as 0.08 and the work hardening rate is low, confirming that the as-received specimen is heavily deformed.  Samples were annealed at 400°C for one hour to recrystallize the microstructure. Figure 7.4.b shows the IPF map of the annealed specimen in the RD-TD plane. Here, the microstructure consists of grains with no or little apparent elongation in this plane with an average aspect ratio of 0.5. The mean grain size is 18 µm which is slightly smaller than that of the as-received sample. Further, the 98  constituent grains have small or no internal misorientation, i.e. 97% of the GOS values are lower than 1°. Both the average grain size and the significant change in internal misorientation indicates that recrystallization had occurred during the annealing treatment. The true stress-true strain curve of the annealed specimen was obtained using a standard tensile specimen at room temperature and is compared with that of the as-received specimen in Figure 7.4.c. Work hardening rate is significantly increased and the total fracture strain increased by 250% in the annealed specimen. The annealed specimen was therefore selected as the starting material to conduct the strain annealing treatment.  Figure 7.4 Inverse Pole Figure (IPF) map in RD-TD plane of the a) as-received and b) annealed conditions at 400°C for one hour. c) True stress-true strain curves of the respective conditions. To apply the required strain for the strain annealing treatment (Figure 4.8), specimens with tapered geometry were deformed uniaxially in tension. During the tensile experiments, plastic strain was measured continuously on the selected area of the gauge, i.e. from position 20 mm to 50 mm using the DIC camera. The test was stopped manually before the fracture point. Figure 7.5 shows the plastic strain measured by the DIC technique along the gauge length at the end of the test. The position zero is located at the 6mm-width side of the gauge section (Figure 4.7). Strain at positions zero to 20 mm was not measured due to the limited region visualized by the camera. The experiment was conducted twice to examine the repeatability of the measurement technique. 99  Plastic strain is uniformly distributed in positions 20 to 40 mm and shows good repeatability. Beyond position 40mm, deformation becomes localized. The entire metallographic and laser ultrasonic measurements were therefore conducted on the first 40 mm of the gauge length. Following the tensile test, specimens were annealed for 14 hours at 225°C and 6 hours at 325°C, to generate a graded microstructure along the gauge length.   Figure 7.5 Engineering plastic strain measured with DIC along the gauge length at the end of the tensile test. IPF and GOS maps of 6 selected positions on the gauge section of strain annealed specimen in the plane normal to the tensile axis are shown in Figure 7.6 and Figure 7.7, respectively. Microstructures consist of polygonal grains and annealing twins with respective EQAD values reported in Table 7.2. The average grain size at position zero is similar to the average grain size for the specimen annealed at 400°C for one hour (Figure 7.4.b). However, the former has a larger grain orientation spread than the latter, i.e. 91% of grains have GOS values below 1° at position zero. The plastic strain at this position is not sufficient to trigger the recrystallization process. The recovery however has already started. At position 6 mm, as small as 19% of grains have GOS values smaller than 1°, this number increases to 29% at position 12 mm and 93% at position 16 mm (Table 7.2). It can be seen that recrystallization due to the strain annealing treatment is in 100  progress and is in a different stage at each of these three positions. In the last two positions 30 mm and 40 mm, 99% of grains have GOS values below 1° indicating a fully recrystallized microstructure. Moreover, the twin length fraction increases from 0.34 at position 6 mm to 0.40 at 16 mm and remains constant after position 30 mm. Dmax/EQAD values continuously increase along the length, i.e. from 4.6 to 8.4 and are substantially higher than those in sheet specimens tabulated in Table 7.1, i.e. 2.9 to 3.7. Higher Dmax/EQAD especially in the last three positions (16, 30, and 40 mm) indicates the presence of a long tail in the grain size distribution.  Figure 7.6 IPF maps of strain annealed specimen at positions of a) 0 mm, b) 6 mm, c) 12 mm, d) 16 mm, e) 30 mm, and f) 40 mm. 101   Figure 7.7 GOS maps of strain-annealed specimen at positions of a) 0 mm, b) 6 mm, c) 12 mm, d) 16 mm, e) 30 mm, and f) 40 mm. Table 7.2 Microstructure parameters measured from EBSD maps shown in Figure 7.6 Position (mm)  EQAD (µm) AWGD (µm) AWGD/EQAD Dmax/EQAD Twin fraction Rex fraction (GOS<1°) 0 24 43 1.8 4.6 0.39 - 6 24 43 1.8 4.9 0.34  0.19  12 26  49 1.9 5.0 0.35  0.29  16 32  74 2.3 6.2 0.40  0.93  30 30  69 2.3 8.0 0.40  0.99  40 27  68 2.5 8.4 0.40  0.99  102  Grain size distribution can be presented as volume fractions of recrystallized and non-recrystallized grains. Accordingly, the grain population in each position from 6 to 40 mm was partitioned into recrystallized (GOS smaller than 1°) and non-recrystallized (GOS larger than 1°) groups. Distributions of recrystallized and non-recrystallized grains are plotted in Figure 7.8. At a position of 6 mm, recrystallization is in the early stages where comparatively small recrystallized grains appear in the microstructure. The small recrystallized grains subsequently grow into rather larger gains at a position of 12 mm. This suggests that the recrystallized grains with a low stored energy have a growth advantage and consume the deformed grains. Further, presence of the large recrystallized grains leads to formation of a tail in the grain size distribution. At positions of 16, 30, and 40 mm, grain size distributions deviate substantially from lognormality as the large recrystallized grains continue to grow further at the end of recrystallization. At these positions, the Area Weighted Grain Diameter (AWGD) is a more suitable representative grain size than a number-averaged value such as EQAD (Table 7.2). AWGD values at positions 16, 30, and 40 mm are linearly correlated with the EQAD values, i.e. 𝐴𝑊𝐺𝐷 = (2.40 ± 0.07) × 𝐸𝑄𝐴𝐷. Here, the correlation factor is higher than that obtained for both sheet copper samples and cobalt superalloys with self-similar grain size distributions. The correlation factor is a function of the Dmax/EQAD ratio or the width of the distribution, the wider the distribution, the higher the correlation factor is. 103   Figure 7.8 Partitioned grain size distribution of strain-annealed specimen at positions of a) 6 mm, b) 12 mm, c) 16 mm, d) 30 mm, and e) 40 mm. 7.4 Laser ultrasonic response in a graded microstructure  Laser ultrasonic scans were conducted in the centre line of the gauge length of tapered samples in the as-received, annealed, deformed (tensile test 1 and 2), and strain annealed states. Here, the 104  reference waveform was acquired from the regular specimen with 3.0 µm grain size. Figure 7.9.a exhibits variation of the attenuation parameter 𝑏 along the gauge length. The vertical error bars inserted in the figure were obtained with the same method as the one used for cobalt superalloys, i.e. the uncertainty in the measurement of parameter 𝑏 at room temperature is 5.6%. Except for the strain annealed specimen, parameter 𝑏 is approximately constant at different positions along the gauge section indicating that the microstructure is homogeneous through the length and that the applied tensile deformation has negligible effect on ultrasonic attenuation. In the strain annealed sample on the other hand, parameter 𝑏 continuously increases from position 0 to 18 mm by a factor of 4 and then remains approximately constant. Based on the metallographic observations, the increase in parameter 𝑏 from position 0 to 18 mm is attributed to the progress of recrystallization, which leads to the formation of a few large grains within the microstructure.   Figure 7.9 a) Square root of 𝑏 parameter measured along the gauge of tapered samples in the as-received, annealed, deformed (tensile test 1 and 2), and strain annealed conditions. Red solid lines are the two linear fits used to apply the lever rule method. b) Evolution of the fraction recrystallized obtained from LUMet parameter 𝑏 (open circles) and metallography (closed circles). 105  The fraction recrystallized can also be obtained by applying a lever rule method on the parameter 𝑏 measured along the gauge length in a first approximation. The first limit required to apply the lever rule was obtained by fitting a constant equation to the 𝑏 measured on the entire gauge length of the annealed sample. The second limit was obtained by applying a constant fit to the 𝑏 measured on the strained annealed sample from position 18 to 40 mm (see Figure 7.9.a). Figure 7.9.b compares the recrystallized fraction obtained in this method and the recrystallized fraction acquired from metallography as reported in Table 7.2. In agreement with the metallographic results, LUMet results show that the evolution of parameter 𝑏 is directly related to the fraction recrystallized. Beyond position 8 mm, the attenuation parameter 𝑏 is mostly governed by the large recrystallized grains, occupying a substantial volume of the probed sample.  Attenuation parameter 𝑏 and average grain size are correlated using the empirical equation (5.1) as shown in Figure 7.10.a and Figure 7.10.b. Here, both EQAD and AWGD are used as a measure of average grain size. 𝑏 values measured on both sheet specimens with a homogeneous microstructure (named homogeneous) and the strain annealed specimen with a graded microstructure (named graded) were used to establish the correlation. For the sake of simplicity the fitting parameter 𝜀 was imposed to be the value obtained in cobalt superalloys, i.e. 0.23, since copper and cobalt have similar single crystal elastic anisotropy factors (3.2 and 2.8, respectively). The fit parameter obtained using EQAD (Figure 7.10.a) is 𝛿 = 0.031 ± 0.002 which is similar to the one acquired in cobalt superalloys, i.e. 𝛿 = 0.024 ± 0.006. When the grain size distribution is self-similar, i.e. Dmax/EQAD is in the range 2.0 to 5.0, this correlation can be used to predict the average grain size with an acceptable precision. The relative precision in LUMet measurement of grain size in terms of EQAD is ±13% [38]. However, when the distribution is non-self-similar, i.e. Dmax/EQAD is above 5.0, the EQAD-based correlation underestimates the value of average grain size. This is due to the fact that when a long tail exists in the distribution, AWGD is a more suitable representative of the effective average grain size rather than the EQAD. In such a scenario, an AWGD-correlation is used to predict the apparent average grain size (Figure 7.10.b). Here, the fit parameter is 𝛿 = 0.020 ± 0.001. The main advantage of the AWGD-based correlation is that a measure of the average grain volume can be obtained with a simple laser ultrasonic experiment 106  straightforwardly without any further data manipulations or any need to conduct the labor-intensive and complicated ex-situ characterizations such as 3D EBSD.   Figure 7.10 Correlation between the relative change in grain size and ultrasonic parameter 𝑏 measured at room temperature on sheet specimens with a homogeneous microstructure and on the strain annealed specimen with a graded microstructure using a) EQAD and b) AWGD as the measure of mean grain size. 7.5 Summary     Laser ultrasonic grain size measurement was extended to pure copper. To this aim, the frequency-dependent attenuation parameter was measured at room temperature on specimens with two different geometries, sheet and tapered. Sheet samples consisted of homogeneous microstructures with mean grain sizes (EQAD) ranging from 3 µm to 115 µm. Tapered samples consisted of a graded microstructure deformed at one end and fully annealed at the other end with a more or less an apparent constant mean grain size (EQAD). It was observed that a graded microstructure can be distinguished from a homogeneous microstructure by measuring the parameter 𝑏 along the sample gauge length. An empirical fit was used to correlate attenuation parameter 𝑏 and metallographic grain size values (EQAD and AWGD). When grain size distribution is self-similar, either EQAD-based or AWGD-based correlation can be used to 107  estimate average grain size. In scenarios where a substantial tail exists in the grain size distribution, the AWGD-based correlation provides a more accurate estimate of average grain size. In general, laser ultrasonics provides a volumetric measure of the average grain size. The LUMet grain size is therefore more consistent with the AWGD values since an area-weighted grain diameter is a better representative of the grain volume compared to a number-averaged grain diameter.   108  Chapter 8  Discussion  8.1 Harmonization of grain size measurements with laser ultrasonics   Empirical correlations have been developed between the frequency dependent attenuation parameter 𝑏 and the mean grain size measured by metallography in weakly textured FCC polycrystalline metals. The general form of these equations is given by 𝛼(𝑓, 𝑇) = Γ(𝑇)𝐷𝑛−1𝑓𝑛. In an attempt to harmonize the existing empirical correlations for different materials, a normalizing parameter associated with the amplitude of grain scattering is needed.  Theoretical investigations  Polycrystalline metals consist of a multitude of grains, each is elastically anisotropic and has a specific crystallographic orientation. Variation of elastic properties from grain to grain causes scattering of ultrasonic waves at grain boundaries. If the grain to grain variation of elastic properties is small and all the crystallographic orientations exist with equal frequency, the material is considered isotropic on the macroscopic scale. In an isotropic polycrystal, Stanke and Kino defined a variable to measure the degree of inhomogeneity in terms of single crystal elastic constants [58]. Under this assumption, attenuation of the longitudinal waves in the Rayleigh asymptote 𝛼𝑙𝑅(𝑓, 𝑇) and the stochastic limit 𝛼𝑙𝑆(𝑓, 𝑇) are defined respectively by Bhatia and Moore [146] and Lifshits and Parkhomovski [147] as: 𝛼𝑙𝑅(𝑓, 𝑇) = 𝜉𝑅(13) × 𝑓4 × 𝑉𝑔 (8.1) 𝛼𝑙𝑆(𝑓, 𝑇) = 𝜉𝑆(13) × 𝑓2 × ?̅? (8.2) Here 𝑓 is the ultrasonic wave frequency, 𝑉𝑔 is the grain volume, and ?̅? is an effective linear grain size. 𝜉𝑅 and 𝜉𝑆 are relative inhomogeneity factors in the Rayleigh and the Stochastic regimes and are defined respectively as: 109  𝜉𝑅 = [(𝜀𝑙𝑅(𝑇))2 × (𝜋𝑣𝑙0(𝑇))4]3 (8.3) 𝜉𝑆 = [(𝜀𝑙𝑆(𝑇))2 × (𝜋𝑣𝑙0(𝑇))2]3 (8.4) where 𝑣𝑙0(𝑇) is the longitudinal wave velocity in an isotropic medium obtained using the Hill averaging method at the temperature 𝑇. 𝜀𝑙𝑅(𝑇) and  𝜀𝑙𝑆(𝑇) are relative elastic inhomogeneity in the Rayleigh and the Stochastic regimes, respectively, that are defined as:  (𝜀𝑙𝑅(𝑇))2 =1375𝜋{𝑐(𝑇)2(𝑐110 (𝑇))2} {1 +32(𝑣𝑙0(𝑇)𝑣𝑠0(𝑇))5} (8.5) (𝜀𝑙𝑆(𝑇))2 =4525{𝑐(𝑇)2(𝑐110 (𝑇))2}   (8.6) where 𝑐(𝑇) = 𝑐11(𝑇) − 𝑐12(𝑇) − 2𝑐44(𝑇) is the invariant anisotropy factor of a cubic crystal at the temperature 𝑇, 𝑐110 (𝑇) is the unweighted average of 𝑐11(𝑇) over all possible orientations, and 𝑣𝑠0(𝑇) is the shear wave velocity in an isotropic polycrystal obtained using the Hill averaging method. By comparing Equations (8.1) and (8.2) with the empirical scattering correlation, i.e. 𝛼(𝑓, 𝑇) = Γ(𝑇)𝐷𝑛−1𝑓𝑛, it can be seen that the relative inhomogeneity factor can be selected as the scaling parameter controlling the amplitude of interaction or scattering of the wave with the constituent grains. In practice, grain size in metals generally ranges from 1 to 500 μm which is smaller than or comparable to the wavelength of laser-generated ultrasound. In this study, the maximum grain size measured was in the order of 250 μm. It is therefore assumed that the studied grain structures are only in the Rayleigh regime and/or in the transition of the Rayleigh to the stochastic regime.  The single crystal elastic constants, i.e. 𝑐11(𝑇),  𝑐12(𝑇) and 𝑐44(𝑇), are required for calculation of 𝜀𝑙𝑅(𝑇), which can be obtained from the previous studies in different metals (Table 8.1.) Wazzan et al. obtained the three independent single crystal elastic constants of FCC cobalt by measuring the velocities of elastic waves, longitudinal and shear, in the temperature range of 0 to 315°C using conventional ultrasonic technique [148]. Linear fits were applied to the Wazzan et al. data points to obtain the temperature dependence of the elastic constants in cobalt as shown in Table 8.1. 110  Single crystal compliance constants of two nickel superalloys in the temperature range of 27 to 1142°C were determined using the dynamic resonant spectroscopy technique [149], [150]. Here, quartic polynomial equations were applied to the experimental data to describe the compliance constants of nickel superalloys with respect to temperature.  Table 8.1 Single crystal elastic constants of FCC cobalt, 𝛾-iron, and copper and compliance constants of nickel superalloys reported in literature Material  Method T (°C) 𝑐11 or 𝑠11 (GPa) 𝑐12 or 𝑠12 (GPa) 𝑐44 or 𝑠44 (GPa) cobalt Ultrasonics [148] 0 to 315 𝑐11= −0.08T + 219 𝑐12= −0.07T + 145 𝑐44= −0.03T + 124 Nickel Dynamic resonant spectroscopy [149], [150] 27 to 1142 𝑠11= 8.02 − 4.17× 10−4𝑇 + 1.61× 10−5𝑇2− 2.71× 10−8𝑇3+ 1.54× 10−11𝑇4 𝑠12= −2.89 − 4.15× 10−4𝑇 − 4.29× 10−6𝑇2+ 8.05× 10−9𝑇3− 5.08× 10−12𝑇4 𝑠44= 7.55 + 2.39× 10−3𝑇 − 1.15× 10−7𝑇2− 1.07× 10−9𝑇3+ 1.63× 10−12𝑇4 Copper Ultrasonics [53], [151-155] -268 to 527  𝑐11= −0.04T + 172 𝑐12= −0.02T + 126 𝑐44= −0.02T + 82 𝛾-iron Neutron diffraction [156] 1155 𝑐11 = 154 ± 14 𝑐12 = 122 ± 13 𝑐44 = 77 ± 8 Several researchers measured the single crystal elastic constants of pure copper. Goens [151] measured the single crystal elastic constants of copper at room temperature (27°C) using conventional ultrasonic technique for the first time. Then, Lazarus et al. [152] obtained the pressure dependence of the elastic constants of pure copper by measuring longitudinal and shear wave velocities in single crystals.  Following a similar methodology as Lazarus et al. [152], Hill [53] and Schmunk et al. [153] respectively obtained the elastic constants of pure copper and copper-nickel alloys with an absolute accuracy of ±0.5%. Temperature dependence of the single crystal elastic properties of copper was first measured by Overton and Gaffney in the temperature range of -268 to 27°C using contact ultrasonics [154]. Later on, Chang et al. [155] extended the temperature range of measurement to 527°C with the same absolute accuracy as the previous 111  studies. Linear regressions were applied to all the reported values to acquire the temperature dependence of the elastic constants in pure copper in the range of -268 to 527°C. In case of γ-iron, Zarestky and Stassis measured the elastic constants at 1155°C using inelastic neutron scattering [156]. Figure 8.1 shows the temperature dependence of the relative inhomogeneity factors in the Rayleigh regime  𝜉𝑅 in cobalt, nickel, 𝛾-iron, and copper in the temperature range available in the literature. Copper has a higher relative inhomogeneity factor as compared to nickel and cobalt at room temperature (27°C). The  𝜉𝑅 of cobalt and nickel are similar in the temperature range of 27 to 315°C. Above 827°C,  the relative inhomogeneity factor of nickel substantially increases to a value as high as 0.124 at 1142°C. The 𝜉𝑅 of 𝛾-iron is lower than that of nickel at 1155°C.   Figure 8.1 Temperature dependence of the relative inhomogeneity factor in the Rayleigh regime for cobalt, nickel, 𝛾-iron, and copper. The data points were calculated using Equations (8.1) and (8.5), and the single crystal elastic constants reported in the respective metals. The attenuation spectrum of a longitudinal ultrasound wave in cobalt, nickel, and copper at 27°C and in 𝛾-iron at 1155°C are calculated using Equation (8.1) in the Rayleigh regime as shown in Figure 8.2.a. The grain size was assumed to be 50 µm. 𝛾-iron has the highest attenuation at each frequency followed by copper, cobalt, and nickel. As discussed earlier, the attenuation is related to the relative inhomogeneity factor, the higher the  𝜉𝑅, the larger the attenuation at each frequency 112  (Table 8.2). Attenuation versus frequency of the studied metals can be scaled into one single spectrum according to the respective inhomogeneity factors introduced earlier (Figure 8.2.b). The Hill averaging scheme was selected for the calculation of the elastic constant and longitudinal velocity, i.e. 𝑐110 (𝑇) and 𝑣𝑙0(𝑇), assuming an isotropic continuum. The isotropic assumption corresponds to a polycrystalline aggregate of randomly oriented grains where a sufficient number of grains exists for the polycrystalline aggregate to be considered elastically isotropic. The inhomogeneity factors were calculated at 27°C for nickel, cobalt, and copper and at 1155°C for 𝛾-iron. The relative inhomogeneity 𝜀𝑙𝑅(𝑇) and longitudinal velocity 𝑣𝑙0(𝑇) used to calculate the  𝜉𝑅 are summarized in Table 8.2.  Figure 8.2 a) Calculated attenuation spectrum in the studied metals in the Rayleigh regime and b) Attenuation spectrum scaled by  𝜉𝑅(13). Table 8.2 𝜀𝑙𝑅(𝑇) and 𝑣𝑙0(𝑇) for nickel, cobalt, and copper at 27°C and for 𝛾-iron at 1155°C Material T (°C) 𝜀𝑙𝑅(𝑇) 𝑣𝑙0(𝑇) (mm/µs) 𝛾-iron 1155 0.135 4.882 Nickel 27 0.094 5.315 Cobalt 0.107 5.587 Copper 0.108 4.750 113  Experimental investigations  The frequency dependent attenuation parameter 𝑏 was measured experimentally in different metals and correlated with the grain size measured by metallography. The following assumptions were considered in all the available correlations constructed for FCC single-phase metals [37], [38], [61], [84], [157]: 1. The attenuation spectrum is measured using the single-echo technique and is defined by the ratio of the amplitude spectrum of an echo measured in the tested sample to the amplitude of an echo acquired in a reference sample in which the attenuation is weak.  2. The parameter 𝑏 is determined by fitting a power law equation (Equation 2.15) to the measured attenuation spectrum imposing a constant value of 3 for exponent 𝑛.  3. The metallographic grain size for nickel, cobalt, and copper is defined as the mean equivalent area diameter (EQAD) and the area weighted grain diameter (AWGD) including twin boundaries in the statistics for grain size distributions, Dmax/EQAD, in the range 2 to 5. for steels in the austenite phase, the metallographic grain size is only determined as EQAD.  The calibration available for steels in the austenite region was developed by Kruger et al. using an amplitude spectrum of a reference measured at room temperature in a sample composed of fine ferrite grains (Reference grain size 𝐷0 < 5 µ𝑚). The square root of 𝑏 was plotted with respect to the absolute value of austenite grain size (Figure 2.8) to obtain the calibration equation as [158]:   √1000𝑏 = 𝐴1 × (𝑇 − 900) + 𝜂 × √|𝐷2 − 𝐷02|1−𝜀 (8.7) where 𝐴1 = 0.0005, 𝜂 = 0.0461, 𝜀 = 0.30870, 𝐷 is the grain size, and 𝑇 is temperature in degree Celsius. Here, 𝐷0 is assumed to be zero. The first part of the equation accounts for temperature and the second part is related to the grain size contribution. Equation (8.7) is illustrated in Figure 8.3.a at three different temperatures. The grain size calibration for nickel superalloys was developed by Garcin et. al  [37], [84] using an amplitude of a reference measured at 1050°C in a sample composed of grains with 20 µm mean grain size (EQAD). 𝑏 was related to the relative change in grain size. The nickel superalloys grain size calibration at 1050°C is 114  shown in Figure 8.3.b. The grain size calibration for cobalt superalloys was develped in the present work using the reference signal measured at room temperature in a sample composed of grains with a mean grain size (EQAD) of 4 µm as shown in Figures 5.5.a and b. The grain size calibration introduced in chapter 5 can be expressed as: √1000𝑏 = 𝛿𝐶𝑜(1 + κ𝑐𝑜 × 𝑇1.5) × √|𝐷2 − 𝐷02|1−𝜀 (8.8) where the fit parameters are 𝛿𝐶𝑜 = 0.024 ± 0.006, κ𝑐𝑜 = (2.453 ± 0.008) × 10−5 °𝐶−3/2, and 𝜀 = 0.23 ± 0.06. The temperature contribution (1 + κ𝑐𝑜 × 𝑇1.5) was measured in the range of 200 to 1200°C. Equation (8.8) is shown in Figure 8.3.c for three different temperatures.  Finally, the calibration for grain size in copper was developed in the present work as shown in Figure 7.9.a using the same approach as the one used for nickel and cobalt superalloys. The reference sample was composed of grains with a mean grain size (EQAD) of 3 µm at room temperature. For the sake of harmonization, only data points measured from samples with a self-similar grain size distribution, i.e. Dmax/EQAD in the range 2 to 5 are considered here. Based on this assumption, 𝛿𝑐𝑢 = 0.030 ± 0.002. Here, the temperature contribution parameter κ𝐶𝑢 is not obtained since the measurements were only conducted at room temperature. Figure 8.3.d shows the copper calibration. It can be seen from Figure 8.3.a to Figure 8.3.d that the temperature contribution to attenuation is different in 𝛾-iron (austenite) and cobalt superalloys. It should also be mentioned that κ𝑁𝑖 was not reported in the previous investigation [37]. 115   Figure 8.3 Correlation between square root of laser ultrasonic parameter 𝑏 and square root of the relative change in square of grain size for a) steels in austenite phase developed at 900, 1050, and 1300°C [158],  b) nickel at 1050°C [37], c) cobalt at 900, 1050, and 1300°C [38], and d) copper at 27°C. The average grain sizes in terms of AWGD are plotted with respect to the EQAD values in Inconel, cobalt superalloys, and copper in Figure 8.4. As stated in chapters 5 and 7, EQAD and AWGD values are linearly correlated in pure copper and cobalt superalloys. The correlation factors for cobalt and copper respectively were 1.58 ± 0.02 and 1.59 ± 0.02 for microstructures with Dmax/EQAD ratio in the range of 2 to 5. In nickel superalloy, EQAD and AWGD values are also linearly correlated and the correlation factor is1.90 ± 0.04 for microstructures with Dmax/EQAD ratio in the range of 3.7 to 7.3 [37]. The AWGD values for austenite were not available in the 116  literature and only the EQAD values were measured [158]. As the Dmax/EQAD ratio studied in steels in the austenite phase was in the range of 2 to 5, it was assumed that the EQAD and the AWGD values are linearly related with a correlation factor of 1.6.   Figure 8.4 Correlation between AWGD and EQAD values measured in a) Inconel, b) cobalt superalloys, and c) copper. Experimental data points measured in steels in the austenite phase, nickel and cobalt superalloys, and copper are compared in Figure 8.5 in terms of variation of square root of ultrasonic parameter 𝑏 with respect to square root of the relative change in square of grain size in terms of AWGD. Here, the error bars are removed for the sake of clarity. At each of the relative grain size values, nickel superalloys have a higher attenuation and attenuation parameter 𝑏 as compared to austenite, copper and cobalt. This behavior can be explained by the higher relative inhomogeneity factor of nickel superalloys at1050°C as compared to austenite and cobalt (see Figure 8.1) which is directly related to the larger attenuation parameter obtained in this material (see Figure 8.3). Moreover, the data points obtained along the gauge length of copper specimen with a long-tailed grain size distribution are included for comparison (labeled graded Cu). As discussed in chapter 7, Dmax/EQAD ratio in graded copper ranges from 6.2 to 8.4 which is significantly higher than that in other metals, i.e. 2 to 5.   117   Figure 8.5 Square root of 𝑏 measured with laser ultrasonics versus square root of the relative change in grain size in terms of AWGD in austenite [158], Inconel [37], cobalt superalloys [38], and copper. 𝑏 values measured in the studied metals can be scaled by the relative inhomogeneity factor 𝜉𝑅 under the isotropic polycrystalline aggregate assumption. The 𝜀𝑙𝑅(𝑇) described in Equation (8.5) can also expressed as: (𝜀𝑙𝑅(𝑇))2 = (𝜀𝑙𝑅(27))2 × 𝛤(𝑇) (8.9) where 𝛤(𝑇) is the experimentally measured temperature contribution function. The 𝛤(𝑇) acquired in cobalt superalloys was 𝛿 + κ𝑐𝑜 × 𝑇1.5 (see Equation (8.8)). The 𝛤(𝑇) obtained here for cobalt superalloys is in agreement with the temperature dependence of the relative inhomogeneity factor reported in the previous investigations. 𝛤(𝑇) for steels and nickel superalloys has not been measured and the temperature contributions were obtained from Equation (8.5) using the existing experimental data in the literature. Square root of the 𝑏 values scaled by the 𝜉𝑅 with respect to square root of the relative change in grain size (AWGD) is shown in Figure 118  8.6. Following the scaling of the 𝑏 values by the 𝜉𝑅, cobalt, nickel, and austenite show similar scattering behaviors.   Figure 8.6 Square root of scaled attenuation parameter 𝑏 with respect to the relative change in square of grain size (AWGD) in the studied metals. The discrepancies observed in the scaled experimental data points of copper from other materials can be due to the state of the reference sample used to obtain the 𝑏 values. Figure 8.7.a to c show the intensity of the orientation distribution function (ODF) at 45° section for Inconel, cobalt superalloy, and copper references, respectively. The maximum ODF intensity in the reference samples of cobalt and nickel superalloys in this section are 1.7 and 1.4, respectively, this indicates that the constituent grains are randomly oriented. The copper reference waveform was obtained from a sample annealed at 360°C for one hour. Before the annealing treatment, the sample was 93% rolled at 77 K. The maximum ODF intensity of copper is 5 which is higher than those in cobalt and nickel superalloys (Figure 8.7.c). This maximum ODF intensity on the other hand is lower than that in a fully recrystallized pure copper reported by Lim and Rollet [159], i.e. 16. The obtained ODF maximum intensity depends on many factors such as the measurement method, i.e. XRD, Neutron diffraction, or EBSD, and the plane in which the scan is conducted. This suggests that the constituent grains in the copper reference sample are not randomly oriented. Further, the 119  relative inhomogeneity factor (Equation (8.3)) was calculated under the isotropic aggregate assumption. According to the EBSD measurements, the isotropic assumption does not stand for the copper reference with a measurable texture, resulting in a different scattering behavior than Inconel and cobalt superalloys with an isotropic reference. Further investigation on the scattering behavior of copper is needed to confirm this explanation. It should be noted that attenuation in textured metals with a cubic structure was also described by explicit mathematical equations [58], [160]. The existing formalisms for textured aggregates are however, more complicated than those for the isotropic assumption by Stanke and Kino [58] and beyond the scope of the present work.  Figure 8.7 45° ODF intensity projections of a) Inconel, b) cobalt, and c) pure copper references. Based on the above considerations, a single linear fit can be applied to the scaled experimental data points of cobalt, nickel, and austenite (Figure 8.6). This means that the fit parameter 𝜀 introduced in Equations (8.7) and (8.8) is assumed zero for the sake of simplicity. This assumption is not far from reality since only the data points in the Rayleigh regime are considered here. Accordingly, variation of parameter 𝑏 with the grain size change (AWGD) in FCC metals can be described by a harmonized equation in the form: √1000𝑏 = 𝜓𝜉𝑅√𝐷2 − 𝐷02 (8.10) where  𝜓 = 4.16 × 106 ± 7 × 104 is a fitting parameter that applies to weakly textured materials, 𝜉𝑅 represents the material properties, and √𝐷2 − 𝐷02 accounts for the scale of the microstructure. 120  For sufficiently small 𝐷0, Equation (8.10) depicts the measured 𝑏. If a large 𝐷0 is used, the measured 𝑏 values are essentially re-scaled. According to Equation (8.10), 𝑏 = 0 for 𝐷 = 𝐷0. For this case, the change in grain size is essentially measured, i.e. whether 𝑏 increases or decreases with respect to the reference 𝑏. If the reference grain size 𝐷0 is known, the absolute value of grain size 𝐷 can also be obtained. Figure 8.8.a to Figure 8.8.e show the evolution of the LUMet grain size (AWGD) in a series of  L605 and CCM superalloys as an example. The LUMet grain size values were calculated using the cobalt calibration (the smoothed LUMet grain size evolution in terms of the EQAD was also shown in Figure 5.7 and Figures 6.2.a to c, respectively) and the harmonized equation. The 𝐷0 in Figure 8.8. a and b is 4 μm and in Figure 8.8. c to e is 12 μm. It can be observed that grain size evolution obtained from the harmonized equation is in good agreement with the evolution acquired from the cobalt calibration.   Figure 8.8 Grain size evolution (AWGD) in a) L605 and b) CCM superalloys during isothermal holding at 1200°C. Grain size evolution in L605 after hot deformation at c) 1000°C, d) 1050°C, and e) 1100°C using the cobalt calibration and the harmonized Equation (8.10). 121  The established harmonized equation can facilitate the measurement of grain size evolution in other metals and alloys. To this aim, the temperature dependence of elastic constants in the studied material is required to calculate the relative inhomogeneity factor. If the obtained relative inhomogeneity factor with respect to temperature is similar to those reported for cobalt, nickel or austenite, Equation (8.10) can be used to determine grain size in the studied material. In such a scenario, only one laser-ultrasonic measurement is needed on a sample with a known grain size (𝐷0). This known grain size can be either small or relatively large and the laser ultrasonic measurement can be conducted at room or an elevated temperature. If the obtained relative inhomogeneity factor versus temperature is different from those reported for cobalt, nickel, or 𝛾-iron, the temperature contribution to attenuation has to be measured at a constant grain size, e.g. the attenuation parameter 𝑏 has to be measured during a continuous heating or cooling scenario, while the average grain size is known and remains constant. It can be seen that the harmonized equation significantly reduces the number of experiments and labor-intensive characterizations required for the design of a new calibration. In some cases, only a few measurements are required to use the harmonized equation. In general, when a new alloy system is being studied, the harmonized equation would provide the expected range of attenuation and therefore eliminates or at least simplifies the design of a new calibration.    8.2 Numerical investigation of grain boundary type effects on ultrasonic attenuation  To investigate the effect of grain boundary type on ultrasonic attenuation, single phase untextured microstructures containing hexagonal grains with narrow grain size distributions were generated using a centroidal Voronoi tessellation [157]. Such a microstructure is shown in Figure 8.9.a. Each grain was defined with a specific crystallographic orientation and assigned with a particular stiffness tensor. The stiffness tensor for each grain was calculated by rotating the single crystal stiffness tensor of copper at room temperature according to the grain crystallographic orientation using CTOME software. First, a random grain boundary misorientation distribution ranging from 0° to 65° was considered for the microstructure. The random misorientation distribution was in agreement with the MacKenzie distribution (Figure 8.9.b). Then, a percent of orientations ranging from 5.9% to 25% was replaced with the twin boundary misorientation, i.e. 60° rotation about <111> direction. In order to substitute a grain boundary by a twin misorientation, the following 122  procedure was applied. A grain "A" is selected randomly in the map, and its nearest neighbor, grain "B" located on the right hand side was identified. The orientation of the neighboring grain B was then recalculated so that the misorientation between grain A and B is defined as 60° rotation about the <111> axis. The operation is repeated until the fraction of boundary replaced reach the desired twin fraction. FE simulation was conducted in Abaqus CAE using 2D dynamic explicit modelling. Dimensions of the FE template was 2 mm in thickness and 10 mm in length and the mesh size was 4 µm in the central region, i.e. 2 by 4 mm, (see Figure 4.15). An ultrasonic pulse was generated along a 2 mm length at the top boundary of the central region. This 2 mm length is in agreement with the diameter of the LUMet laser spot. The vertical displacements of nodes at the generation line were averaged and plotted with respect to time to obtain the simulated waveforms. From the obtained waveforms, the attenuation spectrum was calculated. The reference waveform required to calculate the attenuation spectrum was generated by travelling the FE pulse in the isotropic media. The isotropic media was the area outside the central region and was defined by one stiffness tensor averaged over all orientations in the microstructure using the Voigt averaging method. The FE attenuation spectrum for different amounts of twin misorientation is presented in Figure 8.9.c. The attenuation spectrum is not measurably affected by the presence of twin boundaries up to 25% in the microstructure. These simulations provide a preliminary analysis on the effect of twin boundaries on ultrasonic attenuation. Here, the geometry of twin boundary was assumed to be the same as for other regular high angle grain boundaries. However, a real twin boundary in an FCC metal has rather a faceted geometry.    123   Figure 8.9 a) Microstructure generated by centroidal Voronoi tessellation. b) misorientation distribution of grain boundaries for microstructures with 0% (top) and 14.7% (bottom) twin boundaries. c) FE-simulated attenuation spectra of samples with different percent twin boundaries ranging from 0 to 25% [157]. To investigate the effects of twin boundaries considering both the misorientation angle and the geometry, actual microstructures containing equiaxed grains with a weak texture were taken from the EBSD maps of two cobalt superalloys. The first one was the L606 annealed at 1050°C for 300s (L1050_300s) after deformation (see Figure 6.5.a) and the other one was the L605 annealed at 1100°C for 1000s (L1100_1000s) from a 23% cold rolled state. The methodology of the FE simulation was described in section 4.6.2. Figure 8.10.a to d show the grain maps with and without twin boundaries and their respective grain boundary misorientation distributions for the L1050_300s sample. When twin boundaries are included in the microstructure (Figure 8.10.a and c) , as much as 43% of the boundaries have misorientation angle of the twin boundaries (Table 8.3). By excluding twins (Figure 8.10.b and d), the misorientation distribution becomes equivalent to a MacKenzie random distribution. Similar results were obtained in the sample L1100_1000s (Figure 8.11.a to d). In this case, the percent twin boundary is 29%. Mean grain size and Dmax/EQAD values of these microstructures are summarized in Table 8.3. In order to only investigate the effect of twin boundaries on ultrasonic attenuation, grains were scaled in such a way that microstructures with and without twins had similar mean grain size values.   124  Table 8.4 summarizes the scaling factors and the scaled mean grain sizes in the studied samples.  Figure 8.10 Grain map of the L1050_300s sample a) with and b) without twin boundaries. Grain boundary misorientation distributions c) including and d) excluding twin boundaries. The MacKenzie distribution is also included for comparison.   125   Figure 8.11 Grain map of the L1100_1000s sample a) with and b) without twin boundaries. Grain boundary misorientation distributions c) including and d) excluding twin boundaries. The MacKenzie distribution is also included for comparison. Table 8.3 Microstructure parameters of the L1050_300s and L1100_1000s samples measured from EBSD maps Specimen  EQAD with twins (µm) Dmax/EQAD (with twins) EQAD without twins (µm) Dmax/EQAD (without twins) Twin boundary % L1050_300s 15  4.7 25  3.7  43   L1100_1000s 7 4.3 10  4.5  29     126  Table 8.4 Scaled microstructure parameters used for the FE simulations Specimen  EQAD with twins (µm) EQAD without twins (µm) Scaling factor with (without) twins L1050_300s 44 50 3.0  (2.0) L1100_1000s 32 32 4.5   (3.2)  The FE template was generated with half of each EBSD map using a mirroring process. The EBSD map was translated and/or rotated along the mirror plane. For each map, the position of the mirroring plane was changed while the position of wave generation was kept constant. Translation of the mirror plane in the L1050_300s sample with and without twins can be seen in the velocity maps shown in Figure 8.12.a and b. Longitudinal velocity ranges from 4.3 to 5.2 mm/µs. The FE simulation was conducted four times for each scenario to obtain the average attenuation spectrum. The reference waveform was generated by propagating the FE pulse in the isotropic media.   Figure 8.12 Voigt averaged longitudinal velocity in the y direction for the L1050_300s sample a) with and b) without twins. Position of the mirror plane was changed for each FE simulation.  The FE simulated attenuation spectra in the L1050_300s sample with and without twins are shown in Figure 8.13.a. At low frequencies, attenuation is similar in both cases, but at higher 127  frequencies attenuation in the sample without twins is higher. This difference is due to the fact that the mean grain size value without twins is slightly higher than the mean grain size with twins, i.e. 50 µm versus 44 µm. The larger the average grain size the higher the attenuation. Figure 8.13.b compares the attenuation spectra obtained in the L1100_1000s sample with and without twins. Here, the average grain size is exactly the same in both cases and therefore any differences in their attenuation spectra is only associated with the presence of twin boundaries. However, variation of attenuation versus frequency between the two conditions is insignificant, this suggests that twin boundaries have similar scattering behavior as other regular high angle grain boundaries. The twin fraction therefore cannot be extracted by laser ultrasonic measurements.   Figure 8.13 Comparison between the FE simulated frequency dependence of attenuation for templates with and without twin boundaries for the a) L1050_300s and b) L1100_1000s samples. It should be mentioned that many twin boundaries occur in closely packed pairs such as Figure 6.5. In such a scenario, if the twinned area appears in a large grain, twin boundary must be considered since a substantial grain volume is located between these boundaries. In the Rayleigh regime, number and volume of grains cause scattering rather than the grain boundaries. Further, elastic stiffness tensor of any two grains with the twin misorientation relationship is not similar. For instance, if an arbitrary copper grain with Euler angles of 39°, 66°, 27° is considered, its stiffness tensor can be calculated by rotating the single crystal stiffness tensor of copper as:   128  239 89 88 0 0 0 89 222 106 0 24 0 89 106 222 0 24 0 0 0 0 58 0 24 0 24 24 0 40 0 0 0 0 24 0 40 where the components unit is GPa. Second grain is obtained by rotating the first grain about <111> axis to the amount of 60° (twin misorientation relationship in FCC metals).  Euler angles of the second grain is: 184 119 112 0 24 0 119 222 75 0 14 0 119 75 222 0 14 0 0 0 0 27 0 14 24 14 14 0 64 0 0 0 0 14 0 64 Grains with different stiffness tensors show different scattering behaviors. If the twinned grain is smaller than 5μm, the grain volume between twin boundaries resembles small particles and cannot be measured with the LUMet system.    129  Chapter 9  Conclusions and future work  9.1 Conclusions The Laser Ultrasonics for Metallurgy (LUMet) system was used for in-situ investigation of recrystallization and grain growth in FCC polycrystalline metals during thermo-mechanical treatments. FE modelling approach was used to investigate the effect of grain boundary misorientation distribution on the ultrasonic attenuation. The outcomes of this research are summarized in the following:  1. Evolution of mean grain size in cobalt superalloys, CCM and L605, was monitored in-situ during continuous heating and isothermal holding between 1000°C to 1200°C following cold rolling. To this aim, an empirical correlation was established between frequency-dependent attenuation parameter and metallographic grain size. When the grain size distribution is self-similar, either EQAD-based or AWGD-based correlation can be used to estimate the average grain size. The kinetics of recrystallization is monitored in real time through the refinement of mean grain size. With the established calibration, it is possible to quantitatively compare the recrystallization rates in the studied cobalt superalloys. During the investigated grain growth scenarios, grain size distributions remained self-similar and the relative fraction of annealing twins did not evolve to a measurable extent. The LUMet grain growth rate was described with a power law grain growth model in the studied temperature range.  2. Recrystallization and grain growth following hot compression between 1000°C and 1100°C was monitored in-situ in the L605 cobalt superalloy through the grain size evolution. Here, the established grain size calibration was used to indicate the progress of recrystallization. Analysis of the grain size evolution after deformation revealed the different scenarios encountered in L605 superalloy after deformation in the studied temperature range. For instance, after deformation at 1000°C, grain refinement was less than that after deformation at 1050 and 1100°C. This suggests that recrystallization was only partial after deformation at 1000°C while full recrystallization occurred after 130  deformation at 1050°C and 1100°C. Further, insignificant variation of ultrasonic velocity during the course of recrystallization gave a first indication that recrystallization was not accompanied by major textural changes in L605 superalloy.  3. The LUMet technology was used to characterize samples with homogeneous microstructures (self-similar grain size distributions) and graded samples with varying grain size distributions (from lognormal to long-tailed). An empirical equation has been proposed to correlate the attenuation parameter and the metallographic grain size. It was observed that when a substantial tail exists in the grain size distribution, the AWGD-based correlation provides a better estimation of the representative average grain size than the EQAD-based correlation. Laser ultrasonics is a volumetric measure of properties of the probed material. Thus, LUMet grain size is more consistent with the AWGD values since an area-weighted grain diameter is a better representative of the grain volume compared to a number-averaged grain diameter.  4. For the first time, all the existing empirical correlations developed to laser ultrasonically measure mean grain size in FCC metals, i.e. iron, nickel, cobalt and copper, have been harmonized. It was confirmed that the single crystal elastic constants directly control the amplitude of grain scattering. The established harmonized equation can be potentially used to measure the grain size evolution in other metals and alloys without the need to develop a new calibration, which requires an extensive number of experiments and labor-intensive ex-situ characterizations. To use the harmonized equation, the temperature dependence of elastic constants in the studied material is required to calculate the scaling parameter in the equation (the relative inhomogeneity factor). If the obtained relative inhomogeneity factor versus temperature is similar to those reported for cobalt, nickel or austenite, the harmonized equation proposed here can be directly used to determine the grain size in the studied material. In such a scenario, only one laser-ultrasonic measurement is needed on a sample with a known grain size, i.e. as a reference, which does not have to be a small value. If the obtained relative inhomogeneity factor is different from those reported here, the temperature contribution to attenuation has to be measured at a constant grain size. It can be seen that the harmonized equation at least significantly reduces the number of experiments and ex-situ characterizations required for the design of a new calibration.  131  5. Effects of the grain boundary type on attenuation has been studied with the FE simulation of ultrasonic wave propagation in an FCC polycrystalline metal. Microstructures required to generate the FE template were obtained from the EBSD scans on the selected cobalt superalloys. The simulated attenuation versus frequency showed that there is no significant difference between scattering behavior of twin boundaries and other regular high angle grain boundaries, i.e. the LUMet technique cannot be used to extract the fraction of twin boundaries.        132  9.2 Suggestions for future work  Based on the results presented in this thesis, suggestions for future studies are: 1. Limited qualitative studies have been conducted so far regarding the effects of grain shape on ultrasonic attenuation. A systematic investigation on the effects of grain aspect ratio on ultrasonic properties, velocity and attenuation, should be conducted using both experimental and FE modelling approaches.  2. Grain size evolution following hot compression was studied in a L605 cobalt superalloy. It is interesting to evaluate the grain size evolution after deformation in CCM cobalt superalloy with the LUMet system systematically. This study will help us to see whether the different recrystallization behavior following hot deformation in L605 and CCM superalloys can be distinguished with the LUMet system.  3. In the attempt to harmonize the existing grain size empirical calibrations in FCC metals, the temperature contribution functions of Inconel and steels in the austenite phase were obtained from the temperature dependence of single crystal elastic constants of the respective materials reported in the literature. It can be useful to measure the temperature contribution functions in nickel based superalloys and steels using the LUMet system following the same methodology as the one used for cobalt superalloys. Furthermore, the single crystal elastic constants of FCC metals can be determined accurately in a broad temperature range using laser ultrasonics. To this aim, the wave velocities need to be measured in single crystals of the studied metal in at least two different directions, i.e. <100> and <110>.  4. Grain size values acquired in FCC metals using the LUMet system can be further analyzed using machine learning algorithms.  5. The harmonized grain size calibration can be extended to other systems than FCC. Also a more fundamental inclusion of the scattering theory including textured materials might be useful in the evaluation of LUMet measurements. 6. The established calibrations correlated the ultrasonic attenuation parameter and the metallographic area-weighted grain diameter. Even though the AWGD is a better representative of the grain volume than the EQAD, it is measured from the 2D SEM micrographs. It is therefore interesting to determine the average grain sizes using a 3D 133  characterization technique such as 3D EBSD and use the obtained results in the harmonized calibration.   134  Bibliography [1] F.J. Humphreys and M. Hatherly, Recrystallization and related annealing phenomena, 2nd ed. Elsevier, 2004. [2] M. Dubois, M. Militzer, A. Moreau, and J. F. 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A, vol. 36, no. 4, pp. 895–905, 2005.     151  Appendix 1  True stress-true strain compression curves: correction for friction  In this appendix, the equations used to calculate the friction coefficient and to correct the true stress-strain curves are described. During a high-temperature compression test, friction between the tested sample and the anvils plays an important role in the deformation behavior of the material. Several studies have been conducted to determine or limit the friction coefficient during a high-temperature forging process [161]. Generally, lubricants are used to reduce the friction between the work piece and compression anvils. The friction coefficient however, cannot be decreased to zero especially for deformation at high temperatures. Several attempts have been made to quantitatively determine the friction coefficient associated with a compression test in which a barreled work piece is generated [162-164]. Based on the FE calculations conducted by Li et. al [165], friction coefficient 𝑚 can be obtained by measuring the sample geometry before and following deformation as: 𝑚 =𝑅𝑡ℎ𝑒𝑜𝑟𝑦ℎ𝑓𝑖𝑛𝑎𝑙× 𝑏𝑇𝑟𝑒𝑠𝑐𝑎4√3−2𝑏𝑇𝑟𝑒𝑠𝑐𝑎3√3 (1) where 𝑅𝑡ℎ𝑒𝑜𝑟𝑦 is the theoretical final radius, ℎ𝑓𝑖𝑛𝑎𝑙 is the final height of the cylinder measured after deformation, and 𝑏𝑇𝑟𝑒𝑠𝑐𝑎 is the barreling factor determined under the Tresca criterion as: 𝑏𝑇𝑟𝑒𝑠𝑐𝑎 = 4 ×∆𝑅𝑅𝑡ℎ𝑒𝑜𝑟𝑦×ℎ𝑓𝑖𝑛𝑎𝑙∆ℎ (2) where ∆𝑅 is the difference between the maximum radius and radius at the sample surface and ∆ℎ is the change in height after deformation. 𝑚 value varies in the range of zero to one, being zero in the absence of friction and one in the complete sticking condition. Table 1 summarizes the barreling factors and friction coefficients obtained in the single-hit and double-hit experiments in the present work. The barreling factor ranges from 0.6 to 1.2 and friction coefficient ranges from 0.1 to 0.3. Favre [86] obtained an average barreling factor of 1 and average friction coefficient of 152  0.6. The higher friction coefficient reported by Favre as compared to those obtained in the present work is due to the higher amount of strain (total true strain of 0.8) applied in the former.  Table.1 Sample geometry, barreling factor, and friction coefficient measured after deformation Test T (°C) Diametric strain  𝑅𝑡ℎ𝑒𝑜𝑟𝑦 (mm) ∆𝑅 (mm) ℎ𝑓𝑖𝑛𝑎𝑙 (mm) ∆ℎ (mm) 𝑏𝑇𝑟𝑒𝑠𝑐𝑎 𝑚 𝐶 Single-hit 1000 -0.19 5.2 0.2 12.6 2.5 0.6 0.1 0.08 1050 -0.21 5.3 0.3 12.5 2.6 1.0 0.2 0.14 1100 -0.23 5.3 0.4 12.2 2.9 1.2 0.3 0.17 Double-hit 1000 -0.44 5.9 0.6 10.8 4.3 1.1 0.3 0.20 1050 -0.44 5.9 0.6 10.7 4.3 0.9 0.3 0.17 The corrected true stress for friction 𝜎 can be determined using the following equation [166]: 𝜎 =𝐶2𝑃2[exp(𝐶) − 𝐶 − 1] (3) with  𝐶 =2×𝑚×𝑅0ℎ0. Here 𝑃 is the true stress in absence of friction, 𝑅0 and ℎ0 are the sample radius and height before deformation, respectively. Equation (3) was used to calculate the friction-corrected flow curves assuming that the compression stress remains constant in the entire specimen.    

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