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In-situ evaluation of the hcp to bcc phase transformation kinetics in commercially pure titanium and… Shinbine, Alyssa 2016

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In-situ evaluation of the hcp to bcc phasetransformation kinetics in commercially puretitanium and Ti-5Al-5Mo-5V-3Cr alloy usinglaser ultrasonicsbyAlyssa ShinbineB.Sc, University Of Alberta, 2013A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTSFOR THE DEGREE OFMaster of Applied ScienceinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Materials Engineering)The University of British Columbia(Vancouver)March 2016© Alyssa Shinbine 2016AbstractThis thesis developed and validated laser ultrasonics as an in-situ monitor of phase transformationsin commercially pure titanium and Ti - 5 wt.% Al - 5 wt.% Mo - 5 wt.% V - 3 wt.% Cr (Ti-5553).Three studies (Chapters 5, 6 and 7) were performed to achieve this goal. The first study involvedusing finite element modeling (FEM) to simulate wave propagation through a 2-phase aggregate tounderstand the effects of precipitate arrangement and phase fraction on the velocity signal. Thepredicted ultrasound velocity depended on the geometric configuration of the microstructure andthe relative size of the pulse’s wavelength compared to the microstructural feature size. However,for mixtures of phases with similar elastic properties and densities (such as in α and β titanium), thepossible averaging schemes produce nearly identical velocities, and thus using a rule of mixturesinvolving the α and β velocities was confirmed to be sufficient. The second study showed thatthe ultrasonic velocity is sensitive to the α → β and β → α transformations in commerciallypure titanium, even though the density and elastic modulus of these two phases are very similar.Extraction of the transformation kinetics from the ultrasonic velocity does require, however, theeffects of the strong starting texture and texture evolution during grain growth to be accountedfor. Finally, the third study presented in Chapter 7 took Ti-5553 specimens, solutionized themto the fully β condition, and then held them for varying times at a 700 °C isotherm to monitorprecipitation kinetics with LUMet. The precipitation of α grains could be monitored by using therelative change in velocity and compared to the ex-situ obtained phase fraction.While laser ultrasonics has been previously used to measure the elastic constants in Ti-H alloys[1] and to qualitatively observe the transformation kinetics in Ti-6V-4Al [2] the work presented hererepresents the first fully quantitative assessment of transformation kinetics in pure titanium via laserultrasonics. This is a significant result since ex-situ, metallographic analysis of the transformationin commercially pure titanium is not possible as the high temperature β phase is not stable at roomtemperature, and it paves the way for this technique to be used for microstructure monitoring duringmore complex thermo-mechanical processing paths in the Gleeble thermo-mechanical simulator.Laser ultrasonics was also validated in Ti-5553, where it was used to monitor the precipitation ofα precipitates during an isothermal treatment, and produced comparable kinetics to the kineticsderived from ex-situ metallography.iiPrefaceThe experimental work presented in this thesis was conducted at the University of British Columbiawithin the department of Materials Engineering. This included all the experimental design, materialheat treatments, in-situ observation, sample preparation, and metallography. The initial sampletemplate file used in ABAQUS was created by Quentin Puydt at the University of British Columbia.This template file was modified by the author to contain various microstructural arrangements (viaMATLAB) and material properties in correspondence with the study presented in Chapter 5. Theequilibrium α phase fraction at 700 °C used in Chapter 7 was calculated in the TTTI3 database inThermocalc software by Julien Teixeira, at Institut Jean Lamour - SI2M in Nancy France.Results and discussions pertaining to the hcp to bcc transformation in commercially pure tita-nium have been submitted for publication and is currently under review (A. Shinbine, T. Garcin,C. Sinclair (2015) “In-situ laser ultrasonic measurement of the hcp to bcc transformation in com-mercially pure titanium.” Submitted and Under Review).iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1 General characteristics of the α and β phases in titanium . . . . . . . . . . . . . . . . . . . . 32.1.1 Crystal structures of the allotropic α and β phases . . . . . . . . . . . . . . . . . . . 32.1.2 Temperature dependence of density of α and β phases . . . . . . . . . . . . . . . . . 42.1.3 The effect of alloying elements on α and β phase stability in titanium alloys . . . . . 52.1.4 Microstructural design for mechanical properties in β alloys . . . . . . . . . . . . . . 82.2 Elastic properties of titanium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.1 Temperature dependence of elastic constants . . . . . . . . . . . . . . . . . . . . . . 102.2.2 Elastic anisotropy in titanium hcp and bcc single crystals . . . . . . . . . . . . . . . 122.3 Mechanisms involved in the α↔ β phase transformation . . . . . . . . . . . . . . . . . . . . . 162.3.1 The α to β phase transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3.2 Grain growth in β phase titanium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3.3 The β to α phase transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3.4 Observation of phase transformation kinetics in titanium and Ti-5553 . . . . . . . . 272.4 Laser ultrasonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.4.1 Laser generation and detection of ultrasonic pulse . . . . . . . . . . . . . . . . . . . . 292.4.2 Interpretation of waveform signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.5 Ultrasonic wave propagation in metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.5.1 Calculation of velocity from the Christoffel equation . . . . . . . . . . . . . . . . . . 332.5.2 Ultrasound velocity in an isotropic medium . . . . . . . . . . . . . . . . . . . . . . . 342.5.3 Rotation of elastic stiffness tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.5.4 Elastic properties of polycrystalline aggregates in the isotropic assumption . . . . . . 372.5.5 Ultrasound velocity in a textured polycrystalline aggregate . . . . . . . . . . . . . . 40ivTABLE OF CONTENTS2.5.6 Velocity in the two phase region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 Scope and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 Experimental Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.1 As-received materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.1.1 Commercially pure titanium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.1.2 As-received Ti-5Al-5Mo-5V-3Cr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.2 Ex-situ microstructural characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.2.1 Scanning electron microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.2.2 Electron backscatter diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.3 In-situ microstructural characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.3.1 Laser ultrasonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.3.2 Thermal treatments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.3.3 Interpretation of waveform data in CTOME . . . . . . . . . . . . . . . . . . . . . . . 504.3.4 Interpretation of error in laser ultrasound measurements . . . . . . . . . . . . . . . . 515 Finite Element Modeling of Ultrasonic Wave Propagation in Dual Phase Material . . 545.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.2 Input parameters, construction and assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 545.2.1 Wavelet generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.3 Interpretation of FEM data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.4 Calculation of average velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.5 FEM results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.6 Sensitivity of averaging schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.7 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676 In-situ Laser Ultrasonic Measurement of the HCP to BCC Transformation in Com-mercially Pure Titanium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696.2 Characterization of the as-received material . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706.3 Ex-situ observation of microstructural changes upon thermal cycling . . . . . . . . . . . . . . 706.4 Evaluation of the ultrasonic velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756.5 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 827 In-situ Laser Ultrasonic Measurement During Aging of Ti-5553 . . . . . . . . . . . . . 837.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 837.2 Characterization of the as-received material . . . . . . . . . . . . . . . . . . . . . . . . . . . . 837.3 Ex-situ metallography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 847.4 Evaluation of ultrasonic velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 897.5 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 958 Summary and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 978.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 978.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101A EBSD IPF comboscan maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107B Plotting 3-D Surface Young modulus as a Function of Orientation . . . . . . . . . . . . 109B.1 Young modulus as a function of orientation in the titanium hcp unit cell . . . . . . . . . . . 109B.2 Young modulus as a function of orientation in the titanium bcc unit cell . . . . . . . . . . . 112vList of TablesTable 2.1 Parameters used in the Calphad approach for calculation molar volume, density, and CLE 5Table 2.2 Legend of author based color codes and symbols used in Figures 2.7a and 2.7b to describethe elastic response in hcp and bcc titanium . . . . . . . . . . . . . . . . . . . . . . . . . 10Table 2.3 Comparison of calculated anisotropies for hcp and bcc titanium . . . . . . . . . . . . . . . 13Table 2.4 The effect of cooling rate on formed α microstructures when α + β alloy Ti -6Al-4V [13],commercially pure titanium (cp-Ti) [14, 15], and extremely pure (ep-Ti) [14] is cooledfrom above the transus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Table 5.1 Input parameters defining two isotropic phases used in an explicit 2-D FEM simulationof an ultrasound pulse propagating in a dual phased polycrystalline aggregate structure . 55Table 5.2 Summary of imposed microstructures seeded to the core interaction region of the meshedspecimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Table 5.3 Density (ρ) and P-wave moduli (λ) of polycrystalline α and β titanium at 882 °C . . . . . 66Table 6.1 Microstructural parameters of commercially pure titanium in the: as-received condition,after 1 treatment cycle, after 5 treatment cycles . . . . . . . . . . . . . . . . . . . . . . . 73Table 6.2 Reduced axis/angle pairs for each type of α/α boundaries that can result from the sameparent β grain [88] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Table 6.3 Best fit values of the adjustable parameters (A1 and A2) in equation 6.2 obtained bycomparing during fitting the heating and cooling portions of the experimental data inFigure 6.6 for the 1st, 2nd, and 5th cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . 79Table 7.1 Phase fraction of α (fα) precipitated during isothermal holding at 700 °C . . . . . . . . . . 85viList of FiguresFigure 2.1 Unit cells of a) hexagonal closed packed (hcp) alpha phase titanium and b) body centercubic (bcc) β phase titanium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Figure 2.2 Density of titanium hcp and bcc phases obtained using the CALPHAD method [25] . . . 5Figure 2.3 The effect of alloying elements on equilibrium titanium phase diagram for additions of a)α b) β-eutectoid and c) β-isomorphous stabilizers. The β-transus is highlighted in purple[30]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Figure 2.4 Schematic mapping multicomponent alloy classifications on a β isomorphous pseudo-binary phase diagram. Ms and Mf refer to the martensite start and finish lines, respec-tively [10]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Figure 2.5 Two needles of atom probe tomography (APT) reconstructions demonstrating the pres-ence of homogeneously dispersed ω phases after quenching and ω aging at 300 °C for 8hours. Reprinted and reproduced with permission from Elsevier Publishing Ltd. [35] . . 9Figure 2.6 Results Coakley et al. showing evolution of Vickers micro-hardness of Ti-5553 as afunction of ageing time. The data presented shows the hardness of the quenched material(A-0), 300 °C and 400 °C samples with ageing times between 1 and 8 h, labeled A-1 toA-8 and C-1 to C-8 respectively. The hardness of sample B-2 which received a 300 °C/8h+ 500 °C/2h aging heat treatment is also presented. Reprinted and reproduced withpermission from Elsevier Publishing Ltd. [35] . . . . . . . . . . . . . . . . . . . . . . . . 9Figure 2.7 a) Polycrystalline bulk and shear moduli of the hcp and bcc phases in pure titanium[36–41] and a nearly pure Ti-0.05% H alloy [1]. The red line signifies the transformationtemperature Teq of 882 °C; and b) Compilation of single crystalline elastic moduli ofhcp (data left of Teq = 882 °C) and bcc (data right of Teq = 882 °C) phases in puretitanium [36–39, 41]. The red line signifies the transformation temperature Teq of 882°C. The symbols used here indicate the type of modulus plotted while the color indicatesthe source of the data. See Table 2.2 for details of both. . . . . . . . . . . . . . . . . . . 11Figure 2.8 Temperature dependence of the anisotropy ratio (A) in hcp titanium [36] . . . . . . . . . 14Figure 2.9 Young Modulus as a function of orientation where the radius extending from the originprovides E in GPa and axis units are GPa for a) hcp titanium unit cell at 25 °C, andb) bcc titanium unit cell at 1000 °C. Code for calculating the direction dependence ofYoung modulus is provided in Appendices B.1 and B.2 . . . . . . . . . . . . . . . . . . . 16Figure 2.10 Atomic projections on {0001} hcp planes and {110} bcc showing the changes in packingplanes which result from the Burger’s Orientation Relationship (BOR) observed duringthe hcp to bcc transformation scheme. Reprinted and reproduced with permission fromAmerican Physical Society (APS) [36] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Figure 2.11 Schematic of the competitive growth between two types of β morphology, where theintragranular plates are indicated by P, the allotriomorphs are indicated by A, and thematrix is comprised of α grains. Reprinted and reproduced with permission from ElsevierPublishing Ltd. [11] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19viiLIST OF FIGURESFigure 2.12 Backscatter electron contrast images depicting grain growth of β titanium after a) 30minutes, b) 1 hr, and c) 4 hrs at temperatures of approximately 910 °C. After 30 minutes,(a) surface relief due to thermal etching of the prior α grain boundaries is still visible, andthe β grains are determined only by the orientation contrast. Reprinted and reproducedwith permission from Elsevier Publishing Ltd. [11] . . . . . . . . . . . . . . . . . . . . . 20Figure 2.13 Optical micrographs of cp-Ti (grade 2) at cooling rates of a) 1 °C.s–1, b) 50 °C.s–1, c)91 °C.s–1, d) 356 °C.s–1, e) 651 °C.s–1, and f) 1604 °C.s–1. Aα, Mα, BWα, and α standfor acicular, massive, basketweave, and martensitic α. Obtained by Kim et al. (reprintedand reproduced with kind permission from Springer Science and Business Media) [15] . . 23Figure 2.14 Plots showing a) fraction transformed over time held at each isotherm and b) the timetemperature transformation (TTT) diagram for Ti-5553 obtained by Kar et al. (reprintedand reproduced with permission from Elsevier Publishing Ltd.) [57] . . . . . . . . . . . . 25Figure 2.15 SEM backscatter micrographs of microstructures observed after treatment at 750 °C fora) 5 min and b) 30 min, 650 °C for c) 5 min and d) 30 min, and transmission electronmicroscope images of α plates precipitated after treatment at 550 °C for a) 5 min and b)30 min. Obtained by Kar et al. (reprinted and reproduced with permission from ElsevierPublishing Ltd.) [57] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Figure 2.16 simplified schematic of the optical set-up for two-wave beam mixing method interferom-etry presented in [75] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Figure 2.17 Example of a waveform depicting compressive and shear echo signals in a 3 mm thickcommercially pure titanium specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Figure 4.1 Schematic of cp-Ti specimen in the Gleeble with the normal direction (ND) aligned withthe direction of wave propagation, the rolling direction (RD) aligned with the length ofthe specimen, the transverse direction (TD) aligned with the width of the specimen, andthe EBSD image plane is shown in grey lying in the plane defined by the TD normal. . . 49Figure 4.2 Heat treatment profiles for specimens enduring a) 1 cycle of treatment, b) 2 cycles oftreatment, and c) 5 cycles of treatment where the red line indicates the equilibriumtransformation temperature of 882 °C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50Figure 5.1 (a). 2D generated mesh part with thickness of 2 mm, length of 10 mm and mesh sizeof 4 μm in the small mesh region (4 mm x 2 mm), (b). horizontal plates, (c). verticalplates and (d). randomly dispersed elements. Vertical displacement (uy) is measured asthe direct output of the simulation from nodes in the generation/detection line (blue),and the vertical displacement direction is indicated by the red arrow. . . . . . . . . . . 56Figure 5.2 a). Time dependence of the Ricker pulse amplitude and b). an example of the Gaussianscaling used to normalize the applied amplitude (at t = 125 s) across the generationboundary, where the peak of the Gaussian is centered on the mid line of the specimen . 58Figure 5.3 Example of averaged displacement over time data taken from upper boundary nodesresponsible for generation and detection of the Ricker’s ultrasonic pulse . . . . . . . . . 59Figure 5.4 Example of a) two waveform echos (f and g), and, b) the corresponding amplitude ofcross correlation function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60Figure 5.5 Ultrasonic velocity computed by the six potential averaging schemes listed in Equa-tions 5.4, 5.5, 5.6, 5.9, 5.10, 5.11, as well as FEM simulation results, where vHP, vVP,and vFEM refer to simulation trial 9 (horizontal plate), trial 10 (vertical plate) and trials1 - 8 (dispersed), respectively, as defined in Table 5.2.) . . . . . . . . . . . . . . . . . . 62Figure 5.6 Schematic showing imposed Ricker wavelet a) propagating in horizontal plates where thewavelength is relatively larger than microstructural features, b) propagating in horizontalplates where the wavelength is smaller than microstructural features, c) propagating invertical plates where the wavelength is relatively larger than microstructural features,d) propagating in vertical plates where the wavelength is smaller than microstructuralfeatures. The black lines indicates atoms in the wave, where lines appearing close togethershow compression of the wave, and lines spread apart indicate refraction. Wavelength isthe distance between instances of maximum compression. . . . . . . . . . . . . . . . . . 65viiiLIST OF FIGURESFigure 5.7 Velocity plotted against a changing ratio of P-wave modulus (ρα/ρβ), where λα was fixedat 123.72 GPa, and λβ is variable for a given phase fraction of β a) fβ = 0.1, b) fβ = 0.5,and c) fβ = 0.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Figure 6.1 Normal direction (ND) EBSD IPF map showing equiaxed and recrystallized grains inthe as-received plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70Figure 6.2 Backscatter electron (BSE) images with detailed orientation contrast for the specimensin the a) as-received state (AR), b) after 1 treatment cycle (1-cycle), c) after 5 treatmentcycles (5-cycles) of heat treatment, and low magnification BSE images showing typicalmicrostructures d) after 1 treatment cycle (1-cycle), and e) after 5 treatment cycles (5-cycles) of heat treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Figure 6.3 Representative inverse ND pole figure maps (ND - IPF maps, A = 6.49 x 105 μm2)showing the microstructure and microtexture in the: a) as-received state, b) after 1treatment cycle, c) after 5 treatment cycles. In a), b), and c) the area of observationcorresponds to the region where the ultrasonic pulse was induced and measured. Notethat these regions have been cropped from much larger maps, the overall size of theparent map being given in Table 6.1. Full comboscan maps provided in Appendix A. . . 72Figure 6.4 Grain boundary maps demonstrating: a) the α/α and b) prior β special boundary map ofa specimen after 1 treatment cycle, and c) the α/α and d) prior β special boundary mapof a specimen after 5 treatment cycles. Full comboscan maps provided in Appendix A. . 74Figure 6.5 Pole figures showing texture in the: a) as-received state, b) after 1 treatment cycle, c)after 5 treatment cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75Figure 6.6 Comparison of calculated velocity values to LUMet observations for heating and coolingof the a) 1st treatment cycle, b) 2nd treatment cycle, and c) 5th treatment cycle, alongwith d) a compilation of velocity profiles demonstrating the variation in β velocity duringheating in the 1st cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76Figure 6.7 Overview of the model used to calculate the velocity in a bulk aggregate of titanium.*Δv/ΔT back-calculated from average slopes of experimentally observed velocities andoff-set so model velocity intersects experimental at 1000 °C . . . . . . . . . . . . . . . . . 78Figure 6.8 α → β and β → α phase transformation kinetics obtained from Equation 3 using theparameters given in Table 6.3 for specimens during the 1st, 2nd and 5th cycle, respectively 81Figure 7.1 Secondary Electron Image (SEI) demonstrating the as-forged Ti-5553 microstructure atmagnifications of a) 1000x b) 10 000x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84Figure 7.2 Electron backscatter diffraction (EBSD) inverse pole figure (IPF) map demonstratingglobular primary α precipitates dispersed in a large β grain . . . . . . . . . . . . . . . . . 84Figure 7.3 BSE images taken at 20 kV and at low magnification of a specimen representative of thestarting condition at the onset of the 700 °C isotherm (solutionized for 15 minutes at 900°C, then cooled to 700 °C at 6.7 °C.s–1 and quenched). . . . . . . . . . . . . . . . . . . . 86Figure 7.4 EBSD phase map taken after a specimen was held at 700 °C for 5 minutes where blackand white indicate the α and β phases, respectively . . . . . . . . . . . . . . . . . . . . . 86Figure 7.5 BSE images taken at 20 kV and a magnification of 500x of specimens held at the isother-mal treatment for a) 10, b) 33, c) 53, d) 75 and e) 180 minutes. . . . . . . . . . . . . . . 87Figure 7.6 Triple point β boundary in a specimen aged for 75 minutes at 700 °C demonstrating filmsof αGB forming between the β grains and the growth of αW1 side-plates into the β grainvisualized as a a) backscatter electron image and b) inverse pole figure map . . . . . . . 88Figure 7.7 Measurements of isothermal α precipitation obtained via ex-situ metallography . . . . . 88Figure 7.8 Comparison of a) raw absolute velocities at various isothermal holding times, and b)in-situ ultrasonic observation depicting relative change in velocity (Δv = vt – vβ) . . . . 89Figure 7.9 A comparison of: a) Inverse ND pole figure map (ND - IPF map) showing the microstruc-ture and microtexture in hcp commercially pure titanium after 5 treatment cycles, andb) BSE image of a Ti-5553 specimen solutionized for 15 minutes at 900 °C, then cooledto 700 °C at 6.7 °C.s–1 and quenched to depict the bcc microstructure at the on-set ofthe isothermal treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92ixLIST OF FIGURESFigure 7.10 Schematic showing a) the as-received (AR) 2nd echo reference, fully β (BCC) 2nd echoreference, and an arbitrary and representative high temperature 2nd echo (Current sig-nal), and b) cross-correlation function obtained when the current signal is compared tothe as-received reference (red) and to the fully β reference (black). The dashed linesindicate the maximum amplitude, and corresponding time delay of each cross-correlationfunction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94Figure 7.11 Comparison fraction of α precipitation obtained via ex-situ metallography and normalizedvelocity data during isothermal treatment . . . . . . . . . . . . . . . . . . . . . . . . . . 95Figure A.1 Full EBSD obtained comboscan a) inverse ND pole figure maps (ND - IPF maps) showingthe microstructure and microtexture, b) the α/α special boundary map, and c) prior βspecial boundary map of a specimen after 1 treatment cycle . . . . . . . . . . . . . . . . 107Figure A.2 Full EBSD obtained comboscan a) inverse ND pole figure maps (ND - IPF maps) showingthe microstructure and microtexture, b) the α/α special boundary map, and c) prior βspecial boundary map of a specimen after 5 treatment cycles. . . . . . . . . . . . . . . . 108xList of AbbreviationsAPT atom probe tomographyAR as-receivedbcc body center cubicBOR Burger’s Orientation RelationshipBSE backscatter electronCCT continuous cooling transformation diagramCLE coefficient of linear expansionCPE4R four-node bi-linear quadrilateral plane strain elementscp-Ti commercially pure titaniumCALPHAD computer coupling of phase diagrams and thermochemistryCTOME computational tools for metallurgyEBSD electron backscatter diffractionEQAD equal area diameterFE finite elementFEM finite element modellingFFT fast fourier transformhcp hexagonal closed packedIFFT inverse fast fourier transformIPF Inverse Pole Figure mapJMAK Johnson-Mehl-Avrami-Kolmogorov modelKSR Kurdjumov-Sachs relationshipLUMet laser ultrasonics for metallurgyND normal directionODC orientation distribution coefficientsP-wave modulus pressure wave modulusRD rolling directionSANS small angle neutron scatteringSEM scanning electron microscopySEI secondary electron imagexiList of AbbreviationsSE2 single echo technique, with second echo selected as referenceTEM transmission electron microscopyTi-5553 Ti - 5 wt.% Al - 5 wt.% Mo - 5 wt.% V - 3 wt.% CrTTT time temperature transitionTD transverse directionXRD X-ray diffractionxiiAcknowledgmentsThere are numerous people whose guidance, inspiration, and unwavering support made this thesispossible. I would be honored to take this moment to express my gratitude and respect for mymentors, colleagues, friends and family who have helped me along the way.First and foremost, I wish to thank my supervisors, Dr. Chad Sinclair and Dr. Thomas Garcinfor the support, guidance, and wealth of expertise they provided throughout this work.I am very grateful to members of the LUMet group at the University of British Columbia. Inparticular, I wish to thank Dr. Warren Poole and Dr. Matthias Militzer for their expertise andfeedback throughout my study. I also want to thank Mahsa Keyvani for her willingness to answerquestions and provide insight in the lab.I am very grateful to Quentin Puydt for his mentorship and willingness to tutor me on the useof ABAQUS and Finite Element Modeling.I am very appreciative of the faculty and staff of the Department of Materials Engineering. Inparticular, I wish to thank the machine shop staff (Ross Mcleod, Carl Ng, and David Torok) andJacob Kabel for their expert services, assistance, and uplifting conversations.I would also like to thank Beth Sterling, Michael Wong, and Greg Nelson for their comradery,advice and friendship.I want to thank my family: Mom, Dad, Sarah, Jeremy, and Peter, thank you for your constantencouragement and love.Lastly, I would like to thank NSERC Canada for their contributions to my research.xiiiChapter 1IntroductionTitanium and titanium alloys offer a wide array of benefits when compared to structural steelsand nickel based alloys. Benefits include a high strength to density ratio, relatively high meltingpoint, excellent corrosion resistance and comparable strength to structural steels [3–8]. As such,titanium and titanium alloys are used for structural applications in aerospace, biomedical, energyand marine industries, where the high performance characteristics of titanium and its alloys justifyits relatively high cost.In both commercially pure titanium and Ti-5553, the microstructure and mechanical propertiesare usually optimized by the control of the high temperature solid-state phase transformations thatoccur during high temperature processing. Upon heating, the low temperature hcp-alpha (α) phasetransforms into the high temperature bcc-beta (β) phase at temperatures of about 882 °C[9] and 838°C[10] for pure titanium and Ti-5553, respectively. This allotropic transformation to the β phaseis followed by rapid grain growth [11, 12]. Upon subsequent cooling, the reverse transformationfrom β to α phase results in the formation of different α phase microstructures, eg. polygonalgrains or a martensitic microstructure depending on the cooling conditions[13–15]. These differentmicrostructures impact the final mechanical properties of the part[5, 16]. It is therefore critical tounderstand the mechanisms underlying this transformation in order to further control and optimizethe use of this material for targeted applications.There are a number of challenges associated with the in-situ study of the α-β phase transforma-tion in titanium and titanium alloys. Unlike the austenite-ferrite phase transformation in steels, thevolume change between the α-phase and β-phase is very small rendering the measurement of trans-1Introductionformation kinetics by dilatometry a significant challenge [17, 18]. Like ultra-low alloyed steels, theobservation of the microstructure post mortem is also not a viable option for low alloyed titaniumalloys as the β-phase cannot be retained at room temperature. Direct observation of the transfor-mation performed in-situ at high temperature is possible [11], but limited to the extreme surface ofthe specimen and unsuitable for the measurement of bulk transformation kinetics. In-situ neutronor x-ray diffraction [19, 20] measurements made during continuous heating and cooling are anotheroption but are limited by the acquisition time of the technique (particularly for neutron diffrac-tion) [21] and by the rapid grain growth that occurs in the β-phase at high temperature. In-situelectrical resistivity measurements have been shown to be viable for measurements in commerciallypure titanium but the method becomes more difficult to interpret in alloys where scattering frominterfaces and solute atoms, which redistribute on transformation, can’t be neglected [14, 15].This thesis uses and validates laser ultrasonics as a useful method for the in-situ monitoring ofphase transformations in commercially pure titanium and Ti-5553. In the case of the commerciallypure titanium, a method using laser ultrasound was developed to account for texture changes duringthe transformation and grain growth above the β-transus. It was found that this was essential toproperly account for the observed transformation kinetics. This method was then used to evaluatethe isothermal β → α kinetics for the Ti-5553 alloy. In this case, the ability of laser ultrasonics tomeasure the transformation kinetics could be compared to ex-situ metallography thereby allowingfor validation of this technique.2Chapter 2Literature Review2.1 General characteristics of the α and β phases in titanium2.1.1 Crystal structures of the allotropic α and β phasesTitanium and titanium alloys exist in several crystal structures depending on the temperature,composition, and treatment path (eg. cooling rate). The most important phases in titanium arethe α and β phases. Another important phase is the orthorhombic ω phase that can result atspecific pressures and compositions [13]. This phase is not observed under atmospheric pressureand equilibrium conditions in pure titanium and is not focused on in the current study. The β↔ αtransformation is allotropic and occurs at 882 °C [9]. Titanium exists at room temperature as ahexagonal closed packed (hcp) α phase, as shown in Figure 2.1a. Upon heating above 882 °C the hcplattice will transform into a body centered cubic (bcc) lattice. When considering non-interactingidentical hard spheres packed together in the hcp crystal structure, an ideal hcp cell has a c/a ratioof 1.633 [9]. However, in titanium, a slight atomic distortion occurs resulting in contraction alongthe c-axis [22]. This results in a c/a ratio of:4.685A◦2.951A◦ = 1.5876 < 1.633 (2.1)The insertion of interstitial atoms such as C, N, O or the addition of substitutional α stabilizerswith atomic radii smaller than that of titanium, will result in the increase of the c/a ratio [23, 24].3Literature Reviewa. b.Figure 2.1: Unit cells of a) hexagonal closed packed (hcp) alpha phase titanium and b) body centercubic (bcc) β phase titanium2.1.2 Temperature dependence of density of α and β phasesOne of the properties that is important for determining ultrasonic wave propagation in materials isthe density. The density of hcp and bcc titanium can be calculated using the CALPHAD parameters(Table 2.1) outlined in the work of Lu et al. [25]. The calculated temperature dependences of densityin the hcp and bcc phases are presented in Figure 2.2. This approach makes two assumptions:There are no magnetic effects present, and the properties are calculated at atmospheric pressure.The temperature range was limited to between 298 K and the melting point 1945 K. Under theseassumptions, the density (ρ), the molar volume (Vm) of a non-magnetic material, and the coefficientof linear thermal expansion (CLE) can be expressed as [25]:ρ =MVm(2.2)Vm (T) = Voexp(∫ TTo3CLEdT)(2.3)CLE =13(a + bT + cT2 + dT–2)(2.4)Where, the coefficients a, b, c, d, the molar volume (Vo) at the reference temperature (To = 298K), and the molar mass of titanium (M) are given below in Table 2.1 [25].4Literature ReviewTable 2.1: Parameters used in the Calphad approach for calculation molar volume, density, andCLETi phase a b c d Vo M(10–5K–1) (10–8K–2) (10–11K–3)(K)(10–6m3mol–1) (10–3Kgmol–1)α (hcp) 2.47929 1.03566 0 -0.144379 10.5463 47.876β (bcc) 1.53848 1.86567 0 0 10.5919 47.876In pure titanium, the percent difference in the densities of the α and β phase is 0.18 %, whencalculated at the β-transus temperature of 882 °C. Conversely, the percent difference in density ofaustenite and ferrite in pure iron is 1.2 % at the same temperature, which is an order of magnitudelarger than the percent difference in titanium. It can be seen that the volume change betweenthe α-phase and β-phase is very small rendering the measurement of transformation kinetics bydilatometry a significant challenge compared to the austenite-ferrite phase transformation in steels[17, 18].Figure 2.2: Density of titanium hcp and bcc phases obtained using the CALPHAD method [25]2.1.3 The effect of alloying elements on α and β phase stability in titanium alloysAt a given temperature and pressure, the equilibrium phase of titanium is dependent on the alloyingelements [4]. Alloying elements in titanium alloys are commonly described as being α stabilizers,β-isomorphous stabilizers, and β-eutectoid stabilizers. The α stabilized alloys are alloys that requireincreased temperatures compared to pure titanium to induce the α→ β transformation. The mostcommon α stabilized alloys are Ti-Al based systems. Other α stabilizing elements include, oxygen,carbon, and nitrogen. The effect of introducing α stabilizers was demonstrated in the earliest of theTi-Al phase diagrams, proposed by Molchanova [26]. Molchanova’s phase diagram demonstrated5Literature Reviewthat with increasing addition of an α stabilizing element, the temperature separating the fullybcc β phase region from the α + β region, called the β-transus, will increase [26, 27]. This effectis demonstrated in the simple schematic representing possible titanium alloy phase diagrams fordifferent stabilizer additions in Figure 2.3, where the β-transus is highlighted purple in each case.β-isomorphous stabilizers are elements that decrease the β-transus temperature, or the tempera-ture at which the α→ β transformation occurs (cf. Figure 2.3b). The most common β -isomorphousstabilized alloys are Ti-Mo alloys. For additions of Mo up to 21 wt.%, a near linear increase in theβ transus temperature is exhibited [28]. The solubility of molybdenum in titanium is very limited[28], hence, vanadium is also commonly employed as a β stabilizer since it spans a higher range ofsolubility [29]. Other β-isomorphous stabilizers include niobium and tantalum [10].Finally, the β-eutectoid stabilizing elements also lower the β-transus temperature (cf. Fig-ure 2.3c) . The most common β-eutectoid stabilized alloys are Ti-Cr alloys. However, iron is alsoa fairly common β-eutectoid stabilizer, along with (less commonly used) manganese, cobalt, cop-per, silicon, and hydrogen [10]. Ti-Cr alloys can form the phase TiCr2, which has three possiblecrystal structures, and occurs over a very narrow compositional range of 65-68 wt% chromium [10].Both chromium and iron are termed sluggish eutectoid formers, meaning the presence of theseintermetallic compounds in commercial alloys is unlikely [10].a. b. c.Figure 2.3: The effect of alloying elements on equilibrium titanium phase diagram for additions ofa) α b) β-eutectoid and c) β-isomorphous stabilizers. The β-transus is highlighted in purple [30].Multicomponent alloys, such as alloy Ti-5553, are comprised of both α and β stabilizers. Suchalloys allow for the combination of the good forgeability of the bcc crystal structure with the abilityto be hardened to fairly high strength levels via aging treatments [3]. While other classificationschemes exist, this thesis will make use of the “US Technical Multicomponent Classification” which6Literature Reviewsubdivides titanium alloys into α, β and α + β alloys where α + β can be further classified aseither near-α or near-β alloys. Under this classification, Ti-5553 is described as a β alloy [10].Interpreting the effects of the numerous stabilizing elements discussed above can be representedmore simply by calculating their equivalent α and β stabilizing effects on titanium, where theseeffects are represented as an equivalent aluminum and molybdenum concentration, respectively.The aluminum equivalent concentration in weight percent ([Al]eq) can be calculated based on theweight percent concentrations of Al, Zr, Sn, and O as shown in Equation 2.5 [31].[Al]eq = [Al] +16[Zr] +13[Sn] + 10[O] (2.5)Conversely, the molybdenum equivalent concentration in weight percent ([Mo]eq) can be cal-culated based on the weight percent concentrations of Mo, Ta, Nb, W, V, Cr and Ni as shown inEquation 2.6 [26].[Mo]eq = [Mo] +15[Ta] +13.6[Nb] +12.5[W] +11.5[V] + 1.25[Cr] + 1.25[Ni] (2.6)For the β phase to be retained at room temperature, a minimum [Mo]eq of 10 is needed [10].Ti-5553 has an [Mo]eq of 13 and can retain a fully β structure upon quenching [10]. Conceptually,titanium alloy classification can be visualized as a β isomorphous pseudo-binary phase diagram [32].This scheme orders alloys according to their relative abundance of α and β stabilizing elements interms of [Mo]eq and [Al]eq. This representation is provided in Figure 2.4.In Figure 2.4 the line labeled MS refers to the martensite start temperature. If an alloy isquenched fast enough below this temperature, the β phase will transform martensitically to formα’. For alloys with compositions lying to the left of the Ms line (shown in Figure 2.4), the martensitictransformation will occur if the cooling rate surpasses a critical value. The martensitic transforma-tion involves a collective movement of atoms to change the crystal lattice from a bcc to a hcp crystalstructure. This cooperative motion involves a relatively large number of atoms that are displaced byonly a fraction of the inter-atomic spacing relative to their neighbors and occurs by a shear-based,diffusionless mechanism. To date, only two martensitic structures have been reported in titaniumalloys. These are the hcp α’ structure, common to pure titanium and low alloyed systems, and anorthothrombic α” structure, common to titanium alloy systems alloyed with β stabilizers [33]. The7Literature ReviewMs temperature of Ti-5553 is below room temperature (Figure 2.4). This means that if Ti-5553 isquenched it will form a metastable, fully β microstructure. This is why Ti-5553 is often called ametastable β alloy.Figure 2.4: Schematic mapping multicomponent alloy classifications on a β isomorphous pseudo-binary phase diagram. Ms and Mf refer to the martensite start and finish lines, respectively [10].Given its composition, Ti-5553 has a β transus temperature of 838 °C which is much lowerthan the transformation temperature of pure titanium (882 °C). Figure 2.4 demonstrates that atroom temperature, and given equilibrium conditions, that a mixture of α and β is stable. This isimportant because if Ti-5553 is quenched to form a 100 % β microstructure, it can then undergoan aging heat treatment to precipitate the α phase and create a dual-phase microstructure.2.1.4 Microstructural design for mechanical properties in β alloysThe value of a metastable β alloy such as Ti-5553 is that it can be produced with lesser quantitiesof β stabilizer elements, so as to produce alloys with the highest strength-to-density ratios intitanium alloys [34]. Metastable β alloys have high ductility in the quenched (fully β) state, whichis beneficial in the fabrication of complex parts. After the part has been formed, high strength canbe achieved by aging heat treatments, where the formation of fine α precipitates contributing tothe high strength associated with these materials [34].The nucleation of the α phase occurs heterogeneously on grain boundaries, inclusions and otherprecipitates. This allows the precipitation strengthening due to α formation to be manipulated by8Literature Reviewchanging the density and arrangement of nucleation sites. In Ti-5553 this is done by isothermalaging at low temperature to form a fine dispersion of metastable ωiso (Figure 2.5) [35]. After theformation of the ω phase, the material is subjected to a higher temperature aging treatment duringwhich a fine dispersion of the alpha phase forms preferentially on the ω phase.Figure 2.5: Two needles of atom probe tomography (APT) reconstructions demonstrating thepresence of homogeneously dispersed ω phases after quenching and ω aging at 300 °C for 8 hours.Reprinted and reproduced with permission from Elsevier Publishing Ltd. [35]Coakley et al. [35] found that ω aging followed by α aging treatments resulted a 90 % increasein hardness compared to the quenched material due to the distribution of fine α precipitates formedafter treatment, as shown in Figure 2.6.Figure 2.6: Results Coakley et al. showing evolution of Vickers micro-hardness of Ti-5553 as afunction of ageing time. The data presented shows the hardness of the quenched material (A-0),300 °C and 400 °C samples with ageing times between 1 and 8 h, labeled A-1 to A-8 and C-1 toC-8 respectively. The hardness of sample B-2 which received a 300 °C/8h + 500 °C/2h aging heattreatment is also presented. Reprinted and reproduced with permission from Elsevier PublishingLtd. [35]9Literature Review2.2 Elastic properties of titanium2.2.1 Temperature dependence of elastic constantsLike density, the elastic properties are also important for determining ultrasonic wave propagationin materials. The bulk elastic properties of hcp and bcc titanium have been explored experimentally[1, 36–39] as well as computationally [40, 41]. The bulk (B) and shear (G) modulus are presentedin Figure 2.7a. The bulk and shear modulus of the α phase decreases with increasing temperature.Ogi et al. [38] measured that the longitudinal and bulk moduli of the high temperature β phasedemonstrate limited temperature dependence. The temperature dependence of single crystallineelastic moduli of hcp (data left of Teq = 882 °C) and bcc (data right of Teq = 882 °C) phases inpure titanium are given in Figure 2.7b [36–39, 41]. Numerous data from various sources have beencompiled in Figures 2.7a and 2.7b. The corresponding symbols and color coding used to representthe data and their sources are provided below in Table 2.2.Table 2.2: Legend of author based color codes and symbols used in Figures 2.7a and 2.7bto describe the elastic response in hcp and bcc titaniumElastic ModulusMethod color phase c11 c33 c44 c66* c′* c13 c12 B Gultrasound [36] hcp x x x x x xelectromagnetic hcp/bcc x x x x x x xacoustic resonance [38]ultrasound [37] bcc x x x xcoherent inelastic bcc x x x x x xneutron scattering [39]resonant-ultrasound bcc x x x xspectroscopy [41]laser ultrasound** [1] hcp/bcc x xlaser ultrasound*** [1] bcc x xfirst principle bcc xcalculations [40]first principles bcc xcalculations [42]* Where c66 =12 (c11 – c12) in the hcp phase and c′= 12 (c11 – c12) in the bcc phase** Data for hcp polycrystalline constants for low alloyed titanium (Ti 0.05 % H) [1]*** Data for bcc polycrystalline constants for pure titanium at room temperature, extrapo-lated [1]10Literature Reviewa. b.Figure 2.7: a) Polycrystalline bulk and shear moduli of the hcp and bcc phases in pure titanium [36–41] and a nearly pure Ti-0.05% H alloy [1]. The red line signifies the transformation temperatureTeq of 882 °C; and b) Compilation of single crystalline elastic moduli of hcp (data left of Teq =882 °C) and bcc (data right of Teq = 882 °C) phases in pure titanium [36–39, 41]. The red linesignifies the transformation temperature Teq of 882 °C. The symbols used here indicate the type ofmodulus plotted while the color indicates the source of the data. See Table 2.2 for details of both.In hcp titanium, the 5 independent elastic stiffness constants c11, c44, c12, c13, and c33 (shownin Figure 2.7) have demonstrated temperature dependence in the single phase elastic region [36, 38].The linear regressions for each independent elastic constant (in Pa) obtained from Fisher et al. [36]as a function of temperature (in oC) are as follows:c11(Pa) =(–0.0513(GPa.°C–1)T(°C) + 163.34(GPa))∗ 109(Pa.GPa–1) (2.7)c12(Pa) =(0.0118(GPa.°C–1)T(°C) + 91.313(GPa))∗ 109(Pa.GPa–1) (2.8)c13(Pa) =(0.0006(GPa.°C–1)T(°C) + 68.736(GPa))∗ 109(Pa.GPa–1) (2.9)c33(Pa) =(–0.0363(GPa.°C–1)T(°C) + 181.29(GPa))∗ 109(Pa.GPa–1) (2.10)c44(Pa) =(–0.0199(GPa.°C–1)T(°C) + 46.856(GPa))∗ 109(Pa.GPa–1) (2.11)No temperature dependent profiles of single crystal elastic constants for the bcc titanium phaseare currently available. However, single crystal elastic constants were determined at 1000 °C [37, 41]and at 1020 °C [1]. The values from Fisher and Dever [37] can be used when looking at potential11Literature Revieworientation dependent ultrasound velocities in bcc titanium.c11 = 99 ∗ 109(Pa) (2.12)c12 = 85 ∗ 109(Pa) (2.13)c44 = 33.6 ∗ 109(Pa) (2.14)The first principle calculations of Ahuja et al. [43] demonstrated that the bcc structure intitanium is expected to be unstable at 1 atm and absolute zero temperature. Despite this instability,the bcc crystal structure is stable in titanium at sufficiently high temperatures. This is due to thehigh entropy associated with the bcc crystal structure, and the fact that the structure is stabilizeddue to the lattice vibration modes of titanium at high temperature. Due to this, it is expectedthat β titanium would exhibit some anomalous physical properties [43]. Ahuja et al. [43] showedthat as the pressure increases, so does the stability of the bcc phase, and this is attributed to thedecreasing magnitude of the negative single crystal tetragonal shear constant (c′) of the bcc phasewith decreasing volume until it becomes positive. The tetragonal shear constant in the bcc crystalis given by:c′=12(c11 – c12) (2.15)Fisher and Renken [36] and Ogi et al. [38] measured the single crystal elastic constants of hcptitanium as a function of temperature up until the β-transus temperature of 882 °C. Petry et al.[39], Fisher and Dever [37], and Ledbetter et al. [41] measured the single crystal elastic constantsof bcc titanium at temperatures of 1020 °C, 1000 °C, and 1000 °C, respectively. The single crystalelastic constants presented in the literature [36–39, 41] of the hcp and bcc phases in titanium arepresented in Figure 2.7b.2.2.2 Elastic anisotropy in titanium hcp and bcc single crystalsWhile it may not be initially obvious by referencing Figures 2.7a and 2.7b, the α and β phases aremarkedly different in regards to their elastic response. The calculated anisotropies for each phase12Literature Reviewat selected temperatures are provided in Table 2.3. In particular, the elastic anisotropy of the bccphase is much greater than that of the hcp phase. Firstly, it can be observed that the anisotropyin hcp titanium increases with increasing temperature. The elastic anisotropy of hcp titaniumat absolute zero was close to one (A = 1.17), indicating near elastic isotropy [36]. As the hcptitanium was heated, both Ogi et al. [38] and Fisher and Renken [36] observed increasing valuesin elastic isotropy, and a value of 3.58 was observed at the β-transus (T = 882 °C) by Fisher andRenken [36]. The elastic anisotropy is further increased upon transformation into the bcc phase,where Fisher and Dever [37] and Ledbetter et al. [41] observed values of 4.8 and 5, respectivelyat 1000 °C. Figure 2.8 demonstrates the non-linear increase in anisotropy in the hcp phase as thetransformation temperature is approached.Table 2.3: Comparison of calculated anisotropies for hcp and bcc titaniumTemperature phase A = c44c66 A =c440.5(c11–c12)Reference(°C) (hcp/bcc)-273.15 hcp 1.17 - [36]20 hcp 1.31 - [36]882 hcp 3.58 - [36]1000 bcc - 4.8 [37]1000 bcc - 5 [41]1020 bcc - 3 [39]13Literature ReviewFigure 2.8: Temperature dependence of the anisotropy ratio (A) in hcp titanium [36]Unfortunately, no studies to date have reported the temperature dependence of the single crys-tal elastic constants in bcc titanium, so from the data presented above, it is difficult to discern thetemperature dependence of its elastic anisotropy. However, if one returns to the work of Ahuja etal. [43] outlining the effect of pressure on c′, one can make an inference as to the anisotropy oftitanium at absolute zero and compare the calculated anisotropy to the experimentally observedhigh temperature values [37]. The key observation one can make, is that as the pressure approachesatmospheric pressure, c′will become increasingly negative. A negative c′indicates that the crystalstructure is mechanically unstable. At atmospheric pressure and absolute zero, a c′of approxi-mately -75 GPa is calculated from first principles [43]. Ahuja et al. [43] also calculated the c44at these conditions and achieved a value of 35.8 GPa, which is in relatively good agreement withthe experimentally observed values for bcc titanium presented above in Table 2.3. This indicatesthat c44 in bcc titanium is relatively insensitive to temperature. However, the drastic increasein stiffness calculated at absolute zero [43] and observed at 1000 °C [37] in c′(from -75 GPa to7 GPa) indicates that shear corresponding to c′is strongly altered by vibrational effects at high14Literature Reviewtemperature [43].The anisotropy of the bcc crystal has inherent effects on the observed ultrasound velocity duringmonitoring of the bcc β phase. The elastic response of a crystal along any set of directions may becalculated by resolving the stress state onto the crystal axes [44]. The orientation dependence ofthe Young’s modulus (E) is given by Equation 2.16 and Equation 2.17 for a cubic and hexagonalcrystal, respectively [44]. The coordinate system parameters α, β, γ used in Equation 2.16 andEquation 2.17 are given in terms of the direction indices [hkl] in Miller notation (for both the hcpand bcc cases) by Equations 2.18, 2.19, 2.20, respectively [44].1Ebcc= s11 + (2s12 – 2s11 + s44)(β2γ2 + γ2α2 + α2β2)(2.16)1Ehcp=(1 – γ2)2s11 + γ4s33 + γ2(1 – γ2)(2s13 + s44) (2.17)α =h(h2 + k2 + l2)1/2 (2.18)β =k(h2 + k2 + l2)1/2 (2.19)γ =l(h2 + k2 + l2)1/2 (2.20)Figure 2.9 shows the Young moduli across varied directions in the bcc and hcp unit cell andwas created using Equation 2.16, Equation 2.17, the room temperature single crystal compliancesfor hcp titanium [36], and the compliances available for bcc titanium at 1000 °C [37].15Literature Reviewa. b.Figure 2.9: Young Modulus as a function of orientation where the radius extending from the originprovides E in GPa and axis units are GPa for a) hcp titanium unit cell at 25 °C, and b) bcc titaniumunit cell at 1000 °C. Code for calculating the direction dependence of Young modulus is providedin Appendices B.1 and B.2It can be observed that the elastic response in the hcp crystal is isotropic in the basal planewhich is characteristic of all hcp crystals. The Young’s moduli observed along the c-axis (Ec) andin the basal plane (Ebasal) are 146 GPa and 104 GPa, respectively. Thus, sound will travel fasterwhen the sound wave’s propagation is aligned with the c-axis of the hcp crystal. Conversely, the bcccrystal demonstrates greatest anisotropy when comparing [100] and [111] directions. The minimumand maximum observed Young’s moduli occur along the [100] and [111] directions, respectively,where E[100] = 20.5 GPa and E[111] = 89.6 GPa.2.3 Mechanisms involved in the α↔ β phase transformation2.3.1 The α to β phase transformationCrystallography plays a large role in the transformation from the α to β phase and vice versa.Numerous investigations observing the β ←→ α transformation have described the orientationrelationship between a parent β grain and a resulting daughter α grain as being a Burgers orientationrelationship [45–47]. This orientation relationship also governs the reverse transformation from agiven α grain to a newly formed β grain upon heating. The crystallographic Burgers orientation16Literature Reviewrelation between the β phase and α phase is defined as [48]:(110)β ‖ (0002)α[11¯1]β ‖ [112¯0]α(2.21)Where both the diffusional and martensitic transformation follows this relation. There are6 closest packed planes and 2 slip directions within the β parent grains resulting in a maximumoccurrence of 12 distinct orientations that can be formed in the nucleated α phase for a given parentorientation.Section 2.1.2 had shown the c/a ratio for an hcp titanium unit cell to be approximately 1.59.A c/a < 1.633 implies a larger separation of atoms in the basal planes of the hcp structure thanthat which exists between different planes, as shown in Figure 2.10 [36].Figure 2.10: Atomic projections on {0001} hcp planes and {110} bcc showing the changes in packingplanes which result from the Burger’s Orientation Relationship (BOR) observed during the hcp tobcc transformation scheme. Reprinted and reproduced with permission from American PhysicalSociety (APS) [36]Due to this wide separation, it can be assumed that the thermal energy is most likely going toexcite vibrational modes which will alter these separations [36]. Since shear distortions are easierto produce than compressional distortions [36], thermal vibrational modes corresponding to the c66mode are thought to occur in the hcp lattice when approaching the transformation temperature[36, 49]. Fisher and Renken [36] postulated that, in accordance with the work of Zener [49], thelarge shear vibrations in the basal plane could potentially be vibrating out of phase, contributingto an increased repulsion of the two basal planes. This repulsion would manifest as a highertemperature dependence of the coefficient of linear thermal expansion (CLE) parallel to the c-axis.17Literature ReviewThe studies of Sirota and Zhabko [50], and Spreadborough and Christian [51] demonstrated thatα titanium demonstrates a higher CLE parallel to the c-axis than parallel to the a-axis, and thatthe anisotropy of the CLE in different directions diminishes with increasing temperature [50]. Theonset of positive curvature in the c66 modulus at 146 ° C (cf. Figure 2.7b) combined with thedecreased anisotropy in the CLE with increasing temperature would suggest a decreasing effect oftemperature on the vibrational amplitudes of the lattice. Fisher and Renken [36] also presentedresistivity curves that demonstrated a departure from linear dependence at approximately 146° C, corresponding to changes in electron scattering probability of the lattice. This aligns wellwith the changes observed in CLE [50, 51] and the elastic constant c66 [36]. Fisher and Renken[36] suggested that the combination of Burger’s model for the structural change endured upontransformation from the hcp to bcc lattice, and the extreme temperature dependence of c66 and c′in both the hcp and bcc phases, respectively, can be viewed as a sudden shift in the direction ofthe high amplitude thermal vibrations [36] which is the ultimate driver for this transformation. Asobserved in Figure 2.10, the atoms in the basal planes of the hcp lattice are transformed into thecloser packed {110} planes in the bcc lattice which decreases the amplitudes of the shear modes[36].There are few publications in the literature characterizing the α(hcp)→ β(bcc) transformationin pure titanium. This is due to the requirement for in-situ evaluation of the transformation inpure titanium. In a study of super-heated Ti films, specimens were heated by a laser pulse at a rateof 1011 K.s–1 to just below the melting temperature and it was found that the bcc phase nucleatedat a rate of 1025 m–3.s–1 corresponding to a diffusionless and heterogeneous nucleation scenarioand the resulting plates grew at rates near 1400 m.s–1 [52]. These results are not consistent with adiffusion controlled transformation and rather is in the order of the velocity of elastic shear wavesin hcp titanium (approximately 3000 m.s–1 ) [52]. Conversely, when heated above the transus withlower thermal rates (below 1000 K.s–1 ), the transformation will occur massively and is diffusioncontrolled. A study using in-situ electron backscatter diffraction (EBSD) was able to observethe α(hcp) → β(bcc) transformation of intragranular and allotriomorphic β grains [11]. Sewardet al. [11] found that while the intragranular β grains shared a Burgers orientation relationshipwith the parent α grain and a morphology similar to a military (diffusionless and shear based)transformation, the interface velocities were more consistent with a diffusional transformation. They18Literature Reviewobserved the fastest interfaces traveled at 10–2 m.s–1 [11]. This is far less than the mobility expectedin a diffusionless shear based transformation (3000 m.s–1 ). Conversely, unlike the intragranularplates, the grain boundary allotriomorphs did not have morphologies consistent with a militarytransformation. Instead, the daughter grain is formed via the massive transformation, nucleatingand growing along the α – α grain boundaries. These distinct morphologies are presented in theschematic shown in Figure 2.11.Figure 2.11: Schematic of the competitive growth between two types of β morphology, where theintragranular plates are indicated by P, the allotriomorphs are indicated by A, and the matrix iscomprised of α grains. Reprinted and reproduced with permission from Elsevier Publishing Ltd.[11]Both of these sets of observations [11, 52] are consistent with the mechanism proposed by Fisherand Renken [36] and the data presented above. This is because superheating critically above thetransformation temperature would instantaneously change the vibrational directions and intensitiesin the hcp sublattices, resulting in a shear driven transformation from the hcp lattice to the bcclattice. Conversely, the lower temperature results of Seward et al. [11] shows that while thetransformation may have military characteristics, it has kinetics consistent with a diffusional basedtransformation, which in turn emphasizes the thermal dependence of the changes in vibrations andshear observed over the transformation regime.19Literature Review2.3.2 Grain growth in β phase titaniumGrain growth is the process where the average size of grains within a polycrystalline aggregateincreases. This process is driven by the decrease in free energy that results from a decrease ingrain boundary area [12]. The experiment of Gil et al. took an initial specimen with an equal areadiameter (EQAD) of 54 μm, and isothermally held the specimen at 1000 °C, and 1100 °C. After,EQADs of 358 μm and 429 μm were observed, respectively [12]. The observations of Seward et al.[11] also indicated extreme cases of grain growth (demonstrated in Figure 2.12) in commerciallypure titanium, where after 110 minutes of holding at 910 °C an area weighted average grain sizeof approximately 1.2 mm was observed. This extensive grain growth has a direct influence on thenumber of grains probed during ultrasonic sensing, which leads to an increased contribution ofgrain orientation effects on the velocity signal. Anisotropy in the hcp and bcc phases in regards toelastic response is explored in Section 2.2.2 and a thorough examination on calculating the effectof orientation and anisotropic response is provided in Section 2.5 .a. b. c.Figure 2.12: Backscatter electron contrast images depicting grain growth of β titanium after a)30 minutes, b) 1 hr, and c) 4 hrs at temperatures of approximately 910 °C. After 30 minutes, (a)surface relief due to thermal etching of the prior α grain boundaries is still visible, and the β grainsare determined only by the orientation contrast. Reprinted and reproduced with permission fromElsevier Publishing Ltd. [11]2.3.3 The β to α phase transformationContinuous cooling and the β→ α transformation in pure titanium and titanium alloysThe β to α phase transformation observed upon cooling is also heavily effected by the strong thermaldependence of the c66 and c′shear constant in both the hcp and bcc phases, respectively (discussedextensively above in Section 2.2 and Section 2.3.1), and much like with the transformation observed20Literature Reviewupon heating where the heating rate dictates the character of transformation, the mechanism oftransformation upon cooling is affected by the cooling rate.Depending on the cooling rate from above the β transus, transformations can occur via a varietyof mechanisms in titanium alloys. Studies on the phase transformations present in cooling of theα+β alloy Ti -6Al-4V [13], and on commercially pure titanium [14, 15] have demonstrated the effectof cooling on microstructure in α titanium. Ahmed and Rack [13] observed hexagonal martensiteformed at high cooling rates. The bcc → hcp martensitic transformation can be described as aresult of two basic processes. First, a shuffling of parallel (110) planes in the bcc crystal in theopposite [11¯0] directions by an amount equal to one sixth of the interplanar spacing between the(110) planes happens. This occurs in conjunction with a pure shear along the {112} family planesof the bcc crystal acting in the 〈111¯〉 directions [53]. The shuffling serves to produce the ABABstacking characteristic of hexagonal closed packed structures, and the shear serves to convert theirregular hexagonal structure of the (110) bcc planes to a regular hexagonal atomic arrangementwith a characteristic angle of θ = 120◦. The morphology of martensite can be classified as eitherlath or acicular. Lath martensite refers to a packet martensite structure consisting of 50-100 μmirregular regions containing sub-micron packets of α laths of identical Burgers relationship variantaligned in parallel [10, 54]. Conversely, the acicular structure is characterized as needle-like plates,with adjacent α each having a distinct Burgers relationship variant [10, 54]. Lath martensite hasa greater tendency to form in pure titanium or solute lean titanium alloys. The cooling rates atwhich martensite can be obtained for pure titanium and alloy Ti -6Al-4V are given in Table 2.4.At slightly slower cooling rates α forms via the massive transformation. The massive trans-formation involves the original phase decomposing into a new phase of identical composition butdifferent crystal structure, and is a diffusional transformation. Ahmed and Rack [13] observed thatdecreases in cooling rate corresponded to increases in phase fraction of the observed massive α phase.The massive phase nucleated at grain boundaries and exhibited a blocky appearance. The crystalstructure of the massive phase was hexagonal closed packed, where each grain was comprised of aheavily dislocated substructure [13]. As the cooling rate was reduced further, the heterogeneousnucleation of grain boundary primary α was observed to compete with massively formed α phase[13–15]. This resulted in long primary α Widmansta¨tten plates growing from the grain boundarieswith retained β existing between plates. With further decreases in cooling rate, the Widmansta¨tten21Literature Reviewα grew, reaching the center of each β grain. The nucleation of more Widmansta¨tten plates off ofexisting plates creates the typically observed basketweave morphology [13–15]. A summary of theeffect of cooling rate on observed α morphologies in alloy Ti -6Al-4V and commercially pure ti-tanium is presented in Table 2.4. The optical micrographs depicting the effect of cooling rate onmicrostructure collected by Kim et al. [15] are provided below in Figure 2.13.Table 2.4: The effect of cooling rate on formed α microstructures when α+ β alloy Ti -6Al-4V [13],commercially pure titanium (cp-Ti) [14, 15], and extremely pure (ep-Ti) [14] is cooled from abovethe transusCooling Rate (°C.s–1)Microstructure Ti -6Al-4V cp-Ti cp-Ti ep-Ti[13] (grade-2)[15] (grade-4)[14] [14]Martensite α′ 410 1604 300 1100Massive α 20 - 410 90 - 1604 30 - 300 350 - 1000Acicular α - 58 - 1604 1 - 300 40 - 300Plate-like 1- 20 1 - 90 1 - 30 1 - 40The diffusional transformations responsible for the massive and primary α both occur via het-erogeneous nucleation, either along existing β grain boundaries or on existing α plates. The caseof α precipitation in the near β alloy, Ti-5553, is a complex one, as there are 4 additional alloyingelements which will tend to partition. Aluminum, being an α stabilizer, will tend to segregate to thenewly formed α precipitates. Conversely, molybdenum, vanadium, and chromium are all β stabiliz-ers and will thus prefer to remain in the β matrix. These elements possess substantial differencesin diffusivities relative to one another, leading to significant gradients at the α-β interfaces duringnucleation.Cooling rate have been discussed in the above section as a significant factor in determining themorphology of the transformed microstructure. The cooling rate will also influence compositionalpartitioning since partitioning is a diffusion dependent effect. The diffusivities of molybdenum,vanadium, and chromium in titanium at 1273K are 3.35 x 10–10, 1.18 x 10–9, and 4.47 x 10–9m2s–1, respectively [55]. Thus, while molybdenum will be rejected from the newly forming andgrowing α precipitates, a molybdenum pile-up will occur at α-β interfaces.22LiteratureReviewFigure 2.13: Optical micrographs of cp-Ti (grade 2) at cooling rates of a) 1 °C.s–1, b) 50 °C.s–1, c) 91 °C.s–1, d) 356 °C.s–1, e) 651 °C.s–1,and f) 1604 °C.s–1. Aα, Mα, BWα, and α stand for acicular, massive, basketweave, and martensitic α. Obtained by Kim et al. (reprintedand reproduced with kind permission from Springer Science and Business Media) [15]23Literature ReviewIsothermal aging in near β titanium alloysThe α precipitation kinetics and mechanism in Ti-5Al-5Mo-5V-3Cr have been explored during low-temperature isothermal holding via in-situ resistivity and in-situ high energy x-ray diffraction [56],as well as during high temperature holding treatments via ex-situ examination such as metallogra-phy and hardness testing [57, 58]. Kar et al. [57] solution treated and aged at isotherms held attemperatures of 750 °C, 650 °C, and 550 °C for 30 minutes to create a time temperature transfor-mation (TTT) diagram (see Figure 2.14). They observed α volume fractions of 25.5, 37.5, and 32.7% formed after 30 minutes of undercooled holding at the temperatures 750 °C, 650 °C, and 550°C, respectively [57]. The α precipitates consisted of allotriomorph grains formed along the β – βgrain boundaries and plate-like precipitates observed throughout the β grains [57] (see Figure 2.15).While limited sources [57, 58] were found documenting high temperature isothermal aging in Ti-5Al-5Mo-5V-3Cr, very similar allotriomorph and plate morphologies were observed in other β alloyssuch as Ti 17 [18], β21-Ti alloy [59], Ti-15V-3Cr-3Sn-3Al [60] and Ti-8Al-1Mo-1V alloy [61]. Pre-cipitation during isothermal undercooling treatments in near β titanium alloys have been observedto be heterogeneous, occurring initially along grain boundaries, then as plates growing outwardsfrom existing α – β interfaces, and finally forming plate packets throughout the β grain.24Literature ReviewFigure 2.14: Plots showing a) fraction transformed over time held at each isotherm and b) thetime temperature transformation (TTT) diagram for Ti-5553 obtained by Kar et al. (reprintedand reproduced with permission from Elsevier Publishing Ltd.) [57]25LiteratureReviewFigure 2.15: SEM backscatter micrographs of microstructures observed after treatment at 750 °C for a) 5 min and b) 30 min, 650 °C forc) 5 min and d) 30 min, and transmission electron microscope images of α plates precipitated after treatment at 550 °C for a) 5 min andb) 30 min. Obtained by Kar et al. (reprinted and reproduced with permission from Elsevier Publishing Ltd.) [57]26Literature Review2.3.4 Observation of phase transformation kinetics in titanium and Ti-5553Capturing the kinetics of the α to β and the β to α transformation via commonly employed ex-situ methods (metallography, SEM) in other systems is not a realistic approach in pure titaniumand some low-alloyed titanium materials, since as demonstrated in Sections 2.1.2 and 2.1.3, thetransformation is allotropic in nature, and the β phase cannot be retained when quenching fromabove the transus for many compositions. Furthermore, even quenching to form a martensiticstructure, is not readily feasible for many sample geometries, as cooling rates greater than above1000 °C.s–1 have been reported to be necessary to obtain martensite in pure titanium [14, 15] (cf.Section 2.3.3).Direct SEM/EBSD observation of the α/β transformation in pure titanium performed in-situ athigh temperatures is possible and can give excellent insight into texture changes and morphologicalchanges observed during transformation [11]. However, this technique is limited to viewing asmall area on the extreme surface of the specimen and is unsuitable for measurement of the bulktransformation kinetics. Furthermore, the resolution is limited by the rastering time for the electronbeam to sample the entire area, and, as such, small area sizes must be selected to resolve fasttransformations. In-situ neutron or x-ray diffraction [19, 20] measurements made during continuousheating and cooling are another option but are limited by the acquisition time of the technique(particularly for neutron diffraction) [21] and by the rapid grain growth that occurs in the β-phaseat high temperature, where having a small sample of large grains of particular orientation can causeissues with peak detection. In-situ electrical resistivity measurements have been shown to be viablefor measurements in commercially pure titanium but the method becomes more difficult to interpretin alloys where scattering from interfaces and solute atoms, which redistribute on transformation,can not be neglected [14, 15]. Furthermore, the departure from linearity observed in single phaseα resistivity data corresponding to changes in the scattering probability of the hcp lattice [36] canalso complicate the measurement. Dilatometry is also made difficult due to the anisotropy presentin the coefficients of thermal expansion along the c-axis and a-direction in the hcp phase [50, 51],indicating that the measurement would be texture and grain size sensitive.Conversely, capturing the kinetics of the α to β and the β to α transformation via commonlyemployed ex-situ methods (metallography, SEM) in Ti-5553 is much easier, due to its ability as27Literature Reviewa metastable alloy to retain the β phase upon quenching from above the β-transus temperature.However, as Ti-5553 is intended for landing gear applications, and often undergoes forging andaging treatments [10], a non-contact, in-situ, and industrially feasible method to monitor phasetransformations during complex thermo-mechanical processing steps would be highly beneficial.The need for controlling the α + β microstructure was discussed in Section 2.1.4. Recently, anextensive study conducted by Coakley et al. [35] monitored ω and α aging via ex-situ approachessuch as atom probe tomography (APT), x-ray diffraction (XRD), and transmission electron mi-croscopy (TEM), as well as in-situ small angle neutron scattering (SANS). However, the authorsacknowledged the modeling of SANS data is not simple, and requires complimentary microscopy,as well as a neutron source [35]. Laser ultrasonics could be an additional technique to help monitorthe complex phases and microstructures that develop in Ti-5553 and commercially pure titanium.The basis of this technique is explored in the following section (Section 2.4).2.4 Laser ultrasonicsConventional ultrasound analysis has been used to measure thickness, detect cracks and voids,inspect weld quality, and to characterize materials [62]. This technique involves a piezoelectrictransducer that is capable of emitting and receiving ultrasound pulses. While ultrasonic testinghas been a very successful and cost effective non-destructive technique, it is subject to a varietyof constraints. A key limiting factor is the need to have the specimen either immersed in a tankor have direct contact to achieve ultrasonic coupling [62]. This requirement limits the applicationof ultrasound to low temperature scenarios in the context of having the specimen immersed, or, ifhigh temperatures are required, direct contact must be made between the transducer and specimen.Both of these constraints place impositions on monitoring microstructural changes in-situ duringcomplex thermomechanical processing steps. Furthermore, the bandwidth of the piezo-transducersis limited, which limits the shape of the specimen to pieces with limited concavity.These limitations can be negated via the use of laser light for the generation and detection ofultrasound in a material [62, 63]. Advantages of laser ultrasonic measurement includes the fact thatit is completely non-contact, effective at high temperatures, and that it has a high precision for thecapture of sound wave velocity [63]. Due to the versatility stemming from the non-contact nature,28Literature Reviewlaser ultrasonics has been successfully utilized in numerous applications, including determiningelastic constants [63, 64], observing recrystallization [65–68] and phase transformations [67, 69, 70],measuring grain size [68, 71, 72], and monitoring strip and specimen thicknesses [63, 67]. Thefollowing section describes the principle for generation and detection of ultrasound with lasers, aswell as outline examples of the applications for velocity-based and attenuation-based monitoring.2.4.1 Laser generation and detection of ultrasonic pulseThe generation of ultrasound by a laser source can be either caused by a thermo-elastic mechanismor an ablation mechanism [62]. The thermo-elastic mechanism involves the laser light being ab-sorbed to some depth into the material and is completely non-destructive [62, 73]. This producesheat, causing the material to expand. This expansion induces strain and a corresponding stressthat is the source of the waves propagating through the material and along the material’s surface[62, 73]. Thermo-elastic generation is very successful at generating surface waves when the sourcebeam is focused on the surface as a small point. This creates a circle wave of cylindrical symmetryemitting from the spot [74].By increasing the energy density, the material surface will melt and eventually a thin layerof material at the surface will vaporize. Upon vaporization, the vapor (and surrounding air ifpresent), will ionize. This ionization creates a plasma cloud that expands away from the point ofinteraction. This expansion of plasma creates a reactionary force in the specimen perpendicular tothe surface causing a displacement that propagates through the material as an ultrasound pulsewith primarily longitudinal components. This technique is not purely non-destructive, as with eachinstance of ablation, a thin layer of metal material is removed from the sample surface. Afterprolonged treatment, a small crater of diameter equal to the beam diameter may be visible uponthe sample surface [62].A second laser will illuminate the sample surface continuously or in pulses of sufficient durationto detect the ultrasonic signal of interest. The light is either scattered or reflected upon hittingthe surface and can be collected by an interferometer. The interferometer detects the microscopicchanges in the sample surface caused by the arrival of the ultrasound wave at the detection surface[75]. While there are many interferometry techniques such as the confocal Fabry-Perot interfer-ometer [76] and the Michelson interferometer [75], the two-wave beam mixing method [77] is the29Literature Reviewinterferometer set up used for this thesis work. In the two-wave beam mixing method, the detec-tion beam first strikes the surface and is scattered producing a signal beam (So). The signal beaminterferes with a plane wave beam sourced directly from the laser called the pump beam (Po). Thisinterference occurs within a photo-refractive crystal. The general schematic of an optical set-upsensitive enough to detect the phase shifts in the signal beam produced by the ultrasonic induceddisplacements on the sample surface is provided in figure 2.16.Figure 2.16: simplified schematic of the optical set-up for two-wave beam mixing method interfer-ometry presented in [75]2.4.2 Interpretation of waveform signalAfter collection and demodulation using the active, two beam mixing interferometry approach, theoutput can be displayed as an amplitude (in voltage) plotted over time, as shown in Figure 2.17.The amplitude is proportional to the monitored surface displacement.30Literature ReviewFigure 2.17: Example of a waveform depicting compressive and shear echo signals in a 3 mm thickcommercially pure titanium specimenThe signal presented in Figure 2.17 corresponds to a set up where the detection and generationlasers are directed at the same surface (meaning the wave must travel a distance equal to twicethe thickness of the specimen to be measured). Shortly after generation, the first compressive echocan be observed. The compressive echos correspond to the longitudinal component of the planarwavefront. The appearance of the first echo means after generation, the wavefront has traveledthrough the material to the point where it is being monitored as a surface displacement.The compressive echos provide two essential parameters. The first is the time delay. Whenevaluating longitudinal velocity, the time delay is the time between two compressive echos andcorresponds to the time needed for the wavefront to travel from the original surface, throughthe medium and back again to be measured. By knowing the time delay and the thickness ofthe specimen being observed, the longitudinal sound velocity can be calculated for the material.The second key parameter is the attenuation. Attenuation is the loss in amplitude observed overtime. This is affected by the average grain size in a given material, as well as the texture inanisotropic materials. Attenuation is due to the elastic scattering of the longitudinal wave bygrains [62, 63, 66, 68, 71, 72].31Literature ReviewUltrasonic velocity measurement in observations of texture and phase transformationUltrasonic velocity is a function of the temperature, texture and phase composition of a material,as all three factors will affect the aggregate elastic properties and density of the material.The work of Dubois, Moreau and Bussie`re [70] also investigated extensively the phase trans-formations in steels. They observed that the ferromagnetic-paramagnetic transition that occurs atthe Curie temperature (750 °C) affected the ultrasound velocity noticeably. They were also able toobserve hysteresis in velocity signal depicting transformations in steels with carbon concentrationsabove 0.05 %. However, what is of interest to this particular study is their modeling of velocitybased on textural changes observed in the steel specimens. Dubois et al. [70] calculated the ori-entation distribution coefficients (ODC) from the ultrasonic data of the as-received state. Thiswas then used to compute the bulk texture of the austenite phase, and the returning ferrite phaseafter one cycle of continuous treatment using the Kurdjumov-Sachs relationship (KSR). This modelapproach was built upon the assumption that each of the 24 variants defined under the KSR areequally probable to form and that the number of grains within the probed volume is large enoughto allow for each variant to be selected equally often statistically and is truly a statistically relevantbulk texture. They cited a lack of grain growth during the duration of the treatments to ensurethat the assumption is a good approximation. Under these assumptions, Dubois et al. [70] modeledthe expected velocity based on the as-received texture information and found it to be comparableto the experimentally determined values. Velocity profiles obtained via laser ultrasonics were alsoused to monitor recrystallization and phase transformations in aluminum[65], low carbon steels[67], Ti-6Al-4V [2] and zirconium [78]. To date, laser ultrasound has seen limited usage in thecontext of evaluating titanium and titanium alloys [1, 2]. In particular, this technique is of interestin evaluating transformation kinetics in commercially pure titanium since the high temperature bccphase is not stable at room temperature for ex-situ evaluation.32Literature Review2.5 Ultrasonic wave propagation in metals2.5.1 Calculation of velocity from the Christoffel equationIn an elastic medium the velocity of an ultrasonic pulse is dependent on the elastic stiffness tensor.In particular, the Christoffel equation can be used to calculate elastic wave velocity in an anisotropicmedium and is derived from solving for the characteristic equation of the wave equation aftersubstituting in Hooke’s law [79]. For a unit vector of propagation direction n, the Christoffel tensoris given by [79, 80]:〈T〉 =3∑j=13∑l=1c′ijklnjnl (2.22)Where cijkl is the 4th order crystallographic tensor describing the elasticity in a given media, andn is the propagation direction of the pulse.As the elastic tensor is symmetric, the Christoffel tensor is also symmetric, and thus is invariantupon altering the sign of the propagation direction. The elastic strain energy of a stable crystalis always positive and real, the the eigenvalue solutions to the 3 x 3 Christoffel tensor are alsoreal and positive, and are related to wave velocities propagating through the crystal. The P wavevelocity, S1 wave velocity, and S2 wave velocity are related the square root of the fraction of thefirst, second and third eigenvalue over density, respectively [80].vP =(λ1ρ)0.5(2.23)vS1 =(λ2ρ)0.5(2.24)vS2 =(λ3ρ)0.5(2.25)The three polarization directions, or displacement vectors, are the eiganvalue solutions of theChristoffel tensor. The three displacement vectors are mutually perpendicular since the Christoffeltensor is symmetric. The p-wave displacement direction is parallel to the propagation direction,33Literature Reviewand signifies the longitudinal component of the wave [80]. Conversely, the two S wave componentsare perpendicular to the propagation direction (n), and are termed quasi-S waves [80].2.5.2 Ultrasound velocity in an isotropic mediumThe stiffness matrix of an isotropic continuum ([c]iso) only contains the two independent elasticityparameters c11 and c44, and is defined as [44, 81]:[c]iso =c11 c11 – 2c44 c11 – 2c44 0 0 0c11 – 2c44 c11 c11 – 2c44 0 0 0c11 – 2c44 c11 – 2c44 c11 0 0 00 0 0 2c44 0 00 0 0 0 2c44 00 0 0 0 0 2c44(2.26)The [c]iso matrix is invariant upon rotation, meaning that if rotated by an arbitrary direction,the stiffness matrix will not change.The two independent elasticity parameters are often expressed as Lame´ parameters [81]:λo = c11 – 2c44 (2.27)μ = 2c44 (2.28)The two independent Lame´ parameters are closely related to the Eigenvalues of the elasticitymatrix, where μ can also be referred to as G and is the shear modulus. The elasticity matrix can34Literature Reviewbe rewritten as [81]:[c]iso,Lame´ =λo + 2μ λo λo 0 0 0λo λo + 2μ λo 0 0 0λo λo λo + 2μ 0 0 00 0 0 μ 0 00 0 0 0 μ 00 0 0 0 0 μ(2.29)Upon expansion of [c]iso,Lame´ into its corresponding 4th order elasticity tensor [82] and substi-tution of the elasticity tensor into Equation 2.22, and recalling that the P-wave velocity can bedefined by Equation 2.23, it can be seen the velocity of P-waves is related to elastic moduli inisotropic media by [83]:vP,iso,single =(λo + 2μρ)0.5(2.30)Where λo + 2μ can be defined as the P-wave modulus (λ), and thus, alternatively, velocity canbe defined as:vP,iso,single =(λρ)0.5(2.31)2.5.3 Rotation of elastic stiffness tensorsThe hcp unit cell of titanium has anisotropic elastic properties (cf. Section 2.2.2). Ultrasoundvelocity is approximated to be proportional to the square root of the fraction P-wave modulus ofa material over it’s density. Thus, any anisotropy present in the elastic properties of the materialwill produce a anisotropic response in the velocity. The velocity of a wave in any given directionof the crystal can be evaluated by applying the appropriate rotation to the stiffness tensor prior tocomputation of the P-wave velocity.Each orientation given in Bunge Euler notation describing a discrete volume in the orientation35Literature Reviewdistribution (OD) must be converted to Bond notation. This was accomplished by transformingfirst the three Bunge Euler angles (φ1, φ2, Φ) to Roe angles (ψ, θ, φR) via:ψ =φ1 Φ = 0φ1 +pi2 Φ ≥ piφ1 +3pi2 Φ ≤ pi; θ =Φ Φ = 02pi – Φ Φ ≥ piΦ Φ ≤ pi; φR =φ2 Φ = 0φ2 +3pi2 Φ ≥ piφ2 +pi2 Φ ≤ pi(2.32)Next, it is necessary to convert the above Roe angles to Bond angles:ξz = –φR; η = –θ; ξp = –ψ (2.33)The Bond angle description for each discrete orientation can be used to rotate the stiffnessmatrix [c] to find the corresponding matrix responsible for describing the elastic properties at thatgiven orientation. There are 3 rotations in total applied: one about the z axis, one about the newy axis, and finally one about the newly formed z axis. Provided the absolute value of ξz, η, and ξpare greater than zero, the following rotation can be applied to the stiffness matrix [79]:[c′] = [Mz][My][Mz][c][Mz]T[My]T[Mz]T (2.34)The rotation matrix [Mz] is defined as [79]:[Mz] =cos(ξz)2 sin(ξz)2 0 0 0 sin(2ξz)sin(ξz)2 cos(ξz)2 0 0 0 –sin(2ξz)0 0 1 0 0 00 0 0 cos(ξz) –sin(ξz) 00 0 0 sin(ξz) cos(ξz) 0–sin(2ξz)2sin(2ξz)2 0 0 0 cos(2ξz)(2.35)The rotation matrix [My] is defined as [79]:36Literature Review[My] =cos(η)2 0 sin(η)2 0 –sin(2η) 00 1 0 0 0 0sin(η)2 0 cos(η)2 0 sin(2η) 00 0 0 cos(η) 0 sin(η)sin(2η)2 0–sin(2η)2 0 cos(2η) 00 0 0 –sin(η) 0 cos(η)(2.36)Now, each rotated stiffness matrix [c’] must be converted back to the 4th order crystallographictensor [cijkl], which can then be converted to the Christoffel Tensor 〈T〉 by Equation 2.22 (Sec-tion 2.5.1). The longitudinal, or P-wave, velocity is then given by Equation 2.23 (Section 2.5.1).2.5.4 Elastic properties of polycrystalline aggregates in the isotropic assump-tionIsotropic polycrystalline aggregates of hcp crystalsThe stiffness matrix for an hcp crystal is denoted by [c]hcp throughout the following sections andis given by:[c]hcp =c11 c12 c13 0 0 0c12 c11 c13 0 0 0c13 c13 c33 0 0 00 0 0 c44 0 00 0 0 0 c44 00 0 0 0 0 c66(2.37)Where c66 =c11–c122 .The macroscopic elastic properties of a polycrystal can be calculated under the assumptionof an isotropic polycrystalline aggregate. This corresponds to the aggregate consisting of smallgrains of random orientation, where there is a sufficient grain count for the sample as a whole to beconsidered isotropic in elastic property. Two methods of calculating the polycrystalline averages ofmacroscopic properties are typically employed: the Voight and Reuss methods [82, 84, 85].In 1889, Voight [84] chose to analyze the stiffness constants [c] under the assumption that37Literature Reviewhomogeneous strain was maintained throughout the stressed polycrystal in all directions. Via 3-Dintegration, under the isotropic polycrystalline assumption, Voight derived general equations forthe bulk modulus (BVoight) and the shear modulus (GVoight) [82, 84].The bulk modulus accounts for a volume change with no observed shape change, and is describedas:BVoight =2c11 + c33 + 4c13 + 2c129(2.38)Conversely, the shear modulus accounts for a shape change with no observed volume change,and is described as:GVoight =3.5c11 + c33 – 2c13 – 2.5c12 + 6c4415(2.39)Contrarily, Reuss [85] chose to analyze the compliances [s] under the assumption that homo-geneous stress was maintained throughout the strained polycrystal. Under the isotropic polycrys-talline assumption, Reuss also derived general equations for: the bulk modulus (BReuss) and theshear modulus (GReuss) [82].The Reuss bulk modulus is described as [82, 85]:BReuss = [2s11 + s33 + 4s13 + 2s12]–1 (2.40)Conversely, the shear modulus accounts for a shape change with no observed volume change,and is described as [82, 85]:GReuss = 15 [14s11 + 4s33 – 8s13 – 10s12 + 6s44]–1 (2.41)In practice, experimental measurements suggest that the elastic response lies between thebounds set forth by the Voight [84] and Reuss [85] assumptions, and thus an arithmetic aver-age, called the Hill average(BHill, GHill), of the two is used. The pressure wave (P-wave) moduluscan then be calculated as[83]:38Literature ReviewλHill = BHill +43GHill (2.42)Isotropic polycrystalline aggregates of bcc crystalsThe stiffness matrix for a bcc crystal is denoted by [c]bcc throughout the following sections and isgiven by [44]:[c]bcc =c11 c12 c12 0 0 0c12 c11 c12 0 0 0c12 c12 c11 0 0 00 0 0 c44 0 00 0 0 0 c44 00 0 0 0 0 c44(2.43)The P-wave modulus (λ = λo + 2μ) and shear modulus (μ = G) are given under the Voightassumption that homogeneous strain was maintained throughout the stressed polycrystal as [86]:λVoight = c11 –25(c11 – c12 – 2c44) (2.44)μVoight = c44 +15(c11 – c12 – 2c44) (2.45)Conversely, if it is assumed that homogeneous stress was maintained throughout the polycrystal,the Reuss averaged Lame´ parameters are given as [86]:λReuss =2(s11 + s12 –(s11–s12–0.5s44)5)((s11 + 2s12)(s44 +45 (s11 – s12 – 0.5s44))) (2.46)μReuss =(s44 +45(s11 – s12 – 0.5s44))–1(2.47)39Literature Review2.5.5 Ultrasound velocity in a textured polycrystalline aggregateUntil this section, the effect of temperature and orientation on velocity had only been consideredfor a single crystal or a polycrystalline aggregate in the isotropic assumption (i.e. weakly textured).However, the section through which the planar, longitudinal wavefront travels through in the LUMetscenario contains many grains of varying orientations. Thus, it is prudent to effectively describethe orientation distribution (OD), to determine the volume fraction of grains at a given orientation,and to calculate a weighted average of the Christoffel tensor to compute a velocity representativeof the aggregate’s texture.At each set of Euler angles describing a bin in the OD, the intensity of occurrence can be usedto calculate the volume fraction of grains that possess that current orientation [87].f(i) =IΔφ1Δφ2(cos(Φ – ΔΦ2))pi2(2.48)Vm(i) =f(i)n∑i=1f(i)(2.49)Where each iteration, i, corresponds to a discrete orientation, Δφ1, Δφ2, ΔΦ are equal topi180 .This results in a volume fraction for each discrete orientation occurrence mapped in the OD,where each discrete orientation occurrence can be used to rotate the given stiffness tensor, and thusChristoffel tensor, to one descriptive of the elastic response at that given orientation. This createsa weighted distribution of possible Christoffel tensors, where volume fraction serves as the weight.The polycrystalline averages can be calculated under the Voight, Reuss or Hill convention [80].Under the Voight convention, it is assumed homogenous strain was maintained in all directionsthroughout the polycrystal as it is stressed [82]. This corresponds to the induced tensor (strain)at every position is set to equal the macroscopic induced tensor of that specimen [82, 84]. TheVoight average of the Christoffel tensor, 〈T〉Voight, is defined as the volume weighted average ofthe individual tensors (T) at a given orientation (gm) with a volume fraction of Vm [80]:40Literature Review〈T〉Voight =M∑m=1VmT (gm) (2.50)Conversely, the Reuss average employs the assumption of homogeneous stress [82, 85]. Thisimplies that the applied tensor (stress) at each location is set to equal the macroscopic appliedtensor of the entire specimen. The Reuss average of the Christoffel tensor, 〈T〉Reuss, is then definedas the volume weighted average of the inversed individual tensors (T–1) at a given orientation (gm)with a volume fraction of Vm [80]:〈T〉Reuss =[M∑m=1VmT–1 (gm)]–1(2.51)Empirical observations have demonstrated that tensor of an aggregate material lies in betweenthe Voight and Reuss assumptions, and thus the Hill average can be utilized:〈T〉Hill = 12[〈T〉Voight + 〈T〉Reuss](2.52)Thus, the average Christoffel tensor, and therefore the average longitudinal velocity, of a poly-crystalline aggregate can be computed at a given temperature by substituting the Hill averagedChristoffel tensor (Equation 2.52) into Equations 2.22 and 2.23.2.5.6 Velocity in the two phase regionThe ultrasonic velocity dependence on the phase fraction of a multiphase compound consists of nu-merous inputs, including the morphology of phases, elastic constants, texture effects, and density.However, Kruger and Damm [69] proposed that for constituent phases of similar elastic proper-ties and densities, complex modeling can be simplified to a volume fraction-weighted-average ofindividual constituent velocities independent of morphology and obtained by the rule of mixtures.For a general case considering a dual phase system, this rule of mixtures can be taken on velocity41Literature Reviewas:v = fαvα + fβvβ (2.53)Kruger and Damm applied the ultrasonic lever-rule method to the monitoring of austenitedecomposition into ferrite [69]. During measurement they observed a monotonic increase as thesteel was cooled from the austenite phase. Because the specimen was being continuously cooled,it was necessary to compute the single phase profile for ferrite and austenite region, so that thesecurves could be averaged and compared to the experimentally obtained laser ultrasound results.The austenite fraction over temperature profile was obtained using laser ultrasonics and dilatometry,and were found to be comparable[69]. Section 5 explores this assumption in greater detail using aFinite Element Model and compares various averaging methods.42Chapter 3Scope and ObjectivesAn integral step in ensuring titanium and its alloys meet their demanding performance criteriais controlling the mechanical properties via strict control of the microstructure. In particular, thevarious phase transformations present in titanium were shown to greatly alter the microstructure intitanium and its alloys. Thus, it would be prudent to monitor and evaluate phase transformations intitanium during thermo-mechanical processing. However, phase transformations in titanium occurat a high temperature, and in the context of commercially pure titanium (cp-Ti) and some Ti-alloys, the high temperature phase (β) does not exist at room temperature. As such, conventionalmethods for monitoring and studying phase transformation kinetics are not always viable for thismaterial.As discussed in the previous chapter, Laser ultrasonics for metallurgy (LUMet) provides anoption for monitoring phase transformations during thermo-mechanical processing and, to date,only two studies have attempted to use this technique on titanium alloys [1, 2]. The objective ofthis work is to evaluate the feasibility of LUMet as a non-destructive tool for monitoring the α/βphase transformation kinetics in pure and near-β titanium alloy.In order to achieve this goal, three studies were conducted. Firstly, a finite element model-ing (FEM) approach was used to simulate the ultrasound response in a simulated 2 dimensionalmicrostructure for a 2-phase material, in order to understand the effects of precipitate spatial ar-rangement and increasing phase fraction on the velocity signal. In particular, this initial studyidentified how the measured ultrasonic velocity could be obtained from the phase fraction and therespective properties of the two phases.43Scope and ObjectivesSecondly, cp-Ti specimens were subjected to continuous heating and cooling treatments throughthe β transus (882 °C). Cp-Ti was selected in order to monitor a 100% α→ β transformation, andthus the largest possible change in velocity due phase transformation could be detected and resolvedwith the ultrasound technique. Continuous heating as opposed to isothermal holding was selectedto minimize heating and cooling overshoot error, since the α/β transformation in cp-Ti is allotropicand would begin before the isotherm was established in the system. Specimens were cycled 5 timesto determine repeatability of the velocity response and examine the effects of bulk texture changeson the velocity profile. Ex-situ metallography and electron backscatter diffraction (EBSD) wereemployed to analyze the microstructure and texture of the specimens before and after treatment.Finally, Ti-5553 specimens were subjected to a 15 minute hold at 900oC to fully transformthe alloy to the β phase, cooled at 7 °C.s–1 to 700°C and then held for 5, 10, 33, 53, 75, or 180minutes before being quenched. During each isotherm, LUMet was employed to monitor the αprecipitation kinetics in-situ. Ex-situ metallography and EBSD provided a direct method for ob-serving microstructural evolution, as well as the nucleation and growth kinetics during the precipi-tation process, and was correlated against the LUMet results. The isothermal holding temperature(700°C) for precipitation was selected to provide the fastest rate of transformation according to thepublished time temperature transformation (TTT) diagram [57]. Other isothermal holding temper-atures located either above or below this peak would result in extensive treatment and monitoringtimes needed to reach equilibrium, and can result in signal degradation during LUMet observationdue to surface damage endured over this time.44Chapter 4Experimental Methodology4.1 As-received materials4.1.1 Commercially pure titaniumA commercially pure (99.7 wt% Ti) titanium (cp-Ti) plate purchased from Alfa-Aesar was used forthis study. The primary impurities in this material were oxygen (1800 wt.ppm), iron (500 wt.ppm),nitrogen (100 wt.ppm) and hydrogen (15 wt.ppm). Thin plate specimens having dimensions of 60 x10 x 3 mm were machined from the as-received plate for the laser ultrasonic experiments, with theplate’s normal direction (ND) corresponding to the smallest specimen dimension and the rollingdirection (RD) to the largest (Figure 4.1). The 3 mm thickness of the specimens provided a goodcompromise when optimizing the ultrasonic response in the specimens since it was not thick enoughto cause significant attenuation (where the second and third echos would no longer be resolved)and was not so thin as to cause an overlap between subsequent echos.4.1.2 As-received Ti-5Al-5Mo-5V-3CrThe material used for this study is a near-β titanium alloy (Ti- 5553) provided by UTC AerospaceSystems and developed by Verkhana Salda Metallurgical Production Association of composition:5.34 - 5.43 wt.% Al, 4.84 - 5.18 wt.% V, 4.96 - 5.12 wt.% Mo, and 2.73 - 2.86 wt.% Cr. In addition,the alloy contains approximately 0.3 - 0.34 wt.% Fe, 0.029 - 0.031 wt.% Si, 0.003 - 0.008 wt.% Zr,0.148 - 0.151 wt.% O, 0.0011 - 0.0026 wt.% H2, and 0.006 - 0.007 wt.% C.45Experimental Methodology4.2 Ex-situ microstructural characterization4.2.1 Scanning electron microscopySpecimens for scanning electron microscopy (SEM) and electron backscatter diffraction (EBSD)observation were prepared by surface grinding using 400, 600, 800, and 1200 grit silicon carbidepapers using water as a lubricant. The specimen surfaces were then mechanical polished using 6μm and 1 μm diamond suspension followed by a finishing step using a 5:1 (by volume) mixtureof 0.05 μm colloidal silica to 30 wt % hydrogen peroxide solution. To ensure the specimens werenot heavily contaminated with colloidal sillica particles, the polishing cloth was flooded with wateruntil it was clean and the samples were polished just with water to remove the colloidal sillica fromthe surface. After each grinding step the samples were rinsed under cool tap water for 30 seconds,and then immediately rinsed with denatured ethanol, and immediately dried with compressed air.Additionally, after the final colloidal sillica polishing step, the specimens were ultrasonicated indenatured ethanol for 30 minutes, before being dried using compressed cool air, and being storedunder vacuum to prevent the formation of hydrides and oxides on the sample surface.SEM observations were made using either a Hitatchi S-570 scanning electron microscope with atungsten filament source and an applied voltage of 20 kV, or a Carl Zeiss NTS Ltd. Sigma scanningelectron microscope with a field emission source and an applied voltage of 20 kV. Backscatterelectron contrast, where phase contrast or orientation contrast in single-phased materials, was usedto determine the microstructure. Phase contrast was thresholded to determine the fraction of αproduced after each isothermal holding time, and the standard deviation of all measurements wereused as error descriptors. Grain sizes were taken as the diameter of a circle of equal area (EQAD).4.2.2 Electron backscatter diffractionEBSD observations were made using a Carl Zeiss NTS Ltd. Sigma scanning electron microscopewith a field emission source and an applied voltage of 20 kV and 60 μm aperture size. The EBSDpatterns were collected with a Digiview detector and indexed using the EDAX TSL OrientationImaging Microscopy (OIM) data collection software.Using the EDAX TSL OIM Analysis 6 software, grains were defined as clusters of at least 5points in size and grain boundaries were identified using a disorientation angle of 5◦ or greater. The46Experimental Methodologygrain confidence index standardization method was employed to ensure indexed points with lowerconfidence indices that shared the same orientation as their surrounding grains were not excluded.Finally, an instance of grain dilation was performed and the resulting map was visually checked toensure false grains were not created by the dilation step.The area of each grain was extracted and the grain sizes were taken as the diameter of a circleof equal area (EQAD). The location of the prior β grain boundaries for the as-treated specimenswere estimated using the 6 reduced axis-angle pairs method as described by Wang et al. [88].This approach allows for the approximate reconstruction of the β grain structure from a fully αmicrostructure by identifying special grain boundary disorientations between α grains within atolerance of ± 5 °. The grain boundaries that do not share special axis-angle relationships arisingfrom the Burgers orientation relationship between the α-phase and β-phase are considered as priorβ grain boundaries.EBSD map files, containing, x, y coordinates, and the corresponding Euler angles describingthe orientation of the given pixel, were acquired using the Carl Zeiss NTS Ltd. Sigma ScanningElectron Microscope with a Digiview detector, using EDAX TSL Orientation Imaging Microscopy(OIM) Data Collection software. The MTEX quantitative texture analysis tool was used to createa discretized OD with step size of 5o. An OD was taken for each specimen, and used to predictvelocity for a given texture following Sections 2.5.3 and 2.5.5 [80].4.3 In-situ microstructural characterization4.3.1 Laser ultrasonicsHeat treatments were conducted in a Gleeble 3500 thermomechanical direct resistance heatingfurnace (Dynamic System Inc. Poestenkill, NY) with a window transparent to the excitation laserand infrared laser wavelengths. A series of pumping steps involving a first stage of rough pumpingusing a mechanical pump and a later high vacuum stage involving a diffusion pump were employedto create a vacuum environment within the testing chamber of pressure at most 0.5 Pa. A lowfriction force jaw assembly was used ensuring the treatment was purely thermal. The specimenswere secured on either end by two sets of water cooled copper grips. A current was applied tothe grips and flowed through the specimen, resulting in resistive heating of the specimen. The47Experimental Methodologytemperature of the specimen was measured and controlled with a pair of K-type thermocouplewires spot-welded at the mid-length of the specimen on the opposite side of the laser beams.Laser ultrasonic measurements were conducted in the Gleeble 3500 thermomechanical simulatorusing a Laser Ultrasonics for Metallurgy (LUMet) sensor. The LUMet sensor is attached to the reardoor of the Gleeble chamber. A frequency-doubled Q-switched Nd:YAG laser with a wavelengthof 532 nm is used for the generation of a wide band compressive ultrasonic pulse. The duration ofthe laser pulse is approximately 6 ns, it has a maximum energy of 72 mJ and up to 50 pulses canbe generated per second. The laser pulse produces a broadband ultrasonic pulse by vaporizing asmall quantity of material at the surface (of the order of one micrometer per hundred laser pulses).The ultrasonic pulse propagates back and forth through the thickness of the specimen and itsamplitude decreases by interacting with the material and its microstructure. Successive arrivalsof the ultrasonic pulse at the generation surface are detected with a frequency-stabilized Nd:YAGpulsed laser which illuminates the surface with infrared radiation at a wavelength of 1064 μm anda pulse duration of 90 μs. The infrared detection laser that is reflected on the specimen surface isdemodulated inside a photo-refractive crystal using an active interferometer approach [75]. Afteronly a few oscillations of the ultrasonic pulse between the two parallel surfaces of the specimen,its intensity is only weakly diffused in other directions and the ultrasonic properties measured arerepresentative of the average properties of the material over a volume created by the surface ofthe laser spot (about 2 mm) multiplied by the specimen thickness. Both generation and detectionlaser beams are co-linearly aligned at the center length of the specimen. The time required for theultrasonic pulse to travel back and forth through the specimen thickness is calculated using a crosscorrelation algorithm applied on two echoes collected at the specimen surface. The velocity is thencomputed by the ratio of the propagation distance by the time between two echoes. Cross correlationand velocity calculation was conducted using the software package CTOME (Computational Toolsfor Metallurgy) [89] developed by Thomas Garcin at the University of British Columbia. Figure 4.1shows a schematics illustration of the measurement configuration for the cp-Ti tests. The Ti-5553specimens were oriented identically as shown in Figure 4.1, with the exception that the backscattersurfaces were taken by cutting down the TD plane at the center-line.48Experimental MethodologyFigure 4.1: Schematic of cp-Ti specimen in the Gleeble with the normal direction (ND) alignedwith the direction of wave propagation, the rolling direction (RD) aligned with the length of thespecimen, the transverse direction (TD) aligned with the width of the specimen, and the EBSDimage plane is shown in grey lying in the plane defined by the TD normal.4.3.2 Thermal treatmentsContinuous heat treatments to evaluate phase transformation kinetics in cp-TiContinuous heating and cooling cycles were conducted at a rate of 3 °C.s–1. Three conditionswere selected for monitoring the α/β transformation. As will be shown below, these cyclic thermaltreatments led to three distinct microstructures and textures allowing the technique to be evaluatedfor different conditions. In the first set of tests the as-received material was continuously heatedto 1000 °C followed by cooling to room temperature where the microstructure was observed. In asecond set of tests, specimens were heated to 1000 °C followed by cooling to room temperature. Thissame specimen was then subjected to the same cycle, the transformation kinetics being monitoredin the 2nd heating/cooling cycle and the microstructure observed after the 1st cycle. In the finalset of experiments, specimens were cycled 4 times between room temperature and 950 °C. On the5th cycle the transformation kinetics were monitored, the microstructure being observed followingthe final cooling. In the third set of experiments it was decided to limit the upper temperature to950 °C so as to avoid excessive grain growth. The first two sets of tests were conducted to 1000 °Cso as to allow for the ultrasonic velocity to be monitored over as large a range of temperatures inthe β-phase as possible. If a specimen was to endure repeated cycling, between each cycle, it would49Experimental Methodologybe polished gently to remove any oxides that would prevent attachment of thermocouples or causescattering of the infrared laser. The cycling treatment profiles are shown in Figure 4.2.a. b. c.Figure 4.2: Heat treatment profiles for specimens enduring a) 1 cycle of treatment, b) 2 cycles oftreatment, and c) 5 cycles of treatment where the red line indicates the equilibrium transformationtemperature of 882 °C.Isothermal heat treatments to evaluate precipitation kinetics in Ti-5553Specimens were heated to 900 °C and held for 15 minutes to fully dissolve the α phase. Specimenswere then cooled at 6.7 °C.s–1 to 700 °C. Specimens were held at 700 °C for times of: 5, 10, 33,53, 75, and 180 minutes before being quenched with helium gas to room temperature. A controlspecimen, after being cooled at 6.7 °C.s–1 to 700 °C was quenched immediately to determine thefull β microstrucuture at the onset of the isotherm. Laser ultrasonic measurements were takenduring the isothermal segments of each specimen’s treatment to monitor the α phase precipitationand ex-situ metallographic evaluation was conducted after each quench.4.3.3 Interpretation of waveform data in CTOMEThe velocity is then computed by the ratio of the propagation distance by the time betweentwo echoes. Cross correlation and velocity calculation was conducted using the software pack-age CTOME [89]. Within the CTOME software, velocity was calculated using the single echotechnique (SE2), where the software calculates the cross correlation function between the secondecho of each waveform and the second echo of a waveform in a reference file. Using this technique,the velocity is calculated using the equation:50Experimental Methodologyv =ztR + Δt=2netR + Δt(4.1)Where z is the propagation distance, tR is arrival time of the reference echo, e is the samplethickness, Δt is the delay time calculated from cross-correlating the observed echo with the referenceecho, and n is the echo number. The arrival time of the reference echo, tR, is measured by locatingthe time at the local minimum for the reference echo. The time of travel (τ) of the considered echo,is then tR + Δt.Processing of cp-Ti velocity dataEach data file was compared using the second echo as a reference echo with the SE2 algorithmin CTOME. The reference file (reference C) was taken in the as-received state, under vacuum, atroom temperature, and was kept constant for each data set. The reference waveform used in theanalysis was the 20th, as it was representative and of sufficient quality. The window size was setto be 0.35 μs, the maximum frequency was set to 15 MHz, and the window was centered on theminimum of each waveform.Processing of Ti-5553 velocity dataEach data file was compared using the second echo as a reference echo with the SE2 algorithmin CTOME. The reference file (reference A) was taken in the as-received state, under vacuum, atroom temperature, and was kept constant for each data set. The reference waveform used in theanalysis was the 20th, as it was representative and of sufficient quality. The window size was setto be 0.35 μs, the maximum frequency was set to 15 MHz, and the window was centered on theminimum of each waveform.4.3.4 Interpretation of error in laser ultrasound measurementsThere are two important error descriptors in velocity, including, the absolute error in velocity andthe precision (relative error in velocity) of the velocity measurements.Velocity is typically measured with an accuracy (absolute error) equal to the accuracy of the51Experimental Methodologythickness measurement, i.e. of about 1 %. Velocity is defined as distance traveled (2ne) where e isthe thickness of the piece and n is the echo number, divided by time of travel (τ). The error can becalculated by summation in quadrature since the two quantities are independent. Application ofthe logarithm to both sides of the equation before differentiation and addition in quadrature yields:v =2neτ(4.2)ln (v) = ln (2n) + ln (e) – ln (τ) (4.3)(δvv)2= 0 +(δee)2+(δττ)2(4.4)δv = v((δee)2+(δττ)2)0.5(4.5)Where δe is the error in thickness due to the caliper reading, and is equal to 0.01 mm, andδτ is an estimate of the error in the measurement of the delay that is obtained from the normalregression (Gaussian fit) applied to the cross-correlation function for the localization of the localmaximum in CTOME [89]. The software Gnuplot [90] estimates a standard deviation for eachfitting parameters from the variance-covariance matrix. In this study, the standard deviation forthe value of delay is of the order of 1%. For time delays in titanium of approximately 0.98 μs, theerror is given as ±0.009 μs.With this, relative error is given as:δvv=((δee)2+(δττ)2)0.5(4.6)δvv=((0.01mm3mm)2+(0.009μs0.98μs)2)0.5(4.7)δvv= 0.97% (4.8)This absolute error of approximately 1 % is used to compare velocity profiles of different treat-ments to one another.The precision (repeatability) of the measurement was taken by capturing multiple velocity52Experimental Methodologymeasurements at the same location on a titanium sample, held at room temperature. The standarddeviation of the velocity measurements corresponded to 0.01 % which was used to describe therelative error in velocity, when comparing points on the same velocity profile.53Chapter 5Finite Element Modeling ofUltrasonic Wave Propagation in DualPhase Material5.1 IntroductionThe purpose of this section is to describe the results of a finite element (FE) simulation of ultra-sound propagation in a two dimensional specimen containing two isotropic phases, and perform asensitivity analysis on six possible averaging schemes that could be used to calculate velocity in theaggregate. In particular, this modeling approach intended to evaluate the proper way to predictthe average ultrasonic velocity, as measured in an experiment, from the velocities of the individualphases. The effect of phase fraction, and particle geometry was explored.5.2 Input parameters, construction and assumptionsExplicit, 2-D finite element modelling (FEM) simulations were performed in Abaqus CAE version6.13. Each phase was assumed to be elastically isotropic, and defined by its respective density,Young’s modulus, and Poisson ratio (Table 5.1), where values were decided to be representative ofα and β titanium.54Finite Element Modeling of Ultrasonic Wave Propagation in Dual Phase MaterialTable 5.1: Input parameters defining two isotropic phases used in an explicit 2-D FEM simulationof an ultrasound pulse propagating in a dual phased polycrystalline aggregate structurePhase Density (ρ) Young Modulus (E) Poisson Ratio (υ)(α/β) (kg.m–3) (GPa)α 4502 120 0.3β 4382 70 0.3The 2-D meshed specimen had a thickness of 2 mm, a length of 10 mm, and a mesh size of 4μm in the small meshed region (4 mm by 2 mm) located in the center of the specimen. In order tofix the sample geometry in space during the pulse propagation, additional boundary conditions aredefined for the elements of the mesh located along the left and right vertical edges of the geometrysuch that no rotation and translation are permitted, i.e. U1 = U2 = U3 = UR1 = UR2 = UR3 =0 (Abaqus Boundary Condition: Mechanical-Encastre).The mesh size in the simulation region is adjusted to ensure that the smallest longitudinalwavelength present in the system during the simulation is described by at least 10 elements of themesh. An estimate of the smallest wavelength is obtained by the ratio of the longitudinal velocity(Cd ' 5000 m.s–1) by the maximum frequency of vibration (fmax ' 100 MHz) [91]. The mesh sizemust therefore be smaller than Cd/(fmax*10) = 5000/(100*10) = 5 μm. The element size (Le) isthen set to 4 μm for the set of simulations presented in this chapter.The elements in the small meshed region were defined as four-node bi-linear quadrilateral planestrain (CPE4R) elements, meaning that per element there are four straight element sides with 4nodes, where one node is located at each corner. Conversely, the larger mesh structure outside thesmall meshed region served the purpose of dissipating transverse components of the wavefront, soonly the longitudinal component that returned to the monitoring region would be accounted forwithin the simulation time.The time increment for the simulation is calculated by ensuring that at least 4 time steps arerequired for a longitudinal wave to propagate a distance of one mesh, i.e. Δt = 1/4 (Le/Cd) =1/4*(4/5) = 0.2 ns [91].Scripts to assign the material properties of the elements were generated in MATLAB version2013a. As shown in Figure 5.1, three different geometric arrangements of the two phases were55Finite Element Modeling of Ultrasonic Wave Propagation in Dual Phase Materialconsidered. The plate structures, shown in Figure 5.1b and Figure 5.1c were created with 50 %of the elements being β phase elements. The motivation behind the creation of the horizontaland vertical plate geometries was to compare the simulated results to theoretically determinedultrasonic velocities under series and parallel assumptions, respectively. The dispersed elementstructure shown in Figure 5.1d was modeled at various phase fractions of β phase, ranging from 0% to 100 %. Table 5.2 summarizes the relative phase fraction and seeded microstructure imposedunto the small meshed region of the FEM specimen for each trial.Figure 5.1: (a). 2D generated mesh part with thickness of 2 mm, length of 10 mm and mesh sizeof 4 μm in the small mesh region (4 mm x 2 mm), (b). horizontal plates, (c). vertical plates and(d). randomly dispersed elements. Vertical displacement (uy) is measured as the direct output ofthe simulation from nodes in the generation/detection line (blue), and the vertical displacementdirection is indicated by the red arrow.56Finite Element Modeling of Ultrasonic Wave Propagation in Dual Phase MaterialTable 5.2: Summary of imposed microstructures seeded to the core interaction region of the meshedspecimenTrial fα fβ Imposed microstructure(%) (%)1 0 100 dispersed2 10 90 dispersed3 30 70 dispersed4 50 50 dispersed5 70 30 dispersed6 80 20 dispersed7 90 10 dispersed8 100 0 dispersed9 50 50 horizontal plates10 50 50 vertical plates5.2.1 Wavelet generationAn initial displacement was applied to a region 2 mm in size located at the center of the topboundary of the meshed part (cf. Figure 5.1). The displacement is applied in the form of a Rickerwavelet (frequency = 10 MHz) because it provided a means to center the frequency range withinthe experimental frequency range of LUMet (5 - 20 MHz). The amplitude of a Ricker wavelet isgiven by [92, 93]:Ao = 1.1(1 – 2pi2f2(t – bc)2)e–pi2f2(t–bc)2(5.1)Where b is 125 ns, and serves to center the Ricker peak at 125 ns, and d is 1.1 and applied forscaling the amplitude, f is the frequency (Hz), t is time (s), and A is the displacement amplitude.The absolute amplitude selected for convenience has little effect in the model as the medium wasconsidered with no elastic limit, and the amplitude of the pulse does not effect the measuredresponse (velocity). The reader is referred to the work of Murray and Wagner [94] for a more indepth analysis on modeling displacement amplitudes in the ablative regime. Constants b and c are57Finite Element Modeling of Ultrasonic Wave Propagation in Dual Phase Materialimplemented to scale the Ricker pulse to better match experimental observations from the LUMetmeasurement. The Ricker pulse generated over a span of 250 ns is displayed in Figure 5.2a. Asthe ultrasonic wave is characterized by setting the frequency to 10 MHz and setting the materialproperties of the system which define a velocity range of [4600, 6000] m.s–1 (calculated from the 100% β and α conditions shown in Table 5.1), the range of wavelengths for the ultrasound pulse usedin each trial spans from 0.46 mm to 0.6 mm. The wavelength constitutes approximately a thirdof the specimen’s thickness (2 mm). In order to prevent a discontinuity between the nodes in thegeneration region and the adjacent nodes, a normalized Gaussian function (Figure 5.2b) symmetricabout the centerline, defines the amplitude of the Ricker wavelet spatially along the generation line(2 mm) as:A = Ao ∗ e(x22(0.5)2)(5.2)a. b.Figure 5.2: a). Time dependence of the Ricker pulse amplitude and b). an example of the Gaussianscaling used to normalize the applied amplitude (at t = 125 s) across the generation boundary, wherethe peak of the Gaussian is centered on the mid line of the specimen5.3 Interpretation of FEM dataAlong the boundary length where generation occurred, vertical displacement data was extractedfrom the nodes at the excitation area and averaged at each time step (8 ns) for a total time of 2000ns. The excitation area corresponds to along the 2 mm highlighted in blue in Figure 5.1. Thislength of 2 mm was selected because it is consistent with the diameter of the laser spot used inLUMet. When the averaged displacement data (uy) is plotted against time, the simulated waveform58Finite Element Modeling of Ultrasonic Wave Propagation in Dual Phase Materialis visible. An example of such a waveform is shown in Figure 5.3.Figure 5.3: Example of averaged displacement over time data taken from upper boundary nodesresponsible for generation and detection of the Ricker’s ultrasonic pulseFrom the above waveform, it is then possible to calculate the average velocity of the ultrasonicpulse. This average velocity is calculated as twice the thickness of the specimen(2 ∗ 2 mm = 4 mm)divided by the time it takes for the pulse to travel through the specimen and back to the monitoredsurface (tdelay). The tdelay is found by calculating the maximum amplitude of the cross correlationbetween the first and second echo shown in Figure 5.3. Cross correlation was calculated as:Ax–corr = IFFT(FFT (f) ∗ FFT (g)) (5.3)Where FFT is the fast fourier transform of the echos, IFFT is the inverse fourier transform ofthe echos, and the overbar indicates the complex conjugate.The amplitude of the cross correlation function and respective time delay was calculated usingthe software package CTOME, where a sample output of the cross correlation function for a givenwaveform is provided in Figure 5.4. The maximum of the cross correlation amplitude signifies thetime delay. The two echo technique (TE1) was used to cross-correlate the first echo to the secondecho in the same waveform to find time delay in the FEM study.59Finite Element Modeling of Ultrasonic Wave Propagation in Dual Phase Materiala. b.Figure 5.4: Example of a) two waveform echos (f and g), and, b) the corresponding amplitude ofcross correlation function5.4 Calculation of average velocitiesOne intention of these simulations was to compare the FEM computed average velocity to waysone could compute the average velocity based on the moduli and the densities of the two phases(cf. Equations 5.4, 5.5, 5.6, 5.9, 5.10, 5.11). First the prediction based on approximating theinhomogeneous material as a homogeneous material using either the upper bound averaged elasticmodulus to calculate velocity (v1) or the lower bound averaged elastic modulus to calculate velocity(v2), and the arithmetic average (v3) was tested and are given as:v1 =(1ρ(1 – υ)(1 + υ) (1 – 2υ)(fαEα + fβEβ))1/2(5.4)v2 =(1ρ(1 – υ)(1 + υ) (1 – 2υ)EαEβfβEα + fαEβ)1/2(5.5)v3 =(1ρ(1 – υ)(1 + υ) (1 – 2υ)12((fαEα + fβEβ)+EαEβfβEα + fαEβ))1/2(5.6)Where the density (ρ) calculated at a given phase fraction is given using the rule of mixtures:60Finite Element Modeling of Ultrasonic Wave Propagation in Dual Phase Materialρ = fαρα + fβρβ (5.7)And the P-wave modulus had been expressed in terms of the Young modulus (E) and Poissonratio (υ):λ =(1 – υ)(1 + υ) (1 – 2υ)E (5.8)Alternatively, the predicted velocities obtained by assuming that the wave spends equal time inthe two phases (v4) or that the wave causes equal displacement in the two phases (v5) were tested,and an arithmetic average of the two were taken (v6).v4 = fαvα + fβvβ (5.9)v5 =(fαvα+fβvβ)–1(5.10)v6 =12(fαvα + fβvβ)+(fαvα+fβvβ)–1 (5.11)Where vα and vβ are the velocities observed in the individual α and β phases, respectively:vα =((1 – υ)(1 + υ) (1 – 2υ)Eαρα)1/2(5.12)vβ =((1 – υ)(1 + υ) (1 – 2υ)Eβρβ)1/2(5.13)5.5 FEM resultsIn the literature [69], it has been argued that the averaging scheme is not very important when thedifference in properties (density and elastic moduli) are small. This idea can be demonstrated byusing titanium as an example, which is the intention of the following section.61Finite Element Modeling of Ultrasonic Wave Propagation in Dual Phase MaterialThe FEM simulated velocity (corresponding to trial 1 - 8 of Table 5.2) demonstrated an in-crease in average velocities as additions of α phase particles were randomly added to the β matrix.Figure 5.5 compares the FEM simulated velocities for dispersed, horizontal plate, and vertical platearrangements of the two phases. Also shown are the predicted velocities according to the averagingschemes listed in Equations 5.4, 5.5, 5.6, 5.9, 5.10, and 5.11.Figure 5.5: Ultrasonic velocity computed by the six potential averaging schemes listed in Equa-tions 5.4, 5.5, 5.6, 5.9, 5.10, 5.11, as well as FEM simulation results, where vHP, vVP, and vFEMrefer to simulation trial 9 (horizontal plate), trial 10 (vertical plate) and trials 1 - 8 (dispersed),respectively, as defined in Table 5.2.)It is important to note that predictions from v3 and v6 overlap such that only v3 is visible.It appears that the FEM simulation containing the horizontal plated microstructure (trial 9of Table 5.2) is in complete agreement with the value calculated at fα = 50 % when using theharmonic averaging approach on the Young’s modulus (v2, Equation 5.5). When transverse loading62Finite Element Modeling of Ultrasonic Wave Propagation in Dual Phase Materialis assumed and velocity averaged by Equation 5.10, the velocity (v5) lay between the FEM velocities(Trials 1 - 8, Table 5.2) and the velocity obtained using v2. Conversely, the vVP value obtainedby the FEM simulation containing the vertical plated core interaction region (trial 10 in Table 5.2)lay in between the v1 and the rule of mixtures applied to velocity (v4).One can see that averaging based on the elastic moduli and wave velocities gives distinctlydifferent results in the case of the upper (v1 and v4) and lower (v2 and v5) bound predictions.First, upon examining the FE simulated lower bound containing horizontal plates, it was ob-served that the simulated velocity (vHP) coincided perfectly with v2. To understand why vHP isconsistent with v2 instead of v5, it is important to consider the relative size of the wavelength ofthe applied wavelet, which is between 0.45 and 0.6 mm, compared to the size of the microstructuralfeatures (plates of alternating phases), which are 4 μm in size.When the wavelength of the applied wave is approximately the same size or smaller than thewidth of each microstructural feature (Figure 5.6a), it is possible to solve for the aggregate velocityin the following way:v =dt(5.14)v =Lα + Lβt(5.15)Where t = tα + tβ which corresponds to time, tα and tβ are time spent in the α and β phases,respectively, d is the distance traveled, and Lα and Lβ are the widths of the α and β plates,respectively (as shown in Figure 5.6).Substituting for tα and tβ using Equation 5.14:t =Lαvα+Lβvβ(5.16)And substituting Equation 5.16 into Equation 5.15 yields:v =Lα + LβLαvα+Lβvβ= Lαvα + LβvβLα + Lβ–1 (5.17)63Finite Element Modeling of Ultrasonic Wave Propagation in Dual Phase MaterialWhere Lα + Lβ = Ltotal, and therefore:v =(LαLtotal(1vα)+LβLtotal(1vβ))–1(5.18)v =(fαvα+fβvβ)–1= v5 (5.19)Conversely, if the wavelength of the applied wavelet is larger than the distinct microstructuralfeatures (Figure 5.6b), as observed in the FE simulation, then the above formulation is no longervalid. The waveform no longer lies in one feature at a distinct unit in time but contacts manyfeatures simultaneously. Instead of imagining the wave propagating through the aggregate featureby feature (step by step), it is now more accurate to imagine the waveform moving through onecontinuous medium where the medium’s properties are defined as an average of the properties ineach phase. For the horizontally plated microstructure, this average is calculated in series usingthe harmonic averaging approach on the Young modulus (v2). The fact that v2 matches vHP isdue to the relatively large wavelength of the Ricker pulse when compared to the horizontal platethickness. Thus, the formulation described above in Equations 5.14 through 5.19 is only appropriatefor coarser microstructures.However, comparison of the upper bounds (v1 and v4) to the FE simulated velocity in a verticalplated microstructure (vVP), shows that v4 ≤ vvp ≤ v1. Even when the wavelength of the appliedRicker pulse is less than the size of the width of the vertical plates (Figure 5.6c), the wavefrontcannot separate and thus will spend equal time in each phase (each plate) while propagating in thisconfiguration. The wavefront must remain intact and can only travel at one velocity, and thereforeit is not expected to produce a direct match to v5. It is interesting to see that v1 also did notmatch the simulated velocity, demonstrating that even with a larger wavelength (Figure 5.6d), sincedistinct segments of the wavefront will touch distinct features, the complex interactions involvedto ensure the wavefront remains intact still have an effect.64Finite Element Modeling of Ultrasonic Wave Propagation in Dual Phase Materiala. b.c. d.Figure 5.6: Schematic showing imposed Ricker wavelet a) propagating in horizontal plates where thewavelength is relatively larger than microstructural features, b) propagating in horizontal plateswhere the wavelength is smaller than microstructural features, c) propagating in vertical plateswhere the wavelength is relatively larger than microstructural features, d) propagating in verticalplates where the wavelength is smaller than microstructural features. The black lines indicatesatoms in the wave, where lines appearing close together show compression of the wave, and linesspread apart indicate refraction. Wavelength is the distance between instances of maximum com-pression.Finally, the FE simulated results for wave propagation in a dispersed microstructure (vFEM),appeared to fall between v3, v6 and v5, where v3 and v6 are nearly identical. Given that thevelocity in a randomly dispersed mixture is often approximated as the average of the upper andlower bounds, which when computed using FE simulations were vVP and vHP, and where vVP waslower than v1, it appears intuitive that vFEM is slightly lower than the arithmetic averages v3 and65Finite Element Modeling of Ultrasonic Wave Propagation in Dual Phase Materialv6.5.6 Sensitivity of averaging schemesIn the literature [69], it has been arugued that the averaging scheme is not very important whenthe difference in properties (density and moduli) are small. This is demonstrated using titaniumas an example in the following section by evaluating the sensitivity of averaging schemes v1 - v6to changes in the P-wave modulus (λ) which under a polycrystalline and isotropic assumption isgiven by Equation 5.8.The relevant polycrystalline material properties in pure titanium are calculated at the trans-formation temperature of 882 °C and are provided in Table 5.3.Table 5.3: Density (ρ) and P-wave moduli (λ) of polycrystalline α and β titanium at 882 °CPhase Density (ρ) P-wave Modulus (λ) Reference(α/β) (g.m–3) (GPa)α 4417.19 123.72 [25, 36]β 4409.34 115.3 [25, 38]This corresponds to a state where the elastic modulus of the β phase is 93 % of the elasticmodulus of the α phase (λβ/λα = 93 %), and the density of the β phase is 99.8 % of the densityof the α phase (ρβ/ρα = 99.8 %). If λα is fixed to its polycrystalline value of 123.72 GPa, and λβis variable, the sensitivity of each averaging scheme presented in Equations 5.4, 5.5, 5.6, 5.9, 5.10,and 5.11 can be computed for given phase fractions. Figure 5.7 shows the sensitivity of averagingschemes v1 - v6 for β phase fractions of a. fβ = 0.1, b. fβ = 0.5, and c. fβ = 0.9. Regardless ofphase fraction, it can be observed that in instances where the material properties (λ, ρ) are similar,the selection of an averaging scheme has little effects on the resulting computed velocity.66Finite Element Modeling of Ultrasonic Wave Propagation in Dual Phase Materiala. b.c.Figure 5.7: Velocity plotted against a changing ratio of P-wave modulus (ρα/ρβ), where λα wasfixed at 123.72 GPa, and λβ is variable for a given phase fraction of β a) fβ = 0.1, b) fβ = 0.5, andc) fβ = 0.9Given the similarity in elastic moduli of α phase and β phase titanium, a rule of mixtures onvelocity (Equation 5.9) will suffice when modeling the effects of phase change. The application ofa rule of mixtures has been used before in laser ultrasonic evaluation of phase transformations insteels by Kruger and Damm [69].5.7 Chapter summaryThis chapter examined the propagation of an ultrasonic pulse in different geometric arrangements oftwo phase mixtures using a 2-D, explicit finite element model. The selection of a correct averaging67Finite Element Modeling of Ultrasonic Wave Propagation in Dual Phase Materialscheme to predict velocity was found to not only depend on the geometric configuration of the twophases but also the relative size of the wavelength compared to the width of the microstructuralfeatures. When the FE simulation ran on a series of horizontal plates, the resulting velocity waspredicted by taking the lower bound average of the Young modulus (v2), due to the relativelylarge wavelength of the pulse compared to the plate thickness. Conversely, when vertical plateswere used in the simulation, neither averaging assumptions (v1 and v4) were able to predict theresulting velocity exactly, demonstrating that interactions are at work to ensure the wavefrontremains in tact. Finally, in the context of titanium, predicting velocity via a rule of mixtures overvelocity is sufficient given the similarities in the bulk P-wave moduli and densities of the two phases.68Chapter 6In-situ Laser Ultrasonic Measurementof the HCP to BCC Transformationin Commercially Pure Titanium6.1 IntroductionHere, laser ultrasonic measurements are presented for the continuous heating and cooling of acommercially pure titanium plate with the goal being to provide a quantitative assessment of thetransformation kinetics. By performing experiments that thermally cycled through the β transus,the kinetics of the α to β and β to α-phase transformations could be assessed. Quantitative ex-situmetallography and EBSD have also been used to evaluate grain size and texture of the α-phase postmortem. The purpose of examining specimens after different numbers of thermal treatment cyclesthrough the β transus was to illustrate the importance of grain size (particularly in the β-phase)and crystallographic texture on phase fraction. From this, a systematic method is presented foranalyzing the ultrasonic velocity to obtain the fraction of α-phase and β-phase accounting for theeffects of material property and texture changes on phase transition.69In-situ Laser Ultrasonic Measurement of the HCP to BCC Transformation in Commercially PureTitanium6.2 Characterization of the as-received materialThe as-received microstructure of the commercially pure titanium specimens (composition dictatedin Section 4.1) was composed of polygonal α grains with a mean equivalent area diameter (EQAD)of 42 μm as determined by electron backscatter diffraction (EBSD). The as-received microstructureis shown in Figure 6.1.Figure 6.1: Normal direction (ND) EBSD IPF map showing equiaxed and recrystallized grains inthe as-received plate6.3 Ex-situ observation of microstructural changes upon thermalcyclingFigure 6.2 shows examples of backscatter electron (BSE) images with orientation contrast for thespecimens in the a) as-received state (AR), b) after 1 treatment cycle (1-cycle), and c) after 5treatment cycles (5-cycles) of heat treatment. It can be observed that the as-treated structurescontain arrangements of α packets containing substructures of plates aligned in parallel, growingout from prior β grain boundaries. Figure 6.2d and 6.2e show low magnification BSE images of theplated structure after 1 treatment cycle (1-cycle), and after 5 treatment cycles (5-cycles) of heattreatment, respectively.70In-situ Laser Ultrasonic Measurement of the HCP to BCC Transformation in Commercially PureTitaniumFigure 6.2: Backscatter electron (BSE) images with detailed orientation contrast for the specimensin the a) as-received state (AR), b) after 1 treatment cycle (1-cycle), c) after 5 treatment cycles(5-cycles) of heat treatment, and low magnification BSE images showing typical microstructuresd) after 1 treatment cycle (1-cycle), and e) after 5 treatment cycles (5-cycles) of heat treatmentFigure 6.3 shows representative portions of inverse pole figure maps measured for the specimensin the a) as-received state (AR), b) after 1 treatment cycle (1-cycle), and c) after 5 treatment cycles(5-cycles) of heat treatment. In each case the observations were made on the plane normal to theTD direction at the center of the specimen. Moreover, the location of observation was selected tocoincide with the region sampled by the ultrasonic pulse. Measurements were made in only onetwo-dimensional plane. All the EBSD maps were ≥ 1.5 x 1.5 mm in the plane of wave propagation(cf. Figure 4.1). This includes a large fraction of the grains that interacts with the ultrasonic pulse.This is important so as to have the local texture for velocity prediction.As can be seen in Figure 6.3a, the as-received state is composed of polygonal grains, whosesize is relatively homogeneous. In contrast, the microstructure of the specimens subjected to 171In-situ Laser Ultrasonic Measurement of the HCP to BCC Transformation in Commercially PureTitaniumand 5 treatment cycles (Figure 6.3b and 6.3c) are composed of coarse grains with irregular grainboundaries resulting from the cycling between the α and β-phases. The plate-like substructureof each grain in the as-treated conditions, which was visible under BSE orientation contrast (cf.Figure 6.2), was not observed in Figures 6.3b and 6.3c indicating that the plate packets wereof similar orientation, with disorientations less than 5°. The grain size (EQAD) for the threemicrostructure are reported in Table 6.1 together with the total number of grains sampled for eachspecimen. Note that the total area sampled is much larger than the size of the representative mapsshown in Figure 6.3. After 1 treatment cycle, a 2 fold increase in grain size is measured for theα-phase, while after 5 treatment cycles a grain size 3 times that of the as-received material wasobtained.Figure 6.3: Representative inverse ND pole figure maps (ND - IPF maps, A = 6.49 x 105 μm2)showing the microstructure and microtexture in the: a) as-received state, b) after 1 treatmentcycle, c) after 5 treatment cycles. In a), b), and c) the area of observation corresponds to theregion where the ultrasonic pulse was induced and measured. Note that these regions have beencropped from much larger maps, the overall size of the parent map being given in Table 6.1. Fullcomboscan maps provided in Appendix A.72In-situ Laser Ultrasonic Measurement of the HCP to BCC Transformation in Commercially PureTitaniumTable 6.1: Microstructural parameters of commercially pure titanium in the: as-received condition,after 1 treatment cycle, after 5 treatment cyclesSpecimen EQAD (μm) Grains Sampled Overall Map Size (106 μm2)Cycles of Treatment α β α βas-received 42 - 6371 - 8.671 cycle 70 242 846 60 3.245 cycles 145 289 316 67 5.23Grain boundary disorientations for specimens following 1 treatment cycle (Figure 6.3b) and 5treatment cycles (Figure 6.3c) were calculated from the grain orientations. Each boundary wasselected according to the classification proposed by Wang et al. [88] with a tolerance of ± 5°.The results of this analysis are shown in Figure 6.4a and Figure 6.4b where coloured boundariescorrespond to one of the 6 reduced axis-angle pairs (provided in Table 6.2) that describe α/αboundaries in the case where both crystals nucleated from the same β grain as distinct α variantsdefined by the Burgers orientation relationship. The black grain boundaries do not share a specialorientation relationship and therefore are taken as the boundaries between α grains nucleatedfrom two different β grains. Assuming that the α grains nucleate at β grain boundaries [95], theboundary between these two α grains can be considered as the approximate location of a prior βgrain boundary. The resulting grain boundaries of the parent β-phase are given in Figure 6.4cand Figure 6.4d for specimens after 1 and 5 treatment cycles respectively. This method allows forthe estimation of the mean EQAD of the β grains for these two specimens, which are presentedin Table 6.1. The β grain size is estimated to be very large by this technique. After 1 treatmentcycle, a 6 fold increase in grain size is measured for the β-phase when compared to the as-receivedEQAD. After 5 treatment cycles, a grain size 7 times that of the as-received material was obtained(cf. Table 6.1). This observation suggests extensive grain growth occurring above the β transus,which has been observed before for commercially pure titanium [11].73In-situ Laser Ultrasonic Measurement of the HCP to BCC Transformation in Commercially PureTitaniumTable 6.2: Reduced axis/angle pairs for each type of α/α boundaries that can result from the sameparent β grain [88]Type Axis Angle Color1 I 0°2 [1 1 2¯ 0] 60°3 [1¯8 13 31 8] 60.83°4 [1¯0 5 5 3¯] 63.26°5 [8 1¯9 11 0] 90°6 [0 0 0 1] 10.53°Figure 6.4: Grain boundary maps demonstrating: a) the α/α and b) prior β special boundary mapof a specimen after 1 treatment cycle, and c) the α/α and d) prior β special boundary map of aspecimen after 5 treatment cycles. Full comboscan maps provided in Appendix A.Figure 6.5 shows the pole figures extracted from the full set of EBSD measurements (cf. Ta-ble 6.1) for specimens in the as-received state, after 1 treatment cycle and after 5 treatment cycles.The initial as-received texture shown in Figure 6.5a has a near basal texture with a distinct splitaround the rolling direction, characteristic of a microstructure initially cold-rolled and annealed[11, 96]. The spotty textures observed after 1 cycle and after 5 cycles in Figure 6.5b and Fig-ure 6.5c are characteristic of the coarse α grain size given in Table 6.1.74In-situ Laser Ultrasonic Measurement of the HCP to BCC Transformation in Commercially PureTitaniumFigure 6.5: Pole figures showing texture in the: a) as-received state, b) after 1 treatment cycle, c)after 5 treatment cycles6.4 Evaluation of the ultrasonic velocityFigure 6.6 presents the ultrasonic velocity measured during the 1st, 2nd, and 5th thermal cycles.The velocity in the α-phase is observed to decrease linearly with temperature as expected basedon the change in density [25] and elastic constants [36, 38]. In all cases, the linear portion ofthe heating segment of the curves was observed up until a temperature of approximately 885 °C,which is close to the expected equilibrium β transus temperature for pure titanium (882 °C) [9].Similarly, a linear increase of velocity with decreasing temperature was observed in the α-phase fortemperatures below ∼ 850 °C. At high temperature, the β-phase exhibits a weakly temperaturedependent ultrasonic velocity, which is consistent with previous experiments [38].The velocity in the α-phase is measurably higher at 700 °C in the heating stage of the 1stcycle compared to the value observed at 700 °C after cooling. This difference suggests a strongevolution of the texture during the α → β → α transitions, which is consistent with the textureand grain size changes shown in Figure 6.4a and Figure 6.4b. Aside from as-received specimen, thevelocity measured at 700°C for all other conditions (on heating and on cooling) were found to be thesame within experimental error. This suggests that, while the texture change on the 1st treatmentcycle strongly affects the velocity measurement, subsequent texture changes on heating/coolingdo not. In contrast, the absolute value of the velocity of the β-phase measured, for example,at 920 °C was found to vary strongly from one experiment to another. Repetitions of the 1 cycleexperiment (Figure 6.6d) showed a wide range of final velocities of the β-phase. As will be discussed75In-situ Laser Ultrasonic Measurement of the HCP to BCC Transformation in Commercially PureTitaniumbelow, this specimen-to-specimen variation is indicative of statistical scatter arising from havingfew grains in the measurement volume. The increasing grain size observed upon repeated cyclingwas demonstrated in the ex-situ results, and as a consequence, the number of grains sampled inthe volume scanned by the ultrasonic is also reduced (cf. Table 6.1).a. b.c. d.Figure 6.6: Comparison of calculated velocity values to LUMet observations for heating and coolingof the a) 1st treatment cycle, b) 2nd treatment cycle, and c) 5th treatment cycle, along with d) acompilation of velocity profiles demonstrating the variation in β velocity during heating in the 1stcycleTo extract the phase transformation kinetics from the measurements shown in Figure 6.6, amodel is required that combines the temperature dependence of the density and elastic constants.The temperature variation of the density of the α-phase and β-phase was estimated based on theCALPHAD approach described in Lu et al. [25]. The temperature dependence of the five indepen-dent elastic constants of the α-phase for pure titanium were obtained by linearization of the data76In-situ Laser Ultrasonic Measurement of the HCP to BCC Transformation in Commercially PureTitaniumprovided by Ogi et al. [38], and Fisher and Renken [36]. To calculate the polycrystalline ultrasonicvelocity of the α-phase, the polycrystalline averaged stiffness tensor was calculated using EBSDmeasured texture data (Sections 2.5.3 and 2.5.5) [80]. The Cristoffel tensor was next constructedfrom the polycrystalline stiffness tensor, and the pressure wave modulus, λHill1,α , obtained as theappropriate eigenvalue [79]. The single crystal elastic constants for unalloyed β-phase in Ti havebeen reported by Fisher and Dever [36]. These experiments as well as those of Senkov et al. [1]point to a very low temperature dependence for the β-phase. Notably, the results also show a veryhigh elastic anisotropy with 2C44/ (C11 – C12) ≈ 4.8 [36]. As with the α-phase, one can calculatethe expected ultrasonic velocity knowing the elastic constants only if the crystallographic textureis also known. In the present work, the texture of the β-phase could not be directly evaluated.Lacking this data, the value of the pressure wave modulus for the β-phase was fixed based on ob-served experimental velocity at 1000 °C. The temperature dependence of λpoly1,β was then fit to theexperiments. Averaging the slope of the velocity-temperature plot for all runs within the fully βcondition led to a temperature dependence of -0.0001 mm.μs–1.°C–1. This was therefore used inthe model as the temperature dependence of the velocity in the β-phase. Figure 6.7 provides anoverview of the anisotropy dependent modeling of velocity.77In-situ Laser Ultrasonic Measurement of the HCP to BCC Transformation in Commercially PureTitaniumFigure 6.7: Overview of the model used to calculate the velocity in a bulk aggregate of titanium.*Δv/ΔT back-calculated from average slopes of experimentally observed velocities and off-set somodel velocity intersects experimental at 1000 °CTo calculate the ultrasonic velocity in the regime where two phases co-exist, one must makean assumption regarding the way in which the velocities of the individual phases contribute. Asexplored in Chapter 5, the predicted velocity is only weakly dependent on the assumed averagingscheme when the density and moduli are similar. In the present alloy, the percent differences inthe modulus and densities of the α-phase and β-phase are 6.8% and 0.18 %, respectively when78In-situ Laser Ultrasonic Measurement of the HCP to BCC Transformation in Commercially PureTitaniumcalculated at the β-transus temperature under a polycrystalline aggregate assumption. Given thesesmall differences and the sensitivity curves provided in Section 5.6 a simple linear addition law issufficient to predict the aggregate velocity:vmix =(1 – fβ)vα + fβvβ =(1 – fβ)(λHill1,αρα)1/2+ fβλpoly1,βρβ1/2 (6.1)Using the values of λHill1,α , λpoly1,β , ρα and ρβ as described above, the only unknown parameter isthe volume fraction of the β-phase, fβ. To simplify fitting the experimental data, the variation offβ with temperature was assumed to vary sigmoidally:fβ =1(1 + e(–A1∗(T–A2))) (6.2)Where A1 and A2 are adjustable parameters, and T is temperature in °C. The optimal values ofadjustable parameters for each cycle are shown in Table 6.3.Table 6.3: Best fit values of the adjustable parameters (A1 and A2) in equation 6.2 obtained bycomparing during fitting the heating and cooling portions of the experimental data in Figure 6.6for the 1st, 2nd, and 5th cyclesheating coolingCycle A1 A2 A1 A21st 0.219 908 0.260 8712nd 0.166 903 0.361 8685th 0.166 899 0.260 872Figure 6.6 compares the experimentally measured and numerically predicted ultrasonic velocities(via equation 6.1). Figure 6.8 shows that the transformation kinetics required to fit the experiments(via equation 6.2) are nearly the same for the all of the heating cycles and all of the cooling cycles,as one would expect. The model described above, accounting for the temperature dependence ofthe material properties, texture, and phase fraction, adequately describes the variation of ultrasonicvelocity during cooling in each of the cycles shown in Figure 6.6. In the case of the 5-cycle specimenduring the heating stage one sees a discrepancy between the temperature dependence of the velocity79In-situ Laser Ultrasonic Measurement of the HCP to BCC Transformation in Commercially PureTitaniumin the α-phase during heating. As noted above, for this specimen the texture of the α-phase wasmeasured only after the 5th treatment cycle, not before. This result would suggest that the texturebefore and after the 5th treatment cycle was not the same, leading to the good prediction of thevelocity in the α-phase on cooling but a poor fit on heating. If this was the case, then one mightexpect the same problem in the case of the specimen cycled twice where the texture was measuredprior to, but not after, the thermal cycle. Another factor must be considered in this analysis,however. Examining the data in Table 6.1, one sees that the α grain size increases drasticallybetween the 2nd and 5th treatment cycle. This suggests that the discrepancy in the model fit to theheating and cooling segments of the 5th treatment cycle comes from the sampling of too few grainsin the α-phase. This leads to incomplete statistical sampling of the texture and the unsuitability ofthe texture measured after the cycle as a replacement for the texture before the cycle. Conversely,the α-phase grain size in the case of the specimen after one treatment cycle is only approximatelyhalf of the grain size after 5 treatment cycles.A more drastic discrepancy between the model and experiments occurred during the α → βtransformation on heating during the 1st cycle. In this case the experiments reveal a surprisingvariation in the velocity, its value first dropping before once again increasing. This result wasreproduced on 3 specimens, though as noted above, the final high temperature velocity of the β-phase varied strongly from one specimen to the next (cf. Figure 6.6d). One interpretation of thisresult could be that the α → β transformation does not finish until the end of the strong increasein velocity following the minimum at 922 °C. This, however, would suggest drastically differenttransformation kinetics on the first heating compared to all of the other specimens studied. If,however, one considers the bottom of the minimum in velocity as the end of the transformation,then one finds very close agreement for the transformation kinetics with the other two conditionsduring heating. This observation suggests that another mechanism is more likely to cause the strongincrease in velocity in the β-phase following the end of the transformation. It is notable that thisincrease in velocity continues to the highest temperature monitored, this temperature dependencebeing abnormal compared to the other two specimens.One possible explanation for the rapid increase in velocity is that it is due to the very rapidgrain growth that has been reported in the β-phase just following the completion of the phasetransformation [11, 12]. The α grain size of the as-received material was shown in Table 6.1 to be80In-situ Laser Ultrasonic Measurement of the HCP to BCC Transformation in Commercially PureTitanium42 μm while the back-constructed β grain size after the 1st treatment cycle was estimated to be∼ 242 μm. As shown by the in-situ EBSD observations of Seward et al. [11], the formation ofthe β-phase occurs both within the grain and from the α/α grain boundaries simultaneously. Theywere able to observe directly and in-situ a drastic evolution in both the grain size and texture ofthe β-phase. Significantly, the texture at the end of the α → β transformation has been noted tobe strongly cube-like, this texture having a very low longitudinal wave velocity [11, 97, 98]. Usingthe single crystal elastic constants from Fisher and Dever [36] at 1000 °C a lower-bound velocityof 5.139 mm.μs–1 would be expected for propagation along a 〈100〉 direction. The fact that alltests performed showed very similar values of the minimum in velocity would be consistent withall specimens transforming to a similar cube texture on the 1st heating cycle. The scatter in thevelocities of the β-phase at high temperature would also be consistent with this interpretation asthis is what would be expected to arise from the large elastic anisotropy of the β-phase combinedwith statistical scatter in texture as the number of grains sampled by the ultrasonic wave decreaseswith further grain growth.Figure 6.8: α → β and β → α phase transformation kinetics obtained from Equation 3 using theparameters given in Table 6.3 for specimens during the 1st, 2nd and 5th cycle, respectively81In-situ Laser Ultrasonic Measurement of the HCP to BCC Transformation in Commercially PureTitanium6.5 Chapter summaryThe ultrasonic velocity has been shown to be sensitive to α → β and β → α transformation incommercially pure titanium, despite the similarity of density and elastic modulus of the two phases.Converting this ultrasonic velocity to a phase fraction is, however, complicated by the effect of astrong starting texture and extensive grain growth in the β-phase. Under such conditions it isshown that one needs to accurately incorporate the local texture into the analysis in order for oneto predict the experimentally measured ultrasonic velocity from known material properties. Fortests where the texture is well known, e.g. in cases where we were able to measure the texture oncooling, an excellent agreement between the predicted and measured velocity could be obtained. Incontrast, the variation of ultrasonic velocity in cases where the texture was not well known led toan inability to directly match the simulation results. Most notably, the non-monotonic variation ofthe velocity with temperature during the 1st treatment cycle was attributed to a velocity changecaused first by the phase transformation (decrease in velocity) followed by rapid grain growth in theβ-phase (increase in velocity). This change is qualitatively consistent with the recent observationsof the α→ β transformation by in-situ EBSD [11].82Chapter 7In-situ Laser Ultrasonic MeasurementDuring Aging of Ti-55537.1 IntroductionIn this Chapter, the laser ultrasonic technique is used to monitor the β→ α phase transformation inthe alloy Ti-5Al-5Mo-5V-3Cr. The measurements are conducted during isothermal aging scenariosat 700 °C after solution treating. In particular, the evolution of the longitudinal wave velocityis measured during the precipitation reaction that occurs during the isothermal hold below thetransus. Quantitative ex-situ metallography are correlated to the variation of the velocity observedduring the precipitation sequence. Finally, the absolute change in velocity is modelled to contrastand compare ex-situ obtained information and the in-situ measurement of ultrasonic velocity forthe prediction of the α precipitation kinetics in this material.7.2 Characterization of the as-received materialThe initial microstructure of the forged ingot was composed of large and deformed β grains, withglobular primary α regions. When viewing along the transverse direction, the globular primary αprecipitates are consistently round, with a mean equivalent area diameter (EQAD) of 2.58 μm, asshown in Figure 7.1. The as received condition also demonstrated a relatively large β grain sizerelative to the α precipitates. Figure 7.2 shows globular primary α precipitates dispersed throughout83In-situ Laser Ultrasonic Measurement During Aging of Ti-5553a large β grain.a. b.Figure 7.1: Secondary Electron Image (SEI) demonstrating the as-forged Ti-5553 microstructureat magnifications of a) 1000x b) 10 000xFigure 7.2: Electron backscatter diffraction (EBSD) inverse pole figure (IPF) map demonstratingglobular primary α precipitates dispersed in a large β grain7.3 Ex-situ metallographyFigures 7.3, 7.4 and 7.5 shows the microstructural evolution with time during an isothermal holdingtreatment at 700 °C. Figure 7.3 shows the control specimen, after being cooled at 6.7 °C.s–1 to 700°C and then quenched immediately. A fully β microstructure with an EQAD of 182 μm is observedafter the 15 minute solutionizing treatment. Figure 7.4 shows an EBSD phase map demonstratingprecipitation along β grain boundaries after 5 minutes of treatment. EBSD was employed in thisinstance because the phase amount was very limited (0.3 vol. % α), and the phase contrast wasdifficult to distinguish from the overwhelming orientation contrast between β grains. Conversely,84In-situ Laser Ultrasonic Measurement During Aging of Ti-5553Figure 7.5a through 7.5e were taken using backscatter electron (BSE) contrast. Figure 7.5a showsαW1 Widmansta¨tten plates initiating at the grain boundaries and growing in parallel packets intothe grain after 10 minutes where 2.0 α vol. % has formed. After 33 minutes 12.2 α vol. % hasformed and α plate packets (αW2) are seen nucleating throughout the β grain in addition to growingout from the β grain boundaries (Figure 7.5b). Figure 7.5c shows that after 53 minutes 16.3 % ofthe microstructure was α and demonstrated a mixture of morphologies. After 75 minutes, 31.5 %of the microstructure is composed of the α phase and the majority of the α phase is still presentnear the β grain boundaries (Figure 7.5d). Finally, after 180 minutes 39.4 % of the microstructureis homogeneously distributed α precipitates (Figure 7.5e). The fraction of α observed after eachtime along with the standard deviation (calculated from 5 BSE measurements per specimen) issummarized in Table 7.1.Table 7.1: Phase fraction of α (fα) precipitated during isothermal holding at 700 °CHolding Time fα Standard Deviation(min.) (%)5 0.3 -10 2.0 0.533 12.2 2.553 16.3 3.775 31.5 3.0180 39.4 2.9Figure 7.6 demonstrates the variety of morphologies that result during precipitation of the αphase in an aged specimen. Figure 7.6a presents a BSE image demonstrating grain boundary alpha(αGB), Widmansta¨tten side plates (αW1) growing out from the αGB/β interface, along with platepackets nucleating within the grain (αW2). Figure 7.5 and 7.6 demonstrate that the pre-existingβ grain boundaries are the dominant nucleation sites for the α phase. Similar morphologies havebeen observed during aging treatments of β-stabilized alloys in the literature [60] [58]. Figure 7.6bshows the inverse pole figure (IPF) map of the same area photographed in Figure 7.6a. The αGBis shown as a thin film resulting from the coalescence of α precipitates nucleating between the twoprior existing β grains. Figure 7.6b shows that the plates nucleating near a respective boundaryline of αGB will often share the same orientation as the αGB. However, other αW1 variants areobserved as plates extending from the boundaries.85In-situ Laser Ultrasonic Measurement During Aging of Ti-5553Figure 7.3: BSE images taken at 20 kV and at low magnification of a specimen representative ofthe starting condition at the onset of the 700 °C isotherm (solutionized for 15 minutes at 900 °C,then cooled to 700 °C at 6.7 °C.s–1 and quenched).Figure 7.4: EBSD phase map taken after a specimen was held at 700 °C for 5 minutes where blackand white indicate the α and β phases, respectively86In-situ Laser Ultrasonic Measurement During Aging of Ti-5553a. b.c. d.e.Figure 7.5: BSE images taken at 20 kV and a magnification of 500x of specimens held at theisothermal treatment for a) 10, b) 33, c) 53, d) 75 and e) 180 minutes.87In-situ Laser Ultrasonic Measurement During Aging of Ti-5553a. b.Figure 7.6: Triple point β boundary in a specimen aged for 75 minutes at 700 °C demonstratingfilms of αGB forming between the β grains and the growth of αW1 side-plates into the β grainvisualized as a a) backscatter electron image and b) inverse pole figure mapFigure 7.7 shows the plot of the precipitate phase fraction (fα) evolving over the time held attreatment. A sigmoidal shape is clearly evident with kinetics slowing at the three hour mark, neara volume fraction of 39.4 ± 2.9 % which is in good agreement with the equilibrium value of α,fα,eq = 43%, calculated via the TTTI3 database in Thermocalc at the holding temperature T =700 °C. A sigmoidal fit (the blue line in Figure 7.7) is shown as a guide to the eye.Figure 7.7: Measurements of isothermal α precipitation obtained via ex-situ metallography88In-situ Laser Ultrasonic Measurement During Aging of Ti-55537.4 Evaluation of ultrasonic velocityFigure 7.8a shows that the initial value of velocity for the fully β condition varies from 5.85 to 5.97mm.μs–1. This variation is related to various uncertainties in the absolute measurement of velocityand will be discussed further in this section. When the polycrystalline P-wave moduli computedunder isotropic assumption at 700 °C and densities are used (as described in Section 2.5.4), thecomputed velocities in the α and β phases are vα,poly = 5.48 mm.μs–1 and vβ,poly = 5.107 mm.μs–1which means a change of 0.373 is expected for the complete (100 %) β → α transformation. Thismeans that after 180 minutes, given the ex-situ observed α phase fraction of fα,t=180min = 39.4 %(cf. Table 7.1), the predicted velocity would increase by 0.147 mm.μs–1. This is comparable to theLUMet observed increase in velocity of 0.152 mm.μs–1.When the relative change in velocity (Δv = vt – vβ) is calculated from the laser ultrasounddata for each specimen, the curves collapse on one another as shown in Figure 7.8b. However, theabsolute values of the raw velocities measured in LUMet seem to be substantially higher than whatis predicted using the material properties under a polycrystalline assumption.a. b.Figure 7.8: Comparison of a) raw absolute velocities at various isothermal holding times, and b)in-situ ultrasonic observation depicting relative change in velocity (Δv = vt – vβ)Four possible sources of error were investigated as the cause of the observed offset of absolutevalue in velocity. These sources of potential errors included the effect of texture, composition,thickness anomalies, and asymmetry of the measured waveforms. The two material dependentproperties, texture and composition, can both alter the elastic response in a given material. If the89In-situ Laser Ultrasonic Measurement During Aging of Ti-5553polycrystalline aggregate assumption was inappropriate, and instead of using the polycrystallinevalues one were to compute the velocity by accounting for texture (following Section 2.5.3), onewould find the anisotropy of the hcp and bcc crystals are not sufficient to account for the differencesin the observed and predicted absolute values of velocity. For instance, when velocity is computedas a function of orientation, the hcp unit cell demonstrates a minimum or maximum velocity whena wavefront is propagating within the basal plane or parallel to the c-axis, respectively. At 700 °C,the upper and lower bounds for velocity in an anisotropic hcp titanium unit cell are:(φ1,Φ,φ2) = (0, 0, 0)→ vα = 5.946 mm.μs–1 (7.1)(φ1,Φ,φ2) =(0,pi2, 0)→ vα = 5.376 mm.μs–1 (7.2)Where φ1, Φ, and φ2 are the Euler angles given in Radians. Thus, even in a hypotheticalcondition where all the α crystals were aligned such that their c-axes were all parallel to the wavepropagation direction, a velocity of 5.946 mm.μs–1 would be possible, which is significantly lowerthan the velocities shown in Figure 7.8.For the bcc case, the upper and lower bounds in predicted velocities were given when theultrasonic pulse propagated through {110} 〈11¯1〉 and {101} 〈100〉, respectively. The single crystalelastic constants are only available in the literature at 1000 °C, so the velocities have been computedat 1000 °C. At 1000 °C, the upper and lower bounds for an anisotropic bcc titanium unit cell are:(φ1,Φ,φ2) =(0,pi2, 0)→ vβ = 5.55 mm.μs–1 (7.3)(φ1,Φ,φ2) =(0.61548,pi2,pi4)→ vβ = 4.76 mm.μs–1 (7.4)This prediction is limited by the fact that the single crystal elastic constants only exist forbcc titanium at 1000 °C, and that the experimental velocities were observed at 700 °C. However,considering the Bulk modulus observed by Ogi et al. [38] demonstrated a minute temperaturedependence above the transus, the 300 °C temperature difference between predictive computationand the experiment is unlikely to account for the drastic difference between the predicted and90In-situ Laser Ultrasonic Measurement During Aging of Ti-5553observed velocities (5.55 mm.μs–1 versus 5.83 - 5.97 mm.μs–1). Consequently, neither the anisotropyin the bcc nor hcp phase is large enough to account for the high velocities shown in Figure 7.8a.Upon examining Figure 7.8b, it can be observed that the relative change in velocity (Δv =vt –vβ) of each trial overlapped. The EQAD of the bcc grains at the onset of the treatment was 182μm. In Chapter 6 it was seen that the commercially pure titanium specimen that endured 5 cyclesexhibited an EQAD of only 142 μm (cf. Table 6.1). Based on that initial assessment, one wouldassume orientation effects would be a significant contributor to variation in velocity. However,Figure 7.9 demonstrates the reason why texture is such a significant contributor in Chapter 6and not here. Figure 7.9a shows the EBSD obtained microstructure of hcp commercially puretitanium after 5 cycles of heat treatment. Extremely coarse grains (diameter ≥ 500 μm) are mixedwith smaller grains (diameter ≤ 50 μm). Conversely, Figure 7.9b demonstrates a microstructurewith a fairly uniform distribution of grain size. The area weighted grain diameter (dA) of themicrostructures shown in Figure 7.9a and 7.9b are 538 μm and 289 μm, respectively, where the areaweighted grain diameter (dA) was calculated by:dA =1∑ni=0 Ain∑i=1Aidi (7.5)Where Ai is the area of a given grain, di is the diameter of a given grain, and n is the totalnumber of grains.The dA of the treated commercially pure titanium specimen is significantly larger than itscorresponding EQAD, as well as the dA and EQAD of the Ti-5553 specimen. This suggests, thatthe orientation of large grains, such as the big blue one shown in Figure 7.9a contribute significantlyto the velocity of the wave, and in cases where there are sufficient grains with a smaller area weightedfraction, such as in Figure 7.9b, velocity can be predicted using a polycrystalline assumption. Thisis self consistent with the fact that all the velocity curves collapse when the relative change invelocity (Δv = vt – vβ) is calculated, as shown in Figure 7.8b.91In-situ Laser Ultrasonic Measurement During Aging of Ti-5553a. b.Figure 7.9: A comparison of: a) Inverse ND pole figure map (ND - IPF map) showing the mi-crostructure and microtexture in hcp commercially pure titanium after 5 treatment cycles, and b)BSE image of a Ti-5553 specimen solutionized for 15 minutes at 900 °C, then cooled to 700 °C at6.7 °C.s–1 and quenched to depict the bcc microstructure at the on-set of the isothermal treatmentIf one returns to the work of Fisher and Dever [37], the effect of alloy composition on elasticresponse in the bcc phase can be explored. Their work suggests that a percent change in the P-wavemodulus (observed at 1000 °C) as high as 10.42 % can occur with Cr alloy additions of 14.62 wt.%. However, if one calculates the density of bcc Ti and bcc Cr at 1000 °C [25], and computesthe alloy density using a rule of mixtures, and finally computes the predicted velocity, the largestobserved percent change is 1.68 %. The effect of composition and elastic response in hcp crystalswas explored by the work of Senkov et al. [1] which looked at the addition of interstitial hydrogeninto titanium at 20 °C. Senkov et al. [1] found that an addition of 25 at. % H would increase theP-wave modulus by 4.64 %, which corresponded to a 2.29 % change in velocity. These two studies[1, 37] suggest composition does have a minor impact on velocity. However, how each elementaddition effects velocity can vary because it will affect both the aggregates elastic response anddensity.The material dependent properties, such as texture and composition, alone do not account forthe high observed velocities. Experimental error such as inconsistencies in specimen thickness andpoor waveform quality can also contribute to an offset in absolute velocity. For example, withthickness of 3 mm input, CTOME will compute a velocity of 5.832 mm.μs–1 for the initial fully βcondition (t = 0 min) at 700 °C in the specimen aged for 180 minutes. For the fully β specimento lie within the predicted velocity range of [4.76, 5.55] mm.μs–1 a maximum thickness of 2.85 mm(yields a velocity of 5.54 mm.μs–1) would be necessary. However, the specimens were machined with92In-situ Laser Ultrasonic Measurement During Aging of Ti-5553consistent quality to the pure Ti specimens (used in Chapter 6) which were on par with predictedvalues.The final source of error is inherent to the quality of the waveform itself. Figure 7.10a demon-strates the quality of the echos in the as-received state at room temperature (Reference (AR)), inthe fully β condition at 700 °C (Reference (BCC)) and at an arbitrary, representative point duringthe isothermal holding treatment (Current signal). To obtain the results shown in Figure 7.8a,all echos were compared to the as-received state (cf. Section 4.3.3). While the selected referencecontained symmetric echos with sharp peaks, its shape was inconsistent to the shape observed athigh temperatures, which had deteriorated substantially. The echos captured at 700 °C (eg. Ref-erence (BCC) and Current signal) were asymmetric, and attenuated considerably due to the largebcc grains. Thus, when the cross-correlation algorithm is employed in CTOME [89], a consistentmis-windowing can occur [99]. This can be explained by examining the cross-correlation functionsproduced when the current signal echo is compared to the as-received and fully β reference echos.The two resulting cross-correlation functions are shown in Figure 7.10b. Figure 7.10b shows thatwhen the asymmetric current echo is compared to the symmetric echo (Reference (AR)), the result-ing cross-correlation function is asymmetric. When the algorithm in CTOME selects the maximumamplitude (corresponding to the red dashed line shown on Figure 7.10b), it will underestimate thetime delay, which leads to a higher computed velocity. Conversely, when an equally asymmetricand attenuated waveform (Reference (BCC)) is selected, the resulting cross-correlation function issymmetric, as shown by the black function in Figure 7.10b, where the black dashed line identifiesthe maximum amplitude and corresponding time delay. To ensure this offset was indeed occurringin each dataset, each data set was compared to their own 30th echo (representative of the fully βcondition observed during that particular treatment), and velocity was computed. All specimensobserved a decrease in velocity of approximately 0.3 mm.μs–1, indicating that the error introduceddue to asymmetry of the waveform is both consistent and significant. Correction of this system-atic error is beyond the scope of this thesis. However, since the asymmetry effects at 700 °C areconsistent across the aging treatments for each specimen, the relative change in velocity is a validmeasure of the transformation since this offset is consistently applied to all points.93In-situ Laser Ultrasonic Measurement During Aging of Ti-5553a.b.Figure 7.10: Schematic showing a) the as-received (AR) 2nd echo reference, fully β (BCC) 2nd echoreference, and an arbitrary and representative high temperature 2nd echo (Current signal), and b)cross-correlation function obtained when the current signal is compared to the as-received reference(red) and to the fully β reference (black). The dashed lines indicate the maximum amplitude, andcorresponding time delay of each cross-correlation function.Thus, the relative change in velocity depicted in Figure 7.8b was used to determine the kineticsover the five separate treatments. Any change observed in velocity is due to the formation of αprecipitates throughout the specimen. The experimentally observed relative change in velocity (cf.Figure 7.8b) divided by the relative change predicted for the 100 % β→ α transformation (Δvβ→α= 0.373 mm.μs–1) yields the phase fraction of α precipitated during the isotherm:f =ΔvexperimentalΔvβ→α(7.6)Figure 7.11 compares the phase fraction obtained from normalizing the velocity data to thephase fraction measured via ex-situ metallography. Both the in-situ and ex-situ data sets overlap94In-situ Laser Ultrasonic Measurement During Aging of Ti-5553well demonstrating that the monitoring velocity changes in Ti-5553 is an effective way to monitorprecipitation and aging in this material. LUMet measured a systematically higher α phase fractionduring the first 50 minutes of treatment. This has been attributed to difficulty in quantifying ex-situ the α phase fraction at earlier times due to the difficulty resolving the thin αGB that nucleatedon β-β grain boundaries using both the BSE and EBSD techniques. Overall, there seems to beexcellent agreement between LUMet and metallography.Figure 7.11: Comparison fraction of α precipitation obtained via ex-situ metallography and nor-malized velocity data during isothermal treatment7.5 Chapter summaryThe α precipitation kinetics of Ti-5553 were monitored using in-situ laser ultrasound and comparedto ex-situ metallographic results. Three distinct morphologies of α phase were observed over thecourse of the isothermal holding treatment held at 700 °C. These morphologies consisted of: αGBfound precipitating on the grain boundaries, αW1 found growing as Widmansta¨tten plate packetsinto the grain after initiating along αGB - β interfaces, and finally αW2 which was observed toform last, and consisted of Widmansta¨tten plate packets arranged in crossed bundles nucleating95In-situ Laser Ultrasonic Measurement During Aging of Ti-5553throughout the grain body. Laser ultrasonics provided a fast, responsive and in-situ measure of theα precipitation kinetics in a material that demonstrated complex morphologies and microstructures.96Chapter 8Summary and Future Work8.1 SummaryThis thesis developed strategies for interpreting laser ultrasonic data and monitoring the α/β phasetransformations in cp-Ti and the metastable β alloy Ti-5553. This work validated the feasibilityof LUMet as a non-destructive tool for monitoring the α/β phase transformation kinetics in thesematerials.Three studies (Chapters 5, 6 and 7) were conducted to achieve this end. Firstly, a 2-D FEMsimulation of wave propagation through a 2-phase aggregate was developed to understand the effectsof precipitate spacial arrangement and phase fraction on the velocity signal. Chapter 5 found thatthe selection of a correct averaging scheme to predict ultrasound velocity in a 2-phase aggregatedepended on the geometric configuration and the relative size of the pulse’s wavelength comparedto the microstructural feature size. Furthermore, sensitivity analysis of these results showed thatof mixtures of phases with similar elastic properties and densities (such as in α and β titanium),the possible averaging schemes collapse onto each other and produce similar velocities, and thususing a rule of mixtures is sufficient.The second study presented in Chapter 6 subjected cp-Ti specimens to cycles of continuousheat treatments above the transus temperature. The ultrasonic velocity was shown to be sensitiveto both the α → β and β → α transformations in cp-Ti, despite the similar P-wave moduli anddensities of the two phases. Extraction of the transformation kinetics was shown to be complicatedby a strong starting texture in the as-received condition and extensive β grain growth above the97Summary and Future Worktransus. It was found that under these conditions, the local texture must be incorporated intothe analysis to predict the correct velocity. For tests where texture was well quantified, there wasa strong agreement between predicted and measured velocity. Conversely, if texture was not wellquantified, as was the case in heating of the as-received specimens, the model would be incapable ofpredicting the experimentally observed velocity. The non-monotonic presentation observed here wasattributed to a decrease in velocity caused by the phase transformation, followed by a rapid graingrowth in the β phase that caused an increase in velocity. This change is qualitatively consistent tothe recent in-situ EBSD observations of the α → β transformation from a similar starting texturein cp-Ti [11].Finally, the third study presented in Chapter 7 took Ti-5553 specimens, solutionized themto the fully β condition, and then held them for varying times at a 700 °C isotherm to monitorprecipitation kinetics with LUMet. The precipitation of α grains could be monitored by using therelative change in velocity and one ex-situ obtained phase fraction. This was necessary, becauseunlike the cp-Ti waveforms, the Ti-5553 waveforms were attenuated and asymmetric, and induceda consistent offset into the absolute value of velocity. Even still, using the relative change invelocity was able to provide comparable phase fraction data to the metallographic obtained phasefraction data. Thus, laser ultrasound is able to provide a fast, responsive and in-situ methodto measure α-precipitation kinetics in Ti-5553, even though it is a material which demonstratescomplex morphologies and microstructures.8.2 Future workTo date, while the temperature dependence of the polycrystalline elastic moduli is available [1, 38],the temperature dependence of the single crystal elastic constants in pure titanium has not beenmeasured. Using a procedure and texture based velocity model similar to one utilized and validatedin Chapter 6 it may be possible to design experiments and infer the temperature dependence ofthe bcc elastic constants. However, for this proposal to be successful, an additional model mustbe incorporated into the analysis. A model capable of determining the orientation of the hightemperature β phase from measured EBSD data for the low-temperature phase has been developedin the literature [100].98Summary and Future WorkThe first step would be to design a pre-treatment step that would maximize the bcc grain size,since creating a single crystal of bcc pure titanium is unheard of. This can be accomplished byeither extended holding treatments above the transus, or repeated cycling treatments, and thenverified by performing EBSD after the treatment, and calculating the location of prior-β grainboundaries or by using Glavicic’s approach [100] to approximate the bcc microstructure.The next step would be to perform a continuous heating cycle on a pre-treated specimen whereLUMet monitoring of velocity would occur. Then perform a large EBSD comboscan along thecross section of the wave propagation interaction volume located at the center-line of the specimen.After, use Glavicic’s approach [100] in combination with the bcc single crystal elastic constants at1000 °C [37] and the anisotropy dependent velocity model to approximate the high temperaturebcc structure by fitting it to match the observed velocity in the bcc phase at 1000 °C for that cycle.Originally, his model works by selecting possible variants that minimize the global misorientationof the map. By adding the goal seeking approach to predict the experimentally observed velocity,the variants of each grain in the multicrystal can be more accurately selected. The orientationsof the bcc crystals once determined are now assumed to not change during the high temperaturesegment of the cycle, as this was shown in Chapter 6.Now the problem is defined in such a way that the temperature is changing as an input variable,the texture of the bcc phase is a constant input, the predicted velocity is a modeled output whichis compared to the experimentally observed velocity, and the bcc single crystal elastic constants(c11, c12 and c44) are unknown inputs. Because there are three unknowns, three equations must beanalyzed, and thus, three separate cycle treatments on separate specimens with distinct multicrystaltextures (input) and observed velocities (output) must be performed in order to goal seek for thetemperature dependence of c11, c12 and c44. This approach should then be repeated to validate inthe temperature dependence of c11, c12 and c44. This approach is not perfect, and of course it isalways preferable to perform a directional study on a single crystal, but considering the limitationsof the high temperature bcc phase, it is a good attempt at determining the single crystal elasticconstants.Many possible avenues regarding further study of alloy Ti-5553 exist due to it being relativelynew to western markets and its current demand in landing gear applications. This study focused onthe detection and monitoring of the β → α transformation during an isothermal aging treatment.99Summary and Future WorkHowever, much interest surrounds the detection of the ω phase. Exploring the effect of ω phaseprecipitation on velocity and attenuation to see if the formation of the small nano-sized precipitateswould be resolvable by LUMet would be valuable. Also, now that LUMet has been validated asan effective tool to monitor the β → α phase transformation in this alloy, performing repeatedisothermal and continuous cooling treatments to create a detailed time temperature transformation(TTT) diagram and continuous cooling transformation diagram (CCT) would be a valued additionto the literature.100Bibliography[1] O. N. Senkov, M. Dubois, and J. J. Jonas. Elastic moduli of titanium-hydrogen alloys in the temper-ature range 20C to 1100C. Metallurgical and Materials Transactions A, 27(12):3963–3970, December1996.[2] S. Zamiri, B. Reitinger, H. Grun, J. Roither, S. Bauer, and P. Burgholzer. Laser ultrasonic velocitymeasurement for phase transformation investigation in titanium alloy. pages 683–686. IEEE, July 2013.[3] M. Donachie. Titanium: a technical guide. ASM International, Materials Park, Ohio, 2000.[4] G. Lutjering and J.C. Williams. Titanium. Materials Engineering And Processes. Springer, 2 edition,2007.[5] O.M. Ivasishin, P.E. Markovsky, Yu.V. Matviychuk, S.L. Semiatin, C.H. Ward, and S. Fox. A com-parative study of the mechanical properties of high-strength beta-titanium alloys. Journal of Alloysand Compounds, 457(1-2):296–309, June 2008.[6] V. V. Shevel’kov and L. F. Kratovich. Design strength of VT22 titanium alloy in different structuralconditions. Soviet Materials Science, 26(6):687–691, 1991.[7] A.F. Gerday, M. Ben Bettaieb, L. Ducheˆne, N. Cle´ment, H. Diarra, and A.M. Habraken. Interestsand limitations of nanoindentation for bulk multiphase material identification: Application to the betaphase of Ti-5553. Acta Materialia, 57(17):5186–5195, October 2009.[8] J.C. Fanning. Properties of TIMETAL 555 (Ti-5al-5mo-5v-3cr-0.6fe). Journal of Materials Engineeringand Performance, 14(6):788–791, December 2005.[9] C Leyens, M Peters, John Wiley & Sons, and Wiley InterScience (Online service). Titanium andtitanium alloys fundamentals and applications. Wiley-VCH ; John Wiley] [distributor], Weinheim;Chichester, 2003.[10] R. Panza-Giosa. The effect of heat treatment on the microstructural evolution and mechanical propertiesof Ti-5Al-5V-5Mo-3Cr, and its potential application in landing gears. PhD thesis, McMaster University,2009.[11] G.G.E. Seward, S. Celotto, D.J. Prior, J. Wheeler, and R.C. Pond. In situ SEM-EBSD observations ofthe hcp to bcc phase transformation in commercially pure titanium. Acta Materialia, 52(4):821–832,February 2004.[12] F.X. Gil, D. Rodrguez, and J.A. Planell. Grain growth kinetics of pure titanium. Scripta Metallurgicaet Materialia, 33(8):1361–1366, October 1995.[13] T. Ahmed and H.J. Rack. Phase transformations during cooling in alpha + beta titanium alloys.Materials Science and Engineering: A, 243(1-2):206–211, March 1998.101BIBLIOGRAPHY[14] M. S. Oh, Jai-Y. Lee, and J. K. Park. Continuous cooling beta to alpha transformation behaviors ofextra-pure and commercially pure Ti. Metallurgical and Materials Transactions A, 35(10):3071–3077,October 2004.[15] S. K. Kim and J. K. Park. In-situ measurement of continuous cooling beta to alpha transformationbehavior of CP-Ti. Metallurgical and Materials Transactions A, 33(4):1051–1056, April 2002.[16] G. Ltjering. Influence of processing on microstructure and mechanical properties of (alpha + beta)titanium alloys. Materials Science and Engineering: A, 243(1-2):32–45, March 1998.[17] W. Szkliniarz and G. Smo lka. Analysis of volume effects of phase transformation in titanium alloys.Journal of Materials Processing Technology, 53(1-2):413–422, August 1995.[18] J. Teixeira, B. Appolaire, E. Aeby-Gautier, S. Denis, G. Cailletaud, and N. Spa¨th. Transformationkinetics and microstructures of Ti17 titanium alloy during continuous cooling. Materials Science andEngineering: A, 448(1-2):135–145, March 2007.[19] S Malinov, W Sha, Z Guo, C.C Tang, and A.E Long. Synchrotron X-ray diffraction study of the phasetransformations in titanium alloys. Materials Characterization, 48(4):279–295, June 2002.[20] J.W. Elmer, T.A. Palmer, S.S. Babu, and E.D. Specht. In situ observations of lattice expansion andtransformation rates of alpha and beta phases in Ti6al4v. Materials Science and Engineering: A,391(1-2):104–113, January 2005.[21] D. Ma, A. D. Stoica, K. An, L. Yang, H. Bei, R. A. Mills, H. Skorpenske, and X.-L. Wang. Tex-ture Evolution and Phase Transformation in Titanium Investigated by In-Situ Neutron Diffraction.Metallurgical and Materials Transactions A, 42(6):1444–1448, June 2011.[22] E. S. Makarov and L. M. Kuznetsov. The crystal structure and chemical character of lower oxides oftitanium TiO00.48. Journal of Structural Chemistry, 1(2):156–162, July 1960.[23] Gordon A. Sargent and Hans Conrad. On the strengthening of titanium by oxygen. Scripta Metallur-gica, 6(11):1099–1101, November 1972.[24] R. Montanari, G. Costanza, M.E. Tata, and C. Testani. Lattice expansion of Ti6al4v by nitrogen andoxygen absorption. Materials Characterization, 59(3):334–337, March 2008.[25] X.-G. Lu, M. Selleby, and B. Sundman. Assessments of molar volume and thermal expansion forselected bcc, fcc and hcp metallic elements. Calphad, 29(1):68–89, March 2005.[26] E. K. Molchanova. Phase diagrams of titanium alloys. Israel Program for Scientific Translations (1965),1965.[27] A. S. M. International. Al (Aluminium) Binary Alloy Phase Diagrams. In Alloy Phase Diagrams,volume 3 of ASM Handbook, page 2.42.56. 1992.[28] A. S. M. International. Mo (Molybdenum) Binary Alloy Phase Diagrams. In Alloy Phase Diagrams,volume 3 of ASM Handbook, page 2.2922.298. 1992.[29] A. S. M. International. Ti (Titanium) Binary Alloy Phase Diagrams. In Alloy Phase Diagrams,volume 3 of ASM Handbook, page 2.3783.380. 1992.[30] A. Kamp. Investigation of titanium alpha plates by EBSD analysis. PhD thesis, Delft University ofTechnology, Delft, Netherlands, 2007.[31] H.W. Rosenberg. Titanium alloying in theory and practice. The Science, Technology and Applicationof Titanium, pages 851–859, 1970.102BIBLIOGRAPHY[32] G. Welsch, R. Boyer, and E.W. Collings. Materials Properties Handbook: Titanium alloys. ASMinternational, 1993.[33] C. Lin, G. Yin, Y. Zhao, P. Ge, and Z. Liu. Analysis of the effect of alloy elements on martensitictransformation in titanium alloy with the use of valence electron structure parameters. MaterialsChemistry and Physics, 125(3):411–417, February 2011.[34] N.V. Ageev and L.A. Petrova. The theoretical bases of the development of the hight-strengthmetastable beta alloys of titanium. The Science, Technology and Application of Titanium, pages809–814, 1970.[35] J. Coakley, V. A. Vorontsov, N. G. Jones, A. Radecka, P. A. J. Bagot, K. C. Littrell, R. K. Heenan,F. Hu, A. P. Magyar, D. C. Bell, and D. Dye. Precipitation processes in the Beta-Titanium alloyTi5al5mo5v3cr. Journal of Alloys and Compounds, 646:946–953, October 2015.[36] E.S. Fisher and C.J. Renken. Single-Crystal Elastic Moduli and the hcp to bcc Transformation in Ti,Zr, and Hf. Physical Review, 135(2A), 1964.[37] E.S. Fisher and D. Dever. The single crystal elastic moduli of beta-titanium and titanium-chromiumalloys. The Science, Technology and Application of Titanium, pages 373–381, 1968.[38] H. Ogi, S. Kai, H. Ledbetter, R. Tarumi, M. Hirao, and K. Takashima. Titaniums high-temperatureelastic constants through the hcpbcc phase transformation. Acta Materialia, 52(7):2075–2080, April2004.[39] W. Petry, A. Heiming, J. Trampenau, M. Alba, C. Herzig, H. R. Schober, and G. Vogl. Phonondispersion of the bcc phase of group-IV metals. I. bcc titanium. Physical Review B, 43(13):10933–10947, May 1991.[40] S. Nishitani, H. Kawabe, and M. Aoki. First-principles calculations on bcchcp transition of titanium.Materials Science and Engineering: A, 312(1-2):77–83, August 2001.[41] H. Ledbetter. Elastic constants of body-centered-cubic titanium monocrystals. Journal of AppliedPhysics, 95(9):4642, 2004.[42] M. Sanati, A. Saxena, T. Lookman, and R. C. Albers. Landau free energy for a bcc-hcp reconstructivephase transformation. Physical Review B, 63(22), May 2001.[43] R. Ahuja, J. M. Wills, B. Johansson, and O. Eriksson. Crystal structures of Ti, Zr, and Hf undercompression: Theory. Physical Review B, 48(22):16269–16279, December 1993.[44] W. Hosford. The mechanics of crystals and textured polycrystals, volume 32 of Oxford engineeringsciences series. Oxford University Press, New York, 1993.[45] C. G. Rhodes and J. C. Williams. The precipitation of alpha-phase in metastable beta-phase Ti alloys.Metallurgical Transactions A, 6(11):2103–2114, November 1975.[46] E. S. K. Menon and H.I. Aaronson. Interfacial structure of widmanstatten plates in a TiCr alloy. ActaMetallurgica, 34(10):1975–1981, October 1986.[47] T. Furuhara and H.I. Aaronson. Crystallography and interfacial structure of proeutectoid alpha grainboundary allotriomorphs in a hypoeutectoid TiCr alloy. Acta Metallurgica et Materialia, 39(11):2887–2899, November 1991.[48] W.G. Burgers. On the process of transition of the cubic-body-centered modification into the hexagonal-close-packed modification of zirconium. Physica, 1(7-12):561–586, May 1934.[49] C. Zener. Contributions to the Theory of Beta-Phase Alloys. Physical Review, 71(12):846–851, June1947.103BIBLIOGRAPHY[50] N. N. Sirota and T. E. Zhabko. X-ray study of the anisotropy of thermal properties in titanium.Physica Status Solidi (a), 63(2):K211–K215, February 1981.[51] J. Spreadborough and J. W. Christian. The Measurement of the Lattice Expansions and DebyeTemperatures of Titanium and Silver by X-ray Methods. Proceedings of the Physical Society, 74(5):609–615, November 1959.[52] H. Kleinschmidt, A. Ziegler, G. H. Campbell, J. D. Colvin, and O. Bostanjoglo. Phase transformationanalysis in titanium at nanosecond time resolution. Journal of Applied Physics, 98(5):054313, 2005.[53] M. Grujicic and P. Dang. Atomic-scale analysis of martensitic transformation in titanium alloyed withvanadium Part I: verification of the embedded-atom method model. Materials Science and Engineering:A, 205(1-2):139–152, January 1996.[54] S. Banerjee and P. Mukhopadhyay. Phase transformations: examples from titanium and zirconiumalloys. Elsevier, 1 edition, 2007.[55] S. Nag, R. Banerjee, J.Y. Hwang, M. Harper, and H.L. Fraser. Elemental partitioning between alphaand beta phases in the Ti5al5mo5v3cr0.5fe (Ti-5553) alloy. Philosophical Magazine, 89(6):535–552,February 2009.[56] A. Settefrati, E. Aeby-Gautier, B. Appolaire, M. Dehmas, G. Geandier, and G. Khelifati. Low Tem-perature Transformations in beta-Metastable Ti 5553 Titanium Alloy. Materials Science Forum, 738-739:97–102, January 2013.[57] S. K. Kar, A. Ghosh, N. Fulzele, and A. Bhattacharjee. Quantitative microstructural characterizationof a near beta Ti alloy, Ti-5553 under different processing conditions. Materials Characterization,81:37–48, July 2013.[58] A. Dehghan-Manshadi and R. J. Dippenaar. Development of alpha-phase morphologies during lowtemperature isothermal heat treatment of a Ti5al5mo5v3cr alloy. Materials Science and Engineering:A, 528(3):1833–1839, January 2011.[59] S. Malinov, W. Sha, and P. Markovsky. Experimental study and computer modelling of the beta toalpha + beta phase transformation in beta 21s alloy at isothermal conditions. Journal of Alloys andCompounds, 348(1-2):110–118, January 2003.[60] T. Furuhara, S. Takagi, H. Watanabe, and T. Maki. Crystallography of grain boundary alpha precip-itates in a beta titanium alloy. Metallurgical and Materials Transactions A, 27(6):1635–1646, 1996.[61] S. Malinov, P. Markovsky, and W. Sha. Resistivity study and computer modelling of the isothermaltransformation kinetics of Ti8al1mo1v alloy. Journal of Alloys and Compounds, 333(1-2):122–132,February 2002.[62] J.-P. Monchalin. Laser-Ultrasonics: From the Laboratory to Industry. volume 700, pages 3–31. AIP,2004.[63] M. Falkenstro¨m, M. Engman, E. Lindh-Ulmgren, and B. Hutchinson. Laser ultrasonics for processcontrol in the metal industry. Nondestructive Testing and Evaluation, 26(3-4):237–252, September2011.[64] E. Fraizier, M.-H. Nadal, and R. Oltra. Noncontact determination of the elastic moduli of beta-Sn upand through the melting point. Journal of Applied Physics, 93(1):649, 2003.[65] A. Moreau, C. Bescond, S. Bolognini, M. Lord, S.E. Kruger, and C. S. Man. Inline Measurements ofTexture and Recrystallization on Aluminum Alloys. pages 3–10. Automation & Control/ AdvancedOnLine Metallurgical Models/Sensors Materials, Materials Science & Technology, 2005.104BIBLIOGRAPHY[66] S. Sarkar, A. Moreau, M. Militzer, and W.J. Poole. Evolution of Austenite Recrystallization andGrain Growth Using Laser Ultrasonics. Metallurgical and Materials Transactions A, 39(4):897–907,April 2008.[67] A. Moreau, D. Le´vesque, M. Lord, M. Dubois, J.-P. Monchalin, C. Padioleau, and J.F. Bussie`re.On-line measurement of texture, thickness and plastic strain ratio using laser-ultrasound resonancespectroscopy. Ultrasonics, 40(10):1047–1056, December 2002.[68] M. Militzer, T. Garcin, and W. J. Poole. In-situ Measurements of Grain Growth and Recrystallizationby Laser Ultrasonics. Materials Science Forum, 753:25–30, March 2013.[69] Silvio E. Kruger and Edward B. Damm. Monitoring austenite decomposition by ultrasonic velocity.Materials Science and Engineering: A, 425(1-2):238–243, June 2006.[70] M. Dubois, A. Moreau, and J. F. Bussie`re. Ultrasonic velocity measurements during phase transfor-mations in steels using laser ultrasonics. Journal of Applied Physics, 89(11):6487, 2001.[71] M. Maalekian, R. Radis, M. Militzer, A. Moreau, and W.J. Poole. In situ measurement and modellingof austenite grain growth in a Ti/Nb microalloyed steel. Acta Materialia, 60(3):1015–1026, February2012.[72] M. Dubois. A new technique for the quantitative real-time monitoring of austenite grain growth insteel. Scripta Materialia, 42(9):867–874, April 2000.[73] D. A. Hutchins. Ultrasonic generation by pulsed lasers. In Physical Acoustics, volume 18, pages 21–123.Academic Press W.P. Mason and R.N. Thurston, eds., 1988.[74] C. M. Scala. Time- and frequency-domain characteristics of laser-generated ultrasonic surface waves.The Journal of the Acoustical Society of America, 85(4):1569, 1989.[75] R. K. Ing and J.-P. Monchalin. Broadband optical detection of ultrasound by two-wave mixing in aphotorefractive crystal. Applied Physics Letters, 59(25):3233, 1991.[76] J.-P. Monchalin, R. He´on, P. Bouchard, and C. Padioleau. Broadband optical detection of ultrasoundby optical sideband stripping with a confocal FabryPerot. Applied Physics Letters, 55(16):1612, 1989.[77] G. Hamel de Monchenault and Jp. Huignard. Two-wave mixing with time-modulated signal inBi12sio20 theory and application to homodyne wave-front detection. Journal of Applied Physics,63(3):624, 1988.[78] A. Moreau, P.J. Kielczynski, J.F. Bussie`re, and J. H. Root. Ultrasonic Characterization ofTexture inPure and Alloyed Zirconium. pages 119–128. Nondestructive Characterization of Materials VI, PlenumPress, New York, 1994.[79] B.A. Auld. Acoustic fields and waves in solids, volume 1. John Wiley & Sons, Inc., 1973.[80] D. Mainprice, R. Hielscher, and H. Schaeben. Calculating anisotropic physical properties from tex-ture data using the MTEX open-source package. Geological Society, London, Special Publications,360(1):175–192, January 2011.[81] M. Slawinski. Waves and rays in elastic continua. Creative Commons, 2 edition, 2007.[82] D. Tromans. Elastic anisotrioy of hcp metal crystals and polycrystals. IJRRAS, 6(4):462–483, March2011.[83] J. Bradford and R. Latimer. Advances in Near-surface Seismology and Ground-penetrating Radar,volume 15 of Geophysical Development Series. American Geophysical Union, 2010.105BIBLIOGRAPHY[84] W. Voight. Ueber die Beziechung zwischen den beiden Elasticitatsconstanten isotroper Korper. Ann.d. phys. u. chem., 38:573–587, 1889.[85] A. Reuss. Berechung der Fliebenze von Mischkristallen auf Grund der Plastizitatsbedingung furEinkristalle. Zeit. Angew. Math. Mech., 9:49–58, 1929.[86] M. Hirao. Texture of polycrystalline metals characterized by ultrasonic velocity measurements. TheJournal of the Acoustical Society of America, 81(5):1434, 1987.[87] A.D. Rollett. Orientation Distribution: Definition, discrete forms, examples, September 2011.[88] S.C. Wang, M. Aindow, and M.J. Starink. Effect of self-accommodation on alpha/alpha boundarypopulations in pure titanium. Acta Materialia, 51(9):2485–2503, May 2003.[89] T. Garcin. Computational Tools for Metallurgy (CTOME), 2015. http://ctome.org/.[90] T. Williams, Kelley C., and D. Crawford. Gnuplot 4.6, an interactive plotting program, 2014.http://www.gnuplot.info/docs4.6/gnuplot.pdf.[91] Q. Puydt, T. Garcin, M. Militzer, and W. J. Poole. Us wave propagation in an elastic medium: 2 dfinite element modelling. international workshop on laser ultrasonics for metallurgy, November 2013.[92] T. Garcin, Q. Puydt, W. J. Poole, and M. Militzer. Simulation of ultrasound propagation in anisotropicpolycrystalline media, November 2013.[93] B. Tie, D. Solas, J. The´bault, C. Rey, T. Baudin, and A.-S. Mouronval. Mode´lisation nume´rique de lapropagation des ultrasons dans des milieux polycrystallins. Lyon France, 2010.[94] T.W. Murray and J.W. Wagner. Laser generation of acoustic waves in the ablative regime. Journalof Applied Physics, 85(4):2031–2040, February 1999.[95] C. Jourdan, J. Gastaldi, and G. Grange. Heterogeneous nucleation during the titanium alpha tobeta transformation studied by Synchrotron X-ray topography. Acta Metallurgica, 36(11):2979–2987,November 1988.[96] A.O.F. Hayama and H.R.Z. Sandim. Annealing behavior of coarse-grained titanium deformed by coldrolling. Materials Science and Engineering: A, 418(1-2):182–192, February 2006.[97] Z.S. Zhu, J.L. Gu, R.Y. Liu, N.P. Chen, and M.G. Yan. Variant selection and its effect on phase trans-formation textures in cold rolled titanium sheet. Materials Science and Engineering: A, 280(1):199–203, March 2000.[98] N. Gey and M. Humbert. Characterization of the variant selection occurring during the alpha tobeta to alpha phase transformations of a cold rolled titanium sheet. Acta Materialia, 50(2):277–287,January 2002.[99] D. R. Hull, H. E. Kautz, and Vary A. Ultrasonic velocity measurement using phase-slope and crosscorrelation methods. NASA technical memorandum 83794. Spring Conference of the American Societyfor Nondestructive Testing, 1984.[100] M.G. Glavicic, P.A. Kobryn, T.R. Bieler, and S.L. Semiatin. A method to determine the orientationof the high-temperature beta phase from measured EBSD data for the low-temperature alpha phasein Ti-6al-4v. Materials Science and Engineering: A, 346(1-2):50–59, April 2003.106Appendix AEBSD IPF comboscan mapsa.b.c.Figure A.1: Full EBSD obtained comboscan a) inverse ND pole figure maps (ND - IPF maps)showing the microstructure and microtexture, b) the α/α special boundary map, and c) prior βspecial boundary map of a specimen after 1 treatment cycle107EBSDIPFcomboscanmapsa. b. c.Figure A.2: Full EBSD obtained comboscan a) inverse ND pole figure maps (ND - IPF maps) showing the microstructure and microtex-ture, b) the α/α special boundary map, and c) prior β special boundary map of a specimen after 5 treatment cycles.108Appendix BPlotting 3-D Surface Young modulusas a Function of OrientationB.1 Young modulus as a function of orientation in the titaniumhcp unit cellThe following is the script created in the PyDev module of Eclipse SDK used to plot the elastic responsesurface for an titanium hcp unit cell. The single crystal elastic constants lines were linear regressions of datain the literature [36], and the mathematical approach was taken from Hosford’s textbook [44].’ ’ ’Created on 2015–04–07@author : Alyssa’ ’ ’import reimport mathimport numpy as npimport matp lo t l i b . pyplot as p l tfrom matp lo t l i b import *from s c ipy . cons tant s . cons tant s import alphaimport i t e r t o o l sfrom m p l t o o l k i t s . mplot3d import axes3d , Axes3Dfrom matp lo t l i b . font manager import FontPropert i e sgammagamma = np . arange ( 1 . 0 , 181 . 0 , 1 )T = 882 # put T in C e l s i u s herec11 = ( –0.0513*T + 163 .34) * math .pow( 1 0 . 0 , 9 . 0 )c12 = (0 .0118*T + 91 .313) * math .pow( 1 0 . 0 , 9 . 0 )c13 = (0 .0006*T + 68 .736) * math .pow( 1 0 . 0 , 9 . 0 )c33 = ( –0.0363*T + 181 .29) * math .pow( 1 0 . 0 , 9 . 0 )c44 = ( –0.0199*T + 46 .856) * math .pow( 1 0 . 0 , 9 . 0 )c66 = 0 .5 * ( c11 – c12 )A = c44 / c66print Ac hcp = np . array ( [ [ c11 , c12 , c13 , 0 , 0 , 0 ] ,109Plotting 3-D Surface Young modulus as a Function of Orientation[ c12 , c11 , c13 , 0 , 0 , 0 ] , [ c13 , c13 , c33 , 0 , 0 , 0 ] ,[ 0 , 0 , 0 , c44 , 0 , 0 ] , [ 0 , 0 , 0 , 0 , c44 , 0 ] ,[ 0 , 0 , 0 , 0 , 0 , c66 ] ] )s hcp = np . l i n a l g . inv ( c hcp )t e ra = 1000 .0s11 = s hcp [ 0 , 0 ] * math .pow(10 , 9)s12 = s hcp [ 0 , 1 ] * math .pow(10 , 9)s13 = s hcp [ 0 , 2 ] * math .pow(10 , 9)s33 = s hcp [ 2 , 2 ] * math .pow(10 , 9)s44 = s hcp [ 3 , 3 ] * math .pow(10 , 9)s66 = s hcp [ 5 , 5 ] * math .pow(10 , 9)print s11print s12print s44alpha = [ ]beta = [ ]gamma = [ ]gammagamma = np . arange ( 1 . 0 , 181 . 0 , 1 )x = [ ]y = [ ]z = [ ]r = [ ]zMax = 0xMax = 0for i in range (0 , gammagamma. l e n ( ) ) :g = math . rad ians (gammagamma[ i ] )phi = g ;g = math . cos ( g )Edinv = math .pow( ( 1 . 0 – math .pow( g , 2 ) ) , 2)* s11+ math .pow( g , 4)* s33+ math .pow( g , 2 )* ( 1 . 0 – math .pow( g , 2 ) )* ( 2 . 0* s13 + s44 )i f Edinv == 0 . 0 :Ed = 0 .0else :Ed = 1.0/ Edinv #r a d i u sfor otherAngle in range (1 , 36 1 ) :ang le = math . rad ians ( otherAngle )xCart = Ed*math . cos ( ang le )*math . s i n ( phi ) #xyCart = Ed*math . s i n ( ang le )*math . s i n ( phi )#yzCart = Ed*math . cos ( phi )#zr . append (Ed)x . append ( xCart )y . append ( yCart )110Plotting 3-D Surface Young modulus as a Function of Orientationz . append ( zCart )i f xCart > xMax :xMax = xCarti f zCart > zMax :zMax = zCartprint ’xMax : ’+str (xMax)print ’ zMax : ’+str (zMax)print ’ p l o t t i n g ’f ont = { ’ f ami ly ’ : ’ Times New Roman ’}f i g = p l t . f i g u r e ( f i g s i z e =(10 , 10) )ax = f i g . gca ( p r o j e c t i o n=’ 3d ’ )ax . s e t a u t o s c a l e o n ( Fa l se )ax . a x i s ( [ – 200 , 200 , –200 , 200 ] , f o n t d i c t = font )ax . s e t z l i m ( –200 , 200)ax . xax i s . s e t t i c k s (np . arange ( –200 , 300 , 100))ax . yax i s . s e t t i c k s (np . arange ( –200 , 300 , 100))ax . z a x i s . s e t t i c k s (np . arange ( –200 , 300 , 100))fontprop =font manager . FontPropert i e s ( fami ly=’ Times New Roman ’ , s i z e =28)for t i c k in ax . xax i s . g e t m a j o r t i c k s ( ) :t i c k . l a b e l . s e t f o n t p r o p e r t i e s ( fontprop )for t i c k in ax . yax i s . g e t m a j o r t i c k s ( ) :t i c k . l a b e l . s e t f o n t p r o p e r t i e s ( fontprop )for t i c k in ax . z a x i s . g e t m a j o r t i c k s ( ) :t i c k . l a b e l . s e t f o n t p r o p e r t i e s ( fontprop )ax . s c a t t e r (x , y , z , c=r , cmap=cm. get cmap ( ’ gray ’ ) )ax . s e t x l a b e l ( ’ x ’ , s i z e =28, f o n t d i c t = font )ax . s e t y l a b e l ( ’ y ’ , s i z e =28, f o n t d i c t = font )ax . s e t z l a b e l ( ’ z ’ , s i z e =28, f o n t d i c t = font )p l t . show ( )111Plotting 3-D Surface Young modulus as a Function of OrientationB.2 Young modulus as a function of orientation in the titaniumbcc unit cellThe following is the script created in the PyDev module of Eclipse SDK used to plot the elastic responsesurface for an titanium bcc unit cell at 1000 °C. The single crystal elastic constants were taken from theliterature [37], and the mathematical approach was taken from Hosford’s textbook [44].’ ’ ’Created on 2015–04–07@author : Alyssa’ ’ ’import reimport mathimport numpy as npimport matp lo t l i b . pyplot as p l tfrom matp lo t l i b import *from s c ipy . cons tant s . cons tant s import alphaimport i t e r t o o l sfrom m p l t o o l k i t s . mplot3d import axes3d , Axes3Dfrom matp lo t l i b . font manager import FontPropert i e sfrom matp lo t l i b . pyplot import drawfrom matp lo t l i b . delaunay . t r i a n g u l a t e import Tr iangu la t i onfrom matp lo t l i b . patches import Shadow’ ’ ’Data f o r c11 , c12 , c44taken from Fisher and Dever ,g i ven in GPa at T = 1000C’ ’ ’c11 = 99c12 = 85c44 = 33 .6s11 = ( c11 + c12 )/ ( math .pow( c11 , 2)+ c11* c12 – 2*math .pow( c12 , 2 ) )s12 = ( – c12 )/ ( math .pow( c11 , 2)+ c11* c12 – 2*math .pow( c12 , 2 ) )s44 = 1/ c44E100= 1/ s11E111 = 3 .0 / ( s11 +2.0* s12 +s44 )print E100print E111alpha = [ ]beta = [ ]gamma = [ ]r = [ ]x = [ ]112Plotting 3-D Surface Young modulus as a Function of Orientationy = [ ]z = [ ]h = [ ]zMax = 0xMax = 0for i in range (1 , 18 1 ) :g = math . rad ians ( i )phi = g ;g = math . cos ( g )for j in range (1 , 36 1 ) :h = math . rad ians ( j )theta = hth = math . cos ( g )alpha = math . s i n ( phi )*math . cos ( theta )beta = math . s i n ( phi )*math . s i n ( theta )a2 = math .pow( alpha , 2)b2 = math .pow( beta , 2)g2 = math .pow( g , 2)f =3.0*( b2*g2 + g2*a2 + a2*b2 )Edinv = 1/E100 + f *(1/ E111 – 1/E100 )i f Edinv == 0 . 0 :Ed = 0 .0else :Ed = 1.0/ Edinv #r a d i u sxCart = Ed*math . cos ( theta )*math . s i n ( phi ) #xyCart = Ed*math . s i n ( theta )*math . s i n ( phi )#yzCart = Ed*math . cos ( phi )#zr . append (Ed)x . append ( xCart )y . append ( yCart )z . append ( zCart )i f xCart > xMax :xMax = xCarti f zCart > zMax :zMax = zCartprint ’xMax : ’+str (xMax)print ’ zMax : ’+str (zMax)print x . l e n ( )print ’ p l o t t i n g ’f ont = { ’ f ami ly ’ : ’ Times New Roman ’}113Plotting 3-D Surface Young modulus as a Function of Orientationf i g = p l t . f i g u r e ( f i g s i z e =(10 , 10) )ax = f i g . add subplot (111 , p r o j e c t i o n=’ 3d ’ )ax . s e t a u t o s c a l e o n ( Fa l se )ax . a x i s ( [ – 60 , 60 , –60 , 6 0 ] , f o n t d i c t = font )ax . s e t z l i m ( –60 , 60)ax . xax i s . s e t t i c k s (np . arange ( –60 , 60 , 20) )ax . yax i s . s e t t i c k s (np . arange ( –60 , 60 , 20) )ax . z a x i s . s e t t i c k s (np . arange ( –60 , 60 , 20) )fontprop = font manager . FontPropert i e s ( fami ly=’ Times New Roman ’ , s i z e =28)for t i c k in ax . xax i s . g e t m a j o r t i c k s ( ) :t i c k . l a b e l . s e t f o n t p r o p e r t i e s ( fontprop )for t i c k in ax . yax i s . g e t m a j o r t i c k s ( ) :t i c k . l a b e l . s e t f o n t p r o p e r t i e s ( fontprop )for t i c k in ax . z a x i s . g e t m a j o r t i c k s ( ) :t i c k . l a b e l . s e t f o n t p r o p e r t i e s ( fontprop )print len ( x )print len ( y )print len ( z )ax . s c a t t e r (x , y , z , c=r , cmap=cm. get cmap ( ’ gray ’ ) )ax . s e t x l a b e l ( ’ x ’ , s i z e =28, f o n t d i c t = font )ax . s e t y l a b e l ( ’ y ’ , s i z e =28, f o n t d i c t = font )ax . s e t z l a b e l ( ’ z ’ , s i z e =28, f o n t d i c t = font )p l t . show ( )114

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