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A study on the texture and microstructure development in extruded AA3003 alloys and the relevant mechanical… Chen, Jingqi 2018

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    A Study on the Texture and Microstructure Development in Extruded AA3003 Alloys and the Relevant Mechanical Behaviour  by  Jingqi Chen  B.Eng., University of Science and Technology Beijing, 2008 M.Sc., RWTH Aachen University, 2011  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF   DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Materials Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)   February 2018  © Jingqi Chen, 2018   ii    Abstract  In this study, a model alloy (i.e. AA3003) was used to examine microstructure and texture development in the extrusions and the relevant mechanical response. In particular, two homogenization heat treatments were studied as the initial condition, i.e. at 375 °C for 24 h, which produces a high density of dispersoids, and at 600 °C for 24 h, which produces a condition with almost no dispersoids. Three nearly ideal deformation modes were studied, i.e. (i) axisymmetric extension (the central region of a round bar extrusion), (ii) plane strain deformation (the central region of a strip extrusion), and (iii) simple shear deformation (using torsion tests). Electron backscatter diffraction (EBSD) was the main technique to characterize the texture and microstructure of the materials. For the axisymmetric extension and plane strain deformation, it is proposed that the high density of dispersoids in the material causes a large Zener drag, which inhibits grain boundary migration and thereby maintaining the deformation texture and microstructure in the extrusions. For the materials with almost no dispersoids, it is proposed that continuous dynamic recrystallization (CDRX, which is characterized as the subgrain coarsening or the grain boundary migration) occurs during and after the extrusion, and therefore, the texture and microstructure transform from the as-deformed to the recrystallized state. In contrast, for simple shear deformation (i.e. torsion), there is little difference observed between these two materials. However, with an increase in the level of equivalent strain, the texture changes from a deformation to a recrystallized texture. Geometric dynamic recrystallization (GDRX, which is Abstract  iii  characterized as the phenomenon of the grains continuously pinched off during the deformation) is proposed as the mechanism for the case of the simple shear deformation.  For the mechanical behaviour of the extruded materials, it was found that the surface layer has little effect on the measured stress-strain curves. However, the surface layer has a large effect on the R-values, i.e. a characteristic value of the material formability. It is suggested that the large R-value difference between the surface layer and the central region of the sample is attributed to the different textures.   iv    Lay summary  Aluminum alloys are widely applied as structural materials in the automobile industry, e.g. crash tubes, side rails and other components in body frame. To produce the complex shapes needed for these products, extrusion processes at high temperatures are attractive due to the ability to produce high quality components at a competitive cost. To optimize the material design for the structural materials, which are made by extrusion, in service (e.g. folding and buckling during deformation or crashing), this work provides an improved constitutive model which can be used in the next step mechanical response prediction (e.g. plastic stability and anisotropic mechanical response) using finite element method. The constitutive model provided in this work is directly related to the local microstructures and textures, which are not homogeneous in the extrusion profile.   v    Preface  This dissertation is written based on the original work conducted by the author Jingqi Chen in the Department of Materials Engineering, The University of British Columbia. I was the lead investigator, responsible for the designing the experiments, data collection, analysis and interpretation, modelling and writing this manuscript. My supervisor, Dr. Warren Poole, was involved in all stages of the project, provided guidance and assisted with the manuscript composition. The phase field model used in Chapter 6 was developed by Dr. Benqiang Zhu (previous PhD student in the Department of Materials Engineering, The University of British Columbia). The visco-plastic self-consistent (VPSC) model used in Chapters 6 and 8 was developed by Dr. Carlos Tome (Scientist and Team Leader, Materials Science and Technology Division, Los Alamos National Laboratory) and Dr. Ricardo Lebensohn (Scientist and Team Leader, Materials Science and Technology Division, Los Alamos National Laboratory). The simulation data using the finite-element model in Chapter 7 were provided by Dr. Yahya Mahmoodkhani (previous PhD student in the Department of Mechanical and Mechatronics Engineering, University of Waterloo). A version of Chapters 5 and 6 was published in: J. Chen, W. J. Poole, N. C. Parson, “The effect of homogenization conditions on extrusion texture and microstructure evolution in AA3003”, Materials Science Forum, vol. 794-796, pp. 1127-1132, 2014. A version of Chapters 5 and 6 was published in the conference proceedings: J. Chen, W. J. Poole, L. Grajales, N. C. Parson, “A study of the effects of homogenization scenarios and Preface  vi  extrusion conditions on recrystallization mechanisms via analysis of texture and microstructure evolution in AA3003 alloy”, in TMS 2014, San Diego, California, USA, Feb. 16-20, 2014. A version of Chapters 5 and 6 was published in the conference proceedings: J. Chen, W. J. Poole, N. C. Parson, “The effect of homogenization conditions on extrusion texture and microstructure evolution in AA3003”, in ICAA 14th, Trondheim, Norway, Jun. 15-19, 2014. A version of Chapters 5 and 6 was published in the conference proceedings: J. Chen, B. Zhu, W. J. Poole, “A phase field simulation of the recrystallization process after high temperature deformations of Al-Mn-Fe-Si (AA3XXX) alloys”, in ICME 3rd, Colorado Springs, Colorado, USA, Jun. 1-4, 2015. A version of Chapters 6 and 8 was published in the conference proceedings: J. Chen, W. J. Poole, N. C. Parson, “A quantitative study on the influence of through thickness texture variation on the anisotropic mechanical behaviour of extruded strips”, in ICAA 15th, Chongqing, China, Jun. 12-16, 2016. A version of Chapters 6 and 7 was published in the conference proceedings: J. Chen, W. J. Poole, N. C. Parson, “A study of recrystallization near the surface for AA3003 aluminum alloy extrusions”, in IMPC 2016, Quebec City, Quebec, Canada, Sept. 11-15, 2016.   vii    Table of contents  Abstract .......................................................................................................................................... ii Lay summary ................................................................................................................................ iv Preface ............................................................................................................................................ v Table of contents ......................................................................................................................... vii List of tables.................................................................................................................................. xi List of figures .............................................................................................................................. xiv List of symbols .......................................................................................................................... xxiv List of abbreviations ................................................................................................................ xxxi Acknowledgments .................................................................................................................. xxxiii 1 Introduction ........................................................................................................................... 1 2 Literature review ................................................................................................................... 6 2.1 Introduction ..................................................................................................................... 6 2.2 The as-cast microstructure of 3XXX alloys .................................................................... 6 2.3 The change of microstructures during the homogenizations of 3XXX alloys ................ 8 2.4 Work hardening, the as-deformed state and recrystallization ......................................... 9 2.4.1 Work hardening and the as-deformed state ................................................................ 9 2.4.2 Recrystallization during and after deformation ........................................................ 17 Table of contents  viii  2.4.3 The role of second phase particles on recrystallization ............................................ 18 2.5 Textures and microstructures formed by axisymmetric deformation ........................... 20 2.5.1 Deformation textures and microstructures formed by axisymmetric deformation ... 20 2.5.2 Recrystallization textures and the microstructures formed by axisymmetric deformation ........................................................................................................................... 21 2.6 Textures and microstructures formed by plane strain deformation .............................. 22 2.6.1 Deformation textures and microstructures formed by plane strain deformation ...... 22 2.6.2 Recrystallization textures and microstructures formed by plane strain deformation 25 2.7 Textures and microstructures formed by simple shear deformation ............................. 27 2.7.1 Deformation textures and microstructures formed by simple shear deformation..... 27 2.7.2 Recrystallization textures and microstructures formed by simple shear deformation… ....................................................................................................................... 28 2.8 Mechanical behaviour of materials deformed with plane strain deformation .............. 30 2.9 Models for the simulation of deformation .................................................................... 31 2.10 Models for the simulation of the microstructure evolution during recrystallization .... 34 3 Scope and objective ............................................................................................................. 38 4 Methodology......................................................................................................................... 40 4.1 Introduction ................................................................................................................... 40 4.2 Material ......................................................................................................................... 40 4.3 Homogenization treatments .......................................................................................... 41 4.4 Extrusion trials .............................................................................................................. 42 4.5 Torsion tests .................................................................................................................. 45 4.6 High temperature compression tests ............................................................................. 49 4.7 Removal of sample surface ........................................................................................... 51 4.8 Tensile test .................................................................................................................... 52 4.9 Optical metallography ................................................................................................... 55 4.10 Electron backscatter diffraction (EBSD) ...................................................................... 56 Table of contents  ix  4.11 EBSD map clean-up method ......................................................................................... 58 4.12 Description and inputs for the visco-plastic self-consistent (VPSC) model ................. 58 4.12.1 Inputs of the deformation history for compression tests ....................................... 59 4.12.2 Inputs of the deformation histories for axisymmetric and strip extrusions .......... 59 4.12.3 Inputs of the deformation history for torsion tests ................................................ 60 4.12.4 Inputs of the deformation history for tensile tests at room temperature ............... 61 4.13 Description of phase field model .................................................................................. 62 5 Material initial state ............................................................................................................ 63 5.1 Introduction ................................................................................................................... 63 5.2 Experimental results...................................................................................................... 63 6 Ideal deformation modes: axisymmetric, plane strain and simple shear ....................... 66 6.1 Introduction ................................................................................................................... 66 6.2 Experimental results...................................................................................................... 66 6.2.1 Axisymmetric deformation: the centre of the round bar extrusions ......................... 66 6.2.2 Plane strain deformation: the centre of the strip extrusions ...................................... 84 6.2.3 Simple shear deformation: torsion tests .................................................................... 97 6.3 Discussion and simulations ......................................................................................... 103 6.3.1 Simulations for the textures formed by the nearly ideal deformation modes ......... 103 6.3.2 Discussion of the recrystallized textures for the ideal deformation modes ............ 118 7 Textures and microstructures formed on the surface of the extrusion products and their changes from the surface to centre................................................................................. 131 7.1 Introduction ................................................................................................................. 131 7.2 Textures and microstructures on the surface of the extrusion products and their changes from the surface to centre: the round bar extrusions ............................................................... 131 7.2.1 Experimental results................................................................................................ 131 7.2.2 Discussion ............................................................................................................... 143 Table of contents  x  7.3 Textures and microstructures on the surface of the extrusion products and their changes from the surface to centre: the strip extrusions ....................................................................... 144 7.3.1 Experimental results................................................................................................ 144 7.3.2 Discussion ............................................................................................................... 148 8 Mechanical response of the extrusion products.............................................................. 150 8.1 Introduction ................................................................................................................. 150 8.2 Mechanical response of the axisymmetric extrudates ................................................ 150 8.2.1 Experimental results................................................................................................ 150 8.2.2 Discussion and simulations ..................................................................................... 151 8.3 The anisotropic mechanical response of the as-extruded strip ................................... 160 8.3.1 Experimental results................................................................................................ 160 8.3.2 Discussion and simulations ..................................................................................... 165 9 Summary and future work ............................................................................................... 173 9.1 Summary ..................................................................................................................... 173 9.2 Future work ................................................................................................................. 176 Bibliography .............................................................................................................................. 178 Appendix .................................................................................................................................... 196 A.1       Parametric study for texture calculation in axisymmetric extrusion .......................... 196 A.2       Back-estimate subgrain size using the measured grain boundary densities ............... 197 A.3       Misorientation between the texture components in plane strain deformation ........... 198 A.4       Estimate grain dimension after the strip extrusion ..................................................... 199    xi    List of tables  Table 2.1: Crystal structure of Al6(Mn,Fe) and α-Al(Mn,Fe)Si phases ......................................... 8 Table 2.2: Miller indices and Euler angles of the copper, S and brass orientations ..................... 22 Table 2.3: Miller indices and Euler angles of the texture components in the recrystallization of aluminum and its alloys ................................................................................................................ 26 Table 4.1: Chemical composition of the AA3003 alloy in wt.% (which was measured using inductively coupled plasma mass spectrometry) .......................................................................... 41 Table 4.2: Summary of the details of the extrusion trials (note: the initial diameter of the billet for extrusion is 101.6 mm) ............................................................................................................ 43 Table 4.3: Summary of the details for the partial extrusion trial (note: the initial diameter of the billet for extrusion is 101.6 mm)................................................................................................... 45 Table 4.4: Summary of details for torsion tests ............................................................................ 48 Table 4.5: Grinding procedures for sample surface preparation ................................................... 56 Table 4.6: Polishing procedures for sample surface preparation .................................................. 56 Table 5.1: Summary of the number densities and the volume fractions of the dispersoids and constituent particles formed in the homogenization at 600 °C for 24 h and at 375 °C for 24 h ... 64 Table 6.1: Summary of the calculated parameter Dtex based on the experimentally determined pole figures shown in Figures 6.1c and d and the re-calculated pole figures shown in Figures 6.3a and b .............................................................................................................................................. 71 Table 6.2: Summary of the calculated area / volume fractions of the texture fibre <001> // ED and <111> // ED based on the approaches (a) and (b) for the recrystallized and unrecrystallized conditions ...................................................................................................................................... 72 Table 6.3: Summary of the quantitative microstructure properties counted separated in the texture fibres of <001> // ED and <111> // ED ............................................................................ 81 List of tables  xii  Table 6.4: Summary of the calculated parameter Dtex based on the experimentally determined pole figures shown in Figure 6.9c and d and the re-calculated pole figures shown in Figures 6.11a and 6.11b ............................................................................................................................. 89 Table 6.5: Summary of the identified volume / area fractions of the texture components for the texture in the central region of the strip extrusion for the recrystallization condition (a deviation angle of 10 ° was employed) ......................................................................................................... 90 Table 6.6: Summary of the identified volume / area fractions of the texture components for the texture in the central region of the strip extrusion for the as-deformed condition (a deviation angle of 10 ° was employed) ......................................................................................................... 90 Table 6.7: Summary of the estimated subgrain sizes in equivalent diameter (μm) for each texture components for the deformed condition of the as-extruded strip ................................................. 97 Table 6.8: The summary of the microstructure properties of the torsion conditions for the different equivalent strains .......................................................................................................... 101 Table 6.9: The Voce hardening law parameters used for the simulations as shown in Figure 6.19 (the material homogenized at 375 °C for 24 h and deformed at 350 °C with the strain rate of 0.1 s-1, 1 s-1, and 10s-1) ...................................................................................................................... 105 Table 6.10: Comparison of the experimentally determined texture and the simulated texture (neff = 7) for the texture simulation of the as-deformed condition in the axisymmetric extrusions ... 115 Table 6.11: Comparison of the experimentally determined texture and the simulated texture (neff = 7) for the texture simulation of the as-deformed condition in the strip extrusions.................. 115 Table 8.1: The parameters for the Voce law parameters used to fit the homogenization conditions of 600 °C for 24 h and 375 °C for 24 h ....................................................................................... 153 Table 8.2: Summary of the changes of CMn (estimated by the resistivity measurements) by the extrusion processes and the estimated changes of the yield stress based on Equation 8.1 for the extrudates with the homogenization at 600 °C for 24 h and at 375 °C for 24 h, compared with the as-homogenized conditions ......................................................................................................... 156 Table 8.3: Parameters used in the extended Voce law with a consideration of an increased initial dislocation density of ρ2 = 1.8 1013 m-2 .................................................................................... 159 Table A.1: Summary of the area fractions of the <001> // ED and <111> // ED texture components which are calculated based on different calculation ranks in harmonic series expansion and different deviation angles for the EBSD data ..................................................... 196 List of tables  xiii  Table A.2: Summary of the misorientation angles (°) between the each individual texture components (i.e. the cube, Goss, S, copper and brass texture components) in the plane strain deformations ............................................................................................................................... 198    xiv    List of figures   Figure 1.1: Photograph showing a compressed crash tube which buckled and folded under a compressive loading parallel to the extrusion direction of the product (the blue arrows marked as No. 1 and 2 illustrate that the bending axis can occur at different orientations to the loading direction), and the inset showing the anodized micrograph of a microstructure of an extruded strip which underwent an extreme bending, where the surface was fractured (supplied by Dr. Nick Parson).................................................................................................................................... 1 Figure 2.1: (a) backscatter electron (BSE) image of as-cast microstructure of 3003 alloy (Al-1.15Mn-0.58Fe-0.20Si-0.08Cu in wt.%) [21], and (b) BSE image of the microstructure of an 3003 alloy (Al-1.18Mn-0.58Fe-0.20Si-0.08Cu in wt.%) after 1 h heat treatment at 500 ˚C (the applied heating rate is 20 ˚C/s) [22] ................................................................................................ 7 Figure 2.2: TEM image of dispersoids in an AA3003 alloy (Al-1.15Mn-0.58Fe-0.20Si-0.08Cu in wt.%) homogenized at 375 ˚C for 24 h [10] ................................................................................... 9 Figure 2.3: Stress-strain curves for Al-1Mg (in wt.%) at 400 °C [35] ......................................... 13 Figure 2.4: Graphs of the surface areas, normalized by the initial spherical surface, of ellipsoids distorted by deformation modes as a function of the von Mises equivalent total strain [37] ....... 14 Figure 2.5: TEM micrographs revealing dislocation structures after deformation at (a) room temperature, where dislocation tangles are observed, and (b) at 350 °C, where an equiaxed subgrain structure is observed (Al-1Mg, in wt.%, which was deformed with strain rate of 1 s-1 to a final strain 1.1) [38].................................................................................................................... 15 Figure 2.6: Backscatter electron (BSE) image showing high angle grain boundaries in aluminum alloys deformed at 400 °C in plane strain compression: (a) Al-5Mg (in wt.%), showing serrations at the high angle boundaries, (b) Al-2Cu (in wt.%), in which precipitates prevent boundary serrations from developing. .......................................................................................................... 16 Figure 2.7: TEM observations of PSN in annealing of a 60% cold rolled Al-0.45Cu-0.80Si (in wt.%) alloy at 240 ˚C with 0 s (as deformed microstructure), and with a few seconds [49] ........ 19 List of figures  xv  Figure 2.8: The schematic illustration of such lattice rotation during axisymmetric deformation, where the distributions of crystal orientations are plotted with respect to extension axis and the arrows denote the lattice rotation directions [57] ......................................................................... 21 Figure 2.9: (a) orientation distribution function (ODF) figure showing the hot rolled texture of Al-1Mn-1Mg (wt.%) [65] and (b) schematic representation of texture fibre [39] ....................... 23 Figure 2.10: TEM micrograph of a cube-band in the as-deformed condition of an AA3004 (Al-1.2Mg-0.9Mn, in wt.%) rolled at 330 °C [67] .............................................................................. 24 Figure 2.11: (a) orientation distribution function (ODF) figure showing the recrystallized texture of Al-1Mn-1Mg (in wt.%) which was hot rolled and annealed at 330 °C [65], (b) evolution of the average subgrain sizes of the cube-band and matrix during the annealing (i.e. at 330 °C) of hot rolled Al-1Mn-1Mg (in wt.%) at 330 °C [66], (c) evolution of the volume fraction of the grains having the different texture components during the annealing (i.e. at 325 °C) of the hot rolled AA1050 alloy (i.e. at 325 °C) [76] ............................................................................................... 26 Figure 2.12: Schematic illustration of (a) A, (b) A*, (c) C, and (d) B texture components in simple shear deformation in the {111} pole figures, in which the vertical and horizontal directions are parallel with the shear direction (SD) and the torsion axis (Z), respectively [82] . 27 Figure 2.13: EBSD maps showing the microstructure of the AA6060 deformed in the torsion tests at 486 °C with a strain rate of 13 s-1 and to the final equivalent strain of (a) 5 and (b) 25 [90]....................................................................................................................................................... 29 Figure 4.1: Anodized micrograph showing the as-cast microstructure of the AA3003 alloy in the section normal to the casting direction (from the work of Grajales [18]) .................................... 41 Figure 4.2: Schematic illustration of homogenization treatment temperature profiles ................ 42 Figure 4.3: Schematic illustration of the geometry of (a) zero-bearing die and (b) flat die with 3 mm bearing in cross-section, where the extrusion direction is along with vertical direction ....... 43 Figure 4.4: Schematic illustration of the setup for the extrusion press and the standing wave water tank, and the area where the samples were taken from extrudate for further mechanical tests and characterization of texture and microstructure............................................................... 44 Figure 4.5: High temperature torsion sample dimensions (in inches) .......................................... 45 Figure 4.6: Schematic illustration of the setup for high temperature torsion test ......................... 46 List of figures  xvi  Figure 4.7: (a) Temperature profile showing the whole torsion test (material homogenization: 375 °C - 24 h, equivalent surface strain rate: 10 s-1, final equivalent surface strain: 12), including heating, holding, deformation, and cooling; (b) a magnification of the temperature profiles about the deformation region in Figure 4.7a........................................................................................... 47 Figure 4.8: Schematic illustration of the equivalent strain distribution from the centre to the surface of torsion sample gauge section in transection ................................................................. 48 Figure 4.9: Schematic illustration of the quenching setup for the torsion tests ............................ 49 Figure 4.10: Schematic illustration of the high temperature compression setup .......................... 50 Figure 4.11: (a) Temperature profile showing the whole compression test (material homogenization: 375 °C - 24 h, strain rate: 10 s-1; (b) a magnification of the temperature profiles about the deformation region in Figure 4.11a ............................................................................... 50 Figure 4.12: Plot of the extruded strip thickness as a function of etching time for the extruded strips homogenized at 375 °C for 24 h and at 600 °C for 24 h ..................................................... 51 Figure 4.13: Dimensions of the tensile samples machined from billets (the length is parallel to the CD) .......................................................................................................................................... 52 Figure 4.14: Dimensions for the tensile samples machined from axisymmetric extrudate (the length is parallel with the ED) ...................................................................................................... 52 Figure 4.15: Schematic illustration of the scenario for machining the three types of samples from extruded strip ................................................................................................................................ 53 Figure 4.16: Dimensions for the tensile samples machined from extruded strips ........................ 53 Figure 4.17: An example showing the correlation of the loading values captured by tensile machine which has been aligned with the data captured independently by the DIC system ........ 54 Figure 4.18: Schematic illustration of the section applied for EBSD mapping in (a) axisymmetric extrusion and (b) strip extrusion ................................................................................................... 57 Figure 4.19: Schematic illustration about the two types of sections applied for EBSD mapping in torsion sample ............................................................................................................................... 57 Figure 5.1: Optical micrographs showing the microstructures of the homogenization (a) 600 °C - 24 h and (b) 375 °C - 24 h ............................................................................................................ 64 List of figures  xvii  Figure 5.2: (a) IPF map showing the as-cast microstructure, where the IPF colour is according to the casting direction, and (b) {001}, {011}, and {111} pole figures showing as-cast texture, where the direction normal to the paper is aligned with the casting direction ............................. 65 Figure 6.1: IPF maps showing the microstructures in the centre of the longitudinal plane for the axisymmetric extrudate for the billets homogenized (a) at 600 °C for 24 h (where the white arrows denote the grains having the <111> orientations parallel to the extrusion direction) and (b) at 375 °C for 24 h; and pole figures showing the textures in this region for the materials homogenized (c) at 600 °C for 24 h and (d) at 375 °C for 24 h (the pole figures are equal area projections with the extrusion direction (ED) at the centre of the pole figure and the radial direction (RD) at the outer rim) .................................................................................................... 67 Figure 6.2: Plot of the grain size distribution (in equivalent diameter) for the central region of the extrudate with the homogenization at 600 °C for 24 h (the recrystallization condition), and the experimental distribution is fit with a lognormal distribution ...................................................... 68 Figure 6.3: Plot of the re-calculated pole figures based on the ODF showing the textures in this region for the materials homogenized (c) at 600 °C for 24 h and (d) at 375 °C for 24 h ............. 70 Figure 6.4: (a) IPF map showing the as-extruded microstructure in the centre of the longitudinal plane for the extrudate (i.e. the unrecrystallized condition, with the homogenization at 375 °C for 24 h), (b) illustrating the distributions of LAGBs and HAGBs for the as-extrudate microstructure as shown in Figure 6.4 and the inset showing a magnified view of one representative region, and (c) the misorientation profile showing the misorientation angles between the current point and the original point in the direction normal to the extrusion direction (as indicated by the blue arrow in Figure 6.4b) and the identified grains based on the misorientation analysis .................. 74 Figure 6.5: IPF maps illustrating the region in Figure 6.1.3a having the orientations within 15 ° of the ideal <001> // ED texture component, (b) the region having the orientations within 15 ° of the ideal <111> // ED texture component, and (c) the misorientation profiles showing the misorientation between the current points with the original point along the extrusion direction inside the <001> // ED and <111> // ED texture components (as pointed by the white arrows in Figure 6.5a and b, respectively) .................................................................................................... 76 List of figures  xviii  Figure 6.6: An example of the microstructure (for the homogenization of 375 °C for 24 h)of the <001> // ED texture component: (a) the map with the IPF colour according to the extrusion direction, (b) the map with the IPF colour according to the radial direction, (c) the map showing the distribution of LAGBs and HAGBs, and (d) the orientation projection in the {001} and {111} pole figures for the region in Figure 6.6a; and an example of the microstructure of the <111> // ED texture component: (e) the map with the IPF colour according to the extrusion direction, (f) the map with the IPF colour according to the radial direction, (g) the map showing the distribution of LAGBs and HAGBs, and (h) the orientation projection in the {001} and {111} pole figures for the region in Figure 6.6e ..................................................................................... 78 Figure 6.7: Histogram showing the fractions of LAGBs and HAGBs versus their misorientation angles inside the areas having the orientations with 15 ° of the ideal <001> // ED and <111> // ED components in the as-deformed microstructure (the centre of the extrudate with the homogenization at 375 °C for 24 h), respectively (note: the maximum misorientation between different <001> crystal orientation is 45 ° and the maximum misorientation between <111> crystal orientation is 60 °) ............................................................................................................. 80 Figure 6.8: For the centre of the extrudate homogenized at 375 °C with 24 h: (a) a high resolution of IPF map (step size: 50 nm), where the contrast is combined with IQ; and (b) the plot of the kernel average misorientation histograms for the areas having the orientations within 15 ° of the ideal <001> // ED texture component and that of the ideal <111> // ED texture component ..................................................................................................................................... 83 Figure 6.9: IPF maps showing the as-extruded microstructures in the centre of the ED-ND plane of the strip extrusion (at 350 °C with an extrusion ratio 33:1) for the materials homogenized (a) at 600 °C for 24 h and (b) at 375 °C for 24 h; and pole figures showing the experimentally determined textures in this region for the materials homogenized (c) at 600 °C for 24 h and (d) at 375 °C for 24 ................................................................................................................................. 85 Figure 6.10: Plot of the grain size (in equivalent diameter) distribution for the central region of the extrusion with the homogenization at 600 °C for 24 h (the recrystallization condition), and the experimental distribution is fitted by a lognormal distribution with the certain equation and parameters as shown in the figure ................................................................................................. 87 Figure 6.11: Plot of the re-calculated PFs based on the ODF showing the textures in this region for the materials homogenized (c) at 600 °C for 24 h and (d) at 375 °C for 24 h ........................ 89 Figure 6.12: Plots of the β texture fibre for the recrystallized and unrecrystallized (or deformed) conditions ...................................................................................................................................... 91 List of figures  xix  Figure 6.13: (a) IPF map showing the as-extruded microstructure in the centre of the ED-ND plane for the as-extruded strip, (b) the map showing the LAGB and HAGB distributions in the microstructure as shown in Figure 6.13a, and the misorientation profile showing the misorientation angles between the current points with the original point as pointed by the black arrow in Figure 6.13d .................................................................................................................... 93 Figure 6.14: (a) IPF map showing the a representative example of the as-extruded microstructure in the centre of the as-extruded strip, and maps illustrating the spatial distributions of the (b) S, (c) copper, (d) brass, (d) cube, and (e) Goss texture components (the orientations within 10 ° away from the ideal orientations) ........................................................................................................... 96 Figure 6.15: Plot of the flow curves of the torsion tests for the materials homogenized at 600 °C for 24 h and 375 °C with 24 h ....................................................................................................... 98 Figure 6.16: {111} PFs (without ODF calculations) showing the torsion textures at the equivalent strain (a) 5, (b) 6, and (c) 12 of the material homogenized at 600 °C for 24 h and the torsion textures at the equivalent strain (d) 5, (e) 6, and (f) 12 of the material homogenized at 375 °C for 24 h .............................................................................................................................. 99 Figure 6.17: The maps showing the distributions of the LAGBs and HAGBs for the material homogenized at 600 °C for 24 h in the area at the equivalent shear (a) 5 and (b) 12, and the torsion microstructures of the material homogenized at 375 °C for 24 h for in the area at the equivalent shear (c) 5 and (d) 12 in the torsion tests .................................................................. 100 Figure 6.18: The maps showing the distributions of the LAGBs and HAGBs in the torsion tests at the equivalent strain 1 for the materials with the homogenization (a) 600 °C - 24 h and (b) 375 °C - 24 h, where the red arrows denote the HAGBs which are almost parallel to each other and produced the band-structures ............................................................................................... 102 Figure 6.19: Plot of the compression flow curves at 350 °C of the material homogenized at 375 ° for 24 h with different strain rates 0.1 s-1, 1s-1, and10s-1; and the simulated flow curves using the VPSC model (the parameters in the Voce hardening law are shown in Table. 6.2.1 ................. 104 Figure 6.20: {001} and {111} pole figures showing the simulations based on the neff value of (a) 1, (b) 5, (c) 10 and (d) 15 for the round bar extrusion with the equivalent strain of 4 ............... 106 Figure 6.21: Plots of (a) the volume fractions of specific texture components and (b) the maximum value of texture intensity in pole figures as a function of neff value .......................... 107 Figure 6.22: {001} and {111} pole figures showing the simulations based on the neff value of (a) 1, (b) 5, (c) 10 and (d) 15 for the strip extrusion with the equivalent strain of 4........................ 109 List of figures  xx  Figure 6.23: (a) Plot of the simulated area fraction of the S, copper and brass orientations with the applied the parameters neff in the simulations, and (b) plot of the simulated maximum texture intensities in pole figures with the applied the parameters neff in the simulations ...................... 110 Figure 6.24: Plots of the simulated β texture fibres based on the different neff values and the experimental β texture fibre ........................................................................................................ 110 Figure 6.25: {111} pole figures showing the simulations based on the neff value of (a) 1, (b) 5, (c) 10 and (d) 15 for the torsion test with the equivalent strain of 5 ................................................ 112 Figure 6.26: Plot of the simulated maximum texture intensities in pole figures with the applied the parameters neff in the simulations for torsions ....................................................................... 113 Figure 6.27: {001} and {111} PFs showing the simulated texture (neff = 7) for the axisymmetric extrusion ...................................................................................................................................... 115 Figure 6.28: {001} and {111} PFs showing the simulated texture (neff = 7) for the strip extrusion..................................................................................................................................................... 115 Figure 6.29: {111} PF showing the simulated texture (neff = 7) for the torsion ......................... 116 Figure 6.30: (a) - (e): IPF maps showing the simulated microstructure evolution for the subgrain coarsening process with an input microstructure from the deformation condition as shown in Figure 6.1b, predicted by the phase field model (f) IPF map showing the recrystallized microstructure in the axisymmetric extrusion (i.e. with almost no dispersoids after the homogenization of 600 °C for 24 h) ........................................................................................... 122 Figure 6.31: The microstructure evolutions with time: (a) the area fractions of <001> and <111> grains and (b) the subgrain / grain sizes in diameter .................................................................. 124 Figure 6.32: Schematic illustration of the microstructure changed during the simple shear deformation, in which GDRX occurs ......................................................................................... 129 Figure 7.1: IPF maps showing the microstructures on the surface of the axisymmetric extrusion (at 350 °C with an extrusion ratio 70:1) for the materials homogenized (a) at 600 °C for 24 h and (b) at 375 °C for 24 h in the longitudinal plane; and {001} and {111} pole figures showing the textures in the subsurface area marked in Figure 7.1a for the materials homogenized (c) at 600 °C for 24 h and the surface in Figure 7.1b for the material homogenized (d) at 600 °C for 24 h................................................................................................................................................... 132 List of figures  xxi  Figure 7.2: For the extrudate with the homogenization at 375 °C for 24 h: (a) IPF maps showing the microstructure from the surface of the extrudate to the area 1.2 mm away from surface and also the centre area, (b) Plot of the area fractions of the <001> // ED and <111> // ED texture components from the surface to centre, and (c) {001} and {111} pole figures showing the texture of the area from 0.8 mm to 1.2 mm away from surface; and for the extrudate with the homogenization at 600 °C for 24 h: (d) IPF maps showing the microstructure from the surface of the extrudate to the area 1.2 mm away from surface and also the centre area, (e) Plot of the area fractions of the <001> // ED component from the surface to centre, and (c) {001} and {111} pole figures showing the texture of the area from 0.8 mm to 1.2 mm away from surface ................. 134 Figure 7.3: (a) Plot of the estimated shear strains from the surface of the extrudate to the centre (the data is supplied by Mahmoodkhani in University of Waterloo, and the details about the extrusion simulations based on the FEM are discussed in the ref. [17, 44]), (b) IPF map showing the microstructure from the surface of the extrudate to area 0.8 mm away from the surface, (c) {111} PF showing the texture on the surface of extrudate, and (d) {111} PF showing the texture in the area 0.6 to 0.75 mm away from the surface ...................................................................... 138 Figure 7.4: (a) Optical image showing the microstructures on the surface of the partial extruded sample and an estimated track line of the position 0.5 mm away from the surface of the extrudate (the data was supplied by the by Mahmoodkhani in University of Waterloo, and the details about the extrusion simulations based on the FEM were discussed in the ref. [17, 44]), (b) IPF map showing the microstructure in the position No.1 (just before the die exit), (c) {111} PF showing the texture in the position No.1 in the reference ED-RD, (d) {111} PF showing the texture in the position No.1 in the reference MFD-DNMF, (e) IPF map showing the microstructure in the position No.2 (just after the die exit), (f) {111} PF showing the texture in the position No.2 in the reference ED-RD, and (g) {111} PF showing the texture in the position No.2 in the reference MFD-DNMF ............................................................................................................................... 141 Figure 7.5: For the as-extruded strip with the homogenization at 375 °C for 24 h: (a) IPF maps showing the microstructure from the surface to the centre on the ED-ND plane of the as-extruded strip, (b) {001} and {111} PFs showing the texture of the surface layer, as labelled as (b) in Figure 7.5a, (c) {001} and {111} PFs showing the texture of the area from 0.15 mm to 0.3 mm away from surface, as labelled as (c) in Figure 7.5a, (d) plots of the area fractions of the texture components from the 0.15 mm away from the surface to the centre of the as-extruded strip, and (e) {111} PF showing the texture of the surface layer in the reference of ED-ND, which is aligned with that in the torsion tests ........................................................................................... 145 List of figures  xxii  Figure 7.6: For the as-extruded strip with the homogenization at 600 °C for 24 h: (a) IPF maps showing the microstructure from the surface to the centre on the ED-ND plane of the as-extruded strip, (b) {001} and {111} PFs showing the texture of the PCG layer, as labelled as (b) in Figure 7.6a, (c) {001} and {111} PFs showing the texture of the area just adjacent to the PCG layer, as labelled as (c) in Figure 7.6a, (d) plots of the area fractions of the texture components from the area just adjacent to the PCG layer to the centre of the as-extruded strip, and (e) {111} PF showing the texture of the PCG layer in the reference of ED-ND, which is aligned with that in the torsion tests ........................................................................................................................... 147 Figure 8.1: Plots of the flow curves measured at room temperature for the as-homogenized conditions, the central regions of the extrudates (4.75 mm in diameter from the centre-line of the extrudates), and the extrudates with the full diameter (12.7 mm) for the homogenization at 600 °C for 24 h and at 375 °C for 24 h (note: for all tensile tests in this study, the yield stress was determined using the 0.2 % offset method) ................................................................................ 151 Figure 8.2: Plots of the experimental flow curves of the as-homogenized conditions at 600 °C for 24 h and at 375 °C for 24 h, and the simulated flow curves by the VPSC model with the different Voce hardening law parameters as mentioned in Table 8. ......................................................... 153 Figure 8.3: Plots of the experimental flow curves of the as-extruded conditions with the central region of the extrudates (with the homogenization at 600 °C for 24 h and at 375 °C for 24 h), and the simulated flow curves with the <001> // ED texture as the input texture for the extrudate with the homogenization at 600 °C for 24 h and the simulated flow curves with the <001> - <111> double fibre texture for the extrudate with the homogenization at 375 °C for 24 h ................... 154 Figure 8.4: Schematic illustration of the change due to an increased initial dislocation density in the extended Voce law ................................................................................................................ 158 Figure 8.5: Plots of the experimental flow curves of the as-extruded conditions with the central region of the extrudates (with the homogenization at 600 °C for 24 h and at 375 °C for 24 h), and the simulated flow curves with considerations of the effects of the textures, the solid solution levels, and the microstructures for the as-extruded conditions with the homogenization at 600 °C for 24 h and at 375 °C for 24 h ................................................................................................... 159 Figure 8.6: For the materials homogenized (a) at 600 °C for 24 h and (b) at 375 °C for 24 h: plots of the flow curves measured at room temperature along the ED, at 45 ° to the ED, and the TD for the sample of full thickness of the extruded strip and that with only the central region ............ 160 Figure 8.7: For the materials homogenized (a) at 600 °C for 24 h and (b) at 375 °C for 24 h: plots of the R-value evolutions with the plastic true strain along the ED, at 45 ° to the ED, and the TD for the sample of full thickness of the extruded strip and that with only the central region ....... 162 List of figures  xxiii  Figure 8.8: SE2 images showing the cross-section areas of the tensile sample machined from the extruded strip:  with the homogenization at 600 °C for 24 and measured (a) along the ED, (b) at 45 ° to the ED, and (c) along the TD for the sample with full thickness; and for the sample with the central region and measured (d) along the ED, (e) at 45 ° to the ED, and (f) along the TD; and with the other homogenization at 375 °C for 24 and measured (g) along the ED, (h) at 45 ° to the ED, and (i) along the TD for the sample with full thickness; and for the sample with the central region and measured (j) along the ED, (k) at 45 ° to the ED, and (l) along the TD ................... 164 Figure 8.9: For the materials homogenized (a) at 600 °C for 24 h and (b) at 375 °C for 24 h: plots of the experimental and simulated flow curves along the ED, at 45 ° to the ED, and along the TD for the samples with the central region of the extruded strips .................................................... 165 Figure 8.10: For the materials homogenized (a) at 600 °C for 24 h and (b) at 375 °C for 24 h: plots of the experimental and simulated R-value evolutions along the ED, at 45 ° to the ED, and along the TD for the samples with the central region of the extruded strips .............................. 167 Figure 8.11: For the material homogenized at 600 °C for 24 h: plots of the experimental R-value evolutions for the samples with the central region of the extruded strips and the samples with full thickness and the simulated R-value evolutions based on the texture in the centre and the texture on the surface for the tensile directions (a) along the ED, (b) at 45 ° to the ED, and (c) along the TD ............................................................................................................................................... 169 Figure 8.12: For the material homogenized at 375 °C for 24 h: plots of the experimental R-value evolutions for the samples with the central region of the extruded strips and the samples with full thickness and the simulated R-value evolutions based on the texture in the centre and the texture on the surface for the tensile directions (a) along the ED, (b) at 45 ° to the ED, and (c) along the TD ............................................................................................................................................... 171 Figure A.1: A schematic illustration of an artificial microstructure of the subgrains with a uniform shape and size of hexagons, where the grey lines represent the LAGBs (i.e. their misorientations < 15 °) and the black lines represent the HAGBs (i.e. their misorientations ≥ 15 °) .......................................................................................................................................... 197 Figure A.2: Schematic illustration of the grain shape change in the strip extrusion .................. 199    xxiv    List of symbols  α linear factor in Taylor equation αAl thermal expansion coefficient of pure aluminum γ interfacial energy γGB grain boundary energy γij interfacial energy specifically between grain i and j γ15° interfacial energy for the grain boundary which has a misorientation of 15 ° γ̇s local shear rate in a given grain γ̇0 normalization factor for the local shear rate in a given grain Γ accumulated shear in a grain Γdis equivalent shear accumulated in a grain with a consideration of an increased initial dislocation density εt plastic true strain εTt plastic true strain in the thickness direction of the tensile sample εWt plastic true strain in the width direction of the tensile sample ηij interfacial thickness specifically between grain i and j θ rotation angle in torsion tests θ̇ rotation rate in torsion tests θg misorientation angle of crystallographic orientations List of symbols  xxv  θs0 initial hardening rate in the extended version of Voce hardening law θs1 asymptotic hardening rate in the extended version of Voce hardening law μ interface mobility μij interface mobility specifically between grain i and j μ15° interfacial mobility for the grain boundary which has a misorientation of 15 ° σt true stress σ’kl local deviatoric stress tensor σcomp stress tensor to describe the compression test σtension stress tensor to describe the tensile test σtorsion stress tensor to describe the torsion test Σ’kl macroscopic deviatoric stress tensor ∆σGB yield stress change caused by the change of grain size ∆σss yield stress change caused by the change of solid solution level τs0 initial critical resolved shear stress (CRSS) in a given grain (in the extended version of Voce hardening law) τdis0 initial critical resolved shear stress (CRSS) in a given grain (in the extended version of Voce hardening law) with a consideration of an initial dislocation density τs1 difference between the back-extrapolated CRSS and initial CRSS in a given grain (in the extended version of Voce hardening law) τdis1 difference between the back-extrapolated CRSS and initial CRSS in a given grain (in the extended version of Voce hardening law) with a consideration of an initial dislocation density υ Poisson ratio List of symbols  xxvi  ρ+ increase of dislocation density by the Frank-Read source during deformation (in Kocks and Mecking equation) ρ- decrease of dislocation density by the dynamic recovery during deformation (in Kocks and Mecking equation) ρ1 and ρ2 dislocation densities for the as-homogenized condition and the as-extruded condition (in Taylor equation) ρdis total dislocation density change during deformation (in Kocks and Mecking equation) ρGB  area density of all grain boundaries φ1, Φ, φ2 Euler angles ϕi (ϕi) phase field parameter for the grain i (j) A {111} <110> / { 111 } < 110> texture component in simple shear deformations A* { 111} <112> / {111 } <112>) texture component in simple shear deformations Adisp the dispersion level at which the DZener is equal to the DNu a1, a2, a3 three principle axes in an ellipsoid b Burgers vector bs Burgers vector specifically for the slip system s B {112 } <110> / { 112 } < 110> texture component in simple shear deformations Ba  Bdisp the dispersion level at which Zener pinning is sufficient to fully suppress recrystallization C {001} <110> texture component in simple shear deformations List of symbols  xxvii  CMn weight percent of Mn in solid solution dhex equivalent diameter of the hexagon in the calculation of the subgrain size based on the measured grain boundary density dc-0 initial diameter of the compression sample dc-f final diameter of the compression sample after test D0 initial diameter of the compression sample ∆D measured diameter change of compression sample compared with the initial diameter Dg1 and Dg2 grain sizes of the as-homogenized condition and the as-extruded condition in diameter Dij macroscopic strain rate tensor Dtex the parameter to estimate the difference of two textures E Young’s modulus f volume fraction of particles fa(gi) and fb(gi) the texture intensities for the same orientation gi in textures a and b F force gi crystal orientation for grain i G shear modulus hc-0 initial height of the compression sample hc-f final height of the compression sample after test k1 liner factor for the term of dislocation accumulation (in Kocks and Mecking equation) k2 liner factor for the term of dynamic recovery (in Kocks and Mecking equation) List of symbols  xxviii  k3 dislocation accumulation term due to second phase particles (in Kocks and Mecking equation) KGB linear factor in Hall-Petch equation Kss linear factor in the equation to calculate the change of yield stress based on the change of the change of solid solution level lhex length of the side for the hexagon in the calculation of the subgrain size based on the measured grain boundary density Ls gauge length of torsion sample Laxisy-ext velocity gradient tensor to describe the round bar extrusion Lcomp velocity gradient tensor to describe the compression test Lstrip-ext velocity gradient tensor to describe the strip extrusion LExt length of the extruded grain in the strip extrusion Ltension velocity gradient tensor to describe the tensile test Ltorsion velocity gradient tensor to describe the torsion test m strain rate sensitivity ms (msi and msj) symmetric Schmid tensor associated with the slip system s M Taylor factor Mt torque Msec interaction tensor based on the secant approximation Mtg interaction tensor based on the tangent approximation M̃ (M̃ijkl) interaction tensor n inverse value of the strain rate sensitivity neff homogenization parameter List of symbols  xxix  ns (nsi and nsj) normal vector for the slip system s P {011} <122> texture component in plane strain deformations Pzener Zener drag Q {013} <231> texture component in plane strain deformations R R-value Rav average of the measured R-values RExt extrusion ratio Rhomo radius of the grains after the homogenization treatment Rg {124} <211> texture component in plane strain deformations Rp average of the particle radii Rs specimen radius in the gauge section of torsion sample S {123} <634> texture component in plane strain deformations T temperature TExt thickness of the extruded grain in the strip extrusion UR standard deviations of the measured R-values UW and UW0 the average values of the measured sample widths during testing and the initial sample width Wav and W0-av the average standard deviations of the measured width and initial width WExt width of the extruded grain in the strip extrusion X rate sensitivity part in the calculation of equivalent stress based on the measured torque in torsion tests Y working hardening coefficient in the calculation of equivalent stress based on the measured torque in torsion tests List of symbols  xxx  Z torsion axis   xxxi    List of abbreviations  2-D two-dimension (two-dimensional) 3-D three-dimension (three-dimensional) AU arbitrary unit BCC body-centred cubic BSE backscatter electron CDRX continuous dynamic recrystallization  CI confidence index CRSS critical resolved shear stress DC direct chill casting DDRX discontinuous dynamic recrystallization DIC digital image correlation DNMF direction normal to the material flow EBSD electron backscatter diffraction ED extrusion direction fps frame per second FC full-constraints FCC face-centred cubic GDRX geometric dynamic recrystallization List of abbreviations  xxxii  HAGB high angle grain boundary IPF inverse pole figure IQ image quality KAM kernel average misorientation LAGB low angle grain boundary MC Monte Carlo MFD material flow direction MRD multiples of a random distribution ND normal direction ODF orientation distribution function PCG peripheral coarse grain PF pole figure PSN particle stimulated nucleation RC relaxed-constraints RD radial direction SC simple cubic SEM scanning electron microscopy SFE stacking fault energy TD transverse direction TEM transmission electron microscopy VPSC visco-plastic self-consistent    xxxiii    Acknowledgments  My PhD life is not a smooth journey: I was lost, confused, and even suffered on my journey. Franking speaking, there was a moment when I did want to give up, but I did not because there were so many people encouraged me, helped me, and pushed me. I deeply appreciate all they did for me! First, I would like to thank my supervisor Dr. Warren Poole for giving me a chance to study at the University of British Columbia and work on this project. Without his constant support and guidance, I would never be able to finish this journey. I wish to thank him for supporting me to attend so many high level international conferences and providing me so many opportunities to discuss with the experts in the different fields. Without this experience, I would never be able to open my eye. I wish also to thank him for his patience and the freedom he gave to let me make mistakes. Without this, I would never be able to find my way out. I would also like to thank Dr. Mary Wells, Dr. Nick Parson, and Dr. Yahya Mahmoodkhani. I will save all of my memories about our RTA project group. I wish to thank Dr. Mary Wells for her valuable comments, encouragement and kindness, to thank Dr. Nick Parson for his valuable comments based on his unique industrial view, and to thank Dr. Yahya Mahmoodkhani for the days and nights we spent together to discuss problems and the support of his simulation work. I wish also to thank Rio Tinto Aluminum for funding the project and the materials for my research. The discussions with Dr. Chad Sinclair about some scientific questions and my project are very constructive and appreciated. The discussions with Dr. David Embury about my project and Acknowledgements  xxxiv  some interesting historical stories are very inspirational and acknowledged with my gratitude.  Furthermore, I would like to express my special gratitude to Dr. Hamid Azizi for his constant encouragement to keep me moving forward and for his suggestion about my future path. I would like to appreciate Dr. Benqiang Zhu for his help of the phase field simulation, for our memorable friendship, and for the wonderful time we spent together in Vancouver, in mountains, and on the trips to so many places. Last but not least, I am deeply grateful to my insightful parents. Without their unconditional love and their words of wisdom, which opens my heart and clears my vision, I will be a deserter on this journey.  1    1 Introduction  Aluminum alloys from the 6XXX (Al-Mg-Si-Mn-Fe) family are widely applied as structural materials in the automobile industry, e.g. crash tubes, side rails and other components in body frame. To produce the complex shapes needed for these products, extrusion processes at high temperatures are attractive due to the ability to produce high quality components at a competitive cost. The recent increase in the use of aluminum extrusions (such as in the body of the Ford 150 pickup, Jaguar XE S sedan, and Cadillac CT6 sedan) is driving the need for improved understanding of these alloys, their extrusion processes, and the mechanical properties of the extruded products. For these extruded products, the mechanical properties are very important for determining the response of a vehicle in a crash situation. Figure 1.1 shows an example of a compressed crash tube which underwent buckling and folding during axial compression.   Figure 1.1: Photograph showing a compressed crash tube which buckled and folded under a compressive loading parallel to the extrusion direction of the product (the blue arrows marked as No. 1 and 2 illustrate that the bending axis can occur at different orientations to the loading direction), and the inset showing the anodized micrograph of a microstructure of an extruded strip which underwent an extreme bending, where the surface was fractured (supplied by Dr. Nick Parson) 1   Introduction  2  Figure 1.1 is a good example of the complex-multiaxial deformation that can occur during service (e.g. crash) and is also a representation of the deformation condition during forming of parts. It illustrates two important observations. First, the bending axis can vary with respect to the extrusion direction. The studies of the folding for crash tubes reported by Kyrialkides et al. [1] and Bardi et al. [2] suggest that the anisotropic mechanical behaviour of the material can significantly affect the onset of plastic instabilities. For the prediction of cracking during folding of crash tubes, Beland et al. [3] reported that the consideration of material anisotropic mechanical response was important in the finite element (FE) method simulations in order to obtain good agreement with the experiments. Second, the nature of bending results in larger strains at the surface so that the surface microstructure is particularly relevant to the bending response. For aluminum alloys, their plastically anisotropic mechanical behaviour is dominated by their crystallographic textures [4-9] so that the through thickness differences in crystallographic texture in extrusions may be important for understanding the plastic response of the extrusion during forming or in service. Thus, it is necessary to understand: (i) the texture through the thickness of the extrusions and (ii) the relation between the texture and the anisotropic plastic behaviour. The microstructure of 6XXX series alloys contains particles at different length scales, i.e. (i) constituent particles with sizes of 0.5 - 5 μm, (ii) dispersoids with sizes of 20 - 150 nm, and (iii) Mg-Si metastable precipitates with sizes of 1 - 5 nm. However, for high temperature extrusion of 6XXX alloys, the extrusion temperatures used in industry are designed to be higher than the solvus temperatures of Mg-Si precipitates, i.e. Mg and Si are in solid solution during extrusion. Thus, in the current study, a 3XXX alloy (Al-Si-Mn-Fe) which has the same type of constituent particles and dispersoids but without the Mg-Si metastable precipitates has been chosen as a 1   Introduction  3  model system. The development of the textures and microstructures during extrusion of the 3XXX alloys should not be significantly affected by the absence of Mg-Si metastable precipitates. Furthermore, to understand the relationship between different textures and their relevant anisotropic plastic behaviours, the absence of Mg-Si metastable precipitates removes the complication of precipitation strengthening.  In comparison with 6XXX alloys, 3XXX alloys can produce an even higher density of dispersoids during a low temperature homogenization heat treatments (e.g. at 375 °C [10]), and this will be taken advantage of producing a limiting condition where recrystallization is almost fully inhibited during high temperature extrusion. On the other hand, the use of a high temperature homogenization practice (e.g. at 600 °C) can produce a condition where there are almost no dispersoids [10] and recrystallization readily occurs. It is also important to note that industrial extrusion processes inherently involve complex strain paths, i.e. the simplest example being the extrusion of materials into round bars and strips, where the strain paths are axisymmetric and nearly plane strain, respectively. Moreover, in comparison with the centre of the extrusion, the strain path on the surface is also different, since the friction between the extrudate and the die leads to a large component of simple shear at the surface. To study the role of strain path, three nearly ideal deformation modes were studied: (i) axisymmetric stretching, which corresponds the centre of the round bar extrusion, (ii) plane strain deformation relevant to the centre of the strip extrusion, and (iii) simple shear deformation, which always occurs on the surface of the extrusion products. In summary, the industrial interest is to optimize the material design for crash tubes and other structural components to obtain an improved plastic response, based on microstructure and texture. In this work, it is proposed to study this complex question by using a model alloy which 1   Introduction  4  has been deformed under the three simplified deformation modes at high temperature (i.e. axisymmetric stretching, plane strain deformation, and simple shear deformation). In particular, for the extruded strips, the plastic anisotropy of materials at room temperature has been characterized in terms of the uniaxial stress-strain response and the corresponding R-values for different crystallographic textures. The methodology involves a combination of experiments, industrial extrusion trials and polycrystal plasticity simulations. This work is part of a collaborative project between Rio Tinto Aluminum, The University of British Columbia and the University of Waterloo. In this project, Kubiak [11], Geng [12], and Du et al. [13-15] studied the effect of homogenization heat treatments on microstructures for 3XXX alloys and Babaghorbani [16] studied the effect of homogenization heat treatments on mechanical properties for 3XXX alloys. Mahmoodkhani [17] modelled the axisymmetric extrusion process for 3XXX alloys and Grajales [18] studied the as-extruded microstructures for 3XXX alloys (in axisymmetric extrusions). The goal of this work is to provide an improved model which can be used in the finite element model (FEM) simulations of the buckling and folding that occurs during the deformation of a crush tube. The constitutive model should be directly related to the microstructures and textures in extrusions (which is not homogeneous in the profile). This thesis is organized as follows: Chapter 2 will provide a review of the current knowledge available in the literature. In Chapter 3, the scope and objective of this work will be defined. Chapter 4 will introduce the methodology used for the experimental and simulation work. In Chapter 5, the textures and microstructures of the two homogenization conditions will be introduced. In Chapter 6, the textures and microstructures formed in the three nearly ideal deformation modes examined in this study will be reported. Furthermore, the visco-plastic self-1   Introduction  5  consistent (VPSC) model used to simulate the deformation texture formed in the three deformation conditions will be presented, and in addition, the recrystallization mechanisms for each deformation conditions will be discussed. In Chapter 7, the textures and microstructures from the centre to the surface of the extruded products in the round bar extrusions and strip extrusions will be reported and discussed in detail. In Chapter 8, the mechanical behaviour of the round bar extrudates and the anisotropic mechanical behaviour of the as-extruded strips will be presented, and the anisotropic R-values examined in the as-extruded strips will be also discussed. In addition, the VPSC model used to simulate the effect of textures on the mechanical and the anisotropic mechanical behaviour will be discussed. In Chapter 9, a summary will be presented with an outlook into future work.  6    2 Literature review  2.1 Introduction In this chapter, the microstructures of 3XXX aluminum alloys which are formed after direct chill casting and different homogenization heat treatments will be briefly reviewed. Then, the key work regarding crystallographic texture and microstructure of unrecrystallized (or as-deformed) and recrystallized aluminum alloys formed by axisymmetric, plane strain, and simple shear deformation will be presented. Next, the anisotropic mechanical response of samples formed by rolling (i.e. a plane strain deformation) will be introduced. Finally, two types of models will be reviewed, i.e. (i) models which are used to simulate the deformation texture based on the different strain paths and the corresponding texture dependent plastic response, and (ii) models which are used to simulate microstructure evolution during grain coarsening.    2.2 The as-cast microstructure of 3XXX alloys After direct chill casting (DC) of 3XXX alloys, the as-cast microstructure consists of the primary aluminum dendrites (i.e. matrix) and the interdendritic eutectic phases. The interdendritic secondary eutectic phases are a mixture of Al6(Mn,Fe) and α-Al(Mn,Fe)Si particles with a size of a few microns [12, 19-22] which are shown as the white rod and plate shaped particles in backscatter electron (BSE) image of Figure 2.1a [21]. These particles are referred to as constituent particles. 2    Literature review  7  Li et al. [21] reported that the area fraction of the constituent particles in the as-cast microstructure of a 3003 alloy is ≈ 2.9 %, and most constituent particles were identified as the Al6(Mn,Fe) phase. At a higher magnification (see Figure 2.1b [22]), it can be seen that Al6(Mn,Fe) and α-Al(Mn,Fe)Si constituent particles can be distinguished by their different contrast, i.e. the α-Al(Mn,Fe)Si phase is brighter than Al6(Mn,Fe) due to its higher average atomic number. Dehmas et al. [22] claimed that the duplex phased particle, such as the one shown in Figure 2.1b, was formed by the precipitation of the α-Al(Mn,Fe)Si phase on the Al6(Mn,Fe) intermetallics during solidification.   Figure 2.1: (a) backscatter electron (BSE) image of as-cast microstructure of 3003 alloy (Al-1.15Mn-0.58Fe-0.20Si-0.08Cu in wt.%) [21], and (b) BSE image of the microstructure of an 3003 alloy (Al-1.18Mn-0.58Fe-0.20Si-0.08Cu in wt.%) after 1 h heat treatment at 500 ˚C (the applied heating rate is 20 ˚C/s) [22]  The ratio of Fe/Mn is not a constant value in the α-Al(Mn,Fe)Si and Al6(Mn,Fe) phases, i.e. Mn and Fe elements can substitute for each other in the Al6(Mn,Fe) and α-Al(Mn,Fe)Si structures. For the α-Al(Mn,Fe)Si phase, an increase of the Fe/Mn ratio will change the crystal structure from the simple cubic (SC) to the body-centred cubic (BCC) structure [23, 24]. While in the Al6(Mn,Fe) phase, the change of the Fe/Mn ratio only slightly changes the lattice 2    Literature review  8  parameters, and the crystal structure remains the orthorhombic [25, 26]. The details of the crystal structures of the Al6(Mn,Fe) and α-Al(Mn,Fe)Si phases are summarized in Table 2.1. Finally, after DC casting, the primary dendrites are aluminum supersaturated with Mn [14, 15, 22]. Table 2.1: Crystal structure of Al6(Mn,Fe) and α-Al(Mn,Fe)Si phases Name Crystal structure Space group Cell dimensions (nm) Reference Symbol Constitution a b c α Particle α-AlMnSi sc Pm3 1.265 - - [23]  α-AlFeSi bcc Im3 1.256 - - [24]  α-Al(Mn,Fe)Si bcc/sc  1.256-1.265 - -  Al6(Mn,Fe) Al6Mn orthorhombic Cmcm 0.650 0.755 0.887 [25]  Al6Fe orthorhombic Cmcm 0.646 0.744 0.877 [26]   2.3 The change of microstructures during the homogenizations of 3XXX alloys The purpose of the homogenization heat treatment is to reduce microsegregation and prepare the materials for extrusion. In general, two major microstructural changes take place during the homogenization heat treatments of 3XXX alloys. First, the size, shape and crystal structure of the constituent particles can change. Second, the precipitation, growth and coarsening of 20 - 150 nm sized α-Al(Mn,Fe)Si particles [10, 27] occurs. These α-Al(Mn,Fe)Si particles with sizes of 20 - 150 nm are referred to as dispersoids. Different homogenization heat treatments can result in very different number density of dispersoids. For example, a high density of dispersoids (3.6  1020 m-3) was characterized by Li and co-workers after the homogenization at 375 °C for 24 h in an AA3003 alloy [10], as shown in Figure 2.2. On the other hand, for the homogenization at higher temperatures, e.g. at 600 °C for 24 h, Li and co-workers reported that almost no dispersoids were found using TEM [10]. The reason for the dissolution of the dispersoids during the high temperature homogenization is due to the diffusion of Mn from the dispersoids to the 2    Literature review  9  constituent particles. The diffusion of Mn and dispersoid precipitation was studied in detail by Dehmas et al. [22] and Du et al [13-15]. These two homogenization conditions, i.e. the one with a high density of dispersoids (with the homogenization at 375 °C for 24 h) and the one with almost no dispersoids (with the homogenization at 600 °C for 24 h), will be used as the two initial conditions for the current study.  Figure 2.2: TEM image of dispersoids in an AA3003 alloy (Al-1.15Mn-0.58Fe-0.20Si-0.08Cu in wt.%) homogenized at 375 ˚C for 24 h [10]   2.4 Work hardening, the as-deformed state and recrystallization The following will start with a brief overview of work hardening, the as-deformed state (i.e. unrecrystallized state) and possible recrystallization mechanisms. After this introduction, a more detailed review will be given for three different deformation modes relevant to the current study, i.e. axisymmetric extension; plane strain deformation and simple shear deformation.   2.4.1 Work hardening and the as-deformed state     The plastic deformation of metals is one of the most complex problems in physical / 2    Literature review  10  mechanical metallurgy, and a vast literature exists on this topic. According to Humphreys and Hatherly, after the research of recent 50 years on this subject “our knowledge of these matters is still imperfect [28]”. Although the current work is concerned with large strain deformation (i.e. true strains > 4) at high strain rates (i.e. 1 - 100 s-1) of aluminum alloys at deformation temperatures of ≈ 60 - 70 % of the melting point (300 - 400 °C), our knowledge is still generally qualitative for even these restrictive conditions. In the following, a framework for work hardening and the as-deformed state will be reviewed.   2.4.1.1 The phenomenology of work hardening Plastic deformation in aluminum under the conditions of the interest to the present study occurs by dislocation motion (i.e. slip plane of {111} and Burgers vector of ½ <011> [29]). In the beginning of the plastic deformation (i.e. stage II), the dislocation density strongly increases by the dislocation self-reproduction mechanism, e.g. the Frank-Read source [30]. As deformation proceeds, dislocations will rearrange and annihilate each other during deformation, which is referred to as dynamic recovery (stage III), and there will be a competition between the accumulation and the loss of dislocations, shown as follows: disd d d      (2.1) where dρdis is the dislocation density change during deformation, and dρ+ is the increase of dislocation density due to dislocation accumulation and dρ- is the decrease of dislocation density due to the dynamic recovery. In the simple framework of Kocks and Mecking [31, 32], the evolution of dislocation density with respect to strain can be written as follows: 2    Literature review  11   121 2disd k kd    (2.2) where Γ is the accumulated shear strain of the material, and k1 and k2 are the linear factor for the term of dislocation accumulation (ρ+, the first term on the right side) and that of dynamic recovery (ρ-, the second term on the right side), respectively. It should be mentioned here that the dislocation accumulation term is generally athermal and strain rate independent [32]. In contrast, the term describing dynamic recovery is dependent on the deformation temperature, strain rate and the solid solution level [32]. The overall flow stress of the material is governed by the dislocation interactions on non-parallel planes (i.e. mobile dislocations which intersect sessile dislocations on non-parallel slip planes which are known as forest dislocations). The critical resolved shear stress (CRSS) for this hardening mechanism is given by [33]: 12sdisGb    (2.3) where τs is the CRSS at the slip system level, G is the shear modulus, b is the magnitude of Burgers vector, and α is a constant. To integrate Equation 2.2 and substitute into Equation 2.3 give: 111 expss s os           (2.4) where: 112s k Gbk   (2.5) 102s k Gb   (2.6) Equation 2.4 is known as the Voce equation and it is often modified to include a friction stress 2    Literature review  12  and to empirically account for a small but non-zero hardening rate, i.e. θs1, in some cases at large strains (also known as Stage IV) [34], i.e.:   00 1 111-exp -ss s s ss                (2.7) where τs0, θs0, θs1, (τs0+τs1) are the initial CRSS, the initial hardening rate, the asymptotic hardening rate and the back-extrapolated CRSS, respectively for certain slip system. In Equation 2.7, the athermal friction stress, τs0, has been added and the possibility of constant large strain work hardening rate θs1 has also been included in this extended Voce law [34].      The calculation of the response of polycrystal can be determined by either using an average Taylor factor or a polycrystal plasticity model. To use the average Taylor factor, the true stress, σt, and plastic true strain, εt, can be calculated as follows: st M   (2.8) tM  (2.9) where M is Taylor factor, which is 3.06 for a random oriented set of grains in a face-centred cubic (FCC) material. Polycrystal plasticity models, which will be described in detail in Sections 2.10, can be used to account for a set of grains with non-random crystallographic texture. It should be mentioned here that the Taylor factor, M, depends on the crystallographic texture and could vary during the deformation as the texture evolves (the strain path also plays a critical role in the texture development and this will be discussed in Sections 2.4.2.1 - 2.4.2.3).  For high temperature (i.e. ≈ 60 - 70 % of the melting point, 300 - 400 °C) deformation of aluminum and its alloys, the experimental observations reported by Puchi et al. [35] suggest that when the strain increases, (i) the asymptotic hardening rate θs1 is close to zero (shown in Figure 2    Literature review  13  2.3 [35]), and (ii) a steady state flow stress is achieved, which denotes that the rate of dislocation accumulation equals that of dynamic recovery (shown in Figure 2.3 [35]).  Figure 2.3: Stress-strain curves for Al-1Mg (in wt.%) at 400 °C [35]  When there are non-shearable second phase particles in the alloy system, the effect of these particles on the evolution of dislocation density with respect to strain can be accounted for by adding an extra dislocation accumulation term [36]:  121 2 3disd k k kd     (2.10)  where k3 is the extra term to account for the accumulation of dislocations around non-shearable particles. In general, for the deformation at high temperature, since the dynamic recovery rate is very high [28] (i.e. k2 is normally very large) so that the dislocation accumulation due to second phase particles (i.e. the term of k3) can be neglected.   2.4.1.2 The as-deformed state The as-deformed state can be characterized by two basic changes in the microstructure of the alloy. First, the grain shape and the corresponding surface area of grain boundary per unit 2    Literature review  14  volume are modified. The grain shape change and the change of grain boundary area per unit volume are dependent on the strain path [37]. In detail, Bate et al. [37] showed that deformation with the same equivalent strain but with different strain paths can result in different grain shapes and, therefore, different surface areas of grain boundaries as shown in Figure 2.4.  Figure 2.4: Graphs of the surface areas, normalized by the initial spherical surface, of ellipsoids distorted by deformation modes as a function of the von Mises equivalent total strain [37]  Second, the dislocation substructure within the grains is also changed during the deformation. The dislocation substructure can range widely from a nearly random arrangement of individual dislocation line segments (i.e. dislocation tangles, which are often observed in the deformation at room temperature, shown in Figure 2.5a) to subgrains where the dislocations have rearranged themselves during deformation to form walls with a high density of dislocations (i.e. low angle grain boundaries, LAGBs) which surround the interior of the subgrain which has a lower dislocation density (for example, see Figure 2.5b). 2    Literature review  15     Figure 2.5: TEM micrographs revealing dislocation structures after deformation at (a) room temperature, where dislocation tangles are observed, and (b) at 350 °C, where an equiaxed subgrain structure is observed (Al-1Mg, in wt.%, which was deformed with strain rate of 1 s-1 to a final strain 1.1) [38]  Furthermore, there is another critical change occurring during the deformation and it strongly depends on the strain path, which is crystallographic texture. The crystallographic texture, or simply texture, describes the distribution of the preferred crystallographic orientations, or simply orientations, of the grains in the polycrystal [39]. A detailed review of the different textures according to different strain paths (i.e. axisymmetric extension, plane strain deformation and simple shear deformation) will be presented in Sections 2.5 - 2.7.   2.4.1.3 Effect of second phase particles on the as-deformed state When large second phase particles (a few microns, such as constituent particles in AA3XXX alloys) are introduced into aluminum alloys, Jazaeri et al. [40] stated that the particles perturb the flow of the deformed matrix and subsequently cause a locally random texture around them in the deformation at room temperature. For the small particles (normally less than 100 nm, such as dispersoids), Orowan loops can be formed around such small size particles during the deformation. In deformation at room temperature, Apps et al. [41] observed that an addition of 2    Literature review  16  dispersoids in the Al-0.2Sc alloy (in wt.%) hinders the formation of subgrains and forms dislocation tangles in the material. For the deformation at high temperatures and with a large strain, both the elevated temperature and the large strain can accelerate the rate of dynamic recovery [28], which can help the material form subgrains during the deformation. Therefore, in contrast to the case of deformation at room temperature, even when the material has second phase particles, such as constituent particles and dispersoids, equiaxed subgrains were often observed in the deformation at high temperature [38, 42-44]. It should be mentioned here that when small second phase particles are present, these may prevent local migration of the high angle boundaries in the deformation at high temperatures so that the high angle grain boundaries remain planar as shown in Figure 2.6 [28].  Figure 2.6: Backscatter electron (BSE) image showing high angle grain boundaries in aluminum alloys deformed at 400 °C in plane strain compression: (a) Al-5Mg (in wt.%), showing serrations at the high angle boundaries, (b) Al-2Cu (in wt.%), in which precipitates prevent boundary serrations from developing.[28]   2.4.1.4 The stored energy during deformation Most work expended in deforming a metal is converted to heat and only a very small amount (i.e. ≈ 1 %) remains as energy stored in the material [28]. However, even though the amount of 2    Literature review  17  stored energy is very small, it provides the source for all the property changes in deformed metals, such as static recovery, recrystallization and grain growth. For deformation at high temperatures, almost all of the stored energy is derived from the accumulation of dislocations (which has been described in Section 2.4.1.1) and the increase of grain boundary area per unit volume (which has been described in Section 2.4.1.2 ) [28]. It is worth noting that the increase in boundary area is an important factor in promoting continuous recrystallization during or after large strain deformation, and this will be discussed in detail in Section 2.4.2.   2.4.2 Recrystallization during and after deformation As mentioned in the previous section, the stored energy from the deformation can provide the driving for recrystallization after deformation, i.e. static recrystallization. Alternatively, recrystallization can occur simultaneously in the deformation, which is referred to as dynamic recrystallization. In general, recrystallization involves the formation of new strain-free grains and the subsequent growth of these to consume the deformed microstructure [28]. Static recrystallization, usually occurs during a heat treatment after deformation, is characterized by the formation of low dislocation density regions (one mechanism is by grain boundary bulging [29]), and the growth of these regions to consume high dislocation volumes by grain boundary migration [29]. In the case of dynamic recrystallization in aluminum alloys, two mechanisms have been proposed, i.e. continuous dynamic recrystallization (CDRX) [45, 46] and geometric dynamic recrystallization (GDRX) [47, 48]. For continuous dynamic recrystallization, it is characterized as the transformation of subgrains to grains, or subgrain coarsening process by grain boundary migration where misorientations are accumulated to transfer low angle grain 2    Literature review  18  boundaries (LAGBs) to high angle grain boundaries (HAGBs). On the other hand, for geometric dynamic recrystallization, it is characterized as the grain fragmentation, i.e. the phenomenon where elongated grains are pinched off to form HAGBs.   2.4.3 The role of second phase particles on recrystallization In the recrystallization process, second phase particles can either stimulate nuclei to promote recrystallization, which is referred to as the particle stimulated nucleation (PSN, which is often observed in the heat treatment after the deformation at room temperature [49, 50]), or on the contrary, can inhibit the mobility of grain boundaries and, therefore, slow down or even stop the recrystallization behaviour (especially in high temperature deformation with a large strain), which is referred to as the particle pinning or the Zener drag effect.  The phenomenon of the PSN in aluminum alloys was studied by in-situ TEM observations of Humphreys [49] and Bay et al. [50]. They concluded that: (i) the phenomenon of PSN is usually observed in the cases of the particles with the diameters larger than approximately 1 μm; (ii) the orientations of nuclei produced by PSN is generally different from those produced by other recrystallization mechanisms; (iii) because the interaction of dislocations and particles is temperature dependent, PSN only occurs if the deformation is carried out below a critical temperature or above a critical strain rate; (iv) the formation of nuclei is not necessary on the particle surface, but within the deformation zone which surrounds particles; and (v) the nucleus may stop growing when the deformation zone is consumed. One example of PSN in the annealing at 240 ˚C of a 60% cold rolled Al-0.45Cu-0.80Si (in wt.%) alloy is shown by an in-situ TEM observation in Figure 2.7a, which shows a deformed microstructure including a particle, 2    Literature review  19  and in Figure 2.7b [49], where recrystallization has consumed the deformation zone.  Figure 2.7: TEM observations of PSN in annealing of a 60% cold rolled Al-0.45Cu-0.80Si (in wt.%) alloy at 240 ˚C with 0 s (as deformed microstructure), and with a few seconds [28]  According to these characteristics of PSN, an important industrial application is to control the grain sizes of the recrystallized alloys, and in particular, to produce the fine-grained materials [51]. On the other hand, PSN can change the recrystallized textures due to the new orientations produced by the nucleus. In the single crystals, Humphreys [49] observed that the orientation of nuclei is rotated from the deformation matrix by 30 to 40 ° about the <112> axis. In the case of polycrystals, Vatne et al. [52] and Engler et al. [53, 54] reported that PSN typically results in the cubeND orientation, i.e. {001} <310>, and the P orientation, i.e. {011} <122>. The other effect of second phase particles is to cause the pinning pressure on the grain boundaries, which is referred to as the Zener drag. In recrystallization, both nucleation and grain growth need a sufficiently high mobility of grain boundaries. Thus, the second phase particles, which are normally with small sizes such as dispersoids, can inhibit the recrystallization process. A simple estimation of the Smith-Zener drag can be made using the Zener equation: 2    Literature review  20  32Zener GBPfPR   (2.11) [28] where γGB is the grain boundary energy; Rp and f are the radius and the volume fraction of the second phase particles, respectively. The ratio of f / Rp is referred to as the degree of dispersion.   2.5 Textures and microstructures formed by axisymmetric deformation 2.5.1 Deformation textures and microstructures formed by axisymmetric deformation For axisymmetric deformation, e.g. uniaxial extension or the extrusion of axisymmetric products, a double fibre texture is usually observed for aluminum alloys [55-62], i.e. the <001> and <111> crystallographic orientations are parallel with the extension axis. However, Harsha et al. [57] found that the deformation texture depends on the initial texture. Considering an inverse pole figure (IPF) with respect to the extension axis, the grains with initial orientations in the area between <001>, <113> and <011> will be rotated into <001> after the extension, while those between <110>, <111> and <113> will be rotated to <111> [57]. The schematic illustration of such lattice rotation during axisymmetric extension is shown in an inverse pole figure (IPF) of Figure 2.8, where the distributions of crystal orientations are plotted with respect to the extension axis and the arrows denote the lattice rotation directions during the axisymmetric extension [57]. In detail, a higher intensity (which indicates a larger volume fraction) of the texture component of the <111> orientation aligned with the extension direction (i.e. <111> fibre) was reported in the high temperature extrusions of the aluminum alloys [58, 62], where the ratio of the texture intensity of the <111> orientation aligned with the extension direction to that of the <001> orientation aligned with the extension direction (i.e. <001> fibre) can vary in a range from 3 to 6 [58, 62].  2    Literature review  21   Figure 2.8: The schematic illustration of such lattice rotation during axisymmetric deformation, where the distributions of crystal orientations are plotted with respect to extension axis and the arrows denote the lattice rotation directions [57]  In terms of the as-deformed microstructures (e.g. dislocation density, subgrain shape and size etc.), there is some evidence that the substructure of grains with different orientations can be different. For example, Huang et al. [61] reported that grains in high purity aluminum deformed at room temperature having their <111> directions aligned with the tensile axis were characterized by dense dislocation walls and microbands. On the other hand, grains having their <001> directions aligned with the extension direction were characterized by an equiaxed cell structure. It should be mentioned that based on the best knowledge from the author, there is no work found in literature studying the effect of temperature, strain and strain rate on the fractions of the individual texture components for axisymmetric deformation.   2.5.2 Recrystallization textures and the microstructures formed by axisymmetric deformation As described in the previous section, the axisymmetric extension of FCC metals produces the <001> - <111> double fibre texture. When recrystallization takes places during deformation or 2    Literature review  22  after deformation and before quenching, previous studies [55, 56] reported that recrystallization results in a decrease in the intensity of the <111> fibre and an increase in <001> fibre intensively. However, there is relatively little literature on the recrystallized textures and microstructures after axisymmetric deformation.   2.6 Textures and microstructures formed by plane strain deformation 2.6.1 Deformation textures and microstructures formed by plane strain deformation Rolling, as a typical plane strain deformation, is widely studied by previous researchers for more than half century. For most aluminum alloys, the hot rolling textures are characterized as the copper type texture [63-79], which is a typical deformation texture for the materials with high stacking fault energies in the plane strain deformation. This texture is described in terms of three main texture components, i.e. the copper orientation {112} <111>, the S orientation {123} <634> and the brass orientation {011} <211>, where the crystal plane {hkl} represents the plane parallel to the rolling plane, which is the plane aligned with the rolling and transverse directions of the sample, and the crystal orientation <uvw> represents the direction aligned with the rolling direction. The Miller indices and Euler angles of the copper, S, and brass orientations are summarized in Table 2.2. Table 2.2: Miller indices and Euler angles of the copper, S and brass orientations Designation Miller indices Euler angles {hkl}<uvw> φ1Φφ2 copper {112} <111> 90˚ 30˚ 45˚ S {123} <634> 59˚ 34˚ 65˚ brass {011} <211> 35˚ 45˚ 0˚/90˚  2    Literature review  23  One example of the as-deformed plane strain texture formed in a hot rolled Al-1Mn-1Mg (in wt.%) alloy at 330 °C is shown in Figure 2.9a, which was reported by Daaland et al. [65]. The texture is shown in a projection of the 3-D Euler space onto a series of planes normal to the φ2 axis, which is referred to as the orientation distribution function (ODF) figure, as shown in Figure 2.9a.   Figure 2.9: (a) orientation distribution function (ODF) figure showing the hot rolled texture of Al-1Mn-1Mg (wt.%) [65] and (b) schematic representation of texture fibre [39]  The as-deformed plane strain texture shown in Figure 2.9a can be described in detail as a spread of the orientations from the copper orientation through the S orientation and finally to the brass orientation. In other words, the texture can be described by a continuous orientation tube running through Euler space, as schematically represented in the 3-D Euler space in Figure 2.9b. This orientation spread can be also described in terms of the texture fibres, e.g. the α fibre, which runs from the Goss orientation {011} <100> to the brass orientation (shown in Figure 2.9b), and the β fibre, which runs from the brass orientation through the S orientation and finally to the 2    Literature review  24  copper orientation. For the hot rolling with a large thickness reduction of the aluminum alloys, the β fibre is dominant and the α fibre has a much weaker texture intensity, as shown in Figure 2.9a. For the hot rolled microstructures of Al alloys, Ren et al. [64] and Vante et al. [67] reported that the as-deformed microstructure shows elongated band-structure along the rolling direction, in which equiaxed subgrains can be observed. A detailed study about the different microstructures related with the different texture components (i.e. the cube, Goss, S, copper and brass orientations, and in addition, a deviation angle of 15 ° was used to define the certain texture components in this study) in the hot rolling of AA1050 (Al-0.3Fe-0.1Si, in wt.%) at 350 °C was reported by Samajdar et al. [73]. Two critical observations were claimed by the authors: first, a long range lattice rotation [80] was only observed in the band-structure showing the cube orientation (i.e. cube-band microstructure), second, the subgrains showing the cube orientation have a larger size and a lower interior average misorientations, compared with the subgrains showing other orientations (a similar observation was reported in a cold rolled aluminum alloy by Delannay et al. [81]). One example of the cube-band microstructure of AA3004 alloy (Al-1.2Mg-0.9Mn, in wt.%) rolled at 330 °C is shown in Figure 2.10 [67]  Figure 2.10: TEM micrograph of a cube-band in the as-deformed condition of an AA3004 (Al-1.2Mg-0.9Mn, in wt.%) rolled at 330 °C [67] 2    Literature review  25  2.6.2 Recrystallization textures and microstructures formed by plane strain deformation In the subsequent annealing of the hot rolled aluminum alloys, equiaxed grains will form [64, 76, 78], and the texture will change from the deformation texture (i.e. the copper type texture) to the texture showing a transition from the cube orientation {001}<100> with a strong texture intensity to the Goss orientation [65-67, 73, 76]. One example of the recrystallized texture of the Al-1Mn-1Mg (in wt.%) alloy (which was hot rolled at 330 °C and subsequently annealed at 330 °C) is shown in Figure 2.11a [65]. In the study of the texture and microstructure evolution during the annealing after the hot rolling of this alloy, Daaland et al. [66] reported that during the annealing (i.e. at 330 °C), the subgrains showing the cube orientation grow faster than other subgrains, shown in Figure 2.11b. Furthermore, Alvi et al. [76] reported that during the annealing (i.e. at 325 °C) of the hot rolled AA1050 alloy (i.e. at 325 °C), the volume fraction of grains showing the cube orientation increases and finally the cube orientation becomes the dominant texture, shown in Figure 2.11c. According to the previous studies [65-67, 73, 76], the formation of the strong cube texture component during the annealing process is attributed to the recrystallization from the cube-bands formed in the as-deformed microstructure. As mentioned in Section 2.6.1, the cube-bands have a large subgrains size, a lower interior average misorientation, and a long range lattice rotation [80]. Therefore, the cube-bands have the growth advantage during the annealing process. In addition, the texture components reported in the recrystallized texture are summarized in Table 2.3. 2    Literature review  26   Figure 2.11: (a) orientation distribution function (ODF) figure showing the recrystallized texture of Al-1Mn-1Mg (in wt.%) which was hot rolled and annealed at 330 °C [65], (b) evolution of the average subgrain sizes of the cube-band and matrix during the annealing (i.e. at 330 °C) of hot rolled Al-1Mn-1Mg (in wt.%) at 330 °C [66], (c) evolution of the volume fraction of the grains having the different texture components during the annealing (i.e. at 325 °C) of the hot rolled AA1050 alloy (i.e. at 325 °C) [76]  Table 2.3: Miller indices and Euler angles of the texture components in the recrystallization of aluminum and its alloys Designation Miller indices Euler angles {hkl}<uvw> φ1Φφ2 cube {001} <100> 0˚ 0˚ 0˚/90˚ cubeRD {013} <100> 0˚ 22˚ 0˚/90˚ cubeND {001} <310> 22˚ 0˚ 0˚/90˚ Goss {011} <100> 0˚ 45˚ 0˚/90˚ Rg {124} <211> 53˚ 36˚ 60˚ P {011} <122> 65˚ 45˚ 0˚/90˚ Q {013} <231> 45˚ 15˚ 10˚     2    Literature review  27  2.7 Textures and microstructures formed by simple shear deformation 2.7.1 Deformation textures and microstructures formed by simple shear deformation According to previous studies about the texture variations in the torsion tests for aluminum alloys [82-89], there are 4 types of texture components which were frequently reported, i.e. A ({111} <110> / { 111 } < 110>), A* ({ 111} <112> / {111 } <112>), B ({112 } <110> / { 112 } < 110>), and C ({001} <110>) texture components. The crystal orientation <hkl> represents the direction aligned with the shear direction and the crystal plane {uvw} represents the plane parallel with the shear plane, which is the plane aligned with both the shear direction and torsion axis. The schematic illustration of these 4 types of texture components in simple shear deformation is shown in the {111} pole figures in Figure 2.12, in which the vertical and horizontal directions are parallel with the shear direction (SD) and the torsion axis (Z), respectively.  Figure 2.12: Schematic illustration of (a) A, (b) A*, (c) C, and (d) B texture components in simple shear deformation in the {111} pole figures, in which the vertical and horizontal directions are parallel with the shear direction (SD) and the torsion axis (Z), respectively [82]  2    Literature review  28  Montheillet et al. [82] reported the torsion tests with a commercially pure aluminum at 400 °C. The texture evolution with the increase of the equivalent strain can be summarized as follows: the A texture component was first observed at the equivalent strain of 0.6, and with an increase of the equivalent strain to 2, the A* texture component was observed, and with further increase of the equivalent strain to 5, the C texture component became the dominant. After the torsion test with an extremely large equivalent strain of 30, the B texture component was reported as the predominant texture. Some texture simulation studies based on the relaxed-constraints Taylor model [86] and the viscoplastic polycrystal model [87-89] confirmed that if only the strain path of simple shear is considered, the C texture component is the most stable and the dominant orientation at large shear strains. Therefore, the C texture is proposed to be the deformation texture caused by simple shear deformation. For the as-deformed microstructures, Pettersen et al. [90] reported that at the equivalent strain of 1, the large grains have been pinched off by the formed high angle grain boundaries (HAGBs, whose misorientations ≥ 15 °) and at the equivalent strain of 5, the equiaxed subgrains / grains having the C type texture were observed.    2.7.2 Recrystallization textures and microstructures formed by simple shear deformation On the other hand, the formation of the B type texture (observed at large equivalent strain, e.g. 30) is attributed to the dynamic recrystallization in the torsion tests [83, 84, 91]. In detail, the geometric dynamic recrystallization (GDRX) has been proposed for the case of high temperature 2    Literature review  29  simple shear deformation reported in the previous studies [47, 48, 84, 91-93]. Geometric dynamic recrystallization process is mainly characterized by the phenomenon of grain subdivision, i.e. during the deformation, the elongated grains are pinched off by the formed HAGBs and the thickness of the elongated grains will be finally decreased to an order of subgrains. Pettersen et al. [90] reported that similar sizes of subgrains / grains (as shown in Figure 2.13) were observed at the equivalent strain of 5 in the torsion test, where the C type texture is the dominant texture, and at the equivalent strain of 25, where the B type texture is the dominant texture. This observation suggested that the grains were continuous pinched off by the newly formed HAGBs, and thus with an increase of the equivalent strain, the subgrain / gain sizes remain almost constant. In friction stir welding (another type of the simple shear deformation with the large equivalent strains and at the high temperatures), Prangnell et al. [91] observed the microstructure refinement showing the B type texture in the welding region, and the authors suggested that the formation of the B type texture in the refinement microstructure was due to the geometric dynamic recrystallization process caused by the deformation with a large simple shear.   Figure 2.13: EBSD maps showing the microstructure of the AA6060 deformed in the torsion tests at 486 °C with a strain rate of 13 s-1 and to the final equivalent strain of (a) 5 and (b) 25 [90] 2    Literature review  30  2.8 Mechanical behaviour of materials deformed with plane strain deformation For FCC materials deformed in plane strain deformation, the anisotropic mechanical response has been widely reported [4-9, 94]. For samples after rolling, tensile tests are conducted with different directions within the rolling plane. For most of the cases, the anisotropic mechanical response is dominated by crystallographic texture [4-9]. For the example of yield stress, samples with a dominant cube texture (i.e. the recrystallized texture) have the highest yield stress in the tensile test at 45 ° to the extrusion direction [4, 8], compared with the other directions. On the other hand, for materials with the copper type texture (the deformation texture), the tensile test at 45 ° to the rolling direction has the lowest yield stress [9]. In terms of the R-value (defined as the ratio between the plastic strain along the width and thickness of the deformed sample) was also widely studied for the aluminum alloys [4-8, 94]. In addition, by the modelling simulation, Lequeu et al. [95] systematically studied the effects of the individual texture components (in the rolling of FCC materials) on the anisotropic response of the R-values and yield stresses.  For the effect of second phase particles on the anisotropic mechanical response, Bate at al. [4] systematically studied second phase particles with different sizes, i.e. sizes from hundred nanometers to a couple of microns, and with different shapes, i.e. rods, plates and spheres. Based on their work, spherical particles have little influence on the anisotropic mechanical response. In addition, when the sizes of the particle become large and their number density decreases, the effect of the particles on the anisotropic mechanical behaviours is also very small. In practical sheet forming, such as rolling, the texture distribution is not homogeneous through the thickness of the sheet, and a strong effect of shear on the surface of the sheet has often been reported in previous studies [6, 96-99]. However, even though the texture distribution through the thickness is heterogeneous, the characterization of the material anisotropies is mostly based 2    Literature review  31  on the full thickness of the sheet in the previous studies [6, 98, 99]. For the case of commercially pure aluminum rolling, Engler et al. [97] attempted to predict the R-values based on the textures from the different layers of the sheet, but so far the experimental measurement of the R-values with respect to the different layers is still lacking.    2.9 Models for the simulation of deformation In 1938, Taylor [100] suggested in his pioneering work that the plastic strain rate of all the grains within a polycrystal are the same and equal to the externally imposed macroscopic plastic strain rate. The original Taylor model, which is referred to as the full-constraints (FC) Taylor model, considers that: (i) the fields of strain rate is supposed to be homogeneous within each grain and the same with different grains; (ii) the compatibility of the strain rate is throughout the sample, but the stress equilibrium at grain boundaries is neglected; (iii) the prescribed plastic deformation must be accomplished by crystallographic slip and may also include twinning and martensitic transformation.  In the FC Taylor model, the compatibility of the strain rate is throughout the sample. However, more than 30 years ago, it was suggested that the FC model was too strict and the predictions of the deformation textures can be improved by “relaxing” the geometrical constraints [101, 102], which is referred to as the relaxed-constraints (RC) Taylor model. The basic approach of RC Taylor model is to drop some constraints (which are the shear components in the strain rate tensor) to improve the stress homogeneity.  However, neither the FC Taylor model nor RC Taylor model can provide a complete theoretical solution to the plastic deformation of the polycrystal, because in both theories only 2    Literature review  32  the strain rate compatibility is considered. A model which satisfies both strain rate and stress constraints and combined with the self-consistent approach [103] was developed by Lebensohn et al. [104], i.e. the visco-plastic self-consistent (VPSC) model. The model was inspired by the so-called Eshelby method [105]  for solving the problem of an elastic inclusion in a homogeneous medium. The VPSC model extends the approach to allow for visco-plastic materials and assumes that: (i) each grain is considered as an inclusion and the surrounding grains as a homogeneous matrix; (ii) the inclusion represents a grain which can be characterized by its orientation and the matrix reflects the average homogeneous properties of all the other grains; (iii) the strain rate and stress within the inclusion are assumed to be homogeneous; (iv) the properties of matrix are homogeneous, but the resulting strain rate and stress field are heterogeneous. This model couples the strain rate and stress in each grain (inclusion) and the average strain rate and stress in the surrounding (matrix) by the constitutive relation, which considers crystallographic slip and twinning, shown as follows:  s sij ij ijkl kl klsm D M        (2.12)  where: 0nspq pqssm        (2.13) In Equation 2.12, γ̇s is the local shear rate in a given grain, msij = ½(nsibsj + nsjbsi) is the symmetric Schmid tensor associated with the slip system s, where ns and bs are the normal and the Burgers vector for the slip system s, the term ∑ mijss γ̇s is the local deviatoric strain rate, Dij is the overall macroscopic (system) strain rate and n describes a material property which is inversely related to the strain rate sensitivity, m. Furthermore, σ’kl and Σ’kl are the local and macroscopic deviatoric stress tensors, and M̃ijkl is the interaction tensor, which indicates the local 2    Literature review  33  deviation of the strain rate with respect to the macroscopic magnitude. In Equation 2.13, ?̇?0 is the normalization factor and τs is the critical resolved shear stress (CRSS). The evolution of the CRSS, τs, with the accumulated shear strain in a given grain can be described by the extended Voce law [34] of Equation 2.7. Returning to Equation 2.12, the interaction tensor M̃ is given by:    1 1sec: : : :effeff tgnM n I S S M I S S Mn        (2.14) where: 1sec 0ns s sij kl pq pqs ssm m mM        (2.15) sectgM nM  (2.16) where neff is the homogenization parameter, Msec is the interaction tensor based on the secant approximation [106] (i.e. a condition close to the Taylor condition that all grains tend to be deformed with an equal strain rate), and Mtg is interaction tensor based on the tangent approximation [104] (i.e. a condition tending to uniform stress state or the Sachs condition). By tuning the homogenization parameter neff in a range from 1 to n (n is the inverse of the strain rate sensitivity), the VPSC model can be modified from the Taylor condition (neff = 1) to the Sachs condition (neff = n).  The inputs for the model are (i) a set of grains of known crystal structure, shape and crystallographic orientations (i.e. the starting texture), (ii) a material constitutive law, (iii) the deformation strain path as defined by the velocity gradient tensor (note: this may be a constant or varying), and (iv) a homogenization parameter, neff. In the current study, this model will be used to simulate deformation textures under different scenarios. The details of the model implementation will be described in Chapter 4. 2    Literature review  34  Recently, more factors, such as grain aspect ratio and the grain fragmentation [107] were considered in the visco-plastic self-consistent model. In addition, many researchers have developed crystal plasticity finite element model, which combines a finite element formulation and a viscoplastic constitutive response in the crystal plasticity theory. Such approach (see Beaudoin et al. [108] for one example) is used to simulate localized orientation gradients in a single grain or in polycrystals. Furthermore, there are other crystal plasticity models for polycrystalline which are used to simulate the texture evolution during the deformation, i.e. the advanced “Lamel” model [102, 109] from the group in Leuven and the grain interaction (GIA) model [110, 111] from the group in Aachen. Recently, Van Houtte et al. [109], Zhang et al. [8], and Holmedal et al. [112] compared the simulated textures based on the different models with the experimental textures. It is concluded that for different conditions, the different model has the certain advantages, and on the other hand, the validation for each model is also very important [109].   2.10 Models for the simulation of the microstructure evolution during recrystallization As described in Section 2.4.2, recrystallization is a very complex phenomenon, and it normally includes formation of low dislocation density nuclei which is followed by the motion of high mobility boundaries. For the simulation of the microstructure evolution during grain growth, in 1984 Anderson et al. [113] and Srolovitz et al. [114] developed the Monte Carlo (MC) methods for the simulation of grain growth in 2-D. Later, with the consideration of the different nucleation scenarios, the approach has been also applied to study the primary recrystallization [115-117], abnormal grain growth [118] and dynamic recrystallization [119, 120]. In Monte 2    Literature review  35  Carlo models, the material is divided into a number of blocks, which can represent the microstructures such as grains or subgrains. Each block consists of a number of points which have the same crystal orientation and other physical properties so that there is no subsidiary microstructure inside each block and no thickness for the grain or subgrain boundaries. The model is run by changing the orientation or other physical properties of the considered point to that of its neighbours as follows: expA BEpkT      (2.17) where pAB is the probability of the change from the considered point, A, to its neighbours, B; ∆E is the energy change caused by the microstructure change; T is temperature; and k is Boltzmann constant. If the energy change (∆E) is less than or equal to zero, the change is accepted. However, if ∆E is greater than zero, then the change is accepted with a probability as shown in Equation 2.17. Another model applied for the simulation of grain growth is the Vertex-dynamics model [121-124], which describes the boundary migration by tracking the motion of the triple or quadruple junctions of the boundaries. Furthermore, the interfacial dynamics are governed by the balance between the reduction of the interfacial energy and the dragging pressure on the interface. In the early stage of the Vertex-dynamics model, Kawasaki et al. [121] considered the boundaries as straight edges in 2-D. More currently, Weygand et al. [124] developed the Kawasaki model to a model which can describe an arbitrary boundary morphology by the additionally tracking a series of “virtual vertices” along the boundaries. In general, the Vertex-dynamics model is a more economical method to simulate the grain growth where the microstructure which can be realistically simulated by an array of cells. Finally, Chen [125] and Krill III et al. [126] used the phase field approach to simulate the 2    Literature review  36  grain growth behaviour. The phase field model has been first developed to predict the equilibrium and non-equilibrium phase transformation phenomena, which include the liquid-solid [127, 128] and solid-solid [129-131] types. Similar to the Monte Carlo model, the microstructure is discretized, but in the phase field model, the interface between different phases or grains has a width associate with it. Therefore, the phase field model cannot be applied to sharp interface cases, where the interface thickness is atomic. Microstructure evolution in one commonly used phase field code [130, 132] involves numerically solving the following:  22 222iij ij j i i j i j i j drii j ij ijGt                               (2.18) where ϕi (ϕi) represents the phase field parameter for the grain i (j); μij is the interface mobility between grain i and j; γij and ηij are the interface energy and thickness, and ∆Gdri is the interface driving pressure, respectively. Recently, some simulations [133-135] using the phase field approach have been applied to study microstructure evolution in the grain growth process by considering the boundary energy and mobility as a function of its misorientation angle between adjacent grains using the Reed Shockley [136] and Huang-Humphreys [137] approximations, respectively. For the case of LAGBs (i.e. their misorientations < 15 °), the interfacial energy and mobility as the function of the boundary misorientation angle are defined as follows: 15 1 ln15 15g g         (2.19) 415 1 exp 515g                 (2.20) where γ and μ are the interfacial energy and mobility, and γ15° and μ15° represent the interface 2    Literature review  37  energy and mobility when the misorientation θg is equal to 15 °. For HAGBs (i.e. their misorientation ≥ 15 °), their interfacial energy and mobility were assumed to be the same with the case when the misorientation is equal to 15 ° according to Equations 2.19 and 2.20. Ma et al. [135] suggested that the spatial distribution of different texture components can strongly change the texture evolution during grain growth. In this current study, the phase field model will be also used to simulate the microstructure evolution during the grain growth process. The details of the phase field model will be described in Chapter 4.   38    3 Scope and objective      The AA3003 alloy studied in this work is an industry alloy but also a model of 6XXX alloys, i.e. it has similar constituent particles and dispersoids but no Mg-Si nanometer sized strengthening phases. Two extreme conditions of homogenization heat treatments were used as the initial conditions, i.e. the one with a high density of dispersoids (homogenization at 375 °C for 24 h) and the one with almost no dispersoids (homogenization at 600 °C for 24 h). To simplify the industrial extrusion process which often has a complex strain path, three nearly ideal deformation conditions were studied: (i) axisymmetric extension, (ii) plane strain deformation, and (iii) simple shear deformation. To study the mechanical response of the extrusions, a set of tensile tests at room temperature with the different tensile axes with respect to the extrusion direction was studied. The effect of texture difference between the surface layer and the interior of the extruded product was studied using the two types of samples, i.e. one where the surface layer was removed and one with the full thickness of the extruded product. To characterize the textures and microstructures, electron backscatter diffraction (EBSD) is the major technique used in this work. The overall goal of this work is to provide an improved model which can be used in the finite element model (FEM) simulations of the buckling and folding that occurs during the deformation of a crush tube. The constitutive model should be directly related to the microstructures and textures in extrusions (which is not homogeneous in the profile). Therefore, the main objective of this work is designed to characterize and rationalize the textures and microstructures through the 3    Scope and objective  39  thickness of the extruded round bars and strips and their relevant anisotropic mechanical response. In order to achieve this objective, following sub-objectives have been identified: (i) to characterize the as-extruded textures and microstructures from the centre to the surface of the extruded products for the axisymmetric extrusions and strip extrusions for the two bounding homogenization conditions, i.e. the condition with a high density of dispersoids and the condition with almost no dispersoids (ii) to characterize the evolution of the textures and microstructures with different equivalent strains in torsion tests for the two homogenization conditions  (iii) to simulate the deformation textures of the three deformation modes using the visco-plastic self-consistent (VPSC) model (iv) to characterize the recrystallized state for the three deformation paths and attempt to rationalize how the recrystallization texture is related to the as-deformed state (v) to characterize the room temperature stress-strain response of the as-extruded stip using tensile tests conducted at different orientations to the extrusion direction (vi) to link the texture and to the relevant anisotropic mechanical response using VPSC   40    4 Methodology  4.1 Introduction     In this chapter, the experimental and simulation methods used will be described. The basic characteristics of the starting material (e.g. chemistry, grain structure and providence) will first be introduced in Section 4.2. Following this, the details of homogenization treatments, extrusion trials, torsion tests and high temperature compression tests will be described in Sections 4.3 - 4.6. In Section 4.7, the details of how chemical etching was used to remove the surface layer of the extruded produced will be explained, and in Section 4.8, the details about the mechanical tests at room temperature, i.e. tensile tests, for the extruded products with and without the surface layers will be described. The sample preparation for optical microscopy and the electron backscatter diffraction (EBSD) will be explained in Section 4.9. Further, the details of the electron backscatter diffraction (EBSD) technique will be explained in Sections 4.10 and 4.11. For the simulations, the details of how the visco-plastic self-consistent (VPSC) was used to simulate the formation of deformation textures and the tensile response of materials with different initial textures will be explained in Section 4.12. Finally, phase field which was used to describe sub-grain coarsening will be described in Section 4.13.   4.2 Material Billets of AA3003 were direct chill cast by Rio Tinto Aluminum in Arvida, Quebec with a 4    Methodology  41  dimension of 101.6 mm in diameter and 300 mm in length. The chemical composition of this alloy is listed in Table 4.1. The as-cast microstructure is shown in the anodized micrograph of Figure 4.1 in the section normal to the casting direction taken from the work of Grajales [18]. Table 4.1: Chemical composition of the AA3003 alloy in wt.% (which was measured using inductively coupled plasma mass spectrometry)  Mn Fe Si Ti chemical content in wt.% 1.27 0.54 0.10 0.02   Figure 4.1: Anodized micrograph showing the as-cast microstructure of the AA3003 alloy in the section normal to the casting direction (from the work of Grajales [18])   4.3 Homogenization treatments After casting, two homogenization treatments were conducted using a Carbolite circulating air furnace to produce a low and high density of dispersoids. In the case of the low density of dispersoids, the billets were heated at a rate of 150 °C/h to 550 °C and then the heating rate was reduced to 50 °C/h until the billet reached 600 °C after which it was held for 24 h. To produce a high density of dispersoids, the billet was first heated to 450 °C at a rate of150 °C/h, then heated 4    Methodology  42  to 500 °C at a rate of 50 °C/h to nucleate a high density of dispersoids [11], where upon reaching 500 °C the samples were water quenched to room temperature. Subsequently, the material was transferred into a preheated furnace at 375 °C (e.g., for a rectangular sample of 20  20  200 mm, the heating rate was about 3 °C/s), and then it was held for 24 h to grow the dispersoids and to increase their volume fraction to maximize Zener drag. At the end of the homogenization treatment, all of the billets were water quenched to room temperature. The schematic illustration of the thermal histories for the two homogenization treatments is given in Figure 4.2. The homogenization conditions will subsequently be referred to as 375 °C - 24 h and 600 °C - 24 h.  Figure 4.2: Schematic illustration of homogenization treatment temperature profiles   4.4 Extrusion trials After the homogenization treatments, extrusions trials were conducted using a pilot scale extrusion press at Rio Tinto Aluminum in Arvida, Quebec. Prior to extrusion, billets were heated to the extrusion temperature of 350 °C at a heating rate of 5 - 10 °C/s using an induction heater. Then, the billets were transferred to the extrusion press and extruded. In this study, two types of 4    Methodology  43  extrusion dies were used, i.e. an axisymmetric die and a strip die. For the axisymmetric extrusions, the final extrusion product was an extrusion rod with a diameter of 12.7 mm. For the strip extrusions, the billet was extruded into a shape of a strip with 90 mm in width and 2.9 mm in thickness. The details of the extrusion trials are summarized in Table 4.2. As can be seen in Table 4.2, for the axisymmetric extrusions, a zero-bearing die was used. For the strip extrusions, a die with a 3 mm flat bearing was employed. A schematic illustration of these two types of dies in cross-section is shown in Figure 4.3.  Table 4.2: Summary of the details of the extrusion trials (note: the initial diameter of the billet for extrusion is 101.6 mm) Homogenization condition Extrusion type Die geometry Extrusion temperature Extrusion speed Extrusion ratio Extrudate dimension 375 °C - 24 h axisymmetric extrusion zero-bearing die 350 °C 25 mm/s 70:1 12.7 mm in diameter 600 °C - 24 h 375 °C - 24 h strip extrusion flat die with 3 mm bearing 350 °C 20 mm/s 33:1 90  2.9 mm 600 °C - 24 h   Figure 4.3: Schematic illustration of the geometry of (a) zero-bearing die and (b) flat die with 3 mm bearing in cross-section, where the extrusion direction is along with vertical direction  After extrusion, the extrudate was water quenched in a standing water tank. The distance between the die exit and the quench was ≈ 2 m. This corresponds to a time of ≈ 1.1 and 1.4 s 4    Methodology  44  between the materials leaving the die and entering the water quench for the axisymmetric and strip extrusions, respectively. A schematic illustration of the setup for the extrusion press and the water tank is given in Figure 4.4. For all extrusion trials, the first 10 % and the last 2 m of the extruded products were scrapped, and the remaining part of the extruded product was taken for sampling. For the further mechanical tests and the characterization of the as-extruded microstructures and textures, the samples were taken from the middle of the sample length, as shown in Figure 4.4.   Figure 4.4: Schematic illustration of the setup for the extrusion press and the standing wave water tank, and the area where the samples were taken from extrudate for further mechanical tests and characterization of texture and microstructure  To track the texture and microstructure evolution during axisymmetric extrusion, an extrusion trial was designed for the material with a high density of dispersoids (the homogenization at 375 °C for 24 h) where the extrusion process was stopped midway. After the billet was extruded by approximately 50 %, the extrusion press was stopped and then the extrudate and the materials still remaining inside the feeder and container were taken out from the back of the container. After the material was taken out, it was water quenched and the time between the extrusion press being stopped and water quenching is 1 - 2 min. Note that due to the high difficulty of maintaining the materials together when the extrudate was pulled out for quenching, a smaller 4    Methodology  45  extrusion ratio of 17:1 was used for the partial extrusion. The details of the partial extrusion are summarized in Table 4.3. Table 4.3: Summary of the details for the partial extrusion trial (note: the initial diameter of the billet for extrusion is 101.6 mm) Homogenization condition Extrusion type Die geometry Extrusion temperature Extrusion speed Extrusion ratio Extrudate dimension 375 °C - 24 h axisymmetric extrusion zero-bearing die 450 °C 40 mm/s 17:1 25.5 mm in diameter   4.5 Torsion tests After the homogenization treatments at 600 °C for 24 h and at 375 °C for 24 h, the samples for torsion tests were machined from the billet with its length parallel to the casting direction. The torsion sample dimensions are shown in Figure 4.5.   Figure 4.5: High temperature torsion sample dimensions (in inches)  The torsion tests were conducted using a HTS-100 hot torsion machine from Dynamic Systems Inc. under a rough vacuum of ≈ 200 millitorr (≈ 2.7  10-5 MPa). For the high temperature torsion tests, the torsion sample was held in two grips, as shown in Figure 4.6. The 4    Methodology  46  grip on the left is fixed, while the one on the right can be turned by a rotary servo-hydraulic motor.   Figure 4.6: Schematic illustration of the setup for high temperature torsion test  In order to control the temperature during heating and holding, one thermocouple welded to the centre of the gauge length was applied. A heating rate of 5 °C/s was employed for the temperature ramping to the torsion temperature and the sample was held at this temperature for 18 min prior to the torsion tests. It should be mentioned here that for this torsion system, heating and torsion cannot be conducted simultaneously, which means that there is no heating applied during the deformation. When the sample was twisted, the thermocouple welded to the centre of the gauge length often fell off the sample. Due to this reason, another thermocouple was welded on the shoulder of the sample on the side which does not rotate (as shown in Figure 4.6). One example showing the temperature profiles recorded by the thermocouple welded to the centre of the gauge length and the one welded on the shoulder are shown in Figure 4.7 for the whole test including heating, holding, deformation, and cooling (the material was homogenized at 375 °C for 24 h, the equivalent surface strain rate was 10 s-1 in the torsion test, and the final equivalent surface strain was 12 in the torsion test). 4    Methodology  47   Figure 4.7: (a) Temperature profile showing the whole torsion test (material homogenization: 375 °C - 24 h, equivalent surface strain rate: 10 s-1, final equivalent surface strain: 12), including heating, holding, deformation, and cooling; (b) a magnification of the temperature profiles about the deformation region in Figure 4.7a  For torsion tests, the amount of von Mises equivalent strain εeq and von Mises equivalent strain rate ε̇eq can be calculated using the following equations [138]: 13seqsRL   (4.1) 13seqsRL   (4.2) where Ls is the gauge length of torsion sample, Rs is the specimen radius in the gauge section, θ and ?̇? are the rotation angle and angular velocity. As can be seen from Equations 4.1 and 4.2 and in Figure 4.8, the equivalent strain increases from zero in the centre of the cross-section which is normal to the torsion axis (Z) to the maximum value on the surface. A summary of the different conditions of the torsion tests is listed in Table 4.4. To convert the measured torque values to the equivalent stresses, the following equations [138] were used: 4    Methodology  48  33(3 )2teqsMX YR    (4.3) lnlntMX     (4.4) lnlntMY     (4.5) where X is the strain rate sensitivity, Y is the working hardening coefficient, and Mt is the measured torque. For simplicity, the working hardening coefficient for aluminum alloys in the high temperature deformations was assumed to be closed to be zero (i.e. Y ≈ 0 [138]) and a strain rate sensitivity of X ≈ 0.7 (which will be explained later in Section 6.3.1) was used.  Figure 4.8: Schematic illustration of the equivalent strain distribution from the centre to the surface of torsion sample gauge section in transection  Table 4.4: Summary of details for torsion tests Homogenization condition Heating rate and holding time Torsion temperature Equivalent surface strain rate Final equivalent surface shear strain 375 °C - 24 h 5 °C/s with 18 min 350 °C 10 s-1 12 600 °C - 24 h 350 °C 10 s-1 12  After torsion, the samples were quenched by helium with a pressure ≈ 40 psi (≈ 2.8  10-1 4    Methodology  49  MPa) within 1s. Two nozzles with a diameter of 3 mm were used for quenching. The schematic illustration of the quenching setup in the torsion system is shown in Figure 4.8. The average cooling rate in the temperature window from 350 °C to 200 °C is ≈ 8.1 °C/s, which was measured by the thermocouple welded on the sample shoulder.  Figure 4.9: Schematic illustration of the quenching setup for the torsion tests   4.6 High temperature compression tests To explore the strain rate sensitivity of the material homogenized at 375 °C for 24 h and deformed at 350 °C, the compression tests at 350 °C with the strain rates 0.1 s-1, 1 s-1 and 10 s-1 were conducted using a Gleeble 3500 thermo-mechanical simulator with an ISO-T compression plate. Cylindrical samples (8 mm in diameter and 12 mm in length) were machined from the billet with the length parallel to the casting direction. A schematic illustration of the high temperature compression setup is shown in Figure 4.10. To minimize the friction between the sample and the tungsten carbide anvils, a small amount of nickel paste was used as lubrication. The sample was heated at a rate of 5 °C/s to 350 °C and then held at this temperature for 60 s prior to the compression tests. One example showing the temperature profile for the whole compression test is shown in Figure 4.11 (the material was homogenized at 375 °C for 24 h, and a strain rate of 10 s-1 was used in the compression test).  4    Methodology  50   Figure 4.10: Schematic illustration of the high temperature compression setup   Figure 4.11: (a) Temperature profile showing the whole compression test (material homogenization: 375 °C - 24 h, strain rate: 10 s-1; (b) a magnification of the temperature profiles about the deformation region in Figure 4.11a  As can be seen in Figure 4.10, the change in diameter of the compression sample was measured by a c-strain dilatometer. To calculate the true strain εt and true stress σt based on the diameter changed of the samples in the compression test, the following equations were used: 0 002ln( )AltD TDD D  (4.6) 20( )4tFD D (4.7) where D0 is the initial diameter of the compression sample at room temperature; αAl is the 4    Methodology  51  thermal expansion coefficient of FCC aluminum matrix (i.e. at 350 °C , α = 26.9  10-6 K-1, which was interpolated from the data reported by Hatch [139]); T is the deformation temperature (i.e. 350 °C / 623 K); F is the measured compression force; and ∆D is the measured diameter change of compression sample compared with the initial diameter at room temperature.    4.7 Removal of sample surface     To examine the effect of the surface layer of the extruded products on its mechanical behaviour, the surface layers of the as-extruded samples were removed in some samples. In detail, for the axisymmetric extrudates, the surface layer was machined off using a lathe. For the extruded strips, the surface layer was removed by chemical etching using 100 g/L NaOH solution at a temperature of 50 - 60 °C. The thickness of the sample as a function of the etching time is shown in Figure 4.12 for the two homogenization conditions. The average removal rate was 14.4 μm/min for the material homogenized at 375 °C for 24 h and 15.6 μm/min for the material homogenized at 600 °C for 24 h.  Figure 4.12: Plot of the extruded strip thickness as a function of etching time for the extruded strips homogenized at 375 °C for 24 h and at 600 °C for 24 h 4    Methodology  52  4.8 Tensile test To determine the room temperature mechanical behaviour of the as-homogenized materials, the tensile tests at room temperature were conducted. The samples for the tensile tests were machined from the billets with its length parallel to the casting direction. The dimensions of the tensile samples machined from billets are shown in Figure 4.13.   Figure 4.13: Dimensions of the tensile samples machined from billets (the length is parallel to the CD)  For the case of the axisymmetric extrusions, two conditions were examined in the tensile tests at room temperature, i.e. the sample with the full diameter of the extrudate (12.7 mm) and the sample in which the surface layer was removed using a lathe and its diameter was reduced to 4.75 mm. For the samples with the full diameter of the extrudate, raw extrudates were cut along the extrusion direction with 25 mm in length. For the samples where the surface layer was removed, the sample dimensions are shown in Figure 4.14.   Figure 4.14: Dimensions for the tensile samples machined from axisymmetric extrudate (the length is parallel with the ED)  In the case of the strip extrusions, the tensile samples were machined from the extruded strips parallel to the extrusion direction (ED), at 45° to ED, and perpendicular to the extrusion direction 4    Methodology  53  (transverse direction, TD), as shown in Figure 4.15. The dimensions of the samples are shown in Figure 4.16. Two sample thicknesses were examined, i.e. the full thickness of the extruded strip (i.e. ≈ 2.9 mm) and the half thickness (i.e. ≈ 1.4 mm, the surface layer was removed using chemical etching).  Figure 4.15: Schematic illustration of the scenario for machining the three types of samples from extruded strip   Figure 4.16: Dimensions for the tensile samples machined from extruded strips  For the dog bone samples as shown in Figures 4.13 and 4.14, the tensile tests were conducted using a servo-hydraulic test machine (Instron 8872). The applied strain rate was 2  10-3 s-1, which was achieved by programming a crosshead displacement at a constant speed of 0.025 mm/s. For the case of extruded strips (i.e. the samples as shown in Figures 4.15 and 4.16), the tensile tests were carried out at room temperature using a screw driven tensile test machine (Instron TM-L) with a strain rate of 2.6  10-3 s-1 (a constant crosshead velocity of 0.1 inch/min, 4    Methodology  54  ≈ 0.042 mm/s, was applied). For this case, in order to obtain the R-value evolution during the tensile tests, the width change of tensile sample was recorded by the camera with a capture speed of 1 frame per second (fps), and the captured photos were correlated with the measured flow curves by the software of DaVis digital image correlation (DIC). In detail, the software recorded the loading values (in kN) when a photo was captured and these values were used to be correlated with that measured in flow curves, as shown in Figure 4.17.   Figure 4.17: An example showing the correlation of the loading values captured by tensile machine which has been aligned with the data captured independently by the DIC system  To measure the width change from photos, ImageJ image analysis software was used. According to the measured width change and, R-value was calculated as follows, with an assumption of the elastic isotropy of aluminum alloys [140]: WtTtR  (4.8) 00-ln 1 -W ttW WW E      (4.9) 4    Methodology  55  00-- ln 1 - -T tt tW WW E           (4.10) where εWt and εTt are the plastic true strains in the sample width direction and thickness direction; W0 and W are the initial and changed sample widths; E and υ are Young’s modulus (70 GPa) and Poisson ratio (0.35) of pure aluminum at room temperature; σt and εt are the true stress and plastic true strain in the tensile direction. According to the error propagation principle and only considering the error from the width measurements, the experimental error of the measured R-value can be calculated as: 0220-2WWav avR avtUUW WU R              (4.11) where Rav, Wav, and W0-av are the average values for the measured R-values, measured sample widths during testing, and the initial sample width; UW and UW0 are the standard deviations for the measurements of width and initial width. For all tensile tests in this study, three tests were conducted and the yield stress was determined using the 0.2 % offset method.   4.9 Optical metallography In order to observe dispersoids and constituent particles in the as-homogenized conditions by optical microscope, samples were first manually ground and finally polished using colloidal silica. The details about the grinding and polishing procedures are summarized in Tables 4.5 and 4.6, respectively. After surface preparation, samples were etched with 0.5 % hydrofluoric acid (48 % HF) in distilled water. A Nikon Epiphot 300 optical microscope was used to take optical images using the Clemex Vision PE 6.0 image analysis software. 4    Methodology  56  Table 4.5: Grinding procedures for sample surface preparation Type of grinding paper Grit Speed (rpm) Time (min) Lubricant SiC 400 200 ≈ 2 water SiC 800 200 ≈ 2 water SiC 1200 200 ≈ 5 water  Table 4.6: Polishing procedures for sample surface preparation Type of polishing cloth Speed (rpm) Time (min) Polishing solution Lubricant Leco Imperial (1 μm) 100 10 - 15 1 μm diamond suspension Microid Diamond Compound Extender Buehler Chemomet (0.05 μm) 100 ≈ 5 0.05 μm colloidal silica -   4.10 Electron backscatter diffraction (EBSD)     For the characterization of textures and microstructures using the electron backscatter diffraction (EBSD) technique, the different deformation cases (axisymmetric and strip extrusions and the torsion samples) necessitated the use of different sample sections relative to the deformation axes. For the axisymmetric extrusions, a series of EBSD maps were measured from the centre to surface in the longitudinal plane parallel to the extrusion axis, as shown in Figure 4.18a. For the strip extrusion, a series of EBSD maps were measured from the centre to surface in the centre section of the ED-ND plane, as shown in Figure 4.18b. Finally, for the case of torsion samples, the textures were collected from a series of EBSD maps measured from the centre to surface in the cross-section normal to the torsion axis (Z), as shown in Figure 4.19. For the characterization of the grain structure at the different equivalent strains (according to Equation 4.1) in the torsion tests, the sections normal to the radial direction (RD) in the gauge section but with the different relevant radii were examined, as shown in Figure 4.19. 4    Methodology  57   Figure 4.18: Schematic illustration of the section applied for EBSD mapping in (a) axisymmetric extrusion and (b) strip extrusion   Figure 4.19: Schematic illustration about the two types of sections applied for EBSD mapping in torsion sample  The sample surface preparations for the EBSD measurements were followed by the grinding and polishing procedures as shown in Tables 4.5 and 4.6. All of the EBSD measurements were conducted by a Zeiss-Σigma scanning electron microscope (SEM) using a Nikon high speed camera and EDAX/TSL OIM Data collection (6th edition) software. For the EBSD measurements, an acceleration voltage of 20 kV, an aperture size of 60 μm, and a working distance of ≈ 13 mm was used. For the setting of diffraction pattern capturing by the software of OIM Data collection, a binning size of 8  8 with a capture speed of ≈ 50 fps was applied. For 4    Methodology  58  the diffraction pattern identification, only the FCC phased matrix was indexed during the measurements. A step size of 200 nm was used for most cases, while for the selected high resolution cases, a step size of 50 nm was used.   4.11 EBSD map clean-up method For the post processing of the EBSD data, EDAX / TSL-OIM Analysis (6th edition) software was used. To filter out the data showing the low confidence, all of the EBSD maps were first processed by the Confidence Index (CI) Standardization, which is a function in OIM Analysis. The CI of a measured pixel in the EBSD maps indicates the confidence of the determined crystal orientation from the diffraction pattern. The function of CI standardization can change the CIs of all pixels in a grain to the maximum CI value in this grain. After this procedure, pixels with a CI less than 0.1 were filtered out.    4.12 Description and inputs for the visco-plastic self-consistent (VPSC) model To simulate the mechanical behaviours and the deformation textures, a visco-plastic self-consistent (VPSC) model [104] was used. Note, a detailed description of the VPSC model was provided in Section 2.9. The inputs to this model are the initial material texture, the deformation history (velocity gradient tensor / stress tensor), and the constitutive law. For the texture input, 5000 grains were used to describe the initial texture with each grain having an equal weight factor. For the deformation history, the different boundary conditions were applied for the different cases. The details of the boundary conditions which were used to describe the different 4    Methodology  59  deformation histories in the VPSC model will be explained in Sections 4.12.1 - 4.12.3.   4.12.1 Inputs of the deformation history for compression tests To determine the strain rate sensitivity at 350 °C for the material homogenized at 375 °C for 24 h, the inverse of the strain rate sensitivity, i.e. n (n = 1 / m, where m is the strain rate sensitivity value) was adjusted such that the simulated flow curves using the VPSC model can match the experimental flow curves in the compression tests at the different strain rates. To simulate the flow curves in the compression tests, a mixed boundary condition as shown in Equations 4.12 and 4.13 were used for the each simulation steps. In detail, the normal component at the compression direction was controlled by the velocity gradient component (i.e. L33, 𝜀̇ = 0.1, 1, and 10 s-1). For the other normal components, the normal stresses (σ11 and σ22) were imposed to be zero. Moreover, an equivalent strain increment of 0.003 was used for each simulation steps to the final strain of 0.6. 0 00 00 0       compL s  (4.12) 00       comp MPa  (4.13)   4.12.2 Inputs of the deformation histories for axisymmetric and strip extrusions To simulate extrusion textures, a constant velocity gradient tensor (≈ strain rate tensor) was 4    Methodology  60  imposed on each simulation steps. At the extrusion direction, a normal strain rate (L33) was assumed to be 10 s-1 based on the previous finite-element simulations reported by Mahmoodkhani et al. [17, 141]. For the each simulation step, an equivalent strain increment of 0.04 was used and 100 steps were involved to the final equivalent strain of 4, which was predicted by the finite-element simulations of Mahmoodkhani et al. [17, 141] for the final equivalent strains of the axisymmetric and strip extrusions. Equations 4.14 and 4.15 show the velocity gradient tensors used for the texture simulations of axisymmetric extrusion and strip extrusions, respectively. For the case of axisymmetric extrusion, the directions of L11 and L22 are the equivalent radial directions of extrudates. For strip extrusion, the direction of L11 is parallel to the normal direction and the direction of L22 is parallel to the transverse direction of the extruded strips. 5 0 00 5 00 0 10      axisy extL s  (4.14) 9 0 00 1 00 0 10      strip extL s  (4.15)   4.12.3 Inputs of the deformation history for torsion tests To simulate the torsion textures, a boundary condition of a constant velocity gradient tensor (see Equation 4.16) combined with a constant stress tensor (see Equation 4.17) was used. In detail, the normal components were controlled by the stress states, i.e. the normal stresses parallel to the shear direction (σ11), torsion axis (σ22), and radial direction (σ33) were imposed to 4    Methodology  61  be zero, and the shear component on the shear plane was controlled by the strain rate, i.e. L12 = 10 s-1 for the case of the torsion test at an equivalent surface strain rate of 10 s-1. Furthermore, an equivalent strain increment of 0.04 was used for each simulation steps to the final strain of 5. 10 00 00 0      torsionL s  (4.16) 000       torsion MPa  (4.17)   4.12.4 Inputs of the deformation history for tensile tests at room temperature To simulate the tensile flow curves (samples machined from billets, axisymmetric extrudates, and extruded strips), a mixed boundary condition was used as described by Equations 4.18 and 4.19. In detail, the extension direction in the tensile tests was controlled by the strain rate, L33 = 0.002, and the directions normal to the extension direction were controlled by the stress components, i.e. σ11 = 0 and σ22 = 0. In addition, an equivalent strain increment of 0.002 was used for each simulation steps to the final strain of 0.3. 0 00 00 0 0.002      tensionL s  (4.18) 00       tension MPa  (4.19)  4    Methodology  62  4.13 Description of phase field model To simulate grain growth, an in-house phase field model was used, which was developed by Zhu at the University of British Columbia (an application of this in-house phase field model to the recrystallization in a low-carbon steel was reported by Zhu et al. [142]). The microstructure evolution in the current phase field model is run by Equation 2.18, but in the current condition, there is no extra driving pressure on a grain boundary so that the term of ∆Gdri is equal to zero. In addition, an experimental as-extruded microstructure (i.e. with a size of 150  150 μm and taken from the centre of the axisymmetric extrusion with the homogenization at 375 °C for 24 h, which is shown in Figure 6.1c in Section 6.2.1.1) was used as the starting microstructure for the simulation. The grid size in the simulation domain is 0.2 μm and 6 grids were used to describe the width of the boundaries. The Read - Shockley relationship [136] was used to define the boundary energy as a function of misorientation (see Equation 2.19, where γ15° = 3.24  10-13 J/μm2 , i.e. for high purity aluminum at 450 °C [143]) and Huang - Humphreys relationship was used to define the boundary mobility as a function of misorientation [137] (see Equation 2.20, where μ15° = 1.54  10-16 μm4/Js, and γ15°  μ15° = 5  103 μm2/s, i.e. values were taken from the study for a high purity aluminum at 370 °C [144]).  63    5 Material initial state  5.1 Introduction In this chapter, the crystallographic textures and microstructures of the materials for the two homogenization conditions, i.e. at 600 °C for 24 h and at 375 °C for 24 h examined in this work, will be presented. In addition, the number densities and the area or volume fractions of the dispersoids and constituent particles formed in these two homogenization conditions will be summarized based on literature.    5.2 Experimental results Figures 5.1a and 5.1b illustrate optical micrographs showing the distribution of the dispersoids (i.e. the α-Al(Mn,Fe)Si particles with sizes of 20 - 150 nm [10, 27], which are revealed by etched pits and the small dots) and the constituent particles (i.e. a mixture of Al6(Mn,Fe) and α-Al(Mn,Fe)Si particles with sizes of a few microns [12, 19-22]) formed during homogenization at 600 °C for 24 h and at 375 °C for 24 h, respectively. As can be seen in Figures 5.1a and 5.1b, a high density of dispersoids is found for the homogenization condition of 375 °C for 24 h (shown in Figure 5.2b), but almost no dispersoids can be observed after homogenization at 600 °C for 24 h (shown in Figure 5.1a). This observation is consistent with the previous TEM study [10] about these two homogenization conditions. Furthermore, the number densities and the area or volume fractions of the dispersoids and constituent particles formed in these two homogenization 5   Material initial state  64  conditions are summarized in Table 5.1 based on literature. It should be mentioned here that the number density reported for dispersoids in the condition of homogenization at 375 °C for 24 h [10] (shown in Table 5.1) is a number density in a unit volume (the analysis was based on a thin foil in the characterization using TEM). Furthermore, after the homogenization at 375 °C for 24 h, the constituent particles form a network structure, as shown in Figure 5.1b, so that it is hard to measure their number density and thus, there are no values found in the literature. On the other hand, for the homogenization at 600 °C for 24 h, the number density of the constituent particles was reported based on the analysis of the optical micrographs [12], so that this density is given as number of particles per area.  Figure 5.1: Optical micrographs showing the microstructures of the homogenization (a) 600 °C - 24 h and (b) 375 °C - 24 h  Table 5.1: Summary of the number densities and the volume fractions of the dispersoids and constituent particles formed in the homogenization at 600 °C for 24 h and at 375 °C for 24 h Homogenization condition Dispersoid Constituent particle Number density  (m-3) Volume fraction  (%) Number density  (m-2) Area fraction  (%) 600 °C for 24 h almost no dispersoids 2.8  1010 [12] 3.9 [12] 375 °C for 24 h 3.6  1020 [10] 1.5 [10] - 2.4 [16]  5   Material initial state  65  As discussed in Section 2.3, the major changes of the microstructure during the homogenization treatments are the number densities, fractions and distributions of the second phase particles. However, for the matrix, there are almost no changes in grain size, grain morphology, and their crystallographic texture and as such the values from the as-cast condition can be used to represent the as-homogenized grain characteristics. Figure 5.2a shows an EBSD map (which is coloured by the corresponding inverse pole figure) for the as-cast microstructure, in which only the matrix was indexed in the EBSD map. The as-cast microstructure has an average grain size of 70 μm (in equivalent diameter). Figure 5.2b illustrates the {001}, {011} and {111} pole figures of the as-cast texture. The centre of the pole figures is aligned with the casting direction. As can be seen in Figure 5.2b, an almost random texture is observed in the as-cast microstructure, in which the maximum texture intensity is only 1.3 multiples of a random distribution (MRD).   Figure 5.2: (a) IPF map showing the as-cast microstructure, where the IPF colour is according to the casting direction, and (b) {001}, {011}, and {111} pole figures showing as-cast texture, where the direction normal to the paper is aligned with the casting direction  66    6 Ideal deformation modes: axisymmetric, plane strain and simple shear  6.1 Introduction     In this chapter, the textures and microstructures formed in three nearly ideal deformation modes will be examined in detail, i.e. axisymmetric (the centre of the round bar extrusions), plane strain (the centre of the strip extrusions) and simple shear (torsion tests) deformations. In each deformation mode, two conditions will be studied, i.e. the unrecrystallized (or the as-deformed state) and the recrystallized condition. The visco-plastic self-consistent (VPSC) model will be used to simulate the deformation texture formed in the three deformation conditions, and finally, the recrystallization mechanisms for each deformation conditions will be discussed.    6.2 Experimental results 6.2.1 Axisymmetric deformation: the centre of the round bar extrusions 6.2.1.1 Textures and microstructures in the centre of the round bar extrusions     Figure 6.1 illustrates the microstructures and crystallographic textures in the centre of the longitudinal plane of an axisymmetric extrudate for the materials homogenized at 600 °C for 24 h and at 375 °C for 24 h. The extrusion ratio is 70:1 and the deformation mode in the centre of the extrudate is a pure uniaxial extension with an equivalent strain of 4 [17, 141]. The extrusion direction (ED) in the inverse pole figure (IPF) maps of Figures 6.1a and 6.1b is aligned with the vertical direction. The experimentally determined as-extruded textures are plotted in the {001} 6   Ideal deformation modes: axisymmetric, plane strain and simple shear  67  and {111} pole figures of Figures 6.1c and 6.1d.  Figure 6.1: IPF maps showing the microstructures in the centre of the longitudinal plane for the axisymmetric extrudate for the billets homogenized (a) at 600 °C for 24 h (where the white arrows denote the grains having the <111> orientations parallel to the extrusion direction) and (b) at 375 °C for 24 h; and pole figures showing the textures in this region for the materials homogenized (c) at 600 °C for 24 h and (d) at 375 °C for 24 h (the pole figures are equal area projections with the extrusion direction (ED) at the centre of the pole figure and the radial direction (RD) at the outer rim)  Figure 6.1a shows that after extrusion, the material homogenized at 600 °C for 24 h (i.e. the condition with almost no dispersoids) formed equiaxed grains. This suggests that recrystallization occurred, and as such, this condition will be referred to as the recrystallized condition. Recall that the time in between the extrudate leaving the die and entering the water quench was ≈ 1.1 s. Therefore, it is difficult to assess whether the recrystallization occurred during the deformation or after deformation prior to the quench. The texture of these equiaxed grains indicates that most of the grains have their <001> directions aligned with the extrusion 6   Ideal deformation modes: axisymmetric, plane strain and simple shear  68  direction, as shown in the pole figures of Figure 6.1c. It should be noted, however, that some grains have their <111> directions aligned with the extrusion direction, i.e. these grains are coloured blue in the IPF map and have been identified by the white arrows in Figure 6.1a. The quantitative analysis regarding this texture will be further discussed in the next section. In addition, the size (in equivalent diameter) distribution of the equiaxed grains is shown in Figure 6.2, where the distribution is fit with a lognormal distribution with a mean value of 21 μm and a standard deviation of 23 μm.  Figure 6.2: Plot of the grain size distribution (in equivalent diameter) for the central region of the extrudate with the homogenization at 600 °C for 24 h (the recrystallization condition), and the experimental distribution is fit with a lognormal distribution  For the material homogenized at 375 °C for 24 h (i.e. the condition with a high density of dispersoids), the IPF map is shown in Figure 6.1b. In this case, the grains are extremely elongated along the extrusion direction, and inside these elongated grains, equiaxed subgrains were observed (the low angle grain boundaries, i.e. their misorientations < 15 °, are shown by the grey lines inside the grains). Figure 6.1d shows that this material has a double fibre texture with the <001> and <111> orientations parallel to the extrusion direction. According to the previous 6   Ideal deformation modes: axisymmetric, plane strain and simple shear  69  axisymmetric deformation texture studies [56, 59-61, 80], the observed <001> - <111> double fibre texture is a typical deformation texture in the uniaxial extensions. As such, this condition will be referred to as the unrecrystallized or as-deformed condition. The quantitative analysis of this texture will be further discussed in the next section.   6.2.1.2 Quantitative analysis of the textures in the centre of the axisymmetric extrusions To quantitatively analyze these textures, the fractions of each individual ideal texture components need to be determined. In order do so, two approaches were taken, i.e. first, the area fraction of a specific texture component was calculated directly from the EBSD data (using the OIM Analysis software), and second, the volume fraction of a specific texture component was calculated from the orientation distribution function (using the MTEX software). To calculate the orientation distribution function (ODF), a large area (600  150 μm) EBSD map which covers more than 5000 grains or subgrains and the harmonic series expansion approach with an expansion rank of 16 was used. A deviation angle of 15 ° was applied to identify a certain orientation or an ideal texture component for both cases. The detailed parametric study about the influence of the expansion rank in ODF calculation and the deviation angle on the calculated volume fraction of a certain texture component is shown in Appendix 1. Figure 6.3 shows the re-calculated pole figures based on the experimentally determined ODF for the recrystallized condition and the unrecrystallized condition. In comparison with the experimentally determined pole figures as shown in Figure 6.1c and d, the re-calculated pole figures show the same types of the textures but with lower intensities, i.e. for the recrystallized condition, the maximum intensity is 9.7 MRD for the re-calculated pole figures and 10.3 MRD 6   Ideal deformation modes: axisymmetric, plane strain and simple shear  70  for the experimentally determined pole figures, and for the unrecrystallized condition, the maximum intensity is 16.4 MRD for the re-calculated pole figures and 21.4 MRD for the experimentally determined pole figures.  Figure 6.3: Plot of the re-calculated pole figures based on the ODF showing the textures in this region for the materials homogenized (c) at 600 °C for 24 h and (d) at 375 °C for 24 h  The differences between the experimentally determined and re-calculated pole figures were quantitatively evaluated as the value Dtex [145], which is shown as follows:       222A i B iitexA i B iif g f gDf g f g      (6.1) where gi is the crystal orientation extracted from the pole figures (the orientations were extracted with the equal angle of 5 ° in the pole figures), fA(gi) and fB(gi) are the texture intensities for the same orientation gi but in the different pole figures. Table 6.1 summarizes the calculated value Dtex based on the experimentally determined pole figures shown in Figures 6.1c and 6.1d and the 6   Ideal deformation modes: axisymmetric, plane strain and simple shear  71  re-calculated pole figures shown in Figures 6.3a and 6.3b. Table 6.1: Summary of the calculated parameter Dtex based on the experimentally determined pole figures shown in Figures 6.1c and d and the re-calculated pole figures shown in Figures 6.3a and b  {001} pole figure {111} pole figure Recrystallized condition (homogenization at 600 °C for 24 h) 0.05 0.05 Unrecrystallized condition  (homogenization at 375 °C for 24 h) 0.16 0.25  Based on the study reported by Wright et al. [145], when the parameter Dtex is smaller than 0.4, it was reported that the two pole figures have a high similarity. Therefore, the results shown in Table 6.1 suggest that the difference between the experimentally determined pole figures and re-calculated pole figures are minor, and thus, the calculated ODF accurately represents the experimental texture, and can be used for the further calculation of the volume fraction for the certain texture components. Table 6.2 summarizes the calculated area and volume fractions based on the two approaches described earlier. To estimate the properties of the textures and the microstructures from the EBSD maps (i.e. the properties such as the fractions of the certain texture components, the grain or subgrain sizes and the boundary densities), the estimations will be based on a single EBSD map which covers a large area (600  150 μm). As can be seen in Table 6.2, for both the recrystallized and unrecrystallized conditions, the calculated fractions in a unit area and that in a unit volume of the certain texture components are very similar. 6   Ideal deformation modes: axisymmetric, plane strain and simple shear  72  Table 6.2: Summary of the calculated area / volume fractions of the texture fibre <001> // ED and <111> // ED based on the approaches (a) and (b) for the recrystallized and unrecrystallized conditions  Calculation approach Fraction (%) <001> // ED <111> // ED Others Recrystallized condition (homogenization at 600 °C for 24 h) From EBSD map In area 56 3 41 From ODF In volume 58 2 40 Unrecrystallized condition  (homogenization at 375 °C for 24 h) From EBSD map In area 34 64 2 From ODF In volume 33 66 1  For the recrystallized condition, the texture predominantly shows a single texture component with the <001> directions aligned with the extrusion direction (<001> // ED), and quantitatively, the fraction of this texture component is 56 - 58 %. As discussed before, for this condition, there are a small number of the grains which have their <111> directions aligned with the extrusion direction (<111> // ED). The fraction of these grains is 2 - 3 %. Later, in Section 6.3.2.1, the possible significance of these grains will be discussed. Furthermore, it is worth noting that about 40 % of the orientations in this condition are not accounted for with these two texture components.  For the unrecrystallized case, the textures shows the <001> - <111> double fiber texture. Quantitatively, the <001> // ED texture component has a lower fraction, i.e. 33 - 34 %, while the <111> // ED texture component accounts for 64 - 66 % of the total orientations. Together, these two texture components account for 98 - 99 % of the total orientations. In summary, for the unrecrystallized condition, the <111> // ED texture component is the predominant one and with the <001> // ED texture component having a lower fraction. On the other hand, for the recrystallized condition, the <001> // ED texture component is the dominant one and the fraction of the <111> // ED texture component is minor. In the next section, the microstructure of the unrecrystallized extrudate will be further examined.  6   Ideal deformation modes: axisymmetric, plane strain and simple shear  73  6.2.1.3 Microstructures for the centre of the axisymmetric extrusion: the unrecrystallized case Figure 6.4a shows an IPF map in the centre of the longitudinal plane for the unrecrystallized case (i.e. the extrudate with the homogenization at 375 °C with 24 h), and Figure 6.4b illustrates the distributions of low angle grain boundaries (LAGBs, i.e. their misorientation angles < 15 °) and high angle grain boundaries (HAGBs, i.e. their misorientation angles ≥ 15 °) for the IPF map of Figure 6.4a. As can be seen in Figure 6.4b, most of the HAGBs are aligned with the extrusion direction. Between the HAGBs, nearly equiaxed subgrains which are mostly enclosed by LAGBs can be observed, as shown in the magnified inset in Figure 6.4b.  The thickness of the elongated grains is defined by the average distance between the adjacent HAGBs in the direction normal to the extrusion direction. Figure 6.4c shows the misorientation profile across the elongated grains for the blue arrow in the horizontal direction in Figure 6.4b, where the misorientation angle is defined as the angle between the current point and the original point. When the misorientation angle between the adjacent points is larger than 15 °, a HAGB will be defined, and thus, the grain thickness will be measured as the distance between these HAGBs.  Based on this approach, different grains are labelled in Figure 6.4c, and for a better statistics, 50 misorientation profiles with a length of 600 μm along the direction normal to the extrusion direction were included to estimate the average thickness of the grains at the central region of the extrudate. The estimated average grain thickness is 6.6 μm in 2-D and 8.5 μm in 3-D (to obtain the 3-D grain thickness in equivalent diameter, the 2-D planar grain thickness calculated from the EBSD map will be multiplied by a correction factor of 1.28 [146], with the assumption that all the grains have the cylindrical shapes in 3-D). 6   Ideal deformation modes: axisymmetric, plane strain and simple shear  74   Figure 6.4: (a) IPF map showing the as-extruded microstructure in the centre of the longitudinal plane for the extrudate (i.e. the unrecrystallized condition, with the homogenization at 375 °C for 24 h), (b) illustrating the distributions of LAGBs and HAGBs for the as-extrudate microstructure as shown in Figure 6.4 and the inset showing a magnified view of one representative region, and (c) the misorientation profile showing the misorientation angles between the current point and the original point in the direction normal to the extrusion direction (as indicated by the blue arrow in Figure 6.4b) and the identified grains based on the misorientation analysis 6   Ideal deformation modes: axisymmetric, plane strain and simple shear  75  Similar values for the grain thickness (i.e. 7 - 8 μm in 3-D) with similar extrusion conditions (i.e. with the extrusion ratio of 70:1, extrusion temperature at 400 °C and with the extrusion speeds from 2 - 32 mm/s) for AA3003 homogenized at 500 °C for 8 h (i.e. with a lower density of dispersoids compared with the homogenization at 375 °C for 24 h) were reported in the previous study from Grajales [18]. Furthermore, Grajales [18] also reported that based on a mass balance model, the estimated grain thickness for the extrusion with a ratio of 70:1 is 9.1 μm in 3-D, which is similar to the experimentally determined value for the current unrecrystallized condition (i.e. 8.5 μm in 3-D). This suggests that for the material with a high density of dispersoids (i.e. the homogenization at 375 °C for 24 h), the average thickness of the elongated grains formed in the extrusion is attributed to the deformation. Based on the grain shape and texture measurement, one can conclude that the high density of dispersoids successfully suppresses recrystallization during and after the extrusion. As mentioned in Sections 6.2.1.1 and 6.2.1.2, the elongated grains in the unrecrystallized case show the <001> - <111> double fibre texture. To analyze the microstructures of the grains from these two texture components, the microstructures shown in Figure 6.4a were separated into the one having the orientations within 15 ° of the ideal <001> // ED texture component and another having the orientations within 15 ° of the ideal <111> // ED texture component, as shown in Figure 6.5a and b, respectively. Figure 6.5c show an example of the misorientation profiles within the areas having the <001> and <111> orientations aligned with the extrusion direction for the white arrows shown in the IPF maps of Figures 6.5a and 6.5b (in terms of statistic, 20 misorientation profiles were examined and representative samples are shown in Figure 6.5c). In detail, the misorientation profiles shown in Figure 6.5c describe the misorientation angles between the current point and the original point for 100 μm along the extrusion direction. 6   Ideal deformation modes: axisymmetric, plane strain and simple shear  76   Figure 6.5: IPF maps illustrating the region in Figure 6.1.3a having the orientations within 15 ° of the ideal <001> // ED texture component, (b) the region having the orientations within 15 ° of the ideal <111> // ED texture component, and (c) the misorientation profiles showing the misorientation between the current points with the original point along the extrusion direction inside the <001> // ED and <111> // ED texture components (as pointed by the white arrows in Figure 6.5a and b, respectively)   6   Ideal deformation modes: axisymmetric, plane strain and simple shear  77  As can be seen in Figure 6.5c, in the <111> // ED texture component, the variation of the misorientations is less than 10 ° (the misorientation axis is <111> direction in this case). On the other hand, in the <001> // ED texture component, a much larger range of the misorientations can be observed with some as high as 30 ° (the misorientation axis is <001> direction in this case). This observation suggests that a large misorientation can be accumulated in the grains having the <001> orientations aligned with the extrusion direction. This long range lattice rotation has also been previously discussed in the cube orientation formed by the rolling of FCC materials [73, 80, 81, 147].  Figures 6.6a - 6.6d show an example of the microstructure for an area where one of the <001> orientations is within 15 ° of the ideal <001> // ED texture component (in terms of statistic, 10 areas showing the <001> orientations within 15 ° of the ideal <001> // ED texture component were examined, and representative microstructures are shown in Figures 6.6a - 6.6d). Figure 6.6a shows an IPF map with respect to the extrusion direction. To examine the rotation around the extrusion direction on the IPF map, the data has been replotted with respect to the radial direction (RD) of the extrudate in Figure 6.6b. In addition, the distribution of the LAGBs and HAGBs is illustrated in Figure 6.6c, and the crystallographic orientations for this region are shown in the {001} and {111} pole figures of Figure 6.6d.  6   Ideal deformation modes: axisymmetric, plane strain and simple shear  78   Figure 6.6: An example of the microstructure (for the homogenization of 375 °C for 24 h)of the <001> // ED texture component: (a) the map with the IPF colour according to the extrusion direction, (b) the map with the IPF colour according to the radial direction, (c) the map showing the distribution of LAGBs and HAGBs, and (d) the orientation projection in the {001} and {111} pole figures for the region in Figure 6.6a; and an example of the microstructure of the <111> // ED texture component: (e) the map with the IPF colour according to the extrusion direction, (f) the map with the IPF colour according to the radial direction, (g) the map showing the distribution of LAGBs and HAGBs, and (h) the orientation projection in the {001} and {111} pole figures for the region in Figure 6.6e  6   Ideal deformation modes: axisymmetric, plane strain and simple shear  79  As can be observed in Figure 6.6b, there is a wide range of orientations within this area, indicated by the strong colour change along the extrusion direction. Furthermore, the pole figures shown in Figure 6.6d illustrates that there is an almost random distribution of orientations rotated around the <001> axis. In addition, Figure 6.6c shows that some of the HAGBs are aligned with the direction normal to the extrusion direction, as pointed by the red arrows in Figure 6.6c. It is worth noting that the discontinuities in some of the HAGBs are due to the existence of constituent particles (indicated by the black particles in Figure 6.6c). In conclusion, these observations are consistent with the misorientation profile plotted inside <001> // ED texture component in Figure 6.5c. They all indicate that there are long range lattice rotations occurring inside the <001> // ED grains which result in the formation of the HAGBs aligned with the direction normal to the extrusion direction and the breakup of the elongated grains.  In contrast, Figures 6.6e - 6.6h present the results of a representative area where one of the <111> orientations is within 15 ° of the ideal <111> // ED texture component. Figures 6.6e and 6.6f show the IPF colour according to extrusion and radial directions, respectively (in terms of statistic, 10 areas showing the <111> orientations within 15 ° of the ideal <111> // ED texture component were examined, and representative microstructures are shown in Figure 6.6e - h). In comparison with the Figure 6.6b (the case of the microstructure with the orientation within 15 ° of the ideal <001> // ED texture component), a uniform colour along the extrusion direction in Figure 6.6f is observed, and it indicates that the orientation spreads around the extrusion direction are relatively small for this case. This observation can also be confirmed by the orientation analysis for the each individual grains as shown in Figures 6.6g and 6.6h. In detail, the LAGBs and HAGBs are illustrated in Figure 6.6g, and the orientations in this region are shown in the pole figures in Figure 6.6h. Based on their orientations, two grains are identified 6   Ideal deformation modes: axisymmetric, plane strain and simple shear  80  (G1 and G2) and highlighted by different colours (green and purple). As can be seen in Figure 6.6h, inside the each individual grains, the orientation spreads are within ± 3 °. This observation is consistent with the misorientation profile plotted inside the <111> // ED texture component (shown in Figure 6.5c). Furthermore, no HAGBs aligned with the direction normal to the extrusion direction are observed in this region. In order to obtain a better statistical description of the misorientation distributions inside the areas having the orientations within 15 ° of the ideal <001> // ED and <111> // ED components in the as-deformed microstructure, a large area of 600  150 μm which covers more than 20,000 subgrains was examined and the fractions of subgrain boundaries (i.e. their misorientations ≥ 2 °) as a function of their misorientation angles are separately plotted for these two texture components and shown in Figure 6.7.  Figure 6.7: Histogram showing the fractions of LAGBs and HAGBs versus their misorientation angles inside the areas having the orientations with 15 ° of the ideal <001> // ED and <111> // ED components in the as-deformed microstructure (the centre of the extrudate with the homogenization at 375 °C for 24 h), respectively (note: the maximum misorientation between different <001> crystal orientation is 45 ° and the maximum misorientation between <111> crystal orientation is 60 °)  6   Ideal deformation modes: axisymmetric, plane strain and simple shear  81      As can be seen in Figure 6.7, the <001> // ED texture fibre shows the larger fractions of the boundaries showing the large misorientations (i.e. ≥ 8 °), but the <111> // ED texture fibre shows the larger fractions of the boundaries showing the small misorientations (i.e. < 8 °). Moreover, this phenomenon is also consistent with the misorientation profile observation as shown in Figure 6.5c, where more large misorientation angles can be observed in the texture component of <001> // ED.     Returning to Figure 6.4, there are generally two types of HAGBs in this region, i.e. the one between the two texture components and the one within the two texture components. The density of the boundaries is defined by the ratio of the measured length of the boundaries to the measured area (e.g. the measured area is 600  150 μm in this case). The density of the HAGBs in between the two texture components is 179 mm-1, which is similar to that within the two texture components, i.e. 173 mm-1. Further, according to Figure 6.5, the densities of LAGBs and HAGBs and subgrain sizes can further be separately analyzed for the two texture components (within 15 ° of the ideal orientation) and the values are summarized in Table 6.3. Table 6.3: Summary of the quantitative microstructure properties counted separated in the texture fibres of <001> // ED and <111> // ED  Subgrain size in equivalent diameter (μm) LAGB density (mm-1) HAGB density (mm-1) All grains 2.0 (EBSD) 2.0 (back-estimated by boundary densities) 870 173 <001> // ED 2.4 (EBSD) 2.5 (back-estimated by boundary densities) 653 201 <111> // ED 1.9 (EBSD) 1.8 (back-estimated by boundary densities) 1013 157  To estimate the boundary densities inside the individual texture component, the value is defined by the ratio of the measured length of the boundaries to the area of the certain texture component (e.g. for the <001> // ED texture component, the area is 34 %  600  150 μm). To 6   Ideal deformation modes: axisymmetric, plane strain and simple shear  82  estimate the subgrain sizes from the EBSD maps, a subgrain is defined by an area in the EBSD map which is either fully enclosed by LAGBs or enclosed partially by LAGBs and partially by HAGBs. As shown in Table 6.3, the subgrains of the <001> // ED texture component (within 15 ° of the ideal orientation) have an average size of 2.4 μm (in equivalent diameter), while the average subgrain size of the <111> // ED texture component (within 15 ° of the ideal orientation) is 1.9 μm (in equivalent diameter). Alternatively, the subgrain size can be back-estimated by the LAGB and HAGB densities assuming that all subgrains have a uniform hexagonal shape using the following equations (the detailed description of this approach is shown in Appendix 2): 23GBhexl   (6.2) 2 26 3hex hexd l    (6.3) where, ρGB is the area density of all the boundaries (i.e. ρGB = ρLAGB + ρHAGB, based on the definition of subgrains), lhex is the length of the side for the hexagon, and the dhex is the equivalent diameter of the hexagon. Based on the Equations 6.2 and 6.3, the back-estimated subgrain sizes for the <001> // ED and <111> // ED texture components are shown in Table 6.3. For the two texture components, the back-estimated subgrain sizes are similar to the values directly determined using the EBSD maps.  Furthermore, in the area having the orientations within 15 ° of the ideal <111> // ED texture component, the density of LAGBs is almost twice as high as that in the area having the orientations within 15 ° of the ideal <001> // ED texture component. On the other hand, the density of HAGBs in the <001> // ED texture component is larger than that in the <111> // ED texture component. This observation is consistent with the observations discussed earlier for the 6   Ideal deformation modes: axisymmetric, plane strain and simple shear  83  <001> // ED texture component (see Figures 6.5c, 6.6c and 6.7). Moreover, it should be noted that besides the area densities, the alignment of the HAGBs within the two texture components are also different, i.e. for the <001> // ED texture component, some of the HAGBs are aligned with the direction normal to the extrusion direction (see Figure 6.6c), but for the <111> // ED texture component, most of the HAGBs are aligned with the extrusion direction (see Figure 6.6g).      Additionally, it is useful to characterize the misorientation distributions within the subgrains. To observe the detailed microstructure, a high resolution EBSD map with a step size of 50 nm was measured in the central region of the extrudate, and the high resolution IPF map is shown in Figure 6.8a.  Figure 6.8: For the centre of the extrudate homogenized at 375 °C with 24 h: (a) a high resolution of IPF map (step size: 50 nm), where the contrast is combined with IQ; and (b) the plot of the kernel average misorientation histograms for the areas having the orientations within 15 ° of the ideal <001> // ED texture component and that of the ideal <111> // ED texture component  To clearly observe subgrain morphology, the contrast shown in Figure 6.8a is not only from crystal orientation but also from the image quality (IQ) of diffraction patterns. A value of IQ generally describes the sharpness of diffraction patterns. When an area has a large fraction of 6   Ideal deformation modes: axisymmetric, plane strain and simple shear  84  defects, such as dislocations (or LAGBs), the measured diffraction pattern will become blurred and a lower IQ value showing a darker grey in the contrast will be obtained. Therefore, with the contrast from IQ, the morphology of subgrains can be observed more clearly. As can be seen in Figure 6.8a, nearly equiaxed subgrains can be generally observed for the both texture components. To quantitatively analyze the misorientation distribution within the subgrains, Figure 6.8b illustrates the kernel average misorientation (KAM) histograms which are calculated within the subgrains of the both texture components based on the second nearest neighbours. Calcagnotto et al. [148] reported that in the  case of  an EBSD a step size of 200 nm, using the second nearest neighbour in the KAM determination was appropriate to describe the distribution of geometrically necessary dislocations. As schematically illustrated in the inset of Figure 6.8b, the KAM is defined by the average misorientations between the measured point (the red hexagon in the inset in Figure 6.8b) and all of its second nearest neighbours (the blue hexagons in the inset in Figure 6.8b). As can be seen in Figure 6.8b, for the areas separately within the two types of subgrains, most of KAM values are lower than 0.5 °, which is similar to the experimental orientation resolution of the EBSD system. Based on the analysis of the KAM values within the subgrains, it is suggested that the misorientations formed within the subgrains are minor for both texture components, i.e. the subgrain interiors have low dislocation densities.    6.2.2 Plane strain deformation: the centre of the strip extrusions 6.2.2.1 Textures and microstructures in the centre of the strip extrusions Figure 6.9 illustrates the microstructures and textures in the centre of the as-extruded strips for 6   Ideal deformation modes: axisymmetric, plane strain and simple shear  85  the materials homogenized at 600 °C for 24 h and at 375 °C for 24 h. The extrusion direction (ED) and normal direction (ND) in the IPF maps of Figure 6.9a and c are aligned with the vertical and horizontal directions (on the ED-ND plane), respectively.   Figure 6.9: IPF maps showing the as-extruded microstructures in the centre of the ED-ND plane of the strip extrusion (at 350 °C with an extrusion ratio 33:1) for the materials homogenized (a) at 600 °C for 24 h and (b) at 375 °C for 24 h; and pole figures showing the experimentally determined textures in this region for the materials homogenized (c) at 600 °C for 24 h and (d) at 375 °C for 24  For the strip extrusions, the strain in thickness direction is much larger than that in width direction (i.e. the thickness reduction is much larger than the width reduction), therefore the deformation condition in the centre of the strip is very close to plane strain deformation. The experimentally determined textures are plotted in the {001} and {111} pole figures of Figures 6.9c and 6.9d, and the vertical and horizontal directions in the pole figures are aligned with the extrusion direction and transverse direction (TD), respectively. 6   Ideal deformation modes: axisymmetric, plane strain and simple shear  86  As can be seen in Figure 6.9a, after the strip extrusion, the material homogenized at 600 °C for 24 h (the condition with almost no dispersoids) formed the equiaxed grains. This suggests that recrystallization occurred in this condition. Similar to the axisymmetric extrusions discussed in Section 6.2.1.1, it is difficult to assess whether recrystallization occurred during the deformation or after deformation and prior to the quench, since the time between the material leaving the die and entering the water quench was ≈ 1.4 s for these strip extrusions. The experimentally determined texture in the centre of the as-extruded strip shows a transition from the cube orientation {001} <100> via the cubeRD orientation {013} <100> and finally to the Goss orientation {110} <100>, as shown in the pole figures of Figure 6.9c. Similar textures have been observed for the annealing of cold rolled and hot rolled aluminum alloys [53, 65-67, 76, 149-153], and the formation of this texture is generally accepted to be attributed to the recrystallization from the cube-bands formed in the as-deformed microstructure [65-67, 76, 151-153]. Therefore, this condition will be referred to as the recrystallized condition of the strip extrusion. The quantitative study of this texture will be further discussed in the next section. In addition, Figure 6.10 shows the grain size (in equivalent diameter) distribution for the recrystallized condition in the strip extrusions, where the distribution is fit with the lognormal distribution with a mean value of 7 μm in diameter and a standard deviation of 4 μm. To quantitatively describe the spread of the lognormal distribution, the ratio of the standard deviation to the mean value was used, i.e. a smaller value suggests a distribution with more uniform grain sizes. In comparison with the recrystallized condition of the axisymmetric extrusion, in which the ratio is equal to 1.1, the grains here show a smaller value of 0.6. This suggests that in comparison with the axisymmetric extrusion, the average grain size is smaller and has a narrower distribution. 6   Ideal deformation modes: axisymmetric, plane strain and simple shear  87   Figure 6.10: Plot of the grain size (in equivalent diameter) distribution for the central region of the extrusion with the homogenization at 600 °C for 24 h (the recrystallization condition), and the experimental distribution is fitted by a lognormal distribution with the certain equation and parameters as shown in the figure  For the material homogenized at 375 °C for 24 h (the condition with a high density of dispersoids), Figure 6.9b illustrates that the grains were strongly elongated along the extrusion direction. Their texture (see Figure 6.9d) shows a typically as-deformed plane strain texture (a mixture of the copper orientation {112} <111>, S orientation {123} <634>, and brass orientation {011} <211>), which has been widely reported in the rolling of various aluminum alloys [38, 67, 68, 81, 154]. Therefore, this condition will be referred to as the unrecrystallized or as-deformed condition of the strip extrusion, and the quantitative study of this texture will be further discussed in the next section.   6.2.2.2 Quantitative analysis of the textures in the centre of the strip extrusions To quantitatively analyze the textures, the fractions of the characterized texture components in the strip extrusions will be further determined, i.e. the cube orientation, Goss orientation, copper 6   Ideal deformation modes: axisymmetric, plane strain and simple shear  88  orientation, S orientation and brass orientation (the five major texture components in the plane strain deformations in aluminum alloys [38, 53, 67, 68, 81, 149-155]). Similar to the methods applied for the quantitative texture analysis for the axisymmetric extrusions, two approaches were applied to identify the texture fractions, i.e. first, the area fraction of a specific texture component was directly calculated from the EBSD maps (using the OIM Analysis software), and second, the volume fraction of a specific texture component was calculated from the experimentally determined ODF from the EBSD maps (using the MTEX software). Moreover, to calculate ODF, a large area of 600  150 μm and the harmonic series expansion approach with an expansion rank of 16 was used. A deviation angle of 10 ° was applied to identify the certain orientation or texture component for the cases of the strip extrusions. The detailed study about choosing the proper deviation angle is shown in Appendix 3. Figures 6.11a and 6.11b show the re-calculated textures based on the ODFs for the recrystallized and unrecrystallized conditions in the strip extrusions. In comparison with the experimentally determined pole figures (shown in Figures 6.9c and 6.9d), the re-calculated pole figures show the similar textures but with lower texture intensities. In detail, for the recrystallized condition, the maximum intensity is 10.8 MRD for the re-calculated pole figures and 12.1 MRD for the experimentally determined pole figures, and for the unrecrystallized condition, the maximum intensity is 8.8 MRD for the re-calculated pole figures and 11.4 MRD for the experimentally determined pole figures.  6   Ideal deformation modes: axisymmetric, plane strain and simple shear  89   Figure 6.11: Plot of the re-calculated PFs based on the ODF showing the textures in this region for the materials homogenized (c) at 600 °C for 24 h and (d) at 375 °C for 24 h  The difference between the experimentally determined and re-calculated pole figures was also evaluated as the value Dtex [145]. Table 6.2.4 summarizes the Dtex values calculated based on the experimentally determined pole figures shown in Figures 6.9 c and 6.9d and the re-calculated pole figures shown in Figures 6.11a and 6.11b. The results in Table 6.2.4 show that all the calculated the Dtex values are less than 0.4 [145], which suggests that the difference between the experimentally determined pole figures and the re-calculated pole figures are minor. Therefore, the calculated ODF accurately represents the experimental textures and can be used for the further calculation of the volume fractions for the certain texture components.  Table 6.4: Summary of the calculated parameter Dtex based on the experimentally determined pole figures shown in Figure 6.9c and d and the re-calculated pole figures shown in Figures 6.11a and 6.11b  {001} pole figure {111} pole figure Recrystallized condition (homogenization at 600 °C for 24 h) 0.05 0.04 Unrecrystallized condition  (homogenization at 375 °C for 24 h) 0.12 0.13  6   Ideal deformation modes: axisymmetric, plane strain and simple shear  90  Tables 6.5 and 6.6 summarize the area fractions directly determined from the EBSD maps and the volume fractions calculated from the ODFs for the five major texture components of the recrystallized and unrecrystallized conditions.  Table 6.5: Summary of the identified volume / area fractions of the texture components for the texture in the central region of the strip extrusion for the recrystallization condition (a deviation angle of 10 ° was employed) Calculation approach Fraction (%) cube Goss copper S brass others From EBSD map in area 13 8 5 4 2 68 From ODF in volume 16 9 3 8 5 59  Table 6.6: Summary of the identified volume / area fractions of the texture components for the texture in the central region of the strip extrusion for the as-deformed condition (a deviation angle of 10 ° was employed) Calculation approach Fraction (%) cube Goss copper S brass others From EBSD map in area 2 1 15 33 24 25 From ODF in volume 4 1 13 30 23 32  For the recrystallized condition, the area fractions directly determined from the EBSD map show similar results to the volume fractions calculated from the ODF, i.e. both the cube orientation (i.e. its fraction is 13 - 16 %) and the Goss orientation (i.e. its fraction is 8 - 9 %) are generally the dominant texture components. It should be noted that in the recrystallized condition, there are 59 - 68 % of the total orientations which are not accounted for with the five texture components. It is suggested that the orientation spread in this condition is large, i.e. much larger than the deviation angle (i.e. 10 °) applied for the identification of the texture components.  For the unrecrystallized condition, a good agreement can be also found between the area fractions determined directly from the EBSD map and the volume fractions calculated from the ODF. In detail, the S orientation is the dominant texture component (i.e. its fraction is 30 - 33 %) with smaller fractions of the brass orientation (i.e. 22 - 23 %) and the copper orientation (i.e. 13 - 15 %). Furthermore, 25 - 32 % of the total orientations cannot be accounted for with the five 6   Ideal deformation modes: axisymmetric, plane strain and simple shear  91  texture components in this condition.      For the plane strain deformations, the β texture fibre can be also used to characterize the deformation texture. In detail, the β texture fibre shows the calculated texture intensities in the ODF from the copper orientation and through the S orientation and finally to the brass orientation [39]. Figure 6.12 illustrates the β texture fibres for the recrystallized and unrecrystallized (as-deformed) conditions. For the recrystallized condition, since the cube and Goss orientations are the dominant texture components, the β texture fibre shows low intensities (less than 5 MRD). On the other hand, for the unrecrystallized (as-deformed) condition, the β texture fibre shows stronger intensities (higher than 10 MRD), and it shows the highest intensity around the S orientations (i.e. ≈ 20 MRD) with relatively lower intensities around the copper and brass orientations.   Figure 6.12: Plots of the β texture fibre for the recrystallized and unrecrystallized (or deformed) conditions      In conclusion, for the recrystallized condition of the strip extrusion, the as-extruded texture 6   Ideal deformation modes: axisymmetric, plane strain and simple shear  92  shows a transition from the cube orientation to the Goss orientation. For the unrecrystallized (as-deformed) condition of the strip extrusion, the as-extruded texture shows the typically as-deformed plane strain texture, i.e. a mixture of the copper orientation, S orientation and brass orientation, and based on the quantitative characterization of the plane strain texture, the S orientation is the most dominant texture component and the fractions of the brass and copper orientations are relatively lower.   6.2.2.3 Microstructures for the centre of the strip extrusion: the unrecrystallized case Figure 6.13a shows an IPF map in the centre of the ED-ND plane for the as-extruded strip of the unrecrystallized condition (the material homogenized at 375 °C for 24 h and forming a high density of dispersoids). In addition, Figure 6.13b illustrates the LAGB and HAGB distributions for the microstructure as shown in Figure 6.13a. Figure 6.13c illustrates the misorientation profile for the black arrow in Figure 6.13b, and the misorientation angle in Figure 6.13c is defined by the misorientation between the original point and the current point. As can be seen in Figure 6.13b, similar to the axisymmetric extrusion (see Figure 6.4b), most of the HAGBs are aligned with the extrusion direction, and in between the HAGBs the equiaxed subgrains with small sizes are observed. Nevertheless, the thicknesses of the elongated grains are generally thinner than that in the axisymmetric extrusion, as shown in the magnified inset in Figure 6.13b.  6   Ideal deformation modes: axisymmetric, plane strain and simple shear  93   Figure 6.13: (a) IPF map showing the as-extruded microstructure in the centre of the ED-ND plane for the as-extruded strip, (b) the map showing the LAGB and HAGB distributions in the microstructure as shown in Figure 6.13a, and the misorientation profile showing the misorientation angles between the current points with the original point as pointed by the black arrow in Figure 6.13d  6   Ideal deformation modes: axisymmetric, plane strain and simple shear  94  To quantitatively analyze the grain thickness, based on the misorientation profile shown in Figure 6.13c, when the misorientations are larger than 15 °, a HAGB is defined and thus, the grain thickness will be measured as the distance between these HAGBs (the same approach with the case in axisymmetric extrusion as discussed in Section 6.2.1.3). Based on this approach, the different grains are labelled in Figure 6.13c. As can be seen in Figure 6.13c, the estimated grain thicknesses are much less than that in the axisymmetric extrusion condition (see Figure 6.4c). As shown in the Figure 6.13c, most of the grains, in this case, show a thickness which is only a few microns. To obtain a better statistic for the grain thickness, 50 misorientation profiles with a length of 600 μm along the normal direction (the horizontal direction in the IPF map of Figure 6.13a) were included to estimate the average thickness of the grains in the centre of the as-extruded strip. The estimated average grain thickness is 2.7 μm in 2-D. Grajales [18] reported an approach to estimate the change of grain dimension caused by the geometrical change in the deformation. Based on this approach and with an assumption that the as-extruded grains have a cubic shape with a rectangular cross-section normal to the extrusion direction (since the product is extruded into a strip in this case), the estimated grain thickness is 2.0 μm in 2-D (the detail about the estimation of the grain thickness in the strip extrusion is shown in Appendix 4). In comparison with the experimentally determined grain thickness, i.e. 2.7 μm in 2-D, the estimated grain thickness based on the geometrical change in the deformation has the similar value, i.e. 2 - 3 μm. This suggests that for the material with the high density of dispersoids (i.e. the homogenization at 375 °C for 24 h), the average thickness of the elongated grains formed in the strip extrusion is also attributed to the deformation. It is worth noting that the assumption of the as-extruded grain geometry in this approach is simple (i.e. a cube with a rectangular cross-section normal to the extrusion direction), compared with the practical grain shapes. Bate et al. 6   Ideal deformation modes: axisymmetric, plane strain and simple shear  95  [37] considered the as-deformed grain shape as an ellipsoid and reported the changes of the grain surface areas during the different deformation modes. To improve the estimation of the grain thickness in this condition, a similar approach to that reported by Bate et al. [37] is suggested to be studied in the future study. Figure 6.14 illustrates a representative example of the spatial distributions of the five texture components on the ED-ND plane (the orientations within 10 ° of the ideal orientations are applied to identify the certain texture components). As can be seen in Figure 6.14, generally for all of the five texture components, most of the elongated grains are discontinuous along the extrusion direction. For this reason, the misorientation analysis within each texture component is not possible in contrast to the case of the axisymmetric extrusion. Furthermore, the cube-bands formed in the deformed microstructure were reported in the previous studies [65-67, 76, 151-153], which are considered as the orientation which has the advantage of growth in recrystallization and will finally become the predominant texture component in the recrystallized texture. Some similar microstructures as pointed out by the red arrow in Figure 6.14e can also be observed, but rather than a continuous band-structure along the extrusion direction for the classic cube-band [65-67, 76, 151-153], some discontinuous grain islands showing the cube orientations is observed in this condition.  6   Ideal deformation modes: axisymmetric, plane strain and simple shear  96   Figure 6.14: (a) IPF map showing the a representative example of the as-extruded microstructure in the centre of the as-extruded strip, and maps illustrating the spatial distributions of the (b) S, (c) copper, (d) brass, (d) cube, and (e) Goss texture components (the orientations within 10 ° away from the ideal orientations)  6   Ideal deformation modes: axisymmetric, plane strain and simple shear  97  For each texture component, the subgrain sizes in equivalent diameter are summarized in Table 6.2.7, based on the definition of the subgrain introduced in Section 6.2.1.3 and based on a large area of 600  150 μm (which covers more than 20,000 subgrains in total). As can be seen in Table 6.2.7, the subgrain sizes for the S, copper and brass texture components are almost the same, but the cube orientation shows a large average subgrain size (2.2 μm) and the Goss orientation has a small average subgrain size (1.6 μm). Furthermore, it should be mentioned here that the fractions of the cube and Goss orientations are minor in the unrecrystallized condition, thus the estimated subgrain sizes for the both texture components might contain significant uncertainty.  Table 6.7: Summary of the estimated subgrain sizes in equivalent diameter (μm) for each texture components for the deformed condition of the as-extruded strip  cube Goss copper S brass average Subgrain size in equivalent diameter (μm) 2.2 1.6 1.9 1.9 1.8 1.9   6.2.3 Simple shear deformation: torsion tests In a general case of extrusion, a large amount of shear takes place on the surface of the extruded product due to the friction between the high speed flowing materials and static deformation tools (extrusion feeder and die). Therefore, to further study the textures and microstructures on the extrusion surfaces, simple shear deformation, i.e. torsion, was studied and will be reported in this section and then, further compared with the surface textures and microstructures of the extrusion products in the next chapter. The torsion tests were conducted at 350 °C to an equivalent final strain 12 (based on Equation 4.1) on the surface and with an equivalent surface strain rate of 10 s-1 (based on Equation 4.2). Figure 6.15 plots the flow curves 6   Ideal deformation modes: axisymmetric, plane strain and simple shear  98  for the materials homogenized at 600 °C for 24 h and at 375 °C for 24 h. In the following, the textures and microstructures for the equivalent strains from 5 to 12 will be mainly focused on. Note, Mahmoodkhani has shown using a fully coupled thermal-detection based finite element method model that the drop in the flow stress with strain seen in Figure 6.15 is related to adiabatic heating and the corresponding reduction in the flow stress [156]. It was estimated that the final temperature of the sample at the end of the test had risen by 20 °C [156].  Figure 6.15: Plot of the flow curves of the torsion tests for the materials homogenized at 600 °C for 24 h and 375 °C with 24 h   6.2.3.1 Texture evolution with equivalent strains Figures 6.16a - 6.16c illustrate the textures at the equivalent strains of 5, 6 and 12 for the material homogenized at 600 °C for 24 h (the condition with almost no dispersoids) and Figures 6.16d - 6.16f illustrate the textures at the equivalent strains of 5, 6 and 12 for the material homogenized at 375 °C for 24 h (the condition with a high density of dispersoids). In detail, at an intermediate equivalent strain of 5, the material homogenized at 600 °C for 24 h shows a C type 6   Ideal deformation modes: axisymmetric, plane strain and simple shear  99  texture (see Figure 6.16a, and the schematic illustration of the C type texture has been shown in Figure 2.12c). With the increase of the equivalent strain to 6, the texture changes from the C type to a B type, and with a further increase of the equivalent strain to 12, the texture maintains the B type (see Figures 6.16b and 6.16c, and the schematic illustration of the B type texture has been shown in Figure 2.12d). The same trend of the texture variation with the increase of the equivalent strain was observed in the material homogenized at 375 °C for 24 h. Furthermore, in terms of the texture intensities, little difference can be observed between the conditions at the equivalent strains of 5 and 6 for both materials. In addition, from the equivalent strain of 6 to 12, the maximum intensities of the textures only slightly increased in both materials.   Figure 6.16: {111} PFs (without ODF calculations) showing the torsion textures at the equivalent strain (a) 5, (b) 6, and (c) 12 of the material homogenized at 600 °C for 24 h and the torsion textures at the equivalent strain (d) 5, (e) 6, and (f) 12 of the material homogenized at 375 °C for 24 h   6   Ideal deformation modes: axisymmetric, plane strain and simple shear  100  6.2.3.2 Microstructure evolution with equivalent strains Figure 6.17 illustrates the distribution of the LAGBs and HAGBs in the torsion tests for both of the homogenization conditions at the equivalent strains of 5 and 12. The shear direction (SD) in the IPF maps in Figure 6.17 is aligned with the vertical direction and the torsion axis (Z) is aligned with the horizontal direction. The details about the section scenario for the microstructure characterization of the torsion samples can be seen in Figure 4.19.  Figure 6.17: The maps showing the distributions of the LAGBs and HAGBs for the material homogenized at 600 °C for 24 h in the area at the equivalent shear (a) 5 and (b) 12, and the torsion microstructures of the material homogenized at 375 °C for 24 h for in the area at the equivalent shear (c) 5 and (d) 12 in the torsion tests 6   Ideal deformation modes: axisymmetric, plane strain and simple shear  101  As can be seen in the four conditions listed in Figures 6.17a - 6.17d, in contrast to the round bar and strip extrusions in which the LAGBs (i.e. their misorientations < 15 °) are the predominant microstructure, in the torsion tests at equivalent strains of 5 and 12, the HAGBs (i.e. their misorientations ≥ 15 °) are the predominant microstructure in both materials. Furthermore, a quantitative analysis (based on the area 430  200 μm) of the microstructure properties for the four conditions in Figure 6.17 is summarized in Table 6.8.  Table 6.8: The summary of the microstructure properties of the torsion conditions for the different equivalent strains Homogenization condition Microstructure properties Equivalent strain 5 Equivalent strain 12 600 °C - 24 h LAGB density (mm-1) 478 426 HAGB density (mm-1) 1198 1207 grain size (μm) 2.5 2.4 grain aspect ratio 2.3 2.2 375 °C - 24 h LAGB density (mm-1) 432 396 HAGB density (mm-1) 1194 1232 grain size (μm) 2.5 2.4 grain aspect ratio 2.5 2.4  According to Table 6.8, the HAGBs are the predominant microstructure for all four conditions, where their densities are more than twice of that for LAGBs. Since the HAGBs are the predominant microstructure, the refined microstructures as can be seen in all four conditions in Figure 6.17 are considered to be composed of grains rather than subgrains. Therefore, the grain size in equivalent diameter was considered for all four conditions (a grain is defined by an area enclosed with HAGBs in the EBSD maps). As can be seen in Table 6.8, the average grain size in equivalent diameter shows similar values at the equivalent strains of 5 and 12 for both homogenization conditions (i.e. 2.4 - 2.5 μm). Moreover, the grain aspect ratio which is defined by the ratio of the major axis length to the minor axis length of an ellipse fit to an identified grain 6   Ideal deformation modes: axisymmetric, plane strain and simple shear  102  (the best fit was determined by the least squares approach [157]) is almost the same at the equivalent strains of 5 and 12 for both homogenization conditions. In addition, in comparison with the material homogenized at 600 °C for 24 h (in which their grain aspect ratios are 2.2 - 2.3), the grains of the material homogenized at 375 °C for 24 h are more elongated (with the higher grain aspect ratios, i.e. 2.4 - 2.5) at both the equivalent strain of 5 and 12.  At a lower equivalent strain of 1, Figures 6.18a and 6.18b illustrate the distributions of the LAGBs and HAGBs in the microstructures for the materials homogenized at 600 °C for 24 h and at 375 °C for 24 h, respectively.   Figure 6.18: The maps showing the distributions of the LAGBs and HAGBs in the torsion tests at the equivalent strain 1 for the materials with the homogenization (a) 600 °C - 24 h and (b) 375 °C - 24 h, where the red arrows denote the HAGBs which are almost parallel to each other and produced the band-structures  It can be seen that for both materials, the large initial grains (the initial grain size is ≈ 70 μm) are broken up by the formed HAGBs, and the formed HAGBs are almost parallel to each other and produced the band-structures, pointed out by the red arrows in Figure 6.18. Furthermore, inside the band-structures, equiaxed subgrains can be observed in both cases. In detail, the 6   Ideal deformation modes: axisymmetric, plane strain and simple shear  103  average subgrain sizes for the material homogenized at 600 °C for 24 h is 2.5 μm (which is similar to the grain sizes measured at the equivalent strains of 5 and 12 for this material) and that for the material homogenized at 375 °C for 24 h is 2.3 μm (which is also similar to the grain sizes measured at the equivalent strains of 5 and 12 for this material). Moreover, in a comparison of the two materials with different homogenization conditions, no obvious differences can be found at an equivalent strain of 1.   6.3 Discussion and simulations 6.3.1 Simulations for the textures formed by the nearly ideal deformation modes In this section, the textures of the unrecrystallized conditions (i.e. the condition with a high density of dispersoids) will be quantitatively simulated using the visco-plastic self-consistent (VPSC) model. In order to do so, it is necessary to know the strain rate sensitivity at 350 °C of the material homogenized at 375 °C for 24 h (i.e. the condition with a high density of dispersoids). As such, the compression tests at 350 °C with the different strain rates (0.1 s-1, 1 s-1, and 10s-1) were conducted. Figure 6.19 shows the results for the true stress - true strain flow curves. To examine barreling of the compression samples, the amount of barrelling was characterized by Ba = (hc-0  dc-02) / (hc-f  dc-f2) [158], where hc-0 and dc-0 are the initial height and diameter of the compression sample; hc-f and dc-f are the corresponding dimensions after compression test at the centre of the sample. The barreling values for the cases shown in Figure 6.19 were found in a range of 0.91 - 0.93.  The inverse of the strain rate sensitivity value (i.e. n = 1 / m, where m is the strain rate sensitivity value) was adjusted such that the simulated flow curves using the constitutive law in 6   Ideal deformation modes: axisymmetric, plane strain and simple shear  104  the VPSC model (Equations 2.12 - 2.16) matched the experimental steady state flow stresses (plastic true strain 0.3 to 0.6) at the different strain rates as shown in Figure 6.19. In detail, the simulations used the as-cast texture shown in Figure 5.2b as the initial texture. In addition, for the grain interaction term, neff = 1 was used (the determination of the parameter neff will be further discussed in the next sections). Moreover, a mixed boundary condition shown in Equations 4.12 and 4.13 (which have been described in Section 4.12.1 and are shown as follows) was used to describe the strain and stress paths for the compression tests.  0 00 00 0       compL s  (4.12) 00       comp MPa  (4.13) In detail, the compression direction was controlled by the strain rates (i.e. L33, 𝜀̇ = 0.1, 1, and 10 s-1) and for the other normal components, the stresses (σ11 and σ22) were imposed to be zero.   Figure 6.19: Plot of the compression flow curves at 350 °C of the material homogenized at 375 ° for 24 h with different strain rates 0.1 s-1, 1s-1, and10s-1; and the simulated flow curves using the VPSC model (the parameters in the Voce hardening law are shown in Table. 6.2.1 6   Ideal deformation modes: axisymmetric, plane strain and simple shear  105      In addition, an extended version of the Voce law (see Equation 2.7 [34]) was employed for the hardening behaviours in the simulations (the parameters are shown in Table 6.9). By the fitting, the inverse of the strain rate sensitivity value, n, at 350 °C of the material homogenized at 375 °C for 24 h is determined to be 15, and the strain rate sensitivity (i.e. m or Y in Equation 4.4 is ≈ 0.07). Table 6.9: The Voce hardening law parameters used for the simulations as shown in Figure 6.19 (the material homogenized at 375 °C for 24 h and deformed at 350 °C with the strain rate of 0.1 s-1, 1 s-1, and 10s-1) τ0s (MPa) τ1s (MPa) θ0s (MPa) θ1s (MPa) 19.0 6.2 100 0   6.3.1.1 Simulations for the texture formed in axisymmetric deformation: the centre of the axisymmetric extrusions After the axisymmetric extrusion (round bar extrusion), the unrecrystallized condition shows a <001> - <111> double fibre texture. To simulate this texture, the homogenization parameter neff which is for the determination of the interaction tensor M̃ (Equation 4.14) will be adjusted in a range from 1 to n (i.e. = 15) such that the simulated texture can quantitatively match the experimentally determined as-extruded texture. Furthermore, in the simulations, a constant velocity gradient tensor (Equation 4.14, which has been described in Section 4.12.1 and is shown as follows) was imposed on each simulation steps with an equivalent strain increment of 0.04 to a final equivalent strain of 4. For the case of round bar extrusion, the directions of L11 and L22 in Equation 4.14 are the equivalent radial directions of the extrudate, and the direction of L33 is the extrusion direction. 6   Ideal deformation modes: axisymmetric, plane strain and simple shear  106  5 0 00 5 00 0 10      axisy extL s  (4.14) Figure 6.20 shows the {001} and {111} pole figures of the simulations based on the neff values of 1, 5, 10 and 15 for the axisymmetric extrusion. As can be seen in Figure 6.20, all of the simulated textures with the different neff values show the <001> - <111> double fibre texture but with the different maximum intensities and the different fractions of each texture components. In detail, Figures 6.21a and 6.21b illustrate the area fractions of the specific texture components (within 15 ° of the ideal orientations) and the maximum intensities of the overall textures (without the ODF calculation) with the different neff values, respectively.   Figure 6.20: {001} and {111} pole figures showing the simulations based on the neff value of (a) 1, (b) 5, (c) 10 and (d) 15 for the round bar extrusion with the equivalent strain of 4  6   Ideal deformation modes: axisymmetric, plane strain and simple shear  107   Figure 6.21: Plots of (a) the volume fractions of specific texture components and (b) the maximum value of texture intensity in pole figures as a function of neff value  In can be seen in Figure 6.21a that with the increase of neff from 1 (i.e. close to the Taylor condition) to 5, the fraction of the <111> // ED texture component decreases but the fraction of the <001> texture component increases. With further increase neff to 15 (i.e. close to the Sachs condition), both the fractions of these two texture components slightly decrease. The maximum intensity of the overall texture, shown in Figure 6.21b describes the level of texture concentration, and as can be seen in Figure 6.21b, with the increase of neff from 1 to 15, the simulated texture becomes more scattered.   6.3.1.2 Simulations for the textures formed in plane strain deformation: the centre of the strip extrusions As mentioned in Section 6.2.1.1, the as-extruded texture of the unrecrystallized condition in the strip extrusions shows the as-deformed plane strain texture, in which the S orientation is the most dominant texture component and the fractions of the brass and copper orientations are 6   Ideal deformation modes: axisymmetric, plane strain and simple shear  108  relatively lower. In this section, the as-extruded texture of the unrecrystallized condition (shown in Figure 6.9d) will be simulated by the VPSC model. For the texture simulations, a constant velocity gradient tensor (Equation 4.15, which has been described in Section 4.12.1 and is shown as follows) was imposed on each simulation steps with an equivalent strain increment of 0.04 to a final equivalent strain of 4. For the strip extrusion, the directions of L11 and L22 are the normal direction and the transverse direction of the strip, and the direction of L33 is the extrusion direction. 9 0 00 1 00 0 10      strip extL s  (4.15)     Furthermore, to examine the effect of the interaction tensor M̃ (Equation 4.14) on the simulated textures, a similar approach will be applied as in the simulations for axisymmetric extrusion, i.e. the parameter neff will be adjusted from 1 to 15.  Figure 6.22 shows the {001} and {111} pole figures of simulations based on the neff values of 1, 5, 10 and 15 for the strip extrusion. Figure 6.23a shows the area fractions of the S, copper and brass components (within 10 ° of the ideal orientations) in the simulations with the neff values from 1 to 15, and Figure 6.23b illustrates the maximum intensities of the overall textures (without ODF calculations) in the simulations with the neff values from 1 to 15. Contrary to the simulation for the round bar extrusion, where the general texture shows little change with neff (shown in Figure 6.20), in the simulation for the strip extrusion, the simulated texture shows a large change with neff. In detail, when neff = 1 (i.e. close to the Taylor condition), the simulated result shows a texture with a strong copper texture component, shown in Figures 6.22a and 6.23 (where the area fraction of the copper texture component is the largest when neff = 1). With the increase of neff to 5, the simulated results changes to a texture with a strong S texture component, 6   Ideal deformation modes: axisymmetric, plane strain and simple shear  109  shown in Figure 6.22b and 6.23 (where the area fraction of the S texture component is the largest when neff = 5). With further increase of neff to 15 (i.e. close to the Sachs condition), the simulation shows a texture with a strong brass texture component, shown in Figure 6.22b and 6.23 (where the area fraction of the brass texture component is the largest when neff = 15). On the other hand, as can be seen in Figure 6.23b, the maximum intensity for the overall texture (which describes the level of texture concentration) decreases with neff from 1 to 7 and then increases with neff from 7 to 15. In other words, the simulation with neff = 7 shows the most scattered texture, compared with other simulations. In addition, Figure 6.24 shows the simulated β texture fibres based on the different neff values and the experimentally determined β texture fibre for the unrecrystallized condition. As can be seen in Figure 6.24, only the simulation with neff = 7 shows the similar trend to the experimental β fibre, i.e. the S orientation shows the highest intensity and the copper and brass orientations have the relatively lower intensities.  Figure 6.22: {001} and {111} pole figures showing the simulations based on the neff value of (a) 1, (b) 5, (c) 10 and (d) 15 for the strip extrusion with the equivalent strain of 4  6   Ideal deformation modes: axisymmetric, plane strain and simple shear  110   Figure 6.23: (a) Plot of the simulated area fraction of the S, copper and brass orientations with the applied the parameters neff in the simulations, and (b) plot of the simulated maximum texture intensities in pole figures with the applied the parameters neff in the simulations   Figure 6.24: Plots of the simulated β texture fibres based on the different neff values and the experimental β texture fibre   6   Ideal deformation modes: axisymmetric, plane strain and simple shear  111  6.3.1.3 Simulations for the textures formed in simple shear deformation: torsion tests According to the previous studies about the texture variations in the high temperature torsion tests for aluminum alloys [82-89], the C type texture is the deformation texture caused by simple shear. Therefore, the C type texture as shown in Figure 6.16d (the condition of the material homogenized at 375 °C for 24 and at an equivalent strain of 5) will be simulated using the VPSC model in this section. For the simulations, a boundary condition of a constant velocity gradient tensor (Equation 4.16) combined with a constant stress tensor (Equation 4.17) was used to describe the strain and stress paths of the torsion tests, as shown as follows: 10 00 00 0      torsionL s  (4.16) 000       torsion MPa  (4.17) In detail, the normal components were controlled by the stress states, i.e. the normal stresses at the shear direction (σ11), the torsion axis (σ22), and the radial direction (σ33) were imposed to be zero, and the shear component on the shear plane was controlled by the strain rate, i.e. L12 = 10  s-1 for the case of the torsion test at an equivalent surface strain rate of 10 s-1. Furthermore, on each simulation steps, an equivalent strain increment of 0.04 was used for each simulation steps to the final strain 5. Moreover, to determine interaction tensor M̃ (Equation 4.14), the neff value will be adjusted in a range from 1 to 15. Figure 6.25 shows the {111} pole figures of the simulations based on the neff values of 1, 5, 10 and 15 for the torsion tests. 6   Ideal deformation modes: axisymmetric, plane strain and simple shear  112   Figure 6.25: {111} pole figures showing the simulations based on the neff value of (a) 1, (b) 5, (c) 10 and (d) 15 for the torsion test with the equivalent strain of 5  Based on the simulations shown in Figure 6.25, all of the simulated textures show the predominant C type texture. In detail, when the neff values are in the range of 3 - 10, a very strong C type texture is obtained in the simulations. On the other hand, when the neff value is equal to 1 or 15, some other texture components in the simple shear deformation can be still observed, i.e. A ({111} <110> / { 111 } < 110>) and A* ({ 111} <112> / {111 } <112>) orientations. Since the C type texture is a single crystallographic orientation, i.e. {001} <110>, only the maximum values of the texture intensities were used to adjust the match between the simulated textures and the experimentally determined texture. Figure 6.26 illustrates the function of maximum intensities of the overall textures (without ODF calculations) with the neff values from 1 to 15 in the simulations 6   Ideal deformation modes: axisymmetric, plane strain and simple shear  113   Figure 6.26: Plot of the simulated maximum texture intensities in pole figures with the applied the parameters neff in the simulations for torsions   6.3.1.4 Discussion In Sections 6.3.1.1 - 6.3.1.3, the as-deformed (unrecrystallized) textures (the material with a high density of dispersoids) in the three nearly ideal deformation modes (i.e. the centre of the axisymmetric extrusion, the centre of the strip extrusion and the torsion test) were simulated using the VPSC model. For the simulations in each deformation modes, the neff value, which describes the grain interaction, was systematically varied. In this section, an attempt will be made to find a single neff value which provides reasonable texture predictions for all of the three nearly ideal deformations, rather than having separate neff values for each of the deformation modes. For the texture simulations about the axisymmetric extrusion, based on the area fractions for each texture components in the simulations (see Figure 6.21a), the condition neff = 2 gives the best fit with the experimental texture. On the other hand, based on the maximum intensities of 6   Ideal deformation modes: axisymmetric, plane strain and simple shear  114  the simulated textures (see Figure 6.21b), the condition neff = 7 gives the best match with the experimental texture. For the texture simulations about the strip extrusion, according to the β fibres in Figure 6.24, only when the neff value is equal to 7, the simulated β texture fibre gives the similar trend to the experimental β fibre, i.e. the S orientation shows the highest intensity and the copper and brass orientations have the relatively lower intensities. Moreover, as can be seen in Figure 6.23b, the condition neff = 7 also gives the best fit for the maximum texture intensity. Furthermore, according to the fractions for each texture components shown in Figure 6.23a, neff = 7 also gives a general match between the experiment and the simulation. In detail, the S orientation is also the dominant texture component in the simulation, but compared with the experimental texture, the fraction of the S orientation is overestimated. For the texture simulations of simple shear deformation (torsion), all of the simulated textures show the C type texture, but for all of the simulations, the maximum values of the texture intensities in the simulations (i.e. 11 - 20 MRD) are much higher than that from the experiments, i.e. 5.5 MRD. The reasons for the poor prediction are still not clear and are a recommended topic for future study. Based on the axisymmetric and strip extrusion results, it is proposed that using neff = 7 offers a reasonable compromise. Figures 6.27 - 6.29 show the simulated textures for the as-deformed conditions in the axisymmetric extrusion, strip extrusion and torsion, and Table 6.10 and 6.11 summarize the quantitative comparison between the experimentally determined textures and simulated textures for the axisymmetric and strip extrusions. 6   Ideal deformation modes: axisymmetric, plane strain and simple shear  115   Figure 6.27: {001} and {111} PFs showing the simulated texture (neff = 7) for the axisymmetric extrusion  Table 6.10: Comparison of the experimentally determined texture and the simulated texture (neff = 7) for the texture simulation of the as-deformed condition in the axisymmetric extrusions  <001> // ED (in area fraction %) <111> // ED (in area fraction %) Max. intensity (MRD) Experiments 33 66 21.4 Simulations 43 57 21.0   Figure 6.28: {001} and {111} PFs showing the simulated texture (neff = 7) for the strip extrusion  Table 6.11: Comparison of the experimentally determined texture and the simulated texture (neff = 7) for the texture simulation of the as-deformed condition in the strip extrusions  cube (in area fraction %) Goss (in area fraction %) copper (in area fraction %) S (in area fraction %) brass (in area fraction %) Max. intensity (MRD) Experiments 2 0 15 33 24 11.4 Simulations 0 1 23 53 20 11.8  6   Ideal deformation modes: axisymmetric, plane strain and simple shear  116   Figure 6.29: {111} PF showing the simulated texture (neff = 7) for the torsion  When neff value is equal to 7, for the axisymmetric extrusion, the texture prediction using the VPSC model can quantitatively match the experimentally determined texture. As can be seen in Table 6.10, a similar maximum intensity value to the experiment (i.e. ≈ 21 MRD) can be predicted from the simulation. The differences of the area fractions between the experiment and the simulation are 9 - 10 %, but the <111> // ED texture component is still the predominant one in the simulation, compared with the <001> // ED texture component. A similar simulation work using VPSC model was reported by Poudens et al. [59], that the intensity of the <111> // ED texture component is twice as high as that of the <001> // ED texture intensity.  For the strip extrusion, the simulation with neff = 7 can also quantitatively match the experimentally determined texture. As described earlier, the simulation with the neff = 7 shows the best match of the β fibre and the maximum intensity for the overall texture. In terms of the area fractions of each texture components, the simulation can generally match the experimental results, but a large overestimation of the S texture component (i.e. the difference between the simulation and experiment is ≈ 20 % for the area fraction of the S texture component) is predicted from the simulation. Van Houtte et al. [109] and Holmedal et al. [112] compared the texture predictions based on the different crystal plasticity models for plane strain deformations. 6   Ideal deformation modes: axisymmetric, plane strain and simple shear  117  With the consideration of the different aspects, such as the β fibre or the fractions of the different texture components, the different models will show the best fit due to the different physical assumptions in the different models. Therefore, it is hard to find a single model which can perfectly match all of the aspects of the experimental texture. On the other hand, it is worth noting here that there is small fraction of the grains having the cube orientation observed after the strip extrusion, and these grains are important since the recrystallized texture showing a transition from the strong cube orientation to Goss orientation is generally accepted to be attributed to the recrystallization from these grains [65-67, 76, 151-153]. However, in the simulation shown in Table 6.11, there is no cube orientation found after the plane strain deformation with the equivalent strain of 4. For the case of simple shear deformation (torsion), the general type of the deformation texture (i.e. the C type texture) can be also predicted by the simulation with neff = 7, but the maximum texture intensity from the simulation (i.e. 19.3 MRD) is much higher than that from experiment (i.e. 5.5 MRD). Compared with the simulations for the cases of the axisymmetric and strip extrusions, the difference between simulated and experimental textures for the simple shear deformation is the largest in these three nearly ideal deformation modes. Some previous studies [84, 86, 88, 89] reported that the C type texture in the torsion tests can be generally matched using the different crystal plasticity model, but there is still no quantitative comparison between the simulated and experimental textures reported for the simple shear deformations. Li et al. [159] proposed an approach which introduced a co-rotation of the other grains in the VPSC model, and it can smooth the simulated texture intensities in the simple shear deformation. However, the phenomenon of the co-rotation of grains is not confirmed in experiments. In conclusion, based on the simulations with neff = 7, n = 15, and the parameters shown in 6   Ideal deformation modes: axisymmetric, plane strain and simple shear  118  Table 6.9 for the Voce hardening law, the experimentally determined textures can be quantitatively predicted in the cases of the axisymmetric and strip extrusions, but can only qualitatively predicted in the case of the simple shear deformation (i.e. there is a large difference of the texture intensities between the simulated and experimentally determined textures). Compared with the practical extrusion conditions, the current physical assumption in the VPSC model is still relatively simple. For example, the effects of the grain shape and the critical aspect ratio for grain fragmentation [107] on the texture evolution were not considered in the current simulations using the VPSC model. It is recommended that in the future simulation study, the factors about the grain shape could be considered.   6.3.2 Discussion of the recrystallized textures for the ideal deformation modes 6.3.2.1 Discussion on recrystallization for axisymmetric deformation: the centre of the round bar extrusions     For axisymmetric extrusions, as discussed in Sections 6.2.1.1 - 6.2.1.3, the material with almost no dispersoids (with the homogenization at 600 °C for 24 h) formed a recrystallized texture (i.e. the <001> // ED texture component accounts for 56 - 58 % of the total orientations) and equiaxed grain structure. On the other hand, the material with a high density of dispersoids (with the homogenization at 375 °C for 24 h) kept the as-deformed <001> - <111> double fibre texture and formed the elongated grains along the extrusion direction. Some key observations about the as-deformed state (the material homogenized at 375 °C for 24 h and forming a high density of dispersoids) are summarized as follows: (a) in comparison with the <111> // ED texture component, the <001> // ED texture component only occupies 33 - 34 % of the total 6   Ideal deformation modes: axisymmetric, plane strain and simple shear  119  orientations (see Table 6.1); (b) the subgrains in the <001> // ED texture component have a larger average size (see Table 6.2); (c) the phenomenon of the long range lattice rotation across subgrains can be only observed in the <001> // ED texture component (see Figures 6.4 - 6.6); and (d) in terms of misorientation distributions between subgrains, the <001> // ED texture component has a larger fraction of the boundaries showing large misorientations (i.e. ≥ 8 °, see Figure 6.7).  Based on the above observations, a possible recrystallization mechanism based on the theory of continuous dynamic recrystallization (CDRX) [45, 46] is proposed. The theory of CDRX was reviewed in Section 2.4.2. Briefly, CDRX involves the transformation from subgrains to grains, i.e. subgrain coarsening by the grain boundary migration and the concurrent accumulation of misorientations to transfer LAGBs to HAGBs. In the unrecrystallized condition, the existence of a high density of dispersoids results in the observation that the migration of the grain boundaries is strongly inhibited or even stopped by the large Zener drag. Therefore, the transformation from subgrains to grains cannot be achieved and this is why the elongated grains show the as-deformed <001> - <111> double fibre texture. On the other hand, it is assumed that at this relatively high deformation temperature, the dispersoids have little effect on the deformed state. In the recrystallized condition, it is assumed that the same as-deformed state and texture as for the unrecrystallized condition were formed in extrusion. However, in this case, grain boundaries can migrate due to the fact that there are almost no dispersoids in the material homogenized at 600 °C for 24 h. This may occur dynamically during extrusion or statically during the time (1 - 1.4 s) between the end of extrusion and the quench tank. It is practically not possible to determine when exactly this occurs in the present work. In terms of the texture, the recrystallized condition shows a dominant <001> // ED texture 6   Ideal deformation modes: axisymmetric, plane strain and simple shear  120  component (i.e. 56 - 58 %), compared to 33 -34 % in the unrecrystallized condition. To rationalize these results based on the theory of CDRX, it is suggested that the subgrains showing the <001> // ED texture component have kinetic advantages which allow them to coarsen faster and finally consume other subgrains. In detail, based on the Huang - Humphreys model [137] (see Equation 2.20), all of the HAGBs have the same high interfacial mobility, and the mobility of LAGBs increases with their misorientation angles. According to the Huang - Humphreys model and based on observation (d) the <111> // ED texture component has a higher fraction of high angle boundaries with high mobility compared to the <001> // ED texture component. This could rationalize that the subgrains within the <001> // ED texture component having larger subgrain sizes as observed in Table 6.3. Furthermore, when a boundary is migrating, the long range lattice rotations, as observed in the <001> // ED texture component (see Figure 6.5c), may allow the subgrain to accumulate misorientation and transform the LAGBs to HAGBs and finally result in a grain (as defined by having HAGBs as opposed to a subgrain which has LAGBs). Thus, the growth advantage of the subgrains within the <001> // ED texture component may explain how the smaller fraction of the <001> // ED texture component formed in the as-deformed state could turn into the dominant texture in the recrystallized state. With this hypothesis in mind, a phase field model [131, 132, 142] (Equation 4.26) was used to simulate the subgrain coarsening process (this work was conducted in a collaboration with Benqiang Zhu, a former graduate student in UBC). In detail, Read - Shockley relationship [136] (Equation 2.19) and Huang - Humphreys relationship [137] (Equation 2.20) were used to describe the interfacial energy and mobility as the function of the boundary misorientation angles. Furthermore, the grain boundary driving pressure caused by the difference of the stored dislocation densities on the either side of the boundary is neglected due to the observation that 6   Ideal deformation modes: axisymmetric, plane strain and simple shear  121  for the both types of subgrains, most of the KAM values are lower than the orientation resolution in the EBSD system (i.e. the dislocation density within the subgrains is low). In some previous simulation studies [133-135], a phase field model was also used to study grain growth process by considering the boundary energy and mobility as a function of its misorientation angles, but all of the previous studies were based on the synthetic microstructures. In contrast to the previous work, the experimental microstructure from an EBSD map was used as the input for this simulation. In detail, it is assumed that the microstructure and texture from the unrecrystallized condition is representative of the as-deformed state as such the EBSD map of the unrecrystallized condition as shown in Figure 6.2.1b will be used as the input for the simulation. The simulation domain is an area 150 × 150 μm which covers about 4000 subgrains. In the simulation domain, the initial average subgrain size is 2.0 μm (in equivalent diameter) for the <001> // ED texture component and 2.5 μm for the <111> // ED texture component. The small misorientation deviations within the subgrains were ignored in the the simulations so that there is only one orientation applied to represent the orientation of the subgrain, i.e. the most probable orientation. Furthermore, the grid size in the simulation domain is 0.2 μm and five grids were used to describe the width of the boundaries.  Figure 6.30a - 6.30e shows the results for the evolution of microstructure from the simulation. The simulation was stopped at the time when the average grain size (in equivalent diameter) calculated in the simulation domain was similar to the average grain size measured in the recrystallized condition, i.e. 16 μm in equivalent diameter. At the end of the simulation, the predicted time for the subgrain coarsening is 2 s, which is similar to the time between the moments when material was extruded and the material entered water tank, i.e. ≈ 2 s from the finite element simulation [17].  6   Ideal deformation modes: axisymmetric, plane strain and simple shear  122   Figure 6.30: (a) - (e): IPF maps showing the simulated microstructure evolution for the subgrain coarsening process with an input microstructure from the deformation condition as shown in Figure 6.1b, predicted by the phase field model (f) IPF map showing the recrystallized microstructure in the axisymmetric extrusion (i.e. with almost no dispersoids after the homogenization of 600 °C for 24 h)  6   Ideal deformation modes: axisymmetric, plane strain and simple shear  123  It should be mentioned here that the time scale in the simulation cannot be compared or calibrated with experiments directly for the following reasons. First, the mobility used in this simulation is based on a pure aluminum [143] not an alloy where there is solute present [14, 15, 18] , i.e. the effect of solute drag [28] will slow down the boundary migration. Second, the boundary curvature captured in these 2-D simulations is different with reality in 3-D. As can be seen in the simulated microstructures, the grains with the <001> // ED texture component consume their neighbours which have the <111> // ED texture component, and at the end of the simulation, the <001> // ED texture component becomes the predominant texture. It is worth noting that at the end of the simulation, there are also some grains showing the <111> // ED texture component left, i.e. the blue grains in the IPF map of Figure 6.30e. This prediction is consistent with the observation of the recrystallized microstructure, shown in Figure 6.30f. The evolution of some microstructure properties from the simulation are summarized in Figure 6.31. In detail, as can be seen in Figure 6.31a, the area fraction of the <001> // ED texture component (within 15 ° of the ideal orientation) increases to 85 % at the end of the simulation, and that of the <111> // texture component (within 15 ° of the ideal orientation) decreases to 15 % at the end of the simulation. Based on the texture measurement shown in Table 6.2, in the recrystallized condition, the <001> // ED texture component accounts for 56 -58 % of the total orientations and the <111> // ED only accounts for 2 - 3 % of the total orientations in the experiment. With a comparison between the experiment and the simulation, it suggests that the consumption of the <111> // ED texture component in the experiment is faster than that in the simulation and some grains showing other orientations (40 % of the orientations in this condition are not accounted for with the two texture components, shown in Table 6.2) can also survive after recrystallization. 6   Ideal deformation modes: axisymmetric, plane strain and simple shear  124   Figure 6.31: The microstructure evolutions with time: (a) the area fractions of <001> and <111> grains and (b) the subgrain / grain sizes in diameter  Figure 6.31b shows the average subgrain sizes or grain sizes of the two texture components as the function of time. As can be seen in Figure 6.31b, a clear growth advantage of the subgrains showing the <001> // ED texture component can be observed in the simulations. In conclusion, the simulation shows that based on the proposed CDRX hypothesis, the grains with the <001> // ED texture component can grow to the dominant texture. Furthermore, Ma et al. [135] suggested that the spatial distribution of different texture components can strongly change the texture evolution during grain growth. Therefore, it is recommended that in the future, the effect of the spatial distributions of different texture components and microstructures (such as subgrain sizes, densities of HAGBs and LAGBs) on the subgrain growth process will be systematically studied. In addition, the simulations are based in 2-D, so that some effects due to the 3-D microstructure might be missed. For example, the curvature of subgrains in 2-D is different with the real curvature, i.e. in 3-D, as such the coarsening rate driven by grain boundary curvature might be misestimated in the current simulation based on 2-D. Therefore, in the future, 3-D microstructures could be characterized and the simulations based on the 3-D microstructures 6   Ideal deformation modes: axisymmetric, plane strain and simple shear  125  could be conducted.   6.3.2.2 Discussion on recrystallization for plane strain deformation: the centre of the strip extrusions For the unrecrystallized (as-deformed) condition of the strip extrusion, it is proposed that the large Zener drag due to the dispersoids will prohibit the migration of the boundaries and thus, the observations on the as-extruded strip of the material with a high density of dispersoids are representative of the as-deformed state.  For the recrystallized condition, similar to the axisymmetric extrusions, a recrystallization mechanism based on the theory of continuous dynamic recrystallization (CDRX) [45, 46] is proposed. For the recrystallized condition (the material homogenized at 600 °C for 24, and almost no dispersoid formed after the homogenization), the grain boundaries can migrate and the subgrains can be transformed to the equiaxed grain with a large average size, as shown in Figure 6.9. For the texture transition from the unrecrystallized (as-deformed) to recrystallized conditions, i.e. from the as-deformed plane strain texture to the texture showing a transition from the cube orientation to the Goss orientation, it is generally accepted to be attributed to the recrystallization from the cube-bands formed in as-deformed microstructure [65-67, 76, 151-153]. Moreover, the cube orientation has a larger subgrain size (see Table 6.7), compared with the other texture components. In the detailed studies about the microstructure of the cube-bands [65-67, 80, 81, 147], the phenomenon of the long range lattice rotation was also reported. Taking all of these factors, grains or subgrains showing the cube orientation have a growth advantage after deformation. The large subgrain size can help them coarsen relatively faster at the beginning 6   Ideal deformation modes: axisymmetric, plane strain and simple shear  126  stages of subgrain coarsening, and the long range lattice rotation can help them accumulate misorientations and transform LAGBs to HAGBs later during subgrain coarsening. Therefore, these growth advantages can help the cube orientation become the dominant texture component. Nevertheless, one concern should be mentioned that the fraction of the cube orientation formed in the deformed condition is very small (i.e. 2 - 4 %) so that it might be a challenge for the cube orientation to further transform to the dominant texture component (i.e. 13 - 16 %) in the recrystallized texture. On the other hand, the formation of the cube texture component during the deformation is still not clear, since from the simulation shown in Table 6.11, there is no cube orientation predicted from the deformation. Furthermore, the reason to have the Goss orientation (i.e. 8 - 9 %) in the recrystallized condition is still unclear.   6.3.2.3 Discussion on recrystallization simple shear deformation: torsion tests In the torsion tests, the C type texture is the deformation texture caused by the simple shear. On the other hand, the formation of the B type texture, as can be seen in the range of equivalent strains from 6 to 12 for the both materials, is generally accepted to be attributed to the dynamic recrystallization during the torsion tests [83, 84, 91]. In detail, geometric dynamic recrystallization (GDRX) was proposed as the mechanism previous studies [47, 48, 84, 91-93]. In addition, the GDRX process is mainly characterized by the phenomenon of the grain subdivision, i.e. during the deformation, the elongated grains are pinched off by the formed HAGBs and the thickness of the elongated grains will be finally decreased to an order of subgrains. Some key observations on the torsion tests are summarized as follows: (a) for both the 6   Ideal deformation modes: axisymmetric, plane strain and simple shear  127  materials homogenized at 600 °C for 24 h and at 375 °C for 24 h, the texture evolution with equivalent strains are the same, i.e. the C type texture is the dominant texture at the intermediate equivalent strain of 5, and with the increase of the equivalent strain to 6, the texture changes to the B type, and with a further increase of the equivalent strain to 12, the B type texture remains dominant; (b) from the equivalent strain of 5 to 12, the changes of the average subgrain size, HAGB and LAGB densities are very small and the difference between the two materials are also very small; (c) in comparison with the condition at the intermediate equivalent strain of 5, more equiaxed grains are observed at the equivalent strain of 12 for the both materials; and (d) at an equivalent strain of 1, grain fragmentation by the HAGBs produced inside the pre-existed grains can be observed for the both materials, and some equiaxed subgrain can be also observed in the band structures formed by the HAGBs. Based on the texture and microstructure observations as summarized above and the discussion in the literature [47, 48, 82-89, 91-93], GDRX process is also considered to be the case in these torsion tests. In detail, at a low equivalent strain of 1 for the both materials, some HAGBs were observed to be produced inside the pre-existed grains resulting in fragmentation of those grains. In addition, compared with the condition with an equivalent strain of 5, the subgrains are similar and a slightly more equiaxed grain shape was observed at an equivalent strain of 12 for the both materials. This suggests that in the range of equivalent strains from 5 to 12, the elongated grains were continuously pinched off by the HAGBs produced during the deformation. Furthermore, the measured average grain sizes at equivalent strains between 5 and 12 are 2.4 - 2.5 μm for the both materials. At the equivalent strain of 5, the grain size changed by the simple shear deformation was estimated by the VPSC model. In detail, before the deformation, the grain shape was assumed to be a sphere with a diameter of 70 μm (i.e. the equivalent diameter of the 6   Ideal deformation modes: axisymmetric, plane strain and simple shear  128  average size in as-homogenized condition), and after the deformation, the grain was assumed to be deformed into an ellipsoid. Furthermore, all of the other inputs to the VPSC model kept the same with the simulation for the torsion test with the equivalent strain of 5, which has been addressed in Section 6.3.1.4. Based on the grain morphology change estimated by the VPSC model, the spherical grain is deformed into an ellipsoid where one axis (a1) is strongly elongated to 424.4 μm, one axis (a2) keeps almost the same with the radius of the undeformed spherical grain, i.e. 34.6 μm, and the other axis (a3) is reduced into 2.9 μm, and thus, the estimated grain thickness after the simple shear deformation is defined as 2  a3, i.e. = 5.8 μm in 3-D. Compared with the grain size measured in the experiment (i.e. 2.5 in equivalent diameter in 2-D, shown in Table 6.8, and it can be further transformed into 3.2 μm in 3-D by multiplying a correction factor of 1.28 [146]), the estimated grain thickness (i.e. = 5.8 μm in 3-D) is twice as large as the experimental result (i.e. 3.2 μm in 3-D). This suggests that the grain thickness reduction from the simple shear deformation cannot fully satisfy the microstructure refinement shown in Figure 6.17, or on other words, it is suggested that more HAGBs are introduced during the deformation. A proposed hypothesis based on the GDRX process is schematically illustrated in Figure 6.32. As can be seen in Figure 6.32, during simple shear deformation, some HABGs will be generated to pinch of the elongated grains and thus, equiaxed grains can be formed by the newly formed HAGBs, as shown in the experimental microstructure in Figure 6.17. Therefore, the equiaxed grains having a size smaller than the thickness of the elongated grain can be formed after the simple shear deformation and the concurrent GDRX process. 6   Ideal deformation modes: axisymmetric, plane strain and simple shear  129   Figure 6.32: Schematic illustration of the microstructure changed during the simple shear deformation, in which GDRX occurs  This refined microstructure is characteristic when GDRX occurs [91, 93, 160], and this is similar to that observed in the welding region in the friction stir welding reported by Prangnell et al. [91] (the refined microstructure showing the B type texture was observed in the welding region, and the formation of the B type texture was suggested by GDRX process due to the large simple shear). In conclusion, the above observation suggests that GDRX process happens during the torsion tests for both materials. A large amount of simple shear is dominant in the torsion tests and therefore the recrystallization process is proposed to be caused by the phenomenon of the grain subdivision or the elongated grains pinched off by the formed HAGBs. In terms of the crystallographic texture, McQueen at el. [48] suggested the GDRX process should not affect the texture (i.e. simple model of pinching off of grains). However, the texture changes from the C type to the B type were observed in the torsion tests for the both materials. It is suggested that some nuclei with the B type texture orientations are formed in the range of equivalent strains from 5 to 6. The same observation was reported by Barnett et al. [84], i.e. some nuclei showing the B type orientations formed on the serrated HAGBs in the torsions of aluminum alloys at an equivalent strain of 5. With the further shear deformation, the previous 6   Ideal deformation modes: axisymmetric, plane strain and simple shear  130  simulation works based on the viscoplastic polycrystal model from Tóth et al. [87] reported that the B type texture is very stable. This simulation work may explain the observation that at the equivalent strain of 12, the B type texture is still the dominant texture in the torsion tests for the both materials. In the cases of the axisymmetric and strip extrusions, the existence of the dispersoids can change as-extruded textures and microstructures from the recrystallized conditions to the unrecrystallized conditions, which have been discussed in detail in Sections 6.2.2 and 6.2.3. In contrast for the torsion test, there is almost no difference in the textures and microstructures observed for the dispersoid containing alloy (homogenized at 375 °C for 24 h) and the alloy with almost no dispersoids (homogenized at 600 °C for 24 h). The possible reasons might be that first, different recrystallization mechanisms are suggested for the cases of the extrusions and torsion tests, i.e. CDRX is considered to occur in the cases of axisymmetric and strip extrusions and GDRX is considered to take place in the simple shear deformations (torsions). Second, based on the study reported from Bate et al. [37], the different deformation modes can cause different grain surface areas, i.e. the smallest grain surface area is caused by simple shear deformation, compared with the other deformation modes (see Figure 2.4), and the area of the grain surface might play an important role on recrystallization. Nevertheless, the reason for this observation is not clear so far and would be of interest for further study.  131    7 Textures and microstructures formed on the surface of the extrusion products and their changes from the surface to centre  7.1 Introduction     In this chapter, the textures and microstructures on the surface of the extrusion products (axisymmetric and strip extrusions) will be first reported and then, the textures and microstructures in the region between the surface and the centre will be presented In addition, the textures on the surfaces will be compared with the textures from the torsion tests. On the other hand, with a combination of a simulation study based on the finite-element model (FEM), the texture evolution in the area near the surface of the axisymmetric extrusion will be tracked in a partial extruded sample to characterize the texture change between the positions before and after the material flowing through the die exit.  7.2 Textures and microstructures on the surface of the extrusion products and their changes from the surface to centre: the round bar extrusions 7.2.1 Experimental results 7.2.1.1 Textures and microstructures on the surfaces of the round bar extrusions Figures 7.1a and 7.1b show the microstructures on the surface of the extrudates (in the longitudinal planes of the round bar extrudates with the extrusion ratio of 70:1) for the materials with the homogenization at 600 °C for 24 h and at 375 °C for 24 h, respectively. The extrusion direction (ED) in the IPF maps is aligned with the vertical direction. Figure 7.1c and d illustrate 7   Textures and microstructures formed on the surface of the extrusion products and their changes from the surface to centre  132  their textures on the surface in the {001} and {111} pole figures, where the extrusion direction (ED) is aligned with the centre of the pole figures and the radial direction (RD) is at the outer rim.  Figure 7.1: IPF maps showing the microstructures on the surface of the axisymmetric extrusion (at 350 °C with an extrusion ratio 70:1) for the materials homogenized (a) at 600 °C for 24 h and (b) at 375 °C for 24 h in the longitudinal plane; and {001} and {111} pole figures showing the textures in the subsurface area marked in Figure 7.1a for the materials homogenized (c) at 600 °C for 24 h and the surface in Figure 7.1b for the material homogenized (d) at 600 °C for 24 h  For the material homogenized at 600 °C for 24 h (the case with almost no dispersoids), a layer of peripheral coarse grains (PCGs) with a thickness approximately 100 μm [161] was formed on the surface of the extrudate, as shown in Figure 7.1a. For the area adjacent to the PCG layer, equiaxed grains with an average grain size of 8.4 μm in equivalent diameter were formed. In this case, the texture of the PCG layer will not be studied in detail, but instead, the texture of the area adjacent to the PCG layer, which is marked by the dashed rectangular in Figure 7.1a and with a 7   Textures and microstructures formed on the surface of the extrusion products and their changes from the surface to centre  133  thickness about 200 μm, is shown in Figure 7.1c. On the other hand, for the material homogenized at 375 °C for 24 h (the condition with a high density of dispersoids), there is no PCG layer formed on the surface of the extrudate, shown in Figure 7.1b, and the grains were slightly elongated along with extrusion direction and have an average grain size of 3.6 μm in equivalent diameter.  As discussed in the Section 6.2.1.1, different textures found in the central region of these two extrudates, i.e. the predominant <001> // ED texture is formed in the material homogenized at 600 °C for 24 h (i.e. the recrystallized condition), and the <001> - <111> double fibre texture is formed in the material homogenized at 375 °C for 24 h (i.e. the unrecrystallized or as-deformed condition). For the textures on the surfaces of the extrudates, as can be seen in Figures 7.1c and 7.1d, the similar textures are observed on the surfaces. For the areas close to the surface of the extrudates, a large amount of simple shear takes place due to the friction between the aluminum and the extrusion feeder and die [17, 60]. Therefore, to further study the textures shown in Figures 7.1c and 7.1d, the textures on the surface will be compared with the textures in the torsion tests and it will be further discussed in Section 7.2.1.3.    7.2.1.2 Textures and microstructures through the radius of the axisymmetric extrusions Figures 7.2a - 7.2f illustrate the changes of the textures and microstructures from the surface to the centre of the longitudinal planes for the axisymmetric extrudates (with the extrusion ratio of 70:1) with the materials homogenized at 375 °C for 24 h and at 600 °C for 24 h, respectively. 7   Textures and microstructures formed on the surface of the extrusion products and their changes from the surface to centre  134   Figure 7.2: For the extrudate with the homogenization at 375 °C for 24 h: (a) IPF maps showing the microstructure from the surface of the extrudate to the area 1.2 mm away from surface and also the centre area, (b) Plot of the area fractions of the <001> // ED and <111> // ED texture components from the surface to centre, and (c) {001} and {111} pole figures showing the texture of the area from 0.8 mm to 1.2 mm away from surface; and for the extrudate with the homogenization at 600 °C for 24 h: (d) IPF maps showing the microstructure from the surface of the extrudate to the area 1.2 mm away from surface and also the centre area, (e) Plot of the area fractions of the <001> // ED component from the surface to centre, and (c) {001} and {111} pole figures showing the texture of the area from 0.8 mm to 1.2 mm away from surface 7   Textures and microstructures formed on the surface of the extrusion products and their changes from the surface to centre  135  As discussed in Section 6.2.1.1, for the extrudate with the material homogenized at 375 for 24 h (i.e. the condition with a high density of dispersoids), the texture in the central region shows the as-deformed <001> - <111> double fibre texture. Based on that, Figure 7.2b plots the area fractions of the <001> // ED and <111> // ED texture components (within 15 ° of the ideal orientation) through the radius. In detail, the each point in Figure 7.2b illustrates the fraction measured in an EBSD map with a length of 0.6 mm along the radius, which also means that the each point in Figure 7.2b represents the area from this point to the next one (the distance between the points is 0.6 mm). As can be seen in Figure 7.2b, the texture only changes significantly close to the surface. From the region of 0.6 mm away from the surface to the central region, the area fraction of the <111> // ED texture component decreases from 80 % to 64 % and the area fraction of the <001> // ED component increases from 19 % to 34 %. Furthermore, the texture in the area from 0.8 to 1.2 mm away from the surface is shown in Figure 7.2c. As can be seen in Figure 7.2c, the texture turns to the <001> - <111> double fibre texture in this area. However, it is worth noting here that there are some differences between the texture in the central region (shown in Figure 6.1d) and the texture shown in Figure 7.2c. First, compared with the texture in the central region, the symmetry around the extrusion direction in the texture shown in Figure 7.2c is broken, and a single orientation (i.e. the three poles observed in the {111} pole figure in Figure 7.2c) has been found, rather than the randomly distributed orientations around the extrusion direction (shown in Figure 6.1d). Second, the area fractions of the <001> // ED and <111> // ED texture components are different with that in the central region, as shown in Figure 7.2b. For the extrudate with the material homogenized at 600 °C for 24 h (i.e. the condition with almost no dispersoids), the texture in the central region shows the predominant <001> // ED 7   Textures and microstructures formed on the surface of the extrusion products and their changes from the surface to centre  136  texture component. Based on that, Figure 7.2e plots the area fraction of the <001> // ED texture component (within 15 ° of the ideal orientation) through the radius. As can be seen in Figure 7.2e, similar to the extrusion of the materials with a high density of dispersoids, the texture only changes significantly close to the surface. In addition, by the detailed texture characterization in the areas near the surface (as shown in Figure 7.2f), it is found that in the area from 0.8 to 1.2 mm away from the surface, the texture turns to a similar texture as found in the central region, i.e. the predominant <001> // ED texture component. However, compared with the texture in the central region (shown in Figure 6.1c), an almost single orientation (i.e. the four poles observed in Figure 7.2f) can be found in the texture close to the surface, rather than the randomly distributed orientations around the extrusion direction (shown in Figure 6.1c) In conclusion, for the both conditions, the textures are only strongly changed on the surface areas, but from the area close to the surface (i.e. from 0.8 mm away from the surface), the texture will turn to the similar texture with their central regions. For the detailed studies about the texture change in the region from the surface to the 0.8 mm away from surface, the extrudate with the homogenization of 375 °C for 24 h (with a high density of dispersoids, so that the recrystallization process can be supressed as far as possible) will be examined in detail in the next section.   7.2.1.3 Detailed studies about the textures and microstructures near the surface of the extrudate with the homogenization at 375 °C for 24 h A large amount of simple shear occurs on the surface of the extrusion product. The textures and microstructures formed in the simple shear deformation were studied in the torsion tests (in 7   Textures and microstructures formed on the surface of the extrusion products and their changes from the surface to centre  137  the Section 6.2.3). As observed in the torsion tests, the texture changes with the equivalent strain. At the intermediate level of the equivalent strain 5, the C type is formed, which is determined to be the deformation texture based on the simple shear. With the increase of the equivalent strain to 6, the texture is transformed to the B type, which is proposed to be the recrystallized texture trigged by the simple shear, and with the further increase of the equivalent strain to 12, the B type texture will still remain. In detail, the recrystallization process is proposed to be the GDRX in the torsion tests, which is mainly characterized by the grain fragmentation or the phenomenon that the elongated grains are continuously pinched off by the formed HAGBs during the recrystallization process. Figure 7.3a illustrates the estimated equivalent strains from the surface to the centre of the extrudate with the homogenization at 375 °C for 24 based on the finite-element model (FEM) (the data as shown in Figure 7.3a was supplied by Mahmoodkhani in University of Waterloo, and the details about the extrusion simulations based on the FEM were discussed in the ref. [17, 44]). As can be seen in Figure 7.3a, the shear strain decreases from 12 on the surface to 5 in the area 0.6 mm away from the surface and it further decreases to 0 in the centre. A similar trend of the shear strain distribution from the surface of the extrudate to the centre was observed in the experiments reported by Kaneko et al. [60]. Figure 7.3b shows the microstructures from the surface to the area 0.8 mm away from the surface. Furthermore, Figure 7.3c illustrates the texture on the surface (highlighted by the black dashed rectangular in Figure 7.3b), and Figure 7.3d illustrates the texture in the area from 0.6 mm to 0.75 mm away from the surface (highlighted by the white dashed rectangular in Figure 7.3b. 7   Textures and microstructures formed on the surface of the extrusion products and their changes from the surface to centre  138   Figure 7.3: (a) Plot of the estimated shear strains from the surface of the extrudate to the centre (the data is supplied by Mahmoodkhani in University of Waterloo, and the details about the extrusion simulations based on the FEM are discussed in the ref. [17, 44]), (b) IPF map showing the microstructure from the surface of the extrudate to area 0.8 mm away from the surface, (c) {111} PF showing the texture on the surface of extrudate, and (d) {111} PF showing the texture in the area 0.6 to 0.75 mm away from the surface  7   Textures and microstructures formed on the surface of the extrusion products and their changes from the surface to centre  139  To compare the textures on the surface of the extrudate with the textures in the torsion tests, the sample references in the PFs should be aligned, i.e. the extrusion direction on the surface of the extrudate should be aligned with the shear direction in torsion (ED extrusion // SD torsion), and the radial direction of the extrudate should be aligned with torsion axis (RD extrusion // Z torsion). After the sample reference alignment, the projection of the texture on the surface is transformed from the {111} PF in Figure 7.1d, where the centre is aligned with the extrusion direction, to the {111} PF in Figure 7.3c, where the vertical direction is aligned with the extrusion direction. As can be seen in Figure 7.3c, the texture on the surface of the extrudate shows the B type texture. Based on the further analysis of the texture, the local shear direction can be determined on the surface, which shows a certain degree of rotation (about 50 °) away from the extrusion direction, as pointed by the red dashed arrow in Figure 7.3c (compared to Figure 2.12d). For the area which is 0.6 mm away from the surface, the shear strain decreases to values below 5 and the texture in Figure 7.3d (after the alignment as mentioned above, i.e. that the vertical direction in the PF is aligned with extrusion direction) shows the C type texture. Based on the texture analysis, the local shear direction is also rotated a certain degree (about 30 °) away from the extrusion direction (compared to Figure 2.12c).  In conclusion, on the surface of the extrudate, where the shear strain is about 12, the texture shows the B type texture with the local shear direction rotated about 50 ° away from the extrusion direction. On the other hand, for the area 0.6 mm away from the surface, where the shear strain is about 5, the texture shows the C type and with the local shear direction rotated about 30 ° away from the extrusion direction. To further explain the formation of the rotations, the texture tracking on the surface of the partial extrusion combined with the FE study will be discussed in the next section. 7   Textures and microstructures formed on the surface of the extrusion products and their changes from the surface to centre  140  7.2.1.4 The tracking of the textures and microstructures near the surface of the partial round bar extrusion The optical image in Figure 7.4a shows the microstructures on the surface of the partial extruded sample (the extrusion ratio is 17:1 in this case). The material flow line, which is 0.5 mm away from the surface of the extrudate (as highlighted by the green curve in Figure 9.1), was tracked back by the FE simulation (the data was supplied by the by Mahmoodkhani in University of Waterloo, and the details about the extrusion simulations based on the FEM were discussed in the ref. [17, 44]). Based on the estimated material flow line, the areas just before and after the die exit are labelled as the position No.1 and No.2 in Figure 7.4a, respectively. Furthermore, Figure 7.4b illustrates the IPF map measured in position No.1 and its texture is shown in the {111} PFs in Figures 7.4c and 7.4d. For the position No.2, its microstructure is shown in Figure 7.4e and its texture is shown in Figures 7.4f and 7.4g. As can be seen in Figure 7.4b (for the position just before the die exit), the grains were elongated along the direction with a rotation about 30 ° away from the extrusion direction, and this direction is defined as the material flowing direction (MFD). For this area, the texture as illustrated in Figure 7.4c (the extrusion direction is aligned with the vertical direction in the PF) shows a transition from the C type texture to the B type texture, but the C type texture is more dominant. In detail, the local shear direction in this area is aligned with the extrusion direction. On the other hand, for the position just after the die exit, the microstructure as illustrated in Figure 7.4e shows its material flowing direction is aligned with the extrusion direction, but its texture (as illustrates in Figure 7.4f) shows the local shear direction is rotated about 30 ° away from extrusion direction and the texture is transformed to the B type texture.  7   Textures and microstructures formed on the surface of the extrusion products and their changes from the surface to centre  141   Figure 7.4: (a) Optical image showing the microstructures on the surface of the partial extruded sample and an estimated track line of the position 0.5 mm away from the surface of the extrudate (the data was supplied by the by Mahmoodkhani in University of Waterloo, and the details about the extrusion simulations based on the FEM were discussed in the ref. [17, 44]), (b) IPF map showing the microstructure in the position No.1 (just before the die exit), (c) {111} PF showing the texture in the position No.1 in the reference ED-RD, (d) {111} PF showing the texture in the position No.1 in the reference MFD-DNMF, (e) IPF map showing the microstructure in the position No.2 (just after the die exit), (f) {111} PF showing the texture in the position No.2 in the reference ED-RD, and (g) {111} PF showing the texture in the position No.2 in the reference MFD-DNMF  In a short summary, for the position just before the die exit, the material flowing direction shows a rotation about 30 ° away from the extrusion direction, but its local shear direction is aligned with the extrusion direction. For the position just after the die exit, the material flowing direction is aligned with the extrusion direction, but its local shear direction is rotated about 30 ° 7   Textures and microstructures formed on the surface of the extrusion products and their changes from the surface to centre  142  away from the extrusion direction. Based on the above observation, it is suggested that the rotation of the texture (as observed in Figures 7.3c, 7.3d, and 7.4f) is due to the rigid body rotation occurring just during the material flowing through the die exit. In detail, this rigid body rotation tends to align the material flowing direction with the extrusion direction when the materials flow through the die exit, and therefore the local texture is consequently rotated by the angle between the material flowing direction and the extrusion direction for the moment just before the die exit.  To confirm this hypothesis, the textures as illustrated in Figures 7.4c and 7.4f are projected based on the reference of the material flowing direction (MFD) and the direction normal to the material flow (DNMF) and plotted in Figures 7.4d and 7.4g, respectively. In detail, in this particular projection in Figures 7.4d and 7.4g, the vertical direction and horizontal direction in the pole figure are aligned with the materials flowing direction and the direction normal to the material flow, respectively. It should be mentioned here that for the position just after the die exit, the material flowing direction has been already aligned with the extrusion direction so that no further change of the texture projection is conducted.  After the projection of the textures is aligned to the reference MFD-DNMF, the influence of the rigid body rotation is removed, therefore the local shear direction keeps the same direction (i.e. 30 ° rotation away from the material flowing direction) between the cases before and after the die exit. Furthermore, it is also suggested that the rigid body rotation only occurs in the period of the materials flowing through the die exit and it is followed the alignment of the material flowing direction towards the extrusion direction.   7   Textures and microstructures formed on the surface of the extrusion products and their changes from the surface to centre  143  7.2.2 Discussion As observed in Figure 7.1, the textures on the surfaces of the extrudates with the two homogenization conditions show the similar results. Furthermore, as shown in Figure 7.2, for the both conditions, the textures only strongly change near the surface areas, but from the area close to the surface (i.e. from 0.8 mm away from the surface), the texture will turn to the similar texture with their central regions.  Based on the analysis of the texture variation near the surface areas of the extrudate with the homogenization at 375 °C for 24 h (the condition with a high density of dispersoids), on the surface (where the estimated shear strain is about 12), the texture shows the B type texture with about a rotation of 50 ° away from the extrusion direction. On the other hand, for the area 0.6 mm away from the surface (where the shear strain is about 5), the texture shows the C type and with about 30 ° rotation away from the extrusion direction. This observation suggests that with the increase of the shear strain towards the surface, the GDRX process is stimulated and the texture is changed from the deformation texture of the simple shear (the C type) to the texture indicating the GDRX triggered in the simple shear deformation (the B type). Furthermore, in comparison with the texture evolution in the torsion tests, the same trend of the texture variation with the shear strain was observed, i.e. the deformation texture of the simple shear (the C type) is observed at the shear strain of 5, and the texture indicating the GDRX occurring is observed at the shear strain of 12.  Based on the study of the texture tracking on the surface of the partial extruded sample, it is suggested that the rotations of the textures on the surface areas are due to the rigid body rotation occurs in the period of the materials flowing through the die exit. In detail, this rigid body rotation tends to align the material flowing direction with the extrusion direction, and therefore 7   Textures and microstructures formed on the surface of the extrusion products and their changes from the surface to centre  144  the local texture is consequently rotated by the angle between the materials flowing direction and the extrusion direction for the moment just before the die exit. Based on this hypothesis, the different rotation angles observed in the area on the surface (about 50 °) and 0.6 mm away from the surface (about 30 °) are suggested to be attributed to the different rotation angles of the materials flowing direction with respect to the extrusion direction formed just before the die exit. On the surface, a larger rotation of the texture is formed, which denotes that just before the die exit, the angle of the material flowing direction with respect to the extrusion direction formed on the surface is larger than that in the area 0.6 mm away from the surface. This proposal is consistent with the FE model predictions in the previous studies [17, 44, 141].  Furthermore, for the texture tracking on the surface of the partial extruded sample, it is observed that the texture transforms from that showing a mixture of the C type texture and the B type texture (the C type is more dominant) to the B type texture. This denotes that during the material flowing through the die exit, the local shear strain is increased to the critical value to trigger the GDRX process so that the texture is changed to the B type.   7.3 Textures and microstructures on the surface of the extrusion products and their changes from the surface to centre: the strip extrusions 7.3.1 Experimental results For the as-extruded strip with the homogenization at 375 °C for 24 h (the condition with a high density of dispersoids), Figure 7.5a illustrates the microstructures from the surface to the centre on the ED-ND plane of the as-extruded strip. To characterize the textures in the areas near the surface, the texture of the area from the surface to 0.15 mm away from the surface is plotted 7   Textures and microstructures formed on the surface of the extrusion products and their changes from the surface to centre  145  in Figure 7.5b, and texture of the area from 0.15 mm to 0.3 mm away from the surface is plotted in Figure 7.5c.   Figure 7.5: For the as-extruded strip with the homogenization at 375 °C for 24 h: (a) IPF maps showing the microstructure from the surface to the centre on the ED-ND plane of the as-extruded strip, (b) {001} and {111} PFs showing the texture of the surface layer, as labelled as (b) in Figure 7.5a, (c) {001} and {111} PFs showing the texture of the area from 0.15 mm to 0.3 mm away from surface, as labelled as (c) in Figure 7.5a, (d) plots of the area fractions of the texture components from the 0.15 mm away from the surface to the centre of the as-extruded strip, and (e) {111} PF showing the texture of the surface layer in the reference of ED-ND, which is aligned with that in the torsion tests  As can be seen in Figure 7.5b, the texture on the surface has a different texture from the cases in the centre, but as can be seen in Figure 7.5b, with the distance of 0.15 mm away from the 7   Textures and microstructures formed on the surface of the extrusion products and their changes from the surface to centre  146  surface, the texture turns to the deformation texture of plane strain deformation, i.e. the same texture with that in the central region of the as-extruded strip. To quantitatively analyze the texture variation from the surface to the centre of the as-extruded strip, Figure 7.5d plots the area fractions of the cube, S, copper, brass, and Goss texture components (within 10 ° of the ideal orientations) from 0.15 mm away from the surface to the centre. In detail, each point in Figure 7.5d represents the information from this point to the next one (the EBSD map applied for the texture characterization is 0.15 mm in the length along the thickness, and the distance between the adjacent points is 0.15 mm as well). As can be seen in Figure 7.5d, from 0.15 mm away from the surface to the centre, the textures maintain almost the same texture with the case in the central region(the S orientation is the dominant texture component and there are also fewer amounts of the brass and copper orientations).  As discussed above, a large amount of shear occurs on the surface of the extrusion products. To compare the texture on the surface of the as-extruded strips with that in the torsion tests, the sample reference should be aligned. In detail, the shear direction on the surface should be aligned with the extrusion direction (ED extrusion // SD torsion), and the normal direction on the surface should be aligned with the torsion axis (ND extrusion // Z torsion). After the reference alignment, the texture on the surface of the as-extruded strip is transformed from the {111} PF in Figure 7.5b to that in Figure 7.5e. As can be seen in Figure 7.5e, the texture on the surface shows the B type texture. In detail, the local shear direction on the surface is aligned with the extrusion direction in the strip extrusion.  Figure 7.6a illustrates the microstructures from the surface of the as-extruded strip with the material homogenized at 600 °C for 24 h (the condition with almost no dispersoids) to the centre area on the ED-ND plane.  7   Textures and microstructures formed on the surface of the extrusion products and their changes from the surface to centre  147   Figure 7.6: For the as-extruded strip with the homogenization at 600 °C for 24 h: (a) IPF maps showing the microstructure from the surface to the centre on the ED-ND plane of the as-extruded strip, (b) {001} and {111} PFs showing the texture of the PCG layer, as labelled as (b) in Figure 7.6a, (c) {001} and {111} PFs showing the texture of the area just adjacent to the PCG layer, as labelled as (c) in Figure 7.6a, (d) plots of the area fractions of the texture components from the area just adjacent to the PCG layer to the centre of the as-extruded strip, and (e) {111} PF showing the texture of the PCG layer in the reference of ED-ND, which is aligned with that in the torsion tests  In this condition, a PCG layer with a thickness about 0.3 mm was formed on the surface, as highlighted by the black dashed rectangular and labelled as (b) in Figure 7.6a. In the next chapter, it will be discussed that the PFG layer formed on the surface will play an important role on the anisotropic mechanical behaviours (especially for R-values), so that the texture of the PCG layer will be characterized in detail. Since the PCGs have the large sizes in the order of 100 μm, a map 7   Textures and microstructures formed on the surface of the extrusion products and their changes from the surface to centre  148  which covers 10 mm in length and approximately 200 PCGs was used to estimate the texture of the PCG layer. Figure 7.6b illustrates the texture of the PCG layer, and after the sample reference alignment with that in the torsion tests, the texture in Figure 7.6b is transformed to that in Figure 7.6e. On the other hand, for the area adjacent to the PCG layer, as highlighted by the black dashed rectangular and labelled as (c) in Figure 7.6a, its texture as plotted in Figure 7.6c shows a similar texture to its central region, i.e. a transition from the cube orientation to the Goss orientation. Furthermore, to quantitatively analyze the texture variation from the area just adjacent to the PCG layer to the centre, Figure 7.6d plots the area fractions of the different components (within 10 ° of the ideal orientations) through the thickness. As can be seen in Figure 7.6d, from the area adjacent to the PCG layer, the texture keeps almost the same to the centre (the cube and Goss orientation are always the dominant texture components in this region).   7.3.2 Discussion As shown in Figure 7.6, for the as-extruded strip with the homogenization at 600 °C for 24 h (the condition with almost no dispersoids), a PCG layer with a thickness about 0.3 mm was formed on the surface, and the texture of the PCG layer shows a different result with that in the centre. Furthermore, the reason for the formation of PCGs and their texture is still not clear so far [161, 162]. From the area just adjacent to the PCG layer, the texture turns to the same texture with that in the central region, and the texture variations from this area to the central region of the as-extruded strip are minor. For the as-extruded strip with the homogenization at 375 °C for 24 h (the condition with a 7   Textures and microstructures formed on the surface of the extrusion products and their changes from the surface to centre  149  high density of dispersoids), no PCG was formed on the surface of the as-extruded strip. Furthermore, the surface layer with a thickness about 0.15 mm shows a different texture with that in the central region. From the area of 0.15 mm away from the surface to the centre, the texture is similar to the deformation texture of plane strain deformation and the texture variations through the thickness are also minor. In addition, the surface layer shows the B type texture with the local shear direction aligned with the extrusion direction. This suggests that the GDRX process occurred on the surface during extrusion. In comparison with the surface area of the axisymmetric extrusion product, there is no rigid body rotation found based on the B type texture observed on the surface of the as-extruded strip (e.g. the texture in Figure 7.3c). It is suggested that the difference is caused by the different die geometries used between the axisymmetric extrusions and strip extrusions (as schematically illustrated in Figure 4.3). In detail, for the axisymmetric extrusions, a zero-bearing die was used, which means that after the material flows out from the die, there is no further contact between the die and extrudate. On the other hand, for the strip extrusions, a flat die with a 3 mm bearing was used, which can cause the further friction between the surface of the as-extruded strip and the bearing, and therefore it can lead a further simple shear deformation occurring in this area. Since there is a further simple shear deformation after the material flows out from the die, the local shear direction will be aligned with the extrusion direction again, and therefore no rigid body rotation based on the B type texture (as observed in the texture on the surface of the axisymmetric extrusion product, e.g. in Figure 7.3c) can be found in this case.  150    8 Mechanical response of the extrusion products  8.1 Introduction  In this chapter, the mechanical response of axisymmetric extrudates and as-extruded strips examined in this study will be first discussed. In detail, two conditions were examined for each case; (i) samples with the full radius (for the axisymmetric extrudates) or the full thickness (for the as-extruded strips) and (ii) samples with the surface removed (4.75 mm in diameter from the centre-line of the axisymmetric extrudates or a half thickness for the as-extruded strips). The anisotropy of the as-extruded strips was examined by measuring R-values. In addition, the visco-plastic self-consistent (VPSC) model was used to simulate the effect of texture on the mechanical response.   8.2 Mechanical response of the axisymmetric extrudates 8.2.1 Experimental results     Figure 8.1 plots the stress-strain curves measured at room temperature for the different extrusion conditions. In detail, (i) the dashed lines illustrate the stress-strain curves based on the as-homogenized conditions, (ii) the solid lines are for the centre of the extrudate (4.75 mm in diameter from the centre-line of the extrudate) and (iii) the dotted lines are for the as-extruded samples with the full diameter (12.7 mm). Moreover, the red and blue lines in Figure 8.1 denote the different homogenizations of 600 °C for 24 h and 375 °C for 24 h, respectively. 8   Mechanical response of the extrusion products  151   Figure 8.1: Plots of the flow curves measured at room temperature for the as-homogenized conditions, the central regions of the extrudates (4.75 mm in diameter from the centre-line of the extrudates), and the extrudates with the full diameter (12.7 mm) for the homogenization at 600 °C for 24 h and at 375 °C for 24 h (note: for all tensile tests in this study, the yield stress was determined using the 0.2 % offset method)  As can be seen in Figure 8.1, the stress-strain curve from the central region of the extrudate which was homogenized at 600 °C for 24 h shows a flow stress which is 5 - 10 MPa lower than that of the as-homogenized condition. In addition, a comparison of the results for the extrudate with and without the removal of the surface shows almost identical stress-strain response. For the other homogenization condition (i.e. 375 °C for 24 h), the as-homogenized sample has a yield stress of 35 MPa lower than that of the as-extruded sample, however, at larger plastic strains, the difference between the as-homogenized and the as-extruded sample reduces to ≈ 20 MPa. Similar to the previous case, there is very little difference between the sample with the surface layer removed and the one with full diameter.   8.2.2 Discussion and simulations As discussed in Section 8.2.1, there are some differences in the stress-strain curves between 8   Mechanical response of the extrusion products  152  the as-homogenized and the as-extruded condition. On the other hand, the mechanical response of the sample taken from the centre of the extrudate has a similar behaviour compared to the sample with the full diameter of the extrudate. Therefore, for simplicity, the following discussion and simulations will focus on only the central region of the extrudate for the as-extruded and the as-homogenized condition. In detail, the VPSC [104, 163] model will be used to estimate the texture effect on the mechanical response for the different initial microstructures.    8.2.2.1 The effect of the as-homogenized condition on the mechanical response of the axisymmetric extrudate As discussed in Chapter 5, the different homogenization conditions formed very different sizes and number densities of dispersoids with some minor differences in the constituent particles. Moreover, the solid solution levels for Mn will also be different. A detailed study on the effect of these factors on the mechanical response of AA3003 at room temperature has been reported by Babaghorbani [16]. In the current study, the different as-homogenized conditions will not be studied in detail, but for different as-homogenized conditions, different hardening laws (an extended version of the Voce law in the VPSC model [34]) will be used to describe the different mechanical response. In other words, the influence of the different sizes and number densities of dispersoids and constitutive particles and the different solid solution levels caused by the different homogenization processes will be accounted for in the constitutive law used in the VPSC model. The parameters for the extended Voce law were determined by fitting stress-strain curves for the different homogenization conditions. In detail, the as-cast texture (as shown in Figure 5.2b) 8   Mechanical response of the extrusion products  153  was used as the input texture for the simulations. A strain rate sensitivity value n = 50 (m = 0.02) was used in the simulations since the strain rate sensitivity for aluminum alloys at room temperature is very low [164]. Moreover, a mixed boundary condition as described by Equations 4.18 and 4.19 was used (an equivalent strain increment of 0.002 was used for each simulation steps to the final strain), and neff = 10 was employed in the simulations. Table 8.1 summarizes the parameters for the extended Voce law used to fit the stress-strain curves for the homogenization conditions of 600 °C for 24 h and 375 °C for 24 h. Figure 8.2 plots a comparison between the experimental stress-strain curves and the fit based on the parameters of the extended Voce law, given in Table 8.1.  Table 8.1: The parameters for the Voce law parameters used to fit the homogenization conditions of 600 °C for 24 h and 375 °C for 24 h Homogenization τ0s (MPa) τ1s (MPa) θ0s (MPa) θ1s (MPa) 600 °C - 24 h 19.3 23.1 440.0 32.0 375 °C - 24 h 27.2 27.2 660.0 13.5   Figure 8.2: Plots of the experimental flow curves of the as-homogenized conditions at 600 °C for 24 h and at 375 °C for 24 h, and the simulated flow curves by the VPSC model with the different Voce hardening law parameters as mentioned in Table 8.  8   Mechanical response of the extrusion products  154  8.2.2.2 The effect of initial texture on the mechanical response of the axisymmetric extrudate The texture changes significantly from an almost random texture in the as-cast condition (see Figure 5.2b) to the as-extruded textures, i.e. the <001> // ED texture (see Figure 6.1.c) for the extrudate with a homogenization of 600 °C for 24 h, and the <001> - <111> double fibre texture (see Figure 6.1.d) for the extrudate with a homogenization of 375 °C for 24 h. To study the effect of initial texture on the mechanical response at room temperature, the input textures for the simulations were changed to the as-extruded textures with all other parameters being kept the same as the as-homogenized condition. Figure 8.3 plots the experimental stress-strain curves for the as-extruded conditions and the VPSC simulated stress-strain curves with the corresponding initial textures.  Figure 8.3: Plots of the experimental flow curves of the as-extruded conditions with the central region of the extrudates (with the homogenization at 600 °C for 24 h and at 375 °C for 24 h), and the simulated flow curves with the <001> // ED texture as the input texture for the extrudate with the homogenization at 600 °C for 24 h and the simulated flow curves with the <001> - <111> double fibre texture for the extrudate with the homogenization at 375 °C for 24 h  It can be seen in Figure 8.3 that the VPSC simulated stress-strain curves still show some 8   Mechanical response of the extrusion products  155  differences in mechanical response compared to the experimental results. It is proposed that these differences are related to differences in microstructure (other than texture) between the as-homogenized and as-extruded conditions. As such, some further changes for the constitutive law will be proposed in the next section.   8.2.2.3 The effect of the solid solution level on the mechanical response of the axisymmetric extrudate To study the change in the solid solution level of the alloy elements during the extrusion process, resistivity measurements were used. For AA3003 alloys, Mn is the major alloy element in the solid solution [14, 15, 22, 165], and based on the previous studies on the effect of Mn in solid solution on the yield stress in a binary Al-Mn alloy [166] and in AA3003 alloys [16], a linear relationship between Mn in solid solution and the yield stress was proposed by Babaghorbani [16], i.e. : ss ss MnK C    (8.1) where Kss for Mn is 38 MPa per wt. % [16, 166], and CMn is the weight percentage of Mn in solid solution. Table 8.2 summarizes the changes of CMn (estimated by the resistivity measurements, in which a resistivity coefficient of 33 nΩm/wt. % [27] was used) due to the extrusion process and the estimated changes of the yield stress based on Equation 8.1. It can be seen in Table 8.2 that the change of Mn in solid solution is very small during the extrusion process, and as such, the corresponding change of the yield stress is also very small, i.e. 1 - 2 MPa. 8   Mechanical response of the extrusion products  156  Table 8.2: Summary of the changes of CMn (estimated by the resistivity measurements) by the extrusion processes and the estimated changes of the yield stress based on Equation 8.1 for the extrudates with the homogenization at 600 °C for 24 h and at 375 °C for 24 h, compared with the as-homogenized conditions Homogenization Changes of CMn estimated by the resistivity measurements (wt %) Estimated changes of the yield stress based on Eq. 10.1(MPa) 600 °C - 24 h + 0.027 + 1.0 375 °C - 24 h − 0.058 − 2.2   8.2.2.4 The effect of the microstructure on the mechanical response of the axisymmetric extrudate In addition to the changes of the texture and the Mn in solid solution, the grain structure also changes significantly during extrusion. For the case of the homogenization of 600 °C for 24 h, the as-homogenized condition has an equiaxed grain size of ≈ 70 μm and the as-extruded condition has an equiaxed grain size ≈ 15 μm. To consider the effect of grain size on the yield stress, the Hall-Petch relationship was used: 120GB GB gK D     (8.2) where Dg is grain size in diameter, and KGB is a constant. The value for the Hall-Petch slope was taken as 0.068 MPa∙m1/2 [167] (i.e. the value for high purity aluminum). Based on the Hall-Petch relationship, the change in the yield stress for the different grain sizes can then be given by:  1 12 22 1GB GB g gK D D      (8.3) where Dg1 and Dg2 are grain sizes of the as-homogenized condition and the as-extruded condition. Substituting these values into Equation 8.3 gives a yield stress increase of 9.4 MPa, which can be attributed to the grain size decrease from the 70 μm in the as-homogenized condition to 15 μm in the as-extruded condition. For the case of a homogenization of 375 °C for 24 h, the as-extruded microstructure shows the 8   Mechanical response of the extrusion products  157  elongated grains along the ED (see Figure 6.1b). In addition, within the elongated grains, subgrains with an average size ≈ 2 μm in diameter are formed (see Figure 6.8a). For this microstructure, the LAGBs that bond the subgrains will be considered to be a set of dislocation walls [61, 81, 168, 169]. Based on this assumption, the predominant effect of the subgrains will be represented by an increased dislocation density. Therefore, the Taylor equation [33] (i.e. Equation 2.3, which has been described in Section 2.4.1.1) was used to estimate the effect of dislocation densities on the critical resolved shear stress (CRSS) for certain slip system, as shown as follows: 12sdisGb     (2.3) where α is a constant, i.e. 0.33 for the condition of polycrystal of FCC materials [170]; G is the shear modulus (i.e. at room temperature, G = 26.00 GPa [139]), b is the magnitude of the Burgers vector (i.e. at room temperature, b = 2.86 Å); and ρdis is the dislocation density. Based on the Taylor equation, the change in the CRSS due to the different dislocation densities can be then given by:  1 12 20 2 1dis MGb       (8.4) where ρ1 and ρ2 are the dislocation densities for the as-homogenized condition and the as-extruded condition. Moreover, the change due to an increased initial dislocation density in the extended Voce law is schematically illustrated in Figure 8.4.  8   Mechanical response of the extrusion products  158   Figure 8.4: Schematic illustration of the change due to an increased initial dislocation density in the extended Voce law      Compared to the base stress-strain curve, an increased initial dislocation density will increase the CRSS (i.e. τ0s in the extended Voce law) and decrease the initial hardening rate, θ0s, and the saturation stress, τ1dis, as follows: 0 0 0dis s dis     (8.5) 0 00 0 1 1 11 1expdiss ssdis s s s dis dis ss sdd                          (8.6) 1 1 1 0dis s dis s dis       (8.7)     For the dislocation density of the as-homogenized condition (i.e. ρ1 in Equation 8.4), a small value of 1010 m-2 [29] was used. For the dislocation density of the as-extruded condition (i.e. ρ2 in Equation 8.4), it was used as a fit parameter. In this case, a value of 1.8  1013 m-2 was found to give a good fit to the experimental data. As such, according to ρ1 = 1010 m-2 and ρ2 = 1.8  1013 m-2, ∆τ0dis is equal to 10.4 MPa (based on Equation 8.4), and the equivalent strain accumulated in a grain, Γdis (see in Figure 8.4), is equal to 2 %. Therefore, based on the extended Voce law, the initial hardening rate, θ0s, is decreased to θ0dis and 𝜏1s is decreased to 𝜏1dis. The 8   Mechanical response of the extrusion products  159  parameters used in the extended Voce with a consideration of an increased initial dislocation density are summarized in Table 8.3. Table 8.3: Parameters used in the extended Voce law with a consideration of an increased initial dislocation density of ρ2 = 1.8 1013 m-2 Homogenization τ0dis (MPa) τ1dis (MPa) θ0s (MPa) θ1s (MPa) 375 °C - 24 h 37.6 17.0 415.5 13.5  In conclusion, the effect of the initial texture, the solid solution level, and the grain structure / initial dislocation density was accounted for and then the stress-strain curves were recalculated as shown in Figure 8.5. In comparison with the experimental stress-strain curves of the as-extruded conditions with the central region, the simulated stress-strain curves now show good agreement with the experiment.   Figure 8.5: Plots of the experimental flow curves of the as-extruded conditions with the central region of the extrudates (with the homogenization at 600 °C for 24 h and at 375 °C for 24 h), and the simulated flow curves with considerations of the effects of the textures, the solid solution levels, and the microstructures for the as-extruded conditions with the homogenization at 600 °C for 24 h and at 375 °C for 24 h    8   Mechanical response of the extrusion products  160  8.3 The anisotropic mechanical response of the as-extruded strip 8.3.1 Experimental results 8.3.1.1 Anisotropic mechanical response of the as-extruded strip: stress-strain curves Figures 8.6a and 8.6b plot the stress-strain curves measured at room temperature for the extruded strips with a homogenization of 600 °C for 24 h and 375 °C for 24 h, respectively. The blue, green, and red lines in Figure 8.6 denote the stress-strain curves measured parallel to the extrusion direction (ED), at 45 ° to the ED, and parallel to the transverse direction (TD), respectively. In addition, the dashed and solid lines represent the stress-strain curves measured with the sample with the full thickness of the extruded strip and that where the surface layer was removed (i.e. half thickness, as described in Section 4.7, and it is denoted as the “Centre” in Figure 8.6), respectively.  Figure 8.6: For the materials homogenized (a) at 600 °C for 24 h and (b) at 375 °C for 24 h: plots of the flow curves measured at room temperature along the ED, at 45 ° to the ED, and the TD for the sample of full thickness of the extruded strip and that with only the central region  As can be seen in Figure 8.6a, for the material homogenized at 600 °C for 24 h, the stress-strain curves measured along the different directions show similar mechanical response. 8   Mechanical response of the extrusion products  161  Especially at yield, the difference between the different directions is only about 1 - 2 MPa, which lies within the experimental uncertainty of these measurements. Further, the samples with the surface removed and that with the full thickness of the extruded strip also show the similar stress-strain response to each other for all orientations of testing. A slightly higher flow stress (i.e. ≈ 4 - 6 MPa) can be observed for the samples with full thickness at plastic true strain of 0.2. In contrast, Figure 8.6b shows the results for the material homogenized at 375 °C for 24 h. Here, the test at 45 ° to the ED shows a lower stress-strain response while the tests parallel to the ED and TD have very similar results. In detail, the yield stress for the sample of 45 ° to the ED is ≈ 5 MPa lower than that of ED or TD. However, with a further increase of plastic true strain to 0.2, the flow stress for the sample of 45 ° to the ED is ≈ 35 MPa lower than that of ED or TD. Finally, the samples with the surface removed show similar behaviour to the samples with full thickness. In summary, for the material homogenized at 600 °C for 24 h, the anisotropy of the stress-strain response is low, but for the material homogenized at 375 °C for 24 h, a clear difference between the stress-strain response at 45 ° to the ED and that tested parallel to the ED or TD is observed. The stress-strain curves of the samples with the surface removed and these with full thickness have similar behaviour for both the materials.   8.3.1.2 Anisotropic mechanical response of the as-extruded strip: R-values     Figures 8.7a and 8.7b plot the experimental results for the evolution of the R-value with plastic strain for the tensile tests parallel to the ED, at 45 ° to the ED, and parallel to the TD for the material homogenized at 600 °C for 24 and at 375 °C for 24 h, respectively. Furthermore, the 8   Mechanical response of the extrusion products  162  dashed and solid lines denote the R-value evolutions measured based on the sample with the surface removed and the sample with full thickness, respectively.  Figure 8.7: For the materials homogenized (a) at 600 °C for 24 h and (b) at 375 °C for 24 h: plots of the R-value evolutions with the plastic true strain along the ED, at 45 ° to the ED, and the TD for the sample of full thickness of the extruded strip and that with only the central region  As can be seen in Figure 8.7a, for the material homogenized at 600 °C for 24 h, the R-values show a significant dependence on the orientation of the tensile axis. At a plastic strain of 0.15 (i.e. the industry standard condition), the R-values for the samples with full thickness are ≈ 1.6, ≈ 1.6 and ≈ 0.6 for the case of ED, 45 ° to the ED, and TD. Further, the R-values of the samples with full thickness exhibit a different directional dependence from that of the sample where the surface was removed (denoted as the “Centre” in Figure 8.7). In particular, a large difference of the R-values between the samples with full thickness and that with the surface removed is found for the cases of (i) 45 ° to the ED and (ii) TD. For the tests at 45 ° to the ED, the difference of the R-values is about 1.5. For the tests parallel to the TD, there is a large drop of the R-value from 6.6 at a plastic strain of 0.05 to 1.6 at a plastic strain of 0.15 for the sample with full thickness, but the sample with the surface removed shows only a small drop of R-value with increasing 8   Mechanical response of the extrusion products  163  plastic strain in this test direction. For the tests parallel to the ED, the R-values measured from the sample with full thickness are very similar to that measured from the sample with the surface removed. It should be mentioned here that in the test parallel to the ED and using the sample with full thickness (i.e. the solid green line in Figure 8.7a), there is a large experimental error found at low plastic strains (i.e. 0.05 - 0.10). This is due to the inhomogeneity of sample width formed in the tensile test, and the reason to that is not clear. Figure 8.7b shows the experimental results for the material homogenized at 375 °C for 24 h. In this case, a large level of anisotropy is also observed, i.e. R-value (ED) ≈ 0.5, R-value (at 45 ° to the ED) ≈ 3, and R-value (TD) ≈ 1. However, in this case, there is a relatively small effect of removing the surface layers. Figure 8.8 illustrates the secondary electron (SE2) images of the cross-section areas (normal to the tensile directions) for the samples (one with the surface removed and one with the full thickness) tested with the different orientations. Note, the cross-section areas were taken from a region ≈ 2 - 3 mm away from the fracture surface (for the cases shown in Figures 8.8b, 8.8c, and 8.8h, the necking region is longer than 3 mm so that the cross-section area is still in the necking region, but for other cases, the necking region is shorter so that the cross-section area is out of the necking region). The observations in Figure 8.8 can be grouped into three main types: (i) samples where the surface contracts similarly to the centre and as such, the cross-section remains approximately rectangular (i.e. the cases in Figures 8.8a and 8.8d - 8.8l), (ii) samples where the surface contracts greater than the centre and as such, the cross-section changes to a convex shape (i.e. the case in Figure 8.8b), and (iii) samples where the centre contracts more than the surface and as such, the cross-section changes to a concave shape (i.e. the case in Figure 8.8c). A possible explanation for these results will be discussed in the following section.  8   Mechanical response of the extrusion products  164   Figure 8.8: SE2 images showing the cross-section areas of the tensile sample machined from the extruded strip:  with the homogenization at 600 °C for 24 and measured (a) along the ED, (b) at 45 ° to the ED, and (c) along the TD for the sample with full thickness; and for the sample with the central region and measured (d) along the ED, (e) at 45 ° to the ED, and (f) along the TD; and with the other homogenization at 375 °C for 24 and measured (g) along the ED, (h) at 45 ° to the ED, and (i) along the TD for the sample with full thickness; and for the sample with the central region and measured (j) along the ED, (k) at 45 ° to the ED, and (l) along the TD 8   Mechanical response of the extrusion products  165  8.3.2 Discussion and simulations 8.3.2.1 Simulations for the anisotropic mechanical response of the as-extruded strip: flow curves As discussed in Section 8.3.1.1, little difference was found in the stress-strain response of the sample with full thickness and that with the surface removed. This suggests that the effect of the surface layers on the stress-strain curves is very small so that for the following simulations, it will only focus on the condition where the surface layer was removed. Figure 8.9 plots the experimental stress-strain curves measured on the samples with the surface removed and the simulated stress-strain curves based on the initial textures taken from the central region. In detail, the blue, red, and green lines represent the tensile tests parallel to the ED, at 45 ° to the ED, and parallel to the TD, respectively. Furthermore, the solid and dashed lines denote the experiments and simulations, respectively. The constitutive laws used were based on the parameters shown in Table 8.2 for the case of a homogenization of 600 °C for 24 h and the parameters shown in Table 8.3 for the case of a homogenization of 375 °C for 24 h.   Figure 8.9: For the materials homogenized (a) at 600 °C for 24 h and (b) at 375 °C for 24 h: plots of the experimental and simulated flow curves along the ED, at 45 ° to the ED, and along the TD for the samples with the central region of the extruded strips 8   Mechanical response of the extrusion products  166  As can be seen in Figure 8.9a, for the case of homogenization of 600 °C for 24 h, a relatively weak dependence on test direction is predicted by the simulations, which is similar to the experimental results. The predicted yield stress difference between directions is only 1 - 2 MPa and the magnitude matches the experiments. This weak mechanical anisotropy is also consistent with a recent report from Zhang et al. [8], which showed that based on the cube orientation, the difference of the yield stresses between the different directions is normally within 2 % and no obvious anisotropy in terms of yield stress can be found. With the increase of the plastic true strain, the effect of the test direction is increased in the simulations, but this phenomenon is not observed in the experiments. Figure 8.9b shows the results for the material homogenized at 375 °C for 24 h. Here, the simulations generally show the same directional dependence as the experiments, i.e. the ED and TD have the similar stress-strain response and are harder than the case of 45 ° to the ED. This suggests that the anisotropy of the stress-strain response is mainly due to the copper type texture formed in the central region of the extruded strip, and this is also supported by the previous modelling work from Lequeu et al. [95]. In detail, the results for simulations and tests parallel to the ED and TD show a good agreement, but in the case of 45 ° to the ED, the VPSC model underestimates the stress-strain response.   8.3.2.2 Simulations for the anisotropic mechanical response of the as-extruded strip: R-values Figures 8.10a and 8.10b plot the evolution of R-value with plastic strain for the experiments and simulations for the materials homogenized at 600 °C for 24 h and at 375 °C for 24 h, 8   Mechanical response of the extrusion products  167  respectively. In detail, the blue, red and green lines in Figure 8.10 denote the evolution of R-value for the tensile tests parallel to the ED, at 45 ° to the ED, and parallel to the TD, respectively. The solid and dashed lines in Figure 8.10 represent the experiments and simulations, respectively.  Figure 8.10: For the materials homogenized (a) at 600 °C for 24 h and (b) at 375 °C for 24 h: plots of the experimental and simulated R-value evolutions along the ED, at 45 ° to the ED, and along the TD for the samples with the central region of the extruded strips  As can be seen in Figure 8.10a, for the case of homogenization at 600 °C for 24 h, the simulation of the evolution of R-value generally show the same directional dependence of R-values as the experimental observations, i.e. the TD has the largest R-value, the ED shows a smaller R-value, and the direction at 45 ° to the ED has the smallest R-value. In detail, for the case of TD, the simulated R-values show a good match with the experimental results at plastic strains larger than 0.15. Nevertheless, the large drop of the experimental R-values from a plastic strain of 0.05 to a plastic strain of 0.15 is not found in the simulations. For the case of ED and 45 ° to the ED, the simulations also show the same trend as the experimental results (i.e. the R-values < 1), but the simulations predicted lower R-values than the experiments. Furthermore, in 8   Mechanical response of the extrusion products  168  comparison with previous reports regarding to hot rolled aluminum alloys which have been recrystallized [4, 8], R-values of ≈ 0.8 were reported for the tests parallel to the TD, but in the current experiments, R-values of ≈ 1.6 were found. A possible explanation for this is the strong Goss orientation developed in the current as-extruded strip, compared with the recrystallization textures in rolling which mainly have the cube orientation [4, 8]. Based on the previous modelling study by Lequeu at el. [95], the Goss orientation can dramatically increase the R-value in the test parallel to the TD. This would be consistent with the current simulations where R-values of ≈ 1.6 were predicted. Figure 8.10b shows the results for the extruded strip with a homogenization of 375 °C for 24 h. Here, the simulations show the same directional dependence of R-values as the experiments, i.e. the test at 45 ° to the ED has the largest R-value, the TD shows a smaller R-value, and the ED has the smallest R-value. In detail, for both the cases of TD and 45 ° to the ED, the simulations show a good match to the experiments, but for the case of ED, lower R-values were predicted by the simulations. Furthermore, in comparison with previous studies about the R-values for the as-deformed rolling conditions [6], some similar results were reported. In detail, Sakai et al. [6] reported that the direction at 45 ° to the ED shows the largest R-value (i.e. R-values ≈ 1.5) and both the ED and TD show the similar lower R-values (i.e. R-values < 1). In summary, based on the experiments and simulations, it is suggested that the directional dependence of R-value derives from the underlying texture, i.e. the copper type texture in the extruded strip. Figure 8.11 show the results for the evolution of R-value from the experiments and simulations for the material homogenized at 600 °C for 24 h. In detail, the solid and dashed lines in Figure 8.11 represent the experiments and simulations, respectively. In addition, for the experiments, the red and black lines in Figure 8.11 denote the measurements for the samples 8   Mechanical response of the extrusion products  169  with the surface removed (denoted as the “Centre” in Figure 8.11) and the samples with full thickness, respectively. In the case of the simulations, the red and black lines in Figure 8.11 denote the simulations using the texture in the centre of the extruded strip (see Figure 6.9c) and the texture on the surface (as shown in Figure 7.6b) as inputs to VPSC.   Figure 8.11: For the material homogenized at 600 °C for 24 h: plots of the experimental R-value evolutions for the samples with the central region of the extruded strips and the samples with full thickness and the simulated R-value evolutions based on the texture in the centre and the texture on the surface for the tensile directions (a) along the ED, (b) at 45 ° to the ED, and (c) along the TD  As can be seen in Figure 8.11a, for the case of ED, both the experiments and simulations show that the surface layer deforms similar to the centre of the sample although the absolute value for 8   Mechanical response of the extrusion products  170  the R-values predicted by VPSC are lower than the experiments. For the case of 45 ° to the ED (see Figure 8.11b), there is a large effect in the experiments of removing the surface, i.e. a decrease of the R-value from ≈ 1.6 for the sample with full thickness to ≈ 0.5 for the sample with the surface removed. This observation is reflected in Figures 8.7b when it can be seen that the surface layer contracts greater than the central region and a convex cross-section can be found. It should be highlighted here that the R-values were measured optically on the surface edges, and as such, this biases the experimental R-value measurements towards the surface behaviour. On the other hand, the simulations predicted that R-value of ≈ 3 for the surface and that of ≈ 0.25 for the central region. Qualitatively, the simulations are consistent with the experiments and the observation of the cross-section shown in Figure 8.7b, but quantitatively, the simulations overestimated the difference between the surface and centre. It is worth noting here that the surface layer and the interior cannot deform independently to each other, and this may explain that the R-values measured with the sample with full thickness fall in between the simulated R-values for the surface and that for the central region. For the case of TD (see Figure 8.11c), there is also a large effect of the surface layer, i.e. an increase of R-value of ≈ 1.5 to that of ≈ 3 at a plastic strain of 0.1 and an increase of R-value of ≈ 1.5 to that of ≈ 2 at a plastic strain of 0.15. Moreover, this is reflected in the cross-section in Figure 8.8c, where the interior shrinks greater than the surface layer and a concave cross-section can be observed. For this case, the simulation shows a good agreement for the centre region at plastic strain larger than 0.15, and similar to the case of 45 ° to the ED, the experiments for the sample with full thickness fall in between the simulated R-values for the surface and centre.  For the material homogenized at 375 °C for 24 h, Figure 8.12 shows the experimental evolution of R-value for the samples with the surface removed and the samples with full 8   Mechanical response of the extrusion products  171  thickness. The simulated evolution of R-value based on the texture in the centre (see Figure 6.9d) and the texture on the surface of the as-extruded strip (as shown in Figure 7.5b) is also shown.  Figure 8.12: For the material homogenized at 375 °C for 24 h: plots of the experimental R-value evolutions for the samples with the central region of the extruded strips and the samples with full thickness and the simulated R-value evolutions based on the texture in the centre and the texture on the surface for the tensile directions (a) along the ED, (b) at 45 ° to the ED, and (c) along the TD  For both the cases of ED and TD, similar R-values were predicted based on the textures in the centre and on the surface, and similar experimental R-values were also observed. However, the predicted R-values are substantially lower than the experiments. For case of 45 ° to the ED, a much larger R-value (i.e. ≈ 6) was predicted based on the texture on the surface than that in the 8   Mechanical response of the extrusion products  172  centre (i.e. ≈ 3), but in the experiments, the R-values for the sample with full thickness is only slightly larger than the sample with the surface removed. However, in this case, the surface layer is only about 0.15 mm (i.e. ≈ 10 % in thickness), as shown in Figure 7.5a, so that it may be the case that the surface layer is too thin to substantially modify the R-values for the sample with full thickness. In summary, for the extruded strips with a homogenization of 600 °C for 24 h and 375 °C for 24 h, the directional dependence of both the stress-strain curves and R-values can be mainly rationalized by the textures and texture gradients. Moreover, the different textures formed on the surface of the extruded strips show little effect on the stress-strain curves, but they have a significant effect on R-values. For future work based on the study in this chapter, it is recommended that the stress-strain curves and R-values based on the surface layer and the central region of the extruded strip could be used as the input to the finite element model to simulate the buckling and folding of crash tubes. By doing so, not only the material anisotropic mechanical property but also inhomogeneity of the material microstructure and texture could be considered in the simulation. On the other hand, the large effect of the surface layer on R-value should be further applied to the material design to optimize the plastic stability of crash tubes in buckling and folding.  173    9 Summary and future work  9.1 Summary The end overall of this work is to provide an improved model which can be used in the finite element model (FEM) simulations of the buckling and folding that occurs during the deformation of a crush tube. The constitutive model should be directly related to the microstructures and textures in extrusions (which is not homogeneous in the profile). Therefore, the current work is designed: (i) to quantify the texture through the thickness of the extrusions and explore the origin of this and (ii) analyze the relationship between the texture and the anisotropic plastic response. The texture and microstructure gradients in the axisymmetric extrudates and extruded strips were characterized using the electron backscatter diffraction (EBSD) technique. The observations were rationalized based a combination of carefully designed torsion tests, partial extrusions, and simulations. Moreover, the mechanical response of the axisymmetric extrudates and extruded strips were characterized. In particular, the evolution of R-value was measured in the tensile tests with the different tensile direction with respect to extrusion direction of the extruded strip, and the effect of surface layer was also studied using the sample where the surface layer was removed. These results were rationalized using simulations from the visco-plastic self-consistent (VPSC) model so that the effect of texture on the stress-strain response and R-values could be examined. The key specific results from this work are listed as follows: (i) For the central regions of the axisymmetric extrusion, the material with a high density of dispersoids produces the elongated grains along the extrusion direction with the <001> - <111> 9    Summary and future work  174  double fibre texture (i.e. the unrecrystallized or as-deformed condition), and the material with almost no dispersoids forms equiaxed grains with the predominant <001> // ED texture (i.e. the recrystallized condition). The as-deformed <001> - <111> double fibre texture can be quantitatively calculated by VPSC model simulations. In the case of the recrystallized microstructure, it is proposed that continuous dynamic recrystallization (CDRX, which is characterized as subgrain / grain coarsening) is the relevant mechanism. It is suggested that the texture transformation from the as-deformed <001> - <111> double fibre texture to the recrystallized <001> // ED texture in the CDRX process is due to the growth advantages of the subgrains showing the <001> // ED texture component, i.e. a larger subgrain size and their long range rotation.  (ii) For the central regions of the strip extrusions, the materials with a high density of dispersoids produces the elongated grains along the extrusion direction with a copper type texture (i.e. the unrecrystallized or as-deformed condition), and the material with almost no dispersoids forms equiaxed grains with a texture showing a transition from the cube to Goss orientation (i.e. the recrystallized condition). The unrecrystallized texture (i.e. a copper type texture) can be quantitatively simulated using the VPSC model. CDRX is also proposed to be the mechanism in the recrystallization occurring during or after the extrusion for the material with almost no dispersoids. (iii) In the torsion tests (i.e. a nearly ideal simple shear deformation) conducted to different levels of equivalent strain, there is little difference in the textures and microstructures for the material with a high density of dispersoids and the material with almost no dispersoids. With the increase of the equivalent strain from 5 to 6, the textures change from the C type (i.e. a deformation texture in simple shear deformation) to the B type (i.e. a recrystallized texture in 9    Summary and future work  175  simple shear deformation) texture for both the materials, and with the further increase the equivalent strain to 12, the textures remain as the B type. Furthermore, for both homogenization conditions, there are no obvious changes of the subgrain sizes and grain boundary densities from the conditions at the equivalent strains of 5 to 12. For the simple shear deformation, it is proposed that the geometric dynamic recrystallization (GDRX, which is characterized as the elongated grain subdivision) occurs from the equivalent strain of 5 to 6 in both the materials. (iv) For the axisymmetric extrusions, observations were made on the through thickness texture changes. The textures are very different on the surface areas for both homogenization conditions (i.e. the one with a high density of dispersoids and the one with almost no dispersoids) For the material with a high density of dispersoids, a layer of grains showing the B type texture with a further rigid body rotation is produced on the surface of the extrudate. Based on the texture tracking on the surface of a partially extruded sample, it is proposed that this rigid body rotation is due to alignment of material flow to the extrusion direction when the material is flowing through the die exit, and the formation of the B type texture is due to the GDRX occurring caused by the large simple shear happening on the surface of the extrudate. For the material with almost no dispersoids, a layer of peripheral coarse grains (PCG) with a thickness of 100 μm is formed on the surface of the extrudate, and just below the surface, a layer of equiaxed grains show the same texture of the B type texture. (v) For the strip extrusion for which a flat die with a 3 mm bearing was used, the textures are only strongly changed on the surface areas for both the materials. For the material with a high density of dispersoids, a layer of grains showing the B type texture with a thickness of 150 μm is formed on the surface. For the material with almost no dispersoids, a PCG layer with a thickness of 300 μm is observed on the surface. 9    Summary and future work  176  (vi) For the mechanical behaviours of the axisymmetric extruded products, the surface layer shows little effect on the measured flow curves, but the different as-extruded textures has an effect on the stress-strain curves.  (vii) For the anisotropic mechanical behaviours of the extruded strips, the surface layers show little effect on the measured flow curves but a large effect on the R-values. For the effect of surface on R-value, the results can be grouped into three main types: (i) samples where the surface contracts similarly to the centre and as such, the cross-section remains approximately rectangular, (ii) samples where the surface contracts greater than the centre and as such, the cross-section changes to a convex shape, and (iii) samples where the centre contracts more than the surface and as such, the cross-section changes to a concave shape. Based on the simulation using the VPSC model, it is proposed that the large effect of the surface layers on the R-values is attributed to the textures for the surface layers.   9.2 Future work The following offers some interesting areas for the further studies based on the results of the current study. (i) The stress-strain curves and R-values based on the surface layer and the central region of the extruded strip could be used as the input to the finite element model to simulate the buckling and folding of crash tubes in the future. By doing so, not only the material anisotropic mechanical property but also inhomogeneity of the material microstructure and texture could be considered in the simulation.  (ii) 3-D EBSD measurements to characterize the as-extruded microstructures are suggested for 9    Summary and future work  177  the axisymmetric extrusions and strip extrusion for the material with a high density of dispersoids. In detail, the EBSD system with a laser beam to section the material is suggested to be used to capture a large volume of the microstructure. Based on the 3-D EBSD, it is suggested that phase field simulation of the subgrain coarsening process be conducted.  (iii) A systematic study phase field study using synthetic microstructures is also recommended to study the effect of the spatial distribution of the microstructures on the subgrain of grain coarsening process. In detail, it is suggested that the synthetic microstructures are built based on the different spatial distributions of the subgrains showing the different texture components and the different spatial distributions of the subgrains with the different sizes. (iv) As discussed in Chapter 6, in the axisymmetric extensions and plane strain deformations, it is suggested that the continuous dynamic recrystallization (CDRX) occurs in the material with almost no dispersoids, and for the materials with a high density of dispersoids, the large Zener drag prohibits the migration of grain boundaries and keeps the as-extruded microstructures as the unrecrystallized or the as-deformed microstructures. 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"Theoretical predictions and experimental verification of surface grain structure evolution for AA6061 during hot rolling" Acta Materialia, vol. 56, pp. 6174, 2008. [163] Molinari A, Canova GR, Ahzi S. "A self consistent approach of the large deformation polycrystal viscoplasticity" Acta Metallurgica, vol. 35, pp. 2983, 1987. Bibliography  195  [164] Engler O, Wagner P, Savoie J, Ponge D, Gottstein G. "Strain rate sensitivity of flow stress and its effect on hot rolling texture development" Scripta Metallurgica et Materialia, vol. 28, pp. 1317, 1993. [165] Muggerud AMF, Mørtsell EA, Li Y, Holmestad R. "Dispersoid strengthening in AA3xxx alloys with varying Mn and Si content during annealing at low temperatures" Materials Science and Engineering: A, vol. 567, pp. 21, 2013. [166] Ryen Ø, Holmedal B, Nijs O, Nes E, Sjölander E, Ekström H-E. "Strengthening mechanisms in solid solution aluminum alloys" Metallurgical and Materials Transactions A, vol. 37, pp. 1999, 2006. [167] Embury JD. Strengthening methods in crystals. New York: Wiley, 1970. [168] Bay B, Hansen N, Hughes DA, Kuhlmann-Wilsdorf D. "Evolution of f.c.c. deformation structures in polyslip" Acta Metallurgica et Materialia, vol. 40, pp. 205, 1992. [169] Liu Q, Juul Jensen D, Hansen N. "Effect of grain orientation on deformation structure in cold-rolled polycrystalline aluminium" Acta Materialia, vol. 46, pp. 5819, 1998. [170] Lavrentev FF. "The type of dislocation interaction as the factor determining work hardening" Materials Science and Engineering, vol. 46, pp. 191, 1980.  196    Appendix  A.1       Parametric study for texture calculation in axisymmetric extrusion     Table A.1 summaries the area fractions of the <001> // ED and <111> // ED texture components (in the axisymmetric extrusions) which are calculated based on different calculation ranks in harmonic series expansion and different deviation angles for the EBSD data. Table A.1: Summary of the area fractions of the <001> // ED and <111> // ED texture components which are calculated based on different calculation ranks in harmonic series expansion and different deviation angles for the EBSD data          calculation                                                ranks deviation  angle (degree) 16 20 28 Area fraction of the <001> // ED texture component (% 5 14 14 14 10 30 30 30 15 34 34 34 20 36 36 36 Area fraction of the <111> // ED texture component (%) 5 26 26 26 10 57 57 57 15 64 64 64 20 66 66 66  Based on the parametric study shown in Table A.1, the harmonic series expansion rank in the range from 16 to 28 has no influence on the calculated area fraction, but the applied deviation angle has a strong effect from the applied value of 5 ° to 20 °. When the deviation angle is equal to 20 °, the sum of the two texture fibre volume fractions is larger than 100 %, which means that there is some double counting. Moreover, when a deviation angle of 5 ° is applied, the sum of Appendix  197  area fractions of these two texture components is only 40 % of the total orientations. Only when the deviation angle is equal to 15 °, the total volume fraction is not only closed to 100 % but also a little bit less than that. For the above reason, 15 ° is considered to be a reasonable value of the deviation angle for the definition of the certain texture components in this case.   A.2       Back-estimate subgrain size using the measured grain boundary densities Figure A.1 schematically illustrates an artificial microstructure of the subgrains with a uniform shape and size of hexagons.   Figure A.1: A schematic illustration of an artificial microstructure of the subgrains with a uniform shape and size of hexagons, where the grey lines represent the LAGBs (i.e. their misorientations < 15 °) and the black lines represent the HAGBs (i.e. their misorientations ≥ 15 °)  The average subgrain size can be back-estimated by the measured LAGB and HAGB densities in the EBSD map assuming that all subgrains have a uniform hexagonal shape using the Equations 6.2 and 6.3. 23GBhexl   (6.2) Appendix  198  2 26 3hex hexd l    (6.3) where, ρGB is the area density of all the boundaries (i.e. ρGB = ρLAGB + ρHAGB, based on the definition of subgrains), lhex is the length of the side for the hexagon, and the dhex is the equivalent diameter of the hexagon.    A.3       Misorientation between the texture components in plane strain deformation     Table A.2 summarizes the misorientations between the each individual texture components (i.e. the cube, Goss, S, copper and brass texture components) in the plane strain deformations. Table A.2: Summary of the misorientation angles (°) between the each individual texture components (i.e. the cube, Goss, S, copper and brass texture components) in the plane strain deformations Misorientation angles (°) cube Goss copper S brass cube 0 45 53.6 47.7 56.4 Goss - 0 60 43.7 35 copper - - 0 17.9 35.4 S - - - 0 19.7 brass - - - - 0  As can be seen in Table A.2, the misorientation angles between the copper and S texture components and that between the brass and S texture components are 17.9 ° and 19.7 °, respectively. If a deviation angle which is larger than 10 ° (e.g. 15 ° or 20 °) is used to define a certain texture component, there will be a large amount of double counting between the copper and S texture components and the brass and S texture components. On the other hand, if the deviation angle is too small (e.g. 5 °), it will lead a difficulty to identify the area fraction for the certain texture component. With a compromising between above two points, 10 ° is considered to be the deviation angle to define the certain texture components in the plane strain Appendix  199  deformations. It should be mentioned here that with a deviation angle of 10 °, there is still some double counting between the copper and S texture components and between the brass and S texture components.   A.4       Estimate grain dimension after the strip extrusion  Figure A.2 schematically illustrates the assumption of the change of grain dimensions in the strip extrusions with no recrystallization occurring. As can be seen in Figure A.2, before the extrusion (i.e. after the homogenization heat treatment), the grain shape is assumed to be a sphere with a radius of Rhomo (i.e. 35 μm), and after the extrusion with an extrusion ratio of RExt, it is assumed that the grain is deformed into a strip with the width of WExt (i.e. = 90 mm), thickness of TExt (i.e. = 2.9 mm) and length of LExt.  Figure A.2: Schematic illustration of the grain shape change in the strip extrusion  The dimensions of deformed grains based on the assumption shown in Figure 10.2 can be calculated as follows: 2homo Ext Ext ExtR R W T   (A.1) Appendix  200  3hom43 o Ext Ext ExtR W T L   (A.2)     Based on Equations A.1 and A.2, the estimated thickness of the deformed grain in the strip extrusion is 2.0 μm. 

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