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Mechanical properties of a recovered Al-Mg-Sc alloy Roumina, Reza 2009

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MECHANICAL PROPERTIES OF A RECOVERED Al-Mg-Sc ALLOY by REZA ROUMINA M. Sc., Sharif University of Technology, Iran, 2002 B.Sc., Tehran University, Iran, 2000 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIRMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (MATERIALS ENGINEERING) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) June 2009 © Reza Roumina, 2009 Abstract In this study the effect of recovery on the yield strength and work hardening of a model Al-Mg-Sc alloy in the presence of A13Sc precipitates was investigated. Recovered microstructures containing A13Sc precipitates were obtained through a series of thermo-mechanical treatments including pre-aging, cold rolling and annealing. Recovered microstructures were characterized in terms of precipitate size distribution, subgrain sizes and dislocation structures. Yield strength and work hardening of processed microstructures were examined by tensile testing at 77K. The results show that the effect of precipitates arises directly from precipitation strengthening as well as indirectly from their effectiveness at controlling the recovered microstructure. Physical based models were developed to describe the tensile response of recovered microstructures consistent with previous models on single phase recovered Al-Mg alloys. For the first time, the impact of precipitates on recovery kinetics was captured by coupling recovery models of single phase Al-Mg alloys and creep models. In this new modelling approach a transition stage from dislocation annihilation to a climb controlled mechanism was defined. In addition, the effects of both recovery and precipitates on work hardening have been incorporated in the previous models. Finally, model limitations as well as their potential applications to improve mechanical properties including yield strength, ultimate tensile strength and uniform elongation of Al-Mg-Sc alloys by controlling thermo-mechanical processing and chemical composition were revisited. 11 Table of Contents Abstract.ii Table of Contents iii List of Tables vi List of Figures vii List of Symbols xiii Acknowledgments xix Chapter 1-Introduction 1 Chapter 2-Literature Review 4 2.1. Al-Sc Alloys 4 2.1.1. Precipitation of Al3Sc in Al Alloys 5 2.1.1.1. Effect ofAl3Sc on Mechanical Properties of Al Alloys 7 2.1.1.1.1. Effect of Al3Sc Precipitates on Yield Strength in Al Alloys 7 2.1.1.1.2. Effect ofAl3Sc Precipitates on Work Hardening in Al Alloys 10 2.2. Microstructural Evolution in Deformed Al Alloys 14 2.2.1. Recovery in Al Alloys 15 2.2.1.1. Mechanical Properties of Recovered Al Alloys 17 2.2.1.1.1. Effect of Recovery on Yield Strength 17 2.2.1.1.2. Effect of Recovery on Work Hardening 22 2.2.1.2. Effect of Precipitates on Recovery in Al Alloys 27 2.2.2. Recrystallization in Al Alloys 29 2.3. Summary of Literature Review 36 Chapter 3-Scope and Objectives 38 Chapter 4-Experimental Procedures 40 4.1. Starting Materials 40 4.2. Solution Treatment 40 4.3. Thermo-mechanical Treatments 41 4.3.1. Rolling Experiments 42 4.3.2. Heat Treatments 43 4.3.3. Recrystallization 43 4.3.4. Recovery 44 111 4.3.4.1. Aging Treatments .44 4.3.4.2. Deformation and Annealing Treatments 45 4.4. Tensile Testing 49 4.5. Characterization 50 4.5.1. Optical Microscopy 50 4.5.2. Scanning Electron Microscopy (SEM) 50 4.5.3. Electron Back Scattered Diffraction (EBSD) 51 4.5.4. Transmission Electron Microscopy (TEM) 51 4.5.6. Texture Measurements 52 Chapter 5-Experimental Results 55 5.1. Starting Materials 55 5.2. Solution Treatment of Al-Mg-Sc Alloy 56 5.3. Microstructural Characterization 60 5.3.1. Microstructure of Aged Samples 60 5.3.1.1. TEM Observation of Aged Samples 60 5.3.2. Microstructure of Recovered Samples 64 5.3.2.1. TEM Observations of Recovered Al-Mg-Sc and Al-Mg Samples 64 5.3.2.2. TEM Measurements on Precipitate and Subgrain Size in Recovered Al-Mg-Sc Samples 70 5.3.2.3. EBSD Observations on Microstructure of Recovered Al-Mg-Sc Samples 71 5.3.2.4. Texture Measurements on Al-Mg-Sc Samples 73 5.4. Mechanical Properties 76 5.4.1. The Evolution of the Yield Strength 76 5.4.1.1. Stress-Strain Behaviour 81 Chapter 6-Discussion 83 6.1. Microstructure Characterization 83 6.1.1. Precipitate Size Measurements 83 6.1.2. Recovered Microstructures 86 6.2. Mechanical Properties 88 6.2.1. Modelling Yield Strength 88 iv 6.2.1 .1. The Influence of Recovery on Yield Strength 91 6.2.1.2. A Model for the Effect of Precipitates on Recovery Kinetics 94 6.2.1.3. Comparison Between Model and Experiments 104 6.2.1.4. Overall Yield Strength 111 6.2.2. Modelling of Work Hardening: The Combined Effect of Recovery and Precipitates 114 6.2.2.1. Work Hardening Behavior of the Al-Mg-Sc System 116 6.2.3. Application of Models to other Pre-aging Treatments 127 6.3. Summary 131 Chapter 7-Application of Models to Al-Mg-Sc Alloys 132 7.1. Limitations of the Models 132 7.1.1. Annealing Temperature 132 7.1.2. Strain Level 134 7.1.3. Chemical Composition 134 7.2. Application of Models 135 7.2.1. Opportunities for Modifying Mechanical Properties and Alloy Design 136 Chapter 8-Conclusions 144 References 146 Appendix A-Concurrent Precipitation and Recrystallization 152 Appendix B-Uniform Elongation and Maximum Tensile Strength 155 Appendix C-Models for Precipitation Kinetics and Yield Strength for As-Aged Al-2.8%Mg-0.16%Sc 157 V List of Tables Table 2.1. Effect of scandium on tensile properties of the extruded Al-2%Mg alloy annealed at 350°C for lhr [20] 5 Table 2.2. Activation energy and activation volume values deduced by fitting Equation 2.5 to fit 0d1s for Al-Mg alloys isothermally annealed at indicated temperatures 20 Table 4.1. Chemical composition of Al-Mg-Sc and Al-Mg alloys (Weight Percent) 40 Table 4.2. Experimental rolling schedules corresponding to different total reductions....42 Table 5.1. Taylor factor obtained for as-aged, rolled and recovered Al-Mg-Sc samples .76 Table 6.3. Activation energy and activation volume values deduced by fitting Equation 2.5 djg shown in Figure 6.3 (c) 93 Table 6.4. Parameters used in the recovery model 105 Table 6.5. Work hardening model parameters determined experimentally 120 Table 6.6. Required experiments to determine the adjustable parameters for the work hardening model 124 Table C. 1. Parameters for nucleation and growth rates 159 vi List of Figures Figure 2.1. The Al rich side of the binary Al-Sc phase diagram. The lines are obtained from thermodynamics modelling and the symbols indicate the experimental data, figure reproduced with pennission from [101 6 Figure 2.2. Schematic of stages of a dislocation passing through a set of precipitates (a) bowing of a dislocation between precipitates (b) passage of a dislocation through shearable precipitates (c) passage of a dislocation by bowing around non-shearable precipitates 8 Figure 2.3. Experiments (symbols) and model prediction for the increase in yield strength due to A13Sc precipitation, 0PPT, during aging of an Al-Mg-Sc alloy, figure reproduced with permission from [13] 10 Figure 2.4. (a) Work hardening behavior as a function of ad during uniaxial tensile testing for different aging conditions of Al-2.8%Mg-0.16%Sc samples. Lines represent model calculations and symbols indicate experimental data [13] (b) Tensile and work hardening behavior of an Al-0.2%Sc alloy for over-aged (300°C/bOhr) and under-aged (250°C/i Ohr) samples [39]. Figures reproduced with permission from [13, 39] 14 Figure 2.5. TEM image of an Al-2.5%Mg alloy cold rolled to a true strain of 3 showing microstructural evolution during recovery (a) the deformed state (b) annealed at 160°C for 2Ohr (c) annealed at 160°C for l5Ohr, images reproduced with permission from [74]. 17 Figure 2.6. Effect of (a) temperature and (b) pre-strain by cold rolling on recovery kinetics of AA5754 during isothermal annealing [8] 18 Figure 2.7. Schematic presentation of (a)-(l) continuous, (c)-(d) discontinuous recrystallization from subgrains, figures reproduced with permission from [94] 31 Figure 2.8. The effect of grain size and strain on transition from discontinuous to continuous recrystallization in the AA8006, figure reproduced with permission from [102] 33 Figure 2.9. Isothermal recrystallization kinetics of Al-Sc alloys showing recrystallization start (5%) and finish (95%) as a function of annealing time, figure reproduced with permission from [11] 35 vii Figure 2.10. (a) Grain structure of a deformed Al-0.2%Sc alloy during pre-aging at 350°C for 3hr obtained from electron backscattered diffraction method (EBSD): different colors indicate different grain orientations (b) The effect of annealing time and temperature on grain size of the same alloy, figures reproduced with permission from [53] 36 Figure 4.1. Thermo-mechanical schedules designed for recovery of Al-Mg-Sc and Al-Mg alloys 48 Figure 4.2. Selected surfaces in sheet samples for EBSD, TEM studies and texture measurements 54 Figure 4.3. Illustration of selected pole figure angles for texture measurements 54 Figure 5.1. Optical micrograph of the elongated grain structure in the Al-2.8%Mg- 0.16%Sc alloy 55 Figure 5.2. Backscattered image of coarse grained microstructure of the as cast Al 2.5%Mg alloy 56 Figure 5.3. Isothermal sections of the Al-Mg- Sc ternary phase diagram at different temperatures 57 Figure 5.4. (a) Solubility of scandium in aluminum with and without 2.8%Mg (b) Volume fraction of Al3Sc precipitates versus temperature calculated from the solvus in (a) 58 Figure 5.5. (a) The Al-2 . 8%Mg-0. 1 6%Sc microstructure after solution treatment at 610°C for 8 hr (b) microstructure after solution treatment at 610°C for 24 hr, (c) microstructure after solution treatment at 6 10°C for 7 days, RD stands for rolling direction and ND for normal direction. (d) EDX analysis of the segregation indicated by an arrow in (a), (e) EDX analysis of the sample shown in (a) outside of the segregation bands 59 Figure 5.6. TEM images and diffraction pattern of A13Sc precipitates in samples pre aged at 425°C for 80mm (a) bright field image (b) dark field image using the A13Sc <100>AI super-lattice reflection (c) corresponding diffraction pattern<100>AI zone axis62 Figure 5.7. TEM images and diffraction pattern of A13Sc precipitates in samples pre aged at 425°C for 8 days (a) bright field image (b) dark field image using the A13Sc <100>AJ super-lattice reflection (c) corresponding diffraction pattern<100>AI zone axis63 viii Figure 5.8. Precipitate diameter distributions for (a) Aged-8omin (b) Aged-8days (c) Aged-80min-2Ohr 64 Figure 5.9. Bright field TEM images of recovered Al-Mg-Sc samples annealed at 425°C after 80% cold rolling (a) Rec-80min-40s, close to zone axis <1 ‘°>Al, (b) Rec-80min- 2Ohr, close to zone axis <1 lO>M, (c) Rec-80min-14 days, close to zone axis <lOO>AI ....66 Figure 5.10. Bright field TEM images of recovered Al-Mg-Sc samples annealed at 425°C after 80% cold rolling (a) Rec-8 days-40s, close to zone axis <‘l°>Al, (b) Rec-8 days 2Ohr, close to zone axis <1 lO>AI, (c) Rec-8 days-14 days, close to zone axis <lOO>A1 ....67 Figure 5.11. Bright field TEM images of recovered Al-Mg-Sc and Al-Mg samples annealed at 190°C after 80% cold rolling (a) Rec-80min-14 days, close to zone axis <“°>M, (b) Rec-8 days-14 days, close to zone axis <1 l°>AI, (c) Rec-Al-Mg-14 days, close to zone axis<100>A1 69 Figure 5.12. Precipitate diameter distributions for Aged-80min-2Ohr and Rec-80min-2Ohr samples 71 Figure 5.13. Subgrain size measured by TEM for recovered samples with different aging routes annealed at (a) 425°C and (b) 190°C 71 Figure 5.14. EBSD inverse pole figures maps of recovered Al-Mg-Sc microstructures annealed at 425°C for 2Ohr with initial precipitate diameter of (a) 12 nm and (b) 23 nm. The colors correspond to the crystallographic direction parallel to ND 72 Figure 5.15. { 100 } Pole figure for selected grains, A and B, corresponding to the recovered sample shown in Figure 5.14 (a) with initial precipitates size of 1 2nm 73 Figure 5.16. Pole figures of pre-aged at 425°C for 80 mm followed by 80% cold rolling, contour levels: 0.5, 2, 3.5, 5, and 6.5 75 Figure 5.17. Pole figures of pre-aged at 425°C for 80 mm followed by 80% cold rolling and annealing at 425°C for 20 br, contour levels: 0.5, 2, 3.5, 5, and 6.5 75 Figure 5.18. Pole figures of aged samples at 425°C for 80 mm with fully recrystallized microstructure, contour levels: 0.5, 2, 3.5, 5, and 6.5 75 Figure 5.19. Yield strength evolution for Al-Mg-Sc samples subjected to annealing at 190°C and 425°C. Solid symbols represent materials that were pre-aged, rolled and annealed, while open symbols represent materials that were pre-aged in the same way but not subjected to rolling. (a) Materials pre-aged at 425°C for 80 mm and annealed at ix 190°C (Rec-80min-40s to 136 days) (b) Materials pre-aged at 425°C for 80mm and annealed at 425°C (Rec-80min-40s to 69 days) (c) Materials pre-aged at 425°C for 8 days and annealed at 190°C (Rec-8days-40s to 136 days) (d) Materials pre-aged at 425°C for 8 days, rolled then annealed at 425°C (Rec-8 days-40s to 69 days) 78 Figure 5.20. Yield strength evolution for the recovered Al-Mg samples. The homogenized samples were first rolled to 80% reduction followed by recovery annealing at 190°C for 40s to 136 days 79 Figure 5.21. True stress-stain behavior of recovered and as-aged samples (a) pre-aged for 80 mm and annealed at 425°C (b) pre-aged for 8 days and annealed at 425°C (c) pre-aged for 80 mm and annealed at 190°C (d) pre-aged for 8 days and annealed at 190°C 82 Figure 6.1. Precipitate diameter distributions for (a) Aged-80 mm (b) Aged-8 days, cumulative precipitate volume and average precipitate diameter for (c) Aged-80 mm (d) Aged-8 days 85 Figure 6.2. Comparison of subgrain sizes measured by TEM and predicted via Zener pinning for samples recovered at 425°C 87 Figure 6.3. Evolution of yield stress attributable to forest dislocations (Odjs) for (a) recovered Al-Mg-Sc samples annealed at 190°C (b) recovered Al-Mg-Sc samples annealed at 425°C (c) recovered Al-Mg samples annealed at 190°C. The thin dotted lines indicate the yield strength of the Al-Mg alloy and aged Al-Mg-Sc alloy in as rolled state and the model prediction based on Equation 2.5 is illustrated as heavy dashed lines. 92 Figure 6.4. Dislocation climb over a cube-shaped particle showing (a) local climb with a sharp dislocation bend at A (b) general climb where the high curvature at A is relaxed [1171 98 Figure 6.5. Precipitate radius versus (aging/recovery time) 1/3 at 425°C and fit to LSW coarsening behavior given by Equation 6.19 103 Figure 6.6. Experiments and model prediction for decay of 0djs in recovered Al-Mg-Sc samples annealed at (a) 190°C and (b) 425°C 106 Figure 6.7. Sensitivity of the applied model to n values for samples pre-aged at 425°C for 80 mm and recovered at 425°C 108 x Figure 6.8. Transition terms for Rec-8days-40s to 69 days (a) annealed at 425°C (b) annealed at 190°C and strain rates for the same condition (c) annealed at 425°C (d) annealed at 190°C 110 Figure 6.9. Experiments and model prediction for evolution of yield strength of Al-Mg- Sc samples (a) Rec-80min-40s to 136 days annealed at 190°C (b) Rec-80min-40s to 69 days annealed at 425°C (c) Rec-8 days-40s to 136 days annealed at 190°C (d) Rec-8 days-40s to 69 days annealed at 425°C 113 Figure 6.10. Evolution of experimental work hardening rate for recovered and as-aged samples (a) Pre-aged at 425°C 80mm-recovery at T425°C (b) Pre-aged at 425°C 8 days- recovery at T=425°C (c) Pre-aged at 425°C 80mm- recovery at T = 190°C (d) Pre-aged at 425°C 8 days-recovery at T 190°C 117 Figure 6.11. The effect of GNDs on work hardening behavior: The work hardening rate increases to higher levels for a given level of ad due to presence of GNDs. At high stresses the two curves approach one another and the effect of GNDs decreases 118 Figure 6.12. Evolution of work hardening rate, experiments and model predictions, for as solutionized samples 119 Figure 6.13. Evolution of work hardening rate, comparing experiments and model predictions, for as-aged and recovered samples (a) Aged at 425°C for 80mm and 8 days (b) Pre-aged at 425°C 80mm-recovery at T425°C (c) Pre-aged at 425°C 8 days- recovery at T=425°C (d) Pre-aged at 425°C 80mm- recovery at T = 190°C (e) Pre-aged at 425°C 8 days-recovery at T= 190°C 121 Figure 6.14. Subgrain size measured by TEM and values applied in the work hardening model for recovered samples with different aging routes annealed at (a) 425°C and (b) 190°C 122 Figure 6.15. Precipitate spacing and subgrain sizes for recovered samples pre-aged at 425°C (a) for 80mm recovered at 190°C (b) for 80mm recovered at 425°C (c) for 8 days recovered at 190°C (d) for 8 days recovered at 425°C 126 Figure 6.16. Flow stress, experiments and model predictions, for recovered samples (a) Pre-aged at 425°C 80mm- recovery at T 425°C (b) Pre-aged at 425°C 8 days-recovery at T=425°C (c) Pre-aged at 425°C 80mm-recovery at T190°C (d) Pre-aged at 425°C 8days-recovery at T= 190°C 127 xi Figure 6.17. Flow stress, experiments and model predictions, for samples recovered at 410°C and 425°C 129 Figure 6.18. Modelling results on dislocation contribution to the yield strength and precipitate strength for samples recovered at 410°C and 425°C 130 Figure 7.1. Model predictions of recovered and as aged samples (a) yield strength- uniform elongation (b) maximum tensile strength-uniform elongation 137 Figure 7.2. Model predictions on (a) precipitate size distribution and (b) precipitate strength for peak aged (labeled A in Figure 7.1) and recovered at 425°C for 2Ohr and 410°C for 2mm (labeled B and C in Figure 7.1) conditions (courtesy of Fazeli etal. [14] ) 139 Figure 7.3. Volume fraction of Al3Sc precipitates versus temperature and the minimum volume fraction required to produce A13Sc precipitates with spacing of 0.3 m 143 Figure A. 1. Temperature — time — recrystallized fraction plots for a material cold rolled to a) = 1.4 and b) 8 = 2.65 illustrating the onset and finish of recrystallization (approximately indicated with the dashed lines) 153 Figure A.2. EBSD maps of pattern quality showing samples rolled to a strain of 1.4 and annealed at a) 410°C for 5 minutes (8% recrystallized) and b) 420°C for 3 hours (92% recrystallized). The solid white lines indicate boundaries with misorientations of between 3 and 10° while black lines indicate boundaries with misorientations of greater than 15°. Both samples are viewed in the along the transverse direction, the rolling direction being horizontal 154 xii List of Symbols A Creep constant B Temperature dependent constant for subgrain growth b Burgers vector C Constant in the order of 8.55 c1 Constant Concentration of Sc in A13Sc Sc Instantaneous Sc concentration in the matrix Sc Sc concentration in the matrix at the interface D’ Pre-exponential for lattice diffusion in Al DC Pre-exponential for lattice diffusion in Sc Df Lattice diffusion coefficient for Al DC Lattice diffusion coefficient for Sc Diimit The limiting subgrain size predicted by Zener pining Dsubgrain Cell/subgrain size Do(subgrain) Initial Cell/Subgrain size after deformation prior further annealing d Grain size d. Measured diameter of a precipitate dPP((V) Number average precipitate diameter xiii dPP((v) Volume average precipitate diameter E Elastic modulus FRe The driving force for recovery of a 3 -D dislocation network F(r) The maximum interaction force between a precipitate and dislocation line f Transition probability f5 Fraction of total dislocation contributing to dynamic recovery Volume fraction of precipitates G Shear modulus Driving pressure for nucleation K Efficiency of cell/subgrain walls in dislocation annihilation K Geometric factor for storage of geometric necessary dislocations in theG presence of both precipitates and subgrains K Geometric factor for storage of geometric necessary dislocations in the presence of non-shearable precipitates K Geometric factor for storage of geometric necessary dislocations in thesubgrazn presence of subgrains k Constant for precipitate coarsening kB Boltzmann’s constant k Constant in the Hall-Petch equation k Constant corresponding to the storage of dislocations for a given1 orientation Constant corresponding to dynamic recovery for a given orientation k Constant corresponding to the storage of geometric necessarydislocations for a given orientation xiv Dislocation spacing L1 Precipitate spacing M Taylor factor Mb Subgrainlgrain boundary mobility n Stress exponent Number of atoms per unit volume p.r Nucleation rate Number of dislocation nodes np,, Number of precipitates The exponent for superposition law 11total Total number of observed precipitates Transition parameter ‘net The net driving pressure for subgrainlgrain growth Zener pressure pinning P1 Constant for dislocation storage P2 Constant for dislocation annihilation xv QSD Activation energy for lattice diffusion R The gas constant r Radius of a precipitate Average precipitate radius Initial precipitate radius before second aging/annealing treatment The critical precipitate radius for transition from shearable and non r shearable precipitates T Temperature t Time telimb Time for dislocation climb U Activation energy associated with recovery V Activation volume associated with recovery Iimb Dislocation velocity for climb Volume of a precipitate with a measured diameter of d1 Vtotat Total volume of precipitates w Cell/subgrain boundary thickness Z Zeldovich non-equilibrium factor a Constant in the order of 0.3 a Constant used to determine the intrinsic subgrain!grain boundary m mobility Constant Shape factor in the order of 1 xvi a1 Constant in the order of 2 Numerical constant Attachment rate of single atoms to the critical nucleus True strain Plastic strain rate associated with dislocations Plastic strain rate associated with climb of dislocations Uniform elongation Yb Grainlsubgrain boundary energy YAI/AI3Sc Interfacial energy between A13Sc precipitates and Al matrix F Dislocation line tension The Angle between the adjacent arms of a dislocation when it interacts with the obstacle The critical angle between the adjacent arms of a dislocation when it oc overcomes the obstacle 2 Annihilation length within cell/subgrain boundary 2eff The effective mean planar precipitate spacing 2G The geometric slip distance VD Debye frequency 8d,s The work hardening rate due to dislocation storage and annihilation The initial work hardening rate P Dislocation density xvii Flow stress Back stress Dislocation contributions to flow stress Dislocation contributions to the yield strength depending on precipitates during recovery total Dislocation contributions to the yield strength depending on both dis precipitates and forest dislocation annihilation during recovery cr1 Frictional stress Precipitates contributions to flow stress Scaling stress sol Solid solution contributions to flow stress Yield strength Applied shear stress Shear back stress The critical shear stress required for a dislocation to break from C precipitates viii Incubation time Shear Orowan stress xviii Acknowledgments First of all, I thank God for His mercy on me. I would like to thank my supervisor Dr. Chad W. Sinclair for his excellent support, guidance, encouragement and most importantly his friendship throughout this thesis. He continually helped me out to develop my ability to perform independent research as a graduate student. I also thank Dr. Warren J. Poole for his numerous stimulating discussions and comments at all stages of this challenging work. Many thanks to Dr. David Lloyd for his assistance with chemical analysis experiments and Dr. Haiou Jin for performing texture measurements at Novelis Global Research and Development Centre in Kingston. Dr. Paul Dawson from Cornell University deserves a great deal of thanks for providing Material Point Simulator program and for his help on using the program during his stay at UBC. Especial thanks are extended to Dr. Fateh Fazeli for providing me the computer codes developed during his Post-doc at UBC and for his fruitful discussions which contributed to the success of my work. I would like to express my gratitude to all staff members at Department of Materials Engineering at UBC for their assistance on my research work. My especial thanks to Hamid Azizi-Alizamini, Babak Raeisinia, Fateh Fazeli (for the second time) Sujay Sarkar, Leo Colley, Jayant Jam, Arnaud Weck, David Marechal, Phil Tomlinson, and all other friends at UBC for their help. Natural Sciences and Engineering Research Council of Canada (NSERC) for financial support, as well as the supply of materials for this study by Novelis is greatly acknowledged. xix Finally, this thesis is dedicated to the memory of my father. I thank my mother for her unending love, sacrifice and support throughout my education. This thesis is also dedicated to you. Thank you! xx Chapter 1-Introduction The introduction of solute, precipitates and grain boundaries is a practical approach to the design of mechanical properties, including yield strength, work hardening and uniform elongation of aluminum alloys. The technology surrounding solid solution strengthening and precipitation strengthening are relatively well established, certainly to the point where these strengthening mechanisms can be accounted for via phenomenological models applicable to industrial alloys. This is despite the fact that certain areas, e.g. the effect of precipitation on work hardening rate and failure, are only beginning to receive significant attention [1,2]. More recently, our understanding on the role of grain boundaries in modifying the mechanical properties of aluminum alloys has been elucidated and realized in commercial applications due to development of new novel processing technologies [3,4]. Traditionally, materials designers would consider the use of one or two predominant strengthening mechanisms to tailor the mechanical response for a given application. There has been, however, an increasing trend towards coupling different strengthening mechanisms or to design “hetereogeneous” microstructures to find synergistic effects [5]. For example, one can consider combining grain size refinement and precipitation as routes that could be used to modify yield strength and work hardening rate [6]. Synergies arise here from the fact that the precipitates can act both to achieve and maintain a fine grain size while also modifying the yield strength and perhaps work hardening rate by acting as obstacles to dislocations during deformation. Another independent method available for controlling the mechanical properties of metals is to use them in a “pre-worked” or “wrought” form. In the case of aluminum 1 alloys, can body stock represents an example of a material whose strength is largely dictated by the presence of a high density of forest dislocations. Controlling mechanical properties through the introduction of forest dislocations is a robust way of being able to achieve a very wide range of mechanical properties. The limiting cases are represented by the as-cold rolled material (high strength, low uniform elongation) and the fully recrystallized material (large uniform elongation, low strength). A continuum of properties between these limiting cases is possible simply by controlling the temperature and duration of annealing. Using annealing as a method to control mechanical properties, though relatively simple, is not conventionally used in the same manner as grain size refinement or precipitation hardening to control mechanical response in part due to the potential for the microstructure to evolve at ambient temperature. There is, however, the potential for stabilizing a microstructure consisting of a relatively high density of dislocations even to relatively high homologous temperatures, through the incorporation of a fine dispersion of stable precipitates. As described above for the case of a material consisting of precipitates and a fine grain size, synergistic effects can arise from the combination of a recovered microstructure and precipitates. Moreover, by controlling the annealing treatment used in the recovery of the microstructure it can be possible to have a combination of fine grain size, precipitates and forest dislocations. The objective of this work is to investigate the possibilities for using recovery in the presence of a fine dispersion of precipitates as a means of controlling the yield strength and work hardening of aluminum alloys. For this purpose, an Al-Mg alloy containing Sc has been selected as an alloy for achieving a stable recovered 2 microstructUre. Al-Mg alloys are particularly interesting to use in this work as there exists literature for the kinetics of recovery and the resulting mechanical (tensile) properties in the absence of precipitates [7,8]. The addition of Sc to Al-Mg alloys results in the formation of an extremely fine and homogenously dispersed set of approximately spherical Al3Sc precipitates [9,101. Al-Sc alloys have been shown to have a remarkable ability to retard recrystallization to high homologous temperatures while also having a strong precipitation strengthening effect [9,11,121. These alloys with unrecrystallized microstructures are potential wrought alloys for structural applications [91. This work will specifically focus on the relationship between the microstructures obtainable in the recovered Al-Mg-Sc alloy and mechanical properties in terms of yield strength and work hardening behavior. This work also strongly draws on previous work performed at UBC on the precipitation kinetics, yield strength and work hardening rate of the same alloy in the “as-aged” condition [13,14]. Models have been developed previously to describe the evolution of yield strength and work hardening of Al-Mg alloys during recovery [7,15,16]. However, the effect of precipitates was not included in the previous models. The goal here is to develop models for describing the yield and work hardening response of the Al-Mg-Sc alloy in the recovered state containing precipitates. Having models for both yield strength and work hardening rate allows for prediction of mechanical properties such as yield strength, ultimate tensile strength and uniform elongation. This provides useful knowledge to improve mechanical properties of precipitate containing aluminum alloys including Al-Sc alloys by careful alloy and processing design. 3 Chapter 2-Literature Review There is a large body literature surrounding the relationship between mechanical properties and microstructure in aluminum alloys. This chapter will in particular review the impact of recovery, recrystallization and precipitation on mechanical properties of aluminum alloys with a focus on Al-Mg and Al-Mg-Sc alloys. First, mechanical properties and precipitation in Al-Sc alloys will be summarized. Next, recovery and its effect on yield strength and work hardening in aluminum alloys will be presented. Finally, an overview of recrystallization and grain refinement through controlling recrystallization in Al-Sc alloys will be given. 2.1. Al-Sc Alloys There has been a recent extensive review of Al-Sc alloys by Røyset and Ryum [9]. When added to aluminum, scandium is a powerful dispersoid-strengthener, grain refiner and recrystallization inhibitor [9]. All of these beneficial effects from scandium addition are linked to the presence of fine, coherent Al3Sc precipitates that form during the aging of supersaturated Al-Sc solid solutions. Because of these advantages, efforts have been taken to modif’ aluminum alloys, especially Al-Mg alloys, with scandium. The combination of solid solution, strengthening from Mg, precipitation strengthening from Al3Sc precipitates and grain size refinement can be used to improve the strength of Al-Mg-Sc alloys without severely decreasing elongation [17,18,19] as illustrated in Table 2.1. In addition, due to the effectiveness of Al3Sc in retarding recrystallization, Al-Sc alloys with unrecrystallized microstruetures are potential wrought alloys for structural, sport and aircraft applications [9,17,19]. In the next section, the properties and precipitation hardening effect of Al3Sc in Al-Sc alloys will be discussed. 4 Table 2.1. Effect of scandium on tensile properties of the extruded Al-2%Mg alloy annealed at 350°C for lhr [20]. Alloy Composition Tensile Strength Yield Strength Uniform Elongation (%)(wt.%) (MPa) (MPa) Al-2%Mg 180 80 38.5 Al-2%Mg-0.18%Sc 225 135 27.5 Al-2%Mg-0.3%Sc 255 150 22 Al-2%Mg-0.4%Sc 260 165 23.1 2.1.1. Precipitation of A13Sc in Al Alloys The Al-rich primary solid solution of the Al-Sc binary phase diagram shows a limited solubility for scandium below the eutectic temperature. Below the solvus, the Al solid solution is in equilibrium with the Al3Sc phase, which has a primitive Li2 crystal structure [21]. According to Figure 2.1, the maximum solubility of scandium in aluminum is 0.38 wt.%, occurring at the eutectic temperature (660°C). Decomposition of a supersaturated solid solution of Al-Sc directly results in the precipitation of the equilibrium A13Sc phase without the formation of metastable phases [22,23]. The morphology of the precipitates depends on the aging temperature, the scandium concentration and the aging time. Marquis et al. [10] found that aging of Al-0.2 wt% Sc between 300°C and 450°C results in spherical Al3Sc precipitates with an average radius between 2 and 25 nm. Studies of Al alloys containing between 0.2 and 0.5 wt% Sc have shown that fine, spherical, coherent and homogeneously distributed A13Sc precipitates with a 1.10% lattice parameter misfit at 300°C are formed at temperatures below 3 50°C [24,25,26]. Studies on Al-Sc alloys confirm that precipitation of A13Sc is mainly dominated by homogenous nucleation at low temperatures [23,24]. 5 However, heterogeneous nucleation dominates at low super saturations e.g. high temperatures or low Sc concentration [10]. For example in an Al-0.2%Sc alloy, heterogeneous precipitation was observed when samples were quenched from the solutionizing temperature and aged at temperatures in the range of 3 70-490°C [27]. A13Sc precipitates are very stable with respect to coarsening due to the low diffusivity and solubility of scandium. The coarsening behavior of A13Sc precipitates in an Al-0.2%Sc alloy has been studied by Iwamura et al. [28] between 400-490°C. The precipitate radius at the coherent to semi-coherent transition was determined to be 15-40 nm from TEM observations. Beyond this transition one observes a higher rate of coarsening of the Al3Sc precipitates. AtomIc percent scandium 0 0.2 0.4 0.6 0.8 800 750 700 650 600 550 500 450 400 Figure 2.1. The Al rich side of the binary Al-Sc phase diagram. The lines are obtained from thermodynamics modelling and the symbols indicate the experimental data, figure reproduced with permission from [10]. 1.0 1.2 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Al Weight percent scandium 6 2.1.1.1. Effect of A13Sc on Mechanical Properties of Al Alloys Considerable studies have been dedicated to A13Sc precipitation in Al-Sc alloys. However, fewer studies of the effect of A13Sc precipitates on mechanical properties, including yield strength and work hardening, have been conducted. The focus of this section is to review the effect of Al3Sc precipitates on the yield strength and work hardening of Al-Sc alloys. 2.1.1.1.1. Effect ofAI3Sc Precipitates on Yield Strength in Al Alloys It has been reported that despite the low volume fraction of Al3Sc in Al alloys containing 0.2-0.5%Sc (less than 1 vol. % between 250 and 350°C) scandium provides a significant age-hardening effect [29,30,31]. The mechanisms responsible for the increase in yield strength due to A13Sc precipitates depend on the nature of the precipitate itself and also precipitate spacing. In fact, precipitates act as barriers to dislocation motion and dislocations can pass through precipitates at stress levels higher than those required to move dislocations through the matrix phase thereby increasing the yield stress. Once a dislocation line encounters an array of precipitates with a spacing of shown in Figure 2.2, they will bow through precipitates to some angle, q, and the level of applied stress increases since the line tension of the dislocations produces a resisting force which acts to minimize the dislocation length. As the level of applied stress increases, the bowing angle becomes smaller and eventually at some critical breaking angle, p, the dislocation line passes through the precipitates [32]. The critical shear stress, r required for the dislocation to break free of the precipitates is given by: = 2F cos(-) (2.1) ppt 7 where F is the dislocation line tension, b is the Burgures vector and L1 is the precipitate spacing [32]. When the precipitates are shearable, the critical breaking angle is large and the precipitates will be cut by the movement of the dislocation line (Figure 2.2 (b)). For the case of non-shearable precipitates (Figure 2.2 (c)), strengthening is controlled by the Orowan mechanism [33] where the critical breaking angle becomes zero and the dislocation line can oniy pass the precipitates by leaving a dislocation ioop around them. © © (a) (b) (c) Figure 2.2. Schematic of stages of a dislocation passing through a set of precipitates (a) bowing of a dislocation between precipitates (b) passage of a dislocation through shearable precipitates (c) passage of a dislocation by bowing around non-shearable precipitates Seidman et a?. [34] studied precipitation strengthening mechanisms in binary Al-Sc alloys. They observed that the maximum in hardness during aging occurs for a precipitate radius of about 2 tim. Based on the hardness results, they suggested that when the precipitate radius is smaller than 2 tim, shearing mechanisms including coherency strengthening, order strengthening and modulus mismatch strengthening are responsible for hardening while when the radius is larger than 2 nm; the strengthening effect is due to the Orowan mechanism. p/2 8 In a more recent work by Fazeli et a!. [13] the yield strength of an Al-2.8%Mg-O.16%Sc alloy during aging has been studied through combined experiments and modelling. In this modelling approach a mechanical model of yield strength was combined with a precipitation kinetics model described by a Kampmann-Wagner (KWN) [3 5,36] type model. The yield strength model was based on the approach proposed by A. Deschamps and Y. Brechet [37] where it is assumed that there is a linear relationship between the force required to shear a precipitate and the precipitate size up to the shearable-non shearable transition. In the case of shearable precipitates, several mechanisms can be simultaneously active depending on the nature of precipitate-dislocation interaction. Once precipitates are non-shearable the by pass force for a precipitate is constant as dislocations have to ioop around precipitates regardless of the precipitate size. The precipitate size distribution and the effect of vacancies on precipitation kinetics were also considered in the yield strength model developed by Fazeli eta!. [14]. The critical precipitate radius for transition from shearable to non-shearable precipitates was treated as an adjustable parameter. Comparison of experiments and model led to a critical radius of 3.7 nm, consistent with TEM observations [34]. Figure 2.3 shows the agreement between experiments and model prediction during aging at 300°C, 350°C and annealing at 300°C followed by up-quenching to 425°C studied by Fazeli et a!. The peak strength due to Al3Sc precipitation, o,,, in their study occurs at 300°C in which the majority of precipitates are predicted to be in the shearable region. For samples pre-aged at 425°C precipitates were found to be non-shearable. According to the Figure 2.3, the contribution of precipitates to the yield strength at this temperature 9 declines. This is related to the fact that at 425°C coarsening rate is accelerated compared to the one at 3 00°C and the precipitate spacing becomes larger. 140 120 100 Cu O 80 60 a. 0 40 20 0 Figure 2.3. Experiments (symbols) and model prediction for the increase in yield strength due to Al3Sc precipitation, aPPT, during aging of an Al-Mg-Sc alloy, figure reproduced with permission from [13]. 2.1.1.1.2. Effect of A13Sc Precipitates on Work Hardening in Al Alloys There are a few studies on the effect of precipitates on the work hardening. A few studies on the effect of precipitates on work hardening in Al alloys including Al-Sc alloys [13,2,38,39,401 have been conducted recently. Studies by Røyset et al. [39] and Fazeli et al. [13] have shown that the work hardening behavior in aged Al-Sc alloys depends on the nature of Al3Sc precipitates-dislocation interaction. It is found that the work hardening rate of Al-Sc alloys is enhanced when A13Sc precipitates are non-shearable compared to the case where the precipitates are shearable. The role of non-deformable particles on the work hardening behavior of metals and alloys has been discussed over the past 40 years (e.g. see the studies by Kelly and Nicholson [41], Ashby [42,43], Humphreys [44] and Brown and Ham [45]). The deformation of these alloys is non-homogeneous since the matrix deforms plastically more than the non-deformable particles. As described in the previous section dislocation 101 102 10 10 10 106 time, s 10 loops are left around non-shearable precipitates by the Orowan mechanism. Depending on the shape or size of the precipitates, these Orowan loops can be accumulated around precipitates and produce long range elastic stresses resulting in internal stresses and a high work hardening rate [46,47]. For example in a recent study [38] on an Al-Mg-Si-Cu alloy using Bauschinger testing a large fraction of the work hardening rate of over aged samples containing non-shearable lath shaped precipitates is attributed to the development of internal stresses. In this study the level of internal stress for over aged samples was found to be high enough to eventually fracture the majority of precipitates and thereby facilitate plastic relaxation by creating new surfaces. Several authors have discussed that the un-relaxed plastic strain depends on the number of loops and plastic relaxation can occur by conversion of Orowan loops to prismatic loops through mechanisms such as local climb or double cross slip [46,47,48]. Ashby [43] proposed that these prismatic loops are arrays of dislocations known as “geometrically necessary dislocations”, (GNDs), which accommodate deformation gradients and also maintain continuity between second particles and the matrix. Ashby [43] pointed out that in addition to “statistically stored dislocations” accumulated by random trapping; GNDs also contribute to the work hardening. Assuming the total dislocation density is the sum of both types of dislocations, the flow stress, djs’ arising from total dislocation density, p, can be obtained from the Taylor equation [49]: dis = MaGb -q (2.2) where u is a constant, b is the magnitude of the Burgers vector, M is the Taylor factor, and G is the shear modulus. Ashby’s approach explains that in alloys containing 11 non-shearable precipitates higher work hardening rates can be obtained compared to those free of precipitates due to introduction of GNDs. A framework for modelling the work hardening behavior of materials has been suggested by Kocks and Mecking [50,51]. The net rate of dislocation accumulation and annihilation controls the work hardening rate. The contribution of a second phase to the dislocation storage was also added to the model by Estrin and Mecking [52]. The total dislocation storage rate in this model is given by: 4=M(k1ph/2k2p+1) (2.3) where 2G is the geometric slip distance, M, k1, k2 and Ic3 are constants which can be determined by physical arguments. The first term on the right hand side corresponds to the storage of dislocations; the second term is associated with the dynamic recovery and the third term with the storage of geometrically necessary dislocations due to non-shearable precipitates. In a recent study [13] the work hardening behavior of an Al-2.8%Mg-0.16%Sc alloy was investigated using Kocks-Mecking-Estrin model. Combining Equations 2.2 and 2.3, the work hardening rate due to dislocation storage and annihilation during deformation,0djs’ can be expressed as: — d dEs — . dEs K, f U dEs — — ‘ 0 —) + non — shearable ( . )de S dis r where °0 is the initial work hardening rate at the onset of plastic deformation in the absence of second phases or boundaries, K1 is a geometric factor for storage of geometrically necessary dislocations (GNDs) in the presence of non-shearable A13Sc 12 precipitates with the average radius of F and volume fraction of f and u is a scaling stress. In Equation 2.4, theLratio for non-shearable precipitates was considered since the full size distribution was taken and only those non-shearable precipitates were considered to contribute to work hardening. In the case of A13Sc precipitates internal stresses were assumed to be negligible due to their small size, small volume fraction and spherical shape. The hypothesis behind this assumption is that unlike Al alloys with plate or rod shaped precipitates [38,401 plastic relaxation mechanisms are dominant for A13Sc precipitates. Figure 2.4 (a) illustrates the work hardening rate arising from dislocation storage and annihilation,0djs’ versus stress due to dislocations, d,s’ for under aged, peak aged and over aged Al-Mg-Sc samples. According to Figure 2.4 (a), the work hardening rate for over aged samples where all precipitates are non-shearable is improved compared to under aged and peak aged samples containing shearable A135c precipitates. Figure 2.4 (b) illustrates the results of a similar study by Røyset et al. [391 on the tensile and work hardening behavior of an Al-0.2%Sc alloy indicating that the work hardening rate of over-aged samples (300°C/i OOhr) containing non-shearable precipitates is enhanced compared to the under-aged samples (250°C/i Ohr) with shearable precipitates. 13 4000 a 1400 1200 1000 800 600 400 200 0 0 0.02 0.0.4 (b) Figure 2.4. (a) Work hardening behavior as a function ofo during uniaxial tensile testing for different aging conditions of Al-2.8%Mg-O. 1 6%Sc samples. Lines represent model calculations and symbols indicate experimental data [131(b) Tensile and work hardening behavior of an Al-O.2%Sc alloy for over- aged (300°C/bohr) and under-aged (250°C/lOhr) samples [39]. Figures reproduced with permission from [13, 39] 2.2. Microstructural Evolution in Deformed Al Alloys Combined deformation and annealing treatments in aluminum alloys provide one approach to modify mechanical properties including yield strength and work hardening. These thermo-mechanical treatments result in recovered and/or recrystallized microstructures which can be used to engineer microstructures and mechanical 3000 o Underaged: 20mm 3OOC O Peak strength: 2.5hr 300C t Overaged#1: 8.5hr 300’C+8Omin 425’C o Overaged#2: 8.5hr 3O0C+8day 425CC — Model 1000 0 0 100 200 300 °dls’ MPa (a) 0.06 14 properties. For example, several studies [53,54,55,56] have focused on recovery and recrystallization in aluminum alloys with the aim of developing fine-grained and nanostructured materials via thermo-mechanical processing. Studies on fine grained aluminum alloys have shown that yield strength increases via grain refinement [4,57,58,59]. This section briefly describes recovery and recrystallization in aluminum alloys including Al-Sc alloys. More time is devoted to a description of recovery since the present study strongly focuses on recovery and its impact on mechanical response of an Al-Mg-Sc alloy. 2.2.1. Recovery in Al Alloys Recovery of deformed materials is related to dislocation annihilation and rearrangement by glide, climb and cross slip which results in decreasing the energy of the system. Particularly in high purity aluminum, which has high stacking fault energy, recovery is significant because of rapid climb and cross slip [60]: Studies on rolled high-purity polycrystalline aluminum [61,62,63,64] have shown that accumulated dislocation tangles introduced by deformation arrange into relatively equiaxed cell structures with lower energy. As the level of deformation increases, the dislocation density in cells increases and the size of cells becomes smaller and finally at large strain levels grains can be subdivided into different regions containing cell walls [61]. During isothermal annealing of these deformed structures recovery takes place with sharpening of the microstructure associated with dislocation rearrangement and annihilation within the cells and depending on temperature and time the cell structures eventually transform to sharply defined network regions known as subgrains [60,65,66,67]. These subgrain boundaries are low angle boundaries usually with misorientation angles smaller than 100 15 consisting of individual dislocations. By further annealing, subgrain growth occurs before the onset of recrystallization (discussed in the next section) [68,69,701. The driving pressure for recovery in deformed metals is the stored energy introduced by accumulated dislocations. In high-purity aluminum, rearrangement of dislocations during straining at room temperature results in local softening and reduced stored energy [63,711. This recovery process which occurrs during deformation is known as ‘dynamic recovery’. In solute containing aluminum alloys such as Al-Mg alloys, mobile Mg atoms retard dislocation motion and delay dynamic recovery. Therefore, higher stored energy is maintained in the as-deformed microstructure compared to high-purity aluminum leading to a faster static recovery rate during isothermal annealing. Recovery experiments at temperatures above room temperature on aluminum alloys have demonstrated that the required time for complete recovery is reduced by addition of Mg [72,73]. In Al-Mg alloys, strong solute-dislocation interactions affect the mobility of dislocations and therfore cell structures do not form. Instead relatively uniform distributions of dislocations organize in parallel structures along { 111 } planes as has been observed in Al-5%Mg and Al-2.5%Mg alloys [74,75]. At the microstructural level, similar evolution of dislocations in the recovery of Al-Mg alloys compared to high purity aluminum is observed [73,74]. Figure 2.5 shows microstructural evolution during recovery of an Al-2.5%Mg alloy. Figure 2.5 (a) shows the deformed state with dense dislocation structures. In Figure 2.5 (b) and (c) subgrains appear by dislocation annihilation and coincide with sharpening of subgrain walls followed by subgrain growth. 16 Figure 2.5. TEM image of an Al-2.5%Mg alloy cold rolled to a true strain of 3 showing microstructural evolution during recovery (a) the deformed state (b) annealed at 160°C for 2Ohr (c) annealed at 160°C for 1501w, images reproduced with permission from [74]. 2.2.1.1. Mechanical Properties of Recovered Al Alloys Recovery is an important aspect of microstructural stability which changes the mechanical behavior of metals and alloys. Studies on stress-strain behavior of recovered aluminum alloys have shown that high yield strength with finite work hardening at large stress levels can be achieved [76,77]. This is associated with the fact that during deformation and annealing dislocation structures and subgrain boundaries are introduced which affect the yield strength and work hardening rate and therefore the uniform elongation. In this section the effect of recovery on yield strength and work hardening rate will be reviewed. 2.2.1.1.1. Effect of Recovery on Yield Strength The impact of recovery on yield strength of different materials has been studied since the 1 960s [78]. It has been experimentally demonstrated that there is logarithmic 14 17 time decay in the yield strength of materials during recovery. Increasing the level of pre-strain and annealing temperature leads to accelerated recovery kinetics. Figure 2.6 illustrates the effect of temperature and pre-strain on the decay of yield strength of AA5754 aluminum alloy during recovery. Different modelling approaches have been proposed to quantify the softening behavior and to describe the thermal stability of recovered microstructures based on mechanisms including glide, cross slip, climb and solute drag [15]. The challenge is that dislocation structures formed during deformation and recovery are quite complex, requiring overly simplified assumptions for modelling. 240 300 A 200°C T=200°C • 6=0.58 A Cl 225°C 280 o 6=1.08Q 220 A • 0 Cl 250 C 260 A 6=1.83 o A 6=0.58 .c A A I] A 0) A 200 • • o A 240 u A A (j) • A 220 • Cl 180 Cl >- >200 • 160 180 101 102 10’ 10 101 102 10’ 10 Time (s) Time (s) (a) (b) Figure 2.6. Effect of (a) temperature and (b) pre-strain by cold rolling on recovery kinetics of AA5754 during isothermal annealing [8]. In this section two different physically based modelling approaches applied to aluminum alloys are discussed. The first modelling approach proposed by Verdier et a!. [7] is based on the total dislocation density as a single internal state variable. According to this model recovery is a thermally activated process and internal stress relaxation 18 occurs due to dislocation re-arrangement and annihilation [78]. The decay in the yield strength related to dislocation annihilation during recovery, dis’ is expressed as the following: dudlS = — 64vD .2 exp(_—---)sinh(””) (2.5) dt 9M3a2E dis kBT kBT where U and V are activation energy and volume associated with recovery respectively, E is the elastic modulus, M is the Taylor factor, VD is the Debye frequency, a is a constant and kB is Boltzmann’s constant. This model has been used to fit to experiments on Al-Mg alloys by treating activation energy and volume as adjustable parameters. Table 2.2 summarizes the values of activation energy and volumes for Al-Mg alloys studied by Vedier et al. [7] and Go et a?. [8]. The values for activation energy are found to be higher than the activation energy for climb, cross slip in pure Al or Mg diffusion in Al. This can be related to difficulty of cross slip or climb due to Mg diffusion to dislocations and therefore dynamic recovery is more difficult in Al-Mg alloys compared to pure Al [7]. There is no clear explanation for the value of activation volume which is the volume physically swept by a dislocation from an equilibrium state to an activated state after deformation. Activation volume depends on the activation length which is related to the thermal activation of cross slip and is difficult to quantify [15,79]. One of the main assumptions in this model is that recovery occurs due to the annihilation of randomly distributed forest dislocations. In addition, according to this approach the recovery kinetics are only described by the total dislocation density and do not depend on substructural features such as cell or subgrain size. 19 Table 2.2. Activation energy and activation volume values deduced by fitting Equation 2.5 to fit dis for Al-Mg alloys isothermally annealed at indicated temperatures. Material Pre-Strain Temperature U(kJ/mol) V (b3) Reference Al-2.5%Mg 0.1 160°C 172 46 Al-2.5%Mg 1 160°C 206 22 [7] Al-2.5%Mg 3 160°C 200 22 AA5754 0.58 200-250°C 214 54(Al-3%Mg) AA5754 1.08 200-250°C 214 45 [8](Al-3%Mg) AA5754 1.83 200-250°C 214 39(Al-3%Mg) In the above model, the effect of precipitation on recovery is not considered. In a more recent work by Zurob et al. [80] the effect of concurrent precipitation and recovery in a microalloyed steel is taken into account. This model is applicable when precipitates form heterogeneously on dislocation nodes during recovery. According to this model those dislocation nodes pinned by precipitates are not available for recovery and the recovery rate is modified as: dJd,S — 64vD exp(———)sinh(””)(l —-) (2.6)dt 9MaE kBT kBT n where n is the number of dislocation nodes and is the number of precipitates. The modified version of the model describes now as the number of precipitates increases, the recovery rate reduces. Eventually once n = n, recovery ceases and a plateau appears in the softening kinetics. The plateau continues till the start of precipitate coarsening where dislocation nodes are unpinned by coarsening and dissolution of precipitates. The second modelling approach applied to describe the yield strength of single phase Al alloys has been proposed by Nes [15,81]. This model is based on 20 microstructural parameters such as dislocation density within cells/subgrains and cell/subgrain sizes. In this model, the evolution of microstructure during deformation and annealing is taken into account. According to the Nes model the overall yield strength, is given as: = + aMGb +a1MGb D (2.7) subgrain where o-, is the frictional stress, M is the Taylor factor, G is the shear modulus, b is the Burgers vector, p is the dislocation density inside cells or subgrains, Dsubgralfl is the cell or subgrain size, a and a1 are constants with typical values of 0.3 and 2 respectively. Unlike the Verdier model, subgrain boundaries contribute to the yield strength via the Hall-Petch [82,831 effect. The effectiveness of subgrains with low angle boundaries on the yield strength compared to high angle boundaries is still a question of controversy [81]. In Equation 2.7 it is essential to know the evolution of subgrain size during annealing. Subgrain growth can be treated as a thermally activated process based on coalescence and migration of sub-boundaries and can be described by the following Equation: Dubgrajn = D(subgrain) + Bt (2.8) where Do(subgrajfl is the initial subgrain size after deformation, t is the annealing time and B is a temperature dependent constant [84]. This relationship has been confirmed in Al alloys [85,86] by experimental observations. The Nes model has been applied to describe the softening behavior of pure aluminum and Al-Mg alloys. No attempt has been made to capture the impact of precipitates on recovery kinetics using this model. 21 The Nes model captures the role of subgrain evolution during recovery on the yield strength, on the contrary, according to the Verdier model recovery occurs by re-arrangement and annihilation of randomly distributed forest dislocations and the impact of subgrain size on the yield strength is not taken into account. However, the number of adjustable parameters in the Nes model are larger compared to the Verdier model. In this study, the Verdier approach is considered to understand the effect of recovery on the yield strength of the Al-Mg-Sc alloy. 2.2.1.1.2. Effect of Recovery on Work Hardening In addition to the role of recovery on the yield strength, it is also important to understand the effect of recovered microstructures on the work hardening behavior. Understanding work hardening facilitates an understanding of how to increase strength of materials through recovery while sustaining large uniform elongations. The role of recovery on the work hardening behavior of aluminum alloys has been studied to a limited extent [7,76]. Studies on the work hardening of recovered aluminum alloys are practically limited to annealing temperatures and times where recrystallization does not occur. Verdier et al. [7] applied a model with a single internal state variable, based on Mecking-Estrin [52] model, to recovered microstructures: K )p+ 1 (2.9) de Dsubgraln bDsubgrain This equation builds on Equation 2.3 with the addition of the term K Also Dsubgrain the term has been substituted with 1 In this equation M is the Taylor factor, G bDsubgrain 22 k1, k2 and K are constants, Dbajn is the cell or subgrain size and b is the Burgers vector. The first, second and third terms on the right side represent the storage of forest dislocations, dynamic recovery and storage of geometric necessary dislocations by cell or subgrain boundaries respectively. In this approach it is assumed that dynamic recovery occurs by annihilation of forest dislocations within subgrains as well as in cell walls or subgrain boundaries. The rate of dynamic recovery of boundaries with a thickness of w, is therefore taken as being proportional to the annihilation length within the boundary, 2, times the volume fraction of boundaries, W The term K in Equation 2.9 Dsubgrain Dsubgraln is proportional to the volume fraction of boundaries where K describes the efficiency of the walls in dislocation annihilation: 2w (2.10) The parameter K is assumed to be a function of pre-strain since the thickness of cells or subgrains depends on the level of deformation, the boundaries being sharper with increasing deformation [64]. The details of substructure such as boundary misorientation or shape of cells are not considered in the model. This model has been applied to describe work hardening of single phase A1-2.5%Mg alloys. Combining Equations 2.9 and the Taylor Equation, the work dJd.hardemng rate due to dislocation re-arrangement and annihilation, 8djs = istie obtained: °dis = do-dES = 00(1— -) + — ‘2 o-dis (2.11)de 0S o-d,s 23 where 00, initial work hardening rate in stage II, o, scaling stress, Pj and P2 are constants: P1=M3(aG)2 b (2.12) 2Dsubgrain KM (2.13) 0s 2Dyjjgjj, The term describes the storage of geometrically necessary dislocations by the dis presence of the substructure and P20d1s denotes the total dynamic recovery. In this model, 0‘0sat and K were treated as adjustable parameters and subgrain sizes, Dsubgrain, were measured in the TEM. The parameter K is assumed to be a function of pre-strain and the values obtained for K at different strain levels were found to be consistent with the width of cell walls characterized by TEM observations. Equation 2.11 was used to fit to the work hardening rates extracted from tensile curves of Al-2.5%Mg alloys. The model captures the work hardening behavior of recovered samples annealed at different temperatures using the same values for parameters for a given level of pre-strain. The value of 00 was found to be 1700 MPa after fitting Equation 2.11 to experiments. This is of the right order of magnitude for aluminum alloys as 00 is typically found to be of the order of [511. The work hardening of recrystallized Al-2.5% Mg alloys has been described using the same values for 0 and For recrystallized samples the values ofP1 and P2 in Equation 2.11 are assumed to be zero as there are no contributions from cell/subgrain structures and forest dislocations. 24 In a similar study by Chu et a!. [76] the work hardening behavior of high purity aluminum was investigated based on the Kocks-Mecking model modified to give the following form: &djs ddis = — Cfs dis (2.14)d6 where C is a constant and f. is an adjustable parameter which defines the fraction of the total dislocation that contributes to dynamic recovery. In this model it is assumed that only a fixed fraction of dislocations contribute to dynamic recovery in the presence of subgrains. Equation 2.14 is applied to the work hardening of recrystallized and recovered samples. It was found that f fell between 1 and 0.41 for recrystallized and recovered samples respectively. The values of O (172OMPa) and C (8.55) were fixed for both microstructures. The justification for smaller value ofj in recovered microstructures was stated to arise from the fact that dynamic recovery is retarded due to the presence of subgrain boundaries in contradiction to the Verdier model presented earlier. Moreover, it was assumed that part of the dislocations, defined as geometrically necessary dislocations; do not contribute to dynamic recovery. This modelling approach is similar to the Verdier approach in the sense that extra terms related to the presence of cell or subgrain structures are introduced to the Kocks-Mecking model. However, the model presented by Chu is fit to recovered microstructures annealed only at one temperature. On the other hand, the model proposed by Verdier was fit to recovered samples annealed in the range of 120-220°C. In addition, the terms such as subgrain size or volume fraction of boundaries in Verdier model have more physical meaning and can be verified by TEM observations. In contrastj in Equation 2.14 is treated purely as an adjustable parameter. 25 A three microstructural parameter model including the cell/subgrain size, dislocation density inside cell/subgrains and the sub-boundary misorientations approach has been proposed by Nes [161 to model the work hardening of fcc metals. This modelling approach is not restricted to recovered microstructures and has been applied to describe the work hardening behavior of recrystallized microstructures during deformation to very large strains. In this model the flow stress, o, is described using the same extended Taylor equation as previously described in Equation 2.7 with reference to the Nes model for the yield strength: r=cr1+aMGbJ+aM b 1 (2.15) ‘subgrain The Nes model includes evolution equations for each stage of work hardening describing i) storage of dislocations in cell/subgrain networks ii) boundary misorientation iii) creation of new cell/subgrain boundaries and iv) dynamic recovery by dislocation annihilation. The Nes model is different from the Kocks-Mecking and Mecking-Estrin [50,52] models which are one parameter approaches based on an average dislocation density. Unlike the Kocks-Mecking and Mecking-Estrin approaches where dynamic recovery is assumed to follow a simple linear relation with dislocation density and the dislocation annihilation is only temperature dependent at a constant strain rate, the Nes approach considers more detailed microstructural factors that impact on dynamic recovery. In this approach dynamic recovery has been modeled via evolution equations. The evolution equations describe dynamic dislocation network growth for the dislocation density inside subgrains and also dynamic subgrain growth for the subgrain structures. Moreover, annihilation is treated as a function of both temperature and dislocation density for a given strain rate. 26 The number of adjustable parameters in this model is clearly larger than the Kocks-Mecking-Estrin models owing to its higher degree of complexity. More details can be found in the comprehensive review paper by Nes [161. In this study, the Kocks-Mecking-Estrin modelling approach is applied to the Al-Mg-Sc system due to smaller number of adjustable parameters. 2.2.1.2. Effect of Precipitates on Recovery in Al Alloys There are few studies on the effect of precipitates on recovery in aluminum alloys. Humphreys and Hatherly [60] have discussed how dislocation segments can be pinned by precipitates when the precipitate spacing is equal to or smaller than the cell size thereby delaying recovery. In a force balancing approach, the driving force for the recovery of a 3Dsubgrain dislocation network with a mesh size of Dsubgrajn can be estimated from: FRec = Gb2 (2.16) Dsubgrajn A random distribution of precipitates with a spacing of exerts an opposing force by a magnitude of: FRec = c1Gb2 (2.17) L1 where Cl is a constant. However, most of the work in the literature focuses on the effect of precipitates on subgrain growth since subgrain growth plays an important role on the onset of recrystallization. For example, Jin et al. [55] demonstrated that fine Al6Fe precipitates in aluminum alloys retard heterogeneous recrystallization by stabilizing subgrain boundaries. Progressive evolution of subgrains can then result in the development of 27 ultra-fine grained microstructures. The Zener pinning approach [87] has been used to describe the pinning of subgrain boundaries by precipitates [60]. The net driving pressure, Pnet, for subgrain growth is reduced due to the presence of a random distribution of precipitates: p — ____ fYb ‘21net — Dsubgrain 2r where the first term denotes the driving pressure for subgrain growth with an average size Of Dsubgrain and boundary energy of Yb and a is a constant. The second term is the Zener drag for spherical precipitates with volume fraction of J and average radius of i. According to Equation 2.18, subgrain growth is halted once the net driving pressure is zero and the limiting subgrain size is given by: Diimit = 4c (2.19) At this stage subgrain growth and therefore recovery depends on the precipitate size which can be dictated by precipitate coarsening. This stage of recovery is referred to as extended recovery [88]. It should be noted that the interaction of precipitates with low angle boundaries is more complicated than high angle boundaries in grain growth. For example in a study by Jones et al. [89] the recovery of aluminum alloys containing Al203 particles was investigated by TEM and it was found that the Zener analysis over estimated the limiting subgrain size by several order of magnitudes. TEM observations suggest that in addition to the Zener effect a secondary interaction of precipitates and dislocations also retards recovery through pinning of individual dislocations during migration along the planes of low angle boundaries. In another example the formation of subgrains in an Al-Cu alloy during deformation and annealing was studied by Chang 28 [90]. Concurrent precipitation of 9 particles and recovery results in pinning of subgrains. The obtained limiting subgrain size is found to be smaller than the one estimated by Zener analysis. As described in section 2.2.1.1.1 a modelling approach to the effect of precipitation on recovery kinetics applicable to heterogeneous precipitation on dislocation networks was developed by Zurob et a?. [80]. This modelling approach has also been applied to the study of concurrent precipitation and recovery in AA61 11 alloy by Go et a?. [911. The study on the AA6 111 alloy showed that recovery in the presence of precipitate is completely inhibited when the number of dislocation nodes is about 10% of the total number of precipitates. This retardation stage continues till the start of precipitate coarsening. In the coarsening stage, the recovery rate is then controlled by coarsening kinetics. Pinning of dislocations by precipitates in subgrain boundaries of over aged, cold rolled and annealed AA6 111 samples at 325°C was verified by TEM observations. The pinning of subgrain boundaries by precipitates was also captured in a model on subgrain growth rate, dDsubgraln , proposed by Humpbreys and Hatherly [60]: dDsubgraln = Mbyb( m (2.20)dt Dsubgrain 2 where Mb is the mobility of subgrain boundary, treated as an adjustable parameter and am is a constant. 2.2.2. Recrystallization in Al Alloys Recrystallization of deformed materials is described as the formation of new grains free of dislocations by the formation and migration of high angle grain boundaries 29 with misorientation angles larger than 10l50 [60,92]. Similar to recovery, the driving pressure for recrystallization is the stored energy of dislocations accumulated by deformation. Therefore, recovery and recrystallization can act as competitive mechanisms to lower the energy of the system. The onset of recovery is followed by the recrystallization process, the transition from recovery to recrystallization still being an area of interest [93,94]. Recrystallization involves the growth of subgrains by mechanisms such as migration of low angle boundaries in an orientation gradient or strain induced boundary migration [95]. However, the specifics of the dislocation recovery mechanisms required for the onset of recrystallization is not clear [93]. The most common form of recrystallization is referred to as “discontinuous recrystallization” which can be also described by nucleation and growth stages [94]. Instability in subgrain growth during recovery results in discontinuous recrystallization. During discontinuous recrystallization, there is a sharp increase in the fraction of high angle grain boundaries as the deformed substructure is consumed by the recrystallized grains. Under certain conditions, such as low temperature annealing and high levels of deformation, fine equiaxed grains can evolve gradually from uniform subgrain growth. This type of process is termed “continuous recrystallization” [94]. In continuous recrystallization, recrystallized and unrecrystallized regions of the microstructure can not be distinguished and nucleation and growth stages are not identifiable. A theoretical study by Humphreys [94], showed that the instability of subgrain or grain structures depends on the size, boundary energy and mobility of a particular grain relative to the overall grain structures. It is estimated that instability of a grain or subgrain is facilitated by a high mobility, a 30 large size and small boundary energy. Figure 2.7 illustrates the evolution of continuous and discontinuous recrystallization from subgrain structures. Figure 2.7. Schematic presentation of (a)-(b) continuous, (c)-(d) discontinuous recrystallization from subgrains, figures reproduced with permission from [94]. Recrystallization is an important phenomenon to be considered for designing microstructures with required mechanical properties in aluminum alloys. Studies on aluminum alloys have shown that by controlling continuous and discontinuous recrystallization, fine grained microstructures with grain sizes of 1-2 urn, can be processed to improve the yield strength compared to coarse grained materials [96,97]. Investigations on fine grained aluminum alloys containing precipitates reveal that total tensile elongation is improved extensively via superplastic behavior at high temperatures [98,99]. There is also a potential to process heterogeneous microstructures by controlling recrystallization. For example, a fine grained Al-Mg alloy with duplex grain structures I I (a) (b) (c) (d) 31 and modified tensile response has been processed through thermo-mechanical treatment [5]. When discontinuous recrystallization is inhibited, recovery processes proceed to an extended level so a uniform subgrain structure forms and continuous recrystallization can occur by annihilation and rearrangement of dislocations at subgrains leading to equiaxed fine grain structures. This transition from recovery to continuous recrystallization happens when there is sufficient stored energy in the material after the recovery process. The study by Oscarsson et a!. [100] on recrystallization of aluminum alloys reveals that the transition from continuous to discontinuous recrystallization is achieved by increasing the level of cold reduction. This is also demonstrated by Jazaeri et a!. [101,1021 on recrystallization behavior of aluminum alloys. They also concluded that by decreasing the initial grain size, continuous recrystallization is promoted. This is attributed to the fact that increasing strain levels and smaller initial grain size results in a larger fraction of high angle grain boundaries in the microstructures. Microstructures with larger amounts of high angle grain boundaries are more stable against discontinuous recrystallization since the energies and mobility of high angle grain boundaries do not strongly depend on misorientation compared to low angle grain boundaries. Figure 2.8 shows the effect of initial grain size and strain on the transition from discontinuous to continuous recrystallization in an Al-Fe-Mn alloy. Jazaeri et a!. [101,102] pointed out that the initial deformed structure will only be stable against discontinuous recrystallization when the fraction of high angle boundaries is larger than about 70%. 32 1000 . . . . I • 100 Discontinuous _0 .9 I I • • 0 0 — Iv .9 — Continuous I I 0 0 0 0 1 2 3 4 Strain Figure 2.8. The effect of grain size and strain on transition from discontinuous to continuous recrystallization in the AA8006, figure reproduced with permission from [1021. It is also essential to emphasize the effect of precipitates on recrystallization behavior of aluminum alloys. Humpbreys and Hatherly [601 noted that precipitates can have different effects on recrystallization depending on the nature of precipitates such as size or volume fraction. Precipitates larger than 1 jim can act as nucleation sites for recrystallization through particle-stimulated nucleation (PSN). Precipitates can increase the stored energy during deformation and therefore the driving force for recrystallization. Fine spherical precipitates with volume fraction of f and average radius of exert a pinning pressure, Pz, on grain boundaries with energy of according to the Zener-Smith [87] analysis: = 3fr (2.21) 2 Therefore, precipitates can act as recrystallization inhibitors. In this section the main focus is the retarding effect of precipitates on recrystallization. In a study by Jones and Humphreys [11] the impact of A13Sc precipitates on recrystallization of Al-Sc alloys was investigated. It was found that during annealing of deformed supersaturated 33 Al-0.25%Sc recrystallization is inhibited up to very high homologous temperatures. At temperatures higher than 500°C, precipitates lose their full coherency and Zener pressure is reduced due to precipitate coarsening and therefore recrystallization proceeds. In an alloy containing only 0.12%Sc recrystallization was not completely halted at 375°C and concurrent recrystallization and precipitation results in microstructures with deformed and recrystallized regions. Figure 2.9 illustrates the recrystallization kinetics of Al-Sc alloys containing 0.02%, 0.12% and 0.25% Sc. According to Figure 2.9 recrystallization start and end temperatures for the Al-0.12%Sc are nearly independent of the annealing time which indicates that precipitation starts before the end of recrystallization and complex interaction between recrystallization and precipitation occurs. Similar results in Al-Mg alloys containing non-shearable A13Sc have been observed [103,104,1051. According to Mirua’s study [106], the addition of 0.2%wt scandium to the Al-3%Mg increases the recrystallization start temperature by about 200°C. As these A13Sc precipitates pin high angle grain boundaries and prevent grain growth, it is expected that they are useful for producing stable fine grain microstructures through continuous recrystallization. The experimental results on heavily deformed Al-Mg-Sc-Zr alloys followed by annealing [54] indicate, however, that the fine precipitates have a much more complex effect on the continuous recrystallization behavior of these alloys. It is confirmed that after annealing at 400°C where there is enough boundary mobility for continuous recrystallization, a lamellar structure still exists in the deformed microstructures of these alloys due to boundary impingement by precipitates. Even at higher temperatures and strains, it has been 34 observed that the fine grains are more elongated in these alloys compared to those aluminum alloys free of precipitates. 200 O 0.02%Sc - start • 0.02%Sc - finish100 O0.12%Sc-start •0.12%Sc-flnish O 0.25%Sc - start • 0,25%Sc - finish 0 1 10 100 1000 10000 100000 10000 Time (s) Figure 2.9. Isothermal recrystallization kinetics of Al-Sc alloys showing recrystallization start (5%) and finish (95%) as a function of annealing time, figure reproduced with permission from [11]. Fine grained microstructures in Al-Sc alloys processed via severe plastic deformation up to large strain levels and given annealing treatments at low temperatures have been recently observed [107,108,53]. In these alloys, A13Sc precipitates have been found to be very useful for grain refinement since they provide thermal stability to the deformed microstructures and prevent grain growth up to high temperatures. Gholinia et al. [54] has developed micron-scale grain structures in an Al-3%Mg-0.2%Sc-0. 1 %Zr alloy containing stable A13(Sc,Zr) precipitates by rolling to a strain of 6 and annealing at 300-500°C. In other studies, Ferry et al. [53,107,108] developed refined microstructures in an Al-0.2%Sc alloys by deformation to a strain of 9 using equal channel angular pressing (ECAP) and aging at 3 50°C for times longer than 1000 s. Aging at 3 50°C for 3 hr produced an equiaxed microstructure with 0.8 im grain size and uniform distribution of A135c particles via the advance of continuous 35 recrystallization. Annealing of the pre-aged alloy at temperatures up to 5 00°C results in continuous grain coarsening but at temperatures higher than 500°C, discontinuous grain coarsening occurs generating coarse grained microstructures. Figure 2.10 (a) represents the refined microstructure of an Al-0.2%Sc alloy after aging at 350°C and Figure 2.10 (b) shows the effect of annealing time and temperature on grain size. 1.8 1.4 0.21 0 20 40 60 80 100 120 140 Time (mm) (a) (b) Figure 2.10. (a) Grain structure of a deformed Al-O.2%Sc alloy during pre-aging at 350°C for 3hr obtained from electron backscattered diffraction method (EBSD): different colors indicate different grain orientations (b) The effect of annealing time and temperature on grain size of the same alloy, figures reproduced with permission from [53]. 2.3. Summary of Literature Review The studies on Al-Sc alloys reveal that precipitation of A13Sc results in increasing yield strength and the peak age at low temperatures such as 3 00°C is achieved due to presence of large number density of shearable precipitates. Fewer recent studies on work hardening behavior of Al-Sc alloys show that work hardening is enhanced when A135c precipitates are non-shearable because of storage of additional geometric dislocations. This is described in the frame work of Kocks-Mecking-Estrin modelling approach. A 400°C 450°C 0 500cc 36 The impact of A13Sc precipitates on recrystallization has been also investigated. Precipitation ofA13Sc in deformed and annealed Al-Sc alloys delay recrystallization up to high homologous temperatures due to strong pinning effect on grain boundaries. Fine grained microstructures in Al-Sc alloys can be achieved by precipitation, deformation and annealing treatment due to continuous subgrain growth and continuous recrystallization. The nature of recrystallization (continuous or discontinuous) depends on the stability of subgrain structures. However, mechanical properties including yield strength and work hardening of Al-Sc alloys with stable unrecrystallized microstructures have not been investigated. Different modelling approaches based on dislocation annihilation and subgrain formation are presented to explain the relation between mechanical properties and recovery in the absence of precipitates in alloys such as single phase Al-Mg alloys. No similar attempts have been made to understand the role of precipitates on mechanical properties of recovered microstructures. Al-Mg-Sc alloys are ideally suited to be model alloys for this purpose due to the various properties ofAl3Sc precipitates described here. 37 Chapter 3-Scope and Objectives Based on the previous literature review one can conclude that the yield strength and work hardening of recovered materials containing precipitates are not well understood. Yield strength and work hardening can be improved by both recovery and precipitation occurring during thermo-mechanical processing. The main objective of the present work is therefore to study the effect of recovery on the yield strength and work hardening of a model Al-Mg-Sc alloy in the presence ofA13Sc precipitates. In addition to precipitation hardening, the effectiveness of Al3Sc precipitates at pinning boundaries provides an opportunity for following the progress of recovery without interference from recrystallization when annealing for long times at high homologous temperatures. In order to achieve recovered microstructures containing AI3Sc precipitates, pre-aged samples were exposed to deformation and annealing treatments. Annealing was performed at the same temperatures selected for pre-aging treatments or at temperatures in which re-precipitation is suppressed. These specific processing routes were performed to avoid concurrent precipitation and recovery which adds significant complexity to the microstructural development. Deformation was limited to one level of strain to fix the initial dislocation density and substructure for recovery. The chemical composition of the model alloy was Al-2.8%Mg-O.16%Sc. This alloy was selected since the effect of Al3Sc on the mechanical properties of as-aged materials with the same chemical composition was investigated in a previous study. In addition, the mechanical properties of a recovered single phase Al-Mg alloy with a similar Mg content was studied so as to provide a base for the material behavior without precipitates. 38 This study, aims to provide detailed experimental data on tensile response of recovered microstructures containing A13Sc precipitates. From this data physically based models have been built based upon pre-existing models for systems without precipitates. One of the main challenges here is that there is no detailed literature on the role of precipitates on recovery kinetics and mechanical properties (yield strength and work hardening) of recovered microstructures in aluminum alloys. The results of this work provide the knowledge necessary for achieving different combination of mechanical properties in Al-Sc alloys as well as other aluminum alloys by controlling chemical composition, deformation and thermal histories. 39 Chapter 4-Experimental Procedures The objective of this chapter is to describe experimental procedures for developing and characterizing recovered microstructures. First the materials studied are described. Next, the thermo-mechanical treatments for processing the materials are explained. This is followed by a description of the characterization techniques and mechanical testing applied to the processed materials. 4.1. Starting Materials The Al-Mg-Sc alloy for this study was received from the Novelis Global Technology Centre in Kingston ON Canada as a laboratory cast and hot rolled plate with a thickness of 5mm. A cast Al-Mg alloy with similar Mg content to the Al-Mg-Sc alloy was also obtained. The cast Al-Mg alloy without Sc was received in a solution treated state at 530°C for 14 hours followed by water quenching to room temperature. The chemical compositions of both alloys are given in Table 4.1. Titanium diboride was added to the melt during casting for grain refinement. Table 4.1. Chemical composition of Al-Mg-Sc and Al-Mg alloys (Weight Percent) Alloy Mg Sc Ti Fe Si Others Al Al-Mg-Sc 2.90 0.16 0.13 0.05 0.03 <0.001 RAl-Mg 2.50 - - 0.01 <0.001 <0.001 est 4.2. Solution Treatment The solution treatment of the Al-Mg-Sc alloy was performed to dissolve precipitates produced during laboratory casting and hot rolling, thereby obtaining a supersaturated solid solution. The second purpose of solution treatment was to eliminate 40 the elongated grain structure produced during hot rolling. Solution treatment of the Al-Mg-Sc alloy was carried out at 610°C for 7 days in a box furnace followed by water quenching to room temperature. In order to improve homogeneity of temperature and maximize the heating rate during solution treatment two aluminum plates each with a thickness of 10 mm were pre-heated to 610°C in the box furnace before solution treatment. The Al-Mg-Sc sheets were placed between the plates and the temperature of Al-Mg-Sc sheets was monitored using a Fluke K-type thermometer connected to a chart recorder. The heating rate at the onset of solution treatment was found to be 50°C/mm and the temperature variation during annealing was recorded as ± 5°C during the 7 day solution treatment. The temperature for solution treatment was determined from isothermal ternary sections of the Al-Mg-Sc system which were computed for temperatures in the range of 575-620°C. The assessed heats of mixing for the Al-Sc and Al-Mg systems [109,1101 and the heat of formation for Al3Sc [109] were used to construct the ternary sections assuming that Al3Sc is a stoichiometric compound. Reference Al-Mg-Sc samples were solutionized at the optimum temperature for various times and based on observations of microstructures free of segregation and precipitates, a time for solution treatment was selected. 4.3. Thermo-mechanical Treatments The objective of designing thermo-mechanical treatments was to develop recovered microstructures in the Al-Mg-Sc alloy which contained non-shearable precipitates with a strong resistance to coarsening. In order to process recovered microstructures it was also required to understand the temperature range for the onset of recrystallization. To fulfill the abovementioned objectives a number of deformation and 41 annealing routes were designed. In this section, first the general procedures for deformation and annealing treatments will be described followed by presenting the details of thermo-mechanical routes on recovery and recrystallization. 4.3.1. Rolling Experiments In this study, different levels of strain were introduced to the materials by 70%, 80% and 90% cold rolling corresponding to true von Mises strains of 1.4, 1.85 and 2.65 respectively. Rolling was conducted on a conventional laboratory rolling mill consisting of an electric motor connected to work rolls with a diameter of 56 mm through a gear box. In order to introduce the same strain distribution through the sheet thickness in each rolling pass, various rolling schedules were defmed for different total reductions. Experiments were designed so as to keep the same deformation geometry in each rolling pass as defined by the A factor: H1 /R(H0—1) (4.1) where R is the radius of rolling mills, H0 and H1 are the initial and final thickness in each rolling pass. In this study rolling schedules were designed according to Table 4.2 to introduce A factors smaller than one to help reduce strain inhomogeneity [1111. Table 4.2. Experimental rolling schedules corresponding to different total reductions Total Number of Average Reduction per Maximum Minimum Reduction Passes A Pass A A 70% 5 0.4 14% 0.53 0.31 80% 8 0.4 10% 0.53 0.33 90% 15 0.4 6% 0.53 0.32 42 4.3.2. Heat Treatments Heat treatments including aging and isothermal annealing at temperatures above 250°C for less than one day were performed in molten salt baths (60% potassium nitrate + 40% sodium nitrite). Heat treatments at temperatures lower than 250°C were performed in an oil bath containing silicone oil. In order to minimize temperature gradients, both salt and oil baths were stirred during heat treatments. The temperature of the baths was monitored by a K-type thermometer connected to a chart recorder. The heating rate of samples in salt and oil baths was found to be 50°C/s and temperature variation was recorded as ± 3°C. Heat treatments longer than one day were conducted in a box furnace using the same procedure as described in section 4.2. All heat treatments were followed by water quenching to room temperature. 4.3.3. Recrystallization In order to study concurrent precipitation and recrystallization in the Al-Mg-Sc alloy, the solutionized plate was cold rolled to true von Mises strains of 1.4 and 2.65 corresponding to reductions in thickness of 70% and 90% respectively. The cold rolled material was subsequently annealed in salt baths at various combinations of temperature and time so as to establish the onset and end of recrystallization. Recrystallization was not the main focus of the present study and therfore the aim of this preliminary investigation was to identify the range of temperatures and times over which recrystallization may occur for different levels of deformation. This provides useful information for selecting annealing temperatures suitable for achieving recovered microstructures. The results of concurrent precipitation and recrystallization are presented in Appendix A. 43 4.3.4. Recovery To develop recovered microstructures from solutionized materials, different processing routes were performed to obtain microstructures containing 1) different precipitation states and 2) different levels of recovery. In the Al-Mg-Sc alloy, an aging treatment was performed on the solutionized material so as to produce a distribution of non-shearable precipitates. In the highly overaged condition, A13Sc precipitates are known to have a low rate of coarsening [14], meaning that little evolution of the precipitate size and number density was expected to occur during subsequent annealing. Two highly overaged conditions with different average precipitate size were selected for this study. These alloys were subsequently rolled then annealed (as described below) for various times in order to evaluate their microstructure, yield strength and work hardening rate. Finally, the binary Al-Mg alloy was subjected to similar rolling and recovery annealing so as to facilitate the comparison of recovery kinetics in the absence of precipitates. In the following sections the detailed processing routes used to prepare these two alloys are described. 4.3.4.1. Aging Treatments The Al-Mg-Sc alloy was aged so as to produce two conditions containing highly overaged precipitates. This was achieved by a two-step annealing procedure. First, two batches of Al-Mg-Sc alloy were annealed at 3 00°C for 8 hr and quenched in room temperature water. This results in an initial fine dispersion of coherent A13Sc precipitates that precipitate relatively uniformly throughout the microstructure as characterized by Fazeli et al. [14]. The two batches of material were next separated and submitted to 44 either an isothermal anneal at 425°C for 80 mm or an isothermal anneal at 425°C for 8 days. Based upon previous work [13], it was expected that these two annealing treatments would result in significantly different precipitate size distributions both of which would be well beyond the shearable/non-shearable transition (a precipitate diameter of 8 nm). The choice of using overaged samples was made so as to reduce uncertainty in the interpretation of the mechanical properties of recovered materials that would occur if significant evolution of the precipitation state occurred during the recovery anneals. 4.3.4.2. Deformation and Annealing Treatments The aged Al-Mg-Sc and the solutionized Al-Mg samples were cold rolled to a total reduction of 80% corresponding to true von Mises strain of 1.85. Tensile samples were mechanically punched from the cold rolled sheets. The as-rolled materials in the form of tensile samples were subsequently subjected to isothermal annealing to induce recovery. In the case of the Al-Mg-Sc alloy, preliminary studies on recrystallization of the same material revealed that recrystallization is completely suppressed at high homologous temperatures such as 425°C [12] (See Appendix A). Thus, one set of Al-Mg-Sc samples was subjected to isothermal annealing at 425°C for times ranging from a few seconds to several weeks. Choosing 425°C as the recovery temperature has the dual advantage of being a high homologous temperature, thus ensuring rapid microstructural reorganization during annealing, while also being equivalent to the pre-aging annealing temperature. Therefore, the volume fraction of precipitates formed in the pre-aging treatment is stable at this annealing temperature. 45 Annealing at 425°C results in very rapid recrystallization of the Al-Mg sample, thus a second, lower, temperature was selected for performing recovery anneals on both the Al-Mg and the Al-Mg-Sc alloys. For this purpose an annealing temperature of 190°C was selected. Previous studies on recovery of an Al-Mg alloy with similar composition to that studied here [7] showed that recrystallization does not occur at 190°C up to very long times (several weeks). In the case of the Al-Mg-Sc alloy, further precipitation is possible at this lower temperature owing to the lower solubility at 190°C compared to 425°C. To check for further precipitation, yield strength measurements were made on materials subjected to the two-step aging treatment described in section 4.3.4.1 and then annealed further at 190°C without rolling. No change in yield strength was observed for samples annealed for up to 14 days. Some samples were also annealed at 275°C to investigate if concurrent recovery and precipitation occurs at this temperature. To complete the compliment of samples, a series of the Al-Mg-Sc samples (two different precipitation states) were prepared exactly as the samples described above. However, in this case the samples were not cold rolled. Thus, these aged samples were used to estimate the evolution of the precipitate contribution to the flow stress of the material in the absence of recovery. Finally, a set of as-solutionized Al-Mg-Sc samples and a set of fully recrystallized Al-Mg samples were prepared in order to estimate the solid solution contribution to the flow stress. The as-solutionized Al-Mg-Sc samples were solutionized as described above then rolled and given a 1 Os anneal at 600°C to achieve a fully recrystallized, precipitate free microstructure. 46 Figure 4.1 summarizes all of the different annealing and deformation conditions considered in this section. Through the rest of this thesis, a short form notation for the various different samples will be used. In the case of Al-Mg-Sc, recovered samples have been labeled as Rec-X-Y. In this case X is either 80 mm or 8 days, indicating the duration of the final precipitate heat treatment. The second parameter, Y, indicates the time of the recovery annealed at 425°C or 190°C. Samples not subjected to rolling are referred to as Aged-X-Y, X and Y having the same meaning as for the recovered samples. Recovered Al-Mg samples are labeled as Rec-Al-Mg-Y where Y denotes annealing time at 190°C. Samples which are only subjected to rolling are referred as “Rolled”. 47 80% 40s -136 dayS o1led& 3000C, 5201 ,405_l36day5 pMgSC Plate 8 days__ U40s365 ornoge Primary Aging RecovetY C0ndatY Aging figure 4.1. Ther schedules design for reCOV of Al-Mg and al-Mg alloys 48 4.4. Tensile Testing To characterize the mechanical response including yield strength and work hardening of Al-Mg-Sc and Al-Mg alloys, uniaxial tensile tests were conducted at 77K. The reason for performing tests at 77K was to avoid dynamic strain aging common in Al- Mg alloys at room temperature [112]. Tensile samples were punched out of cold rolled sheets with a thickness of 1 mm using a manual die. An extensometer with a gauge length of 25.4 mm was attached to the reduced cross section of samples to measure the elongation. Tensile tests were carried out at a strain rate of 103s’ using an Instron machine with a load cell of 5 kN. The samples and the attached extensometer were immersed in a liquid nitrogen bath during testing, the start of tests commencing once the sample had equilibrated at the bath temperature. Load and displacement were recorded during tensile testing and were converted to engineering stress-engineering strain curves. From engineering stress-strain curves true stress-true strain plots were extracted and the yield strength was determined applying 0.2% offset. The work hardening rate was obtained by numerically differentiating the true stress with respect to true strain followed by smoothing using polynominal fitting to successive segments of the curve. For each condition of thermo-mechanical treatment two tensile tests were conducted and the yield strength was determined by averaging the yield strength of two tests. It was also assured that tensile curves for studying work hardening in each condition were reproducible with a difference less than 5% in flow stress. 49 4.5. Characterization 4.5.1. Optical Microscopy A number of as received hot rolled Al-Mg-Sc samples were selected for optical microscopy to observe the initial grain structure. Small samples with a length of 25mm in the rolling direction were cut from the as received hot rolled plates and were cold mounted on the transverse direction (TD) surface, parallel to the rolling direction. The TD surface was polished using 400 jim, 600 jim and 1200 jim grinding papers followed by polishing to 0.05 jim finish. Samples were then anodized in the Barker’s reagent (200 ml distilled H20 + 6 ml HBF4 (5Owt%)) to reveal the microstructure. The microstructure was then studied using a Nikon EPIPHOT 300 optical microscope under polarized illumination. 4.5.2. Scanning Electron Microscopy (SEM) The microstructure of as-solutionized, deformed and annealed samples was studied with a Hitachi electron microscope S-3000N operating at 20 keV. The microstructure of reference samples exposed to solution treatments for various times were studied by SEM using backscattered electron (BSE) images to understand the optimum time for solution treatment thereby removing segregation in the as-received samples. Deformed and annealed samples were studied to investigate the occurrence of recrystallization. The TD surface of all samples was mechanically polished up to 1200 jim and electropolished in a solution of 20% perchloric acid (60%) in methanol at -3 0°C and viewed in the SEM using backscattered electron contrast. The degree of recrystallization was assessed by the point fraction method according to ASTM E 562. In 50 this method square grid points were superimposed on the BSE images and the fraction of points falling within recrystallized grains was calculated in each field and was averaged over 5 fields for each deformation and annealing condition. 4.5.3. Electron Back Scattered Diffraction (EBSD) In order to verify that no recrystallization occurred in recovered samples a number of deformed and annealed samples were selected for Electron Back Scattered Diffraction (EBSD) study using a Hitachi scanning electron microscope (SEM) at 20 keV. The TD surface shown in Figure 4.2 was mechanically polished up to 1200 tm and electropolished in a solution of 20% perchioric acid (60%) in methanol at -30°C. The samples were inserted in the SEM chamber and the sample holder was tilted 70° towards the detector. The HKL Channel 5 software was used to obtain the EBSD patterns using a step size of 0.5 rim. The boundary between low and high angle boundaries was selected as 15° [60]. 4.5.4. Transmission Electron Microscopy (TEM) Recovered and as-aged microstructures were studied by Transmission Electron Microscopy (TEM) to measure the size of precipitates and subgrains. The as received sheet was also studied using Scanning Transmission Electron Microscopy (STEM) to analyze the phases present after hot rolling by Electron Dispersive X-ray (EDX) analysis. Thin foils were prepared by the conventional method through mechanical polishing up to thickness of 80-100 j.im, punching samples from the sheet and jet polishing using an electrolyte consisting of 5% percholoric acid in methanol operating at -30°C and 20 V. The foils were examined in a Hitachi H-800 scanning-transmission electron microscope 51 operating at 200 kV. To measure precipitate diameters dark field images of recovered and aged samples were taken. The images of precipitates were digitized and precipitate diameters were determined for 3000 precipitates in each case. Subgrain sizes of recovered samples were measured from bright field images using the intercept method [1121. Recovered microstructures were examined by tilting samples close to the (100) Al or (110) Al zone axis. This allowed the dislocation substructures to be revealed at the same time as the A13Sc precipitates since a distinct cube-cube orientation relationship exists between the super-lattice of the Al3Sc precipitates and thefcc aluminum matrix [9]. The surface for TEM study is shown in Figure 4.2. 4.5.6. Texture Measurements Texture measurements on rolled, recrystallized and recovered Al-Mg-Sc samples were conducted at the Novelis Global Research and Development Centre in Kingston. A selected series of recovered and aged samples were mechanically polished on surfaces parallel to the rolling plane to 1/4 and 1/2 of the sheet thickness followed by electropolishing in the solution of 20% perchloric acid (60%) in methanol at -30°C. The surface selected for texture measurements is illustrated in Figure 4.2. X-ray pole figures were measured on the electropolished surfaces using Cu K radiation. For each condition four pole figures, {111}, {200}, {220} and {311} were obtained. For every pole figure, tilt angles (cL shown in Figure 4.3) from 0° to 80° corresponding to 17 sets was recorded. With each tilt angle, the specimen was rotated 360°, 3 in Figure 4.3. Each obtained datum is the ratio between measured intensity and the intensity of a random aluminum powder. Orientation Distribution Functions (ODFs) were computed using the ODF/PF Processing Library developed by P.D. Dawson 52 [113,114]. Taylor factors were calculated from the ODFs assuming { 111 } <110> slip using the Material Point Simulator available from the Deformation Process Laboratory at Cornell University (http://anisotropy.mae.cornell.edu). 53 Surface for TEM Study and Texture Measurements 7 NB RD Surface for EBSD Study Figure 4.2. Selected surfaces in sheet samples for EBSD, TEM studies and texture measurements Figure 4.3. Illustration of selected pole figure angles for texture measurements \13 I RD TD 54 Chapter 5-Experimental Results This chapter describes the experimental results obtained from microstructural characterization and from mechanical property measurements of materials subjected to the processing routes presented in Figure 4.1. As explained in section 4.3, understanding the mechanical response of recovered microstructures containing Al3Sc precipitates requires investigations of as-aged samples for comparison. First, characterization of the materials is presented. This is followed by presentation of results on the tensile response of the alloy. 5.1. Starting Materials Figure 5.1 illustrates an optical image of the initial microstructure of hot rolled Al-Mg-Sc alloy on the TD surface. The elongated grain structure parallel to the rolling direction shows that neither partial nor complete recrystallization took place after hot rolling. I I p I I r I jimi --p Figure 5.1. Optical micrograph of the elongated grain structure in the A1-2.8%Mg-O. 16%Sc alloy 55 Figure 5.2 shows the microstructure of the as received cast ingot of A1-2.5%Mg, solution treated at 530°C for 14 hr. The microstructure is fuiiy recrystallized with an average grain size of about 1 mm determined by the linear intercept method. Figure 5.2. Backscattered image of coarse grained microstructure of the as cast A1-2.5%Mg alloy, black features are characterized as voids formed during casting. 5.2. Solution Treatment of Al-Mg-Sc Alloy Figure 5.3 shows ternary sections of the Al-Mg-Sc system calculated using the data from [109,110]. Here it is assumed that A13Sc is a stoichiometric compound. The alloy for this study (Al-2.8%Mg-0.16%Sc) is indicated by a solid square. Error bars are shown to illustrate the effect of a ± 10% fluctuation in the chemical composition which may arise from segregation due to casting. According to this figure the highest temperature in the Al single phase region, 610°C, is selected for the solution treatment of the alloy. The ternary section at 620°C (not presented here) shows that at 620°C the alloy shifts into the Al-Liquid region indicating the possibility for incipient melting. 56 0.30 0.25 0.20 0.15 0 (1) 0.10 0.05 0.00 AI3Sc+AI . AI3Sc+AI+L I -‘ A — — — - AAI+L —--‘a.— 575CC C 600C a 610C A 0 1 2 3 4 5 6 7 Mg (%wt) Figure 5.3. Isothermal sections of the Al-Mg- Sc ternary phase diagram at different temperatures The solvus line for an Al-Sc binary alloy and the pseudo binary Al-2.8%Mg-0.16%Sc alloy are also shown in Figure 5.4. It can be seen that the addition of 2.8%Mg does not significantly change the solvus position, particularly at low temperatures. The equilibrium volume fraction of A13Sc for the Al-2.8%Mg-0.16%Sc alloy calculated from Figure 5.4 (a) is shown in Figure 5.4 (b). At temperatures below 400°C, the volume fraction remains almost constant at 0.5%. Above 450°C the volume fraction decreases sharply until it reaches almost zero at 595°C. 57 800 0.006 0 0.005 3600 3 0.004 400 0.003 I) L1 E 0.002 i200 0.001 0 0.0 0.1 0.2 0.3 0.4 0.5 Sc(%wt) (a) (b) Figure 5.4. (a) Solubility of scandium in aluminum with and without 2.8%Mg (b) Volume fraction of A13Sc precipitates versus temperature calculated from the solvus in (a). Figure 5.5 illustrates the microstructure of the reference Al-Mg-Sc sample after solution treatment at 6 10°C, for 8 hr, 24 hr and 7 days. As shown in Figure 5.5 (a) in the sample solution treated for 8 hr some regions appear in the form of elongated narrow bands containing particles parallel to the rolling direction (RD). According to Figure 5.5 (d), EDX analysis of the segregation bands (indicated by an arrow in Figure 5.5 (a)) indicates higher than average levels of Sc and Ti. Figure 5.5 (e) illustrates EDX analysis of the sample shown in Figure 5.5 (a) outside of the segregation bands indicating the absence of excess Sc and Ti. Figures 5.5 (b) and (c) show that after 24 hr, most bands are removed and after 7 days they are completely eliminated. As mentioned in section 4.2 solutionizing at 6 10°C for 7 days is selected as the optimum condition to dissolve all Sc and Ti particles. This long time solution treatment resulted in substantial grain growth. The average grain size of the starting microstructure after solution treatment (Figure 5.5 (c)) was found to be approximately 150 jtm. LiquidI Liquid Al+Al3Sc Al-Sc —a— Al-2.8%Mg-Sc — -— AI-2.8%Mg-0.16%Sc 100 150 200 250 300 350 400 450 500 550 600 650 Temperature (CC) 58 (e) Figure 5.5. (a) The Al-2.8%Mg-0. 1 6%Sc microstructure after solution treatment at 610°C for 8 hr (b) microstructure after solution treatment at 610°C for 24 hr, (c) microstructure after solution treatment at 610°C for 7 days, RD stands for rolling direction and ND for normal direction. (d) EDX analysis of the segregation indicated by an arrow in (a), (e) EDX analysis of the sample shown in (a) outside of the segregation bands. Counts Energy 3e’J) Counts (d) Energye)) 59 5.3. Microstructural Characterization In this section characterization of the microstructures resulting from the processing routes in Figure 4.1 are presented. The main focus here is on recovered and aged samples. In particular, microstructural observations have been focused on identifying size and distribution of precipitates in aged and aged and recovered materials as well as making qualitative (comparative) observations of the microstructural evolution associated with recovery. 5.3.1. Microstructure of Aged Samples As described in the literature review, aging of the Al-Sc system at sufficiently low temperatures results in a very fine distribution of small precipitates which are relatively uniformly distributed within individual grains indicative of the fact that heterogeneous nucleation does not dominate [14,26]. Based on Figure 4.1, pre-aging was first performed at 3 00°C to set the initial distribution of precipitates to have these characteristics. Performing the full pre-aging treatment at 425°C would have resulted in substantial amounts of heterogeneous precipitation [14]. Thus, the as-aged samples have a relatively uniformly distributed set of precipitates despite the high final aging temperature and large final size of the precipitates. 5.3.1.1. TEM Observation of Aged Samples Figures 5.6 and 5.7 illustrate TEM observations on the as-aged materials (prior to rolling) in dark and bright field images. Bright field images show the Ashby-Brown [115] contrast of Al3Sc precipitates indicating the coherency strain contrast and the diffraction patterns which demonstrate the super-lattice reflections arising from 60 precipitates. Using dark field images similar to those in Figures 5.6 and 5.7, the precipitate size (diameter) distribution has been measured. The results of these measurements are presented in Figure 5.8. From these size distributions the number average precipitate diameter, d,,, was calculated: - d. = (5.1) total where tota1 is the total number of observed precipitates, d1 is the measured diameter of each precipitates. For the samples pre-aged for 80 mm at 425°C, d was found to be 12 nm. A number average diameter of 24 nm was found for the samples pre-aged for 8 days at 425°C. It is observed that the distribution of precipitate diameter is wider in samples pre-aged at 425°C for 8 days compared to those pre-aged at 425°C for 80 mm. The dark field images also demonstrate that precipitates pre-aged at 425°C for 8 days have a much larger inter precipitate spacing compared to those pre-aged at 425°C for 80 mm. Figure 5.8 (c) shows the precipitate size distribution for samples pre-aged at 425°C for 80mm followed by further annealing at 425°C for 2Ohr. The number average size for this condition was found to be 14 nm compared to 12 nm found for the samples pre-aged at 425°C for 80 mm. It can be seen that further annealing up to 20 hr does not greatly affect the size distribution of precipitates, reflecting the low rate of coarsening of these precipitates. 61 .7, •0 . •,‘-. 4 ...., • rt • ..,, (c) Figure 5.6. TEM images and diffraction pattern of A13Sc precipitates in samples pre-aged at 425°C for 80mm (a) bright field image (b) dark field image using the A13Sc <100)’M super-lattice reflection (c) corresponding diffraction pattern <IOO>Al zone axis 62 (a) (b) Figure 5.7. TEM images and diffraction pattern of A13Sc precipitates in samples pre-aged at 425°C for 8 days (a) bright field image (b) dark field image using the A13Sc<100>Al super-lattice reflection (c) corresponding diffraction pattern <100>AI zone axis I :L ,,44” ;: 45C (a) (b) (c) 63 40 40 _._.35 30 30 C) C) C25 25 D D 2O u_ u_ 15 15 E 10 2 10 D z z 0 60 50 > 40 a) 0 ) 30 U 20 2 z 10 0 80 100 120 Figure 5.8. Precipitate diameter distributions for (a) Aged-8Omin (b) Aged-8days (c) Aged-8Omin-2Ohr 5.3.2. Microstructure of Recovered Samples The microstructures of recovered samples were characterized by various techniques. The results of these observations are presented in this section. 5.3.2.1. TEM Observations of Recovered Al-Mg-Sc and Al-Mg Samples Figures 5.9 shows TEM observation of samples pre-aged at 425°C for 80 mm, cold rolled and annealed at 425°C for 40s, 2Ohr and 14 days. The progress of recovery is clear in Figure 5.9. Figure 5.9 (a) indicates that after annealing for 40s subgrain formation has started, however, dislocation tangles are still observed. According to 0 20 40 60 80 100 120 Precipitate Diameter (nm) (a) 0 0 20 40 60 80 100 120 Precipitate Diameter (nm) (b) 0 20 40 60 Precipitate Diameter (nm) (c) 64 Figure 5.9 (b), annealing for 2Ohr at 425°C leads to a highly recovered microstructure with clean subgrains. Precipitates are observed to be distributed inside the subgrains as well as on subgrain boundaries. Figure 5.9 (c) shows that after annealing up to 14 days the recovered microstructure is maintained without substantial subgrain growth. Similar microstructures are observed for recovered samples that were pre-aged for 8 days as shown in Figure 5.10. Figure 5.10 verifies that the microstructure with larger precipitates is also retained in the recovered state after long annealing at 425°C. 65 Figure 5.9. Bright field TEM images of recovered Al-Mg-Sc samples annealed at 425°C after 80% cold rolling (a) Rec-80min-40s, close to zone axis <1 lO>Al, (b) Rec-80min-2Ohr, close to zone axis <llO>M, (c) Rec-8Omin-14 days, close to zone axis <lOO>AI 66 Figure 5.10. Bright field TEM images of recovered Al-Mg-Sc samples annealed at 425°C after 80% cold rolling (a) Rec-8 days-40s, close to zone axis <1 lO>M, (b) Rec-8 days-20hr, close to zone axis <1 lO>A1, (c) Rec-8 days-14 days, close to zone axis <lOO>A1 67 Figures 5.11 (a) and 5.11 (b) present the microstructures of the pre-aged Al-Mg-Sc samples after 80% cold rolling and annealing at 190°C for 14 days. In comparison to Figures 5.9 (c) and 5.10 (c), one observes a low level of recovery as evidenced by the high residual dislocation content. Figure 5.11 (c) shows the microstructure of the Al-Mg alloy 80% cold rolled and annealed at 190°C for 14 days. It is observed that well recovered subgrains with a very low dislocation density have formed similar to the microstructures formed in the Al-Mg-Sc samples annealed at 425°C between 2Obr and 14 days. 68 (c) Figure 5.11. Bright field TEM images of recovered Al-Mg-Sc and Al-Mg samples annealed at 190°C after 80% cold rolling (a) Rec-8omin-14 days, close to zone axis <1 ‘°>At, (b) Rec-8 days-14 days, close to zone axis <11 O>A1, (c) Rec-Al-Mg- 14 days, close to zone axis <1 OO>Al 69 5.3.2.2. TEM Measurements on Precipitate and Subgrain Size in Recovered Al-Mg-Sc Samples The precipitate size distribution in selected recovered samples was studied by TEM observations. The objective was to investigate whether the rolling and recovery significantly affected the size and distribution of A13Sc precipitates. For this purpose, measurements were made on recovered samples which were pre-aged at 425°C for 80 mm followed by 80% cold rolling and annealing at 425°C for 2Obr. Theses were compared to samples that were aged at 425°C for 80 mm followed by further annealing at 425°C for 2Ohr (Figure 5.8 (c)). The comparison of size measurements on recovered and aged samples are shown in Figure 5.12. The two size distributions in Figure 5.12 are similar. In addition the average precipitate diameter in the recovered sample based on Equation 5.1 was found to be 14 nm which is the same as the average precipitate diameter for the corresponding aged sample. Figure 5.13 shows subgrain size measurements made from TEM observations on recovered samples annealed at 425°C and 190°C based on the linear intercept method. It can be observed that subgrain sizes for recovered samples annealed at 425°C with different initial precipitate diameters are similar. According to TEM observations, the subgrain size of samples annealed at 190°C up to 14 days do not exceed 1 pm and are the same for recovered samples with different initial precipitate sizes. 70 60 50 >, 40 II) 11) 30 U 20 E z 10 0 0 20 40 60 80 100 120 Precipitate Diameter (nm) Figure 5.12. Precipitate diameter distributions for Aged-8Omin-2Ohr and Rec-80min-2Ohr samples E3 CD N DlU) Figure 5.13. Subgrain size measured by TEM for recovered samples with different aging routes annealed at (a) 425°C and (b) 190°C 5.3.2.3. EBSD Observations on Microstructure of Recovered Al-Mg-Sc Samples EBSD maps of samples pre-aged, 80% cold rolled and annealed at 425°C for 2Ohr with different initial precipitate size, are illustrated in Figure 5.14 (a) and (b). It is observed that cold rolled microstructures with elongated grains are preserved after long time annealing at high homologous temperatures. These observations are in agreement with the TEM results presented Figures 5.11(b) and 5.12 (b) in terms of the fact that Aged-80min-2Ohr Rec-80min-2Ohr Annealing at 425°C 4 E ID N (I) 1 D C,) 40s lhr 2Ohr 14 days 4 I I • TEM-Rec - 80mm - 40s to 14 days TEM-Rec- 8days - 40s to 14 days 3 A 10’ 102 10’ 10 lO 10 102 Annealing Time (s) at 425°C (a) 40s lhr 2Ohr 14 days • TEM-Rec-80 mm 4Osto 14 days TEM-Rec- 8 days 40s to 14 days 0 , , 101 102 10’ 10 10° 106 10 Annealing Time (s) at 190°C (b) 71 there is no evidence of recrystallization after annealing. Figure 5.15 shows selected individual grains for the recovered samples in Figure 5.14 (a) and the corresponding { 1 OO} pole figure indicating that the orientation spread within the recovered grains is small. Figure 5.14. EBSD inverse pole figures maps of recovered Al-Mg-Sc microstructures annealed at 425°C for 2Ohr with initial precipitate diameter of (a) 12 nm and (b) 23 nm. The colors correspond to the crystallographic direction parallel to ND. 72 IRD Halt width:5 Cluster size:5 Densities [mud): ti= o.oo;M=1 0.00 8 _____________ Halt idth:5 Cluster size:5 Densities [mud): Min=0.0OMatr1O.0O 4 6 8 Figure 5.15. { 100}Pole figure for selected grains, A and B, corresponding to the recovered sample shown in Figure 5.14 (a) with initial precipitates size of l2nm. 5.3.2.4. Texture Measurements on Al-Mg-Sc Samples Pole figures of as rolled, recovered and fuiiy recrystallized microstructures pre-aged at 425°C for 80 mm are presented in Figures 5.16 to 5.18. It is noted that rolled (a) { i00 } (b) 73 samples (Figure 5.16) have strong textures consistent with cold rolled aluminum alloys [601 and that the textures in well recovered samples annealed at 425°C for 20 hr (Figure 5.17) are similar, however, reduced in contour intensity. Figure 5.18 shows the weak texture in aged samples having a fully recrystallized microstructure. Similar pole figures (not presented here) in samples pre-aged at 425°C for 8 days were observed. 74 RD RD RD TD ill pole figure 200 pole figure 220 pole figure Figure 5.16. Pole figures of pre-aged at 425°C for 80 mm followed by 80% cold rolling, contour levels: 0.5, 2, 3.5, 5, and 6.5. ‘TD ill pole figure 200 pole figure 220 pole figure Figure 5.17. Pole figures of pre-aged at 425°C for 80 mm followed by 80% cold rolling and annealing at 425°C for 20 hr, contour levels: 0.5, 2, 3.5, 5, and 6.5. 111 pole figure 200 pole figure 220 pole figure Figure 5.18. Pole figures of aged samples at 425°C for 80 mm with fully recrystallized microstructure, contour levels: 0.5, 2, 3.5, 5, and 6.5. TD TD RD RD RD 75 Table 5.1 tabulates the Taylor factor (M) values calculated from ODFs using the Material Point Simulator for rolled, recovered and as-aged microstructures pre-aged at 425°C for 80 mm and 8 days based on the textures shown in Figures 5.16 to 5.18. According to these results, the Taylor factor for all microstructures is in the same range with a difference of less than 5%. In the following analysis of the mechanical response of deformed and recovered materials an average Taylor factor of 3.06 will be used for all conditions. Table 5.1. Taylor factor obtained for as-aged, rolled and recovered Al-Mg-Sc samples Condition Taylor Factor (M) Rolled-80 min!8 days 3.03 Rec-80 minl8 days-20 hr 3.03 Aged-80 minl8 days 3.06 5.4. Mechanical Properties In this section the results obtained on the tensile response of recrystallized, recovered and deformed microstructures measured at 77K are presented. The goal is to understand the effect of recovery on the yield strength and work hardening of the Al-Mg-Sc alloy with special attention to the role of Al3Sc precipitates on recovery kinetics. First the evolution of yield strength with annealing time is shown, followed by the stress strain curves. 5.4.1. The Evolution of the Yield Strength The yield strength evolution for all conditions studied is summarized in Figures 5.19 and 5.20. In Figure 5.19, the yield strength of the Al-Mg-Sc alloy is shown with two sets of data presented in each figure. The solid symbols show the evolution of yield strength for materials which were pre-aged for either 80 mm or 8 days at 425°C, then 76 rolled and finally annealed to achieve recovery. The open symbols represent the yield strength for samples that were subjected to exactly the same thermal treatment without the intermediate rolling. The open symbols thus represent the yield strength of materials that have been only aged. Each data point represents the average of at least two tensile tests, the error bars representing the maximum and minimum values of the measurements. The yield strengths of the as-rolled materials are shown as bounding lines on each of the graphs. 77 Pre-Aged 80mm 2mm lhr 2Ohr 136 days 450 ..,// I I I I RoIIed-80min Rec-B0min-40s to 136 days ! / Ii Aged-80min-40s to 136 days I I icc 50 As Solutionized 10510-510-’i00 101 102 10’ 10’ 10’ 10’ 10’ 10’ 10’ 100 10” 102 Annealing Time (s) at 190°C (a) Pre-Aged 8 days 2mm lhr 2Ohr 136 days 450 ___/, I I I 4cc 350 Rolled -8 days 3cc Rec-Bdays-40s to 136 days 250 / 2cc 150 Aged-8days-4Osto 136 days K,100 50 As Solulionized 0 ---// io-’io-’io-’io° 10’ 102 10’ 10’ 10’ 106 10’ 10’ 10’ 10’°lO” 1012 Annealing Time (s) at 190°C (c) Age:-8d:ys-40s to 69 days — ——-— ---- 7 As Solutionized icc 50 0 10-’1o-’1o-1o’ 101 102 10’ 1o 10’ 10’ 10’ 10’ 10’ 1060 10111012 Annealing Time (s) at 425°C (d) Figure 5.19. Yield strength evolution for Al-Mg-Sc samples subjected to annealing at 190°C and 425°C. Solid symbols represent materials that were pre-aged, rolled and annealed, while open symbols represent materials that were pre-aged in the same way but not subjected to rolling. (a) Materials pre-aged at 425°C for 80 mm and annealed at 190°C (Rec-80min-40s to 136 days) (b) Materials pre-aged at 425°C for 80mm and annealed at 425°C (Rec-80min-40s to 69 days) (c) Materials pre-aged at 425°C for 8 days and annealed at 190°C (Rec-8days-40s to 136 days) (d) Materials pre-aged at 425°C for 8 days, rolled then annealed at 425°C (Rec-8 days-40s to 69 days). 400 -w0 Th250 C 2co 150 2mm lhr 2Ohr 69 days (0 0) C a> U) a> (0 C a) U) 0 a) >- 2mm lhr 2Ohr 69 days 450 400 350 300 250 2cc 150 450 —/, 400 Rolled -80mm 350 Rec-SOmin -40s to 69 days -c 250 C 2I3J C!) 150 100 Aged -80mm -40sto 69 daysI I 50 As Solutionized 0 —,—// 106 10-s 104l00 10’ jQ2 10’ 10’ 10’ 10’ 101 10’ 10’ 10b010111012 Annealing Time (s) at 425°C (b) Rolled -8 days Rec-8 days -40s to 69 days 78 2mm lhr 2Ohr 136 days 450 I I I I 400 350 Rolled -Al-Mg 300 250 I / 150 Rec-AI-Mg-40s to 136 days - As Solutionized : 1 - 0 ‘/ 106 10-s 10-10° 101 102 10 10 10 106 10 10 10 101010111012 AnneaHng Time (s) at 190°C Figure 5.20. Yield strength evolution for the recovered Al-Mg samples. The homogenized samples were first rolled to 80% reduction followed by recovery annealing at 190°C for 40s to 136 days. The as-rolled yield strengths of the Al-Mg-Sc samples pre-aged for 80 mm and 8 days are similar (350 MPa versus 330 MPa, respectively), however, one can see that the kinetics of softening are different for the two samples with the samples pre-aged for 80 mm having a slower rate of softening at both 190°C and 425°C. In the case of the aged materials with no deformation (open symbols), one sees that there is no evolution of the yield strength on annealing at 190°C. This is consistent with previous calculations [13] which predicted no significant coarsening of the precipitates at this temperature. More importantly, this also indicates that there was very little, if any, re-precipitation due to the lower solubility of Sc in Al at 190°C compared to the pre-aging temperature of 425°C. Performing the aging at 425°C as opposed to 190°C did, however, result in some continued coarsening of the A13Sc precipitates as indicated by the decreasing yield strength of the un-deformed samples shown in Figures 5.19 (b) and (d). In the case of the material pre-aged for 8 days at 425°C, continued aging at this temperature results in a relatively small drop in yield strength. This is due to the fact that 79 the precipitation strengthening effect is very small in these very highly overaged samples. The same is not true for the material pre-aged for 80 mm at 425°C. In this case, there is a substantial precipitation strengthening in the pre-aged material. Continued aging at 425°C thus results in a significant drop in yield strength. At the very longest aging times the coarsening of the precipitates leads to a convergence of the yield strengths towards that of the solutionized material. Based on the results from the pre-aged, rolled and annealed samples (solid symbols) some similarities and differences between the different conditions studied can be distinguished. In the samples annealed at the lower temperature (190°C) one observes a nearly monotonic logarithmic decay in the yield strength with annealing time. This is very similar in appearance to the decay in yield strength observed for the binary Al-Mg sample shown in Figure 5.20. This is to be compared with the decay of the yield strength in the samples annealed at 425°C. In this case the yield strength does not appear to decay monotonically. Instead, one observes two separate stages in the yield strength evolution. It is important to note that the yield strength of the annealed samples remains significantly higher than those of the aged samples considered at equal annealing/aging times even after very long annealing times. This is indicative of the fact that under the experimental conditions represented by Figure 5.19 (a)-(d) recrystallization of the samples, partial or complete, was never achieved. 80 5.4.1.1. Stress-Strain Behaviour Figure 5.21 illustrates tensile curves of recovered and aged samples obtained from uniaxial tensile testing. Due to coarsening of precipitates at 425°C, the tensile behavior of recovered samples annealed for 14 days is compared to the tensile curves of aged samples which were further aged for 14 days. The small difference between the tensile curves of Aged-8 days and Aged-8 days-14 days implies that coarsening kinetics for aged samples pre-aged at 425°C for 8 days is negligible compared to the one pre-aged at 425°C for 80 mm. Figure 5.21 shows that for both aging routes and annealing temperatures the flow stress of recovered samples lies above the flow stress for the corresponding aging conditions. In addition, the flow stress of recovered samples annealed at 190°C is observed to be higher than that of samples annealed at 425°C. 81 500 500 Rec-8Omin-1 hr450 Rec-8Omn-40s Aged-aomin 450 Rec-Bdays-40s Rec-8 days-lhr ec-80min-l4days —— Aged-8 days a) a) 300 300 — _-8days-14days edI4days Rec-8days -1 4days .250 - .250 (I) Cl) 200 (1)200 2 1— 150 I— 150 100 Annealing at 425°C 100 Annealing at 425°C 50 50 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 True Strain True Strain (a) (b) 500 500 Reo-aomin-2min Rec-80min -1 hr Rec-80min-14 days450 450 Rec-8 d5 °ReC-8 days-lhr 350 - . 300 (0300 200 a) 200 Aged-8 days 400 ed80min 400 Rec-8days-14days a) a) .b 250 250 (I) Cl) 2 l—• 150 1— 150 100 100Annealing at 190°C Annealing at 190°C 50 50 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.00 0.05 010 0.15 0.20 0.25 0.30 0.35 0.40 True Strain True Strain (c) (d) Figure 5.21. True stress-stain behavior of recovered and as-aged samples (a) pre-aged for 80 mm and annealed at 425°C (b) pre-aged for 8 days and annealed at 425°C (c) pre-aged for 80 mm and annealed at 190°C (d) pre-aged for 8 days and annealed at 190°C In this chapter, the results on microstructures characterization and mechanical response of recovered and as-aged materials were presented. In the next chapter, the role of recovery on mechanical properties based on the obtained results will be discussed in the frame work of modelling. 82 Chapter 6-Discussion The results of yield strength measurements on the recovered Al-Mg-Sc alloy illustrate that precipitates have a significant impact on microstructural evolution and on determining the mechanical response of two-phase precipitate containing alloys during recovery. h order to understand the effectiveness of precipitates on recovery, it is necessary to develop quantitative descriptions of the yield strength and work hardening as a function of precipitate and recovery state. In this chapter, the results from microstructural characterization are first discussed followed by the development of quantitative models describing the yield strength and work hardening behavior in a manner consistent with microstructure observations. 6.1. Microstructure Characterization In this section TEM observations on recovered microstructures are discussed. An approach for calculating average precipitate diameter from the size distribution applied in the yield strength and work hardening models is presented. 6.1.1. Precipitate Size Measurements In order to explain the method selected for calculating average precipitate diameter in this study, dark field images of precipitates and precipitate size distribution are re-presented in Figure 6.1 (a) to (d). In these over aged samples, the number density of precipitates in the regions containing larger precipitates is different relative to regions containing smaller precipitates. This is most significant for samples aged for 8 days in this study. According to Figure 6.1 (a) and (b), precipitates are more uniformly distributed based on number density in samples aged for 80 mm compared to samples 83 aged for 8 days. In samples aged for 8 days the number density of precipitates and precipitate spacing vary significantly through the microstructure due to the wide precipitate size distribution observed. This includes some very large precipitates having diameters larger than 40 nrn. The large precipitates represent a small fraction of the total number density of precipitates but they consume a large fraction of the total precipitate volume fraction (Figure 6.1(b) and (d)). The number density average precipitate size described previously by Equation 5.1: - d. = ‘ (5.1) total tends to weight the smaller precipitates more heavily in this average since the number density of the large precipitates is small. As an alternative one could use a volume fraction weighted average precipitate size (dR,t (1/)). The volumetric average applied is given by: dppt(V) =( ).d1 (6.1) total where —--— is the volume fraction of spherical precipitates within the class of precipitates T”total of diameter d1 and Vtotai is the total volume of precipitates. The volume weighted average precipitate size is larger than the number average size since the larger precipitates are weighted more heavily in this case. 84 Volume Average Figure 6.1. Precipitate diameter distributions for (a) Aged-80 mm (b) Aged-8 days, cumulative precipitate volume and average precipitate diameter for (c) Aged-80 mm (d) Aged-8 days Figure 6.1(c) and (d), show that based on a number density average, precipitates smaller than the average consume less than 20% of the total precipitate volume. Conversely, based on volumetric averaging, precipitates smaller than the average consume about 80% of the total precipitate volume. Following Fazeli et al. [13] the volume average precipitate size is used to characterize the precipitate size here. Aged-8Omin 40 35 C) 25 aj 20 U 15 E 10 Z 0 0 20 40 60 80 Precipitate Diameter (nm) (c) 100 a) E 80 D 0 > G) 60 -I-, 0 0 40 2 0 a) > 20 E 0 ‘ 100 120 100 a) E 80 0 > a) 60 0. 0 40 0 a) > 20 D E 100 120 40 35 30 0 25 D a 20 U 15 .0 E 10 Z3 z5 0 (b) 0 20 40 60 80 Precipitate Diameter (nm) (d) 85 Volume average precipitate diameters of 27 mu and 77 nm were obtained for samples pre-aged at 425°C for 80 mm and 8 days respectively. 6.1.2. Recovered Microstructures TEM observations shown in Figures 5.9 and 5.10 show that the pre-aged and rolled Al-Mg-Sc samples were recovered after annealing at 425°C up to 14 days. The stability of the microstructure against recrystallization can be attributed to the pinning of low angle boundaries by A13Sc precipitates consistent with previous observation on recrystallization of Al-Sc alloys [11]. Recrystallization is prevented till the pinning pressure is larger than the driving pressure for recrystallization due to coarsening of precipitates. According to a previous study on Al-Sc alloys [28] the coarsening rate is accelerated when complete loss of coherency occurs. The critical radius for coherency loss is reported to be above 40 urn [28]. In the present study, the average precipitates radius is below 40 nm for both pre-aging conditions. EBSD observations in Figure 5.14 also verify the presence of low angle boundaries in the recovered microstructures. Figure 6.2 compares the subgrain size measured by TEM observations and the limiting subgrain size through Zener pinning effect [60] for each pre-aging conditions. The limiting subgrain size is obtained from: Diimit = (2.19) Figure 6.2 demonstrates that the Zener model over estimates subgrain sizes measured experimentally. For the case of Al-Mg-Sc alloy, cell or subgrain structures form after 80% reduction in thickness by cold rolling (equivalent to true von Mises strain of 1.85) and the initial subgrain size, about 0.5 im measured from TEM, is determined 86 by the level of strain rather than initial precipitate spacing. This is consistent with previous work on deformed and recovered Al alloys [64]. Once the cell or subgrain boundaries form by rolling, then during annealing they are pinned by precipitates and their growth depends on the coarsening kinetics of the precipitates which, as stated above, is slow. The limiting subgrain size in Equation 2.19 is the upper bound for recovered samples. As explained in section 2.2.1.2 similar observations of Equation 2.19 over predicting subgrain size have been previously reported in other recovered Al alloys containing precipitates. This has been argued to arise from additional interaction of dislocations with precipitates during recovery [89]. 40s lhr 2Ohr 14 days 9 I I I I • TEM-Rec - 80mm - 40s to 14 days B TEM-Rec- 8days - 40s to 14 days 7 Zener Pmnning-Rec-8omin-40s to 14 days 6 — — — Zener Pin ning-Rec-8 days-40s to 14 days 101 102 10 i0 10 108 10 Annealing Time (s) at 425°C Figure 6.2. Comparison of subgrain sizes measured by TEM and predicted via Zener pinning for samples recovered at 425°C Figure 5.11 compares the microstructure of Al-Mg and Al-Mg-Sc samples after annealing for 14 days at 190°C. In the Al-Mg alloy well defined subgrains are observed indicating that extensive recovery has occurred. Conversely, a high dislocation density is observed in Al-Mg-Sc samples annealed at the same time and temperature. 87 Figures 5.11(a) and (b) show that re-arrangement of dislocations is still in process in the Al-Mg-Sc sample even after 14 days annealing. Comparison of microstructures of Al-Mg and Al-Mg-Sc samples annealed at 190°C qualitatively suggests that Al3Sc precipitation affects the recovery kinetics. In the following sections, a quantitative approach to describing the microstructure and its relation to the mechanical properties is presented. 6.2. Mechanical Properties The results of mechanical properties measurements shown in Figures 5.19-5.21 are complex owing to the presence of multiple obstacles to dislocations, such as precipitates and subgrain boundaries. In this section the yield strength and work hardening behavior of recovered microstructures are analyzed in an attempt to separate the various contributions to the flow stress. 6.2.1. Modelling Yield Strength In order to proceed to develop a model for yield strength of the Al-Mg-Sc alloy, it is necessary to carefully separate the different contributions to the yield stress. For the recovered Al-Mg-Sc alloy there are multiple sets of obstacles such as precipitates and dislocations with different densities and strengths. As these different obstacles are located at different locations in the slip plane and the glide resistance of these obstacles is proportional to the square root of aerial density of obstacles, it has been described that a superposition law must be used [1161. In this study, the superposition law for recovered microstructures is applied as follows assuming a negligible contribution from grain boundary strengthening due to the large grain size: 88 = + (u + ) (6.2) Equation 6.2 describes the contributions to the yield stress arising from solid solution strengthening precipitation strengthening (cr) and strengthening due to forest dislocations (crd,S). In the case of recovered microstructures the situation is complex due to the presence of both precipitates and dislocations. For simplicity, the exponent s here is assumed to be 2. This is consistent with the value found for the as-aged materials containing non-shearable Al3Sc precipitates [13]. In the present study the solid solution contribution to the flow stress is taken as constant. It is assumed here that magnesium does not participate in the precipitation process and therefore that the magnesium content of the matrix remains fixed. As noted in the literature review, it is possible for some magnesium to replace aluminum in the Al3Sc precipitates but only to a limited extent [117]. Thus, the value of 82 MPa, has been taken from the yield strength of as-solutionized samples. The contribution of precipitates to the yield strength of the recovered samples is more difficult to obtain from experimental measurements on recovered materials owing to the fact that it is not possible to directly identify which portion of the macroscopic yield strength is associated with precipitates and which part results from remaining forest dislocations. While in the as-aged materials shown in Figures 5.6 and 5.7 precipitates are distributed in a matrix that is nearly free of forest dislocations, the recovered microstructures in Figures 5.9 and 5.10 illustrate that precipitates exist both within subgrains containing forest dislocations as well as on subgrain boundaries. The similarity of the size distributions for precipitates in both the as-aged and recovered samples 89 (Figure 5.9) strongly suggest that there is no significant increase in the coarsening rate of precipitates in the recovered material compared to the as-aged material. Moreover, most precipitates were observed to be located inside subgrains rather than on subgrain boundaries. Based on these observations it has been assumed here that the precipitate strengthening is the same for recovered and aged samples subjected to the same thermal history. With this assumption, the precipitate contribution to the yield strength, o,, can be estimated from the as-aged materials as: = 0y — 0soi (6.3) where o is the as-aged yield strength. Equation 6.2 can be re-arranged to obtain the forest hardening contribution to the yield stress in terms of other obtained quantities. [(y — soi )2 — 2 11/2 (6.4) The values of yield strength, o, for recovered and aged samples are the solid and open symbols in Figure 5.19. Equation 6.4 then allows dis to be estimated as a function of annealing time. In the case of the single phase Al-Mg samples, there are no precipitates and thus the only other contributions to the yield strength are the constant solid solution strengthening and forest hardening. The forest hardening contribution, d,s’ for the recovered Al-Mg alloy is calculated from the following: dis = y — ° sol (6.5) 90 where as01 =68 MPa is taken as the yield strength of the fully recrystallized material with a grain size of approximately 200 pm. 6.2.1.1. The Influence of Recovery on Yield Strength For the purposes of modelling the yield strength of recovered samples, the decay in dis due to dislocation rearrangement and annihilation during annealing has to be considered. The contribution of dislocations to the yield strength for different annealing times is calculated from Equation 6.4 for both alloys and plotted in Figure 6.3. 91 2mm 1ir 2Ohr 136 days 2mm 1ir 2Ohr 69 days 320 —// 280 — Rolled-SOmin 240 — Rolled-8 days 200 ,1 . Reo-SOmin-40sto 136 days / -- i / 160 Verdier model f ,7 •--L. ‘ Rec-8days-40s to 136 days “i... 40 0 —,—// 10610 10’10° 10’ 102 10’ 10’ 10’ 106 10’ 10’ Annealing Time (s) at 190°C -V Rolted-O0min - •% Rolled-8 days Rec80min-4Os to 69 days II Rec-8 days-40s to 69 days VerWerModel “-... 0 __,_7/ / 10-’10-’10106 10’ 102 10’ 10’ 10’ 106 10’ 106 Annealing Time (s) at 425°C (a) (b) 2mm lhr 2Ohr 136 days (13 0 U) 0 320 280 240 200 160 120 80 40 CO 0 U) .tz b 320 —/, 280 Rolled-Al-Mg 240 \ ‘-.._ Rec-AI-Mg-4Osto 136 days 200 160 Verdier model ‘4- I: 4° o—,—// 10-610-510-21a 10’ 106 10’ 10’ 10’ 106 10’ 108 Annealing Time (s) at 190°C (c) Figure 6.3. Evolution of yield stress attributable to forest dislocations (u,) for (a) recovered Al-Mg-Sc samples annealed at 190°C (b) recovered Al-Mg-Sc samples annealed at 425°C (c) recovered Al-Mg samples annealed at 190°C. The thin dotted lines indicate the yield strength of the Al-Mg alloy and aged Al-Mg-Sc alloy in as rolled state and the model prediction based on Equation 2.5 is illustrated as heavy dashed lines. 92 In the case of single phase aluminum alloys as described in chapter 2, Verdier et a?. [7] have reformulated the original Friedel model [78] for recovery based on the relaxation of internal stresses associated with the strain fields of the stored dislocations giving: dod,S — 64vD exp(___)sinh(0sV) (2.5) dt 9M3a2E ‘ kT kBT where U and V are activation energy and volume associated with recovery respectively, E is the elastic modulus, M is the Taylor factor, VD is the Debye frequency, a is a constant of the order of 0.3 and kB is the Boltzmann’s constant. As a starting point, this model has been applied to the data in Figure 6.3. Activation energy and volume values were obtained by fitting Equation 2.5 to djs for the Al-2.5%Mg alloy studied as shown in Figure 6.3 (c). The values of U and V (Table 6.3) are within the range of values found in Al-Mg alloys of similar composition [7,8] illustrated in Table 2.2. Table 6.3. Activation energy and activation volume values deduced by fitting Equation 2.5 cYdIS shown in Figure 6.3 (c). U (kJ/mol) V(b3) Pre-strain 6=1.6 Pre-strain 6=1.85 M a T= 190°C T= 190°C 205 30 3.06 0.3 Based on the Verdier model, the softening behavior of the single phase Al-Mg alloy is captured using consistent values for activation energy and volume. In order to understand the role of Al3Sc precipitates on the recovery kinetics, the softening behavior of the Al-Mg alloy is compared to the one for the Al-Mg-Sc alloy. The Verdier model is 93 also applied to the Al-Mg-Sc alloy during recovery. While this model predicts a logarithmic softening of the Al-Mg alloy, it is not capable of explaining the behavior of the Al-Mg-Sc alloy unless the activation energy and/or activation volume for recovery are made to be adjustable parameters for different precipitate sizes. Indeed, comparing the results of Equation 2.5 with the experimental data in Figure 6.3 using an activation energy and volume consistent with the Al-Mg alloy of similar composition to the Al-Mg-Sc alloy studied here shows that the model and experimental results diverge after only very short annealing times at 425°C and after longer annealing times at 190°C. It should also be noted that the precipitate containing materials shows a much lower rate of softening compared to the model predictions. 6.2.1.2. A Model for the Effect of Precipitates on Recovery Kinetics There have been few previous attempts at capturing the effect of precipitates on static recovery kinetics within the framework of a physically based model. As explained in the literature review one notable exception is the work of Zurob et al. [80] who considered the effect of concurrent recovery and NbC precipitation on the softening behavior of microalloyed steels. In the case of NbC precipitation in steels, Zurob et al. [80] modified Equation 2.5 to account for the effect of concurrent precipitation on dislocation nodes and recovery, dcr’ = dcrdlS (‘ — (6 6) di’ dt n) d ppf Here n is the number of dislocation nodes, n the number of precipitates, is the dcrd. . recovery rate controlled by precipitates and dt ‘ is the recovery rate independent of 94 precipitates given by Equation 2.5. This model predicts a stasis in the recovery kinetics n for a critical ratio of —-. In contrast to the situation considered by Zurob et al., the nc results shown in Figure 6.3 correspond to a material containing a fixed volume fraction of precipitates which have not been formed on dislocation nodes. According to Figure 6.3, the experimental data for Al-Mg-Sc diverges from the Verdier model (Equation 2.5) after very short times at 425°C, e.g. 2 mm, and longer times, e.g. 2Ohr, at 190°C. These results, suggest that in addition to annihilation of forest dislocations as described by Equation 2.5, a precipitate dependent mechanism contributes to the recovery kinetics of the Al-Mg-Sc alloy. As a starting point for developing a new model it is assumed that the annihilation of forest dislocations (i.e. Equation 2.5) is the rate controlling mechanism when the spacing of dislocations is smaller than the precipitate spacing. Thus in the early stages of recovery when dislocation density is high, the recovery kinetics should be observed to be independent of the presence of precipitates. Once the spacing between dislocations, LdIS, becomes larger than precipitate spacing, there will be a high probability that dislocations will have to interact with precipitates before annihilation can occur. The simplest way to explain this transition is by defining the transition probability: = (6.7) ppt where n’ is taken as an adjustable parameter of the order of unity and the maximum value offis one. The overall decay in 0dj is thus described as: 95 d;al = dudIS - f) + dut nL1S <1 (6.8-a) d total ddis = dis dis 1, f = 1 (6 8-b)dt dt dcrd. where dt ‘ is the yield stress reduction associated to annihilation of forest dislocations da7t independent of precipitates and dt expresses the precipitate controlled decay of dis The spacing between dislocations is calculated as: 1 LdS = (6.9) Jp where p is the dislocation density which can be obtained from the Taylor Equation: dis = aGbM.4J (2.2) As discussed above, the recovery kinetics is delayed in Al-Mg-Sc samples compared to Al-Mg samples implying that the role of precipitates on recovery has to be taken into account. There are many different approaches one could consider to describe the dislocation-precipitate interaction. For example Zurob et al. [80] assumed that dislocation nodes have to unpin from precipitates for recovery to proceed. This approach is not applicable in the case studied here since the samples were pre-aged. An alternative way by which precipitates could hinder dislocation recovery is if dislocations were required to surmount precipitates prior to annihilation. Given sufficient thermal energy, 96 dislocations may be able to surmount precipitates by a combination of climb and cross slip. Rosier and Arzt [118] have described creep by dislocation climb over precipitates under an applied external stress. In this model the equilibrium shape of a dislocation during climb is calculated based on minimizing the line tension of dislocation assuming that general climb is accomplished by bulk diffusion of vacancies along the dislocation line. In the general climb mechanism there is no constraint on the dislocation shape and sharp bends in the dislocation can be relaxed by diffusion unlike in local climb (Figure 6.4). The sharp bend in a dislocation associated with local climb (Figure 6.4 (a)) is unstable. Recently a detachment mechanism has been proposed in place of the local climb mechanism [118,119]. This model will be discussed below. The predicted time for general climb, telimb, as a function of volume average precipitate diameter (dPP,(V)), temperature (I) and material constants has the following form: Gb4 2D — - tClimb k Tb A• I (6.10) ppt(V) B Vpp ) where G, b, kB, D’ are well known quantities. t is the applied shear stress, Vb is the back stress arising from creation of new dislocation line due to the presence of precipitates. Parameters A and 11 are model constants. The magnitude of rb is found to be in the range of [118,119,120]. This model has been applied to quantify the creep behavior of precipitation-hardened alloys including Al-Sc alloys [118,34,121] subjected to an applied stress. 97 Figure 6.4. Dislocation climb over a cube-shaped particle showing (a) local climb with a sharp dislocation bend at A (b) general climb where the high curvature at A is relaxed [1181. In this study, the general climb mechanism controlled by diffusion is assumed to be the dominant mechanism involved in precipitate dislocation interaction rather than the detachment mechanism proposed by Rosler, Arzt and Wilkinson [122,123,124]. The detachment mechanism has been suggested for the creep behavior of materials containing incoherent precipitates or dispersoids referred to as “interacting particles” [122]. It is shown that the elastic interactions between dislocations and incoherent second phases can 98 be relaxed by a diffusional process and as a result dislocations on glide planes intersecting precipitates are attracted towards them. Therefore the detachment of dislocations from precipitates at the departure side is the rate controlling mechanism in these cases [125,122]. In the general climb mechanism, it is assumed that there is no long range interaction between dislocations and particles referred to as “not interacting particles”. This mechanism is suggested to be dominant when particles or precipitates are coherent [118,119]. An in situ high resolution TEM study by Clark et al. [126] on the interaction of Al3Sc precipitates with subgrain boundaries at high temperatures revealed that for semi-coherent Al3Sc precipitates in the range of 125 to 225 nm detachment is the rate controlling mechanism. However, in the present study the average precipitate diameter is below the range of precipitate size reported in the study by Clark et al. suggesting that most precipitates maintain their coherency with the matrix (see Figure 5.8). This is consistent with TEM observations on the Al-0.2%Sc alloy which showed that the precipitate radius at the coherent to semi-coherent transition is 15-40 nm [28]. According to TEM observations in Figures 5.6 and 5.7, Ashby-Brown contrast appears around the spherical precipitates indicating that precipitates for the selected aging treatments maintain, at least partial, coherency. Previous studies have treated the creep behavior of Al-Sc alloys containing coherent precipitates by the general climb mechanism [29]. The studies of binary Al-0.2%Sc and A1-2%Mg-0.2%Sc alloys at 3 00°C have shown that creep resistance of Sc containing alloys is significantly improved compared to pure aluminum suggesting a strong interaction between Al3Sc precipitates and mobile dislocations. The threshold 99 stress for these alloys increases with increasing average precipitate radius from 2 to 25 nm. The dependence of the threshold stress on precipitate size for coherent precipitates up to radius of 12 nm has been explained based on the general climb mechanism in the presence of elastic interactions between dislocations and coherent precipitates [29,34,121]. In the present work in order to consider the effect of precipitates on recovery kinetics, the time required for a dislocation to climb over a precipitate is calculated via Equation 6.10. Using a similar concept to that used by Verdier the relaxation of the internal stress is taken to be proportional to the strain rate (é,,) associated with climb of dislocations over precipitates: ppt dt = —E6 (6.11) where E is the elastic modulus of aluminum. Using the Orowan Equation: Mñ12, = pbV (6.12) where V is the dislocation velocity which can be derived from [118]: L V = (6.13) tolimb It should be noted that in the present study, unlike in creep experiments, there is no externally applied stress. Instead, relaxation of internal stresses due to annihilation and rearrangement of dislocations is considered to provide the stress for climb. Therefore, in Equation 6.10 the applied stress is replaced by the internal stress, Jdl$: 100 tclimb — Gb4 27rDA kB Tb where the volume average precipitate dPPf(v) diameter is used. By combining Equations 6.11-6.14 and using Equation 2.5 and the Taylor dut dud. Equation, dt and dt in Equations 6.8-a and 6.8-b are reorganized to give: duZ’ = di’ —Eé=—E (6.15-a) Gb4 2icD1’ kBTbppt ( V) dudIS = = — 64vD ( tta1 )2 exp( — U sinh( thS V) dt 9M3a2E dis kBT kBT (6.15-b) where é ‘° and are the plastic strain rates associated with dislocations that annihilate with and without interacting with precipitates respectively. By substitution of 6.15-a and 6.15-b into 6.8-a and 6.8-b an expression is obtained for the overall decay of cYdIS for the Al-Mg-Sc alloy in which the effect of A13Sc precipitates is taken into account. In this approach, the recovery process is assumed as a sequential process where those dislocations interacting with precipitates must first climb over precipitates prior to annihilation. The annihilation and climb processes each occur Id (6.14) M 101 at their own characteristic rate. Here it is assumed that climb is the rate controlling mechanism for those dislocations interacting with precipitates. The precipitate spacing for spherical precipitates, [32) is expressed by: L1 = (6.16) In this study, it is assumed that Mg in solid solution has a small effect on the equilibrium solubility of Sc in Al (See Figure 5.4). Therefore the equilibrium volume fraction ofAl3Sc precipitates is extracted from the binary Al-Sc phase diagram. Precipitate diameters of aged samples for the 3 different conditions shown in Figure 5.8 were measured. In order to obtain precipitate sizes over the entire range of annealing times, the precipitate radius were fit to an equation consistent with the Lifshitz, Slyozov and Wagner (LSW) model of coarsening [127,1281: —i = kt (6.17) In Equation 6.17 is the volume average precipitate radius at time t and k is a fitting constant. Figure 6.5 illustrates the fit of Equation 6.17 to aging experiments performed at 425°C. 102 50 40 (0 30 (U (U 20 C) 10 0 0 0 20 40 60 80 100 (Aging/Recovery time) 1/3 (s)3 Figure 6.5. Precipitate radius versus (aging/recovery time) 1/3 at 425°C and fit to LSW coarsening behavior given by Equation 6.17. To summarize the overall concept of the present model: • Recovery kinetics are considered to be driven by internal stress relaxation. • When there is no interaction between dislocations and precipitates, the recovery kinetics are controlled by the Verdier model (Equation 2.5). • Once a substantial fraction of dislocations must interact with precipitates prior to annihilation, general climb is assumed to be the rate controlling mechanism. • The ratio of the dislocation spacing to the precipitate spacing is assumed to determine the rate controlling mechanism. • The following Equations are used to model the recovery kinetics: = dJdI3 (1 f) + dcr’ nLdIS <1 (6.8-a) = duJt 1, f = 1 (6.8-b) ppt TEM measurements LSW model T=425°C r3-r0=kt r0=7nm, k=O.35nm3/s 103 f’ total °dis lb dot = —Eé° = —E aGbM L Gb4 2,rD A.ppt ppt d2 1(St A” “ppl(V) “B (6.15-a) total T7 “dzs —En = — VD (;tal )2 exp( — ) sinh( d,s ) (6.15-b)dt ° 9M3a2E kBT kBT 6.2.1.3. Comparison Between Model and Experiments The model described above was fit to experiments with least regression using A, n and n’ as adjustable parameters. Figure 6.6 compares the experiments to the fit model for the Al-Mg-Sc alloy. Table 6.4 summarizes the parameters applied in the model for samples annealed at 425°C and 190°C. The values for activation energy and volume were taken to be the same as those of the Al-Mg alloy. The value for the back stress was fixed as where the Orowan stress (a) was obtained from calculation and experiments for as-aged samples (see the section 6.2.1.4). The Taylor factor (M) was also fixed at 3.06 according to the measurements in Table 5.1. The stress exponent (n), transition parameter (n *) and A are adjustable parameters in the model. The dimensionless parameter A is reported to have a very wide range of values (from less than 1 up to 1015) for metals and alloys under creep conditions [1291. The value for A here is found to be very small compared to ones for creep experiments indicating that stress relaxation for recovery occurs at strain rates smaller than the strain 104 rate for creep under external stress. The stress exponent, n, is found to be 2 which is smaller but close to the range, n=3-4, normally applied in the general climb model [1181. Table 6.4. Parameters used in the recovery model Parameter Value Comment Fixed values obtained from Activation energy (U) 205 (KJ/mol) Al-Mg alloys with similar composition Fixed values obtained from Activation volume V(b3) 30 Al-Mg alloys with similar composition Stress exponent (n) 2 - n—5 (425°C 80mm) Transition parameter (n *) = (425°c 8 days) - Numerical constant (A) l.5x101’ - D =D1exp(-QSD/RT) From [1301 Pre-exponential 1 .7x 1 0, m2/s For lattice diffusion ( Df) Activation energy 142, kJ/mol From [130] For lattice diffusion (QSD) Back stress (ab) From [118) Taylor factor (M) 3.06 - 105 2mm lhr 2Ohr 136 days I I I Rolled-8omin Rolled-8 days 200 Reo-8Omin-40s to 136 days 160 120 Rec-8days-40s to 136 days 80 40 Symbols: Experiments Lines: Model o --//-- 1o61051011o0 101 102 io 108 io 108 1o 108 108 1012101 1012 Annealing Time (s) at 190°C (a) 2mm lhr 2Ohr 69days I I I320 -————/, Symbols: Experiments 280 Lines: Model — Rolled-80min 240 RoIled-8 days 200 Reo-80min-40s to 69 days 160 / 120 807 40 Rec-8 days-40s to 69 days 0 —,--—// 10-6 iO- 104102 101 102 10 10 10 106 1o 108 i0 101010111012 Annealing Time (s) at 425°C (b) Figure 6.6. Experiments and model prediction for decay of dEs in recovered Al-Mg-Sc samples annealed at (a) 190°C and (b) 425°C. The adjustable parameter n * can be considered as a correction factor for either the dislocation spacing, LdIS, or precipitate spacing, or both since there are questions that arise for the calculation of these two terms. It was expected that the value of n should be close to one and the same for the two different aging conditions studied here. In fitting the experimental data, however, it was found that both sets of data could not be fit using a single value for this parameter. The best fit values for n were found to be *=5 for 80 mm pre-aging and n*=l1 for 8 days pre-aging. As illustrated by Figure 6.7, the model is quite sensitive to the value of ,,* selected. There are two features which must be discussed with respect to this result. The first is the fact that n is greater than one in both cases and the second is the fact that n is required to be approximately two times larger for the more highly overaged samples. 320 280 240 (0 0 U, -V b (0 0 (0 V b 106 The dislocation spacing, LdIS, is assumed to be proportional to However, this assumes a regular organization of the dislocations. As recovery takes place and subgrain boundaries form, the dislocation distribution becomes increasingly non-random. On the other hand, based on the TEM images shown in Figures 5.9 and 5.10 showing similar microstructures for the two different pre-aged samples, LdjS is not expected to be different significantly for the two pre-aging treatments. It is more likely that n should be viewed as a “correction” for the calculation of the precipitate spacing, Since f = n * then one could consider the effective ppt precipitate spacing as: L L1 (6.18) Given that n*>1 in both cases this would imply that our calculation of over-predicts the true value of This is perhaps related to the fact that the volumetric average precipitate size results in a significantly larger precipitate size (and spacing) compared to other methods such as a number average spacing. On this basis it is not surprising that n*>l. Following with this idea, the difference in n’ for two pre-aging conditions may be related to the fact that the precipitate size distribution is significantly wider for the 8 day aged samples (Figure 6.1). Owing to this large difference in the width of the size distribution then would tend to need to be corrected more for the samples pre-aged for 8 days compared to those pre-aged for 80mm. Another factor is the fact that the precipitate size distribution evolves with time in the case of the 80mm 107 pre-aged material. In this case the size distribution evolves towards that of the sample pre-aged for 8 days at 425°C. Therefore, n generally may be a function of time. However, it is observed here that f —* 1 at short recovery times. Oncef= 1 the rate of recovery becomes independent of variations in n and thus it is unlikely that this is a significant contributor to the different n values found here. The difficulty of describing an average precipitate spacing is similar to that observed in models developed to consider precipitate spacing. As has been reviewed by Ardell [32] there are several geometrical methods for calculating precipitate spacing. Developing and testing different methods for calculating precipitate spacing is beyond the scope of this thesis and thus is proposed as an opportunity to extend this work in the future. 2mm lhr 2Ohr 69 days 280 I I Rolled-8Omin240 Symbols: Experiments Lines: Models 200 Rec-SOmin-40s to 69 days Cu 160 n*6 *_5 io I0-’1O-10° 101 102 10’ 10 10’ 106 10’ 106 l0 101010111012 Annealing Time (s) at 425°C Figure 6.7. Sensitivity of the applied model to n values for samples pre-aged at 425°C for 80 mm and recovered at 425°C. It is important to analyze the dependence of the present model at high (425°C) and low (190°C) temperatures as the softening behavior shown in Figure 6.6 for these temperatures is different. In addition, the hypothesis made for the rate controlling 108 mechanism will be revisited here. For these purposes Equations 6.8-a and 6.8-b are re-written as the following: d°1 =_E[(1_f)+fñpj], *L <1 (6.19-a) p11 total *dj. . nL. dt = —E6, L 1S 1, f = 1 (6.19-b) p1J1’ The transition probability terms, f and 1 — f, are plotted in Figures 6.8 (a) and 6.8 (b) for samples pre-aged at 425°C for 8 days and recovered at 425°C or 190°C. For short annealing times, e.g. 2 mm, at 425°C, as illustrated in Figure 6.8 (a), f is larger than 0.8 and it increases to 1 before 1 hr indicating that all dislocations have to climb before annihilation. On the contrary, in samples recovered at 190°C for short times such as 2 mm, f is only 0.4 and even after annealing for 136 days a complete transition to the climb controlled mechanism does not occur. Therefore, the recovery kinetics is more sensitive to the presence of precipitates at high temperatures (425°C) rather than at lower temperatures (190°C). This is also consistent with the initial analysis in Figure 6.3 which shows a deviation from the Verdier model after only 40s for samples recovered at 425°C. Whereas the samples recovered at 190°C show only small deviation even after long times. The plastic strain rates for the two mechanisms, ° andé, are plotted as a function of annealing time for samples pre-aged at 425°C for 8 days and recovered at 425°C or 190°C in Figures 6.8 (c) and 6.8 (d). It should be noted that as strain rates for each mechanism are a function ofJdIS, a plateau also appears in strain rates. It is evident 109 that é ° is always larger thani consistent with the assumption that the waiting time for climb is longer than that for annihilation. 2mm Iho 2Ohr 69days 2mm lhr 2Ohr 69days 1.0 I I I 10-I I I I I Rec.8days-40s to 69 days / 10.2 Rec-8.days-40s to 69 days 1 o-O.8 / 10-i ;:- 10-5 zo.6 —— n iO 10-’ -D 2Q. 04 10-’ — \ _% ‘ 9 1O’° N 02 ——— 1-f \ 1o.12 \ \\ _______________________________________________ 10 I— \ f io-’ 0.0 1o.10-510-1o-310-210-1 106 10’ 102 10’ 102 100 10’ 10’ 108 10’ 10” 10-15 10.16 Annealing Time (s) at 425°C io-41o-’io-2-’ 100 101 102 10’ 1o 10’ 106 10’ 108 109101010111012 Annealing Time (s) at 425°C (a) (c) 2mm lhr 2Ohr 136 days 2mm lhr 2Ohr 136 days —>—//7 10-1 I I10 I I I I _________________________________________________________ Rec-8days-4OsIo 136 days 10-2 Rec-8days-40s to 136 days 10-’ I i0- 0.6 ‘U) 10-6(6 - 10-’ 2 10-8 10-’C .2 9 100° Cl) 0.2 1-f 10 •p6 10-1 PP((6 f 10-” 0.0 -, 10 1010-51010010.210.1 100 10’ 102101102100101100 106 10’102° 10-15 10-” Annealing Time (s) at 190°C io-4 10-510-210-110010k 102100 10 10’ 106108 108 10’ 101010111012 Annealing Time (s) at 190°C (b) (d) Figure 6.8. Transition terms for Rec-8days-40s to 69 days (a) annealed at 425°C (b) annealed at 190°C and strain rates for the same condition (c) annealed at 425°C (d) annealed at 190°C 110 6.2.1.4. Overall Yield Strength The overall strength of recovered samples containing non-shearable precipitates can be calculated from Equation 6.2 once the contribution from dislocations, precipitates and solid solution are known. The contribution from dislocations was developed in the last section. As explained before, the contribution from solid solution is time independent and equal to 82MPa for the Al-Mg-Sc alloy. As described in section 6.2.1, the contribution of precipitates has been assumed to be the same as that for the as-aged samples. Thus to develop a general expression for the precipitate contribution to the yield strength the approach proposed by Fazeli et a?. [13) is followed and the Orowan stress is calculated as: 0.8M (2F) = bL (6.20) ppt where is the precipitate spacing, M is the Taylor factor, b is the Burgers vector, and F is the line tension of dislocations interacting with precipitates [32]: (‘,rd F = Gb 2/3Ln J (6.21) where d (V) is the volume average precipitate diameter and 13 is a constant depending on the nature of dislocations. Here, 13 is taken as 0.1 which is the average of 0.16 for pure screw dislocations and 0.04 for pure edge dislocations. Figure 6.9 illustrates the excellent agreement between experiments and the model developed here for the yield strength of both recovered and as-aged samples. To summarize, the present model extends the previous model of Verdier to account for the effect of precipitates on recovery and yield strength. The model has 5 111 fitting parameters. The activation energy and volume for dislocation annihilation are obtained from a single phase Al-Mg alloy. The remaining 3 parameters (A, n and n*) could be obtained from two further tests on pre-aged and recovered samples. The variation of n with pre-aging treatment is felt to be an artifact of the precipitate size distribution. As assumption that needs to be confirmed in future work. In addition to the yield strength, it is also important to study the flow stress which depends on the work hardening rate. In the next section, the work hardening behavior of these materials will be investigated. 112 400 Q350 30O 250 0) -U 81) >- 450 350 3O0 250 C 250 C’) >- Symbols: Experiments Lines: Model RoIled-8 days Rec-8days-40s to 136 days / K Aed-8days-40s to 136 days150 150— 1 - LI 50 As Solutionized 0 —.—// 10-10-10-10° 101 102 10’ 10 l0 106 10’ 108 10 10U 1062 Annealing Time (s) at 190°C Rec-80min-40s to 69 days 150Z\ 150 Aged-BOrno,-40s 50 A As Solutionized 0 -r--/’ 10-610510-8100 10’ 10’ 10 l0 106106 10’ 100 10 10101011 1012 Annealing Time (s) at 425°C (b) 2mm lhr 2Ohr 69 days 450 400 Lines: Model 350 Rolled-S days Rec-8 days-40s to 69 days 250 days-40s to 69 250 150 150 — 50 A As Solutionized0 --// 10-610-h10100 10’ 102 10’ 10 10 106 10’ 108 10° 1010 loll 1012 Annealing Time (s) at 425°C (d) Figure 6.9. Experiments and model prediction for evolution of yield strength of Al-Mg-Sc samples (a) Rec-80min-40s to 136 days annealed at 190°C (b) Rec-80min-40s to 69 days annealed at 425°C (c) Rec-8 days-40s to 136 days annealed at 190°C (d) Rec-8 days-40s to 69 days annealed at 425°C. 113 2mm 1r 2phr 16 days 2mm 1-ir 2phr 6days 450 —/ Symbols: Experiments Rolled-S0min Lines: Model 403 350 30) 250 200 450 Symbols: Experirroents - Lines: ModelRolled-80min Rec0mEn0s to 136in days 150 — i I I IX I 11)) Aged-Sommn-40s to 136 days 50 A As Solutioned 0 10-610-510-0100101102100106 10 106 10’ 108 l0 101010111012 Annealing Time (s) at 190°C (a) 2mm lhr 2Ohr 136 days 450 I I I I 0 C 81) Cl) 0) >- (81 UI C 81) C,) II) >- (c) 6.2.2. Modelling of Work Hardening: The Combined Effect of Recovery and Precipitates In a previous study on the work hardening of recovered single phase Al-Mg alloys Verdier et a!. [7] used a variant of the work hardening model developed by Kocks-Mecking-Estrin [50,521 based on the contribution of forest dislocations and subgrain boundaries to work hardening. Similarly, Fazeli et a!. [13] has applied the Kocks-Mecking-Estrin approach to model the effect of precipitates on the work hardening behavior of the Al-Mg-Sc alloy in the as-aged state. To explain the tensile response of recovered materials containing non-shearable precipitates, where precipitates contribute both to work hardening directly as well as via their effects on recovery, methods have been sought to incorporate both effects in the Kocks-Mecking-Estrin model. As a starting point, the work hardening of a recovered single phase Al-Mg alloy is considered. The internal state variable model applied to the work hardening rate of recovered Al-Mg alloys due to dislocation storage and annihilation during deformation, 0dis = dcrd,S was given by Verdier eta!. (see chapter 2, Equation 2.11) as: d — dcr — ds Ksubgrain 1 IvIK dis — — , ( — —) + — dis ( )d6 dis Dsubgraln 2Dsubgrain The first term on the right hand side of Equation 6.22 corresponds to the work hardening rate of a fully recrystallized material. Here °0 is the initial work hardening rate at the onset of plastic deformation, o is a scaling stress and ødLS is the forest dislocation contribution to the flow stress as given by the Taylor Equation [49]. 114 The second and third terms in Equation 6.22 corresponds to the contribution from subgrains which are assumed to contribute to dynamic recovery as well to the storage of dislocations via geometrically necessary dislocations required to maintain compatibility across sub-grain boundaries. In Equation 6.22 Kbajn is a geometric constant associated with storage of geometrically necessary dislocations at subgrain boundaries with a subgrain size of Dsubgrazn. The constant K describes the contribution of dislocations in cell walls or subgrain boundaries to dynamic recovery as explained in section 2.2.1 .1.2. In addition to the effect of recovery on work hardening, the contribution of A13Sc precipitates has to be taken into account. The study by Fazeli et al. [13) revealed that the work hardening response of as-aged Al-Mg-Sc samples can be enhanced due to the presence of non-shearable A13Sc precipitates. In the case of as-aged materials containing non-shearable precipitates an additional term, where L1 is the precipitate spacing, ppt is added to the Kocks-Mecking-Estrin model to account for geometrically necessary dislocations. This ignores the possibility of kinematic hardening arising from plastic incompatibility at the precipitate-matrix [13,38]. In the present study, this term is added to Equation 6.22 to account for the additional generation of geometrically necessary dislocations by precipitates in the recovered microstructure. Thus, an overall expression for the work hardening response of a recovered material containing precipitates of spacing and subgrains of size Dsubgrain, is given by: 6dis = dudIS = (1— + [-- (J_) + Ksubgraln 1 — MK (6.23)d6 s L dis cidis Dsubgram 2Dsubgraen 115 In this case is a geometric constant related to dislocation storage at precipitates which has a similar meaning as Ksubgrajn. The precipitate spacing was calculated from Equation 6.16 where the precipitate diameter was obtained from the precipitate size distribution based on volumetric averaging. 6.2.2.1. Work Hardening Behavior of the Al-Mg-Sc System In order to compare the experimental work hardening behavior with Equation 6.23, it is best to plot 8d,s versus ° dis Therefore, the contribution of forest dislocations (O•dis) to the total flow stress has to be extracted from the experimental stress-strain curves. The contribution of dislocations to the flow stress, ci, is given by: djs = — — ci 1/2 (6.24) This equation is similar to Equation 6.4 except that is replaced byci, the flow stress. The other terms in this equation have been described in section 6.2.1. The work hardening rate associated with dislocation storage, 0dig = dci-d,S , of dE recovered and as-aged samples was extracted from true stress-true strain curves by calculating cYdIS using Equation 6.24 at each strain level and numerical differentiating dis with respect to true strain followed by smoothing polynomial fits to segments of the curve. In Figure 6.10 8djs versus are plotted comparing work hardening rate of recovered samples with those of as-aged samples. 116 o dis (MPa) 0 dis (MPa) (c) (d) Figure 6.10. Evolution of experimental work hardening rate for recovered and as-aged samples (a) Pre-aged at 425°C 80mm-recovery at T=425°C (b) Pre-aged at 425°C 8 days-recovery at T=425°C (c) Pre-aged at 425°C 80mm- recovery at T = 190°C (d) Pre-aged at 42 5°C 8 days-recovery at T= 190°C The work hardening rate of recovered materials at a given level of stress, o dis’ iS higher compared to those in as-aged samples. This is related to the storage of GNDs due to the presence of cell and/or subgrain boundaries in the recovered microstructures. The role of GNDs on the work hardening behavior in a OdisOdis plot is schematically illustrated in Figure 6.11. (3) 4000 3000 2000 Ce V 1000 0 Annealing at 425°C (1) Rec-8 days-40s (2) Rec-8 days-lhr (3) Aged-S days 4000 3000 (‘3 ci 2000 (0 1000 0 (U 0 0, V 0 100 200 300 (2) ° dis (MPa) a dis (MPa) (a) (b) 400 0 100 200 300 400 (U U, V 100 200 300 400 0 100 200 300 400 117 Odis of GNDs Figure 6.11. The effect of GNDs on work hardening behavior: The work hardening rate increases to higher levels for a given level of adS due to presence of GNDs. At high stresses the two curves approach one another and the effect of GNDs decreases. By comparing curves (1) and (3) in Figures 6.10 (a) and (b) it can be understood that the effect of substructure on work hardening for samples with a large precipitate size (pre-aged for 8 days) is more significant than the one with the smaller precipitate size re-aged for 80mm) when recovery occurs at 425°C. However, comparing curves (1) and (3) in Figures 6.10 (c) and (d) shows that substructures are similarly effective for both pre-aging conditions when recovery occurs at 190°C. This is associated with the precipitate spacing relative to subgrain spacing, a feature that can be understood based on the work hardening model described below. The influence of precipitates on the work hardening rate is further elucidated in Figure 6.10 as samples with larger precipitate size have a lower work hardening rate than those with smaller precipitate size; though (as shown in Figure 5.10) the subgrain size is similar for the two samples annealed under similar conditions. Elastic-Plastic Transition 118 The work hardening model described by Equation 6.23 was fit to experiments using least square regression for both recovered and aged samples. In this model there are several adjustable parameters which are listed in Table 6.5. The values of 0 and cr should be independent of the state of recovery andJor the state of precipitation [511. The constants 00 ando have been obtained directly from the work hardening behavior of the as solutionized material shown in Figure 6.12. The initial hardening rate, 00, is found to be 1750 MPa which is consistent with the previous value reported for Al-Mg alloys [7]. The constant 05 is a scaling factor which is expected to be temperature and strain rate dependent [51]. 4000 Experiments Model 3000 (0 0 2000 100: AsSnized 0 50 100 150 200 250 300 dis (MPa) Figure 6.12. Evolution of work hardening rate, experiments and model predictions, for as solutionized samples The value of was determined directly from as-aged samples. Again, the value of this parameter should be independent of precipitate size and volume fraction. Therefore a single value was used to fit the curves shown in Figure 6.13 (a). Finally, the values of Kb,-0I and K were determined from a best fit to all of the recovered 119 conditions. Figure 6.13 illustrates the good agreement between experiments and model predictions for as-aged and recovered samples annealed at 425°C and 190°C. Subgrain size was used as an adjustable parameter to fit the model. Fitting was constrained so as to give values situated within the range predicted experimentally as shown in Figure 6.14. Table 6.5. Work hardening model parameters determined experimentally. Condition 8 (MPa) o (MPa) (MPa2.m) Ksubgrain (MPa2.m) K (m) As-Aged 1750 370 0.026 - - Recovered 1750 370 0.026 0.026 - 120 4000 Experiments 3000 edOmln2000 Ce 1000 Aged-8 days 0 0 100 200 300 400 dis (MPa) (a) 4000 o Annealing at 425°C Annealing at 425°C 3000 Open Symbols: Experiments 3000 Open Symbols: ExperimentsDashed Lines: Model Dashed Lines: Model0 CL 2000 . 2000 Rec-8 days-40s - Rec 8-o 1000 - -l4day 1000 Rec 80 ays-14 days 0 0 0 100 200 300 400 0 100 200 300 400 a dis (MPa) a dis (MPa) (b) (c) 4000 4000 8 Annealing at 190°C Annealing at 19000 0 Open Symbols: Experiments 0 Open Symbols: Experiments 3000 Dashed Lines: Model 3000 Dashed Lines: Model 8 0 0 0 2000 2000 0 S •0 Rec8days-2 __ !ysdaysm10001000 Rec 80 Rec-8 0 0 0 100 200 300 400 0 100 200 300 400 a dis (MPa) a dis (MPa) (d) (e) Figure 6.13. Evolution of work hardening rate, comparing experiments and model predictions, for as-aged and recovered samples (a) Aged at 425°C for 80mm and 8 days (b) Pre-aged at 425°C 80mm-recovery at T=425°C (c) Pre-aged at 425°C 8 days-recovery at T=425°C (d) Pre-aged at 425°C 80mm- recovery at T = 190°C (e) Pre-aged at 42 5°C 8 days-recovery at T= 190°C 121 40s 4 I 40s 4 I lhr 2Ohr 14 days Figure 6.14. Subgrain size measured by TEM and values applied in the work hardening model for recovered samples with different aging routes annealed at (a) 42 5°C and (b) 190°C In fitting the data it was found that the terms related to dislocation storage at precipitates (K,) and at subgrains (Ksubgrajfl) were approximately equal’. The value of K in Equation 6.23 is found to be very small (on the order of 10-8) and the ratio is found to be several orders of magnitudes smaller than that reported by Verdier [7]. For the case of Al-Mg-Sc recovered at 190°C and 425°C where subgrains K. are in an order of 1 im or larger, the impact of the term in Equation 6.23 is Dsubgrain found to be negligible. Therefore, the effect of dislocations in cell wall or subgrain boundaries on dynamic recovery for the investigated recovery conditions is negligible. 1 The K1 value for precipitates here is different from the one obtained by Fazeli et al. [13] since in their r study the geometric slip distance is described by the ratio — while here it is represented by the precipitate iv spacing (Equation 6.24). E3 :1 52 N (0 0) DlCl) lhr 2Ohr 14 days • TEM-Rec - 80mm - 40s to 14 days TEM-Rec- 8days -40s to 14 days Work Hardening Model 0 101 102 10 10 10 106 10 Annealing Time (s) at 425°C (a) E3 a)2 (0 0) DlCl) 0 • TEM-Rec- 80 mm 40s to 14 days TEM-Rec- 8 days 40s to 14 days — Work Hardening Model 101 102 10 10 10° 106 10 Annealing Time (s) at 190°C (b) 122 Based on the above comments, Equation 6.23 can be simplified by removing the annihilation at subgrain boundaries and using one geometric parameter (KG) for geometric storage at subgrains and precipitates: Odis = dcrd(S = 00(1— .-) + !L (_L_ + 1 (6.25)d6 °S °dis Dsubgrain where the effective geometric slip distance now becomes: 1 )_1 (6.26) Dsubgrain The value of the parameter KG is directly related to the efficiency of dislocation storage at subgrain boundaries and precipitates. According to the Kocks-Mecking-Estrin model, dislocation storage due to work hardening, neglecting the effect of subgrains on dynamic recovery, for recovered Al-Mg-Sc samples is given by: = =M(k1f,_2p+L( 1 (6.27) d6 dy b Dsubgrajfl Comparing Equations 6.25, 6.27 and the Taylor Equation k3 was back calculated from KG and was found to be 1. This is within the range of 1-4 reported in previous studies for different particle geometries [43, 131] Table 6.6 summarizes the sequence of required tensile testing experiments to obtain and verify the adjustable parameters in the work hardening model. For the case of recovered Al-Mg-Sc alloys there are three adjustable parameters since it has been found here that the geometric factors , Ksubgraffl) for A13Sc precipitates and subgrains are the same. As Table 6.6 shows, three experiments are required to obtain the parameters in this model for recovered samples. Initially it is necessary to obtain the contribution of precipitates, chemical composition and strain rate on work hardening rate. This can be 123 fulfilled by conducting one tensile test on an as-solutionized sample to fit 0 and o. parameters. Then a tensile test on an as-aged sample should be conducted to capture the contribution of precipitates by obtaining Finally, tensile tests on recovered samples are required to understand the role of recovery on the work hardening rate and thereby fitting Ksubgrajn. In this study, the work hardening model was successfully applied to recovered microstructures containing two different precipitate sizes which were recovered at 190°C and 425°C for 40s, 2mm, lhr, 20 hr and 14 days using the parameters given in Table 6.6. Table 6.6. Required experiments to determine the adjustable parameters for the work hardening model. Sequence of Tensile Experiments to Adjustable Parameter Determine Parameters 1) As-solutionized condition 0 2) As-aged condition K1 3) Recovered condition Ksubgrain In Figure 6.15 the precipitate spacing calculated based on Equation 6.16 is compared to the measured subgrain sizes. This is relevant to the term (_!_ + 1 ) and therefore to understanding the relative importance of d,s Dsubgrain precipitates and subgrains to work hardening. For a given initial precipitate diameter, the work hardening rate of recovered samples annealed at 190°C is maintained at higher stress levels compared to the ones annealed at 42 5°C as shown in Figure 6.10. This behavior can be explained based on the fact that recovery at 190°C leads to smaller subgrain and precipitate sizes in Equation 6.26, therefore, the work hardening rate shifts to higher values at a given stress level. It 124 is concluded that in samples annealed at 190°C, subgrains and precipitates have nearly equal effect on work hardening since the subgrain size and precipitate spacing are approximately the same over the range of annealing conditions studied (Figure 6.15 (a) and (c)). Samples annealed at 425°C, however, have larger subgrain sizes relative to precipitate spacing after 2Ohr annealing and thus in these samples it is the precipitates which tend to control the work hardening rate (Figure 6.15 (b) and (d)). In examining the effect of the starting precipitate diameter for both recovery temperatures, Figure 6.10 demonstrates that for samples with an average precipitate diameter of 77 nm (pre-aged for 8 days), subgrains dominate the work hardening rate (Figure 6.15(c) and (d)) while in the case of an average precipitate diameter of 27 nm (pre-aged for 80mm) precipitates are more efficient since the precipitate spacing is smaller than the subgrain size (Figure 6.15 (a) and (b)). It is noted that after annealing at 425°C for long times such as 20 hr the subgrain sizes are larger than the precipitate spacing even for pre-aging for 8 days and the work hardening is therefore controlled by the precipitates (Figure 6.15 (d)). 125 40s lhr 2Ohr 14 days I I __I__ I 40s 1-ir 2Ohr 14 days 3.0 2.5 [.5 1.0 a) -J 0.5 f 3.0 2.5 2.0 [5 c 1.0 a) -J 0.5 0.0 101 10° 10’ 102 10° 10 10° Annealing Time (s) at 425°C Figure 6.15. Precipitate spacing and subgrain sizes for recovered samples pre-aged at 425°C (a) for 80mm recovered at 190°C (b) for 80mm recovered at 425°C (c) for 8 days recovered at 190°C (d) for 8 days recovered at 425°C To complete this section on the modelling of the work hardening rate, the stress-strain curves for the recovered samples were calculated through the integration of Equation 6.25 and calculation of o via Equation 6.24. Figure 6.16 presents the comparison between the experiments and models for the total flow stress. Given the ability to model both the yield strength and work hardening rate with a relatively small number of fit parameters it is interesting to move to using the model as a predictive tool for the microstructure-mechanical properties relationship. — Precipitate Spacing • Subgrain Size Rec-8omin-40s to 14 days I 0.0 2.0 Precipitate Spacing • Subgrain Size 1.5 Rec-8Omin-40sto 14 days —‘ 0.5 0.0 . f. f, 101 10° 101 102 10° 10 106 106 10’ Annealing Time (s) at 1 900C (a) 40s lhr 2Ohr 14 days 2.0 I • Subgrain Size Precipitate Spacing 1.5 Rec-8 days-40s to 14 days a) c 1.0 .c 0) C a) — 0.5 + 101 10° 101 102 10° 10 10° 106 10’ Annealing Time (s) at 1900C 101 10° 101 102 10° 10 10° 10 10’ Annealing Time (s) at 425°C (b) 40s lhr 2Ohr 14 days • Subgrain Size Precipitate Spacing Rec-8 days-40s to 14 days (c) 106 10’ (d) 126 500 500 100 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 True Strain 400 0 U) 300 U) CL) D I— 200 (a) 100 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 True Strain (b) Figure 6.16. Flow stress, experiments and model predictions, for recovered samples (a) Pre-aged at 425°C 80mm- recovery at T = 425°C (b) Pre-aged at 425°C 8 days-recovery at T=425°C (c) Pre-aged at 425°C 80mm-recovery at T= 190°C (d) Pre-aged at 425°C 8days-recovery at T= 190°C 6.2.3. Application of Models to other Pre-aging Treatments In order to examine the ability of the yield strength and work hardening models to predict the mechanical response for a material subjected to another pre-aging condition, and therefore containing a different precipitate distribution, a new aging and annealing procedure was examined. 400 0 U) 300 Cl) CL) F 200 400 U, 300 U) CL) 200 500 Rec-80min-14 days Rec-8 days-2min Rec-8 days-14 days Annealing at 190°C Solid Lines: Experiments Dashed Lines: Model 100 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 True Strain (c) 500 400 0 U) 300 U) CL) f 200 Annealing at 190°C Solid Lines: Experiments Dashed Lines: Model 100 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 True Strain (d) 127 The Al-Mg-Sc alloy was pre-aged at 3 00°C for 8 hr followed by pre-aging at 410°C for 10 mm. This leads to a different precipitate size distribution compare to that obtained in samples pre-aged at 425°C. The pre-aged material was then cold rolled to 80% reduction followed by annealing at 410°C for 1 hr. By selecting 410°C as the recovery temperature and 80% cold rolling as the deformation level, it is assumed that the subgrain size is the same as the one obtained during recovery at 425°C. As before, pre aging and recovery were conducted at the same temperature to inhibit concurrent precipitation and recovery. A KWN model developed by Fazeli et al. [13] which has been calibrated for the Al-Mg-Sc material studied here was used to predict the precipitate size distribution. The KWN model has been linked to a mechanical model by Fazeli et a!. which allows the contribution of precipitates to the yield strength of the Al-Mg-Sc alloy, o,,, to be determined at any arbitrary annealing temperature. The precipitate strength for pre-aging at 410°C was also determined using the abovementioned model. The KWN and mechanical models developed by Fazeli et a!. are briefly summarized in Appendix C. Knowing the precipitate size distribution and precipitate strength, the yield strength and work hardening rate of the sample recovered at 410°C have been predicted by Equations 6.4, 6.8, 6.15 and 6.25. The same adjustable parameters obtained from the samples recovered at 425°C for lhr (i.e. those given in Tables 6.4 and 6.5) have been applied here. Thus, the only change between the calculations made here and those for samples annealed at 425°C is the precipitate size distribution. Figure 6.17 shows the agreement between the model and the experiment for this condition. In addition, Figure 6.17 compares the tensile response of the sample recovered 128 at 410°C for lhr with the one recovered at 425°C for lhr. According to Figure 6.17, the work hardening and yield strength of the sample recovered at 410°C is higher than that of the sample recovered at 425°C. 600 Rec-1 0mm-i hr 500 Annealing at 410°C Solid Lines: Experiments Dashed Lines: Model 100 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 True Strain Figure 6.17. Flow stress, experiments and model predictions, for samples recovered at 410°C and 425°C. Figure 6.18 compares predictions of dislocation and precipitate contributions to the yield strength for samples recovered at 410°C and 425°C. According to Figure 6.18 (a), 0dis at yield for both recovery temperatures is similar; however, the contribution from precipitates is 40 MPa higher for recovery at 410°C compared to the one recovered at 425°C. As the subgrain size for both conditions is assumed to be the same, it suggests that the contribution from recovery on yield strength and work hardening at 410°C is similar to that at 425°C. On the contrary, the contribution from precipitates is different as the average precipitate sizes and precipitate spacing are different. The volume average precipitate diameter for samples recovered at 4 10°C for lhr is 11 nm while the volume average precipitate diameter for samples recovered at 425°C is 27 nm. Therefore, the enhanced tensile response for recovery at 410°C shown in Figure 6.17 is predominately 129 associated with the pre-aging treatment. This suggests that the same contribution to mechanical properties from recovery for a range of precipitates sizes can be obtained. Thus, one way to tailor the mechanical response would be to keep the same level of recovery but with different precipitate size distributions. 140 120 (V 0- 100 -c 80 ci) Recovery at 425°C Recovery at 410°C (a) 120 100 80 0 60 _••J_ b 40 20 0 —i--— Recovery at 425°C Recovery at 41 0°C (b) Figure 6.18. Modelling results on dislocation contribution to the yield strength and precipitate strength for samples recovered at 4 10°C and 425°C. 130 6.3. Summary The recovered microstructure in the Al-Mg-Sc alloy is found to be stable after very long time annealing at 425°C. This is associated with pinning of low angle boundaries by Al3Sc precipitates. For the first time, the role of precipitates on recovery kinetics and thereby softening behaviour was captured by coupling Verdier model on single phase Al-Mg alloys and creep models. In this new modelling approach a transition stage from dislocation annihilation to a climb controlled mechanism was defined. In addition, the effects of both recovery and precipitates on work hardening have been incorporated in the Kocks-Mecking-Estrin model. These models explain the interaction of dislocations with precipitates and subgrain boundaries during deformation. It is suggested that yield strength and work hardening behavior depend on the length scale of microstructure during recovery, specifically the subgrain size, precipitate size and dislocation spacing. These models have been successfully applied to recovered microstructures processed by specific thermo-mechanical treatments. There are, however, limitations to the models developed here. In the next section, these limitations, as well as future potential application of models will be discussed. 131 Chapter 7-Application of Models to Al-Mg-Sc Alloys In this section the models developed in chapter 6 are revisited to examine their limitations as well as their potential application to the design of microstructure-mechanical properties in Al-Mg-Sc alloys. The possibilities of obtaining different combinations of mechanical properties in recovered materials will be examined and compared against the properties of as-aged materials. Finally, opportunities for modifying the mechanical properties through alloy design in Al-Mg-Sc alloys will be discussed in relation to the models developed here. 7.1. Limitations of the Models While the models developed here are physical and have a minimum number of fitting parameters, there are some remaining limitations to their use. In this section the limitations of these models related to annealing temperature, strain level and chemical composition are described. 7.1.1. Annealing Temperature The yield strength model is limited to thermal treatments where the recovery temperature is the same as the pre-aging temperature or recovery occurs at low temperatures where re-precipitation or dissolution of precipitates does not occur during recovery. For the case of recovery at a temperature different from the pre-aging temperature and higher than 190°C (e.g. 275°C) re-precipitation can take place and interfere with recovery. In this case, due to a low super saturation of Sc after pre-aging at 425°C, it is expected that heterogeneous precipitation on dislocations and grain boundaries will occur. To modify the model for the above mentioned thermal history, in 132 addition to the time for dislocation climb, the pinning of dislocation nodes by precipitates during concurrent precipitation-recovery has to be taken into account. One way of achieving this in the modelling frame work would be to combine the approach developed here with the approach proposed by Zurob et al. [801. Such a combination of models needs further investigation. To describe the work hardening behavior of recovered microstructures, the subgrain size enters as an important parameter. As explained in section 6.1.2 subgrain sizes in the case of recovery are not predicted by A13Sc spacing through the Zener analysis and therefore measurement of subgrain size evolution was necessary for a given annealing temperature. In this study, subgrain size was measured for recovery at 425°C (subgrain size in the range of 1-3 tim) and 190°C (subgrain size in the range of 0.3-1 im). Therefore without further measurements on subgrain size, the work hardening model is limited to temperatures similar to 425°C and 190°C where subgrain sizes are expected to be similar to those measured here. In addition, the application of the work hardening model to temperatures that lead to heterogeneous precipitation on dislocations and subgrain boundaries is not examined here and needs further investigation. Developing detailed models which would allow for accurate prediction of subgrain size would be necessary to extend this work further. 133 7.1.2. Strain Level The yield strength model was tuned to recovered microstructures cold rolled to 80% in reduction corresponding to true von Mises strain of 1.85. Once the yield strength of the as rolled material is determined, the evolution of yield strength due to dislocation annihilation and stress relaxation can be explained by the model at other strain levels. However, in order to apply the model for another strain level, it may be necessary to adjust activation energy and activation volume since a range of activation energy and volume is reported for different strain levels for recovery of Al-Mg alloys [7,8]. In this study the role of low angle subgrain boundaries with misorietation smaller than 150 was applied to the models. This was related to the fact that there was a limitation in the amount of imposed strain, 1.85 equivalent strain, due to initial thickness of as received hot rolled plates. However, according to the study by Jazaeri and Humphreys [102], high angle grain boundaries can be achieved at strain levels larger than 3. For the case of high angle grain boundaries the geometric factor, KG, needs to be readjusted as the efficiency of high angle boundaries on increasing the work hardening rate can be different from the one for the low angle boundaries [811. Similarly, in the yeild strength model the role of high angle boundaries can be included based on the model proposed by Nes [15] where a Hall-Petch type of equation is applied. Therefore, further study is requied for tuning the developed models for large strain levels. 7.1.3. Chemical Composition In the present work the models are applied to the A1-2.8%Mg-0.16%Sc alloy. Previous studies have shown that heterogeneous precipitation can be dominant in Al-Sc alloys with lower Sc contents annealed at similar temperatures to those studied 134 here [25]. As described in section 7.2.1 the role of heterogeneous precipitation is not included in the yield strength and work hardening models. Therefore, these models are limited to those Al-Mg-Sc alloys where homogeneous precipitation occurs. There has been a great interest in addition of Ti and Zr to Al-Sc alloys as they can partition to AI3Sc precipitates and form Al3(SciZr) or Al3(SciTi) precipitates which have similar strengthening and pinning effect [132,133,134]. Since Zr and Ti have a lower cost than Sc, the same precipitate volume fraction could be retained while applying a smaller amount of Sc [132,135]. Segregation of Zr and Ti at the cL-Al/A13Sc interface retards coarsening kinetics and can affect the volume fraction of precipitates. The role of other additional elements such as Ti and Zr is not considered in the present models. In order to extend the models developed here these effects would hence to be considered through models including ternary diffusion and non-stoichiometric precipitates. 7.2. Application of Models Despite the limitations of the models discussed in the previous section, the models developed in this thesis do provide useful information on tensile response and describe the possibility of achieving different combination of mechanical properties via controlling precipitation and recovery. A minimal set of parameters capable of capturing important aspects of the stress strain curve of a material are yield strength, maximum true tensile strength and uniform elongation. Maximum true tensile strength and uniform elongation can be extracted from the work hardening model using the Considére criteria. According to Considére criteria [112], the onset of necking occurs when: 135 (7.1) tIE where both and the flow stress, cr, can be calculated from the work hardening model. dE The results can then be integrated to determine the uniform elongation (See Appendix B). Using this approach yield strength-uniform elongation and maximum tensile strength- uniform elongation correlations for recovered microstructures can be developed. These are compared with the peak aged condition for the same alloy taken from previous studies [13,141. 7.2.1. Opportunities for Modifying Mechanical Properties and Alloy Design Yield strength-uniform elongation and maximum tensile strength-uniform elongation correlations for different processing conditions based on the recovery and precipitation models have been plotted in Figure 7.1. The mechanical models developed in this thesis have been calibrated for experiments on samples pre-aged at 425°C and recovered at 190°C or 425°C. The mechanical properties of these samples recovered for various time are (indicated by bubbles with solid lines) compared to the ones for the peak aged fully recrystallized condition at 3 00°C in Figure 7.1. The peak aged sample results in a higher combination of yield strengthlmaximum tensile strength and uniform elongation compared to samples pre-aged and recovered at 425°C. To elaborate this comparison, the peak aged condition is labeled as “A” while the sample recovered at 425°C for 20 hr is labeled as “B” in Figure 7.1. 136 350 Pre-Aged 41 0°C-Rec 190°C \•\\ / 11300 C -C 1-’ 0) a 250 ,,/“€? \ \ Pre-Aged 41 0°C-Rec 410°Ca) \ \\/ 0 Pre-Aged 425°C-Rec 190C”\ Peak Aged 0200 5: Pre-Aged 425°C-Rec 425°C ‘ B 150 I I I P 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 Uniform Elongation (a) ‘ 600 ft. i\ Pre-Aged 410°C-Rec 190°C\\ •\\%/ 550 \ \ -c °C-Rec 410°C0) \ PreAged4lOC I I. \ Q \/wSOO \ \ -C,) .S 450 A Peak Aged C a) F— 400 Pre-Aged 425°C-Rec 190°C S D 350 Pre-Aged 425°C-Rec 425°C Cr’ 300 I I I I 0.16 0.18 0.20 0.22 024 0.26 0.28 0.30 Uniform Elongation (b) Figure 7.1. Model predictions of recovered and as aged samples (a) yield strength-uniform elongation (b) maximum tensile strength-uniform elongation at 77K 137 It is suggested that the improved mechanical properties of sample A compared to sample B with a well recovered microstructure is associated with the precipitate size distribution. In order to justify this explanation, simulation results on precipitate size distribution and precipitate strength for sample A (the peak aged condition at 3 00°C) and the ones for sample B (recovered at 425°C for 20 hr) are shown in Figure 7.2. The transition from shearable to non-shearable precipitates is indicated as a vertical line in Figure 7.2 (a) showing that all precipitates in sample A are shearable while in sample B they are non-shearable. According to Figure 7.2 (a) and (b), it can be understood that annealing at 425°C results in significant precipitate coarsening compared to 300°C and thereby a substantial drop in precipitate strength. This explains the lower yield strength/maximum tensile strength of sample B compared to that for sample A while both samples have similar uniform elongation. 138 8140 120 100 80 60 40 20 Figure 7.2. Model predictions on (a) precipitate size distribution and (b) precipitate strength at 77K for peak aged (labeled A in Figure 7.1) and recovered at 425°C for 201w and 4 10°C for 2mm (labeled B and C in Figure 7.1) conditions (courtesy of Fazeli et a!. [14]) To investigate the possibility of achieving other combination of mechanical properties by processing recovered microstructures containing precipitates, the models developed here have been applied to another pre-aging condition which results in smaller precipitate sizes compared to the ones for samples pre-aged at 425°C. As explained in co6 U)(U C-) I- a) >C-) a) U I-. LI 0 A PeakAge(A) A — — — Pre-Aged 425°C-Rec 2Ohr 425°C (B) Il Pre-Aged 410°C-Rec 2mm 4100C (C) - Shearable - Non Shearable Transition - ,/ \\ / I / \\ 0 10 20 30 40 50 60 Precipitate Diameter (nm) (a) Co 0 0 Peak Aged Recovery Recovery (A) at 425°C (B) at 410°C (C) (b) 139 section 6.2.3 pre-aging at 300°C for 8 hr followed by pre-aging at 410°C for 10 mm can be one alternative. The developed models here were calibrated only for one set of experiments on samples pre-aged at 410°C followed by recovery at 410°C for 1 hr. However, the model calculations have been extended to pre-ageing at 410°C followed by recovery at 410°C and 190°C for different times. The results on mechanical properties for these conditions (indicated by dashed bubbles) are given in Figure 7.1. Figure 7.1 illustrates that a better combination of yield strength/maximum tensile strength and uniform elongation can be obtained by pre-aging at 410°C than the ones pre-aged at 425°C. For example the sample pre-aged at 410°C followed by recovery at 410°C for 2 mm is labeled as “C” in Figure 7.1. Simulation results on precipitate size distribution and precipitate strength for sample C are given in Figure 7.2. Figure 7.2 (a) shows that for sample C precipitate sizes (mostly in shearable region) are smaller than the ones for sample B and according to Figure 7.2 (b) the precipitate strength for sample C is quite similar to the one for the peak aged condition. From Figure 7.1(b) it can be understood that the maximum tensile strength of sample C is improved by 25% with only 22% reduction in uniform elongation compared to the maximum tensile strength and uniform elongation of the peak aged condition. It should be noted that recovery at 410°C for longer than 2mm (e.g. 15 mm or lhr) results in a combination of shearable and non-shearable precipitates (not presented). Therefore, mechanical properties of recovered microstructures can be modified further by introducing shearable or a combination of shearable and non-shearable precipitates. It is suggested that there can be other possibilities to enhance the combination of mechanical properties of recovered microstructures compared to the as-aged condition. 140 One way can be increasing the level of imposed strain and thereby increasing the driving pressure for recrystallization. This can facilitate achieving partially recrystallized microstructures and improving uniform elongation without significant reduction in yield strength or ultimate tensile strength. Increasing the level of strain can also result in fine grained microstructures with high angle boundaries and a grain size of (1-5 jim) similar to subgrain sizes obtained in this study [53,54]. These fine grained microstructures can result in higher yield strength but not improved work hardening rate or uniform elongation [4,136,137]. Another alternative can be designing microstructures with duplex grain structure. A previous study on a single phase Al-Mg alloy has shown that combination of yield strength and uniform elongation of microstructures with duplex grain structure processed by asymmetric rolling and annealing can be improved compared to the ones for fine or course grained microstructures [5]. It would be of interest to develop Al-Mg-Sc alloys with duplex grain structure containing only shearable or shearable and non-shearable Al3Sc precipitates and compare their mechanical properties with those for recovered microstructures studied here. It is suggested that the introduction of coarse grains and non-shearable precipitates enhances work hardening while fine grains and shearable precipitates improve yield strength. These approaches have the potential for improving mechanical properties of Al alloys containing Al3Sc precipitates and need further investigation. Experimental and modelling results in this work show that in well recovered microstructures precipitate spacing is smaller than the subgrain size. This can be understood from TEM observations, e.g. see Figure 5.9 (b) and (c), in which a large number of precipitates are inside subgrains and may not play a significant role in pinning 141 low angle boundaries. In other words, thermally stable recovered microstructures similar to the ones processed in this study or fined grained Al-Sc alloys developed in other studies [53,54] can be achieved by reducing the volume fraction of A13Sc precipitates. This can be important in the design of Al-Sc alloys due to the high cost of Sc [91. The equilibrium volume fraction of precipitates in different Al-Sc alloys is calculated and plotted in Figure 7.3 and the minimum required volume fraction to produce precipitates with a spacing of 0.3 tm similar to the initial subgrain size after 80% rolling is plotted as a horizontal line. Figure 7.3 illustrates that the minimum volume fraction is also obtained in lower Sc contents such as 0.05%Sc containing alloys. However, heterogeneous precipitation may be dominant in lower Sc content alloys [25]. The role of heterogeneous precipitation was not studied here. Therefore, parametric studies on Al-Sc alloys with lower Sc contents in the frame work of experiments and modelling would appear to be important for looking at the potential of commercial Al-Mg-Sc alloys. 142 0.006 0.005 0.004 U 0.003 0.002 o.oo 0.000 100 150 200 250 300 350 400 450 500 550 600 Temperature (°C) Figure 7.3. Volume fraction of A13Sc precipitates versus temperature and the minimum volume fraction required to produce Al3Sc precipitates with spacing of 0.3 m. 143 AI-0.1 6%Sc AI-0.1%Sc - AI-0.05%Sc 0.3tm Precipitate Spacing — — Chapter 8-Conclusions The impact of recovery on the tensile response of a model Al-2.8%Mg-O. 1 6%Sc alloy has been investigated. Recovered microstructures containing non-shearable precipitates have been achieved by pre-aging, deformation and annealing at different temperatures. The A13Sc precipitates are found to be very effective at sustaining recovered microstructures after annealing up to high homologous temperatures. Yield strength and work hardening behavior of recovered microstructures have been examined by tensile testing. Based on these results physically based models have been developed which allow for a self consistent description of the yield strength and work hardening rate as a function of subgrain size, precipitate state and thermal history. It has been shown that the recovery kinetics, and therefore the yield strength of the material, is strongly affected by the presence of precipitates. A physically based model which accounts for this precipitate size dependence via a climb model has been developed to describe the reduction of yield strength on annealing. This model has been coupled to a previous model appropriate for describing the yield strength of recovered single phase Al-Mg alloys. The model also describes a transition from kinetics dominated by dislocation annihilation without the effect of precipitates to a dislocation climb mechanism. It is argued that, in the very early stages of recovery when the dislocation spacing is much smaller than the precipitate spacing, the rate of recovery should be nearly precipitate size independent. Once the dislocation spacing and precipitate spacing become similar the influence of precipitates is predicted to become significant, consistent with the experimental results obtained in this study. 144 The work hardening rate of the recovered Al-Mg-Sc alloy was accurately captured using a Kocks-Mecking-Estin internal state variable model. The effect of the recovered state enters through 1) the unrecovered forest dislocations density 2) the presence of low angle subgrain boundaries and 3) non-shearable precipitates. Precipitates and subgrain boundaries contribute to the storage of additional geometrically necessary dislocations. It is concluded that the role of subgrain boundaries on work hardening depends on the subgrain size relative to precipitate spacing. In samples annealed at lower temperatures where subgrain sizes are similar to precipitate spacing, the work hardening rate is increased due to the presence of subgrains compared to those annealed at high homologous temperature where the subgrain sizes are larger than precipitate spacing. The model developed here shows how precipitates may be used to tune the mechanical response of recovered or fine grained materials. 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The results of these experiments were summarized in a paper published in the proceeding of the International Conference on Aluminum Alloys [1] and are summarized here. The recrystallization response of the alloy after rolling to von Mises strains of 1.4 and 2.65 are presented in Figure A. 1. Unlike the bulk of the samples studied in this thesis, as-solutionized samples were used as the starting material for these studies. Thus recrystallization and precipitation occur concurrently in these experiments. The recrystallized fraction of deformed and annealed samples was measured from SEM images using point fraction method. The times corresponding to <20% and >80% recrystallized fraction were considered as the start and finish times. Figure A. 1 shows the weak dependence of the recrystallization start temperature with time. This result is in good agreement with the observations of Jones and Humphreys [2] on the recrystallization of a binary Al-0. 1 2wt%Sc alloy. 152 450 450 0 0400 400 350 350 E E 0 0 I— 300 I— 300 250 ....... 250 0.01 0.1 1 10 100 1000 0.01 0.1 1 10 100 1000 Time (Minutes) Time (Minutes) (a) (b) Figure A.1. Temperature — time — recrystallized fraction plots for a material cold rolled to a) = 1.4 and b) e = 2.65 illustrating the onset and finish of recrystallization (approximately indicated with the dashed lines). From the samples illustrated in Figure A. 1, some conditions were selected for more detailed study by EBSD. Figure A.2 illustrates the results of EBSD observation on two conditions corresponding to the onset and final stages of recrystallization at 410°C and 420°C respectively for the material rolled to a strain of 1.4. The unrecrystallized areas in Figure A.2 are well recovered as indicated by the high indexing rate. • . • . •• -‘---- 70% Cold Rolling ( 1.4) Recrystallized 0 -20% • Recrystallized 80 -100% Recrystallization Start — — — Recrystallization End • • 90% Cold Rolling (E 2.65) . Recrystallized 0-20% • Recrystallized 80-100% Recrystallization Start — — — Recrystallization End 153 Figure A.2. EBSD maps of pattern quality showing samples rolled to a strain of 1.4 and annealed at a) 410°C for 5 minutes (8% recrystallized) and b) 420°C for 3 hours (92% recrystallized). The solid white lines indicate boundaries with misorientations of between 3 and 10° while black lines indicate boundaries with misorientations of greater than 150. Both samples are viewed in the along the transverse direction, the rolling direction being horizontal. References 1. R. Roumina, C.W. Sinclair, F. Fazeli, Mater. Sci. Forum, 519-521, 2006,1647. 2. M.J. Jones, F.J. Humphreys,Acta Mater., 2003, 51, 2149. (a) (b) 154 Appendix B-Uniform Elongation and Maximum Tensile Strength Uniform elongation and maximum tensile strength, in uniaxial tensile testing can be calculated based on the Considére criteria: (B.1) d6 In this study the work hardening rate related to dislocations,0d,s = do- dis , is dE investigated: 8dis = dudIS = O(1— Udis) KG i + 1 (B.2)d6 OS L Dsubgrain In order to apply the Consider criteria, a correlation between dci dis and must be d8 d6 established. The flow stress arising from dislocations, cid,s, in the recovered microstructures during deformation is given by: o-dis = — u0, )2 — (B.3) By differentiating Equation B.3 with respect to strain, s, the following is obtained: dud. (u—cr0) dciIS = (B.4) d6 ,J(cr - cr0)2 — d6 Or: dci J(0_uo)2_0pt dudlS (B5) ds (u—u0) ds - Combining Equations B.2, B.3 and B.5 gives: 155 dci = — — (1— — crpj + KG (L + 1 d6 (ci — u0) 0 —u0)2 — -2 L1 Dsubgrain (B.6) By substituting from B.6 into the Considére criteria, the maximum tensile strength at necking is obtained. Using Equation B.3, the value of d,s corresponding to the yield strength (ci),) and the value of dis at maximum tensile strength, is determined. Now the uniform elongation, s, is calculated by integration of Equation B.2 in the following form: dis at M axim urn Tensile Strengthç dudIS — Udisatyietd 80(1 — dis ) + KQ 1 + 1 dis Dsubgrain (B.7) 156 Appendix C-Models for Precipitation Kinetics and Yield Strength for As-Aged A1-2.8%Mg-O.16%Sc C.1. Precipitation Kinetics Model A model based on the Kampmann-Wagner (KWN) approach [1] has been developed by Fazeli et a!. [21 to predict precipitation kinetics and precipitate size distribution of the A1-2.8%Mg-O. 1 6%Sc alloy during isothermal and non-isothermal heat treatments. In the KWN modelling approach, nucleation, growth and coarsening of precipitates are considered to occur concurrently. The transition from nucleation to growth and growth to coarsening are implicitly accounted for based on the evolution of precipitate size distribution during the process. According to the KWN modelling approach: • Nucleation is described based on the classical nucleation theory where nucleation rate is given by: = ZfiN exp [ exp - (C.1) In the study by Fazeli et al. only homogeneous nucleation has been considered consistent with TEM observations for the selected aging conditions. • The growth rate is described by a diffusional growth law which for spherical A13Sc precipitates is given by: dr — D c’ — c cit — r cAt3’ — c (C.2) The parameters for nucleation and growth are summarized in Table C. 1. 157 • The coarsening process is implicitly captured in the KWN model. As the fraction of solute in the matrix decreases, the critical radius of Al3Sc precipitates increases due to reduction in driving pressure for nucleation and growth. Therefore, precipitates in the size distribution with a radius smaller than the critical radius shrink and eventually that size class is removed from size distribution. Conversely, precipitates in the size distribution with radius larger than the critical radius will have a positive growth rate and continue to increase in size. Several recent studies [3,4] on Al-Sc alloys failed to capture the precipitation kinetics using a single value for interfacial energy and thus the interfacial energy (YA1IAISC) was treated as a function of temperature, time and precipitate size. However, in the study by Fazeli et al. the interfacial energy YAI/A1,Sc has been set at a single value of 127 mJm2. This was accomplished by considering the role of excess vacancies on diffusivity of Sc. 158 Table C.1. Parameters for nucleation and growth rates Parameter Explanation Parameter Explanation Zeldovich non-equilibrium r Precipitate radiusZ factor (0.05) Attachment rate of single DSC Effective Sc diffusivity atoms to the critical nucleus eff Number of atoms per unit Sc Instantaneous ScN volume concentration in the matrix Sc concentration in the Driving pressure for Sc matrix at the interfaceAG nucleation r (adjusted based on capillarity effect) r Incubation time (2Zfy’ a Constant CA13Sc Concentration of Sc in A13Sc YAIIA13Sc Interfacial energy C.2. Yield Strength Model A mechanical model for the yield strength of the Al-2.8%Mg-0.16%Sc alloy has been developed by Fazeli et.al. [5] and linked to the precipitation model described in section C. 1. In this model the precipitation strengthening, o , predicted at any arbitrary isothermal and non-isothermal aging treatment is given by: M F(r) (C.3) = where2eff is the effective mean planar precipitate spacing and F(r) is the maximum interaction force between a precipitate and dislocation line. The effective mean planar precipitate spacing is calculated based on average precipitate size distribution obtained from the precipitation kinetics model. For shearable precipitates a linear relation between the maximum precipitate strength and precipitate size is assumed [6]: F(r)=2Fr For rr (C.4) 159 where F is the average dislocation line tension and r is the critical precipitate radius for transition from shearable and non-shearable precipitates. Above this transition, the maximum precipitate strength is obtained given by: F(r) = 2F For r > r (C.5) In this study the shearable/non-shearable precipitate transition was treated as an adjustable parameter which was found to be 3.7 nm consistent with TEM observations [71. References 1. R. Wagner and R. Kampmann, Mater. Sci. Technol., 5,1991, 213. 2. F. Fazeli, C. W. Sinclair, T. Bastow, Metal?. Mater. Trans. A, 2008, 39, 2297. 3. R.W. Hyland, Metal?. Trans. A, 23A, 1992, 1947. 4. J.D. Robson, M.J. Jones, P.B. Prangnell, Acta Mater., 20038, 51, 1453. 5. F. Fazeli, W.J. Poole, C. W. Sinclair, Acta Mater., 56, 2008, 1909. 6. A. Deschamps, Y. Brechet, Acta Mater., 1999, 47, 293. 7. D.N. Seidman, E.A. Marquis, D.C. Dunand, Acta Mater., 2002, 50, 4021. 160

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