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A Field Dislocation Mechanics Approach to Emergent Properties in Two-Phase Nickel-Based Superalloys

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A Field Dislocation Mechanics Approach to Emergent Properties in Two-Phase Nickel-Based Superalloys A Field Dislocation Mechanics Approach To Emergent Properties In Two-Phase Nickel-Based Superalloys By James Jordan Little A thesis submitted to the University of Birmingham for the degree of DOCTOR OF ENGINEERING School of Metallurgy and Materials College of Engineering and Physical Sciences University of Birmingham March 2019 University of Birmingham Research Archive e-theses repository This unpublished thesis/dissertation is copyright of the author and/or third parties. The intellectual property rights of the author or third parties in respect of this work are as defined by The Copyright Designs and Patents Act 1988 or as modified by any successor legislation. Any use made of information contained in this thesis/dissertation must be in accordance with that legislation and must be properly acknowledged. Further distribution or reproduction in any format is prohibited without the permission of the copyright holder. Abstract The objective of this study is the development of a theoretical framework for treating the flow stress response of two-phase alloys as emergent behaviour arising from fundamental dislocation interactions. To this end a field dislocation mechanics (FDM) formulation has been developed to model heterogeneous slip within a computational domain representative of a two-phase nickel- based superalloy crystal at elevated temperature. A transport equation for the statistically stored dislocation (SSD) field is presented and implemented within a plane strain finite element scheme. Elastic interactions between dislocations and the microstructure are explicitly accounted for in this formulation. The theory has been supplemented with constitutive rules for dislocation glide and climb, as well as local cutting conditions for the γ0 particles by the dislocation field. Numerical simulations show that γ0 precipitates reduced the effective dislocation mobility by both acting as discrete slip barriers and providing a drag effect through line tension. The effect of varying microstructural parameters on the crystal deformation behaviour is investigated for simple shear loading boundary conditions. It is demonstrated that slip band propagation can be simulated by the proposed FDM approach. Emergent behaviour is predicted and includes: domain size yield dependence (Hall-Petch relationship), γ0 volume fraction yield dependence (along with more complex γ0 dispersion-related yield and post-yield flow stress phenomena), and hardening related to dislocation source distribution at the grain boundary. From these simulations, scaling laws are derived. Also, the emergence of internal back stresses associated with non-homogeneous plastic deformation is predicted. Prediction of these back stresses, due to sub-grain stress partitioning across elastic/plastic zones, is an important result which can provide useful information for the calibration of phenomenological macroscale models. Validation for the presented model is provided through comparison to experimental micro-shear tests that can be found in published literature. Acknowledgements For the the invaluable support I received during the writing of this thesis I must offer my sincerest thanks to my partner Helen Tatlow. She stood by me through thick and thin, and instilled in me a patience and confidence that was wholly required to bring this work to a conclusion. I dread to think what the last year would have been like without her. Secondly, I offer my humblest respects to my supervisor and good friend, Professor Hector Ba- soalto, who worked tirelessly to assist me throughout my studies. I must thank him particularly for the bleary-eyed evenings, lost Sundays and 1 a.m. phone calls that we both went through as a consequence of this work. If I have seen further it is by standing on the shoulders of Chileans. This project was funded by the Engineering and Physical Sciences Research Council (EPSRC) and the Manufacturing Technology Centre (MTC), through the Manufacturing Technology Engineering Doctorate Centre (MTEDC) in Nottingham - where I spent a happy year completing modules in 2014. From these bodies I would like to particularly acknowledge my industrial supervisor at the MTC, Dr Anas Yaghi, who warmly welcomed me to the manufacturing modelling world and has always been a source of friendly support. Finally I thank my parents for setting me on the path towards academia from the day I was born and never for a moment questioning whether I would succeed. 1 Contents 1 Introduction 6 1.1 Heterogeneity in Micro-scale Plasticity . 6 1.2 Aims of this Work . 7 2 Literature Review 9 2.1 Nickel-based Superalloy Overview . 9 2.1.1 Brief introduction to Nickel-based Superalloys . 9 2.1.2 γ/γ0 microstructures . 10 2.1.3 Characterising deformation . 13 2.1.4 Influence of particle dispersions on emergent properties . 15 2.1.5 Deformation in CMSX-4 type alloys under shear loading . 17 2.2 Dislocations and Plasticity . 19 2.2.1 Dislocations within crystals . 19 2.2.2 Dynamics of dislocations . 21 2.2.3 Dislocation interactions . 22 2.2.4 Continuum dislocation theory . 24 2.3 Computational Material Modelling . 27 2.3.1 Constitutive models . 27 2.3.2 Deformation kinematics - CP . 28 2.3.3 Modelling dislocation dynamics . 30 3 Field Dislocation Mechanics I: General Formulation 35 3.1 Development of General Theory . 35 2 3.1.1 The Burgers Vector . 35 3.1.2 Evolution of Burgers Vector through the Time Derivative . 38 3.1.3 Deformation Kinematics of the Dislocation Field . 41 3.1.4 The Dislocation Flux . 43 3.1.5 Transport Equation of the Dislocation Field . 44 3.2 Elastic interactions from the Dislocation field . 44 4 Field Dislocation Mechanics II: Application to Nickel-based Superalloys 48 4.1 Adapting the Transport Equation to Plane Strain . 48 4.2 Dislocation Velocity Contributions . 52 4.2.1 Separation into principle directions . 52 4.2.2 Glide Velocity . 52 4.2.3 Climb Velocity . 57 4.3 Calculation of the 2D Stress Field . 57 4.3.1 Force on the dislocation field . 59 4.3.2 Line tension force from dislocation pinning . 60 4.4 Shearing of γ0 Precipitates . 65 4.4.1 Background to the precipitate strengthening effect . 65 4.4.2 Approach to modelling particle shear . 66 4.5 Reaction Terms . 70 4.6 Crystal Plasticity Framework for the UMAT subroutine . 71 5 Numerical Implementation 74 5.1 Constitutive Model Framework and Boundary Value Problem . 74 5.2 RVE Finite Element Implementation . 76 5.2.1 RVE model and boundary conditions . 77 5.3 Solving PDEs Numerically with Finite Difference . 80 5.3.1 Discretising the domain and approximating the differentials . 80 5.3.2 Error tolerance and iteration within Fortran . 84 5.3.3 Avoiding density conservation problems . 86 5.4 Environmental Boundary Conditions For FDM . 87 3 5.4.1 Applied strain rate and timesteps . 88 5.4.2 Edge conditions . 89 5.4.3 Microstructure using Obstacle files . 90 5.4.4 Dislocation sources . 91 5.5 Material Parameters . 95 5.5.1 Stiffness of Matrix and Particles . 95 5.5.2 Dislocation transport parameters . 97 6 Full-field Simulations I: Pure Matrix 99 6.1 Penetrable and Impenetrable Grain Boundaries . 100 6.2 Effect of Dislocation Sources on Flow Stress Behaviour . 102 6.2.1 Number of sources . 102 6.2.2 Distribution of sources . 107 6.3 Effect of Jog Spacing . 110 6.4 Effect of Temperature . 114 6.5 Effect of Strain Rate . 114 6.6 Effect of Grain Size . 116 7 Full-field Simulations II: Particle-Matrix 125 7.1 Unimodal Fine Particle Distribution . 126 7.2 Unimodal Large Particle Distribution . 129 7.2.1 Effect of γ0 volume fraction . 129 7.2.2 Effect of Particle Size . 134 7.3 Bimodal Distributions of Particles . 140 7.3.1 Effect of large particle volume fraction . 140 7.3.2 Effect of large particle size . 146 7.3.3 Trimodal dispersions . 149 7.3.4 Influence of temperature . 153 7.4 Benchmarking Model Against Published Experiments . 159 8 Emergent Behaviour I: Backstress Development 165 8.1 Backstresses in γ-only Domains . 167 4 8.1.1 Jog spacing . 167 8.1.2 Source distribution . 172 8.2 Backstresses in γ-γ0 Domains . 179 8.2.1 Volume fraction . 179 8.2.2 Particle radius . 183 9 Emergent Behaviour II: Microstructure-Property relations 188 9.1 Yield Dependence on Microstructure . 188 9.1.1 Varying particle distributions at constant temperature . 188 9.1.2 Varying particle distributions with increasing temperature . 196 9.2 Post-yield Flows Stress Dependence on Microstructure . 198 9.2.1 Opening/closing domain boundaries to dislocation flux . 198 9.2.2 Source distribution . 199 9.2.3 Calculating Hall-Petch relationship constants . 203 9.3 Limitations of Current Model . 204 10 Conclusions and Future Work 206 10.1 Conclusion . 206 10.2 Future work . 210 10.2.1 Considerations for extension to multiple slip systems . 211 5 Chapter 1 Introduction \The general feature exposed by the electron microscopic observations is that work hardened dislocation structures are heterogeneous on a mesoscopic scale. This set two new problems for the theory: to deduce the flow stress.
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