Computational Materials Engineering is an advanced introduction to the computer-aided modeling of essential material properties and behavior, including the physical, thermal and chemical parameters, as well as the mathematical tools used to perform simulations. Its emphasis will be on crystalline materials, which includes all metals. The basis of Computational Materials Engineering allows scientists and engineers to create virtual simulations of material behavior and properties, to better understand how a particular material works and performs and then use that knowledge to design improvements for particular material applications. The text displays knowledge of software designers, materials scientists and engineers, and those involved in materials applications like mechanical engineers, civil engineers, electrical engineers, and chemical engineers.
Readers from students to practicing engineers to materials research scientists will find in this book a single source of the major elements that make up contemporary computer modeling of materials characteristics and behavior. The reader will gain an understanding of the underlying statistical and analytical tools that are the basis for modeling complex material interactions, including an understanding of computational thermodynamics and molecular kinetics; as well as various modeling systems. Finally, the book will offer the reader a variety of algorithms to use in solving typical modeling problems so that the theory presented herein can be put to real-world use.
- Balanced coverage of fundamentals of materials modeling, as well as more advanced aspects of modeling, such as modeling at all scales from the atomic to the molecular to the macro-material
- Concise, yet rigorous mathematical coverage of such analytical tools as the Potts type Monte Carlo method, cellular automata, phase field, dislocation dynamics and Finite Element Analysis in statistical and analytical modeling
Graduate Engineering Students in computational materials modeling and related courses in materials and materials processing simulation; Graduate Students in related science disciplines who may wish to take a materials course elective, including students in chemistry, physics and the life sciences; Professional Engineers and Scientists working in materials research, including all areas of materials science and engineering, generally, as well as chemical engineering, metallurgy, and biomaterials
1 Introduction 1.1 Microstructures Defined 1.2 Microstructure Evolution 1.3 Why simulate Microstructure evolution? 1.4 Further Reading 1.4.1 On Microstructures and their evolution from a noncomputational point of view 1.4.2 On what is not treated in this book 2 Basic Thermodynamics 2.1 Reversible and Irreversible Thermodynamics 2.1.1 The first law of thermodynamics 2.1.2 The Gibbs energy 2.1.3 Molar quantities and the chemical potential 2.1.4 Entropy production and the Second Law of Thermodynamics 2.1.5 Driving force for internal processes 2.1.6 Conditions for thermodynamic equilibrium 2.2 Solution thermodynamics 2.2.1 Entropy of mixing 2.2.2 The ideal solution 2.2.3 Regular solutions 2.2.4 General solutions in multi-phase equilibrium 2.2.5 The dilute solution limit – Henry’s and Raoult’s law 2.2.6 The chemical driving force 2.2.7 Influence of curvature and pressure 2.2.8 General solutions and the CALPHAD formalism 2.2.9 Practical evaluation of multi-component thermodynamic equilibrium 5 6 CONTENTS 3 Monte Carlo Potts Model 3.1 Introduction 3.2 Two state Potts model (Ising model) 3.2.1 Hamiltonians 3.2.2 Dynamics (Probability Transition Functions) 3.2.3 Lattice Type 3.2.4 Boundary Conditions 3.2.5 The Vanilla Algorithm 3.2.6 Motion by Curvature 3.2.7 The Dynamics of Kinks and Ledges 3.2.8 Temperature 3.2.9 Boundary Anisotropy 3.2.10 Summary 3.3 Q-State Potts Model 3.3.1 Uniform Energies and Mobilities 3.3.2 Self-Ordering Behaviour 3.3.3 Boundary Energy 3.3.4 Boundary Mobility 3.3.5 Pinning Systems 3.3.6 Stored Energy 3.3.7 Summary 3.4 Speed-Up Algorithms 3.4.1 The Boundary-Site Algorithm 3.4.2 The N-Fold Way Algorithm 3.4.3 Parallel Algorithm 3.4.4 Summary 3.5 Applications of the Potts Model 3.5.1 Grain Growth 3.5.2 Incorporating Realistic Textures and Misorientation Distributions 3.5.3 Incorporating Realistic Energies and Mobilities 3.5.4 Validating the Energy and Mobility Implementations 3.5.5 Anisotropic Grain Growth 3.5.6 Abnormal Grain Growth 3.5.7 Recrystallisation 3.5.8 Zener Pinning 3.6 Summary 3.7 Final Remarks 3.8 Acknowledgements CONTENTS 7 4 Cellular Automata 4.1 A Definition 4.2 A One Dimensional Introduction 4.2.1 One Dimensional Recrystallization 4.2.2 Before moving to higher dimensions 4.3 +2D CA Modeling of recrystallization 4.3.1 CA-Neighborhood Definitions in 2D 4.3.2 The Interface Discretization Problem 4.4 +2D CA Modeling of Grain Growth 4.4.1 Approximating Curvature in a Cellular Automaton Grid 4.5 Mathematics of Cellular Automata 4.6 Irregular and Shapeless Cellular Automata 4.6.1 Irregular Shapeless Cellular Automata for Grain Growth 4.6.2 In the Presence of Additional Driving Forces 4.7 Hybrid Cellular Automata Modeling 4.7.1 Principle 4.7.2 Case example 4.8 Lattice Gas Cellular Automata 4.8.1 Principle – Boolean lgca 4.8.2 Boolean lgca – Example of Application 4.9 Network Cellular Automata 4.9.1 Combined Network Cellular Automata 4.9.2 CNCA for Microstructure Evolution Modeling 4.10 Further Reading 5 Solid-state diffusion 5.1 Diffusion mechanisms in crystalline solids 5.2 Microscopic diffusion 5.2.1 The principle of time reversal 5.2.2 A random walk treatment 5.2.3 Einstein’s equation 5.3 Macroscopic diffusion 5.3.1 Phenomenological laws of diffusion 5.3.2 Solutions to Fick’s second law 5.3.3 Diffusion forces and atomic mobility 5.3.4 Interdiffusion and the Kirkendall effect 5.3.5 Multi-component diffusion 5.4 Numerical solution of the diffusion equation 8 CONTENTS 6 Modelling precipitation 6.1 Statistical theory of phase transformation 6.1.1 The extended volume approach - KJMA kinetics 6.2 Solid-state Nucleation 6.2.1 Introduction 6.2.2 Macroscopic treatment of nucleation - CNT 6.2.3 Transient nucleation 6.2.4 Multi-component nucleation 6.2.5 Treatment of interfacial energies 6.3 Diffusion-controlled precipitate growth 6.3.1 Problem definition 6.3.2 Zener’s approach for planar interfaces 6.3.3 Quasi-static approach for spherical precipitates 6.3.4 Moving boundary solution for spherical symmetry 6.4 Multi-particle precipitation kinetics 6.4.1 The numerical Kampmann-Wagner model 6.4.2 The SFFK model - A mean-field approach for complex systems 6.5 Comparing the growth kinetics of different models 7 Phase-field Modelling 7.1 A Short Overview 7.2 Phase-field Model for Pure Substances 7.2.1 Anisotropy formulation 7.2.2 Material and Model Parameters 7.2.3 Application to Dendritic Growth 7.3 Case Study 7.3.1 Phase-field Equation 7.3.2 Finite Difference Discretization 7.3.3 Boundary Values 7.3.4 Stability Condition 7.3.5 Structure of the Code 7.3.6 Main computation 7.3.7 Parameter file 7.3.8 MatLab visualization 7.3.9 Examples 7.4 Model for Multiple Components and Phases 7.4.1 Model formulation 7.4.2 Entropy Density Contributions 7.4.3 Evolution Equations CONTENTS 9 7.4.4 Non-dimensionalization 7.4.5 FD discretization and staggered grid 7.4.6 Optimization of the Computational Algorithm 7.4.7 Parallelization 7.4.8 Adaptive Finite Element Method 7.4.9 Simulations of phase transitions and microstructure evolution 7.5 Acknowledgements 8 Discrete Dislocations 8.1 Basics of Discrete Plasticity Models 8.2 Linear Elasticity Theory for Plasticity 8.2.1 Introduction 8.2.2 Fundamentals of Elasticity Theory 8.2.3 Equilibrium Equations 8.2.4 Compatibility Equations 8.2.5 Hooke’s Law—the Linear Relationship between Stress and Strain 8.2.6 Elastic Energy 8.2.7 Green’s Tensor Function in Elasticity Theory 8.2.8 The Airy Stress Function in Elasticity Theory 8.3 Dislocation Statics 8.3.1 Introduction 8.3.2 2D Field Equations for Infinite Dislocations in an Isotropic Linear Elastic Medium 8.3.3 2D Field Equations for Infinite Dislocations in an Anisotropic Linear Elastic Medium 8.3.4 3D Field Equations for Dislocation Segments in an Isotropic Linear Elastic Medium 8.3.5 3D Field Equations for Dislocation Segments in an Anisotropic Linear Elastic Medium 8.4 Dislocation Dynamics 8.4.1 Introduction 8.4.2 Newtonian Dislocation Dynamics 8.4.3 Viscous and Viscoplastic Dislocation Dynamics 8.5 Kinematics of Discrete Dislocation Dynamics 8.6 Dislocation Reactions and Annihilation 10 CONTENTS 9 FEM 9.1 Fundamentals of Differential Equations 9.1.1 Introduction to Differential Equations 9.1.2 Solution of Partial Differential Equations 9.2 Introduction to the finite element method 9.3 Finite Element Methods at the Meso–Macroscale 9.3.1 Introduction and Fundamentals 9.3.2 The Equilibrium Equation in FE Simulations 9.3.3 Finite Elements and Shape Functions 9.3.4 Assemblage of the Stiffness Matrix 9.3.5 Solid-State Kinematics for Mechanical Problems 9.3.6 Conjugate Stress–Strain Measures
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- © Academic Press 2007
- 3rd August 2007
- Academic Press
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Scientist, Paul Scherrer Institute, Villigen PSI, Switzerland
Director and Executive, Max-Planck-Institut für Eisenforschung GmbH