Marc Bernacki is Professor of Physical and Computational Metallurgy at MINES Paris PSL, France. He leads the DIGIMU consortium, which develops new numerical methods for modeling microstructure evolutions in the context of metal forming and their applications.Samuel Forest is Research Director at the CNRS, Professor of Continuum Mechanics at MINES Paris PSL, France, and a member of the French Academy of Sciences. His work focuses on modeling and simulation in the mechanics of materials.

• Preface ixMarc BERNACKI and Samuel FORESTChapter 1 Dislocation-based Mechanics: The Various Contributions of Dislocation Dynamics Simulations 1Sylvain QUEYREAU1.1. Introduction 11.2. Overview of discrete dislocation dynamics 21.2.1. Initial configurations and periodic boundary conditions 41.2.2. Mobility functions 51.2.3. Forces on dislocations 71.2.4. Topological changes 111.2.5. Boundary conditions 121.3. Mesoscale plasticity 151.3.1. Forest interactions 171.3.2. Statistical investigations of dislocation mechanisms 231.3.3. Lattice friction 291.3.4. Combination of multiple strengthening mechanisms 321.3.5. Toward polycrystalline plasticity 331.3.6. Cyclic deformations 401.4. Conclusion and future work 431.5. Acknowledgments 441.6. References 44Chapter 2 Statistical Approach to the Representative Volume Element Size of Random Composites 55Dominique JEULIN and Samuel FOREST2.1. Introduction 552.2. Elements of numerical homogenization of heterogeneous media 562.2.1. Examples of physical properties of heterogeneous media 562.2.2. Change of scale in heterogeneous media 572.2.3. Principle of calculation of apparent properties 582.2.4. Homogenization of random media by numerical simulations 632.3. Statistical definition of the RVE 642.3.1. Variance of estimation and integral range 652.3.2. Definition of the statistical RVE of apparent properties 672.3.3. Practical determination of the size of the RVE 692.3.4. Scaling of the bias, or systematic error 702.3.5. Determination of the size of the RVE in the nonlinear case 722.3.6. Remarks on the reduction of variance 732.4. Examples of application 732.4.1. Linear elastic properties and thermal conductivity 742.4.2. Elastic and viscoelastic properties of nanocomposites 892.4.3. Optical properties of nanocomposites 902.4.4. Fluid flow and permeability 942.4.5. Acoustic properties and wave propagation in fibrous media 952.4.6. Nonlinear properties 972.5. Conclusion 1042.6. References 105Chapter 3 Analytical Micromechanical Methods for Elasto-Viscoplastic Composites and Polycrystals 113Stéphane BERBENNI and Samuel FOREST3.1. Introduction 1133.2. Translated field method 1163.2.1. Field equations and integral equation 1163.2.2. TF-based method: theoretical foundations 1183.2.3. TF method for elasto-viscoplastic two-phase composite materials 1193.2.4. TF method for elasto-viscoplastic polycrystals using the SC scheme 1253.3. The β-model dynasty for multiphase elastoviscoplastic polycrystals 1273.4. Applications of the analytical micromechanical methods 1313.4.1. Applications of the TF method to composites and polycrystals 1313.4.2. Applications of β-models to elastoviscoplastic polycrystals at finite deformations 1413.5. Concluding remarks 1473.6. References 148Chapter 4 Vertex and Front-Tracking Methods for the Modeling of Microstructure Evolution at the Solid State 155Marc BERNACKI4.1. Introduction 1554.2. Vertex frameworks 1564.3. From Vertex to front-tracking then to enriched Vertex frameworks 1604.4. Enrichment of the Vertex approach 1634.5. Other front-tracking frameworks for the modeling of microstructure evolution 1694.6. Conclusion 1734.7. References 173Chapter 5 Phase Field: Theory, Numerical Implementation and Applications 177Ingo STEINBACH and Oleg SHCHYGLO5.1. Introduction 1775.2. The soliton solution of a propagating wave front 1795.3. Thermodynamically consistent derivation of the phase-field equations 1835.4. The multi-phase fields approach 1845.5. Multi-component alloy transformation 1875.6. Multi-phase field with elasticity 1915.7. Numerical treatment and applications 1925.8. Examples 1935.8.1. Martensitic transformation 1935.8.2. Recrystallization and coarsening 1945.8.3. Multicomponent solidification 1985.9. References 199Chapter 6 Level-Set Method for the Modeling of Microstructure Evolution 203Marc BERNACKI6.1. Introduction 2036.2. Kinetics equations of microstructure evolution at the mesoscopic scale 2056.3. Level-set function, description of polycrystal and meshing adaptation 2086.4. Isotropic framework for LS modeling of ReX and GG 2136.5. Anisotropy of GB properties and CDRX modeling 2186.5.1. LS formulations in context of anisotropic GB properties 2186.5.2. CDRX modeling 2256.6. Static/evolutive SPPs 2286.7. Modeling of diffusive solid-state phase transformation 2356.8. Solute drag aspect 2436.9. Conclusion 2466.10. References 248Chapter 7 Resolution Methods for Digital Materials – Recent Developments of Cellular Automaton Method 261Lukasz MADEJ and Mateusz SITKO7.1. Introduction 2617.2. Cellular automaton and applications 2627.3. CA front tracking model within the static recrystallization case study 2687.4. Calculation times and perspectives 2817.5. References 284List of Authors 291Index 293

Marc Bernacki is Professor of Physical and Computational Metallurgy at MINES Paris PSL, France. He leads the DIGIMU consortium, which develops new numerical methods for modeling microstructure evolutions in the context of metal forming and their applications.Samuel Forest is Research Director at the CNRS, Professor of Continuum Mechanics at MINES Paris PSL, France, and a member of the French Academy of Sciences. His work focuses on modeling and simulation in the mechanics of materials.

• Preface ixMarc BERNACKI and Samuel FORESTChapter 1 Dislocation-based Mechanics: The Various Contributions of Dislocation Dynamics Simulations 1Sylvain QUEYREAU1.1. Introduction 11.2. Overview of discrete dislocation dynamics 21.2.1. Initial configurations and periodic boundary conditions 41.2.2. Mobility functions 51.2.3. Forces on dislocations 71.2.4. Topological changes 111.2.5. Boundary conditions 121.3. Mesoscale plasticity 151.3.1. Forest interactions 171.3.2. Statistical investigations of dislocation mechanisms 231.3.3. Lattice friction 291.3.4. Combination of multiple strengthening mechanisms 321.3.5. Toward polycrystalline plasticity 331.3.6. Cyclic deformations 401.4. Conclusion and future work 431.5. Acknowledgments 441.6. References 44Chapter 2 Statistical Approach to the Representative Volume Element Size of Random Composites 55Dominique JEULIN and Samuel FOREST2.1. Introduction 552.2. Elements of numerical homogenization of heterogeneous media 562.2.1. Examples of physical properties of heterogeneous media 562.2.2. Change of scale in heterogeneous media 572.2.3. Principle of calculation of apparent properties 582.2.4. Homogenization of random media by numerical simulations 632.3. Statistical definition of the RVE 642.3.1. Variance of estimation and integral range 652.3.2. Definition of the statistical RVE of apparent properties 672.3.3. Practical determination of the size of the RVE 692.3.4. Scaling of the bias, or systematic error 702.3.5. Determination of the size of the RVE in the nonlinear case 722.3.6. Remarks on the reduction of variance 732.4. Examples of application 732.4.1. Linear elastic properties and thermal conductivity 742.4.2. Elastic and viscoelastic properties of nanocomposites 892.4.3. Optical properties of nanocomposites 902.4.4. Fluid flow and permeability 942.4.5. Acoustic properties and wave propagation in fibrous media 952.4.6. Nonlinear properties 972.5. Conclusion 1042.6. References 105Chapter 3 Analytical Micromechanical Methods for Elasto-Viscoplastic Composites and Polycrystals 113Stéphane BERBENNI and Samuel FOREST3.1. Introduction 1133.2. Translated field method 1163.2.1. Field equations and integral equation 1163.2.2. TF-based method: theoretical foundations 1183.2.3. TF method for elasto-viscoplastic two-phase composite materials 1193.2.4. TF method for elasto-viscoplastic polycrystals using the SC scheme 1253.3. The β-model dynasty for multiphase elastoviscoplastic polycrystals 1273.4. Applications of the analytical micromechanical methods 1313.4.1. Applications of the TF method to composites and polycrystals 1313.4.2. Applications of β-models to elastoviscoplastic polycrystals at finite deformations 1413.5. Concluding remarks 1473.6. References 148Chapter 4 Vertex and Front-Tracking Methods for the Modeling of Microstructure Evolution at the Solid State 155Marc BERNACKI4.1. Introduction 1554.2. Vertex frameworks 1564.3. From Vertex to front-tracking then to enriched Vertex frameworks 1604.4. Enrichment of the Vertex approach 1634.5. Other front-tracking frameworks for the modeling of microstructure evolution 1694.6. Conclusion 1734.7. References 173Chapter 5 Phase Field: Theory, Numerical Implementation and Applications 177Ingo STEINBACH and Oleg SHCHYGLO5.1. Introduction 1775.2. The soliton solution of a propagating wave front 1795.3. Thermodynamically consistent derivation of the phase-field equations 1835.4. The multi-phase fields approach 1845.5. Multi-component alloy transformation 1875.6. Multi-phase field with elasticity 1915.7. Numerical treatment and applications 1925.8. Examples 1935.8.1. Martensitic transformation 1935.8.2. Recrystallization and coarsening 1945.8.3. Multicomponent solidification 1985.9. References 199Chapter 6 Level-Set Method for the Modeling of Microstructure Evolution 203Marc BERNACKI6.1. Introduction 2036.2. Kinetics equations of microstructure evolution at the mesoscopic scale 2056.3. Level-set function, description of polycrystal and meshing adaptation 2086.4. Isotropic framework for LS modeling of ReX and GG 2136.5. Anisotropy of GB properties and CDRX modeling 2186.5.1. LS formulations in context of anisotropic GB properties 2186.5.2. CDRX modeling 2256.6. Static/evolutive SPPs 2286.7. Modeling of diffusive solid-state phase transformation 2356.8. Solute drag aspect 2436.9. Conclusion 2466.10. References 248Chapter 7 Resolution Methods for Digital Materials – Recent Developments of Cellular Automaton Method 261Lukasz MADEJ and Mateusz SITKO7.1. Introduction 2617.2. Cellular automaton and applications 2627.3. CA front tracking model within the static recrystallization case study 2687.4. Calculation times and perspectives 2817.5. References 284List of Authors 291Index 293

Marc Bernacki is Professor of Physical and Computational Metallurgy at MINES Paris PSL, France. He leads the DIGIMU consortium, which develops new numerical methods for modeling microstructure evolutions in the context of metal forming and their applications.Samuel Forest is Research Director at the CNRS, Professor of Continuum Mechanics at MINES Paris PSL, France, and a member of the French Academy of Sciences. His work focuses on modeling and simulation in the mechanics of materials.

• Preface ixMarc BERNACKI and Samuel FORESTChapter 1 Dislocation-based Mechanics: The Various Contributions of Dislocation Dynamics Simulations 1Sylvain QUEYREAU1.1. Introduction 11.2. Overview of discrete dislocation dynamics 21.2.1. Initial configurations and periodic boundary conditions 41.2.2. Mobility functions 51.2.3. Forces on dislocations 71.2.4. Topological changes 111.2.5. Boundary conditions 121.3. Mesoscale plasticity 151.3.1. Forest interactions 171.3.2. Statistical investigations of dislocation mechanisms 231.3.3. Lattice friction 291.3.4. Combination of multiple strengthening mechanisms 321.3.5. Toward polycrystalline plasticity 331.3.6. Cyclic deformations 401.4. Conclusion and future work 431.5. Acknowledgments 441.6. References 44Chapter 2 Statistical Approach to the Representative Volume Element Size of Random Composites 55Dominique JEULIN and Samuel FOREST2.1. Introduction 552.2. Elements of numerical homogenization of heterogeneous media 562.2.1. Examples of physical properties of heterogeneous media 562.2.2. Change of scale in heterogeneous media 572.2.3. Principle of calculation of apparent properties 582.2.4. Homogenization of random media by numerical simulations 632.3. Statistical definition of the RVE 642.3.1. Variance of estimation and integral range 652.3.2. Definition of the statistical RVE of apparent properties 672.3.3. Practical determination of the size of the RVE 692.3.4. Scaling of the bias, or systematic error 702.3.5. Determination of the size of the RVE in the nonlinear case 722.3.6. Remarks on the reduction of variance 732.4. Examples of application 732.4.1. Linear elastic properties and thermal conductivity 742.4.2. Elastic and viscoelastic properties of nanocomposites 892.4.3. Optical properties of nanocomposites 902.4.4. Fluid flow and permeability 942.4.5. Acoustic properties and wave propagation in fibrous media 952.4.6. Nonlinear properties 972.5. Conclusion 1042.6. References 105Chapter 3 Analytical Micromechanical Methods for Elasto-Viscoplastic Composites and Polycrystals 113Stéphane BERBENNI and Samuel FOREST3.1. Introduction 1133.2. Translated field method 1163.2.1. Field equations and integral equation 1163.2.2. TF-based method: theoretical foundations 1183.2.3. TF method for elasto-viscoplastic two-phase composite materials 1193.2.4. TF method for elasto-viscoplastic polycrystals using the SC scheme 1253.3. The β-model dynasty for multiphase elastoviscoplastic polycrystals 1273.4. Applications of the analytical micromechanical methods 1313.4.1. Applications of the TF method to composites and polycrystals 1313.4.2. Applications of β-models to elastoviscoplastic polycrystals at finite deformations 1413.5. Concluding remarks 1473.6. References 148Chapter 4 Vertex and Front-Tracking Methods for the Modeling of Microstructure Evolution at the Solid State 155Marc BERNACKI4.1. Introduction 1554.2. Vertex frameworks 1564.3. From Vertex to front-tracking then to enriched Vertex frameworks 1604.4. Enrichment of the Vertex approach 1634.5. Other front-tracking frameworks for the modeling of microstructure evolution 1694.6. Conclusion 1734.7. References 173Chapter 5 Phase Field: Theory, Numerical Implementation and Applications 177Ingo STEINBACH and Oleg SHCHYGLO5.1. Introduction 1775.2. The soliton solution of a propagating wave front 1795.3. Thermodynamically consistent derivation of the phase-field equations 1835.4. The multi-phase fields approach 1845.5. Multi-component alloy transformation 1875.6. Multi-phase field with elasticity 1915.7. Numerical treatment and applications 1925.8. Examples 1935.8.1. Martensitic transformation 1935.8.2. Recrystallization and coarsening 1945.8.3. Multicomponent solidification 1985.9. References 199Chapter 6 Level-Set Method for the Modeling of Microstructure Evolution 203Marc BERNACKI6.1. Introduction 2036.2. Kinetics equations of microstructure evolution at the mesoscopic scale 2056.3. Level-set function, description of polycrystal and meshing adaptation 2086.4. Isotropic framework for LS modeling of ReX and GG 2136.5. Anisotropy of GB properties and CDRX modeling 2186.5.1. LS formulations in context of anisotropic GB properties 2186.5.2. CDRX modeling 2256.6. Static/evolutive SPPs 2286.7. Modeling of diffusive solid-state phase transformation 2356.8. Solute drag aspect 2436.9. Conclusion 2466.10. References 248Chapter 7 Resolution Methods for Digital Materials – Recent Developments of Cellular Automaton Method 261Lukasz MADEJ and Mateusz SITKO7.1. Introduction 2617.2. Cellular automaton and applications 2627.3. CA front tracking model within the static recrystallization case study 2687.4. Calculation times and perspectives 2817.5. References 284List of Authors 291Index 293