Jean Salençon is a Member of the French Academy of Sciences, the French Academy of Technologies, and Academia Europaea. He is also an Honorary/Foreign Member of the Academies of Hungary, Milan and Lisbon. He is also Senior Fellow of the Hong Kong Institute for Advanced Study (HKIAS). His research interests include continuum mechanics, structural analysis and soil mechanics.

• Preface xiNotations xiiiIntroduction xxvChapter 1. Elastic Domains: Yield Conditions 11.1. Introductory remarks 11.2. An overview of the model 41.2.1. The infinitesimal transformation framework 41.2.2. Time variable 41.3. One-dimensional approach 51.3.1. Uniaxial tension test 51.3.2. Uniaxial tension-compression test 81.3.3. The Bauschinger effect 91.3.4. Other one-dimensional experiments 101.4. Multidimensional approach 111.4.1. A multidimensional experiment 111.4.2. Initial elastic domain 121.4.3. Work-hardening 131.4.4. Perfectly plastic material 141.4.5. Bui’s experimental results 141.5. Yield conditions 161.5.1. Initial yield condition and yield function 161.5.2. Loading function and work-hardening 181.5.3. Simple work-hardening models 191.6. Yield criteria and loading functions 221.6.1. Convexity 221.6.2. Isotropy 231.6.3. The Tresca yield criterion 241.6.4. The von Mises yield criterion 271.6.5. Other yield criteria for metals 301.6.6. Yield criteria for anisotropic materials 311.6.7. Yield criteria for granular materials 351.7. Final comments 41Chapter 2. The Plastic Flow Rule 432.1. One-dimensional approach 432.1.1. Work-hardening material 432.1.2. Perfectly plastic material 452.2. Multidimensional approach for a work-hardening material 462.2.1. Loading and unloading processes 462.2.2. General properties of the plastic flow rule 492.2.3. Plastic potential: associated plasticity 512.2.4. Principle of maximum plastic work 542.2.5. Validation of the principle of maximum plastic work 552.2.6. Piecewise continuously differentiable loading functions 572.3. Multidimensional approach for a perfectly plastic material 592.3.1. Loading and unloading processes 592.3.2. Application of the principle of maximum plastic work 612.3.3. Drucker’s postulate 622.4. Plastic dissipation 642.4.1. Plastic dissipation per unit volume 642.4.2. Plastic dissipation and support function of the elastic domain 642.4.3. Plastic velocity jumps in the case of perfectly plastic materials 652.5. Generalized standard materials 662.6. Mises’, Tresca’s and Coulomb’s perfectly plastic standard materials 692.6.1. Mises’ perfectly plastic standard material 692.6.2. Tresca’s perfectly plastic standard material 712.6.3. Coulomb’s perfectly plastic standard material 722.6.4. About edge and vertex regimes 74Chapter 3. Elastoplastic Modeling in Generalized Variables 773.1. About generalized variables 773.2. Elastic domains 783.2.1. Initial elastic domain 783.2.2. Work-hardening and perfect plasticity 793.3. The anelastic flow rule 803.3.1. Anelasticity or plasticity? 803.3.2. Principle of maximum work 813.3.3. The work-hardening anelastic flow rule 823.3.4. The “perfectly plastic” anelastic flow rule 833.3.5. Anelastic dissipation 843.4. Generalized continua 843.4.1. Curvilinear generalized continuum 843.4.2. Planar generalized continuum 90Chapter 4. Quasi-static Elastoplastic Processes 1014.1. Quasi-static loading processes 1014.1.1. Mechanical evolution within the SPH framework 1014.1.2. Quasi-static loading process within the SPH framework 1044.1.3. Statically admissible and kinematically admissible fields 1044.1.4. Parametric problems 1054.2. Quasi-static elastoplastic loading processes 1084.2.1. Problematics 1084.2.2. Existence and uniqueness theorems 1114.2.3. Uniqueness theorems for stress rates and strain rates 1154.3. Response of a system made from an elastic and standard perfectly plastic material 1164.3.1. Initial elastic domain of the system 1164.3.2. Existence of the solution to the elastoplastic evolution problem 1184.3.3. Solution to the elastoplastic evolution problem 1194.3.4. Limit loads for the system 1204.3.5. Linear elastic response of the system 1214.3.6. Anelastic response of the system 1224.3.7. Taking geometry changes into account 1364.4. Response of a system made from a standard work-hardening elastoplastic material 1394.4.1. Initial elastic domain of the system 1394.4.2. Residual stress rates, residual strain rates 1404.4.3. Maximum work theorem 1404.4.4. Summing up... 141Chapter 5. Quasi-static Elastoplastic Processes: Minimum Principles 1435.1. Elastic and standard perfectly plastic constituent material 1435.1.1. Minimum principle for the stress rate field 1435.1.2. Minimum principle for the velocity field 1465.1.3. Other expressions of the minimum principles 1525.2. Elastic and standard positive work-hardening constituent material 1535.2.1. Revisiting the constitutive equation 1535.2.2. Minimum principle for the stress rate field 1555.2.3. Minimum principle for the velocity field 1565.2.4. Other expressions of the minimum principles 1585.2.5. Historical comments 1585.3. Minimum principles for the stress and strain fields 1595.3.1. Colonnetti’s theorem 1595.3.2. Other expressions of Colonnetti’s minimum principles 160Chapter 6. Limit Loads: Limit Analysis 1616.1. Limit loads and yield design (1) 1616.2. Static approach, first plastic collapse theorem 1636.2.1. Safe loads, interior approach 1636.2.2. Lower bound theorem 1646.3. Kinematic approach, second plastic collapse theorem 1656.3.1. Plastically admissible velocity fields 1656.3.2. Kinematic necessary condition to be satisfied by safe loads 1676.3.3. Exterior approach, upper bound theorem 1696.4. Combining static and kinematic approaches 1706.4.1. Determination of a limit load 1706.4.2. Association theorem 1726.4.3. Duality 1736.5. Limit analysis and the rigid, perfectly plastic material concept 1736.5.1. Rigid and standard perfectly plastic model 1736.5.2. The connection with limit loads 1746.6. Limit loads and yield design (2) 1766.6.1. Fundamentals of the yield design theory 1766.6.2. Resistance of the constituent material 1776.6.3. Potentially safe loads, interior approach and lower bound theorem 1786.6.4. Maximum resisting rate of work, exterior approach and upper bound theorem 1786.6.5. Matching limit load and yield design theories 1816.7. Two-dimensional limit analysis 1826.7.1. Plane strain limit analysis problems 1826.7.2. Partial static solutions to plane strain limit analysis problems 1836.7.3. Complete static solutions to plane strain limit analysis problems 1836.7.4. Complete kinematic solutions to plane strain limit analysis problems 1846.7.5. “Incomplete” solutions to plane strain limit analysis problems 1886.7.6. Complete solutions to plane strain limit analysis problems 1896.7.7. Comments 1926.7.8. Plane stress limit analysis 1946.7.9. Axially symmetric problems 1966.8. Implementation 2006.8.1. Analytical solutions 2006.8.2. Analytical/numerical solutions 2026.8.3. Numerical solutions 2036.8.4. The example of a tantalizing problem 2046.8.5. Final comments 208References 211Index 231

Jean Salençon is a Member of the French Academy of Sciences, the French Academy of Technologies, and Academia Europaea. He is also an Honorary/Foreign Member of the Academies of Hungary, Milan and Lisbon. He is also Senior Fellow of the Hong Kong Institute for Advanced Study (HKIAS). His research interests include continuum mechanics, structural analysis and soil mechanics.

• Preface xiNotations xiiiIntroduction xxvChapter 1. Elastic Domains: Yield Conditions 11.1. Introductory remarks 11.2. An overview of the model 41.2.1. The infinitesimal transformation framework 41.2.2. Time variable 41.3. One-dimensional approach 51.3.1. Uniaxial tension test 51.3.2. Uniaxial tension-compression test 81.3.3. The Bauschinger effect 91.3.4. Other one-dimensional experiments 101.4. Multidimensional approach 111.4.1. A multidimensional experiment 111.4.2. Initial elastic domain 121.4.3. Work-hardening 131.4.4. Perfectly plastic material 141.4.5. Bui’s experimental results 141.5. Yield conditions 161.5.1. Initial yield condition and yield function 161.5.2. Loading function and work-hardening 181.5.3. Simple work-hardening models 191.6. Yield criteria and loading functions 221.6.1. Convexity 221.6.2. Isotropy 231.6.3. The Tresca yield criterion 241.6.4. The von Mises yield criterion 271.6.5. Other yield criteria for metals 301.6.6. Yield criteria for anisotropic materials 311.6.7. Yield criteria for granular materials 351.7. Final comments 41Chapter 2. The Plastic Flow Rule 432.1. One-dimensional approach 432.1.1. Work-hardening material 432.1.2. Perfectly plastic material 452.2. Multidimensional approach for a work-hardening material 462.2.1. Loading and unloading processes 462.2.2. General properties of the plastic flow rule 492.2.3. Plastic potential: associated plasticity 512.2.4. Principle of maximum plastic work 542.2.5. Validation of the principle of maximum plastic work 552.2.6. Piecewise continuously differentiable loading functions 572.3. Multidimensional approach for a perfectly plastic material 592.3.1. Loading and unloading processes 592.3.2. Application of the principle of maximum plastic work 612.3.3. Drucker’s postulate 622.4. Plastic dissipation 642.4.1. Plastic dissipation per unit volume 642.4.2. Plastic dissipation and support function of the elastic domain 642.4.3. Plastic velocity jumps in the case of perfectly plastic materials 652.5. Generalized standard materials 662.6. Mises’, Tresca’s and Coulomb’s perfectly plastic standard materials 692.6.1. Mises’ perfectly plastic standard material 692.6.2. Tresca’s perfectly plastic standard material 712.6.3. Coulomb’s perfectly plastic standard material 722.6.4. About edge and vertex regimes 74Chapter 3. Elastoplastic Modeling in Generalized Variables 773.1. About generalized variables 773.2. Elastic domains 783.2.1. Initial elastic domain 783.2.2. Work-hardening and perfect plasticity 793.3. The anelastic flow rule 803.3.1. Anelasticity or plasticity? 803.3.2. Principle of maximum work 813.3.3. The work-hardening anelastic flow rule 823.3.4. The “perfectly plastic” anelastic flow rule 833.3.5. Anelastic dissipation 843.4. Generalized continua 843.4.1. Curvilinear generalized continuum 843.4.2. Planar generalized continuum 90Chapter 4. Quasi-static Elastoplastic Processes 1014.1. Quasi-static loading processes 1014.1.1. Mechanical evolution within the SPH framework 1014.1.2. Quasi-static loading process within the SPH framework 1044.1.3. Statically admissible and kinematically admissible fields 1044.1.4. Parametric problems 1054.2. Quasi-static elastoplastic loading processes 1084.2.1. Problematics 1084.2.2. Existence and uniqueness theorems 1114.2.3. Uniqueness theorems for stress rates and strain rates 1154.3. Response of a system made from an elastic and standard perfectly plastic material 1164.3.1. Initial elastic domain of the system 1164.3.2. Existence of the solution to the elastoplastic evolution problem 1184.3.3. Solution to the elastoplastic evolution problem 1194.3.4. Limit loads for the system 1204.3.5. Linear elastic response of the system 1214.3.6. Anelastic response of the system 1224.3.7. Taking geometry changes into account 1364.4. Response of a system made from a standard work-hardening elastoplastic material 1394.4.1. Initial elastic domain of the system 1394.4.2. Residual stress rates, residual strain rates 1404.4.3. Maximum work theorem 1404.4.4. Summing up... 141Chapter 5. Quasi-static Elastoplastic Processes: Minimum Principles 1435.1. Elastic and standard perfectly plastic constituent material 1435.1.1. Minimum principle for the stress rate field 1435.1.2. Minimum principle for the velocity field 1465.1.3. Other expressions of the minimum principles 1525.2. Elastic and standard positive work-hardening constituent material 1535.2.1. Revisiting the constitutive equation 1535.2.2. Minimum principle for the stress rate field 1555.2.3. Minimum principle for the velocity field 1565.2.4. Other expressions of the minimum principles 1585.2.5. Historical comments 1585.3. Minimum principles for the stress and strain fields 1595.3.1. Colonnetti’s theorem 1595.3.2. Other expressions of Colonnetti’s minimum principles 160Chapter 6. Limit Loads: Limit Analysis 1616.1. Limit loads and yield design (1) 1616.2. Static approach, first plastic collapse theorem 1636.2.1. Safe loads, interior approach 1636.2.2. Lower bound theorem 1646.3. Kinematic approach, second plastic collapse theorem 1656.3.1. Plastically admissible velocity fields 1656.3.2. Kinematic necessary condition to be satisfied by safe loads 1676.3.3. Exterior approach, upper bound theorem 1696.4. Combining static and kinematic approaches 1706.4.1. Determination of a limit load 1706.4.2. Association theorem 1726.4.3. Duality 1736.5. Limit analysis and the rigid, perfectly plastic material concept 1736.5.1. Rigid and standard perfectly plastic model 1736.5.2. The connection with limit loads 1746.6. Limit loads and yield design (2) 1766.6.1. Fundamentals of the yield design theory 1766.6.2. Resistance of the constituent material 1776.6.3. Potentially safe loads, interior approach and lower bound theorem 1786.6.4. Maximum resisting rate of work, exterior approach and upper bound theorem 1786.6.5. Matching limit load and yield design theories 1816.7. Two-dimensional limit analysis 1826.7.1. Plane strain limit analysis problems 1826.7.2. Partial static solutions to plane strain limit analysis problems 1836.7.3. Complete static solutions to plane strain limit analysis problems 1836.7.4. Complete kinematic solutions to plane strain limit analysis problems 1846.7.5. “Incomplete” solutions to plane strain limit analysis problems 1886.7.6. Complete solutions to plane strain limit analysis problems 1896.7.7. Comments 1926.7.8. Plane stress limit analysis 1946.7.9. Axially symmetric problems 1966.8. Implementation 2006.8.1. Analytical solutions 2006.8.2. Analytical/numerical solutions 2026.8.3. Numerical solutions 2036.8.4. The example of a tantalizing problem 2046.8.5. Final comments 208References 211Index 231

Jean Salençon is a Member of the French Academy of Sciences, the French Academy of Technologies, and Academia Europaea. He is also an Honorary/Foreign Member of the Academies of Hungary, Milan and Lisbon. He is also Senior Fellow of the Hong Kong Institute for Advanced Study (HKIAS). His research interests include continuum mechanics, structural analysis and soil mechanics.

• Preface xiNotations xiiiIntroduction xxvChapter 1. Elastic Domains: Yield Conditions 11.1. Introductory remarks 11.2. An overview of the model 41.2.1. The infinitesimal transformation framework 41.2.2. Time variable 41.3. One-dimensional approach 51.3.1. Uniaxial tension test 51.3.2. Uniaxial tension-compression test 81.3.3. The Bauschinger effect 91.3.4. Other one-dimensional experiments 101.4. Multidimensional approach 111.4.1. A multidimensional experiment 111.4.2. Initial elastic domain 121.4.3. Work-hardening 131.4.4. Perfectly plastic material 141.4.5. Bui’s experimental results 141.5. Yield conditions 161.5.1. Initial yield condition and yield function 161.5.2. Loading function and work-hardening 181.5.3. Simple work-hardening models 191.6. Yield criteria and loading functions 221.6.1. Convexity 221.6.2. Isotropy 231.6.3. The Tresca yield criterion 241.6.4. The von Mises yield criterion 271.6.5. Other yield criteria for metals 301.6.6. Yield criteria for anisotropic materials 311.6.7. Yield criteria for granular materials 351.7. Final comments 41Chapter 2. The Plastic Flow Rule 432.1. One-dimensional approach 432.1.1. Work-hardening material 432.1.2. Perfectly plastic material 452.2. Multidimensional approach for a work-hardening material 462.2.1. Loading and unloading processes 462.2.2. General properties of the plastic flow rule 492.2.3. Plastic potential: associated plasticity 512.2.4. Principle of maximum plastic work 542.2.5. Validation of the principle of maximum plastic work 552.2.6. Piecewise continuously differentiable loading functions 572.3. Multidimensional approach for a perfectly plastic material 592.3.1. Loading and unloading processes 592.3.2. Application of the principle of maximum plastic work 612.3.3. Drucker’s postulate 622.4. Plastic dissipation 642.4.1. Plastic dissipation per unit volume 642.4.2. Plastic dissipation and support function of the elastic domain 642.4.3. Plastic velocity jumps in the case of perfectly plastic materials 652.5. Generalized standard materials 662.6. Mises’, Tresca’s and Coulomb’s perfectly plastic standard materials 692.6.1. Mises’ perfectly plastic standard material 692.6.2. Tresca’s perfectly plastic standard material 712.6.3. Coulomb’s perfectly plastic standard material 722.6.4. About edge and vertex regimes 74Chapter 3. Elastoplastic Modeling in Generalized Variables 773.1. About generalized variables 773.2. Elastic domains 783.2.1. Initial elastic domain 783.2.2. Work-hardening and perfect plasticity 793.3. The anelastic flow rule 803.3.1. Anelasticity or plasticity? 803.3.2. Principle of maximum work 813.3.3. The work-hardening anelastic flow rule 823.3.4. The “perfectly plastic” anelastic flow rule 833.3.5. Anelastic dissipation 843.4. Generalized continua 843.4.1. Curvilinear generalized continuum 843.4.2. Planar generalized continuum 90Chapter 4. Quasi-static Elastoplastic Processes 1014.1. Quasi-static loading processes 1014.1.1. Mechanical evolution within the SPH framework 1014.1.2. Quasi-static loading process within the SPH framework 1044.1.3. Statically admissible and kinematically admissible fields 1044.1.4. Parametric problems 1054.2. Quasi-static elastoplastic loading processes 1084.2.1. Problematics 1084.2.2. Existence and uniqueness theorems 1114.2.3. Uniqueness theorems for stress rates and strain rates 1154.3. Response of a system made from an elastic and standard perfectly plastic material 1164.3.1. Initial elastic domain of the system 1164.3.2. Existence of the solution to the elastoplastic evolution problem 1184.3.3. Solution to the elastoplastic evolution problem 1194.3.4. Limit loads for the system 1204.3.5. Linear elastic response of the system 1214.3.6. Anelastic response of the system 1224.3.7. Taking geometry changes into account 1364.4. Response of a system made from a standard work-hardening elastoplastic material 1394.4.1. Initial elastic domain of the system 1394.4.2. Residual stress rates, residual strain rates 1404.4.3. Maximum work theorem 1404.4.4. Summing up... 141Chapter 5. Quasi-static Elastoplastic Processes: Minimum Principles 1435.1. Elastic and standard perfectly plastic constituent material 1435.1.1. Minimum principle for the stress rate field 1435.1.2. Minimum principle for the velocity field 1465.1.3. Other expressions of the minimum principles 1525.2. Elastic and standard positive work-hardening constituent material 1535.2.1. Revisiting the constitutive equation 1535.2.2. Minimum principle for the stress rate field 1555.2.3. Minimum principle for the velocity field 1565.2.4. Other expressions of the minimum principles 1585.2.5. Historical comments 1585.3. Minimum principles for the stress and strain fields 1595.3.1. Colonnetti’s theorem 1595.3.2. Other expressions of Colonnetti’s minimum principles 160Chapter 6. Limit Loads: Limit Analysis 1616.1. Limit loads and yield design (1) 1616.2. Static approach, first plastic collapse theorem 1636.2.1. Safe loads, interior approach 1636.2.2. Lower bound theorem 1646.3. Kinematic approach, second plastic collapse theorem 1656.3.1. Plastically admissible velocity fields 1656.3.2. Kinematic necessary condition to be satisfied by safe loads 1676.3.3. Exterior approach, upper bound theorem 1696.4. Combining static and kinematic approaches 1706.4.1. Determination of a limit load 1706.4.2. Association theorem 1726.4.3. Duality 1736.5. Limit analysis and the rigid, perfectly plastic material concept 1736.5.1. Rigid and standard perfectly plastic model 1736.5.2. The connection with limit loads 1746.6. Limit loads and yield design (2) 1766.6.1. Fundamentals of the yield design theory 1766.6.2. Resistance of the constituent material 1776.6.3. Potentially safe loads, interior approach and lower bound theorem 1786.6.4. Maximum resisting rate of work, exterior approach and upper bound theorem 1786.6.5. Matching limit load and yield design theories 1816.7. Two-dimensional limit analysis 1826.7.1. Plane strain limit analysis problems 1826.7.2. Partial static solutions to plane strain limit analysis problems 1836.7.3. Complete static solutions to plane strain limit analysis problems 1836.7.4. Complete kinematic solutions to plane strain limit analysis problems 1846.7.5. “Incomplete” solutions to plane strain limit analysis problems 1886.7.6. Complete solutions to plane strain limit analysis problems 1896.7.7. Comments 1926.7.8. Plane stress limit analysis 1946.7.9. Axially symmetric problems 1966.8. Implementation 2006.8.1. Analytical solutions 2006.8.2. Analytical/numerical solutions 2026.8.3. Numerical solutions 2036.8.4. The example of a tantalizing problem 2046.8.5. Final comments 208References 211Index 231