Jacky Mazars is Emeritus Professor at the Polytechnic Institute – Grenoble Alpes University, France, Associate Professor at the University of Sherbrooke, Canada and Honorary Director of the 3SR laboratory in Grenoble. He was one of the initiators of applying damage mechanics to concrete and has led many original developments towards understanding the behavior of structures under severe loadings.Stéphane Grange is Professor at INSA Lyon, France, and Director of the GEOMAS Civil Engineering Laboratory. His research focuses on the dynamic behavior of structures in interaction with their environment and on nonlinear dynamic numerical modeling.

• Foreword by Franz-Josef Ulm ixForeword by Pierre Labbé xiList of Notations xiiiIntroduction xixChapter 1 Mechanisms of Deformation and Failure of Concrete  11.1 Concrete: a material that is both widespread and misunderstood 11.2 Composition and behavior of concrete at an early age 41.2.1 Concrete curing 51.2.2 Consequences of curing and phenomena related to the aging of concrete 81.3 Main aspects of the mechanical behavior of concrete 111.3.1 Concrete under uniaxial loading 111.3.2 Concrete under multiaxial loading 19Chapter 2 Damage Concept and Its Applicability to Concrete 292.1 Damage concept 292.1.1 Miner cumulative damage law 302.1.2 Katchanov’s progressive damage law 322.1.3 Elasticity-damage coupling 342.2 Theoretical bases of damage mechanics 372.2.1 Elasticity-damage coupling 372.2.2 Isotropic damage theory 402.2.3 Damage threshold and notion of loading surface 42Chapter 3 Damage Modeling 453.1 Mazars models for monotonous loadings 453.1.1 Constitutive equations 453.1.2 Using the model (uniaxial case) 493.1.3 Strengths and weaknesses of the Mazars model 513.2 Model for cyclic loadings: the μ model 533.2.1 Concept of effective damage variable 533.2.2 Constitutive equations 553.2.3 Response of the μ model under various types of loading 583.2.4 Adaptation of the μ model to the case of confined loadings 66Chapter 4 Numerical Calculation of Damage 714.1 Reminders on concepts governing the use of finite elements 714.2 Principle flowcharts 734.2.1 Flowchart for the Mazars model 734.2.2 Flowchart for the μ model 744.3 Data preparation 754.3.1 Identification of modeling parameters 754.4 Concrete fracturing energy 804.4.1 Intrinsic and extrinsic energy 814.4.2 Crack band concept, Hillerborg regularization 844.5 Non-local damage concept 84Chapter 5 Applications to Common Reinforced Concrete Structural Elements Cases 895.1 2D FE calculation 895.1.1 Details of the experimental program 905.1.2 Numerical processing 915.1.3 Results 925.2 Calculations by Timoshenko enriched beam elements 955.2.1 Strengths and weaknesses of the multifiber beam description 965.2.2 Bias of results from the choice of material parameters 975.2.3 Multifiber beams and strain localization 985.2.4 Enriched multifiber description and use of suitable parameters 1005.2.5 Simulations based on the enriched multifiber description 1055.3 Multifiber calculations and access to cracking indicators 1065.3.1 Damage fields 1065.3.2 Opening of cracks 107Chapter 6 Modeling of Situations Related to Specific Loadings 1116.1 Simulation of velocity effects 1116.1.1 Analysis resulting from the experiment 1116.1.2 High-velocity loading: application to Spalling tests 1156.1.3 Loading at medium velocity: impact on a reinforced concrete beam 1186.2 Simulation of the effects of concrete maturation 1216.2.1 Problems posed by the behavior of concrete at an early age 1226.2.2 Case of a beam in a situation of restrained shrinkage 1226.2.3 Thermomechanical model of concrete at an early age 1246.2.4 RG8 test: application and results 133Chapter 7 Structures Combining Beams and Planar Elements 1457.1 Simulation of the behavior of a reinforced concrete wall 1457.1.1 Model for structural walls: equivalent reinforced concrete 1467.1.2 Application to the SAFE experiment shear wall case 1477.2 Application to a structure combining walls, beams and columns 1527.2.1 Enriched ERC modeling 1537.2.2 Modeling the response of the SMART model 1557.3 Calculation combining 2D finite elements and multifiber beams 1577.3.1 Case study: loss of bearing capacity of a column in a structure 1587.3.2 Calculation-experiment comparison results 1607.4 Conclusion 162Chapter 8 Assessment of Cracking by Limit Analysis 1658.1 Characterization of cracking: case of homogeneous fields of tensile elements 1658.1.1 Limit analysis and yield design theory 1658.1.2 Case of reinforced concrete beams in bending 1678.2 Tie rod cracking 1708.2.1 Localized cracking and diffuse damage 1708.2.2 Behavioral law for concrete in the diffuse scheme 1768.2.3 Application to an experiment on tie rods carried out at EPFL 1788.3 Homogeneous field created by concrete maturation within a cylindrical wall 1818.3.1 VeRCoRs program and model 1818.3.2 Mesh of the gusset and temperature conditions 1848.3.3 Creep, shrinkage and mechanical properties 1868.3.4 Mechanical calculation 1878.3.5 Principal results and comparisons with in situ measurements 1888.4 Conclusion 193Chapter 9 Exercises and Supplements 1959.1 Determining mechanical characteristics from experimental curves 1959.2 Mazars model: axisymmetric triaxial loading under compression 1979.3 Local and non-local damage 1999.3.1 Example of a concrete bar under direct tension 1999.3.2 Local model response: impact of the number of elements 2009.3.3 Non-local damage problem 2019.3.4 Objective calculation with a local model: Hillerborg method 2039.3.5 Conclusion 2059.4 On the μ model 2069.4.1 Reaching the damage threshold, load-unload criterion 2069.4.2 On the stress triaxiality factor 2079.4.3 Response to triaxial axisymmetric compression loading 2099.5 On the restraint degree R in situations of restrained shrinkage 2159.6 Solving a simple structure using the PVP* 2189.6.1 Problem position 2189.6.2 Using PVP* (assembling the contributions of the elements): preliminary comments for solving 218Appendix: Prerequisites in Solid Mechanics and Finite Element Methods 223References 255Index 265

Jacky Mazars is Emeritus Professor at the Polytechnic Institute – Grenoble Alpes University, France, Associate Professor at the University of Sherbrooke, Canada and Honorary Director of the 3SR laboratory in Grenoble. He was one of the initiators of applying damage mechanics to concrete and has led many original developments towards understanding the behavior of structures under severe loadings.Stéphane Grange is Professor at INSA Lyon, France, and Director of the GEOMAS Civil Engineering Laboratory. His research focuses on the dynamic behavior of structures in interaction with their environment and on nonlinear dynamic numerical modeling.

• Foreword by Franz-Josef Ulm ixForeword by Pierre Labbé xiList of Notations xiiiIntroduction xixChapter 1 Mechanisms of Deformation and Failure of Concrete  11.1 Concrete: a material that is both widespread and misunderstood 11.2 Composition and behavior of concrete at an early age 41.2.1 Concrete curing 51.2.2 Consequences of curing and phenomena related to the aging of concrete 81.3 Main aspects of the mechanical behavior of concrete 111.3.1 Concrete under uniaxial loading 111.3.2 Concrete under multiaxial loading 19Chapter 2 Damage Concept and Its Applicability to Concrete 292.1 Damage concept 292.1.1 Miner cumulative damage law 302.1.2 Katchanov’s progressive damage law 322.1.3 Elasticity-damage coupling 342.2 Theoretical bases of damage mechanics 372.2.1 Elasticity-damage coupling 372.2.2 Isotropic damage theory 402.2.3 Damage threshold and notion of loading surface 42Chapter 3 Damage Modeling 453.1 Mazars models for monotonous loadings 453.1.1 Constitutive equations 453.1.2 Using the model (uniaxial case) 493.1.3 Strengths and weaknesses of the Mazars model 513.2 Model for cyclic loadings: the μ model 533.2.1 Concept of effective damage variable 533.2.2 Constitutive equations 553.2.3 Response of the μ model under various types of loading 583.2.4 Adaptation of the μ model to the case of confined loadings 66Chapter 4 Numerical Calculation of Damage 714.1 Reminders on concepts governing the use of finite elements 714.2 Principle flowcharts 734.2.1 Flowchart for the Mazars model 734.2.2 Flowchart for the μ model 744.3 Data preparation 754.3.1 Identification of modeling parameters 754.4 Concrete fracturing energy 804.4.1 Intrinsic and extrinsic energy 814.4.2 Crack band concept, Hillerborg regularization 844.5 Non-local damage concept 84Chapter 5 Applications to Common Reinforced Concrete Structural Elements Cases 895.1 2D FE calculation 895.1.1 Details of the experimental program 905.1.2 Numerical processing 915.1.3 Results 925.2 Calculations by Timoshenko enriched beam elements 955.2.1 Strengths and weaknesses of the multifiber beam description 965.2.2 Bias of results from the choice of material parameters 975.2.3 Multifiber beams and strain localization 985.2.4 Enriched multifiber description and use of suitable parameters 1005.2.5 Simulations based on the enriched multifiber description 1055.3 Multifiber calculations and access to cracking indicators 1065.3.1 Damage fields 1065.3.2 Opening of cracks 107Chapter 6 Modeling of Situations Related to Specific Loadings 1116.1 Simulation of velocity effects 1116.1.1 Analysis resulting from the experiment 1116.1.2 High-velocity loading: application to Spalling tests 1156.1.3 Loading at medium velocity: impact on a reinforced concrete beam 1186.2 Simulation of the effects of concrete maturation 1216.2.1 Problems posed by the behavior of concrete at an early age 1226.2.2 Case of a beam in a situation of restrained shrinkage 1226.2.3 Thermomechanical model of concrete at an early age 1246.2.4 RG8 test: application and results 133Chapter 7 Structures Combining Beams and Planar Elements 1457.1 Simulation of the behavior of a reinforced concrete wall 1457.1.1 Model for structural walls: equivalent reinforced concrete 1467.1.2 Application to the SAFE experiment shear wall case 1477.2 Application to a structure combining walls, beams and columns 1527.2.1 Enriched ERC modeling 1537.2.2 Modeling the response of the SMART model 1557.3 Calculation combining 2D finite elements and multifiber beams 1577.3.1 Case study: loss of bearing capacity of a column in a structure 1587.3.2 Calculation-experiment comparison results 1607.4 Conclusion 162Chapter 8 Assessment of Cracking by Limit Analysis 1658.1 Characterization of cracking: case of homogeneous fields of tensile elements 1658.1.1 Limit analysis and yield design theory 1658.1.2 Case of reinforced concrete beams in bending 1678.2 Tie rod cracking 1708.2.1 Localized cracking and diffuse damage 1708.2.2 Behavioral law for concrete in the diffuse scheme 1768.2.3 Application to an experiment on tie rods carried out at EPFL 1788.3 Homogeneous field created by concrete maturation within a cylindrical wall 1818.3.1 VeRCoRs program and model 1818.3.2 Mesh of the gusset and temperature conditions 1848.3.3 Creep, shrinkage and mechanical properties 1868.3.4 Mechanical calculation 1878.3.5 Principal results and comparisons with in situ measurements 1888.4 Conclusion 193Chapter 9 Exercises and Supplements 1959.1 Determining mechanical characteristics from experimental curves 1959.2 Mazars model: axisymmetric triaxial loading under compression 1979.3 Local and non-local damage 1999.3.1 Example of a concrete bar under direct tension 1999.3.2 Local model response: impact of the number of elements 2009.3.3 Non-local damage problem 2019.3.4 Objective calculation with a local model: Hillerborg method 2039.3.5 Conclusion 2059.4 On the μ model 2069.4.1 Reaching the damage threshold, load-unload criterion 2069.4.2 On the stress triaxiality factor 2079.4.3 Response to triaxial axisymmetric compression loading 2099.5 On the restraint degree R in situations of restrained shrinkage 2159.6 Solving a simple structure using the PVP* 2189.6.1 Problem position 2189.6.2 Using PVP* (assembling the contributions of the elements): preliminary comments for solving 218Appendix: Prerequisites in Solid Mechanics and Finite Element Methods 223References 255Index 265

Jacky Mazars is Emeritus Professor at the Polytechnic Institute – Grenoble Alpes University, France, Associate Professor at the University of Sherbrooke, Canada and Honorary Director of the 3SR laboratory in Grenoble. He was one of the initiators of applying damage mechanics to concrete and has led many original developments towards understanding the behavior of structures under severe loadings.Stéphane Grange is Professor at INSA Lyon, France, and Director of the GEOMAS Civil Engineering Laboratory. His research focuses on the dynamic behavior of structures in interaction with their environment and on nonlinear dynamic numerical modeling.

• Foreword by Franz-Josef Ulm ixForeword by Pierre Labbé xiList of Notations xiiiIntroduction xixChapter 1 Mechanisms of Deformation and Failure of Concrete  11.1 Concrete: a material that is both widespread and misunderstood 11.2 Composition and behavior of concrete at an early age 41.2.1 Concrete curing 51.2.2 Consequences of curing and phenomena related to the aging of concrete 81.3 Main aspects of the mechanical behavior of concrete 111.3.1 Concrete under uniaxial loading 111.3.2 Concrete under multiaxial loading 19Chapter 2 Damage Concept and Its Applicability to Concrete 292.1 Damage concept 292.1.1 Miner cumulative damage law 302.1.2 Katchanov’s progressive damage law 322.1.3 Elasticity-damage coupling 342.2 Theoretical bases of damage mechanics 372.2.1 Elasticity-damage coupling 372.2.2 Isotropic damage theory 402.2.3 Damage threshold and notion of loading surface 42Chapter 3 Damage Modeling 453.1 Mazars models for monotonous loadings 453.1.1 Constitutive equations 453.1.2 Using the model (uniaxial case) 493.1.3 Strengths and weaknesses of the Mazars model 513.2 Model for cyclic loadings: the μ model 533.2.1 Concept of effective damage variable 533.2.2 Constitutive equations 553.2.3 Response of the μ model under various types of loading 583.2.4 Adaptation of the μ model to the case of confined loadings 66Chapter 4 Numerical Calculation of Damage 714.1 Reminders on concepts governing the use of finite elements 714.2 Principle flowcharts 734.2.1 Flowchart for the Mazars model 734.2.2 Flowchart for the μ model 744.3 Data preparation 754.3.1 Identification of modeling parameters 754.4 Concrete fracturing energy 804.4.1 Intrinsic and extrinsic energy 814.4.2 Crack band concept, Hillerborg regularization 844.5 Non-local damage concept 84Chapter 5 Applications to Common Reinforced Concrete Structural Elements Cases 895.1 2D FE calculation 895.1.1 Details of the experimental program 905.1.2 Numerical processing 915.1.3 Results 925.2 Calculations by Timoshenko enriched beam elements 955.2.1 Strengths and weaknesses of the multifiber beam description 965.2.2 Bias of results from the choice of material parameters 975.2.3 Multifiber beams and strain localization 985.2.4 Enriched multifiber description and use of suitable parameters 1005.2.5 Simulations based on the enriched multifiber description 1055.3 Multifiber calculations and access to cracking indicators 1065.3.1 Damage fields 1065.3.2 Opening of cracks 107Chapter 6 Modeling of Situations Related to Specific Loadings 1116.1 Simulation of velocity effects 1116.1.1 Analysis resulting from the experiment 1116.1.2 High-velocity loading: application to Spalling tests 1156.1.3 Loading at medium velocity: impact on a reinforced concrete beam 1186.2 Simulation of the effects of concrete maturation 1216.2.1 Problems posed by the behavior of concrete at an early age 1226.2.2 Case of a beam in a situation of restrained shrinkage 1226.2.3 Thermomechanical model of concrete at an early age 1246.2.4 RG8 test: application and results 133Chapter 7 Structures Combining Beams and Planar Elements 1457.1 Simulation of the behavior of a reinforced concrete wall 1457.1.1 Model for structural walls: equivalent reinforced concrete 1467.1.2 Application to the SAFE experiment shear wall case 1477.2 Application to a structure combining walls, beams and columns 1527.2.1 Enriched ERC modeling 1537.2.2 Modeling the response of the SMART model 1557.3 Calculation combining 2D finite elements and multifiber beams 1577.3.1 Case study: loss of bearing capacity of a column in a structure 1587.3.2 Calculation-experiment comparison results 1607.4 Conclusion 162Chapter 8 Assessment of Cracking by Limit Analysis 1658.1 Characterization of cracking: case of homogeneous fields of tensile elements 1658.1.1 Limit analysis and yield design theory 1658.1.2 Case of reinforced concrete beams in bending 1678.2 Tie rod cracking 1708.2.1 Localized cracking and diffuse damage 1708.2.2 Behavioral law for concrete in the diffuse scheme 1768.2.3 Application to an experiment on tie rods carried out at EPFL 1788.3 Homogeneous field created by concrete maturation within a cylindrical wall 1818.3.1 VeRCoRs program and model 1818.3.2 Mesh of the gusset and temperature conditions 1848.3.3 Creep, shrinkage and mechanical properties 1868.3.4 Mechanical calculation 1878.3.5 Principal results and comparisons with in situ measurements 1888.4 Conclusion 193Chapter 9 Exercises and Supplements 1959.1 Determining mechanical characteristics from experimental curves 1959.2 Mazars model: axisymmetric triaxial loading under compression 1979.3 Local and non-local damage 1999.3.1 Example of a concrete bar under direct tension 1999.3.2 Local model response: impact of the number of elements 2009.3.3 Non-local damage problem 2019.3.4 Objective calculation with a local model: Hillerborg method 2039.3.5 Conclusion 2059.4 On the μ model 2069.4.1 Reaching the damage threshold, load-unload criterion 2069.4.2 On the stress triaxiality factor 2079.4.3 Response to triaxial axisymmetric compression loading 2099.5 On the restraint degree R in situations of restrained shrinkage 2159.6 Solving a simple structure using the PVP* 2189.6.1 Problem position 2189.6.2 Using PVP* (assembling the contributions of the elements): preliminary comments for solving 218Appendix: Prerequisites in Solid Mechanics and Finite Element Methods 223References 255Index 265