Editors A. Srikantha Phani, University of British Columbia, CanadaMahmoud I. Hussein, University of Colorado Boulder, USA

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• List of Contributors xiiiForeword xvPreface xxv1 Introduction to Lattice Materials 1A. Srikantha Phani andMahmoud I. Hussein1.1 Introduction 11.2 Lattice Materials and Structures 21.2.1 Material versus Structure 31.2.2 Motivation 31.2.3 Classification of Lattices and Maxwell’s Rule 41.2.4 ManufacturingMethods 61.2.5 Applications 71.3 Overview of Chapters 8Acknowledgment 10References 102 Elastostatics of Lattice Materials 19D. Pasini and S. Arabnejad2.1 Introduction 192.2 The RVE 212.3 Surface Average Approach 222.4 Volume Average Approach 252.5 Force-based Approach 252.6 Asymptotic Homogenization Method 262.7 Generalized Continuum Theory 292.8 Homogenization via BlochWave Analysis and the Cauchy–Born Hypothesis 322.9 Multiscale Matrix-based Computational Technique 342.10 Homogenization based on the Equation of Motion 362.11 Case Study: Property Predictions for a Hexagonal Lattice 382.12 Conclusions 42References 433 Elastodynamics of Lattice Materials 53A. Srikantha Phani3.1 Introduction 533.2 One-dimensional Lattices 553.2.1 Bloch’s Theorem 573.2.2 Application of Bloch’s Theorem 593.2.3 Dispersion Curves and Unit-cell Resonances 593.2.4 Continuous Lattices: Local Resonance and sub-Bragg Band Gaps 613.2.5 Dispersion Curves of a Beam Lattice 623.2.6 Receptance Method 643.2.7 Synopsis of 1D Lattices 673.3 Two-dimensional Lattice Materials 673.3.1 Application of Bloch’s Theorem to 2D Lattices 673.3.2 Discrete Square Lattice 703.4 Lattice Materials 723.4.1 Finite Element Modelling of the Unit Cell 753.4.2 Band Structure of Lattice Topologies 773.4.3 Directionality ofWave Propagation 843.5 Tunneling and EvanescentWaves 853.6 Concluding Remarks 873.7 Acknowledgments 87References 874 Wave Propagation in Damped Lattice Materials 93Dimitri Krattiger, A. Srikantha Phani andMahmoud I. Hussein4.1 Introduction 934.2 One-dimensionalMass–Spring–DamperModel 954.2.1 1D Model Description 954.2.2 Free-wave Solution 96State-spaceWave Calculation 97Bloch–Rayleigh Perturbation Method 974.2.3 Driven-wave Solution 984.2.4 1D Damped Band Structures 984.3 Two-dimensional Plate–Plate Lattice Model 994.3.1 2D Model Description 994.3.2 Extension of Driven-wave Calculations to 2D Domains 1004.3.3 2D Damped Band Structures 101References 1045 Wave Propagation in Nonlinear Lattice Materials 107Kevin L.Manktelow,Massimo Ruzzene andMichael J. Leamy5.1 Overview 1075.2 Weakly Nonlinear Dispersion Analysis 1085.3 Application to a 1D Monoatomic Chain 1145.3.1 Overview 1145.3.2 Model Description and Nonlinear Governing Equation 1145.3.3 Single-wave Dispersion Analysis 1155.3.4 Multi-wave Dispersion Analysis 116Case 1. GeneralWave–Wave Interactions 117Case 2. Long-wavelength LimitWave–Wave Interactions 1195.3.5 Numerical Verification and Discussion 1225.4 Application to a 2D Monoatomic Lattice 1235.4.1 Overview 1235.4.2 Model Description and Nonlinear Governing Equation 1245.4.3 Multiple-scale Perturbation Analysis 1255.4.4 Analysis of Predicted Dispersion Shifts 1275.4.5 Numerical Simulation Validation Cases 129Analysis Method 130Orthogonal and Oblique Interaction 1315.4.6 Application: Amplitude-tunable Focusing 133Summary 134Acknowledgements 135References 1356 Stability of Lattice Materials 139Filippo Casadei, PaiWang and Katia Bertoldi6.1 Introduction 1396.2 Geometry, Material, and Loading Conditions 1406.3 Stability of Finite-sized Specimens 1416.4 Stability of Infinite Periodic Specimens 1426.4.1 Microscopic Instability 1426.5 Post-buckling Analysis 1456.6 Effect of Buckling and Large Deformation on the Propagation Of Elastic Waves 1466.7 Conclusions 150References 1517 Impact and Blast Response of Lattice Materials 155Matthew Smith,Wesley J. Cantwell and Zhongwei Guan7.1 Introduction 1557.2 Literature Review 1557.2.1 Dynamic Response of Cellular Structures 1557.2.2 Shock- and Blast-loading Responses of Cellular Structures 1577.2.3 Dynamic Indentation Performance of Cellular Structures 1587.3 Manufacturing Process 1597.3.1 The Selective Laser Melting Technique 1597.3.2 Sandwich Panel Manufacture 1607.4 Dynamic and Blast Loading of Lattice Materials 1617.4.1 ExperimentalMethod – Drop-hammer Impact Tests 1617.4.2 ExperimentalMethod – Blast Tests on Lattice Cubes 1627.4.3 ExperimentalMethod – Blast Tests on Composite-lattice Sandwich Structures 1637.5 Results and Discussion 1657.5.1 Drop-hammer Impact Tests 1657.5.2 Blast Tests on the Lattice Structures 1677.5.3 Blast Tests on the Sandwich Panels 170Concluding Remarks 173Acknowledgements 174References 1748 Pentamode Lattice Structures 179Andrew N. Norris8.1 Introduction 1798.2 Pentamode Materials 1838.2.1 General Properties 1838.2.2 Small Rigidity and Poisson’s Ratio of a PM 1858.2.3 Wave Motion in a PM 1868.3 Lattice Models for PM 1878.3.1 Effective PM Properties of 2D and 3D Lattices 1878.3.2 Transversely Isotropic PM Lattice 188Effective Moduli: 2D 1908.4 Quasi-static Pentamode Properties of a Lattice in 2D and 3D 1928.4.1 General Formulation with Rigidity 1928.4.2 Pentamode Limit 1948.4.3 Two-dimensional Results for Finite Rigidity 1958.5 Conclusion 195Acknowledgements 196References 1969 Modal Reduction of Lattice Material Models 199Dimitri Krattiger and Mahmoud I. Hussein9.1 Introduction 1999.2 Plate Model 2009.2.1 Mindlin–Reissner Plate Finite Elements 2009.2.2 Bloch Boundary Conditions 2029.2.3 Example Model 2039.3 Reduced Bloch Mode Expansion 2049.3.1 RBME Formulation 2049.3.2 RBME Example 2059.3.3 RBME Additional Considerations 2079.4 Bloch Mode Synthesis 2089.4.1 BMS Formulation 2089.4.2 BMS Example 2109.4.3 BMS Additional Considerations 2109.5 Comparison of RBME and BMS 2129.5.1 Model Size 2129.5.2 Computational Efficiency 2139.5.3 Ease of Implementation 214References 21410 Topology Optimization of Lattice Materials 217Osama R. Bilal and Mahmoud I. Hussein10.1 Introduction 21710.2 Unit-cell Optimization 21810.2.1 Parametric, Shape, and Topology Optimization 21810.2.2 Selection of Studies from the Literature 21810.2.3 Design Search Space 21910.3 Plate-based Lattice Material Unit Cell 22010.3.1 Equation of Motion and FE Model 22110.3.2 Mathematical Formulation 22210.4 Genetic Algorithm 22310.4.1 Objective Function 22310.4.2 Fitness Function 22410.4.3 Selection 22410.4.4 Reproduction 22410.4.5 Initialization and Termination 22510.4.6 Implementation 22510.5 Appendix 226References 22811 Dynamics of Locally Resonant and Inertially Amplified Lattice Materials 233Cetin Yilmaz and Gregory M. Hulbert11.1 Introduction 23311.2 Locally Resonant Lattice Materials 23411.2.1 1D Locally Resonant Lattices 23411.2.2 2D Locally Resonant Lattices 24111.2.3 3D Locally Resonant Lattices 24311.3 Inertially Amplified Lattice Materials 24611.3.1 1D Inertially Amplified Lattices 24611.3.2 2D Inertially Amplified Lattices 24811.3.3 3D Inertially Amplified Lattices 25311.4 Conclusions 255References 25612 Dynamics of Nanolattices: Polymer-Nanometal Lattices 259Craig A. Steeves, Glenn D. Hibbard,Manan Arya, and Ante T. Lausic12.1 Introduction 25912.2 Fabrication 25912.2.1 Case Study 26212.3 Lattice Dynamics 26312.3.1 Lattice Properties 264Geometries of 3D Lattices 264Effective Material Properties of Nanometal-coated Polymer Lattices 26512.3.2 Finite-elementModel 266Displacement Field 266Kinetic Energy 268Strain Potential Energy 269Collected Equation of Motion 27012.3.3 Floquet–Bloch Principles 271Generalized Forces in Bloch Analysis 272Reduced Equation of Motion 27412.3.4 Dispersion Curves for the Octet Lattice 27512.3.5 Lattice Tuning 277Bandgap Placement 277Lattice Optimization 27712.4 Conclusions 27812.5 Appendix: Shape Functions for a Timoshenko Beam with Six Nodal Degreesof Freedom 279References 280Index 283

Editors A. Srikantha Phani, University of British Columbia, CanadaMahmoud I. Hussein, University of Colorado Boulder, USA

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• List of Contributors xiiiForeword xvPreface xxv1 Introduction to Lattice Materials 1A. Srikantha Phani andMahmoud I. Hussein1.1 Introduction 11.2 Lattice Materials and Structures 21.2.1 Material versus Structure 31.2.2 Motivation 31.2.3 Classification of Lattices and Maxwell’s Rule 41.2.4 ManufacturingMethods 61.2.5 Applications 71.3 Overview of Chapters 8Acknowledgment 10References 102 Elastostatics of Lattice Materials 19D. Pasini and S. Arabnejad2.1 Introduction 192.2 The RVE 212.3 Surface Average Approach 222.4 Volume Average Approach 252.5 Force-based Approach 252.6 Asymptotic Homogenization Method 262.7 Generalized Continuum Theory 292.8 Homogenization via BlochWave Analysis and the Cauchy–Born Hypothesis 322.9 Multiscale Matrix-based Computational Technique 342.10 Homogenization based on the Equation of Motion 362.11 Case Study: Property Predictions for a Hexagonal Lattice 382.12 Conclusions 42References 433 Elastodynamics of Lattice Materials 53A. Srikantha Phani3.1 Introduction 533.2 One-dimensional Lattices 553.2.1 Bloch’s Theorem 573.2.2 Application of Bloch’s Theorem 593.2.3 Dispersion Curves and Unit-cell Resonances 593.2.4 Continuous Lattices: Local Resonance and sub-Bragg Band Gaps 613.2.5 Dispersion Curves of a Beam Lattice 623.2.6 Receptance Method 643.2.7 Synopsis of 1D Lattices 673.3 Two-dimensional Lattice Materials 673.3.1 Application of Bloch’s Theorem to 2D Lattices 673.3.2 Discrete Square Lattice 703.4 Lattice Materials 723.4.1 Finite Element Modelling of the Unit Cell 753.4.2 Band Structure of Lattice Topologies 773.4.3 Directionality ofWave Propagation 843.5 Tunneling and EvanescentWaves 853.6 Concluding Remarks 873.7 Acknowledgments 87References 874 Wave Propagation in Damped Lattice Materials 93Dimitri Krattiger, A. Srikantha Phani andMahmoud I. Hussein4.1 Introduction 934.2 One-dimensionalMass–Spring–DamperModel 954.2.1 1D Model Description 954.2.2 Free-wave Solution 96State-spaceWave Calculation 97Bloch–Rayleigh Perturbation Method 974.2.3 Driven-wave Solution 984.2.4 1D Damped Band Structures 984.3 Two-dimensional Plate–Plate Lattice Model 994.3.1 2D Model Description 994.3.2 Extension of Driven-wave Calculations to 2D Domains 1004.3.3 2D Damped Band Structures 101References 1045 Wave Propagation in Nonlinear Lattice Materials 107Kevin L.Manktelow,Massimo Ruzzene andMichael J. Leamy5.1 Overview 1075.2 Weakly Nonlinear Dispersion Analysis 1085.3 Application to a 1D Monoatomic Chain 1145.3.1 Overview 1145.3.2 Model Description and Nonlinear Governing Equation 1145.3.3 Single-wave Dispersion Analysis 1155.3.4 Multi-wave Dispersion Analysis 116Case 1. GeneralWave–Wave Interactions 117Case 2. Long-wavelength LimitWave–Wave Interactions 1195.3.5 Numerical Verification and Discussion 1225.4 Application to a 2D Monoatomic Lattice 1235.4.1 Overview 1235.4.2 Model Description and Nonlinear Governing Equation 1245.4.3 Multiple-scale Perturbation Analysis 1255.4.4 Analysis of Predicted Dispersion Shifts 1275.4.5 Numerical Simulation Validation Cases 129Analysis Method 130Orthogonal and Oblique Interaction 1315.4.6 Application: Amplitude-tunable Focusing 133Summary 134Acknowledgements 135References 1356 Stability of Lattice Materials 139Filippo Casadei, PaiWang and Katia Bertoldi6.1 Introduction 1396.2 Geometry, Material, and Loading Conditions 1406.3 Stability of Finite-sized Specimens 1416.4 Stability of Infinite Periodic Specimens 1426.4.1 Microscopic Instability 1426.5 Post-buckling Analysis 1456.6 Effect of Buckling and Large Deformation on the Propagation Of Elastic Waves 1466.7 Conclusions 150References 1517 Impact and Blast Response of Lattice Materials 155Matthew Smith,Wesley J. Cantwell and Zhongwei Guan7.1 Introduction 1557.2 Literature Review 1557.2.1 Dynamic Response of Cellular Structures 1557.2.2 Shock- and Blast-loading Responses of Cellular Structures 1577.2.3 Dynamic Indentation Performance of Cellular Structures 1587.3 Manufacturing Process 1597.3.1 The Selective Laser Melting Technique 1597.3.2 Sandwich Panel Manufacture 1607.4 Dynamic and Blast Loading of Lattice Materials 1617.4.1 ExperimentalMethod – Drop-hammer Impact Tests 1617.4.2 ExperimentalMethod – Blast Tests on Lattice Cubes 1627.4.3 ExperimentalMethod – Blast Tests on Composite-lattice Sandwich Structures 1637.5 Results and Discussion 1657.5.1 Drop-hammer Impact Tests 1657.5.2 Blast Tests on the Lattice Structures 1677.5.3 Blast Tests on the Sandwich Panels 170Concluding Remarks 173Acknowledgements 174References 1748 Pentamode Lattice Structures 179Andrew N. Norris8.1 Introduction 1798.2 Pentamode Materials 1838.2.1 General Properties 1838.2.2 Small Rigidity and Poisson’s Ratio of a PM 1858.2.3 Wave Motion in a PM 1868.3 Lattice Models for PM 1878.3.1 Effective PM Properties of 2D and 3D Lattices 1878.3.2 Transversely Isotropic PM Lattice 188Effective Moduli: 2D 1908.4 Quasi-static Pentamode Properties of a Lattice in 2D and 3D 1928.4.1 General Formulation with Rigidity 1928.4.2 Pentamode Limit 1948.4.3 Two-dimensional Results for Finite Rigidity 1958.5 Conclusion 195Acknowledgements 196References 1969 Modal Reduction of Lattice Material Models 199Dimitri Krattiger and Mahmoud I. Hussein9.1 Introduction 1999.2 Plate Model 2009.2.1 Mindlin–Reissner Plate Finite Elements 2009.2.2 Bloch Boundary Conditions 2029.2.3 Example Model 2039.3 Reduced Bloch Mode Expansion 2049.3.1 RBME Formulation 2049.3.2 RBME Example 2059.3.3 RBME Additional Considerations 2079.4 Bloch Mode Synthesis 2089.4.1 BMS Formulation 2089.4.2 BMS Example 2109.4.3 BMS Additional Considerations 2109.5 Comparison of RBME and BMS 2129.5.1 Model Size 2129.5.2 Computational Efficiency 2139.5.3 Ease of Implementation 214References 21410 Topology Optimization of Lattice Materials 217Osama R. Bilal and Mahmoud I. Hussein10.1 Introduction 21710.2 Unit-cell Optimization 21810.2.1 Parametric, Shape, and Topology Optimization 21810.2.2 Selection of Studies from the Literature 21810.2.3 Design Search Space 21910.3 Plate-based Lattice Material Unit Cell 22010.3.1 Equation of Motion and FE Model 22110.3.2 Mathematical Formulation 22210.4 Genetic Algorithm 22310.4.1 Objective Function 22310.4.2 Fitness Function 22410.4.3 Selection 22410.4.4 Reproduction 22410.4.5 Initialization and Termination 22510.4.6 Implementation 22510.5 Appendix 226References 22811 Dynamics of Locally Resonant and Inertially Amplified Lattice Materials 233Cetin Yilmaz and Gregory M. Hulbert11.1 Introduction 23311.2 Locally Resonant Lattice Materials 23411.2.1 1D Locally Resonant Lattices 23411.2.2 2D Locally Resonant Lattices 24111.2.3 3D Locally Resonant Lattices 24311.3 Inertially Amplified Lattice Materials 24611.3.1 1D Inertially Amplified Lattices 24611.3.2 2D Inertially Amplified Lattices 24811.3.3 3D Inertially Amplified Lattices 25311.4 Conclusions 255References 25612 Dynamics of Nanolattices: Polymer-Nanometal Lattices 259Craig A. Steeves, Glenn D. Hibbard,Manan Arya, and Ante T. Lausic12.1 Introduction 25912.2 Fabrication 25912.2.1 Case Study 26212.3 Lattice Dynamics 26312.3.1 Lattice Properties 264Geometries of 3D Lattices 264Effective Material Properties of Nanometal-coated Polymer Lattices 26512.3.2 Finite-elementModel 266Displacement Field 266Kinetic Energy 268Strain Potential Energy 269Collected Equation of Motion 27012.3.3 Floquet–Bloch Principles 271Generalized Forces in Bloch Analysis 272Reduced Equation of Motion 27412.3.4 Dispersion Curves for the Octet Lattice 27512.3.5 Lattice Tuning 277Bandgap Placement 277Lattice Optimization 27712.4 Conclusions 27812.5 Appendix: Shape Functions for a Timoshenko Beam with Six Nodal Degreesof Freedom 279References 280Index 283