Dynamics of Lattice Materials

List of Contributors xiii Foreword xv Preface xxv 1 Introduction to Lattice Materials 1 A. Srikantha Phani andMahmoud I. Hussein 1.1 Introduction 1 1.2 Lattice Materials and Structures 2 1.2.1 Material versus Structure 3 1.2.2 Motivation 3 1.2.3 Classification of Lattices and Maxwell’s Rule 4 1.2.4 ManufacturingMethods 6 1.2.5 Applications 7 1.3 Overview of Chapters 8 Acknowledgment 10 References 10 2 Elastostatics of Lattice Materials 19 D. Pasini and S. Arabnejad 2.1 Introduction 19 2.2 The RVE 21 2.3 Surface Average Approach 22 2.4 Volume Average Approach 25 2.5 Force-based Approach 25 2.6 Asymptotic Homogenization Method 26 2.7 Generalized Continuum Theory 29 2.8 Homogenization via BlochWave Analysis and the Cauchy–Born Hypothesis 32 2.9 Multiscale Matrix-based Computational Technique 34 2.10 Homogenization based on the Equation of Motion 36 2.11 Case Study: Property Predictions for a Hexagonal Lattice 38 2.12 Conclusions 42 References 43 3 Elastodynamics of Lattice Materials 53 A. Srikantha Phani 3.1 Introduction 53 3.2 One-dimensional Lattices 55 3.2.1 Bloch’s Theorem 57 3.2.2 Application of Bloch’s Theorem 59 3.2.3 Dispersion Curves and Unit-cell Resonances 59 3.2.4 Continuous Lattices: Local Resonance and sub-Bragg Band Gaps 61 3.2.5 Dispersion Curves of a Beam Lattice 62 3.2.6 Receptance Method 64 3.2.7 Synopsis of 1D Lattices 67 3.3 Two-dimensional Lattice Materials 67 3.3.1 Application of Bloch’s Theorem to 2D Lattices 67 3.3.2 Discrete Square Lattice 70 3.4 Lattice Materials 72 3.4.1 Finite Element Modelling of the Unit Cell 75 3.4.2 Band Structure of Lattice Topologies 77 3.4.3 Directionality ofWave Propagation 84 3.5 Tunneling and EvanescentWaves 85 3.6 Concluding Remarks 87 3.7 Acknowledgments 87 References 87 4 Wave Propagation in Damped Lattice Materials 93 Dimitri Krattiger, A. Srikantha Phani andMahmoud I. Hussein 4.1 Introduction 93 4.2 One-dimensionalMass–Spring–DamperModel 95 4.2.1 1D Model Description 95 4.2.2 Free-wave Solution 96 State-spaceWave Calculation 97 Bloch–Rayleigh Perturbation Method 97 4.2.3 Driven-wave Solution 98 4.2.4 1D Damped Band Structures 98 4.3 Two-dimensional Plate–Plate Lattice Model 99 4.3.1 2D Model Description 99 4.3.2 Extension of Driven-wave Calculations to 2D Domains 100 4.3.3 2D Damped Band Structures 101 References 104 5 Wave Propagation in Nonlinear Lattice Materials 107 Kevin L.Manktelow,Massimo Ruzzene andMichael J. Leamy 5.1 Overview 107 5.2 Weakly Nonlinear Dispersion Analysis 108 5.3 Application to a 1D Monoatomic Chain 114 5.3.1 Overview 114 5.3.2 Model Description and Nonlinear Governing Equation 114 5.3.3 Single-wave Dispersion Analysis 115 5.3.4 Multi-wave Dispersion Analysis 116 Case 1. GeneralWave–Wave Interactions 117 Case 2. Long-wavelength LimitWave–Wave Interactions 119 5.3.5 Numerical Verification and Discussion 122 5.4 Application to a 2D Monoatomic Lattice 123 5.4.1 Overview 123 5.4.2 Model Description and Nonlinear Governing Equation 124 5.4.3 Multiple-scale Perturbation Analysis 125 5.4.4 Analysis of Predicted Dispersion Shifts 127 5.4.5 Numerical Simulation Validation Cases 129 Analysis Method 130 Orthogonal and Oblique Interaction 131 5.4.6 Application: Amplitude-tunable Focusing 133 Summary 134 Acknowledgements 135 References 135 6 Stability of Lattice Materials 139 Filippo Casadei, PaiWang and Katia Bertoldi 6.1 Introduction 139 6.2 Geometry, Material, and Loading Conditions 140 6.3 Stability of Finite-sized Specimens 141 6.4 Stability of Infinite Periodic Specimens 142 6.4.1 Microscopic Instability 142 6.5 Post-buckling Analysis 145 6.6 Effect of Buckling and Large Deformation on the Propagation Of Elastic Waves 146 6.7 Conclusions 150 References 151 7 Impact and Blast Response of Lattice Materials 155 Matthew Smith,Wesley J. Cantwell and Zhongwei Guan 7.1 Introduction 155 7.2 Literature Review 155 7.2.1 Dynamic Response of Cellular Structures 155 7.2.2 Shock- and Blast-loading Responses of Cellular Structures 157 7.2.3 Dynamic Indentation Performance of Cellular Structures 158 7.3 Manufacturing Process 159 7.3.1 The Selective Laser Melting Technique 159 7.3.2 Sandwich Panel Manufacture 160 7.4 Dynamic and Blast Loading of Lattice Materials 161 7.4.1 ExperimentalMethod – Drop-hammer Impact Tests 161 7.4.2 ExperimentalMethod – Blast Tests on Lattice Cubes 162 7.4.3 ExperimentalMethod – Blast Tests on Composite-lattice Sandwich Structures 163 7.5 Results and Discussion 165 7.5.1 Drop-hammer Impact Tests 165 7.5.2 Blast Tests on the Lattice Structures 167 7.5.3 Blast Tests on the Sandwich Panels 170 Concluding Remarks 173 Acknowledgements 174 References 174 8 Pentamode Lattice Structures 179 Andrew N. Norris 8.1 Introduction 179 8.2 Pentamode Materials 183 8.2.1 General Properties 183 8.2.2 Small Rigidity and Poisson’s Ratio of a PM 185 8.2.3 Wave Motion in a PM 186 8.3 Lattice Models for PM 187 8.3.1 Effective PM Properties of 2D and 3D Lattices 187 8.3.2 Transversely Isotropic PM Lattice 188 Effective Moduli: 2D 190 8.4 Quasi-static Pentamode Properties of a Lattice in 2D and 3D 192 8.4.1 General Formulation with Rigidity 192 8.4.2 Pentamode Limit 194 8.4.3 Two-dimensional Results for Finite Rigidity 195 8.5 Conclusion 195 Acknowledgements 196 References 196 9 Modal Reduction of Lattice Material Models 199 Dimitri Krattiger and Mahmoud I. Hussein 9.1 Introduction 199 9.2 Plate Model 200 9.2.1 Mindlin–Reissner Plate Finite Elements 200 9.2.2 Bloch Boundary Conditions 202 9.2.3 Example Model 203 9.3 Reduced Bloch Mode Expansion 204 9.3.1 RBME Formulation 204 9.3.2 RBME Example 205 9.3.3 RBME Additional Considerations 207 9.4 Bloch Mode Synthesis 208 9.4.1 BMS Formulation 208 9.4.2 BMS Example 210 9.4.3 BMS Additional Considerations 210 9.5 Comparison of RBME and BMS 212 9.5.1 Model Size 212 9.5.2 Computational Efficiency 213 9.5.3 Ease of Implementation 214 References 214 10 Topology Optimization of Lattice Materials 217 Osama R. Bilal and Mahmoud I. Hussein 10.1 Introduction 217 10.2 Unit-cell Optimization 218 10.2.1 Parametric, Shape, and Topology Optimization 218 10.2.2 Selection of Studies from the Literature 218 10.2.3 Design Search Space 219 10.3 Plate-based Lattice Material Unit Cell 220 10.3.1 Equation of Motion and FE Model 221 10.3.2 Mathematical Formulation 222 10.4 Genetic Algorithm 223 10.4.1 Objective Function 223 10.4.2 Fitness Function 224 10.4.3 Selection 224 10.4.4 Reproduction 224 10.4.5 Initialization and Termination 225 10.4.6 Implementation 225 10.5 Appendix 226 References 228 11 Dynamics of Locally Resonant and Inertially Amplified Lattice Materials 233 Cetin Yilmaz and Gregory M. Hulbert 11.1 Introduction 233 11.2 Locally Resonant Lattice Materials 234 11.2.1 1D Locally Resonant Lattices 234 11.2.2 2D Locally Resonant Lattices 241 11.2.3 3D Locally Resonant Lattices 243 11.3 Inertially Amplified Lattice Materials 246 11.3.1 1D Inertially Amplified Lattices 246 11.3.2 2D Inertially Amplified Lattices 248 11.3.3 3D Inertially Amplified Lattices 253 11.4 Conclusions 255 References 256 12 Dynamics of Nanolattices: Polymer-Nanometal Lattices 259 Craig A. Steeves, Glenn D. Hibbard,Manan Arya, and Ante T. Lausic 12.1 Introduction 259 12.2 Fabrication 259 12.2.1 Case Study 262 12.3 Lattice Dynamics 263 12.3.1 Lattice Properties 264 Geometries of 3D Lattices 264 Effective Material Properties of Nanometal-coated Polymer Lattices 265 12.3.2 Finite-elementModel 266 Displacement Field 266 Kinetic Energy 268 Strain Potential Energy 269 Collected Equation of Motion 270 12.3.3 Floquet–Bloch Principles 271 Generalized Forces in Bloch Analysis 272 Reduced Equation of Motion 274 12.3.4 Dispersion Curves for the Octet Lattice 275 12.3.5 Lattice Tuning 277 Bandgap Placement 277 Lattice Optimization 277 12.4 Conclusions 278 12.5 Appendix: Shape Functions for a Timoshenko Beam with Six Nodal Degrees of Freedom 279 References 280 Index 283