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

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

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