Cover

Title Page

Copyright

Dedication

Contents

List of Contributors

Foreword

Preface

Chapter 1 Introduction to Lattice Materials

1.1 Introduction

1.2 Lattice Materials and Structures

1.2.1 Material versus Structure

1.2.2 Motivation

1.2.3 Classification of Lattices and Maxwell's Rule

1.2.4 Manufacturing Methods

1.2.5 Applications

1.3 Overview of Chapters

Acknowledgment

References

Chapter 2 Elastostatics of Lattice Materials

2.1 Introduction

2.2 The RVE

2.3 Surface Average Approach

2.4 Volume Average Approach

2.5 Force-based Approach

2.6 Asymptotic Homogenization Method

2.7 Generalized Continuum Theory

2.8 Homogenization via Bloch Wave Analysis and the Cauchy-Born Hypothesis

2.9 Multiscale Matrix-based Computational Technique

2.10 Homogenization based on the Equation of Motion

2.11 Case Study: Property Predictions for a Hexagonal Lattice

2.12 Conclusions

References

Chapter 3 Elastodynamics of Lattice Materials

3.1 Introduction

3.2 One-dimensional Lattices

3.2.1 Bloch's Theorem

3.2.2 Application of Bloch's Theorem

3.2.3 Dispersion Curves and Unit-cell Resonances

3.2.4 Continuous Lattices: Local Resonance and sub-Bragg Band Gaps

3.2.5 Dispersion Curves of a Beam Lattice

3.2.6 Receptance Method

3.2.7 Synopsis of 1D Lattices

3.3 Two-dimensional Lattice Materials

3.3.1 Application of Bloch's Theorem to 2D Lattices

3.3.2 Discrete Square Lattice

3.4 Lattice Materials

3.4.1 Finite Element Modelling of the Unit Cell

3.4.2 Band Structure of Lattice Topologies

3.4.3 Directionality of Wave Propagation

3.5 Tunneling and Evanescent Waves

3.6 Concluding Remarks

3.7 Acknowledgments

References

Chapter 4 Wave Propagation in Damped Lattice Materials

4.1 Introduction

4.2 One-dimensional Mass-Spring-Damper Model

4.2.1 1D Model Description

4.2.2 Free-wave Solution

State-space Wave Calculation

Bloch-Rayleigh Perturbation Method

4.2.3 Driven-wave Solution

4.2.4 1D Damped Band Structures

4.3 Two-dimensional Plate-Plate Lattice Model

4.3.1 2D Model Description

4.3.2 Extension of Driven-wave Calculations to 2D Domains

4.3.3 2D Damped Band Structures

References

Chapter 5 Wave Propagation in Nonlinear Lattice Materials

5.1 Overview

5.2 Weakly Nonlinear Dispersion Analysis

5.3 Application to a 1D Monoatomic Chain

5.3.1 Overview

5.3.2 Model Description and Nonlinear Governing Equation

5.3.3 Single-wave Dispersion Analysis

5.3.4 Multi-wave Dispersion Analysis

Case 1. General Wave-Wave Interactions

Case 2. Long-wavelength Limit Wave-Wave Interactions

5.3.5 Numerical Verification and Discussion

5.4 Application to a 2D Monoatomic Lattice

5.4.1 Overview

5.4.2 Model Description and Nonlinear Governing Equation

5.4.3 Multiple-scale Perturbation Analysis

5.4.4 Analysis of Predicted Dispersion Shifts

5.4.5 Numerical Simulation Validation Cases

Analysis Method

Orthogonal and Oblique Interaction

5.4.6 Application: Amplitude-tunable Focusing

Summary

Acknowledgements

References

Chapter 6 Stability of Lattice Materials

6.1 Introduction

6.2 Geometry, Material, and Loading Conditions

6.3 Stability of Finite-sized Specimens

6.4 Stability of Infinite Periodic Specimens

6.4.1 Microscopic Instability

6.5 Post-buckling Analysis

6.6 Effect of Buckling and Large Deformation on the Propagation Of Elastic Waves

6.7 Conclusions

References

Chapter 7 Impact and Blast Response of Lattice Materials

7.1 Introduction

7.2 Literature Review

7.2.1 Dynamic Response of Cellular Structures

7.2.2 Shock- and Blast-loading Responses of Cellular Structures

7.2.3 Dynamic Indentation Performance of Cellular Structures

7.3 Manufacturing Process

7.3.1 The Selective Laser Melting Technique

7.3.2 Sandwich Panel Manufacture

7.4 Dynamic and Blast Loading of Lattice Materials

7.4.1 Experimental Method - Drop-hammer Impact Tests

7.4.2 Experimental Method - Blast Tests on Lattice Cubes

7.4.3 Experimental Method - Blast Tests on Composite-lattice Sandwich Structures

7.5 Results and Discussion

7.5.1 Drop-hammer Impact Tests

7.5.2 Blast Tests on the Lattice Structures

7.5.3 Blast Tests on the Sandwich Panels

Concluding Remarks

Acknowledgements

References

Chapter 8 Pentamode Lattice Structures

8.1 Introduction

8.2 Pentamode Materials

8.2.1 General Properties

8.2.2 Small Rigidity and Poisson's Ratio of a PM

8.2.3 Wave Motion in a PM

8.3 Lattice Models for PM

8.3.1 Effective PM Properties of 2D and 3D Lattices

8.3.2 Transversely Isotropic PM Lattice

Effective Moduli: 2D

8.4 Quasi-static Pentamode Properties of a Lattice in 2D and 3D

8.4.1 General Formulation with Rigidity

8.4.2 Pentamode Limit

8.4.3 Two-dimensional Results for Finite Rigidity

8.5 Conclusion

Acknowledgements

References

Chapter 9 Modal Reduction of Lattice Material Models

9.1 Introduction

9.2 Plate Model

9.2.1 Mindlin-Reissner Plate Finite Elements

9.2.2 Bloch Boundary Conditions

9.2.3 Example Model

9.3 Reduced Bloch Mode Expansion

9.3.1 RBME Formulation

9.3.2 RBME Example

9.3.3 RBME Additional Considerations

9.4 Bloch Mode Synthesis

9.4.1 BMS Formulation

9.4.2 BMS Example

9.4.3 BMS Additional Considerations

9.5 Comparison of RBME and BMS

9.5.1 Model Size

9.5.2 Computational Efficiency

9.5.3 Ease of Implementation

References

Chapter 10 Topology Optimization of Lattice Materials

10.1 Introduction

10.2 Unit-cell Optimization

10.2.1 Parametric, Shape, and Topology Optimization

10.2.2 Selection of Studies from the Literature

10.2.3 Design Search Space

10.3 Plate-based Lattice Material Unit Cell

10.3.1 Equation of Motion and FE Model

10.3.2 Mathematical Formulation

10.4 Genetic Algorithm

10.4.1 Objective Function

10.4.2 Fitness Function

10.4.3 Selection

10.4.4 Reproduction

10.4.5 Initialization and Termination

10.4.6 Implementation

10.5 Appendix

References

Chapter 11 Dynamics of Locally Resonant and Inertially Amplified Lattice Materials

11.1 Introduction

11.2 Locally Resonant Lattice Materials

11.2.1 1D Locally Resonant Lattices

11.2.2 2D Locally Resonant Lattices

11.2.3 3D Locally Resonant Lattices

11.3 Inertially Amplified Lattice Materials

11.3.1 1D Inertially Amplified Lattices

11.3.2 2D Inertially Amplified Lattices

11.3.3 3D Inertially Amplified Lattices

11.4 Conclusions

References

Chapter 12 Dynamics of Nanolattices: Polymer-Nanometal Lattices

12.1 Introduction

12.2 Fabrication

12.2.1 Case Study

12.3 Lattice Dynamics

12.3.1 Lattice Properties

Geometries of 3D Lattices

Effective Material Properties of Nanometal-coated Polymer Lattices

12.3.2 Finite-element Model

Displacement Field

Kinetic Energy

Strain Potential Energy

Collected Equation of Motion

12.3.3 Floquet-Bloch Principles

Generalized Forces in Bloch Analysis

Reduced Equation of Motion

12.3.4 Dispersion Curves for the Octet Lattice

12.3.5 Lattice Tuning

Bandgap Placement

Lattice Optimization

12.4 Conclusions

12.5 Appendix: Shape Functions for a Timoshenko Beam with Six Nodal Degrees of Freedom

References

Index

EULA