Geometric Control of Fracture and Topological Metamaterials (eBook)

Foreword 7
Preface 10
Acknowledgments 11
Parts of This Thesis Have Been Published in the Following Journal Articles 12
Contents 13
1 Introduction 16
1.1 Curvature and Geometry 18
1.2 From Geometry to Topology 19
1.3 From Mathematics to Mechanics 21
1.3.1 Curvature and Elasticity in Thin Sheets 22
1.3.2 Berry Curvature, Chern Numbers, and Topological Mechanics 23
1.4 Scope of This Book 28
Part I Gaussian Curvature as a Guide for Material Failure 30
2 Fracture in Sheets Draped on Curved Surfaces 31
2.1 Gaussian Curvature as a Tool 31
2.2 Fracture Onset: Griffith Lengths and Crack Kinking 32
2.2.1 Griffith Length for a Small Crack 34
2.2.2 Crack Kinking 35
2.3 Crack Trajectories 37
2.3.1 Perturbation Theory Prediction of Crack Paths 37
2.3.2 Phase-Field Model on Curved Surfaces 39
2.4 Crack Arrest 42
2.5 Controlling Cracks with More Complex Surfaces 42
2.6 Conclusion 44
3 Conforming Nanoparticle Sheets to Surfaces with Gaussian Curvature 45
3.1 Gaussian Curvature and Nanoparticle Sheets 46
3.2 Experimental Procedure 48
3.3 Monolayer Morphology: Coverage, Cracks, and Folds 49
3.4 Energy Scaling 50
3.4.1 Energy Costs to Conform: Bending and Stretching 52
3.4.2 Alternatives to Elastic Conformation: Avoiding Adhesion, Plastic Deformation, and Folding 52
3.4.3 Three Regimes Arise from Energy Scaling 53
3.5 Bending and Adhesion 54
3.6 Strain Analysis 55
3.6.1 Image Analysis 55
3.6.2 Spring Network Simulations 58
3.6.3 Comparison with Incompressible Solution 60
3.6.4 Azimuthal Cracks in Simulations 60
3.7 Plastic Deformation 61
3.7.1 Formation of Dislocations 61
3.7.2 Formation of Azimuthal Cracks 63
3.8 Formation of Folds at Large Sphere Sizes 64
3.9 Conclusion 64
Part II Topological Mechanics in Gyroscopic Metamaterials 66
4 Realization of a Topological Phase Transition in a GyroscopicLattice 67
4.1 Topological Phase Transitions 67
4.2 Experimental Setup 68
4.3 Broken Symmetries in the Honeycomb Lattice 70
4.4 Breaking Inversion Symmetry in Experiment 70
4.5 Measuring the Topological Phase Transition 72
4.6 Competing Broken Symmetries 74
4.7 Conclusion 76
5 Tunable Band Topology in Gyroscopic Lattices 77
5.1 Gyroscopic Lattices 77
5.2 The Equations of Motion 78
5.3 Twisted Spindle Lattice 80
5.4 Time Reversal Symmetry and Topological Bandgaps 84
5.5 Competing Symmetries in Topological Gyroscopic Systems 85
5.6 Towards Topological Design 87
5.7 Conclusion 89
6 Topological Insulators Constructed from Random Point Sets 90
6.1 Gyroscopic Metamaterials as a Model System 90
6.2 Amorphous Voronoi Networks 91
6.3 Interpretation of the Real-Space Chern Number 93
6.4 Local Geometry Controls Band Topology 95
6.5 Spectral Flow Through Adiabatic Pumping 96
6.6 Broken Time Reversal Symmetry 100
6.7 Conclusion 103
7 Conclusions and Outlook 104
A Creation of Surfaces of Revolution with Prescribed Gaussian Curvature 107
A.1 Governing Equations 107
A.2 Equations for the Surface of a Pseudosphere 108
A.3 Obtaining Geodesic Circles 109
B Stretching Energy in Stamped Sheets on Spherical Surfaces 112
C Symplectic Structure of Gyroscopic Motion 114
D Interpretation of Real-Space Chern Number 117
References 123