Geometric Control of Fracture and Topological Metamaterials

Foreword
Preface
Acknowledgments
Parts of This Thesis Have Been Published in the Following Journal Articles
Contents
1 Introduction
	1.1 Curvature and Geometry
	1.2 From Geometry to Topology
	1.3 From Mathematics to Mechanics
		1.3.1 Curvature and Elasticity in Thin Sheets
		1.3.2 Berry Curvature, Chern Numbers, and Topological Mechanics
	1.4 Scope of This Book
Part I Gaussian Curvature as a Guide for Material Failure
	2 Fracture in Sheets Draped on Curved Surfaces
		2.1 Gaussian Curvature as a Tool
		2.2 Fracture Onset: Griffith Lengths and Crack Kinking
			2.2.1 Griffith Length for a Small Crack
			2.2.2 Crack Kinking
		2.3 Crack Trajectories
			2.3.1 Perturbation Theory Prediction of Crack Paths
			2.3.2 Phase-Field Model on Curved Surfaces
		2.4 Crack Arrest
		2.5 Controlling Cracks with More Complex Surfaces
		2.6 Conclusion
	3 Conforming Nanoparticle Sheets to Surfaces with Gaussian Curvature
		3.1 Gaussian Curvature and Nanoparticle Sheets
		3.2 Experimental Procedure
		3.3 Monolayer Morphology: Coverage, Cracks, and Folds
		3.4 Energy Scaling
			3.4.1 Energy Costs to Conform: Bending and Stretching
			3.4.2 Alternatives to Elastic Conformation: Avoiding Adhesion, Plastic Deformation, and Folding
			3.4.3 Three Regimes Arise from Energy Scaling
		3.5 Bending and Adhesion
		3.6 Strain Analysis
			3.6.1 Image Analysis
			3.6.2 Spring Network Simulations
			3.6.3 Comparison with Incompressible Solution
			3.6.4 Azimuthal Cracks in Simulations
		3.7 Plastic Deformation
			3.7.1 Formation of Dislocations
			3.7.2 Formation of Azimuthal Cracks
		3.8 Formation of Folds at Large Sphere Sizes
		3.9 Conclusion
Part II Topological Mechanics in Gyroscopic Metamaterials
	4 Realization of a Topological Phase Transition in a GyroscopicLattice
		4.1 Topological Phase Transitions
		4.2 Experimental Setup
		4.3 Broken Symmetries in the Honeycomb Lattice
		4.4 Breaking Inversion Symmetry in Experiment
		4.5 Measuring the Topological Phase Transition
		4.6 Competing Broken Symmetries
		4.7 Conclusion
	5 Tunable Band Topology in Gyroscopic Lattices
		5.1 Gyroscopic Lattices
		5.2 The Equations of Motion
		5.3 Twisted Spindle Lattice
		5.4 Time Reversal Symmetry and Topological Bandgaps
		5.5 Competing Symmetries in Topological Gyroscopic Systems
		5.6 Towards Topological Design
		5.7 Conclusion
	6 Topological Insulators Constructed from Random Point Sets
		6.1 Gyroscopic Metamaterials as a Model System
		6.2 Amorphous Voronoi Networks
		6.3 Interpretation of the Real-Space Chern Number
		6.4 Local Geometry Controls Band Topology
		6.5 Spectral Flow Through Adiabatic Pumping
		6.6 Broken Time Reversal Symmetry
		6.7 Conclusion
	7 Conclusions and Outlook
A Creation of Surfaces of Revolution with Prescribed Gaussian Curvature
	A.1 Governing Equations
	A.2 Equations for the Surface of a Pseudosphere
	A.3 Obtaining Geodesic Circles
B Stretching Energy in Stamped Sheets on Spherical Surfaces
C Symplectic Structure of Gyroscopic Motion
D Interpretation of Real-Space Chern Number
References