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Analytical and Computational Modeling of Mechanical Waves in Microscale Granular Crystals: Nonlinearity and Rotational Dynamics

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Analytical and Computational Modeling of Mechanical Waves in Microscale Granular Crystals: Nonlinearity and Rotational Dynamics Analytical and Computational Modeling of Mechanical Waves in Microscale Granular Crystals: Nonlinearity and Rotational Dynamics Samuel P. Wallen A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Washington 2017 Supervisory Committee: Nicholas Boechler, Chair Bernard Deconinck, GSR Ashley Emery, Reading Committee Member I. Y. (Steve) Shen Duane Storti, Reading Committee Member Program Authorized to Offer Degree: Mechanical Engineering c Copyright 2017 Samuel P. Wallen University of Washington Abstract Analytical and Computational Modeling of Mechanical Waves in Microscale Granular Crystals: Nonlinearity and Rotational Dynamics Samuel P. Wallen Chair of the Supervisory Committee: Assistant Professor Nicholas Boechler Mechanical Engineering Granular media are one of the most common, yet least understood forms of matter on earth. The difficulties in understanding the physics of granular media stem from the fact that they are typically heterogeneous and highly disordered, and the grains interact via nonlinear contact forces. Historically, one approach to reducing these complexities and gaining new insight has been the study of granular crystals, which are ordered arrays of similarly-shaped particles (typically spheres) in Hertzian contact. Using this setting, past works explored the rich nonlinear dynamics stemming from contact forces, and proposed avenues where such granular crystals could form designer, dynamically responsive materials, which yield beneficial functionality in dynamic regimes. In recent years, the combination of self-assembly fabrication methods and laser ultrasonic experimental characterization have enabled the study of granular crystals at microscale. While our intuition may suggest that these microscale granular crystals are simply scaled-down versions of their macroscale counterparts, in fact, the relevant physics change drastically; for example, short-range adhesive forces between particles, which are negligible at macroscale, are several orders of magnitude stronger than gravity at microscale. In this thesis, we present recent advances in analytical and computational modeling of microscale granular crystals, in particular concerning the interplay of nonlinearity, shear interactions, and particle rotations, which have previously been either absent, or included separately at macroscale. Drawing inspiration from past works on phononic crystals and nonlinear lattices, we explore problems involving locally-resonant metamaterials, nonlinear localized modes, amplitude-dependent energy partition, and other rich dynamical phenomena. This work enhances our understanding of microscale granular media, which may find applicability in fields such as ultrasonic wave tailoring, signal processing, shock and vibration mitigation, and powder processing. TABLE OF CONTENTS Page List of Figures . v Chapter 1: Introduction . 1 1.1 Motivation . 1 1.2 Macroscale Granular Crystals . 3 1.2.1 Hertzian Contact . 3 1.2.2 One-Dimensional Granular Crystals . 5 1.2.3 Two- and Three-Dimensional Granular Crystals . 7 1.3 Influence of Shear Interactions and Particle Rotations . 9 1.3.1 Shear Contact Interactions . 9 1.3.2 Rotational Waves in the Linear Regime . 10 1.3.3 Theory of Generalized Continua . 11 1.4 Microscale Granular Crystals . 12 1.5 Organization of this Thesis . 15 Chapter 2: A Self-Assembled Metamaterial for Lamb Waves . 17 2.1 Introduction . 17 2.2 Theoretical Model . 18 2.3 Derivation of the Lamb Wave Dispersion Relation . 19 2.4 Discussion . 21 2.5 Conclusion . 23 Chapter 3: Discrete Breathers in a Mass-in-mass chain with Hertzian Local Resonators 24 3.1 Introduction . 24 3.2 Model . 27 3.2.1 Motivating Physical Scenario . 27 3.2.2 Hamiltonian 1D Discrete Model . 28 i 3.2.3 Parameter Fitting . 29 3.3 Numerical Investigation of Breathers . 33 3.4 Numerical Simulations . 36 3.4.1 Hamiltonian Case . 36 3.4.2 Effects of Energy Leakage . 38 3.5 Conclusion . 43 Chapter 4: Dynamics of a Monolayer of Microspheres on an Elastic Substrate . 45 4.1 Introduction . 46 4.2 Model . 47 4.2.1 Contact Stiffness . 49 4.2.2 Equations of Motion of the Spheres . 51 4.2.3 Effective Medium Approximation . 51 4.3 Dispersion Relations . 52 4.3.1 Rigid Substrate . 52 4.3.2 Elastic Substrate . 55 4.4 Numerical Results and Discussion . 57 4.4.1 Rigid Substrate . 57 4.4.2 Elastic Substrate . 60 4.5 Experimental Implications . 65 4.6 Conclusion . 66 4.7 Related Experimental Work . 67 Chapter 5: Nonlinear Dynamics of a 1D Granular Crystal Adhered to a Substrate 69 5.1 Introduction . 69 5.2 Model . 70 5.2.1 Normal contact: Hertz/DMT model . 70 5.2.2 Shear contact: Mindlin Model . 71 5.3 Equations of Motion . 71 5.3.1 Nondimensionalization . 72 5.3.2 Expansion of Nonlinear Forces . 72 5.3.3 Dimensionless Equations of Motion . 72 5.4 Dispersion Relation . 72 ii 5.5 Derivation of the Nonlinear Schrodinger Equation . 74 5.5.1 Slow Modulation of an Envelope . 74 5.5.2 Quasi-Discreteness Approximation . 74 5.5.3 Ansatz . 74 5.5.4 Solvability Conditions . 75 5.5.5 Focusing and Defocusing Behavior . 76 5.5.6 Breather Solutions . 77 5.6 Numerical Results . 78 5.7 Conclusions . 80 Chapter 6: Shear to Longitudinal Mode Conversion via Second Harmonic Generation in a 2D Microscale Granular Crystal . 81 6.1 Introduction . 82 6.2 Theory . 83 6.2.1 Model . 83 6.2.2 Quasi-linear Regime . 86 6.2.3 Weakly-nonlinear Regime: Second Harmonic Generation and Nonlinear Mode Conversion . 87 6.3 Simulations . 95 6.3.1 Numerical Setup . 95 6.3.2 Numerical Results . 95 6.4 Conclusion . 97 6.5 Contact Spring Displacements . 98 Chapter 7: Energy Partition in a 2D Microscale Granular Crystal Subject to an Impulsive Point Load: Multibody Dynamics Simulation . 100 7.1 Introduction . 100 7.2 Multibody Dynamics Simulation . 101 7.3 Numerical Setup . 103 7.4 Results and Discussion . 104 7.4.1 Spatial Distribution of Kinetic Energy . 105 7.4.2 Amplitude Dependence of Propagation and Particle Velocity . 107 7.5 Conclusion . 110 iii Bibliography . 112 Appendix A: Acknowledgments for Technical Content . 124 Appendix B: Computational Methods . 126 B.1 Solution of Ordinary Differential Equations . 126 B.1.1 Fourth-Order Explicit Runge-Kutta Method . 126 B.1.2 Step Size Selection . 128 B.2 Solution of Dispersion Equations . 129 B.3 On the Use of Mathematica, etc. for Algebraic Manipulation . 129 Appendix C: Details for Chapter 5: Nonlinear Dynamics of a 1D Granular Crystal Adhered to a Substrate . 131 C.1 Dimensionless Equations of Motion . 132 C.2 Solvability Conditions for the NLS Derivation . 133 C.2.1 Solution at O("1)............................. 133 C.2.2 Solution at O("2)............................. 134 C.2.3 Solution at O("3)............................. 135 iv LIST OF FIGURES Figure Number Page 1.1 Images of example macroscale granular crystals. (a) One-dimensional. (b) Two-dimensional, hexagonal-close-packing. (c) Three-dimensional, hexagonal- close-packing. 3 1.2 Schematic of a 1D granular crystal. Crosses denote the equilibrium positions of the sphere centers, and dots denote the current positions. The deformed contact regions are highlighted in red. 5 1.3 (a-c): Spatiotemporal plots of particle velocity for a simulated chain of 80 microscale spheres, for (a) near-linear, (b) weakly-nonlinear, and (c) strongly- nonlinear cases. (d-f): Time histories of velocity of the 30th sphere correspond- ing to (a-c). Dashed black lines in (a-c) indicated the lattice site whose time history is shown in (d-f). 8 1.4 Macroscopic view of a microscale GC, fabricated using a convective self- assembly technique. ..
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