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Copyright by Stephanie Gabrielle Konarski 2014 Copyright by Stephanie Gabrielle Konarski 2014 The Thesis Committee for Stephanie Gabrielle Konarski Certifies that this is the approved version of the following thesis: Nonlinear Acoustic and Dynamic Response of Heterogeneous Materials Containing Snapping Acoustic Metamaterial Inclusions APPROVED BY SUPERVISING COMMITTEE: Mark F. Hamilton, Supervisor Michael R. Haberman, Co-Supervisor Nonlinear Acoustic and Dynamic Response of Heterogeneous Materials Containing Snapping Acoustic Metamaterial Inclusions by Stephanie Gabrielle Konarski, B.S. THESIS Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE IN ENGINEERING THE UNIVERSITY OF TEXAS AT AUSTIN August 2014 Acknowledgments First and foremost, I would like to thank my advisor, Dr. Michael R. Haberman, and secondary advisor, Dr. Mark F. Hamilton, whose guidance is invaluable and patience is very much appreciated. I would also like to acknowledge the Applied Research Laboratories at the University of Texas at Austin for awarding me the McKinney Fellowship in Acoustics. which has allowed me the opportunity to pursue an interesting research topic that I love. Finally, thank you to all my friends who have kept me sane over the years and provided the helpful advice and random distractions I needed to write this thesis. iv Nonlinear Acoustic and Dynamic Response of Heterogeneous Materials Containing Snapping Acoustic Metamaterial Inclusions Stephanie Gabrielle Konarski, M.S.E. The University of Texas at Austin, 2014 Supervisors: Mark F. Hamilton Michael R. Haberman Acoustic metamaterials are sub-wavelength structures designed to over- come limitations in the material properties of conventional materials. The present research focuses on the nonlinear acoustic and dynamic response of a specific type of engineered microstructure called a snapping acoustic metama- terial (SAMM). Snapping of these elements is defined as large, rapid deforma- tions induced by infinitesimal perturbations in the time-varying external pres- sure. Snapping behavior in SAMM elements results from their non-monotonic stress-strain response, which displays regimes of positive and negative stiff- ness. This work presents a modeling study of the nonlinear behavior of both individual SAMM elements and a heterogeneous material containing a dilute concentration of SAMM elements embedded in a nearly incompressible vis- coelastic solid. Two different scenarios are considered: (i) nonlinear wave v propagation in the heterogeneous medium, and (ii) forced nonlinear dynam- ics of inclusions embedded in a viscoelastic medium. The nonlinearity of the SAMM elements is represented by a cubic pressure-volumetric strain relation- ship based on finite element model results from previous work. The effective nonlinear response of a heterogeneous mixture of SAMM elements embedded in a matrix, characterized by the parameters B=A and C=A, is then deter- mined using both a nonlinear mixture law and a nonlinear Hashin-Shtrikman approach. The former estimate is limited to matrix materials with zero shear modulus, which cannot stabilize SAMM inclusions in regimes of negative stiff- ness. The augmented Hashin-Shtrikman method, however, includes nonlinear elasticity and the shear modulus of the matrix material. It therefore provides accurate estimates of the homogenized material when SAMM elements display negative stiffness and enhanced acoustical nonlinearity. The distortion of an acoustic wave propagating through the effective medium is studied through nu- merical solution of a nonlinear evolution equation that includes both quadratic and cubic nonlinearity. Finally, the forced nonlinear dynamic response of both a single SAMM element in a matrix and a domain of effective medium material embedded in matrix is considered. This behavior is of interest for generating enhanced absorption of acoustic wave energy because snapping leads to large hysteresis in the stress-strain response. A generalized Rayleigh-Plesset anal- ysis is adapted to model the large-deformation dynamics associated with the system. vi Table of Contents Acknowledgments iv Abstract v List of Tables x List of Figures xi Chapter 1. Introduction 1 1.1 Motivation . .1 1.2 Metamaterials . .2 1.2.1 Negative Stiffness . .7 1.3 Related Examples and Applications . .9 1.3.1 Linear Acoustic Metamaterials . 10 1.3.1.1 Wave Manipulation Due to Resonant Effects . 10 1.3.1.2 Damping Due to Structural Negative Stiffness . 12 1.3.2 Nonlinear Systems . 13 1.3.2.1 Vibration Isolation . 13 1.3.2.2 Nonlinear Wave Manipulation . 15 1.4 Objective . 16 1.5 Thesis Overview . 17 Chapter 2. Governing Behavior of Snapping Acoustic Metama- terial Inclusions 21 2.1 Nonlinear Stiffness . 23 2.1.1 Nonlinear Springs . 24 2.1.2 Nonlinear Stress-Strain Response . 27 2.2 Constitutive Relationship . 31 2.3 Representative SAMM Inclusion Response . 36 vii 2.3.1 Overall Behavior . 37 2.3.2 Local Coefficients . 41 2.4 Summary . 47 Chapter 3. Acoustic Nonlinearity 48 3.1 Acoustic Nonlinearities . 49 3.1.1 State Equations . 49 3.1.2 Parameters of Nonlinearity . 52 3.2 Homogenization . 56 3.2.1 Mixture Law Homogenization Approach . 58 3.2.1.1 Theoretical Formulation . 59 3.2.1.2 Mixture Law Homogenization Results . 60 3.2.2 Hashin-Shtrikman Approach . 67 3.2.2.1 Theoretical Formulation . 68 3.2.2.2 Nonlinear Hashin-Shtrikman Homogenization Re- sults . 74 3.3 Nonlinear Macroscopic Behavior . 81 3.3.1 Comparison with Conventional Materials . 86 3.4 Propagation Model . 89 3.4.1 Evolution Equation . 89 3.4.2 Shock Formation Distance . 91 3.4.3 Dimensionless Evolution Equation . 99 3.4.4 Simulations of Waveform Distortion . 100 Chapter 4. Nonlinear Dynamics 109 4.1 Dissipation of Energy . 109 4.2 Inclusion Dynamics of Rayleigh-Plesset Form . 112 4.2.1 SAMM Inclusion . 114 4.2.2 Nonlinear Effective Medium Response . 118 4.3 Nonlinear Dynamics Results . 122 4.3.1 Single SAMM Inclusion . 124 4.3.2 Effective Medium with SAMM Inclusions . 132 4.3.3 Comparing Effective Medium with SAMM Inclusions and Air Filled Voids . 137 4.4 Discussion . 139 viii Chapter 5. Conclusion 142 5.1 Summary . 142 5.2 Future Work . 146 Appendices 148 Appendix A. Derivation of Lossy Wave Equation with Quadratic and Cubic Nonlinearity 149 A.1 Wave Equation . 149 A.2 Evolution Equation . 154 Appendix B. Harmonic Materials 158 Appendix C. Time Domain Algorithm Including Cubic Nonlin- earity 164 C.1 Nonlinear Distortion . 164 C.2 Thermoviscous Absorption . 165 Appendix D. Derivation of Rayleigh-Plesset Type Equation 168 Bibliography 172 ix List of Tables 3.1 Bulk modulus, B=A, C=A and C=B for example fluids, solids, and mixtures containing bubbles from the references denoted in the last column unless otherwise noted in the table cells. 54 x List of Figures 1.1 Examples of acoustic metamaterials in the literature. .4 2.1 Diagram of the SAMM inclusion of interest. 22 2.2 Force versus displacement for linear, hardening and softening springs. 25 2.3 Amplitude versus frequency for (a) linear spring, (b) hardening spring and (c) softening spring . 26 2.4 Example nonlinear stress-strain constitutive curve. 29 2.5 Fourth order strain energy density function versus displacement in mm for a single SAMM inclusion from FEM. 37 2.6 Pressure in MPa as a function of volume corresponding to six deformed states of SAMM inclusion that demonstrate the snap- ping response. 39 2.7 Hydrostatic pressure, Pi0 , for small perturbations about a vol- umetric pre-strain imposed on the SAMM incluison. 42 2.8 Local linear stiffness, Ki, for small perturbations about a volu- metric pre-strain imposed on the SAMM inclusion. 43 0 2.9 Local second order modulus, Ki , for small perturbations about a volumetric pre-strain imposed on the SAMM inclusion. 45 00 2.10 Local third order modulus, Ki , for small perturbations about a volumetric pre-strain imposed on the SAMM inclusion. 46 3.1 Multiscale model for an effective medium containing a dilute concentration of inclusions showing the macroscale, mesoscale and microscale. 58 3.2 (B=A)∗ derived from the mixture law with φ = 1% SAMM inclusions in water. 62 3.3 (C=A)∗ derived from the mixture law with φ = 1% SAMM inclusions in water. 63 3.4 (C=B)∗ derived from the mixture law with φ = 1% SAMM inclusions in water. 64 3.5 K∗ derived from the mixture law with φ = 1% SAMM inclusions in water. 65 xi 3.6 Schematic of homogenization using a Hashin-Shtrikman homog- enization scheme. 68 3.7 K∗ derived from the Hashin-Shtrikman approach with φ = 1% SAMM inclusions for Km = 2 GPa and νm = 0:5, ν = 0:4995, νm = 0:4991, and νm = 0:4989. 76 3.8 (B=A)∗ derived from the Hashin-Shtrikman approach with φ = 1% SAMM inclusions for Km = 2 GPa and νm = 0:5, ν = 0:4995, νm = 0:4991, and νm = 0:4989. 77 3.9 (C=A)∗ derived from the Hashin-Shtrikman approach with φ = 1% SAMM inclusions for Km = 2 GPa and νm = 0:5, ν = 0:4995, νm = 0:4991, and νm = 0:4989. 78 3.10 (C=B)∗ derived from the Hashin-Shtrikman approach with φ = 1% SAMM inclusions for Km = 2 GPa and νm = 0:5, ν = 0:4995, νm = 0:4991, and νm = 0:4989. 80 3.11 Pressure-volume relationship for the overall macroscopic response as a function of the local perturbations on the microscale with φ = 1% SAMM inclusions for Km = 2 GPa and νm = 0:5, ν = 0:4995, νm = 0:4991, and νm = 0:4989. 83 3.12 Pressure-volume relationship near zero strain for the overall macroscopic response as a function of the local perturbations on the microscale with φ = 1% SAMM inclusions for Km = 2 GPa and νm = 0:5, ν = 0:4995, νm = 0:4991, and νm = 0:4989.
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