UC San Diego

Herein we study novel wave phenomena found in topological mechanical metamaterials. Both zero and finite frequency topological mechanical metamaterials are studied. Previous works demonstrating the ability to manipulate mechanical energy utilize sub-wavelength scale patterns. We first expand the linear topological theory, and demonstrate that under a proper change of coordinates, previously unknown topological features arise in certain materials. We take a classic example of a mass spring model with alternating masses, and show that by studying the strains in the system, instead of the displacements, hidden symmetries arise that provide topological protection to edge modes present in the system. However unlike the fixed boundary conditions necessary in a stiffness dimer, the mass dimer requires free boundary conditions to preserve the necessary symmetries for topological protection. We further demonstrate that this framework is not restricted to mechanical topological systems but can be used in any system characterized by a second order differential equation that can be modeled in a reduced order mass spring network.

We then focus on the interplay between a local resonance in a structure and finite frequency topological edge modes. We introduce an effective stiffness dimer that has an effective frequency dependent spring. We find the new system is able to support two sets of edge modes, one set in the Bragg gap, and one set in the local resonance induced gap. We also demonstrate the tunability of the topological modes introduced by local resonance, allowing the edge modes to move inside of the bandgap due to the local resonators. We show for a special case, when the edge mode exists at a critical frequency where the effective stiffness is zero, the system demonstrates singular modes, where the amplitude is purely localized at one unit cell and the decay rate into the bulk is theoretically infinite.

We then move our studies to zero frequency topological mechanical metamaterials. We study exact geometric solutions to the zero modes of a twisted topological Kagome Lattice. Nonlinear wavelike phenomena is observed, and a mapping is made between the 2+0 dimensional zero modes, and a 1+1 dimension time varying system. We observe special nonlinear effects such as spatial harmonic generation, localized topological switching, and solitary waves. Finally in the fifth chapter, we finally extend the study by introducing weak torsional springs into the system, changing the 2+0 dimensional wave equation into a locally 2+1 dimensional ultrahyperbolic wave propagation in the topological Maxwell lattice under specific excitation conditions. Using a continuum model we demonstrate the hyperbolic nature of the zero energy modes, before raising the modes to finite frequency creating a locally ultrahyperbolic equation. We demonstrate strong wave beaming and directional confinement that is tied to the topological polarization of the lattice.