Dynamic Homogenization Of Linear Waves In Periodic Media And Origami-Inspired Structures

Abstract

This dissertation aims to establish a comprehensive analytical framework for dynamic homogenization of wave motion at arbitrary frequency in (i) “perforated” periodic con- tinua, and (ii) periodic origami-inspired structures described via “bar-and-hinge” com- putational paradigm. For a given spectral (i.e. frequency-wavenumber) content the body force acting on the structure, the “activated” Bloch eigenstates of the lattice are iden- tified and classified depending on the multiplicity of participating energy levels. In the vicinity of an isolated dispersion surface (single energy level), an effective field equation with homogenized source term is formulated (via projection onto the dominant Bloch eigenstate) to obtain the leading- and second-order approximations of both macroscopic, i.e. “mean”, and microscopic wave motion. When the activated spectral neighborhood features more than one dispersion surface, the zeroth- and first-order systems of effec- tive field equations with homogenized source terms are formulated, covering a variety of topological configurations such as Dirac points, avoided crossings, and near-Dirac points. On setting the source term to zero, the featured system of equations degenerates to a low-order algebraic eigenvalue problem that accurately captures the local geometry of (a cluster of) dispersion surfaces. The proposed homogenization framework is veri- fied by comparing the asymptotic approximation of the dispersion relationship with its numerically-evaluated counterpart and deployed to approximate the total and effective motion in: (i) two-dimensional (2D) Kagome lattice featuring nearly-hexagonal Neu- mann exclusions, (ii) 2D square lattice of circular Dirichlet obstacles, (iii) 2D Miura-ori origami structure, and (iv) 1D Miura tube. Specifically, the asymptotic model is shown to approximate the dispersion relationship in the neighborhood of isolated dispersion surfaces and tight clusters thereof with equal fidelity. It is also found that the homog- enized model is capable of accurately capturing the body-force induced waveforms in a lattice, both in terms of macroscopic i.e. effective motion and microstructural motion when higher-order models are considered.