We investigate the dynamics of one-dimensional elastic lattices with disorder in the form of random non-local connectivites according to the “small-world” model of network connections. We present preliminary investigations on their dynamics, and illustrate the formation of spectral gaps which are formed for increasing degrees of disorder. These gaps are shown to persist across multiple lattice realizations and different sizes, signaling a robust property of the disorder systems. We also discuss possible experimental realizations on acoustic waveguides. These preliminary findings illustrate intriguing possibilities for wave manipulation and transport properties that are enabled via disorder in elastic and acoustic systems with random network connections.
Owing to smooth surface transitions and open-cell connectivity, minimal surfaces can be conveniently designed to form singly, doubly, triply periodic, and quasi-periodic structures, that provide new opportunities for novel metamaterial design of potential engineering relevance. Prior studies have demonstrated band gaps and engineered mechanical properties in minimal surface structures. However, their potential for the exploitation of topological phenomena through elastic waves remains largely unexplored. In this work, we design periodic and quasi-periodic minimal surface metamaterials and explore their topological properties. We start with the construction of 1D/2D periodic minimal surfaces with dimerized-like parametrizations. The designs facilitate the opening of band gaps upon breaking the inversion symmetry of the unit cells through a band inversion process that defines distinct topological phases. Simulations demonstrate the existence of 0D localized modes in 1D interfaced structures, and robust waveguiding along a valley-type 1D interface separating distinct 2D domains. These designs are fabricated with additive manufacturing technologies and tested with laser vibrometry, confirming the presence of the predicted topological states. We then investigate 1D quasi-periodic minimal surfaces through a quasi-periodic modulation of the dimerization parameter. Such structures support topological gaps forming a fractal spectrum that resemble the Hofstadter butterfly. The existence of non-trivial gaps and localized modes in quasiperiodic systems extend the avenues for wave localization and transport exploring higher dimensional topologies. With the growth of additive manufacturing techniques, the presented framework of minimal surfaces provides remarkable design freedom to explore a variety of symmetries in 1D, 2D, and 3D domains, enabling a variety of other wave physics and topological phenomena to be explored.
Topological metamaterials are a new class of materials that support topological modes such as edge modes and interface modes, which are commonly immune to scattering and imperfections. This novelty has been the subject of extensive research in many branches of physics such as electronics, photonics, phononics, and acoustics. The nontrivial topological properties related to the presence of topological modes are tipically found in periodic media. However, it was recently demonstrated that structures called quasicrystals may also exhibit nontrivial topological behavior attributed to dimensions higher than that of the quasicrystal. While quasiperiodicity has received a lot of attention in the fields of crystallography and photonics, research into quasiperiodic elastic structures has been scarce. In this paper, we show how the concepts of quasiperiodicity may be applied to the design of topological mechanical metamaterials. We start by investigating the boundary modes present in quasiperiodic 1D phononic lattices. These modes have the interesting property of being localized at either one of the two different boundaries depending on the value of an additional parameter, which is remnant of the higher dimension. A smooth variation of this parameter in either time or a spatial dimension can lead to a robust transfer of energy between two sites of the structure. We present an idealized mechanical system composed by an array of coupled rods that may be used as a platform for realizing this kind of robust transfer of energy. These are preliminary investigations into a entirely new class of structures which may lead to novel engineering applications.
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