The modern theory of charge polarization in solids is based on a generalization of Berry's phase. The possibility of the quantization of this phase arising from parallel transport in momentum space is essential to our understanding of systems with topological band structures. Although based on the concept of charge polarization, this same theory can also be used to characterize the Bloch bands of neutral bosonic systems such as photonic or phononic crystals. The theory of this quantized polarization has recently been extended from the dipole moment to higher multipole moments. In particular, a two-dimensional quantized quadrupole insulator is predicted to have gapped yet topological one-dimensional edge modes, which stabilize zero-dimensional in-gap corner states. However, such a state of matter has not previously been observed experimentally. Here we report measurements of a phononic quadrupole topological insulator. We experimentally characterize the bulk, edge and corner physics of a mechanical metamaterial (a material with tailored mechanical properties) and find the predicted gapped edge and in-gap corner states. We corroborate our findings by comparing the mechanical properties of a topologically non-trivial system to samples in other phases that are predicted by the quadrupole theory. These topological corner states are an important stepping stone to the experimental realization of topologically protected wave guides in higher dimensions, and thereby open up a new path for the design of metamaterials.

In most macroscale robotic systems, propulsion and controls are enabled through a physical tether or complex onboard electronics and batteries. A tether simplifies the design process but limits the range of motion of the robot, while onboard controls and power supplies are heavy and complicate the design process. Here, we present a simple design principle for an untethered, soft swimming robot with preprogrammed, directional propulsion without a battery or onboard electronics. Locomotion is achieved by using actuators that harness the large displacements of bistable elements triggered by surrounding temperature changes. Powered by shape memory polymer (SMP) muscles, the bistable elements in turn actuate the robot's fins. Our robots are fabricated using a commercially available 3D printer in a single print. As a proof of concept, we show the ability to program a vessel, which can autonomously deliver a cargo and navigate back to the deployment point.

We consider two-dimensional phononic crystals formed from silicon and voids, and present optimized unit cell designs for (1) out-of-plane, (2) in-plane and (3) combined out-of-plane and in-plane elastic wave propagation. To feasibly search through an excessively large design space (∼10 40 possible realizations) we develop a specialized genetic algorithm and utilize it in conjunction with the reduced Bloch mode expansion method for fast band structure calculations. Focusing on highsymmetry plain-strain square lattices, we report unit cell designs exhibiting record values of normalized band-gap size for all three categories. For the combined polarizations case, we reveal a design with a normalized band-gap size exceeding 60%.Phononic crystals (PnCs) are periodic materials that exhibit distinct frequency characteristics such as the possibility of formation of band gaps. In general, it is most advantagous to have the frequency range of a band gap maximized while pulling its midpoint as low as possible in order to keep the unit cell size to a minimum. Selecting the topological distribution of the material phases inside the unit cell provides a a powerful means towards reaching this target, and this has been the focus of numerous research studies not only on PnCs but also photonic crystals (PtCs).The exploration for optimal unit cell designs was initiated by Cox and Dobson in 1999 [15] (in the context of PtCs). The articles by Burger et al. [16] and Jensen and Sigmund [17] provide a review of subsequent studies concerned with band-gap widening in PtCs. In the area of PnCs, the problem has been treated in a variety of settings and using several techniques. For example, unit cells have been optimized in one-dimension [18,19] and in two-dimensions (2D) [20][21][22][23][24][25], using gradient-based [21][22][23] as well as non-gradient-based [24,25] techniques. Interest in band-gap size maximation has also been treated outside the scope of the unit cell dispersion problem [21,26]. In all these optimization studies the focus has been primarily on PnCs based on an infinite thickness model and a material composition consisting of two or more solid (or solid and fluid) phases with the exception of a few investigations that considered thin-plate singlephase models [22,23]. Recognizing the practical significance of solid-and-air PnCs with relatively large crosssectional thickness, some studies considered the configuraton of a 2D solid matrix with periodic cylindrical voids * Corresponding author; mih@colorado.edu.-modeled under 2D plain-strain conditons [27] or as a three-dimensional continuum with free surface boundary conditions [28] -and investigated the dependence of band-gap size upon the void radius. For combined out-of-plane and in-plane waves in 2D infinite-thickness PnCs formed from silicon and a square lattice of circular voids, it has been shown that the band-gap size normalized with respect to the mid-gap frequency cannot exceed 40% [27]. In this letter we utlize a specialized optimization algorithm in pursuit of ...

We investigate the dispersion characteristics of locally resonant elastic metamaterials formed by the erection of pillars on the solid regions in a plate patterned by a periodic array of holes. We show that these solid regions effectively act as springboards leading to an enhanced resonance behavior by the pillars when compared to the nominal case of pillars with no holes. This local resonance amplification phenomenon, which we define as the trampoline effect, is shown to cause subwavelength band gaps to increase in size by up to a factor of 4. This outcome facilitates the utilization of subwavelength metamaterial properties over exceedingly broad frequency ranges.Phononic crystals and locally resonant acoustic/elastic metamaterials have been the focus of extensive research efforts in recent years due to their attractive dynamical characteristics, such as the possibility of exhibiting band gaps. In a phononic crystal, band gaps are generated by Bragg scattering for which an underlying constraint is that the wavelength has to be on the order of the lattice spacing. 1,2 In a locally resonant acoustic/elastic metamaterial, on the other hand, band gaps may be generated by the mechanism of hybridization between local resonances and the dispersion properties of the underlying medium, and this in turn may take place at the subwavelength regime. 3 While in principal periodicity is not a necessity in a metamaterial, the introduction of the locally resonant elements in a symmetric fashion enables intrinsic, unitcell based, description of the wave propagation characteristics, in addition to attaining the benefits of order and compactness. The presence of periodicity, in itself, produces direction-dependent frequency bands and band gaps (caused by Bragg scattering). These unique dispersion properties may sufficiently be utilized in numerous applications involving wave filtering, 4-6 localization, 7,8 guiding, 8,9 focusing, 10-12 collimation, 13,14 among others. The added feature of local resonance, however, gives rise to a qualitatively different type of dynamical behavior, such as negative effective elastic moduli and/or density, 15-18 along with the possibility of generation of subwavelength band gaps. 3 The applications of locally resonant acoustic/elastic metamaterials are, in turn, far from conventional, e.g., subfrequency wave isolation, 19 subwavelength focusing and imaging, 20,21 and cloaking, 22 to name a few. Crossing the boundaries of acoustics and elasticity, our group at CU-Boulder has recently proposed the utilization of locally resonant metamaterials for the control of heat in a semiconducting thin-film. 23 Regardless of the application, one of the generally desirable characteristics is for the metamaterial to exhibit a large band gap. While the problem of unit-cell optimization for maximum band-gap size has been actively pursued a) mih@colorado.edu for phononic crystals, 6,24,25 less work has been done in exploring new approaches for band-gap enlargement in locally resonant acoustic/elastic metamaterials.One...