39 research outputs found
Non-singular and singular flat bands in tunable acoustic metamaterials
Dispersionless flat bands can be classified into two types: (1) non-singular
flat bands whose eigenmodes are completely characterized by compact localized
states; and (2) singular flat bands that have a discontinuity in their Bloch
eigenfunctions at a band touching point with an adjacent dispersive band,
thereby requiring additional extended states to span their eigenmode space. In
this study, we design and numerically demonstrate two-dimensional thin-plate
acoustic metamaterials in which tunable flat bands of both kinds can be
achieved. Non-singular flat bands are achieved by fine-tuning the ratio of the
global tension and the bending stiffness in triangular and honeycomb lattices
of plate resonators. A singular flat band arises in a kagome lattice due to the
underlying lattice geometry, which can be made degenerate with two additional
flat bands by tuning the plate tension. A discrete model of the continuum
thin-plate system reveals the interplay of geometric and mechanical factors in
determining the existence of flat bands of both types. The singular nature of
the kagome lattice flat band is established via a metric called the
Hilbert-Schmidt distance calculated between a pair of eigenstates
infinitesimally close to the quadratic band touching point. We also simulate an
acoustic manifestation of a robust boundary mode arising from the singular flat
band and protected by real-space topology in a finite system. Our theoretical
and computational study establishes a framework for exploring flat-band physics
in a tunable classical system, and for designing acoustic metamaterials with
potentially useful sound manipulation capabilities.Comment: 11 pages (excluding references) and 7 figure
Space-time symmetry and parametric resonance in dynamic mechanical systems
Linear mechanical systems with time-modulated parameters can harbor
oscillations with amplitudes that grow or decay exponentially with time due to
the phenomenon of parametric resonance. While the resonance properties of
individual oscillators are well understood, identifying the conditions for
parametric resonance in systems of coupled oscillators remains challenging.
Here, we identify internal symmetries that arise from the real-valued and
symplectic nature of classical mechanics and determine the parametric resonance
conditions for periodically time-modulated mechanical metamaterials using these
symmetries. Upon including external symmetries, we find additional conditions
that prohibit resonances at some modulation frequencies for which parametric
resonance would be expected from the internal symmetries alone. In particular,
we analyze systems with space-time symmetry where the system remains invariant
after a combination of discrete translation in both space and time. For such
systems, we identify a combined space-time translation operator that provides
more information about the system than the Floquet operator does, and use it to
derive conditions for one-way amplification of traveling waves. Our results
establish an exact theoretical framework based on symmetries to engineer exotic
responses such as nonreciprocal transport and one-way amplification in
space-time modulated mechanical systems, and can be generalized to all physical
systems that obey space-time symmetry.Comment: 14 pages, 4 figure
Non-dispersive one-way signal amplification in sonic metamaterials
Parametric amplification -- injecting energy into waves via periodic
modulation of system parameters -- is typically restricted to specific
multiples of the modulation frequency. However, broadband parametric
amplification can be achieved in active metamaterials which allow local
parameters to be modulated both in space and in time. Inspired by the concept
of luminal metamaterials in optics, we describe a mechanism for one-way
amplification of sound waves across an entire frequency band using
spacetime-periodic modulation of local stiffnesses in the form of a traveling
wave. When the speed of the modulation wave approaches that of the speed of
sound in the metamaterial -- a regime called the sonic limit -- nearly all
modes in the forward-propagating acoustic band are amplified, whereas no
amplification occurs in the reverse-propagating band. To eliminate divergences
that are inherent to the sonic limit in continuum materials, we use an exact
Floquet-Bloch approach to compute the dynamic excitation bands of discrete
periodic systems. We find wide ranges of parameters for which the amplification
is nearly uniform across the lowest-frequency band, enabling amplification of
wavepackets while preserving their speed, shape, and spectral content. Our
mechanism provides a route to designing acoustic metamaterials which can
propagate wave pulses without losses or distortion across a wide range of
frequencies.Comment: 12 pages, 7 figures; v2: compressed images for faster renderin
Fluctuating shells under pressure
Thermal fluctuations strongly modify the large length-scale elastic behavior
of crosslinked membranes, giving rise to scale-dependent elastic moduli. While
thermal effects in flat membranes are well understood, many natural and
artificial microstructures are modeled as thin elastic {\it shells}. Shells are
distinguished from flat membranes by their nonzero curvature, which provides a
size-dependent coupling between the in-plane stretching modes and the
out-of-plane undulations. In addition, a shell can support a pressure
difference between its interior and exterior. Little is known about the effect
of thermal fluctuations on the elastic properties of shells. Here, we study the
statistical mechanics of shape fluctuations in a pressurized spherical shell
using perturbation theory and Monte Carlo computer simulations, explicitly
including the effects of curvature and an inward pressure. We predict novel
properties of fluctuating thin shells under point indentations and
pressure-induced deformations. The contribution due to thermal fluctuations
increases with increasing ratio of shell radius to thickness, and dominates the
response when the product of this ratio and the thermal energy becomes large
compared to the bending rigidity of the shell. Thermal effects are enhanced
when a large uniform inward pressure acts on the shell, and diverge as this
pressure approaches the classical buckling transition of the shell. Our results
are relevant for the elasticity and osmotic collapse of microcapsules.Comment: To appear in PNAS; accepted version including Supplementary
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Cooperativity, Fluctuations and Inhomogeneities in Soft Matter
This thesis presents four investigations into mechanical aspects of soft thin structures, focusing on the effects of stochastic and thermal fluctuations and of material inhomogeneities. First, we study the self-organization of arrays of high-aspect ratio elastic micropillars into highly regular patterns via capillary forces. We develop a model of capillary mediated clustering of the micropillars, characterize the model using computer simulations, and quantitatively compare it to experimental realizations of the self-organized patterns. The extent of spatial regularity of the patterns depends on the interplay between cooperative enhancement and history-dependent stochastic disruption of order during the clustering process. Next, we investigate the influence of thermal fluctuations on the mechanics of homogeneous, elastic spherical shells. We show that thermal fluctuations give rise to temperature- and size-dependent corrections to shell theory predictions for the mechanical response of spherical shells. These corrections diverge as the ratio of shell radius to shell thickness becomes large, pointing to a drastic breakdown of classical shell theory due to thermal fluctuations for extremely thin shells. Finally, we present two studies of the mechanical properties of thin spherical shells with structural inhomogeneities in their walls. The first study investigates the effect of a localized reduction in shell thickness—a soft spot—whereas the second studies shells with a smoothly varying thickness. In both cases, the inhomogeneity significantly alters the response of the shell to a uniform external pressure, revealing new ways to control the strength and shape of initially spherical elastic capsules.Engineering and Applied Science
Theory of Interacting Dislocations on Cylinders
We study the mechanics and statistical physics of dislocations interacting on cylinders, motivated by the elongation of rod-shaped bacterial cell walls and cylindrical assemblies of colloidal particles subject to external stresses. The interaction energy and forces between dislocations are solved analytically, and analyzed asymptotically. The results of continuum elastic theory agree well with numerical simulations on finite lattices even for relatively small systems. Isolated dislocations on a cylinder act like grain boundaries. With colloidal crystals in mind, we show that saddle points are created by a Peach-Koehler force on the dislocations in the circumferential direction, causing dislocation pairs to unbind. The thermal nucleation rate of dislocation unbinding is calculated, for an arbitrary mobility tensor and external stress, including the case of a twist-induced Peach-Koehler force along the cylinder axis. Surprisingly rich phenomena arise for dislocations on cylinders, despite their vanishing Gaussian curvature.Engineering and Applied SciencesMolecular and Cellular BiologyPhysic
The influence of explicit local dynamics on range expansions driven by long-range dispersal
22 pagesRange expansions are common in natural populations. They can take such forms as an invasive species spreading into a new habitat or a virus spreading from host to host during a pandemic. When the expanding species is capable of dispersing offspring over long distances, population growth is driven by rare but consequential long-range dispersal events that seed satellite colonies far from the densely occupied core of the population. These satellites accelerate growth by accessing unoccupied territory, and also act as reservoirs for maintaining neutral genetic variation present in the originating population, which would ordinarily be lost to drift. Prior theoretical studies of dispersal-driven expansions have shown that the sequential establishment of satellites causes initial genetic diversity to be either lost or maintained to a level determined by the breadth of the distribution of dispersal distances. If the tail of the distribution falls off faster than a critical threshold, diversity is steadily eroded over time; by contrast, broader distributions with a slower falloff allow some initial diversity to be maintained for arbitrarily long times. However, these studies used lattice-based models and assumed an instantaneous saturation of the local carrying capacity after the arrival of a founder. Real-world populations expand in continuous space with complex local dynamics, which potentially allow multiple pioneers to arrive and establish within the same local region. Here, we evaluate the impact of local dynamics on the population growth and the evolution of neutral diversity using a computational model of range expansions with long-range dispersal in continuous space, with explicit local dynamics that can be controlled by altering the mix of local and long-range dispersal events. We found that many qualitative features of population growth and neutral genetic diversity observed in lattice-based models are preserved under more complex local dynamics, but quantitative aspects such as the rate of population growth, the level of maintained diversity, and the rate of decay of diversity all depend strongly on the local dynamics. Besides identifying situations in which modeling the explicit local population dynamics becomes necessary to understand the population structure of jump-driven range expansions, our results show that local dynamics affects different features of the population in distinct ways, and can be more or less consequential depending on the degree and form of long-range dispersal as well as the scale at which the population structure is measured