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Register a new userConical Wave Propagation and Diffraction in 2D Hexagonally Packed Granular Lattices
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Added by
Christopher Chong Dr.
Publication date
2015
fields
Physics
and research's language is
English
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Linear and nonlinear mechanisms for conical wave propagation in two-dimensional lattices are explored in the realm of phononic crystals. As a prototypical example, a statically compressed granular lattice of spherical particles arranged in a hexagonal packing configuration is analyzed. Upon identifying the dispersion relation of the underlying linear problem, the resulting diffraction properties are considered. Analysis both via a heuristic argument for the linear propagation of a wavepacket, as well as via asymptotic analysis leading to the derivation of a Dirac system suggests the occurrence of conical diffraction. This analysis is valid for strong precompression i.e., near the linear regime. For weak precompression, conical wave propagation is still possible, but the resulting expanding circular wave front is of a non-oscillatory nature, resulting from the complex interplay between the discreteness, nonlinearity and geometry of the packing. The transition between these two types of propagation is explored.
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We study wave propagation in two-dimensional granular crystals under the Hertzian contact law consisting of hexagonal packings of spheres under various basin geometries including hexagonal, triangular, and circular basins which can be tiled with hexagons. We find that the basin geometry will influence wave reflection at the boundaries, as expected, and also may result in bottlenecks forming. While exterior strikers the size of a single sphere have been considered in the literature, it is also possible to consider strikers which impact multiple spheres along a boundary, or to have multiple sides being struck simultaneously. It is also possible to consider obstructions or even strikers in the interior of the hexagonally packed granular crystal, as previously considered in the case of square packings, resulting in the basin geometry no longer forming a convex set. We consider various configurations of either boundary or interior strikers. We shall also consider the case where a granular crystal is composed of two separate crystals of differing material, with a single interface between the two distinct materials. Depending on the relative material properties of each type of sphere, this can result in a trapping of most of the wave energy within one of the two regions. While repeated reflections from the boundaries will cause the systems we study to fall into disorder for large time, there are a number of interesting wave structures and patters that emerge as transients at intermediate timescales.
We conduct an extensive study of nonlinear localized modes (NLMs), which are temporally periodic and spatially localized structures, in a two-dimensional array of repelling magnets. In our experiments, we arrange a lattice in a hexagonal configuration with a light-mass defect, and we harmonically drive the center of the chain with a tunable excitation frequency, amplitude, and angle. We use a damped, driven variant of a vector Fermi- Pasta-Ulam-Tsingou lattice to model our experimental setup. Despite the idealized nature of the model, we obtain good qualitative agreement between theory and experiments for a variety of dynamical behaviors. We find that the spatial decay is direction-dependent and that drive amplitudes along fundamental displacement axes lead to nonlinear resonant peaks in frequency continuations that are similar to those that occur in one-dimensional damped, driven lattices. However, driving along other directions leads to the creation of asymmetric NLMs that bifurcate from the main solution branch, which consists of symmetric NLMs. When we vary the drive amplitude, we observe such behavior both in our experiments and in our simulations. We also demonstrate that solutions that appear to be time-quasi-periodic bifurcate from the branch of symmetric time-periodic NLMs.
We describe the dynamic response of a two-dimensional hexagonal packing of uncompressed stainless steel spheres excited by localized impulsive loadings. After the initial impact strikes the system, a characteristic wave structure emerges and continuously decays as it propagates through the lattice. Using an extension of the binary collision approximation (BCA) for one-dimensional chains, we predict its decay rate, which compares well with numerical simulations and experimental data. While the hexagonal lattice does not support constant speed traveling waves, we provide scaling relations that characterize the power law decay of the wave velocity. Lastly, we discuss the effects of weak disorder on the directional amplitude decay rates.
We investigate the singularities of the trace of the half-wave group, $mathrm{Tr} , e^{-itsqrtDelta}$, on Euclidean surfaces with conical singularities $(X,g)$. We compute the leading-order singularity associated to periodic orbits with successive degenerate diffractions. This result extends the previous work of the third author cite{Hil} and the two-dimensional case of the work of the first author and Wunsch cite{ForWun} as well as the seminal result of Duistermaat and Guillemin cite{DuiGui} in the smooth setting. As an intermediate step, we identify the wave propagators on $X$ as singular Fourier integral operators associated to intersecting Lagrangian submanifolds, originally developed by Melrose and Uhlmann cite{MelUhl}.
We study solitary wave propagation in 1D granular crystals with Hertz-like interaction potentials. We consider interfaces between media with different exponents in the interaction potential. For an interface with increasing interaction potential exponent along the propagation direction we obtain mainly transmission with delayed secondary transmitted and reflected pulses. For interfaces with decreasing interaction potential exponent we observe both significant reflection and transmission of the solitary wave, where the transmitted part of the wave forms a multipulse structure. We also investigate impurities consisting of beads with different interaction exponents compared to the media they are embedded in, and we find that the impurities cause both reflection and transmission, including the formation of multipulse structures, independent of whether the exponent in the impurities is smaller than in the surrounding media. We explain wave propagation effects at interfaces and impurities in terms of quasi-particle collisions. Next we consider wave propagation along Hertz-like granular chains of beads in the presence of disorder and periodicity in the interaction exponents present in the Hertz-like potential, modelling, for instance, inhomogeneity in the contact geometry between beads in the granular chain. We find that solitary waves in media with randomised interaction exponents (which models disorder in the contact geometry) experience exponential decay, where the dependence of the decay rate is similar to the case of randomised bead masses. In the periodic case of chains with interaction exponents alternating between two fixed values, we find qualitatively different propagation properties depending on the choice of the two exponents. In particular, we find regimes with either exponential decay or stable solitary wave propagation with pairwise collective behaviour.
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