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Register a new userControl of Rayleigh-like waves in thick plate Willis metamaterials
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We explore interactions of elastic waves propagating in plates (with soil parameters) structured with concrete pillars buried in the soil. Pillars are 2 m in diameter, 30 m in depth and the plate is 50 m in thickness. We study the frequency range 5 to 10 Hz, for which Rayleigh wave wavelengths are smaller than the plate thickness. This frequency range is compatible with frequency ranges of particular interest in earthquake engineering. It is demonstrated in this paper that two seismic cloaks configurations allow for an unprecedented flow of elastodynamic energy associated with Rayleigh surface waves. The first cloak design is inspired by some approximation of ideal cloaks parameters within the framework of thin plate theory. The second, more accomplished but more involved, cloak design is deduced from a geometric transform in the full Navier equations that preserves the symmetry of the elasticity tensor but leads to Willis equations, well approximated by a homogenization procedure, as corroborated by numerical simulations. The two cloakss designs are strikingly different, and the superior efficiency of the second type of cloak emphasizes the necessity for rigor in transposition of existing cloakss designs in thin plates to the geophysics setting. Importantly, we focus our attention on geometric transforms applied to thick plates, which is an intermediate case between thin plates and semi-infinite media, not studied previously. Cloaking efficiency (reduction of the disturbance of the wave wavefront and its amplitude behind an obstacle) and protection (reduction of the wave amplitude within the center of the cloak) are studied for ideal and approximated cloaks parameters. These results represent a preliminary step towards designs of seismic cloaks for surface Rayleigh waves propagating in sedimentary soils structured with concrete pillars.
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Acoustic bianisotropy, also known as the Willis parameter, expands the field of acoustics by providing nonconventional couplings between momentum and strain in constitutive relations. Sharing the common ground with electromagnetics, the realization of acoustic bianisotropy enables the exotic manipulation of acoustic waves in cooperation with a properly designed inverse bulk modulus and mass density. While the control of entire constitutive parameters substantiates intriguing theoretical and practical applications, a Willis metamaterial that enables independently and precisely designed polarizabilities has yet to be developed to overcome the present restrictions of the maximum Willis bound and the nonreciprocity inherent to the passivity of metamaterials. Here, by extending the recently developed concept of virtualized metamaterials, we propose acoustic Willis metamaterials that break the passivity and reciprocity limit while also achieving decoupled control of all constitutive parameters with designed frequency responses. By instituting basis convolution kernels based on parity symmetry for each polarization response, we experimentally demonstrate bianisotropy beyond the limit of passive media. Furthermore, based on the notion of inverse design of the frequency dispersion by means of digital convolution, purely nonreciprocal media and media with a broadband, flat-response Willis coupling are also demonstrated. Our approach offers all possible independently programmable extreme constitutive parameters and frequency dispersion tunability accessible within the causality condition and provides a flexible platform for realizing the full capabilities of acoustic metamaterials.
An array of ten broadband stations was installed on the Popocatepetl volcano (Mexico) for five months between October 2002 and February 2003. 26 regional and teleseismic earthquakes were selected and filtered in the frequency time domain to extract the fundamental mode of the Rayleigh wave. The average dispersion curve was obtained in two steps. Firstly, phase velocities were measured in the period range [2-50] s from the phase difference between pairs of stations, using Wiener filtering. Secondly, the average dispersion curve was calculated by combining observations from all events in order to reduce diffraction effects. The inversion of the mean phase velocity yielded a crustal model for the volcano which is consistent with previous models of the Mexican Volcanic Belt. The overall crustal structure beneath Popocatepetl is therefore not different from the surrounding area, and the velocities in the lower crust are confirmed to be relatively low. Lateral variations of the structure were also investigated by dividing the network into four parts and by applying the same procedure to each sub-array. No well-defined anomalies appeared for the two sub-arrays for which it was possible to measure a dispersion curve. However, dispersion curves associated with individual events reveal important diffraction for 6 s to 12 s periods which could correspond to strong lateral variations at 5 to 10 km depth.
In [AIP Advances 6, 121707 (2016)], a soil structured with concrete columns distributed within two specially designed seismic cloaks thanks to a combination of transformational elastodynamics and effective medium theory was shown to detour Rayleigh waves of frequencies lower than 10 Hz around a cylindrical region. The aforementioned studies motivate our exploration of interactions of surface elastic waves propagating in a thick plate (with soil parameters) structured with concrete pillars above it. Pillars are 40 m in height and the plate is 100 m in thickness, so that typical frequencies under study are below 1 Hz, a frequency range of particular interest in earthquake engineering. We demonstrate that three seismic cloaks allow for an unprecedented flow of elastodynamic energy. These designs are achieved by first computing ideal cloaks parameters deduced from a geometric transform in the Navier equations that leads to almost isotropic and symmetric elasticity (4th order) and density (2nd order) tensors. To do this we extend the theory of Non-Euclidean cloaking for light as proposed by the theoretical physicists Leonhardt and Tyc. In a second step, ideal heterogeneous nearly isotropic cloaks parameters are approximated by averaging elastic properties of sets of pillars placed at the nodes of a bipolar coordinate grid, which is an essential ingredient in our Non-Euclidean cloaking theory for elastodynamic waves. Cloaking effects are studied for a clamped obstacle (reduction of the disturbance of the wave wavefront and its amplitude behind a clamped obstacle). Protection is achieved through reduction of the wave amplitude within the center of the cloak.These results represent a first step towards designs of Non-Euclidean seismic cloaks for surface (Rayleigh and Love) waves propagating in semi-infinite elastic media structured with pillars.
Recent static experiments on twist effects in chiral three-dimensional mechanical metamaterials have been discussed in the context of micropolar Eringen continuum mechanics, which is a generalization of Cauchy elasticity. For cubic symmetry, Eringen elasticity comprises nine additional parameters with respect to Cauchy elasticity, of which three directly influence chiral effects. Here, we discuss the behavior of the static case of an alternative generalization of Cauchy elasticity, the Milton-Briane-Willis equations. We show that in the homogeneous static cubic case only one additional parameter with respect to Cauchy elasticity results, which directly influences chiral effects. We show that the Milton-Briane-Willis equations qualitatively describe the experimentally observed chiral twist effects, too. We connect the behavior to a characteristic length scale.
Additional electromagnetic waves and additional boundary conditions (ABCs) in non-local materials attracted a lot of attention in the past. Here we report the possibility of additional propagating and evanescent waves in local anisotropic and bi-anisotropic linear materials. We investigate the possible options for ABCs and describe how to complement the conventional 4 Maxwells boundary conditions in the situations when there are more than 4 waves that need to be matched at the boundary of local and linear quartic metamaterials. We show that these ABCs must depend on the properties of the interface and require the introduction of the additional effective material parameters describing this interface, such as surface conductivities.
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