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We study the dynamic surface response of neolithic stone settlements obtained with seismic ambient noise techniques near the city of Carnac in French Brittany. Surprisingly, we find that menhirs (neolithic human size standing alone granite stones) with an aspect ratio between 1 and 2 periodically arranged atop a thin layer of sandy soil laid on a granite bedrock, exhibit fundamental resonances in the range of 10 to 25 Hz. We propose an analogic Kelvin-Voigt viscoelastic model that explains the origin of such low frequency resonances. We further explore low frequency filtering effect with full wave finite element simulations. Our numerical results confirm the bending nature of fundamental resonances of the menhirs and further suggest additional resonances of rotational and longitudinal nature in the frequency range 25 to 50 Hz. Our study thus paves the way for large scale seismic metasurfaces consisting of granite stones periodically arranged atop a thin layer of regolith over a bedrock, for ground vibration mitigation in earthquake engineering., Comment: 6 pages, 7 figures

Diverting, and controlling, elastic vibrations impacting upon infrastructure is a major challenge for seismic hazard mitigation, and for the reduction of machine noise and vehicle vibration in the urban environment. Seismic metamaterials (SMs), with their inherent ability to manipulate wave propagation, provide a key route for overcoming the technological hurdles involved in this challenge. Engineering the structure of the SM serves as a basis to tune and enhance its functionality, and inspired by split rings, swiss-rolls, notch-shaped and labyrinthine designs of elementary cells in electromagnetic and mechanical metamaterials, we investigate altering the structure geometries of SMs with the aim of creating large bandgaps \textcolor{black}{in a subwavelength regime}. We show that square stiff inclusions, perform better in comparison to circular ones, whilst keeping the same filling fraction. En route to enhancing the bandgap, we have also studied the performance of SMs with different constituent materials; we find that steel columns, as inclusions, show large bandgaps, however, the columns are too large for steel to be a feasible material in practical or financial terms. Non-reinforced concrete would be preferable for industry level scaling up of the technology because, concrete is cost-effective, easy to cast directly at the construction site and easy to provide arbitrary geometry of the structure. As a part of this study, we show that concrete columns can also be designed to exhibit bandgaps if we cast them within a soft soil coating surrounding the protected area for various civil structures like a bridge, building, oil pipelines etc.

Liquid wave studies have shown that under certain constructive interference conditions, an abnormal size wave could be generated at a specific point. This type of wave, described long ago in the literature, was named 'Draupner Wave' in 1995. It was the first rogue wave, more than ten meters high, to be detected by a measuring instrument, occurring at the Draupner platform in the North Sea off the coast of Norway. For very soft and thick soils in continental sedimentary basin, with high water content, low shear wave velocity and wave bouncing effects on the edges of the basin, one can question the theoretical existence of such a type of freak waves that may have come unnoticed amongst surface seismic waves. This short communication presents recent observations and the conditions and limits of the analogy between gravity and seismic waves., Comment: 13 pages, 7 figures

Some properties of electromagnetic metamaterials have been translated, using some wave analogies, to surface seismic wave control in sedimentary soils structured at the meter scale. Two large scale experiments performed in 2012 near the French cities of Grenoble [Brule et al., PRL 112, 133901, 2014] and Lyon [Brule et al., Sci. Rep. 7, 18066, 2017] have confirmed the usefulness of this methodology and its potential influence on soil-structure interaction. We present here a new perspective on the in-situ experiment near Lyon, which unveils energy corridors in the seismic lens. We further introduce a concept of time-modulated seismic metamaterial underpined by an effective model based on Willis's equations. As a first application, we propose that ambient seismic noise time-modulates structured soils that can be viewed as moving media. In the same spirit, a design of an analogous seismic computer is proposed making use of ambient seismic noise. Seismic signals transmitted between remote seismic computers can help forming an Internet of Things. We further recall that ancient Roman theaters and forests of trees are examples of large scale structures that behave in a way similar to electromagnetic metamaterials: invisibility cloaks and rainbows, respectively. Seismic metamaterials can thus not only be implemented for shielding, lensing and cloaking of potentially deleterious Rayleigh waves, but they also have potential applications in energy harvesting and analogous computations using ambient seismic noise and this opens new vistas in seismic energy harvesting via natural or artificial soil structuring., Comment: 27 pages, 8 figures

Metre size inertial resonators located in the ground have been theoretically shown to interact with a seismic wave (attenuation, band gaps) to enable protection of surface structures such as buildings. The challenge for Civil Engineering is to both reduce the size of these resonators and to increase their efficiency. Here we explore steel spheres, connected to a concrete bulk medium, either by a coating of rubber, or rubber and steel ligaments, or air and steel ligaments. We show that for a cubic lattice periodicity of 1 metre, we achieve stop bands in the frequency range 14 to 20 Hz; by splitting spheres in 2 and 8 pieces, we tune down the stop bands frequencies and further increase their bandwidth. We thus demonstrate we are able to provide a variety of inertial resonators with stop bands below 10 Hz i.e., in the frequency range of interest for earthquake engineering.

The discovery of photonic crystals thirty years ago, in conjunction with research advances in plasmonics and metamaterials, has inspired the concept of decameter scale metasurfaces, coined seismic metamaterials, for an enhanced control of surface (Love and Rayleigh) and bulk (shear and pressure) elastodynamic waves. The powerful mathematical tools of coordinate transforms, effective medium and Floquet-Bloch theories, which have revolutionized nanophotonics, can be translated in the language of civil engineering and geophysics. Experiments on seismic metamaterials made of buried elements in the soil demonstrate that the fore mentioned tools make possible a novel description of complex phenomena of soil-structure interaction during a seismic disturbance. But the concepts are already moving to more futuristic concepts and the same notions developed for structured soils are now used to examine the effects of buildings viewed as above surface resonators in megastructures such as metacities. But this perspective of the future should not make us forget the heritage of the ancient peoples. Indeed, we finally point out the striking similarity between an invisibility cloak design and the architecture of some ancient megastructures as antique Gallo-Roman theaters and amphitheatres., Comment: 13 pages, 16 figures

We outline some recent research advances on the control of elastic waves in thin and thick plates, that have occurred since the large scale experiment [Phys. Rev. Lett. 112, 133901, 2014] that demonstrated significant interaction of surface seismic waves with holes structuring sedimentary soils at the meter scale. We further investigate the seismic wave trajectories in soils structured with buildings. A significant substitution of soils by inclusions, acting as foundations, raises the question of the effective dynamic properties of these structured soils. Buildings, in the case of perfect elastic conditions for both soil and buildings, are shown to interact and strongly influence elastic surface waves; such site-city seismic interactions were pointed out in [Bulletin of Seismological Society of America 92, 794-811, 2002], and we investigate a variety of scenarios to illustrate the variety of behaviours possible., Comment: 15 pages, 9 figures

Structured soils with regular meshes of metric size holes implemented in first ten meters of the ground have been theoretically and experimentally tested under seismic disturbance this last decade. Structured soils with rigid inclusions embedded in a substratum have been also recently developed. The influence of these inclusions in the ground can be characterized in different ways: redistribution of energy within the network with focusing effects for seismic metamaterials, wave reflection, frequency filtering, reduction of the amplitude of seismic signal energy, etc. Here we first provide some time-domain analysis of the flat lens effect in conjunction with some form of external cloaking of Rayleigh waves and then we experimentally show the effect of a finite mesh of cylindrical holes on the ellipticity of the surface Rayleigh waves at the level of the Earth's surface. Orbital diagrams in time domain are drawn for the surface particle's velocity in vertical (x,z) and horizontal (x,y) planes. These results enable us to observe that the mesh of holes locally creates a tilt of the axes of the ellipse and changes the direction of particle movement. Interestingly, changes of Rayleigh waves ellipticity can be interpreted as changes of an effective Poisson ratio. However, the proximity of the source is also important to explain the shape of the ellipses. We analyze these observations in terms of wave mode conversions inside the mesh and we propose to broaden the discussion on the complexity of seismic wave phenomena in structured soils such as soils foundations and on the coupling effects specific to the soil-structure interaction., Comment: 7 pages, 5 figures

Following the advent of electromagnetic metamaterials at the turn of the century, researchers working in other areas of wave physics have translated concepts of electromagnetic metamaterials to acoustics, elastodynamics, as well as to heat, mass and light diffusion processes. In elastodynamics, seismic metamaterials have emerged in the last decade for soft soils structured at the meter scale, and have been tested thanks to full-scale experiments on holey soils five years ago. Born in the soil, seismic metamaterials grow simultaneously on the field of tuned-resonators buried in the soil, around building's foundations or near the soil-structure's interface, and on the field of above-surface resonators. In this perspective article, we quickly recall some research advances made in all these types of seismic metamaterials and we further dress an inventory of which material parameters can be achieved and which cannot, notably from the effective medium theory perspective. We finally envision perspectives on future developments of large scale auxetic metamaterials for building's foundations, forests of trees for seismic protection and metamaterial-like transformed urbanism at the city scale., Comment: 12 pages, 11 figures

We introduce the concept of transformation crystallography which consists of the application of geometric transforms to periodic structures. We consider motifs with three-fold, four-fold and six-fold symmetries according to the crystallographic restriction theorem. Furthermore, we define motifs with five-fold symmetry such as quasi-crystals generated by a cut-and-projection method. We analyze elastic wave propagation in the transformed crystals and (Penrose-type) quasi-crystals with the finite difference time domain freeware SimSonic. We consider geometric transforms underpinning the design of seismic cloaks with square, circular, elliptical and peanut shapes in the context of triangular, square and honeycomb crystals. Interestingly, the use of morphing techniques leads to the design of cloaks with interpolated geometries reminiscent of Victor Vasarely's artwork. Employing the case of transformed graphene-like (honeycomb) structures allows one to draw useful analogies between large-scale seismic metamaterials such as soils structured with columns of concrete or grout with soil and nanoscale biochemical metamaterials. We further point out similarities between cloaks for elastodynamic and hydrodynamic waves and cloaks for diffusion (heat or mass) diffusion processes, notably with respect to invisibility and protection. Experimental data extracted from field test analysis of soil structured with boreholes demonstrates the application of bio-inspired seismic metamaterials. We conclude that these novel materials hold strong applications in biophysics and geophysics., Comment: 25 pages, 14 figures

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 cloaks's designs are strikingly different, and the superior efficiency of the second type of cloak emphasizes the necessity for rigor in transposition of existing cloaks's 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., Comment: V2: Major changes. Quantitative study carried out. New figures. Investigation of elastic trapped modes and resonances. Latex, 13 pages, 8 pages. Pacs: 41.20.Jb,42.25.Bs,42.70.Qs,43.20.Bi,43.25.Gf. The last version will appear at AIP Advances

A prerequisite for achieving seismic invisibility is to demonstrate the ability of civil engineers to control seismic waves with artificially structured soils. We carry out large-scale field tests with a structured soil made of a grid consisting of cylindrical and vertical holes in the ground and a low frequency artificial source (< 10 Hz). This allows the identification of a distribution of energy inside the grid, which can be interpreted as the consequence of an effective negative refraction index. Such a flat lens reminiscent of what Veselago and Pendry envisioned for light opens avenues in seismic metamaterials to counteract the most devastating components of seismic signals.

We investigate the elastic stop band properties of a theoretical cubic array of iron spheres connected to a bulk of concrete via iron or rubber ligaments. Each sphere can move freely within a surrounding air cavity, but ligaments couple it to the bulk and further facilitate bending and rotational motions. Associated low frequency local resonances are well predicted by an asymptotic formula. We find complete stop bands (for all wave-polarizations) in the frequency range $[16,21]$ Hertz (resp. $[6,11]$ Hertz) for $7.4$-meter (resp. $0.74$-meter) diameter iron spheres with a $10$-meter (resp. $1$-meter) center-to-center spacing, when they are connected to concrete via steel (resp. rubber) ligaments. The scattering problem shows that only bending modes are responsible for damping and that rotational modes are totally overwritten by bending modes. Regarding seismic applications, we further consider soil as a bulk medium, in which case the relative bandwidth of the low frequency stop band can be enlarged through ligaments of different sizes that allow for well separated bending and rotational modes. We finally achieve some damping of elastodynamic waves from $8$ to $49$ Hertz (relative stop band of $143$ percent) for iron spheres $0.74$-meter in diameter that are connected to soil with six rubber ligaments of optimized shapes. These results represent a preliminary step in the design of seismic shields placed around, or underneath, the foundations of large civil infrastructures.

We propose that wave propagation through a class of mechanical metamaterials opens unprecedented avenues in seismic wave protection based on spectral properties of auxetic-like metamaterials. The elastic parameters of these metamaterials like the bulk and shear moduli, the mass density, and even the Poisson ratio, can exhibit negative values in elastic stop bands. We show here that the propagation of seismic waves with frequencies ranging from 1Hz to 40Hz can be influenced by a decameter scale version of auxetic-like metamaterials buried in the soil, with the combined effects of impedance mismatch, local resonances and Bragg stop bands. More precisely, we numerically examine and illustrate the markedly different behaviors between the propagation of seismic waves through a homogeneous isotropic elastic medium (concrete) and an auxetic-like metamaterial plate consisting of 64 cells (40mx40mx40m), utilized here as a foundation of a building one would like to protect from seismic site effects. This novel class of seismic metamaterials opens band gaps at frequencies compatible with seismic waves when they are designed appropriately, what makes them interesting candidates for seismic isolation structures.

Materials engineered at the micro- and nano-meter scale have had a tremendous and lasting impact in photonics and phononics, with applications ranging from periodic structures disallowing light and sound propagation at stop band frequencies, to subwavelength focussing and cloaking with metamaterials. Here, we present the description of a seismic test held on a soil structured at the meter scale using vibrocompaction probes. The most simplistic way to interact with a seismic wave is to modify the global properties of the medium, acting on the soil density and then on the wave velocity. The main concept is then to reduce the amplification of seismic waves at the free surface, called site effects in earthquake engineering. However, an alternative way to counteract the seismic signal is by modifying the distribution of seismic energy thanks to a metamaterial made of a grid of vertical, cylindrical and empty inclusions bored in the initial soil, in agreement with numerical simulations using an approximate plate model., Comment: 9 pages, 5 figures