Stéphane Brûlé

Control of Rayleigh-like waves in thick plate Willis metamaterials

Recent advances in control of anthropic seismic sources in structured soil led us to 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 strickingly different, and the superior efficiency of the second type of cloak emphasizes the necessity for rigour 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.

Metamaterial-like transformed urbanism

Viewed from the sky, the urban fabric pattern appears similar to the geometry of structured devices called metamaterials; these were developed by Physicists to interact with waves that have wavelengths in the range from nanometers to meters (from electromagnetic to seismic metamaterials). Visionary research in the late 1980s based on the interaction of big cities with seismic signals and more recent studies on seismic metamaterials, made of holes or vertical inclusions in the soil, has generated interest in exploring the multiple interaction effects of seismic waves in the ground and the local resonances of both buried pillars and buildings. Here, we use techniques from transformational optics and theoretically validate, by numerical experiments, that a district of buildings could be considered as a set of above-ground resonators, purely elastic, interacting with an incident seismic signal. We hope that our proposal will contribute to all theoretical and experimental efforts in design of cities of the future, from a metamaterial standpoint.

Flat lens effect on seismic waves propagation in the subsoil

We show that seismic energy simulated by an artificial source that mainly propagates Rayleigh surface waves, is focused in structured soil made of a grid of holes distributed in the ground. 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 a dynamic anisotropy akin to 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 partially or totally the most devastating components of seismic signals.

Spanning the scales of mechanical metamaterials using time domain simulations in transformed crystals, graphene flakes and structured soils

We begin with a brief historical survey of discoveries of quasi-crystals and graphene, and then 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 from periodic structures in higher-dimensional space. 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 honeycomb crystals that can be viewed as scaled-up versions of graphene. Interestingly, the use of morphing techniques leads to the design of cloaks with interpolated geometries reminiscent of Victor Vasarelys 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 identify similarities in designs of cloaks for elastodynamic and hydrodynamic waves and cloaks for diffusion (heat or mass) processes, as these are underpinned by geometric transforms. Experimental data extracted from field test analysis of soil structured with boreholes demonstrates the application of crystallography to large scale phononic crystals, coined as seismic metamaterials, as they might exhibit low frequency stop bands. This brings us to the outlook of mechanical metamaterials, with control of phonon emission in graphene through extreme anisotropy, attenuation of vibrations of suspension bridges via low frequency stop bands and the concept of transformed meta-cities. We conclude that these novel materials hold strong applications spanning different disciplines or across different scales from biophysics to geophysics.

Seismic metamaterials: Shielding and focusing surface elastic waves in structured soils

Phononic crystals and metamaterials are man-made structures (with periodic heterogeneities typically a few micrometers to centimeters) that can control sound in ways not found in nature. Whereas the properties of phononic crystals derive from the periodicity of their structure, those of metamaterials arise from the collective effect of a large array of small resonators. These effects can be used to manipulate acoustic waves in unconventional ways, realizing functions such as invisibility cloaking, subwavelength focusing, and unconventional refraction phenomena (such as negative refractive index and phase velocity). Recent work has started to explore another intriguing domain of application: using similar concepts to control the propagation of seismic waves within the surface of the Earth. Our research group at the Aix-Marseille University and French National Center for Scientific Research (CNRS) has teamed up with civil engineers at an industrial company, Menard, in Nozay, also in France, and carried out ...

Consideration of Grain Size Distribution and Interfacial Transition Zone in the Prediction of Elastic Properties of Cementitious Composites

The purpose of the current article is a micromechanical-based model predicting the elastic properties of cementitious composites exhibiting low elasticity moduli. The model is generalized to composites made of a matrix in which are embedded various spherical concentric inclusions of different radi and properties.