Abstract
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.
Highlights
In a recent theoretical proposal,[1] control of body waves was numerically demonstrated with a spherical shell consisting of an anisotropic heterogeneous elasticity tensor without the minor symmetries, and an heterogeneous density
Since the first cloak design, which we propose is deduced from elements of thin plate theory, we look at the frequencies from 3 Hz to 5 Hz, which correspond to a transition between plate and Rayleigh waves
We have conclusively shown that transformed Navier equations of the Willis type with a symmetric transformed elasticity tensor, allows for approximate cloak’s designs getting rid of
Summary
The contrast between the soil and concrete column’s parameters makes it possible to achieve uncommon effective material parameters, such as with strong artificial anisotropy and dispersion, and even artificial inertia and viscosity that can be interpreted as rank-3 tensors in an effective Willis equation This leads to a markedly enhanced control of surface Rayleigh wave trajectories compared to our earlier work.[14] comparisons between numerical results demonstrate that one can achieve some cloaking of surface (Rayleigh-like) wave trajectories, where the wave field is almost unperturbed outside the seismic cloak, whereas it simultaneously nearly vanishes in its center.
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