Locally Resonant Metasurfaces for Shear Waves in Granular Media

Rachele Zaccherini, Andrea Colombi, Antonio Palermo, Vasilis K. Dertimanis, Alessandro Marzani, Henrik R. Thomsen, Bozidar Stojadinovic, and Eleni N. Chatzi

Phys. Rev. Applied 13, 034055 – Published 23 March 2020

Abstract

In this paper, the physics of horizontally polarized shear waves traveling across a locally resonant metasurface in an unconsolidated granular medium is experimentally and numerically explored. The metasurface is comprised of an arrangement of subwavelength horizontal mechanical resonators embedded in a granular layer made of silica microbeads. The metasurface supports a frequency-tailorable attenuation zone induced by the translational mode of the resonators. The experimental and numerical findings reveal that the metasurface not only backscatters part of the energy but also redirects the wave front underneath the resonators, leading to a considerable amplitude attenuation at the surface level, when all the resonators have similar resonant frequency. A more complex picture emerges when using resonators arranged in a so-called graded design, e.g., with a resonant frequency increasing or decreasing throughout the metasurface. Unlike the mechanism observed in a bilayered medium, shear waves localized at the surface of the granular material are not converted into bulk waves. Although a detachment from the surface occurs, the depth-dependent velocity profile of the granular medium prevents the mode conversion and the horizontally polarized shear wave front returns to the surface. The outcomes of our experimental and numerical studies allow for understanding the dynamics of wave propagation in resonant metamaterials embedded in vertically inhomogeneous soils and, therefore, may be valuable for improving the design of engineered devices for ground-vibration and seismic wave containment.

Received 31 October 2019
Revised 9 February 2020
Accepted 4 March 2020

DOI:https://doi.org/10.1103/PhysRevApplied.13.034055

Physics Subject Headings (PhySH)

Acoustic metamaterials Acoustic wave phenomena Granular packing Shear deformation

Granular solids Surfaces

Resonance techniques Surface acoustic wave

Particles & FieldsCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Rachele Zaccherini ^1,*, Andrea Colombi ¹, Antonio Palermo ², Vasilis K. Dertimanis ¹, Alessandro Marzani ², Henrik R. Thomsen ³, Bozidar Stojadinovic ¹, and Eleni N. Chatzi ¹

¹Department of Civil, Environmental and Geomatic Engineering, ETH Zürich, Zürich 8093, Switzerland
²Department of Civil, Chemical, Environmental and Materials Engineering—DICAM, University of Bologna, Bologna 40136, Italy
³Department of Earth Sciences, ETH Zürich, Zürich 8092, Switzerland

^*zaccherini@ibk.baug.ethz.ch

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Vol. 13, Iss. 3 — March 2020

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Images

Figure 1
The experimental setup and mechanical resonators. (a) The experimental setup, involving a wooden box filled with granular material, a 3D laser Doppler vibrometer, an array of subwavelength horizontal oscillators and a three-axis piezoelectric actuator. (b) A model of the elementary unit cell, including a squared brass mass, four trusslike springs, four ligaments acting as vertical supports, and an outer casing. The first translational mode of the resonant unit together with an illustration of the assembled 3D-printed resonator.
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Figure 2
Dispersion analysis. (a) The experimental, numerical (black line), and analytical (red dashed line) dispersion curves of the first two low-order SH modes localized at the surface of the granular medium, together with the corresponding shape modes for a value of $k = 20 rad / m$ . (b) The experimental and numerical (black line) dispersion curves of the first two low-order SH modes propagating through the resonant metasurface, together with the shape mode of ${SH}_{2 m}$ for a value of $k = 20 rad / m$ . The shaded light-blue area identifies the frequency range in which the surface displacement is negligible. This is defined as $u_{y, h r} / u_{y, max} < 0.3$ , where $u_{y, h r}$ is the shear horizontal displacement calculated at the resonator embedding depth $h_{r}$ and $u_{y, max}$ is the maximum displacement along the depth. (c) Experimental dispersion curves of SH waves propagating through the nonresonant casings surface. The lateral inset shows the average particle velocity measured in the frequency domain after one line of resonators (red line) and after one line of casings (blue line). (d) A drawing of the 3D unit cells with and without the resonators developed in comsol multiphysics^®.
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Figure 3
The experimental and numerical displacement field with and without the metasurface. (a) The experimental displacement field of SH waves propagating at the free surface of the granular medium (top snapshots) and through the resonant metasurface (bottom snapshots) at three sequential time instants ( $t_{1} = 0.0210 s$ , $t_{2} = 0.0260 s$ , and $t_{3} = 0.0312 s$ ). (b) The computational domain characterized by the granular substrate and the resonant metasurface excited by a Ricker pulse centered at 300 Hz. The inset shows an enlargement of the adaptive mesh. (c) The numerical displacement field of SH waves propagating at the free surface of the granular medium (top snapshots) and through the resonant metasurface (bottom snapshots) at three sequential time instants ( $t_{1} = 0.0210 s$ , $t_{2} = 0.0260 s$ , and $t_{3} = 0.0312 s$ ). (d) Vertical cross sections through the center of the model, showing the numerical displacement field of SH waves traveling at the free surface (top snapshots) and through the metasurface (bottom snapshots) at $t_{4} = 0.0485 s$ .
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Figure 4
The experimental velocity field within graded metasurfaces. (a) An outline of the experimental domain with resonant metasurfaces of gradually increasing and decreasing frequency. (b) The experimental particle velocity as a function of the frequency and the distance from the source for the case of a graded metasurface of decreasing frequency. (c) The same as (b) but for a graded metasurface of increasing frequency.
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Figure 5
Full-scale time-domain numerical simulations. (a) The computational domain modeled for the reference case characterized by the granular substrate excited by a harmonic line source (350 and 370 Hz). (b) The displacement field of harmonic SH waves traveling at the free surface of the granular substrate at two different excitation frequencies (350 and 370 Hz). (c) The computational domain characterized by the granular substrate and the graded metasurface excited by a harmonic line source (350 and 370 Hz). (d) The displacement field of harmonic SH waves traveling through the graded metasurface at two different excitation frequencies (350 and 370 Hz).
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