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Supplemental Material for Experiments on Seismic Metamaterials: Molding Surface Waves

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Supplemental Material for Experiments on Seismic Metamaterials: Molding Surface Waves Supplemental Material for Experiments on seismic metamaterials: Molding surface waves S. Br^ul´e1, E.H. Javelaud1, S. Enoch2, S. Guenneau2 1 M´enard, 91 620 Nozay, France and 2 Aix−Marseille Universit´e,CNRS, Centrale Marseille, Institut Fresnel, UMR 7249, 13013 Marseille, France We provide additional details on construction and characterization of the seismic metamaterial led near the French city of Grenoble in summer 2012. Additional numerical simulations based on an approximate plate model solved using Finite Elements suggest that the type of source (plane wave or point force), the overall shape of the metamaterial and the viscoelastic nature of the soil matter in the control of surface Rayleigh waves. Notably, cavity-like effects might counteract shielding efficiency due to some localization of elastic energy within the center of the structured soil. I. BACKGROUND WORK Let us first describe in more details the construction and experimental setup of the seismic metamaterial proof of concept led near the French city of Grenoble in summer 2012. Following some workshop between research scientists in Fresnel Institute and the Menard's Earthquake Engineering Team during the fall of 2011, some design of a first seismic metamaterial was completed in the summer of 2012. The Menard company subsequently selected a few possible locations in France for the experiment, and opted for a relatively homogeneous soil in a construction site near the city of Grenoble, which also had the advantage of being only mildly viscoelastic (nevertheless elastic waves within this soil still propagate with notable absorption, but the interpretation of shielding effect from the seismic metamaterial dominates enough the damping effect within the non-structured soil for the experiment to be meaningful). II. CONSTRUCTION OF THE SEISMIC METAMATERIAL One of the constraints we faced in the construction of the meter scale metamaterial is the changeable nature of the soil with weather forecast (e.g. a wet soil is much more viscoelastic than a dry soil, so clear sky is preferable). Moreover, it transpires from two photos of a typical borehole of the large scale metamaterial in the left panel of Fig.1 that the structured soil will have some imperfections (such as heterogeneity of soil). The air inclusion is clearly not perfectly cylindrical (rough surface and rather conical shape). However, in our numerical model we assumed such imperfections can be neglected (in order to keep our model two-dimensional for computational easiness during the shape optimization algorithm leading to an optimal structure for shielding effect in a given frequency range). Hence we should expect some slight discrepancy between experiments and numerics. Nevertheless, drilling rig used to drill the 5 meters deep air holes, see photos in middle panel of Fig. 1, and good precision in the drilling technique schematically shown in right panel of Fig. 1, make it possible to construct a large scale periodic structure, the result of which appears in photo of Fig. 2. Let us now turn to the characterization of the seismic metamaterial. III. EXPERIMENT AND MEASUREMENTS The experiment was carried out in mid-August 2012 notably for good weather forecast reasons (dry soil being less viscoelastic) and also because the earthquake engineering team of the Menard's company had a three-day break from a large civil engineering work. It took three days to build the large metamaterial and perform the measurements with a team of ten people and one crane. The experimental setup can be seen in Fig. 2 wherein the green, brown and colored regions respectively show the sensors, the boreholes and the seismic source areas. Importantly, before altering the land in any way, we needed to know how a seismic wave traveled through the land. We thus generated a 50 Hz seismic surface wave and measured how these so-called Rayleigh waves traveled through the land using an array of 20 sensors. We conducted four trials and obtained a good sense of how the waves traveled through the ground: In this way, we had a benchmark. We then created the metamaterial by drilling thirty one holes in three rows of ten holes each, and one hole slightly shifted with respect to the array. The rows were not lined up. Each hole was 5 meters deep and had a diameter of 0:32 meters. The holes were placed 1:73 meters apart. This spacing was selected based on the wavelength of seismic 2 Figure 1: Construction of the large scale metamaterial: Photos of a typical borehole of the structured soil (Left Panel); Photos of the drilling rig used to make the boreholes (Middle panel); Vibroprobe used for the seismic test (Right panel). A movie is also available at [1]. Figure 2: Photo of the experiment: The Sensors are located in the green colored area. The evenly spaced bore holes (with center to center spacing of 1:73m) are located in the brown colored region. The seismic wave is produced by the rotation (at angular frequency of 50 Hertz) of a vibroprobe in the red colored region. surface waves, which were 1:56 meters in our experiment. With the 20 detectors all in place, we again set off a seismic surface wave and made our measurements. The seismic surface waves were almost nonexistent after the first row of boreholes, and nearly immeasurable after the second, as it transpires from Fig. 3b). In order to map completely the energy field, the sensors were set four times on site (green, blue, pink and orange grids in Fig. 3a) before and after carrying out the boreholes (Fig. 3a). The fact that the elastic energy is 2:3 times larger at the source point when it is in presence of the metamaterial in Fig. 3b), is reminiscent of the Local Density 3 of States obtained for a source placed near a mirror in optics [2]. The downside to this is that the waves are reflected backwards, meaning that if there were any buildings on the wrong side of this type of seismic cloak, they would experience an increase in seismic activity. However, this is not a concern for isolated structures such as oil refineries, power plants, nuclear reactors, and some monuments. Figure 3: Experimental setup and measurements for a monochromatic source: (a) Seismic test metamaterial map with the borehole pattern and four sets of velocimeters (green, blue, pink and orange grids); (b) Experimental results map after inter- polation between sensors (absolute change of the energy): Difference of the measured energy's field (arbitrary units) after and before carrying out the boreholes. Black rectangles symbolize sensors. Note the large elastic energy enhancement around the source in the presence of boreholes. Figure 4: Experimental results's map after interpolation between sensors (relative change of the energy): Difference of the measured energy field (arbitrary units) normalized by the energy field after we carried out the boreholes, with image magnifi- cation on the structured soil. Black rectangles symbolize sensors, white circles, the boreholes and red cross, the source. Blue (resp. red) color corresponds to regions where the elastic energy is lower (resp. larger) after we carried out boreholes, see accompanying text for more details. One should note that the sensor located at point (0; 5) measures 6 times less elastic energy after we carried out the boreholes, see accompanying text. 4 It is enlightening to map the relative change of energy ∆ = (J2 − J1)=J2, where J1 and J2 are respectively the measured energy's field before and after we carried out the boreholes, see Fig. 4: This allows one to have a finer estimate of the elastic energy's change than in Fig. 3. One notes that in the deep blue regions, −5 ≤ ∆ ≤ −0:2, hence 1:2 ≤ J1=J2 ≤ 6 (i.e. the elastic energy is up to 6 times larger in the absence of seismic metamaterial, hence one can say that the structured soil offers a good protection against the surface wave generated at 50 Hz in the blue region). Regarding the red region around the source, relative change of energy varies in the range [0:2; 0:4], that is to say that J2=J1 varies in the range [1:25; 1:67]. This range is still in reasonably good agreement with the well-known value of the Local Density of States obtained for a source placed near a mirror in optics [2]. Finally, white regions correspond to ∆ = 0 that is J1=J2 = 1 (no change of elastic energy after we carried out the boreholes) and values of ∆ larger than 0:6 far behind and on the sides of the seismic metamaterial are measured at distances from the source larger than 10 meters, that is when measurements become less reliable (due to the viscoelastic nature of the soil). IV. NUMERICAL MODELS OF PLATES In this section, we propose further numerical illustrations of the experiment. We first compare dispersion curves computed with the COMSOL MULTIPHYSICS package using a Mindlin model for plates of moderate thickness (equation 1 of the accompanying letter) with those computed with the Kirchhoff model for thin plates (equation 2 of the accompanying letter). In this way, we can confirm the existence of the partial stop band around the frequency of 50 Hz used in the experiment. We then look at a viscoelastic kirchhoff model, which shows attenuation of the elastic signal should not be neglected. A. Kirchhoff thin plate versus Mindlin moderately thick plate models We report in Fig. 5 the dispersion diagrams computed with the two plates' models taking the same parameters as in the accompanying letter. We observe that the first partial stop band is not at the exact same location in the two models, but the second, most important, one remains centered around 50 Hz.
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