Supplemental Material for Experiments on seismic metamaterials:Molding surface waves

S. Brule1, E.H. Javelaud1, S. Enoch2, S. Guenneau2

1 Menard, 91 620 Nozay, France and2 Aix−Marseille Universite, CNRS, Centrale Marseille,Institut Fresnel, UMR 7249, 13013 Marseille, France

We provide additional details on construction and characterization of the seismic metamaterialled near the French city of Grenoble in summer 2012. Additional numerical simulations based on anapproximate plate model solved using Finite Elements suggest that the type of source (plane waveor point force), the overall shape of the metamaterial and the viscoelastic nature of the soil matterin the control of surface Rayleigh waves. Notably, cavity-like effects might counteract shieldingefficiency 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 ofconcept led near the French city of Grenoble in summer 2012. Following some workshop between research scientistsin Fresnel Institute and the Menard’s Earthquake Engineering Team during the fall of 2011, some design of a firstseismic metamaterial was completed in the summer of 2012. The Menard company subsequently selected a few possiblelocations in France for the experiment, and opted for a relatively homogeneous soil in a construction site near the cityof Grenoble, which also had the advantage of being only mildly viscoelastic (nevertheless elastic waves within thissoil still propagate with notable absorption, but the interpretation of shielding effect from the seismic metamaterialdominates 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 ofthe 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.1that the structured soil will have some imperfections (such as heterogeneity of soil). The air inclusion is clearly notperfectly cylindrical (rough surface and rather conical shape).

However, in our numerical model we assumed such imperfections can be neglected (in order to keep our modeltwo-dimensional for computational easiness during the shape optimization algorithm leading to an optimal structurefor shielding effect in a given frequency range). Hence we should expect some slight discrepancy between experimentsand 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 constructa large scale periodic structure, the result of which appears in photo of Fig. 2. Let us now turn to the characterizationof 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 lessviscoelastic) and also because the earthquake engineering team of the Menard’s company had a three-day break froma large civil engineering work. It took three days to build the large metamaterial and perform the measurements witha team of ten people and one crane. The experimental setup can be seen in Fig. 2 wherein the green, brown andcolored 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 throughthe land using an array of 20 sensors. We conducted four trials and obtained a good sense of how the waves traveledthrough 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 slightlyshifted with respect to the array. The rows were not lined up. Each hole was 5 meters deep and had a diameter of0.32 meters. The holes were placed 1.73 meters apart. This spacing was selected based on the wavelength of seismic

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Figure 1: Construction of the large scale metamaterial: Photos of a typical borehole of the structured soil (Left Panel); Photosof the drilling rig used to make the boreholes (Middle panel); Vibroprobe used for the seismic test (Right panel). A movie isalso available at [1].

Figure 2: Photo of the experiment: The Sensors are located in the green colored area. The evenly spaced bore holes (withcenter to center spacing of 1.73m) are located in the brown colored region. The seismic wave is produced by the rotation (atangular 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 seismicsurface wave and made our measurements. The seismic surface waves were almost nonexistent after the first row ofboreholes, 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 orangegrids in Fig. 3a) before and after carrying out the boreholes (Fig. 3a). The fact that the elastic energy is 2.3 timeslarger at the source point when it is in presence of the metamaterial in Fig. 3b), is reminiscent of the Local Density

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of States obtained for a source placed near a mirror in optics [2]. The downside to this is that the waves are reflectedbackwards, meaning that if there were any buildings on the wrong side of this type of seismic cloak, they wouldexperience 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 theborehole 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 andbefore carrying out the boreholes. Black rectangles symbolize sensors. Note the large elastic energy enhancement around thesource in the presence of boreholes.

Figure 4: Experimental results’s map after interpolation between sensors (relative change of the energy): Difference of themeasured 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, seeaccompanying text for more details. One should note that the sensor located at point (0, 5) measures 6 times less elastic energyafter we carried out the boreholes, see accompanying text.

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It is enlightening to map the relative change of energy ∆ = (J2 − J1)/J2, where J1 and J2 are respectively themeasured energy’s field before and after we carried out the boreholes, see Fig. 4: This allows one to have a finerestimate 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, henceone can say that the structured soil offers a good protection against the surface wave generated at 50 Hz in the blueregion). Regarding the red region around the source, relative change of energy varies in the range [0.2, 0.4], that is tosay that J2/J1 varies in the range [1.25, 1.67]. This range is still in reasonably good agreement with the well-knownvalue of the Local Density of States obtained for a source placed near a mirror in optics [2]. Finally, white regionscorrespond 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 sourcelarger 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 curvescomputed 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 ofthe accompanying letter). In this way, we can confirm the existence of the partial stop band around the frequency of50 Hz used in the experiment. We then look at a viscoelastic kirchhoff model, which shows attenuation of the elasticsignal 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 asin the accompanying letter. We observe that the first partial stop band is not at the exact same location in the twomodels, but the second, most important, one remains centered around 50 Hz. This confirms the stop band origin ofthe reflection of the elastic signal in the experiment.

B. Thin elastic versus thin viscoelastic plate models

Let us start the numerical part with the non-viscoelastic model (equation 1 of the accompanying letter). Allcomputations were performed with the commercial finite element package COMSOL MULTIPHYSICS [3]. Accordingto the 50 Hz source, the wavelength of the signal propagated at the source is 1.56 m compared to the 1.73 m separatingeach borehole’s center, hence we expect to be in the range of frequencies for stop bands, as confirmed by the dispersionrelations shown in Figure 2 of the accompanying letter. The center of the vibroprobe is located at 1.5 m from thefirst line of the grid, in the perpendicular direction, hence we set up a point force there in the computational model(which is a square of 50 m ×50 m surrounded by perfectly matched layers which further attenuate the signal before itreaches the edges of the domain where we set up stress-free boundary conditions). In the soft soil of the experiment,the attenuation is significant, but not dominant over the domain where we compute the interaction between seismicwave and structured soil (the signal to noise ratio is over two at 10 m from the source, so the wave indeed propagatesthrough the set of boreholes). Fig. 1b indeed shows a concentration of energy in a 10 m wide zone strongly attenuatedtowards the structure centered around the vibroprobe, in good agreement with panel (a) of Fig. 6. In this area andnear the source, the difference between the measured energy field after and before carrying out the boreholes canreach twice the initial value confirming a strong reflection of surface waves by the seismic metamaterial. The signalhardly exceeds the second row of boreholes in Fig. 3b) and Fig. 4, showing the efficiency of this device, for thisgeometry and a 50 Hz source in soft soils, here again in good agreement with Fig. 6(a).

Since our numerical model is shown to be in good agreement with experimental data, it seems interesting tonumerically explore possible extensions of our seismic metamaterial. In Fig. 6 we look into various configurations offlat seismic metamaterials, wherein the source is a point force (a), (b), or a plane wave (c), and even when the array istilted through an angle of π/4, which leads to the lensing effect due to some form of negative refraction. In Fig. 7(d),we show that one can achieve some focusing effect when additional boreholes are drilled in the soil, with the samediameter (0.32m) and center-to-center spacing (1.73m), but importantly the color scale has been saturated at thelocation of the point source as the wave amplitude is weak behind the lens (this means in a more realistic, viscoelasticmodel, we cannot see the image any more since the wave signal is strongly attenuated at the image location). In panel(a), an image is formed behind an array of 46 boreholes, but its intensity is very low (lower than in panel (d)). We

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Figure 5: Simulated dispersion curves for Kirchhoff (upper panel) and Mindlin (lower panel) plate models: Frequency versusBloch wavenumber describing the reduced Brillouin zone of vertices Γ = (0, 0), X = (π/d, π/d), M = (π/d, 0) for a periodicplate of pitch d = 1.73m and thickness 5m, with inclusions of diameter 0.32m and density 1/1000th that of the surroundingmedium (soil). The quasi linear behavior of the acoustic band in the Mindlin model can be attributed to the large thickness ofthe plate versus the array pitch, which is not reflected in the Kirchhoff (thin-plate) model. It seems plausible to assume thatthe real dispersion of Rayleigh waves in the experiment (which would require solving a full three-dimensional Navier equation)is similar to what we achieve with the two approximate plate models. One notes that both models predict a partial stop bandaround the frequency of 50 Hz which is consistent with the plot of surface wave intensity for a forcing at that frequency in theinset of the upper panel.

therefore decided to increase the number of boreholes (Fig. 7b), which shows no improvement in the intensity of theimage, see panel (b). We then played with the overall shape of the seismic metamaterial in order to further enhanceits intensity. We found that a ring-type geometry is optimal for the focusing effect, and the location of the sourcecan be shifted all around its perimeter, see Fig. 7(c) and (d). Such a numerical investigation now requires furtherexperimental evidences before some conclusions are drawn (note for instance that our numerical modeling does notaccount for viscoelastic features of the soil which will damp the elastic signal, as transpires from Fig. 7), but onecan already argue that attempting to shield seismic waves with a simple design of a ring-type seismic metamaterialmight not prove to be an ideal solution, see Fig.7 (c) and (d). A concept of seismic cloak to protect a building fromearthquakes would at least require in-depth analysis to achieve some judicious, not necessarily periodic, arrangementof boreholes.

Previous solutions from the thin plate model do not include viscoelastic effects. Isotropic or orthotropic elasticand isotropic viscoelastic plate vibrations can be found in [4, 5]. Wave propagation in an orthotropic thin plate withviscoelastic behavior can be described by several rheological models (Voigt-Kelvin, Maxwell, Zener standard model,etc.). For the derivation of equation describing the vertical vibration of a thin viscoelastic orthotropic rectangularplate for Maxwell’s model, the constitutive equations take the following integral form

σij = Bijklεkl −∫ τ

0

Lijkl(t− τ)εkl(τ) dτ = Bijklεkl − Lijkl ? εkl , (1)

where i, j, k, l = 1, 2, 3, ? denotes the time convolution product, σ is the rank-two stress tensor, B and L are rank-fourtensors and ε is the rank-two strain tensor which satisfies the strain-displacement relation:

εij =1

2

(∂ui∂xj

+∂uj∂xi

). (2)

One can then derive essentially four viscoelastic plate models. In the first of them (Kirchhoff model), the plate isassumed to be in the state of pure bending in which plane cross-sections of the plate, to remain plane and perpendicular

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to the midplane of the plate. Thus, shear deformation is not included in this model. A state of plane stress isalso supposed and the effects of rotary inertia are neglected. In the second model (Rayleigh model), the effects ofrotary inertia are also included without any shear deformation. The third model (Flugge model) incorporates sheardeformation, but not rotary inertia effects. The fourth model (Timoshenko-Mindlin model) takes into account bothrotary inertia effects and shear deformations We opt for the first model (that of Kirchhoff), and following [5] we get

ρ−1 ?∇ ·(E1/2 ?∇ρ−1 ?∇ ·

(E1/2 ?∇Ψ

))− β4 ∂

2Ψ

∂t2= 0 , (3)

where ? denotes the time convolution product. Here Ψ(x, y) is the amplitude of displacement along the z-axis in Fig.1b of the accompanying letter, and ρ is the heterogeneous density of the plate: In the soil ρ = 1500 kg/m3 and in boredholes ρ = 1.2 kg/m3. The diameter of the inclusions is 0.32 m. Moreover, the parameter β4 = ρh/D, with h the platethickness (assumed to be 5 m), and D the plate rigidity, which in the present case is D = Eh3/(12(1 − ν2)). Fromthe analysis of the investigations carried out on site, the soil characteristics implemented in the model are E = 100MPa and ν = 0.3. We consider a transient source at the frequency 50 Hz and solve the integro-differential equation(3) over a time period of 120 s using the COMSOL MULTIPHYSICS commercial software.

The results of seismic wave interaction with the structured soil without (Fig. 6 and Fig. 7) and with (Fig. 8)viscoelasticity clearly demonstrate that this can deeply affect the nature of wave propagation. Indeed, one can see forinstance that the non-viscoelastic model does not account for the fact that a fourth row of boreholes deeply attenuatesthe seismic wave in Figure 6, panels (a) and (b), unlike for the viscoelastic counterpart in Figure 8, panels (a) and(b), wherein we used some viscoelasticity parameters which fit the experimental data showing strong elastic waveattenuation over a distance of 10 meters: The signal to noise ratio is over two at 10 m from the source (note ourpreliminary numerical results are only for qualitative comparisons).

Figure 6: Simulated lensing effect for two seismic metamaterials. Plot of flexural wave (i.e. displacement in xz-plane in Fig.1 b of accompanying letter) intensity for a seismic source at 50 Hz and a seismic metamaterial with (a) same parameters asin Fig. 3 (notably 31 boreholes) and (b)-(c) 41 boreholes and (d) 78 boreholes: Shielding for a forcing with (a) 31 boreholes,(b) 41 boreholes; Shielding for an incident flexural wave (Gaussian beam) for 41 boreholes; (d) Lensing for 78 boreholes for aforcing at 50 Hz when the square array is tilted by an angle π/4.

V. CONCLUDING REMARKS AND PERSPECTIVES

Seismic waves have tremendous applications in civil engineering [6]. Their enhanced control in structured soilscould lead to earthquake protections with improved efficiency or life expectancy (passive mass dampers suffer from

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Figure 7: Simulated lensing effect for three ‘ring-type’ seismic metamaterials. Plot of flexural wave (i.e. displacement inxz-plane in Fig. 1 b of accompanying letter) intensity for a forcing at 50 Hz and a seismic metamaterial with same parametersas in Fig. 3, but with more boreholes: Focusing with (a) 46 boreholes, (b) 68 boreholes, (b) 128 boreholes, (d) 128 boreholeswith a forcing of a shifted location.

Figure 8: Simulated shielding effect for four viscoelastic seismic metamaterials. Plot of flexural wave (i.e. displacement inxz-plane in Fig. 1 b of accompanying letter) intensity for a seismic source (plane wave) at 50 Hz and a seismic metamaterialwith (a) same parameters as in Fig. 3 (notably 31 boreholes) and (b)-(c) 41 boreholes and (d) 78 boreholes: Shielding for aforcing with (a) 31 boreholes, (b) 41 boreholes.

inherent spring fatigue) since drilling air holes or putting concrete pillars with a specific (but not necessarily periodic)pattern in a soil can detour, or reflect, surface waves from an area to be protected (nuclear plant, airport, monument

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or even city). Back in 1968, R.D. Woods already realized in situ tests, however with a 200 to 350 Hz source (so wellbeyond the seismic wave spectrum involved in destruction of human infrastructures [6, 7]), in order to demonstratecertain protection with circular or linear empty trenches [8], essentially based on some Bragg reflection phenomenon.These results were compared 20 years afterwards against numerical modeling studies performed by P.K. Banerjee [9].More recently, a concept of concentric multilayers to detour surface seismic waves was proposed by some of us [10].On the other hand, S.H. Kim and M.P. Das proposed to damp the seismic waves by converting the energy carriedby surface waves into another type such as sound and heat [11], which is reminiscent of passive-mass dampers, butusing the concept of locally resonant structures [12], now known as acoustic metamaterials [13], rather than springsand masses, thus avoiding fatigue of the structure. Alternatively, one can also embed a phononic crystal within thebuilding foundation [14] thereby prohibiting propagation of elastic waves along the building, but this is an invasivetype of protection, not applicable to certain structures such as historical monuments.

S.G. is thankful for European funding through ERC Starting Grant ANAMORPHISM. S.B. and E.J. thank theMENARD’s Earthquake Engineering Team.

[1] See Supplemental Material at [http://link.aps.org/ supplemental/10.1103/PhysRevLett.000.000000] for a movie on theconstruction and characterization of the seismic metamaterial.

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Science 1(2), 120-123 (2009).[6] Semblat, J.F. and Pecker, A. Waves and vibrations in soils: earthquakes, traffic, shocks, construction works, (IUSS Press,

Pavia, 2009).[7] http://physicscentral.com/explore/action/cloaking-earthquakes.cfm[8] Woods, R.D. Screening of surface waves in soils (Tech. Rep. IP-804, University of Michigan, 1968).[9] Banerjee, P.K., Ahmad, S. and Chen, K. Advanced application of BEM to wave barriers in multi-layered three-dimensional

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(2000).[13] Craster, R. V. and Guenneau, S. Acoustic Metamaterials: Negative Refraction, Imaging, Lensing and Cloaking (Springer-

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