Towards the design of a feasible seismic metabarrier using multi-mass resonators

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1 Towards the design of a feasible seismic metabarrier using multi-mass resonators Antonio Palermo, Matteo Vitali, Sebastian Krödel, Chiara Daraio, Alessandro Marzani DICAM University of Bologna Bologna May 15, 2017

2 Metamaterials: Periodic and locally resonant media Periodic Media Locally resonant materials Liu, Science (2000) Maldovan Nature (2013) Image courtesy of L. Bonanomi Wave filtering at wavelength λ a a periodicity length Wave filtering at subwalenght scale f f r resonance frequency

3 Periodic and locally resonant media: Multi-scale applications nm MHz Lee, Adv.Mater.(2012) mm KHz Wu, Phys. Rev. B (2009) m Hz Casadei, J.Appl. Phys (2012) Kim, Phys. Rev. B (2012) Brule, PRL (2014)

Benedettini et al. Bull. Earth Eng.(2014) 0.

4 Acceleration (g) Seismic excitation & Building Natural Frequencies Northridge earthquake spectrum (1994) [6] Hz λ = m F. Benedettini et al. Bull. Earth Eng.(2014) Frequency (Hz) (2015) Wave s control at subwalength scale

Seismic Rayleigh Waves Displacement seismograph http://web.ics.purdue.edu/~braile/edumod/waves/rwave.

5 Seismic Rayleigh Waves Displacement seismograph 22:20:00 :30 :40 :50 Time[hr:min:sec]

6 Rayleigh waves and local resonances: Surface waves deflection by forest trees ~40Hz Colombi et al. Sci. Rep. (2016) Colombi et al. Sci. Rep. (2016)

7 The idea: a Seismic Metabarrier for Rayleigh waves f r A. Palermo et al, Sci. Rep (2016)

8 Seismic Metabarrier: Resonator Design 2-10 m m

9 Modeling approach Analytical Approach Soil Numerical Model Experimental Verification

1 ω2 k 2 c L 2 +i(ωt kz) 1 ω2 k 2 c S 2 +i(ωt kz) Wave equation: ω k = c < c

10 Analytical model: 3-Mode Resonators 1. horizontal 2. vertical 3. rotational Rayleigh waves Pressure and shear potentials: kz φ = Ae kz ψ = Be 1 ω2 k 2 c L 2 +i(ωt kz) 1 ω2 k 2 c S 2 +i(ωt kz) Wave equation: ω k = c < c S K H K R 2 φ = 1 c L 2 2 φ t 2 2 ψ = 1 2 ψ 2 c S t 2 K v Boundary conditions: z Soil x

11 Analytical model: Dispersion Relation c S c R c L Bulk Shear waves Band Gap c c S Bulk shear wave c S

12 Analytical model: Band Gap limits Upper and lower edge Band Gap Limits: Vertical mode: f : k f = f v f + : k = ω c S f + = f v (β + β 2 + 1) Non-Dimensional parameter: β = mω v 2Aρ s c S,soil 1 c 2 S,soil 2 c L,soil Ω(β) f + f ΔΩ = 2 β + β (β + β )

13 Analytical model: FE validation Seismic Metabarrier Harmonic Excitation [1 Hz] y z x

14 Analytical model: FE validation Seismic Metabarrier Harmonic Excitation [3 Hz] y z x

15 Case Study: Analytical prediction Resonator f v = 4. 9 Hz m = 6.6 ton 2m c L,soil c S,soil 1.1 m Soil = 230 m s = 120 m s β = 0.37

16 Case Study: FEM simulation 12 Resonators Attenuation 60%

17 Scaled experimental setup: measurements 12 Resonators β exp = 0.37 BG = khz

18 Seismographs: free propagation vs. resonators measurements line x wave front free propagation resonators boundary reflections boundary reflections resonators c r initial pulse initial pulse

19 Frequency analysis Initial pulse Transmitted pulse Transmitted frequency content Res no Res. BG khz

20 Performance of the metabarrier m A BG c s BG

Multi-mass metabarrier for broad band attenuation Multi-mass resonators for multi-frequency attenuation Rayleigh waves Pressure and shear

21 Multi-mass metabarrier for broad band attenuation Multi-mass resonators for multi-frequency attenuation Rayleigh waves Pressure and shear potentials: kz φ = Ae kz ψ = Be 1 ω2 k 2 c L 2 +i(ωt kz) 1 ω2 k 2 c S 2 +i(ωt kz) Boundary conditions: σ zz = σ zzres Modal approach σ xz = 0

22 Multi-mass metabarrier for broad band attenuation Multi-mass resonators for multi-frequency attenuation m 2 = 3.5 ton m 1 = 7 ton

23 Multi-mass metabarrier for broad band attenuation: Performance DM 2SM BGs Lres Identical total mass with half lenght

24 Rainbow trapping and metawedge Rainbow trapping Krodel et al, Ext. Mech. Lett. (2014) Metawedge Colombi et al. Sci. Rep. (2016)

25 A Multi-mass metawedge f r,0 Single-Mass Metawedge Lres f r,n f r,0 Lres f r,0 = 2. 4 Hz ff r,0 r,n = Hz 4 Hz f r,n = 13.4 Hz f r,n Double-Mass Metawedge K f r,m Lres/2 f r,n 2 nd Mode f r,0 f r,m 1 st Mode

26 Optimal multi-mass metabarrier for broad band attenuation Structural analysis f r,1,, f r,n Input f r,1,, f r,n Δf r,1,, Δf r,n GA Optimization Output Min ( m) Δf r,1,, Δf r,n

27 Optimal multi-mass metabarrier for broad band attenuation m 4 = 3.0 ton m 3 = 4.0 ton m 2 = 5.6 ton m 1 = 4.7 ton m/a < 20 ton/m 2 f r,1 = 2.27 Hz f r,2 = 4.48 Hz f r,3 = 7.33Hz f r,4 = Hz

28 Main results Main results Design of seismic metabarrier for surface wave attenuation Proof of concept validation on a scaled down model Analytical and numerical studies show achievable large attenuation (60%) in the low frequency range Hz with compact single and multi-mass metabarriers Limitations «Ideal soil»: - linear elastic; -isotropic/homogenous; - flat/regular surface The complex propagation pattern of seismic waves in real soils and the full interaction between soil and metamaterial cannot be captured. Elastic resonator: energy dissipation and material nonlinearities can influence the dynamic response of the resonator.

29 Acknowledgments: Prof. Alessandro Marzani Prof. Chiara Daraio Sebastian Krödel Matteo Vitali Thanks for your attention! Antonio Palermo DICAM

SURFACE WAVE MODELLING USING SEISMIC GROUND RESPONSE ANALYSIS

43 SURFACE WAVE MODELLING USING SEISMIC GROUND RESPONSE ANALYSIS E John MARSH And Tam J LARKIN SUMMARY This paper presents a study of surface wave characteristics using a two dimensional nonlinear seismic