hers (Inst. Fresnel d), M. Clark, V. Ageeva . Colquitt (Liverpool), P. R. Craster, Paris ———————— December 2017 – p. 1/30 Richard Craster [email protected]

Seismic metamaterials Department of Mathematics, Imperial College London (Nottingham), P. Sebbah, G. Levebvre (Inst Langevin) and ot Roux (ISTerre, Grenoble), S. Guenneau, Y. Achaoui, S. Enoch joint with A. Colombi, B. Maling, O. Schnitzer (Imperial), D Marseille), T. Antonakakis (Multiwave AG), S. Brule (Menar / S wave mode R. Craster, Paris ———————— December 2017 – p. 2/30

Challenges Very low frequency and long waves 0-10 Hz High material mismatch of soil, rock andSoil building mediation materials and seismic site effects Elasticity - full vector system, surface Rayleigh waves and P Three routes/ concepts: : Seismic mode converters and the:Elastic elastic surface rainbow. wave lenses. :Elastic phononic crystals with ultra-broad stopbands. conversion and coupling at interfaces. face to bulk waves, ollaborators R. Craster, Paris ———————— December 2017 – p. 3/30

Seismic mode conversion Can we divert waves somewherecan “safe”? this Mode be conversion achieved from using sur A sub-wavelength forest resonators? metamaterial !? See the recent experiments of Roux and c https://metaforet.osug.fr/ ex lenses? Can this be R. Craster, Paris ———————— December 2017 – p. 4/30

Seismic gradient index lenses Can we steer surface wavesachieved around using objects sub-wavelength using soil gradient mediation? ind A Luneburg lens arrangement similar to that in optics. tructures that operate R. Craster, Paris ———————— December 2017 – p. 5/30 l overlaying bedrock) from

Seismic “shields” Hz - V. hard to achieve. 10 − 0 Can we design structures thatseismic protect waves? buildings Important (on to soft note soi at that we need sub-wavelength s J. Mech. Phys. n a red-blue colorscale, . Guenneau R. Craster, Paris ———————— December 2017 – p. 6/30 transforms and the Bloch (proportions are not to scale). The inset depicts the source 2 u

Surface sub-wavelength resonators Volume 99, 379-393, 2017] Solids problem gives the dispersion curves. [D. J. Colquitt, A. Colombi, R. V. Craster, P. Roux, and S. R. L time function generating the monochromaticThis Rayleigh can wave. be solved exactly for an infinite array using Fourier The 2D computational domain forthe the vertical half-space displacement also showing, i ayleigh wave. R. Craster, Paris ———————— December 2017 – p. 7/30 ing Bloch theory. Note the

Dispersion curves The dispersion curves for anvery infinite narrow periodic stop-band, array the found slow us waves and hybridisation of R represents λ on avefield magnitude ower branch, ree half-space are also nity of a resonance, whilst R. Craster, Paris ———————— December 2017 – p. 8/30 branches and the first alf-space. (b) illustrates the

Below the stop band the wavelength. Elastic streamlines arecolorcode. superimposed The to field the in w theAn resonators animated is version of not this shown. figure Start animation Start animati (a) A zoom-in of dispersionresonance. curves, P, highlighting S the and first Rayleigh two waveannotated dispersion in curves different for colours. the (b,c)stop f Wavefield band inside behaviour the of h the(c) array shows for the surface usual waves in pass the band vici behaviour associated with the l on above the Rayleigh R. Craster, Paris ———————— December 2017 – p. 9/30 apparent bandgap. rsection of the dispersion tes the weak transmission of

Above the stop-band The upper branch of theRayleigh dispersion waves (blue) for line. frequency-wavenumber combinations (b) just sound-line. illustra (c) demonstrates the behaviourcurves close with to the the Rayleigh inte sound-lineAn and animated responsible version for of the this figure Start animation Start animati rt Rayleigh waves into reate a seismic filter. R. Craster, Paris ———————— December 2017 – p. 10/30

Surface wave mode converter - Metawedge Start animation Here we designbulk a waves graded (this surface is array a to genuine conve seismic metamaterial....) or to c , 6 , Scientific Reports R. Craster, Paris ———————— December 2017 – p. 11/30

The dispersion curves andlength dependence upon resonator 27717,2016] [A. Colombi, D. Colquitt, P. Roux, S. Guenneau, R. V. Craster awedge does this over a R. Craster, Paris ———————— December 2017 – p. 12/30

Broadband performance very broad range of frequencies. Absolutely key is to obtain broadband performance, and the met Scientific orests as a natural resonances” R. Craster, Paris ———————— December 2017 – p. 13/30

A natural metamaterial? , 2016] 6 , Reports [A. Colombi, P. Roux, S. Guenneau, P. Gueguen and R. Craster “F seismic metamaterial: Rayleigh wave bandgaps induced by local ium block created by R. Craster, Paris ———————— December 2017 – p. 14/30

Ultrasonic experiments - scaled down in size Experimental setup - laser dopplermicro-milling. vibrometer scanning alumin R. Craster, Paris ———————— December 2017 – p. 15/30

Scaled down in size but up in frequency ve (filtered at 450-600 Hz) R. Craster, Paris ———————— December 2017 – p. 16/30

Elastic rainbow Experimental results showing the reflection of the surface wa R. Craster, Paris ———————— December 2017 – p. 17/30

Mode conversion Experiments and numerics teer” Rayleigh waves erg lens. R. Craster, Paris ———————— December 2017 – p. 18/30

Designer elastic surfaces Here we are trying tousing design ideas elastic from surfaces metamaterials/transformation so optics that - we Luneb can “s Start animation Start animation Use the Maxwell-Garnett effective medium approach.

v2 − v2 v2 v2 − ( se s0)( s0 + i ) f =1 2 2 2 2 ; (vse + vs0)(vi − vs0) to tune the effective “refractive” index.

R. Craster, Paris ———————— December 2017 – p. 19/30 (a) Maps of the elastic energy distribution at the surface (z=0) for the homogenous case. Same as (b) but for the lenses case. (c) Spectral density of the rooftop motion of the building in dB. In the upper panel, the blue trace is calculated for the homogeneous case while the black is obtained when the lenses enclose the building. The lower panel is identical, but now for the vertical component.

R. Craster, Paris ———————— December 2017 – p. 20/30 (a) Plot of the differential energy between reference and protected case vs. incidence angle of the wavefront with respect to the lenses. (b) Same as Fig. 3b but for a plane wave approaching the lenses with a 45◦ incidence angle.

R. Craster, Paris ———————— December 2017 – p. 21/30 tructures that operate R. Craster, Paris ———————— December 2017 – p. 22/30 l overlaying bedrock) from

Seismic “shields” Hz - V. hard to achieve. 10 − 0 Can we design structures thatseismic protect waves? buildings Important (on to soft note soi at that we need sub-wavelength s R. Craster, Paris ———————— December 2017 – p. 23/30

A phononic shield under construction (Brule - Menard) cible Brillouin zone ith no columns (b) a ssumed on all its boundary) R. Craster, Paris ———————— December 2017 – p. 24/30

Idealised upper/lower bounds . XM Dispersion curves for idealised cases:clamped (a) perfectly clamped rigid bedrock column w (i.e.of with radius zero 0.15 displacement m. a TheseΓ are shown around the edges of the irredu m 6 . 0 s of the irreducible R. Craster, Paris ———————— December 2017 – p. 25/30 m linked to their nearest al columns of radius 3 . 0 XM Γ

Real designs Dispersion curves for realistic cases: (a) steel cylindric clamped to the bedrock (b) steel columns of radius neighbours via steel plates. TheseBrillouin are zone shown around the edge mented by m (upper curve) and 6 . e same filling fraction, are 0 e value of the vertical ency (in Hz) through an R. Craster, Paris ———————— December 2017 – p. 26/30 s

Transmission spectra Attenuation (in dB) of incomingarray Rayleigh with waves 10 versus rows frequ of clamped cylindrical inclusions of radiu clamped cross-shaped inclusions (lower curve).displacement The at absolut 9 Hz forattachments (a) to the its steel neighbours; column the andshown geometries, (b) as which a insets. share column th aug 10 . 4189 − = 11 T R. Craster, Paris ———————— December 2017 – p. 27/30 ource placed centrally in the ictions. ive medium tensor ; (c) 138 kHz creating a + shape with 74 ; (b) 122 kHz hyperbolic behaviour following from . = 311 = 1025 22 22 T T = . and 11 51 . 17 T .

Kirchhoff plate approximation/ HFH 180 = 19 − 22 T = 11 Dispersion curves, frequencies to investigate(a-c) and the HFH flexural pred wave fieldarray. predicted (a) for 94 a kHz, point isotropic frequencycomponents response s as predicted by effect T and =0.33) ν , E=74 GPa, . Plates of duraluminium a R. Craster, Paris ———————— December 2017 – p. 28/30 3 kg/m = 2789 ρ

Experiments (pinned elastic plates) 10 cm 0.5 mm-thick vibrating plate ( × These are delicate experiments, highly frequency10 dependent cm (with Sebbah, Lefebvre, Antonakakis, Achaoui, Guenneau) full FDTD at by a forcing of fixed R. Craster, Paris ———————— December 2017 – p. 29/30 (f) (b) (d) (c) (a) (e) kHz of spectral bandwidth kHz of spectral bandwidth. (a,c,e) are experiments at 10 1 kHz with kHz with 128 119 , ,

FDTD simulations versus experimental data 105 110 , , Ω=90 Numerical versus experimental plots offrequency. flexural (b,d,f) waves are excited snapshots inΩ=95 the time domain taken from s and theory. hysics olls-Royce, AMEC s for mitigating ns, edited books etc. pproaches and steering on the R. Craster, Paris ———————— December 2017 – p. 30/30

Concluding remarks Departments providing fast, accurate numerics,complementary asymptotic insight. a (in NDE work) or government DSTL. Maling et al Wave Motion 2016 for asymptotics re Maxwell. We have here illustrate three novel metamaterial based design Much of this work is done in conjunction with Engineering and P Some is done with industry: Menard (Civil Engineering), EDF, R Academic papers in these areas, a coupleFor of the patent dynamic applicatio anisotropy: Lefebvre et al PRL 2017 experiment Whispering Bloch Modes covered in Maling et al 2016a,b. seismic waves and ground vibration - diverting into the bulk, surface or shielding the structure.