Unveiling Extreme Anisotropy in Elastic Structured Media

G. Lefebvre1, T. Antonakakis2, Y. Achaoui3, R. V. Craster4, S. Guenneau3, P. Sebbah1,5

1 Institut Langevin, ESPCI ParisTech CNRS UMR7587, 1 rue Jussieu, 75238 Paris cedex 05, France2 Multiwave Technologies, Switzerland

3 Aix-Marseille Univ., CNRS, Centrale Marseille, Institut Fresnel, Marseille, France4 Department of Mathematics, Imperial College London, London SW7 2AZ, UK and

5 Department of Physics, The Jack and Pearl Resnick Institute forAdvanced Technology, Bar-Ilan University, Ramat-Gan 5290002, Israel

Periodic structures can be engineered to exhibit unique properties observed at symmetry points,such as zero group velocity, Dirac cones and saddle points; identifying these, and the nature of theassociated modes, from a direct reading of the dispersion surfaces is not straightforward, especiallyin three-dimensions or at high frequencies when several dispersion surfaces fold back in the Bril-louin zone. A recently proposed asymptotic high frequency homogenization theory is applied to achallenging time-domain experiment with elastic waves in a pinned metallic plate. The predictionof a narrow high-frequency spectral region where the effective medium tensor dramatically switchesfrom positive definite to indefinite is confirmed experimentally; a small frequency shift of the pulsecarrier results in two distinct types of highly anisotropic modes. The underlying effective equationmirrors this behaviour with a change in form from elliptic to hyperbolic exemplifying the high degreeof wave control available and the importance of a simple and effective predictive model.

PACS numbers: 41.20.Jb,42.25.Bs,42.70.Qs,43.20.Bi,43.25.Gf

The analysis of periodic media underpins advances inelectronic properties, wave transport in photonics andacoustics, as well as in interference phenomena through-out many fields of physical and engineering sciences.High symmetry points on dispersion diagrams where thegroup velocity tends to zero and the density of states ishigh, are often associated with unusual effective prop-erties such as slow-light and ultra-directivity [1, 2] whenthe bands display vanishing curvature and even all-angle-negative refraction [3]. More recently, photonics researchhas focused on Dirac cones following the rise of graphene[4, 5] as well as on saddle points in the search for hyper-bolic type features [6, 7].

However, identifying the remarkable symmetry pointsand the nature of their associated mode from a directreading of the dispersion surfaces is not straightforward.An asymptotic theory that easily identifies the natureof these singularities has been developed [8], that doesnot require extensive computation of the Bloch dispersionsurfaces. Indeed, in contrast to most standard methodssuch as semi-analytic plane-wave [9, 10] and multipole ex-pansions [11], numerical finite element or finite differencetime domain methods [12], the so-called high frequencyhomogenization (HFH) approach [8] is purely algebraic.Because it only focuses on the apexes of the Brillouinzone, HFH captures the essential features without inten-sive computation and in an intuitive manner.

The basic methodology has been validated in the mi-crowave regime [13] but for elastic waves the tensorialequations make any interpretation and prediction ex-tremely difficult and the situation is far more demanding.By considering thin plates, the problem becomes moretractable and several achievements have been reportedrecently, including broadband cloaking [14, 15], negative

refraction [16] and lensing [17, 18] effects which paralleleffects observed earlier in electromagnetic metamaterials[19, 20]. Topical large-scale applications to seismic meta-materials [21] and experiments using pillars [22] or sub-wavelength resonators attached to elastic plates [23] pro-vide further motivation. The Kirchhoff-Love (KL) plateequation provides the simplest model that captures flexu-ral waves and for structured media such as phononic crys-tals and metamaterials the underpinning theory is beingadvanced accordingly [5, 24–26]. The range of validityof KL plate theory, as used for structured media, versususing full elasticity is unclear; for homogeneous platesthe KL theory is only valid for thicknesses up to around1/20th of a wavelength [27] and we operate outside this.Furthermore, the experimental behaviour in the time do-main, and the potential for hyperbolic behaviour (whichis hard to find experimentally as it is highly dependentupon frequency), all remain unexplored. The full elasticsystem is awkward to disentangle and, given its complex-ity, the simplicity of the KL theory and HFH are appeal-ing; we carefully justify the use of the simplified modelin [28].

In this Letter, the utility of this simplified viewpointwhere the elastic wave physics is captured using the sim-ple plate model, and the frequency-dependent anisotropyidentified by the HFH asymptotic approach is tested inthe most challenging case of vector elastic waves andguides the design of a doubly-periodic clamped elasticplate capable of exhibiting the highly directive featuresof both elliptic and hyperbolic points. Such a constrainedelastic plate has a zero-frequency stop band that imme-diately excludes any possibility of using a quasi-statichomogenization method. We show that HFH theory effi-ciently and accurately models such a system at arbitrarily

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high frequencies. It predicts the conditions for the obser-vation of a narrow high-frequency spectral region wherethe effective medium tensor dramatically switches frompositive definite to indefinite. Using heterodyne laser in-terferometry, we measure the time evolution of an elasticGaussian pulse with central frequency, Ω0, within thepinned elastic plate. At the remarkable frequencies Ω0

identified by the HFH theory, we observe highly direc-tive emission and show how a small frequency shift ofthe pulse carrier results in two distinct types of highlyanisotropic modes.

FIG. 1: Asymptotic prediction of local mode structure andrespective dispersion surfaces: Effective medium tensor com-ponents govern the parabolic (|T11| |T22|), hyperbolic(T11T22 < 0), or elliptic (T11T22 > 0) character of Eq. (1))and capture the extreme anisotropic features of waves propa-gating within a doubly periodic medium shown in (a,c,e). Thecorresponding local dispersion surfaces are shown in (b,d,f).The algebraic criteria for critical points straightforwardly ex-tend to n-dimensional periodic systems and reveal effectiveelliptic (T11T22...Tnn > 0) or hyperbolic (T11T22...Tnn < 0)features without resorting to visualization of hyper-surfaces.

For waves propagating through an infinite periodicstructure, one invokes Bloch’s theorem [29, 30] and con-sider an elementary cell; the Bloch wave-vector κ =(κ1, κ2) characterizes the phase-shift going from one cellto the next and the dispersion relation connects it with

FIG. 2: Experimental apparatus for imaging pinned elas-tic plate vibrations. The heterodyne laser interferometer(Thales SH-140) probes the out-of-plane vibration at the sur-face of a 10 cm × 10 cm large, 0.5 mm-thick Duraluminiumplate, clamped on a square periodic lattice by 3-mm diametercolumns screwed between the thin plate and a metallic base(see inset). 3 mm-thick adhesive rubber is sandwiched be-tween the plate’s edges and a square metallic frame attachedto the base, to reduce unwanted elastic edge reflections.

frequency. As is well-known in solid state physics [30]only a limited range of wavenumbers need normally beconsidered, namely the wavenumbers along the edges ofthe irreducible Brillouin zone. The HFH theory focuseson the vicinity of the vertices of the Brillouin zone, whichin a square lattice reduce to 3 high symmetry points Γ,M and X located at the apex of a right-angled trian-gle. Within this framework, the wavefield, u, can befactorized as the product f(x)U(x; Ω0), [25] of the Blochsolution U(x; Ω0) at the vertex at frequency Ω0 and anenvelope function f(x) given by the effective equation

Tij(Ω0)∂2f

∂xi∂xj+ (Ω2

0 − Ω2)f = 0, (i, j = 1, 2) (1)

with x = (x1, x2). Eq. (1) is a partial differential equa-tion (PDE) of second order which is obtained from anasymptotic analysis of the KL equation which is fourth-order, see [28] and [25] for the full derivation; Ω is thewave frequency and Ω0 is the standing wave frequency atthe vertex of the Brillouin zone. In essence, the factor-ization f(x)U(x; Ω0) captures the small scale variationof the wavefield and overcomes the limitation of stan-dard homogenization theories, which consider the wave-field uniform at the lattice scale. At the same time, theHFH theory does not require infinite periodic structureand can describe finite-size systems. The nature of thesolutions is critically determined by the dynamic effectivemedium tensor Tij . For a lattice cell with reflectional androtational symmetry (here, a square cell with a circularclamped defect at the center), the off-diagonal terms are

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zero and this is just

T11(Ω0)∂2f

∂x21+ T22(Ω0)

∂2f

∂x22+ (Ω2

0 − Ω2)f = 0. (2)

We can immediately see that having both T11, T22 of thesame sign leads to the PDE in (2) being of elliptic char-acter; conversely opposite signs lead to hyperbolic be-haviour. This change of character is transparently cap-tured by the model and rapidly detected using the HFHtheory. Similarly one can have T11 T22 or vice-versaleading to strong directional behaviour along one axisor the other and this behaviour too is predicted. Thesethree types of PDE behaviour, parabolic (|T11| |T22|),hyperbolic (T11 = −T22) and elliptic (T11 ' T22), are il-lustrated in Fig. 1 where the associated modes are shownside-by-side with the corresponding local dispersion sur-faces. The HFH theory therefore reduces the difficultproblem of interpreting often intricate isofrequency sur-faces to the straightforward comparison of the diagonalterms of the tensor, here just two numbers, Tii.

Here we want to investigate the predictive capabilityof the HFH theory in the framework of elastic waves.Therefore, we consider pinned (so-called) platonic crys-tals (PPCs) created by an array of clamped points, withthe objective of achieving highly directive emission of aperiodically oscillating source located therein. PPCs arethe elastic plate analogue of photonic crystals with infi-nite conducting rods [31]. Indeed, PPCs display a zerofrequency stop band so that flexural waves with wave-lengths arbitrarily long compared to the array pitch can-not propagate.

The system we want to study is shown in Fig. 2. Threemm-diameter and 5 mm-high cylinders have been ma-chined in a Duraluminium block to form a 10 by 10 squarearray of supports with 1 cm lattice spacing. Each sup-port has been threaded to enable a 10 cm × 10 cm 0.5mm-thick vibrating plate (ρ = 2789 kg/m3, E=74 GPa,ν=0.33) to be attached securely to it; in the languageof elastic plate theory these are clamped conditions. Al-though the clamped area is localised to the pin positions,the pins are not precisely point-like and so the clampingarea is accounted for in the theory.

The KL thin plate theory is often quoted, as in say[27], as only being accurate for wavelengths greater than20 times the plate thickness and this condition is not metat the frequencies we consider. However, it is the wave-length in the periodic system that actually matters, andthat can be large compared to the plate thickness, evenat high frequencies; we justify this technical point fur-ther by giving analogous computations for the full vectorelastic system in [28] .

Within this thin plate framework, the explicit formof the effective medium tensor Tij for the PPC geom-etry is given in [25]. The HFH theory is able to dis-criminate between interesting and uninteresting symme-try points and capture a rich array of behaviours for the

given experimental parameters. In our case, we selectthree frequencies of interest at which modes with remark-able properties are predicted. At 122 kHz near symme-try point X, the coefficients Tii in the effective tensorof the HFH theory have opposite sign, T11 = −180.172and T22 = 311.7432, and are close enough to reflecthyperbolic behaviour analog to Fig. 1(b). For excita-tion at the center of the system, the predicted theoreti-cal wavefield distribution at this particular frequency isshown in Fig. 3(b): The mode is highly directive and isaligned along the diagonals of the system. At 138 kHz,strongly differing Tii coefficients predict another very dis-tinct anisotropy within the effective material (Fig. 3(c)),|T22/T11| ∼ 0.005 leading to the effective equation beingparabolic. The mode is expected now to radiate alongthe lattice axes as shown in Fig. 1(a). For complete-ness we discuss the situation of equal T11, T22 as shownin Fig. 1(c), here we expect isotropy and elliptic effec-tive equations. The HFH prediction extends beyond thesymmetry points with a dispersion relation local to theband edge given in terms of the effective tensor

Ω ∼ Ω0 −Tij(Ω0)

2Ω0κiκj , (3)

where κi = Ki − di and di = 0,−π, π depending on theband edge chosen.

In the experiment, the ultrasound emission is pro-duced by a broadband piezoelectric transducer (Pana-metrics M109). A silica needle is stuck on the trans-ducer and put in contact with the bottom surface of theplate. We generate a single-oscillation short pulse exci-tation synthesized by an arbitrary waveform generator(Agilent 33120A). The out-of-plane displacement is mea-sured point by point with a broadband heterodyne in-terferometric laser probe (Thales SH140). The probe isscanned on a square grid with 0.68 mm step resolutionand the spatio-temporal distribution of the vibration fieldat the surface of the plate is reconstructed. The timesignal for each point is averaged ten times as a com-promise to eliminate most of the noise and yet conservea reasonable measurement time. The data are subse-quently filtered to obtain the elastic response to a narrowbandwidth (∆Ω0) Gaussian pulse with central frequency,Ω0, easily adjustable around theoretical prediction. Toreduce elastic reflections at the edges of the plate andmimic a finite array, surrounded by perfectly matchedlayers, a 3 mm-thick layer of adhesive rubber is sand-wiched between the edges of the vibrating plate and arigid metallic frame (see Fig. 2). This method efficientlyabsorbs elastic waves in the kHz range and reduces un-wanted reflections [18]; we measured an attenuation of 8dB after reflection from the absorbing boundaries. Theexperimental setup is shown in Fig. 2.

The experimental results are presented in Fig. 4, to-gether with time domain simulations of the pinned-plateusing the SIMSONIC package [34] which solves the full

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100

120

140

160

180

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(b)

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FIG. 3: HFH asymptotics: (a-c) the flexural wave field pre-dicted for a point frequency source placed centrally in thearray. (a) 94 kHz, isotropic response as predicted by ef-fective medium tensor components T11 = T22 = 1025; (b)122 kHz hyperbolic behaviour following from T11 = −180.17and T22 = 311.74; (c) 138 kHz creating a + shape withT11 = −4189.10 and T22 = 19.51. (d) Bloch dispersion di-agram using the irreducible Brillouin zone ΓXM (shown in-set) with frequencies used in (a-c) as red dots. (e-g) log-logvalidate the HFH asymptotics in the neighbourhood of thefrequencies used in (a-c). Circles are HFH asymptotics of thedispersion curves for the XΓ path; dashed curves are from fi-nite element numerics for the KL model and the vertical scaleis the difference of frequency from that at the band edge.

3D Navier system with the Finite Difference Time Do-main (FDTD) method with absorbing boundary condi-tions. In contrast to the modal representation shownin Fig. 1, here we show snapshots of the time evolutionof the elastic field. The complete temporal build-up ofthis mode is shown as a movie in [28]. Nevertheless, thehyperbolic and elliptic behaviour are well captured withhighly directional radiation at frequencies close to theprediction. Figures 4c & d show the X-shape obtainedat Ω0=105 kHz (∆Ω0 = 10 kHz), which is characteristicof hyperbolic dispersion. This contrasts with + shapeemission obtained at Ω0 = 128 kHz (∆Ω0 = 10 kHz) andshown in Figs. 4e & f. The elliptic isotropic behaviouris at Ω0 = 90 kHz (∆Ω0 = 10 kHz) as in Figs. 4a &b. Resonant frequencies found numerically (95, 110 and119 kHz) differ from the experimental frequencies by lessthan 10 %. The small difference can be attributed to theabsence of residual reflections in the simulations whereperfectly absorbing boundaries allow full establishmentof the modes. Note also that the numerical distributionsare sharper as the bandwidth used in the simulations is

(a)

(c)

(e)

(b)

(d)

(f)

FIG. 4: Snapshots of flexural wavefield evolution from apulse excitation at the center of the system with central fre-quency Ω and bandwidth ∆Ω. (a,c,e) experiment at Ω =90, 105, 128 kHz and ∆Ω = 10 kHz, see also the videos in[28]. We observe wavelengths at 90 kHz : 9 mm, at 105 kHz: 27.9 mm, and at 128 kHz : 6.8 mm which are at least tentimes the scanning resolution.(b,d,f) numerical simulations,FDTD solving the full 3D Navier system, at Ω = 95, 110, 119kHz with ∆Ω = 1 kHz. The minor difference in frequency isattributed to finite boundary effects in the experiments.

only ∆Ω0 = 1kHz, leading to better spectral selectivity.Actually, the choice of the pulse bandwidth in the ex-periment was crucial and delicate as a trade-off was tobe found between sharp modal selectivity (narrow band-width) and early establishment of the mode (short pulse)before edges are reached and reflections perturb the modedistribution. This problem is less of an issue in the nu-merics although ∆Ω0 is kept reasonably large to limitmode establishment time and computational time. As aresult, a residual trace of the + shape mode is seen ontop of the X shape mode e.g. in Fig. 4d. The minorfrequency offset between experiment and theory is alsoan expected consequence of modelling the plate vibra-tion with KL plate theory; this offset being also seen inthe detailed comparisons with the full elastic dispersiondiagram of [28].

In this Letter, the recently introduced HFH was ap-plied to the KL thin plate model, in a counter-intuitivehigh frequency limit where it might conventionally not beexpected to hold, to predict critical frequencies at whichstriking, strong anisotropic, wave patterns are generatedwithin a periodically pinned plate. Experimental obser-

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vations confirm these asymptotic predictions, as do timedomain computations.

We were drawn to this more intuitive, asymptotic ap-proach as the available alternatives require heavy compu-tational resources and, at high frequencies, produce re-sults hard to interpret as illustrated in [28]. These issuesare shared with many other complex doubly and triplyperiodic media, such as photonic and phononic crystals.The techniques developed, and illustrated herein, are alsovaluable in these contexts.

The applications are to slow wave waveguiding in delaylines (note the possibility to achieve ’U-turns’ in PPCswithout line defects, as shown in [28]), superlensing [6]and most importantly to energy harvesting through thePurcell effect taking advantage of the high local densityof states at the critical frequencies [6, 35, 36]. Interest-ingly, a pristine plate could mimic a periodically pinnedplate making use of an array of magnets or alternativelyof electrodes deposited on top and bottom surfaces of apiezoelectric plate.

R.V.C. thanks the EPSRC (UK) for support. S.G. isthankful for an ERC starting grant (ANAMORPHISM).P.S. is thankful to the Agence Nationale de la Recherchesupport under grant ANR PLATON (No. 12-BS09-003-01), the LABEX WIFI (Laboratory of Excellencewithin the French Program Investments for the Future)under reference ANR-10-IDEX-0001-02 PSL* and thePICS-ALAMO. This research was supported in part byThe Israel Science Foundation (Grant No. 1781/15 and2074/15).

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