Graded metasurface for enhanced sensing and energy harvesting
Jacopo M. De Ponti, Andrea Colombi, Raffaele Ardito, Francesco Braghin, Alberto Corigliano, Richard V. Craster
GGraded metasurface for enhanced sensing and energy harvesting
Jacopo M. De Ponti ∗ Dept. of Civil and Environmental Engineering, Politecnico di Milano,Piazza Leonardo da Vinci, 32, 20133 Milano, Italy andDept. of Mechanical Engineering, Politecnico di Milano,Via Giuseppe La Masa, 1, 20156 Milano, Italy
Andrea Colombi
Dept. of Civil, Environmental and Geomatic Engineering,Stefano-Franscini-Platz 5, 8093 Z¨urich, Switzerland
Francesco Braghin
Dept. of Mechanical Engineering, Politecnico di Milano,Via Giuseppe La Masa, 1, 20156 Milano, Italy
Raffaele Ardito and Alberto Corigliano
Dept. of Civil and Environmental Engineering, Politecnico di Milano,Piazza Leonardo da Vinci, 32, 20133 Milano, Italy
Richard V. Craster
Dept. of Mechanical Engineering, Imperial College London,South Kensington Campus, London, U.K. andDept. of Mathematics, Imperial College London,South Kensington Campus, London, U.K. (Dated: July 23, 2019) a r X i v : . [ phy s i c s . c l a ss - ph ] J u l bstract In elastic wave systems, combining the powerful concepts of resonance and spatial grading withinstructured surface arrays enable resonant metasurfaces to exhibit broadband wave focusing, modeconversion from surface (Rayleigh) waves to bulk (shear) waves, and spatial frequency selection.Devices built around these concepts allow for precise control of surface waves, often with struc-tures that are subwavelength, and utilise rainbow trapping that separates the signal spatially byfrequency. Rainbow trapping yields large amplifications of displacement at the resonator positionswhere each frequency component accumulates. We investigate whether this amplification, and theassociated control, can be used to create energy harvesting devices; the potential advantages anddisadvantages of using graded resonant devices as energy harvesters is considered.We concentrate upon elastic plate models for which the A mode dominates, and take advantageof the large displacement amplitudes in graded resonant arrays of rods, to design innovative meta-surfaces that focus waves for enhanced piezoelectric sensing and energy harvesting. Numericalsimulation allows us to identify the advantages of such graded metasurface devices and quantifyits efficiency, we also develop accurate models of the phenomena and extend our analysis to thatof an elastic half-space and Rayleigh surface waves. ∗ Corresponding author e-mail: [email protected] . INTRODUCTION Metamaterials, in their modern guise, emerged about two decades ago in optics [1] and theirpotential in creating artificial media, with properties not found in nature, and the conse-quent opportunities in wave physics, have triggered intense research activity. Metamaterialconcepts have now become a paradigm for the control of waves across much of physics andengineering and are widely used in electromagnetism [1–4], acoustics [5, 6] and elasticity [7].Whilst many theoretical studies have made the basic concepts well-established, manufactur-ing complex 3D bulk metamaterials remains a challenge. Attention has therefore focussedon metasurfaces because wave propagation is often dominated by scattering from, transmis-sion through, or waves guided along, surfaces; surfaces are also easier to pattern, print, ormanufacture. Metamaterials are available in a variety of shapes and sizes, but all featurearrangements of local heterogeneities are often repeated periodically.In elasticity, metamaterials achieved popularity with ideas based around Bragg scattering,phononic crystals, and material contrast to create band-gaps [8–10] that often draws uponideas from the photonic crystal community. In geophysical settings, and for applications inMechanical and Civil engineering, there has been a drive to obtain broader band-gaps at alow frequency [11–14] and to use locally resonant metamaterials [15–17]. Locally resonantmetamaterials are characterised by resonating elements, analogous to Helmholtz resonators[18] in acoustics and Fano resonances [19], enabling the creation of low-frequency band-gapsin structures relatively small compared to the wavelength; these ideas can be augmented bytuning or spatially grading the resonators to broaden or decrease band-gaps. At a smallerscale, and consequently higher frequency, wave redirection and protection [20–23] promiseadvantages in reducing vibrations in manufacturing, or in laboratories, where high tolerancesor precision measurements (e.g. interferometry) are required; similar motivations occur inultrasonic inspection to increase the signal to noise ratio. To capitalize on these recentmetamaterial designs, that can improve energy focusing over a wide frequency band, energyharvesting is another attractive application; at the millimetric, or centimetric, lengthscalesconcerned these would operate at frequencies in the kHz range.Fuelled by demands to reduce the power consumption of small electronic components,vibration-based energy harvesting has received considerable attention over the last twodecades; it is attractive to power devices using existing vibrational energy that reduces,3r removes, costs associated with periodic battery replacement and the chemical waste ofconventional batteries. Many applications arise: wireless sensor networks for civil infras-tructure monitoring, unmanned aerial vehicles, battery-free medical sensors implants, andlong-term animal tracking sensors [24]. Among the various possible energy harvesting meth-ods, piezoelectric materials offer several advantages due to their large power densities andease of application [25–27]. Piezoelectric energy harvesting is driven by the deformationof the host structure due to mechanical or acoustic vibrations that convert to an electricalpotential via embedded piezoelectric materials. To increase harvester efficiency, using ideasbased around structuring surfaces, several approaches such as creating a parabolic acousticmirror, point defects in periodic phononic crystals and acoustic funnels have been employed[28–31]; lenses to concentrate narrow band vibrations have been proposed using phononiccrystals [32–34] and resonant metamaterials endowed with piezoelectric inserts have ap-peared very recently [35]. Our aim here is to complement these studies by using a gradedarray to create a metawedge [38], and introduce piezoelectric elements into the array, thisaddresses one of the main challenges in energy harvesting which is to achieve broadbandenergy production from ambient vibration spectra. At present, multimodal harvesting, orexploiting non-linear behaviour, partly addresses this challenge [24]. Multimodal harvestingrequires multiple bending modes with close, and effective, resonant peaks thereby leadingto a broadband device; these systems can be affected by two problems: low overall powerdensity (power/volume or power/weight) and complex interface circuits [24]. Similarly toharvesting, resonant metamaterial devices have also been affected by a limited bandwidth.However recent research efforts have proven how this can be enlarged by adopting nonlinearresonators [36, 37] and, above all, graded array designs [38].In this article, our approach is to use graded resonator arrays to concentrate energy exploit-ing the properties of the rainbow trapping device shown in Fig. 1. Because such systemsalready contain a collection of resonators, the inclusion of vibrational energy harvesters isstraightforward leading to truly multifunctional meta-structures combining vibration insu-lation with harvesting. In section II we recall the resonant metawedge design principlesfor an elastic plate and elastic halfspaces. Section III presents the harvester design and weshow how it can be tuned by altering the rod length, grading angle and spacing in orderto maximise the focussing. Here we also introduce a simplified analytical model for theharvester so that optimal design parameters are rapidly found. 3D numerical simulations of4he electromechanical device installed on a plate and on a half-space are presented in sectionIV. Finally, we draw together some concluding remarks in section V.
II. THE GRADED RESONANT METASURFACE FOR ELASTIC WAVES
A cluster of rods attached to an elastic substrate whether acting as a phononic crystal, forshort pillars [39–41], or as a resonant metamaterial, for long rods [42], creates a versatilesystem for controlling elastic waves and lends itself well to the fabrication of graded designs.The physics of the resonant metasurface is described through a Fano-like resonance [19]; asingle rod attached to an elastic surface readily couples with the motion of either the A modein a plate, or the Rayleigh wave on a thick elastic substrate (half-space), and the coupling isparticularly strong at the longitudinal resonance frequencies of the rod. Mathematically, theeigenvalues of the equations describing the motion of the substrate and the rod are perturbedby the resonance, mode repulsion occurs, and complex roots arise leading to the formationof a band-gap [43, 44]. When the resonators are arranged on a subwavelength cluster (i.e.with λ , the wavelength, (cid:29) than the resonator spacing), as in the metasurface we use, theresonance of each rod acts constructively enlarging the band-gap until, approximately, therod’s anti-resonance [42, 45]. Because the resonance frequency of the rod, which dependson rod height, determines the band-gap position, then by simply varying the length of therods one gets an effective band-gap that is both broad and subwavelength; the rod lengthis therefore a key parameter for metasurface tunability. However both periodicity and/orthe distribution of the rods cannot be overlooked as they also shape the dispersion curvesleading to a zone characterised by dynamic anisotropy and negative refraction [46, 47]. Theband-gap frequency f is calculated approximately starting from the resonator length, h , andthe well-known formula [20, 47]: f = 14 h (cid:115) Eρ , (1)where E is the Young’s modulus of the material and ρ its density. Eq. (1) holds provided thesubstrate is sufficiently firm so that one end of the rod is effectively clamped. Conversely,an overly soft substrate (e.g. a very thin plate) may result in a rod behaving as a rigidbody connected with a spring thus compromising the amplification. In practical terms,5his provides bounds on the plate thickness and requires cross-checking, via finite elementsimulations, the modal shape of the longitudinal eigenmode. Similar considerations are validfor the half-space, that, again, cannot be too soft when compared to the resonators [45]. FIG. 1. The elastic metasurface used for energy harvesting (a) and conventional dispersion curves(b) for a periodic array of identical resonators of fixed height (here we take the resonator to haveheight 581 mm, just before that endowed with piezo-patches in the graded wedge of (a)). Panel (c)illustrates the trapping mechanism for the axial mode due to the metasurface grading by showingthe relevant dispersion curve at fixed frequency but varying resonator height; the rod is showninset at 581 mm.
A schematic showing the resonant metawedge is depicted in Fig. 1(a). We place a gradedarray of rods atop an elastic plate, and these rods have resonances (both flexural and lon-gitudinal); if we assume the rods have constant height then these naturally appear in thedispersion curves of Fig. 1(b). These dispersion curves are computed along the 2D irre-ducible Brillouin zone [48] using finite elements, [49] in a physical cell containing a singleresonator, that incorporate the Bloch phase shift via Bloch-Floquet periodic boundary con-ditions. Fig. 1(b) shows the longitudinal resonance (dashed line), and additionally a largenumber of flexural resonances are also visible. The impact of these flexural resonances onthe wave propagation is negligible [50], at least in terms of the band-gaps. An unconven-tional dispersion curve representation, where the frequency is fixed and the resonator heightvaried, as in Fig. 1(c), shows how rod height affects the array. By grading the array fromshort to tall resonators, Fig. 1(c), we see that the dispersion curve hits the bottom of theband-gap and subsequently the energy must be reversed and propagate back through the6rray; equally importantly the group velocity tends to zero at this turning point and thewave slows down and therefore spends time in the vicinity of the resonator feeding energyinto it.The resonant metawedge can also be attached onto a deep elastic substrate where it iscapable of mode conversion or the trapping and reflection of Rayleigh waves. The effectis underpinned by the hybridisation between the longitudinal resonance of the rods, andthe vertical component of Rayleigh waves [38, 42, 45]; the type of interaction, conversionor trapping, depends on the direction of wave propagation across the metawedge. Whenmoving towards increasing rod lengths we obtain trapping, conversely by reversing the wavedirection, we obtain Rayleigh to shear, S, wave conversion. Because each rod has a differentresonant frequency, the turning point (where conversion or trapping take place) varies spa-tially according to the frequency content of the incident wave (the so-called rainbow effect[51]). Trapping is by far the most interesting property for energy harvesting as it leads tohigher amplification on the resonator responsible for the trapping. For the elastic plate, asthe substrate, there is no conversion to a bulk mode as this requires a deep substrate, butthe concept of trapping carries across to the antisymmetric A Lamb mode.
III. HARVESTER DESIGN
Concentrating energy at a known spatial position, as the graded array does, is only part of therequirement for harvesting: It is also necessary to design an arrangement for the piezoelectricpatches that takes full advantage of the displacements induced within the array and uponthe rods. To understand the mechanisms involved we will take a single piezo-augmentedresonator and place it within the graded array (Fig. 1).Recalling that the dominant mode of interest is the longitudinal one, the harvester we useis the rod, with four cantilever beams arranged in a cross-like shape placed upon the top, asdepicted in Fig. 2(a); each beam embeds a piezoelectric patch and their motion in harvestingmode is shown in Fig. 2(b). In our elastic modelling we consider the plate and rods to bemade from aluminium ( E a = 70 GPa, ν a = 0 .
33 and ρ a = 2710 kg / m ) while the piezoelectricpatches are made of PZT-5H ( E p = 61 GPa, ν p = 0 .
31 and ρ p = 7800 kg / m ).The harvester design, Fig. 2 (a), has strong dynamic coupling between the rod and thecantilever beams as we carefully design it to work in the double amplification regime coupling7he axial fundamental mode with the flexural one of the beam (see Fig. 2 (b)) i.e. the lengthsof the cantilever beams are not arbitrary. Fig. 2 (c) illustrates the transmission spectra fora single cantilever beam, the harvester we use and a simple although very accurate modelof a rod with beams and enables us to rapidly tune the system. FIG. 2. Harvester shown with the piezoelectric patches (a), the harvesting fundamental mode (b)showing the double amplification occurring when the cantilever beams fundamental mode and therod axial mode are tuned to operate together. (c) shows the transmission spectrum for a singlebeam, the simple spring-mass model (shown inset) of a rod with beams and analytical transmissionof the harvester.
Provided the axial and flexural frequencies are matched, an amplification over a wide fre-quency range is obtained. We design the harvester to maximize this interaction at the trap-ping frequency of the metasurface in a range between 2 and 2 . A r = 25 mm (square cross section), A b = 10 mm (same width as the rod, i.e.5mm), l r = 460 mm, l b = 25 mm and h p = 0 . w r ( z, t ) and w b ( x, t )the vertical displacement of the rod and the beam respectively, the effective mass is obtainedwriting the kinetic energy of the system for both the axial and flexural mode: T r ( t ) = 12 M b (cid:20) ∂w r ( z, t ) ∂t | z = l r (cid:21) + 12 (cid:90) z = l r z =0 ρ a A r (cid:20) ∂w r ( z, t ) ∂t (cid:21) dz = 12 (cid:20) M b + 13 ρ a A r l r (cid:21) ˙ w r ( t )(2) T b ( t ) = 12 (cid:90) x = l b x =0 ( ρ a A b + ρ p A p ) (cid:20) ∂w b ( x, t ) ∂t (cid:21) dx = 12 (cid:20) ρ a A b + ρ p A p ) l b (cid:21) ˙ w b ( t ) (3)while elastic and electric lumped coefficients directly follow from the internal energy defini-tion: U r ( t ) = 12 (cid:90) z = lrz =0 (cid:90) A ( T zz S zz ) dAdz = 12 k r w r ( t ) (4) U b ( t ) = 12 (cid:90) x = l b x =0 (cid:90) A ( T xx S xx + T xz S xz − D zz E zz ) dAdx = 12 k b w b ( t ) − θw b ( t ) v b ( t ) − C p v b ( t )(5)with a separation of variables taken as: w r ( z, t ) = w r ( t ) z/l r and w b ( x, t ) = [3( x/l b ) − ( x/l b ) ] w b ( t ), M b = 4( ρ a A b + ρ p A p ) the total mass of the four beams, T and S thestress and strain fields, θ the electromechanical coupling coefficient and C p the internalpiezoelectric capacitance. Lumped coefficients are then defined as: k r = E a A r /l r , k b =1 / ( l b E a I b + l b G a A ∗ b ), θ = − e a p h p l p ( h p + 2 b b h p − y n h p ), C p = ¯ (cid:15) a p l p /h p , m r = 1 / ρ a A r l r + M b , m b = (33 / ρ a A b + ρ p A p ) l b and G a = E a / ν a ) being the aluminium shear modulus, a p the patch width, b b the beam thickness, y n the neutral axis of the composite beam, e the considered piezoelectric coefficient and ¯ (cid:15) the constant-stress dielectric constant. Thedynamics, of the electromechanical problem, are now succinctly described by three linear9oupled ordinary differential equations: m r ¨ w r + k r ( w r − w ) + 4 k b ( w r − w b ) = 0 m b ¨ w b + k b ( w b − w r ) − θv b = 0 C p ˙ v b + ˜ Gv b + θ ˙ w b = 0 (6)where ˜ G is the electrical conductance. Fourier transforming (6) gives the transfer function:˜ T ( ω ) = k r k b ( − m r ω + k r + 4 k b )( − m b ω + k b + iωθ iωC p + ˜ G ) − k b (7)from which the voltage and power follow directly as:˜ V ( ω ) = − iωθiωC p + ˜ G ˜ T ( ω ) w , ˜ P ( ω ) = ˜ G (cid:34) iωθiωC p + ˜ G ˜ T ( ω ) w (cid:35) with w the imposed harmonic displacement amplitude. For brevity we do not incorporatedamping, but it can be easily included in (6) by introducing a term linearly dependent on thevelocity ˙ w . The main contributor to damping in this system is provided by the piezoelectricmaterial since the quality factor of aluminium is very high. However we neglected dampinghere since the thickness of the patch is very small with respect to that of the beam. Thissimplified model provides surprisingly accurate predictions in the frequency range of interest(see Fig. 3 (a) and (b)), with just a 0 .
7% of error in the natural frequency prediction.
FIG. 3. Comparison between the analytical and numerical transmission spectrum in short circuit(a) and voltage in open circuit (b). C p / ˜ G (i.e. RC p ) defines the time constant of the circuit τ RC providing measure ofthe time required to charge the capacitor through the resistor. The time constant is relatedto the cutoff circular frequency which is an alternative parameter of the RC circuit: ω RC = 1 τ RC = ˜ GC p (8)The values of ˜ G maximising the electric power at each excitation frequency are obtained byimposing its stationarity with respect to ˜ G (the dashed white line in Fig. 4(a)): ˜ G opt. ( ω ) = k r ωθ +4 k b ωθ − m r ω θ − C p k r m b ω − C p k b m r ω − C p k b m b ω + C p m r m b ω + C p k r k b ω − k r k b + k r m b ω + k b m r ω +4 k b m b ω − m r m b ω (9)The optimal electric conductance is then obtained from the intersection of (8) and (9) (theintersection of black and white dashed lines in Fig. 4 (a)) . The corresponding optimalelectrical resistance is R opt = 7 . k Ω. FIG. 4. Power output vs. electrical conductance and frequency (a). Dashed white line shows theoptimal loading at each excitation frequency. Dashed black line the cutoff frequency for differentvalues of conductance. (b) show the corresponding electrical resistance for optimal power (normal-ized with respect to the gravitational acceleration g ) (blue curve) compared with other resistancevalues. Now that we have obtained an harvester optimised for longitudinal rod resonance, it can beintroduced into the metasurface array to assess whether its performance is increased whenrainbow trapping occurs. The considerable prior research with this metamaterial in thefrequency range of interest [21, 42] provide essential insights to the numerical simulations.11
V. NUMERICAL RESULTS
We present numerical simulations of the graded harvester on an elastic plate, the plate hasthe form of a finite width strip to reduce computational complexity [38, 50] whilst stillpreserving the physics of the resonant metasurface. The array is attached to an aluminiumstrip 30 mm wide and of thickness of 10 mm which is sufficiently stiff to avoid anomalousresonances in the rod. The array is composed of 30 rods, each with square cross section ofarea 25 mm , a linear height gradient ranging from 250 to 650 mm and a constant spacingbetween the rods of 15 mm; this results in a ∼ ◦ slope angle. Equation (1) suggests thatthe metasurface will have the longitudinal fundamental mode in the range 2 to 5 kHz androd number 26 (with the rod numbering in the array having 1 the shortest resonator and 30the longest), with length 460 mm, is the harvester designed in section III.To benchmark the graded harvester, and to highlight the advantage of the elastic wavefocussing, a time domain simulation with a Gaussian excitation centered at 2.15 kHz (fre-quency of the main mode of the harvester) is performed. This is compared to an isolatedharvester on the same plate with, and without, the metasurface (see Fig. 5 (a) and (b)). FIG. 5. Harvester on a plate strip with (a), and without (b), the metasurface at time t = 10 ms .Electrical power for a maximum input acceleration of 1 g with, and without, the metasurface (c). p , on a surface S = 650 mm , appliedin the same position which generates an A Lamb wave. The amplitude of the pressureis such that a maximum acceleration of 1 g is imposed at the input. FEM simulations onthese 3D structures are performed in ABAQUS CAE 2018 introducing piezoelectric elementson the top of the harvester, with zero voltage boundary conditions on the bottom face ofthe piezoelectric patches. Electrodes are modelled introducing a voltage constraint on thepiezo-faces in order to define equipotential surfaces. The electrodes on top are connected inseries and attached to the previously defined optimal resistance; to model this, a customisedFortran subroutine has been implemented in ABAQUS, according to [52, 53]. Absorbingboundary conditions are imposed with a parametric analysis assigning different values ofdamping at the finite elements along x at the edge of the strip. Damping is computedadopting a cubic law function with c max = 10 , according to [54]. It can be clearly seenthat the metasurface (see Fig. 5 (c)) provides a strong amplification of the electrical power,and when the harvester is embedded in the metasurface, due to the lower group velocity ofthe waves interacting with the array grading, the power generation peak occurs at a latertime. The wave velocity decreases along the grading, reaching the harvester with a nearlyzero value. For short propagation time (e.g. 10 ms), the grading has a detrimental effect onthe power production. Conversely if the excitation is sufficiently long, energy stops in theposition of the harvester loading it continuously and, in this case, the energy productionpeaks at about 20 ms; after this time, waves are backscattered and the power productiondecreases. It is worth noticing that not only is the peak higher, but globally the area belowthe curve is higher: this further demonstrates that energy is trapped inside the metasurface.We recall that the metasurface also provides strong vibration attenuation over the entirefrequency range due to band-gap generated by the longitudinal resonances of the rods. Thebehaviour is strongly subwavelength, as can be noticed comparing the rod spacing with thewavelength of the A mode in the frequency range we consider.It is possible to increase the harvesting bandwidth by introducing other harvesters in differ-ent positions. In this way, for broadband input, energy is almost uniformly trapped insidethe grading and feeds different harvesters. The harvesters introduced maintain the samebeams on top and only differ in terms of the rod length. From a practical, and economicalperspective, this solution is more feasible since the piezo patches are equal and so is the opti-13al resistance R opt . Theoretically, a beam grading induces higher power production for theisolated harvesters, as explained in section III. However, since the harvesters are electricallyconnected in series, the probability of out of phase responses and hence charge cancellationincreases. Therefore, unless one aims at powering multiple devices accommodating different R opt , keeping the same piezoelectric beam size leads to a simpler and more controllable sys-tem. The rod length of the harvesters is defined adopting a linear variation starting fromresonator 26, with the same metasurface angle. As for the previous case, the piezoelectricpatches of each harvester are connected in series to the same electrical resistance. The inputvibration is a zero mean coloured noise with a frequency content mainly in the metasurfacerange between 2 and 5 kHz. This is applied with a uniform pressure on the same surface S , such that a maximum 1 g acceleration is imposed at the input. The optimal electri-cal resistance in this case is obtained with a parametric analysis of the entire system. Thismeans that (9), derived for an harmonic regime, cannot be used here because of the dynamicinteraction between harvester of different length, the broadband and transient nature of theinput signal. By performing different numerical analyses, a value of 1.18 k Ω is adopted. Thecase with isolated harvesters is then compared with the one where they are introduced insidethe metasurface (Fig. 6 (a) and (b)). As in the previous case, it can be clearly seen that themetasurface (Fig. 6 (c)) strongly amplifies the electrical power. The global behaviour shownin Fig. 6 (c) is really similar to the one reported in Fig. 5 (c). The longer time necessary toreach the power production peak points to the same energy trapping mechanism discussedfor the single harvester. Finally, we remark a phase synchronisation across the harvestersenhanced by the presence of the grading that partially avoids charge cancellation when theyare connected in series. 14
IG. 6. Six harvesters on a plate strip with (a) and without (b) the metasurface at time t = 10 ms . Electrical power for a maximum input acceleration of 1 g with and without the metasurface(c). The same mechanism is found when placing the same metasurface on an elastic half-space.As for the plate we took a finite width strip, but now of infinite depth with plane strainboundary conditions applied on these faces along the y direction. This set-up representsthe case where the metasurface is built directly on the ground or on a thick mechanicalcomponent to harvest ambient noise. We use the same coloured noise excitation adoptedfor the plate strip, adopting a uniform pressure able to impose a maximum 1 g accelerationat the input. The same harvesters are placed inside, and outside, the metasurface (see Fig.7 (a) and (b)) and connected in series with an electrical resistance equal to the previousone (it is assumed to power the same device). The metasurface is able to provide a strongfocusing due to rainbow trapping, inducing, at the same time, a Rayleigh wave band-gap(Fig. 7 (a) and (b)). 15 IG. 7. Six harvesters on an half-space strip with (a) and without (b) the metasurface (equal tothe one on the plate strip) at time t = 10 ms . Electrical power for a maximum input accelerationof 1 g with and without the metasurface (c). V. CONCLUDING REMARKS
We have numerically demonstrated the advantage of combining graded resonant metama-terials with multiple piezoelectric harvesters both in terms of band widening and overallefficiency. This is demonstrated for both a plate, and a half-space, with a graded arrayresonators; this exploits the wave trapping mechanism, increasing the performance of theintroduced piezoelectric energy harvesters. To exploit this behaviour we find that a suffi-ciently long excitation is required as the wave group velocity decreases along the gradingreaching the value of zero in the position of the harvester. In this manner, before beingbackscattered, the waves enjoy a longer interaction with the harvester enhancing its powerproduction when compared against the case of isolated harvesters. Additional features arethat the system can be frequency-tuned simply by adding masses on the top of the rods and16n this way it is possible to match the longitudinal mode of the rod and the flexural one ofthe piezoelectric patch even at very low frequencies; this is important from an applicationpoint of view where one wish to exploit actual low frequency ambient spectra ( (cid:28)
ACKNOWLEDGEMENTS
RVC thanks the UK EPSRC for their support through Programme Grant EP/L024926/1and the Physics of Life grant EP/T002654/1. RVC also acknowledges the support of theLeverhulme Trust. Al.Cor. thanks the national project PRIN15 2015LYYXA8. An.Col.thanks the SNSF for their support through the Ambizione fellowship PZ00P2-174009. JMDPthanks Imperial College for its hospitality and Politecnico di Milano for the scholarship on”Smart Materials and Metamaterials for industry 4.0”.17
1] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart. Magnetism from conductors andenhanced nonlinear phenomena.
IEEE Transactions on Microwave Theory and Techniques ,47(11):2075–2084, 1999.[2] D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire. Metamaterials and negative refractiveindex.
Science , 305:788–792, 2004.[3] S. A. Ramakrishna and T. M. Grzegorczyk.
Physics and applications of negative refractiveindex materials . CRC Press, 2008.[4] D. H. Werner.
Broadband Metamaterials in Electromagnetics: Technology and Applications .Pan Stanford Publishing, 2016.[5] Z. Liu, X. Zhang, Y. Mao, Y.Y. Zhu, Z. Yang, C.T. Chan, and P. Sheng. Locally resonantsonic materials.
Science , 289(5485):1734–1736, 2000.[6] R. V. Craster and S. Guenneau.
Acoustic Metamaterials: Negative Refraction, Imaging,Lensing and Cloaking . London: Springer, 2012.[7] R Craster and S Guenneau.
World Scientific Handbook of Metamaterials and Plasmonics:Volume 2: Elastic, Acoustic and Seismic Metamaterials . World Scientific, 2017.[8] S. Brˆul´e, E. H. Javelaud, S. Enoch, and S. Guenneau. Experiments on seismic metamaterials:Molding surface waves.
Phys. Rev. Lett. , 112:133901, 2014.[9] T. Antonakakis, R. V. Craster, and S. Guenneau. Moulding and shielding flexural waves inelastic plates.
Euro. Phys. Lett. , 105:54004, 2014.[10] D. Torrent, Y. Pennec, and B. Djafari-Rouhani. Omnidirectional refractive devices for flexuralwaves based on graded phononic crystals.
J. Appl. Phys. , 116(22):224902, 2014.[11] Y. Achaoui, T. Antonakakis, S. Brule, R. Craster, S. Enoch, and S. Guenneau. Clampedseismic metamaterials: Ultra-low frequency stop bands.
New J. Phys. , 19:063022, 2017.[12] M. Miniaci, A. Krushynska, F. Bosia, and N. M Pugno. Large scale mechanical metamaterialsas seismic shields.
New J. Phys. , 18(8):083041, 2016.[13] L. D’Alessandro, R. Ardito, F. Braghin, A. Corigliano, Low frequency 3D ultra-wide vibrationattenuation via elastic metamaterial.
Sci. Rep. , 9:8039, 2019.[14] L. D’Alessandro, B. Bahr, L. Daniel, D. Weinstein, R. Ardito Shape optimization of solidairporous phononic crystal slabs with widest full 3D bandgap for in-plane acoustic waves.
Journal f Computational Physics , Vol. 344, 2017.[15] V.K. Dertimanis, I.A. Antoniadis, and E.N. Chatzi. Feasibility analysis on the attenuation ofstrong ground motions using finite periodic lattices of mass-in-mass barriers. J. Engng Mech. ,142(9):04016060, 2016.[16] G. Finocchio, O. Casablanca, G. Ricciardi, U. Alibrandi, F. Garesc, M. Chiappini, andB. Azzerboni. Seismic metamaterials based on isochronous mechanical oscillators.
Appl.Phys. Lett. , 104(19):191903, 2014.[17] G. Carta, I.S. Jones, N. V. Movchan, A. B. Movchan, and M. J. Nieves. Gyro-elastic beamsfor the vibration reduction of long flexural systems.
Proc. R. Soc. Lond. A , 473:2017.0136, 072017.[18] F. Lemoult, M. Fink, and G. Lerosey. Acoustic resonators for far-field control of sound on asubwavelength scale.
Phys. Rev. Lett. , 107:064301, 2011.[19] A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar. Fano resonances in nanoscale structures.
Rev. Mod. Phys. , 82:2257–2298, 2010.[20] A. Colombi, V. Ageeva, R. Smith, R. Patel, M. Clark, R. Craster, A. Clare, S. Guenneau, andP. Roux. Enhanced sensing and conversion of ultrasonic rayleigh waves by elastic metasurfaces.
Sci. Rep. , 7:6750, 02 2017.[21] A. Colombi. Resonant metalenses for flexural waves.
J. Acoust. Soc. Am. , 140(5):EL423, 2016.[22] K.H. Matlack, A. Bauhofer, S. Kr¨odel, A. Palermo and C. Daraio, Composite 3D-printedmetastructures for low-frequency and broadband vibration absorption.
Proc. Natl. Acad. Sci. ,113(30):8386–8390, 2016.[23] D. Cardella, P. Celli, and S. Gonella. Manipulating waves by distilling frequencies: a tunableshunt-enabled rainbow trap.
Smart Materials and Structures , 25(8):085017, 2016.[24] A. Erturk and N. Elvin.
Advances in Energy Harvesting Methods . Springer, 2013.[25] A. Erturk and D. J. Inman.
Piezoelectric Energy Harvesting . John Wiley and Sons, Ltd, 2011.[26] S. R. Anton and H. A. Sodano. A review of power harvesting using piezoelectric materials(2003-2006).
Smart Materials and Structures , 16(3):R1–R21, May 2007.[27] R. Ahmed, F. Mir, and S. Banerjee. A review on energy harvesting approaches for renewableenergies from ambient vibrations and acoustic waves using piezoelectricity.
Smart Materialsand Structures , 26(8):085031, July 2017.
28] M. Carrara, M. R. Cacan, J. Toussaint, M. Leamy, M. Ruzzene, and A. Erturk. Metamaterial-inspired structures and concepts for elastoacoustic wave energy harvesting.
Smart Materialsand Structures , 22:065004, 04 2013.[29] H. Lv, X. Tian, M. Y. Wang, and D. Li. Vibration energy harvesting using a phononic crystalwith point defect states.
Appl. Phys. Lett. , 102:034103, January 2013.[30] L.-Y. Wu, L.-W. Chen, and C.-M. Liu. Acoustic energy harvesting using resonant cavity of asonic crystal.
Appl. Phys. Lett. , 95:013506, June 2009.[31] S. Gonella, A. C. To, and W. K. Liu. Interplay between phononic bandgaps and piezoelectricmicrostructures for energy harvesting.
J. Mech. Phys. Solids , 57:621633, November 2009.[32] S. Tol, F. L. Degertekin, and A. Erturk. Phononic crystal luneburg lens for omnidirectionalelastic wave focusing and energy harvesting.
Appl. Phys. Lett. , 111:013503, July 2017.[33] S. Tol, F. L. Degertekin, and A. Erturk. Gradient-index phononic crystal lens based enhance-ment of elastic wave energy harvesting.
Appl. Phys. Lett. , 109:064902, May 2016.[34] A. Zareei, A. Darabi, M. J. Leamy, and M.-R. Alam. Continuous profile flexural GRIN lens:Focusing and harvesting flexural waves.
Appl. Phys. Lett. , 112:023901, January 2018.[35] C Sugino and A Erturk. Analysis of multifunctional piezoelectric metastructures for low-frequency bandgap formation and energy harvesting.
J. Phys. D: Appl. Phys , 51:215103, May2018.[36] L. Cveticanin and M. Zukovic. Negative effective mass in acoustic metamaterial with nonlinearmass-in-mass subsystems.
Communications in Nonlinear Science and Numerical Simulation ,51:89 – 104, 2017.[37] A. Casalotti, S. El-Borgi, and W. Lacarbonara. Metamaterial beam with embedded nonlinearvibration absorbers.
International Journal of Non-Linear Mechanics , 98:32–42, 2018.[38] A. Colombi, D. Colquitt, P. Roux, S. Guenneau, and R. V. Craster. A seismic metamaterial:The resonant metawedge.
Sci. Rep. , 6:27717, 2016.[39] T-T. Wu, Z.-G. Huang, T.-C. Tsai, and T.-C. Wu. Evidence of complete band gap andresonances in a plate with periodic stubbed surface.
Appl. Phys. Lett. , 93(11):111902, 2008.[40] Y. Pennec, B. Djafari-Rouhani, H. Larabi, J. O. Vasseur, and A. C. Hladky-Hennion. Low-frequency gaps in a phononic crystal constituted of cylindrical dots deposited on a thin ho-mogeneous plate.
Phys. Rev. B , 78:104105, Sep 2008.
41] Y. Achaoui, A. Khelif, S. Benchabane, L. Robert, and V. Laude. Experimental observation oflocally-resonant and bragg band gaps for surface guided waves in a phononic crystal of pillars.
Phys. Rev. B , 83(10):10401, 2011.[42] M. Rupin, F. Lemoult, G. Lerosey, and P. Roux. Experimental demonstration of ordered anddisordered multi-resonant metamaterials for Lamb waves.
Phys. Rev. Lett. , 112:234301, 2014.[43] N. C. Perkins and C. D. Mote. Comments on curve veering in eigenvalue problems.
J. SoundVib. , 106:451–463, 1986.[44] L. D. Landau and E. M. Lifshitz.
Quantum Mechanics non-relativistic theory . PergamonPress, 1958.[45] A. Colombi, P. Roux, S. Guenneau, P. Gueguen, and R.cV. Craster. Forests as a natural seis-mic metamaterial: Rayleigh wave bandgaps induced by local resonances.
Sci. Rep. , 5(5):19238,2016.[46] N. Kaina, F. Lemoult, M. Fink, and G. Lerosey. Negative refractive index and acousticsuperlens from multiple scattering in single negative metamaterials.
Nature , 525:77–81, 2015.[47] A. Colombi, R. Craster, D. Colquitt, Y. Achaoui, S. Guenneau, P. Roux, and M. Rupin.Elastic wave control beyond band-gaps: Shaping the flow of waves in plates and half-spaceswith subwavelength resonant rods.
Frontiers in Mechanical Engineering , 3, 05 2017.[48] C. Kittel.
Introduction to Solid State Physics . Wiley, Hoboken, NJ, 2005.[49] COMSOL. , 2012.[50] E. G. Williams, P. Roux, M. Rupin, and W. A. Kuperman. Theory of multiresonant meta-materials for A lamb waves. Phys. Rev. B , 91:104307, 2015.[51] K. L. Tsakmakidis, A. D. Boardman, and O. Hess. Trapped rainbow storage of light inmetamaterials.
Nature , 450:397–401, 2007.[52] G. Gafforelli, R. Ardito, and A. Corigliano. Improved one-dimensional model of piezoelectriclaminates for energy harvesters including three dimensional effects.
Composite Structures ,127:369–381, 2015.[53] R. Ardito, A. Corigliano, G. Gafforelli, C. Valzasina, F. Procopio, R. Zafalon. Advanced modelfor fast assessment of piezoelectric micro energy harvesters.
Frontiers in Materials , Vol. 3,2016.[54] P. Rajagopal, M. Drozdz, E.A Skelton, Lowe M. J S, and R.V. Craster. On the use ofabsorbing layers to simulate the propagation of elastic waves in unbounded isotropic media sing commercially available finite element packages. NDT E International , 51:30 – 40, 2012., 51:30 – 40, 2012.