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Controllable Wave Propagation of 2D Hybrid Dispersive Medium

Author(s): Parra, Edgar A.F.; Bergamini, Andrea E.; Ermanni, Paolo

Publication Date: 2016

Permanent Link: https://doi.org/10.3929/ethz-a-010878235

Rights / License: In Copyright - Non-Commercial Use Permitted

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Controllable Wave Propagation of 2D Hybrid Dispersive Medium

Edgar Flores Parra1,∗ Andrea Bergamini2, Paolo Ermanni1

1ETH Zurich, Composite Materials and Adaptive Structures Laboratory, Zurich, Switzerland2Empa, Laboratory for Mechanical Integrity of Energy Systems, Dubendorf, Switzerland

Abstract

In this paper, we report on the wave transmission characteristics of a hybrid two dimensional (2D)medium. The hybrid characteristic is the result of the coupling between a 2D mechanical waveguide inthe form of an elastic plate, supporting the propagation of transverse waves, and a discrete network ofinterconnected electrical transmission lines, consisting of a series of inductors connected to ground throughcapacitors. The capacitors correspond to a periodic array of piezoelectric patches that are bonded to the platecoupling the two waveguides. The coupling leads to a hybrid medium that is characterized by a coincidencecondition for the frequency/wavenumber value corresponding to the intersection of the dispersion curvesof the electrical and mechanical waveguides. In the frequency range centered at coincidence, the hybridmedium features strong attenuation of wave motion as a result of the energy transfer towards the electricalnetwork. The distinct shape of the dispersion curves at the intersection of the waveguides suggests thatenergy transfer results from weak coupling eigenvalue phenomena. The additional mechanical degrees,of expanding from 1D to 2D, are investigated by calculating the dispersion relations for the piezoelectricelements arranged following the five 2D Bravais lattices. In addition, the ability to conveniently tune thedispersion properties of the interconnected network of electrical transmission lines is exploited by studyingthe dispersion a square geometry lattice with diatomic inductances. The diatomic inductances allow foranisotropic wave propagation by means of directional tunning of the electrical network. The anisotropywas investigated by looking at the transmittance of a finite plate with 13 by 13 piezoelectric elements. Themedium consisting of mechanical, piezoelectric, and analog electronic elements can be easily interfacedwith digital devices to offer a novel approach to smart materials.

1. INTRODUCTION

Control of elastic waves with arrays of periodic piezoelectric shunts for attenuation of mechanical vi-brations has attracted increased interest. Most of the work has focused on 1D arrays of passive or activeshunted piezoelectric elements. This paper reports on an extension of the functionality of the unit cell of2D periodic structures with interconnected piezoelectric elements. Here, the connectivity of the unit cellsis defined by the propagation of both mechanical and electrical waves in two dimensions. As we will showin this article, this extension has a remarkable effect on the overall dispersive properties of the resultingmedium. Forbidden frequency ranges in the dispersion curves of solid media through phononic crystals(PC) and mechanical metamaterials (MM) have been reported in literature. In PCs, bandgaps result from[1, 2] periodic modulations of the mass density and/or elastic constants of the material resulting from thebasis of the crystal (e.g. diatomic materials [1, 3]). Such band gaps exist for wavelengths on the order ofthe unit cell size and can be complete [4], i.e. for any direction of propagation, or partial, i.e. directionspecific [5]. In metamaterials, on the other hand, the inclusion of suitably designed locally resonating unitsallows for the sub-wavelength modification of the dispersive properties of a medium, as reported by Liu in

∗ edgarf@ethz.ch

ICAST2016: 27th International Conference on Adaptive Structures and TechnologiesOctober 3-5, 2016, Lake George, New York, USA

the mechanical domain [6]. Waves at frequencies corresponding to wavelengths substantially larger than theunit cell size can be attenuated by local resonators. However, metamaterials comprised of arrays of identi-cal linear resonators typically address narrow frequency ranges [7]. Other interactions between modes canaffect the propagation of waves and induce attenuation. As discussed by Mace et al. [8], mode veering andlocking occur, due to weak coupling between two modes. In the vicinity of the crossing frequency betweenthe uncoupled modes these eigenvalue phenomena lead to exchange of energy. The manifestation of onephenomenon or the other depends on the product of the group velocities of the interacting modes. A neg-ative product yields locking while a positive product yields veering. In veering, two modes may approacheach other, but instead of crossing, they veer away and diverge. In locking two modes with opposing slope,interact by creating a pair of attenuating waves [8] with a reduced absolute group velocity. In both casesreduced transmittance can be observed over the corresponding frequency range.

Both, PCs and mechanical metamaterials, as reported in surveyed literature, consist of a mechanicalmedium, through which waves propagate and interact with “inclusions” that either scatter them to generatedestructive interference at certain wavenumbers or that absorb and dissipate energy through local reso-nances. The selection and modification of the elastic components defining the properties of media made ofartificial atoms provide additional degrees of freedom in the design of these systems. In many of the re-ported materials, the nature of the inclusions is purely mechanical [4, 6, 10], while in some cases, adaptivematerials are exploited to modify the geometry of the unit cell [11], to tune the properties of the locally res-onating units [12–14] or to modify the connectivity of a PC [15]. In the latter cases, what could be definedas the electric domain of the unit cell is self-contained and only exchanges energy with the mechanical do-main within the unit cell, thus it can be regarded as an inclusion in the mechanical medium. The mechanicalcomponent of the unit cell is the only pathway for the exchange of energy with neighboring cells.

Additionally, electrical media are intensively investigated objects in physics as well as in communicationscience, with the lumped (or discrete) transmission line brought as a classical example for the investigationof electromagnetic wave propagation in media made of artificial atoms. Albeit, there is sparse literature on2D electrical networks comprised from interconnecting such transmission lines. The effect of the electricalinterconnection of the piezoelectric elements on the dynamic behavior of a plate is explored by dell’Isola etal. [16–18] to control structural vibrations.

In this work we will discuss the effect of the interaction between wave modes in the electrical andmechanical domain on the propagation of mechanical waves in the proposed 2D hybrid medium.

2. 2D HYBRID MEDIA

The medium investigated in this work is characterized by periodicity in both the electrical and mechan-ical domain. The mechanical component consists of an aluminum plate Fig. 6 onto which piezoelectricelements are mounted. The electrical component is realized by combining inductors with piezoelectric el-ements serving the dual purpose of capacitors and of electro-mechanical transducers offering an interfacebetween the electrical and mechanical domains of the medium. The resulting electrical medium is definedas a network of interconnected transmission lines. The periodicity of the mechanical domain is spatial asit is defined by the arrangement of the piezoelectric elements. Conversely, the periodicity of the electricaldomain is defined by the inductance values of the interconnected transmission lines but also by the spatialperiodicity of the piezoelectric elements on the aluminum substrate that form the discrete structure of theelectrical network [20].

The appeal of a 2D hybrid electro-mechanical dispersive medium lays in the capability to modify theproperties of the electrical domain of the unit cell and thus of the dispersion modes by exploiting the tun-ability of the electrical components. Given the interaction between the mechanical and electrical domains ofthe hybrid medium, the electrical medium will affect the propagation of waves in the mechanical medium.As we will show, based on its dispersion properties, the system operates in a sub-wavelength regime, i.e. the

ICAST2016: 27th International Conference on Adaptive Structures and TechnologiesOctober 3-5, 2016, Lake George, New York, USA

unit cell size is substantially smaller than the length of the waves it affects. As such, it is deemed appropriateto regard this system as an equivalent hybrid medium in the metamaterial sense [21]. The 2D interconnectedelectrical components of the medium offer more complex dynamics than the 1D interconnected due to theadditional geometrical and electrical degrees of freedom. These added degrees of freedom enable modify-ing the unit cell symmetry by varying the physical position of the piezoelectric elements and the electricaldomain, by introducing features such as diatomic lattices, comprised of alternating inductances.

3. METHODS

In the following section we describe the numerical methods for studying this medium as well as thederivation of the differential equations describing the electrical domain.

3.1. Numerical Calculation of the Dispersion Curves of the Hybrid Medium.

We calculated the dispersion curves of the 2D hybrid medium for the five Bravais lattices that exist intwo dimensions using a monoatomic inductance. The study was performed using the finite element method(COMSOL Multiphysics) by analyzing the eigenfrequencies of the unit cell modeled considering Floquet-Bloch boundary conditions. A similar implementation, for a purely mechanical system, is presented forexample in [4]. In the two-dimensional hybrid medium model discussed in this work, periodic boundaryconditions were imposed on both, the mechanical and the electrical domain. For the mechanical domain, theFloquet-Bloch boundary conditions were implemented by COMSOL on each pair of parallel faces that formthe unit cell parallelogram of each Bravais lattice, such that ur = ule

−ia1(kx+ky) and ub = ute−ia2(kx+ky),

where ur, ul, ub, and ut are respectively the mechanical degrees of freedom on the right, left, bottom, topsides of the unit cell, and a1,2 are the primitive lattice vectors as seen in Fig. 3. The geometry of theunit cell was defined according to the five different Bravais lattices studied. The lattice constant for themonoatomic unit cell was a =10 mm, and the subtrate had thickness hs =1 mm. The square piezoelectricelement had sides, lp=8 mm, and a thickness hp =0.5 mm. We modeled the unit cell as a three-dimensionalstructure as seen in Fig 3. The material properties were the ones of aluminum described in the COMSOLmaterials library, while the properties of the STEMiNC SM112 implemented were the ones provided bythe supplier for the proprietary material. For the electrical domain of the monoatomic inductance, theperiodicity was implemented directly by imposing Eq. 6 using the Global ODEs and DAEs physics ofCOMSOL. Eq. 6 is derived by first relating the voltage across the N th piezoelectric element, VN , andthe voltages across the adjacent piezoelectric elements VNx−1, VNx+1, VNy−1, VNy+1 through the Floquet-Bloch boundary conditions, Eq. 1-4. In Eq. 6, L is the value of the inductors in the unit cell, IlN , IrN ,IbN ,ItN are the currents flowing in and out of the unit cell (from left to right and bottom to top, see Fig. 1,kx and ky are the wavenumbers in the X, Y directions respectively while a1, a2 are the lattice vectors in thedirect space. Lastly, the current, IN resultant from the charges qN on the top electrodes of the piezoelectricelements are related through the first time derivative IN = IlN − IrN + IbN − ItN = ˙qN .

ICAST2016: 27th International Conference on Adaptive Structures and TechnologiesOctober 3-5, 2016, Lake George, New York, USA

VNx−1 = VNeia1(kx+ky) (1)

VNy−1 = VNeia2(kx+ky) (2)

VNx+1 = VNe−ia1(kx+ky) (3)

VNy+1 = VNe−ia2(kx+ky) (4)

L(IlN + IbN − IrN − ItN ) = VN (e−iakx + eiakx + e−iaky + eiaky − 4) (5)

LqN = VN (e−iakx + eiakx + e−iaky + eiaky − 4) (6)

Figure 1. Electrical network for unit cell of square lattice with monoatomic inductuance configuration andcorresponding Bloch-Floquet boundary conditions

We then exploited the electrical degrees of freedom by calculating the dispersion curves of the 2Dhybrid medium for the square Bravais lattice using a diatomic inductances. The study was performedfollowing the same methodology as was done for the previously described monoatomic unit cell. However,in the case of the 2D diatomic unit cell, deriving the ODEs to describe the electrical medium was morecomplex as the number of variables increased. The full derivation, found the APPENDIX, leads to Eq.7-10. These equations relate the charges and voltages of the four piezoelectric elements, comprising thediatomic unit cell, to the inductance values and the wavenumbers kx and ky, as seen in Fig. 2. Note,this diatomic configuration is just one of many that can be achieved by varying the inductance values.

ICAST2016: 27th International Conference on Adaptive Structures and TechnologiesOctober 3-5, 2016, Lake George, New York, USA

Figure 2. Electrical network for unit cell of square lattice with diatomic inductance configuration andcorresponding Bloch-Floquet boundary conditions.

Figure 3. General model for calculating the dispersion of a monoatomic unit cell of a Bravais lattice.Bloch-Floquet boundary conditions are applied to the two pairs of parallel faces

ICAST2016: 27th International Conference on Adaptive Structures and TechnologiesOctober 3-5, 2016, Lake George, New York, USA

Figure 4. Illustration of the five Bravais lattices that exist in 2D and their corresponding k-paths. From leftto right and top to bottom, hexagonal, face centered, rectangular, square and oblique.

0

5000

10000

Γ K M H Γ0

5000

10000

Γ K M Γ

0

5000

10000

Freq

uenc

y [H

z]

Γ K M H Γ0

5000

10000

Γ K M Γ

0

5000

10000

Γ K M H Γ

Figure 5. Dispersion curves for interconnected monoatomic unit cell configurations of the 5 Bravaislattices. From left to right and top to bottom, hexagonal, face centered, rectangular, square and oblique.

ICAST2016: 27th International Conference on Adaptive Structures and TechnologiesOctober 3-5, 2016, Lake George, New York, USA

0 = (e−ilc((kx+ky))((((L1((L2((q2 + q3 + q4)) + 2((V2 + V3 + V4)))) + L2((−2V1 + V2 + V4))))eilc((kx+ky))−L1V3e

ilc((2kx+ky)) − L1V3eilc((kx+2ky)) + L1((V1 + V2))((−eilckx))− L1((V1 + V4))eilcky )))L−1

2

(7)

0 = (e−ilc((kx+ky))((((L1((L2((q1 + q3 + q4)) + 2((V1 + V3 + V4)))) + L2((V1 − 2V2 + V3))))eilc((kx+ky))−L1((V2 + V3))eilc((2kx+ky)) − L1V4e

ilc((kx+2ky)) + L1((V1 + V2))((−eilckx))− L1V4eilcky )))L−1

2

(8)

0 = (e−ilc((kx+ky))((((L1((L2((q1 + q2 + q4)) + 2((V1 + V2 + V4)))) + L2((V2 − 2V3 + V4))))eilc((kx+ky))−L1((V2 + V3))eilc((2kx+ky)) − L1((V3 + V4))eilc((kx+2ky)) − L1V1e

ilckx − L1V1eilcky )))L−1

2

(9)

0 = (e−ilc((kx+ky))((((L1((L2((q1 + q2 + q3)) + 2((V1 + V2 + V3)))) + L2((V1 + V3 − 2V4))))eilc((kx+ky))−L1V2e

ilc((2kx+ky)) − L1((V3 + V4))eilc((kx+2ky)) − L1V2eilckx − L1((V1 + V4))eilcky )))L−1

2

(10)To investigate the dispersion curves of the 5 Bravais lattices of the interconnected hybrid medium, kx

and ky were imposed such that they traced the k-path of the Irreducible Brillouin Zone of each lattice.

Table 1. Primitive lattice vectors and geometric conditions

Lattice a1[mm] a2[mm] ConditionOblique 10x 12x+7.5y |a1| 6= |a2| ,

α 6= π/2Rectangular 7x 10y |a1| 6= |a2| ,

α = π/2Centered Rectangular 10x 5x+7.5y |a1| 6= |a2| ,

α 6= π/2Hexagonal 10x 5x+8.33y |a1| = |a2| ,

α = (2/3)πSquare 10x 10y |a1| = |a2| ,

α = π/2

3.2. Numerical Calculation of the Transmittance of the 2D Hybrid Medium.

The transmittance of the finite 2D hybrid medium was calculated in COMSOL Multiphysics by modelinga plate with 13 by 13 piezoelectric elements. We considered a three-dimensional model of an aluminumplate with a thickness of 1 mm, a length of 205 mm, and width of 135 mm as seen in Fig. 6.

In the center portion of the plate, the 2D hybrid medium with unit cell size of 10 mm was added, consist-ing of 169 unit cells. The piezoelectric elements were the same as those used in the dispersion calculations.We modeled the inductive and resistive components using the electric circuit physics of COMSOL. Theselected value for the inductance L, as well as for the geometry of the unit cell and the material propertieswere the same as for the calculation of the dispersion curves in order to establish a direct correlation be-tween the dispersion curve of the hybrid medium and the transmittance of a finite hybrid medium for eachof the configurations. The mechanical transmittance of the hybrid medium was calculated by taking theratio of the spatial average of the velocity amplitudes, over a region 40 mm long by 135 mm width, beforeand after the periodic arrangement:

ICAST2016: 27th International Conference on Adaptive Structures and TechnologiesOctober 3-5, 2016, Lake George, New York, USA

Figure 6. Numerical COMSOL model of finite plate with interconnected elements for square Bravaislattice configuration

T (ω) =va,outva,in

=(va(ω, x ∈ [500, 600], y ∈ [500, 600], z ∈ [0, hs]))

(va(ω, x ∈ [200, 300], y ∈ [500, 600], z ∈ [0, hs]))(11)

The system was excited mechanically by a line load applied along the edge of the plate, marked by thered line in Fig 6, to induce a plane wave along the X direction. The steady state frequency response of thesystem was calculated over the frequency range of interest.

4. RESULTS AND DISCUSSION

4.1. Monoatomic Unit Cell

We shall now direct our attention to the behavior of a finite sample of the proposed hybrid medium.We numerically investigated the transmission of mechanical waves through a finite sample of the materialconsisting of 13 by 13 unit cells for the square lattice geometry. The transmittance was only investigatedfor the square lattice as the dispersion curves for all the geometries indicate mode veering to occur overapproximately the same frequency range. Moreover, at low frequency/wavenumber considered the wave-length spans several unit cells thus the influence of the piezoelectric placement is not as paramount as thechanges that can be induced by varying the electrical parameters on the wave propagation properties. Thetransmittance results, shown in Fig. 7, show that attenuation occurs over the frequency range of modeveering as expected.

4.2. Diatomic Unit Cell

To further exploit the wave attenuation capabilities of the hybrid dispersive medium, where the electricaldomain has the potential of being readily modified, we introduce a diatomic cell characterized by anisotropicwave propagation capable of directionally tunning the electrical network based on the chosen inductancevalues. All material and geometric parameters are kept unchanged with respect to the previously studied

ICAST2016: 27th International Conference on Adaptive Structures and TechnologiesOctober 3-5, 2016, Lake George, New York, USA

1000

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3000

4000

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ΓKMΓ−15 −10 −5 0 5

1000

2000

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4000

5000

Transmittance [dB]

Freq

uenc

y [H

z]

Figure 7. Dispersion and transmittance for square monoatomic unit cell configuration L=10 H. The dashedgreen line in the transmittance plots represents the purely structural response (circuit off) while the solid

red line represent that of the hybrid medium (circuit on).

Figure 8. Diatomic network configurations i and ii

square monoatomic unit cell, except for the unit cell length which is now 2a to account for the increase inspatial periodicity.

The 2D diatomic cell of the hybrid medium allows for a much more complex response than itsmonoatomic counterpart. Given that the monoatomic cell is composed of only one piezoelectric ele-ment with the same inductance value along both the X and Y directions. As a result, for the monoatomiccell, the wave propagation response along the X and Y can not be independently tailored. Conversely, forthe 2D square diatomic cell two different inductance values can be chosen for any of the twelve differentinductance locations, as seen in Fig. 2, theoretically yielding 4096, (L1;2)

12, possible network configura-tions to for the diatomic cell. However, it is not practical nor of interest to study all possible combinations.In this paper we study the two configurations seen in Fig. 8.

Fig. 9 and 10 show the calculated dispersion and transmittance for a finite hybrid medium consistingof 6.5 by 6.5 diatomic unit cells. Each unit cell was composed of four piezoelectric elements and twelveinductors. In the Fig. 9a,b) and 10c) the grey shading in the figures shows the qualitative frequency rangecorrespondence between the attenuation regions and mode veering observed in the dispersion curves. More-over, in the dispersion curves of Fig. 9 and 10, the uncoupled electrical modes (cyan) and mechanical modes(magenta) have been overlaid on the dispersion curves of the coupled system. The uncoupled electricalmodes were obtained by assuming harmonic oscillating charges, qN = QN sin(ωt), VN = QN sin(ωt)/Cp

ICAST2016: 27th International Conference on Adaptive Structures and TechnologiesOctober 3-5, 2016, Lake George, New York, USA

0

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Γ X M Γ−15 −10 −5 0 5

1000

20003000

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50006000

a)

Transmittance [dB]

Freq

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y [H

z]

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Γ X M Γ−15 −10 −5 0 50

2000

4000

6000

Transmittance [dB]

Freq

uenc

y [H

z]

b)

Figure 9. a) Dispersion and transmittance for monoatomic unit cell configuration i with L1=10 H andL2=10 H, b) Dispersion and transmittance for diatomic unit cell configuration i with L1=1 H and L2=10 H.The dashed green lines in the transmittance plots represent the purely structural response (circuit off) whilethe solid red lines represent that of the hybrid medium (circuit on). The cyan lines in the dispersion plotsrepresent the uncouples electrical modes, the magenta represent the uncoupled mechanical modes and the

blue represent the coupled mechanical and electrical modes.

and solving for the roots of the determinant of matrix A, found in the APPENDIX. Plotting the uncoupledmodes elucidates which dispersion curves are electrical and mechanical in origin. The dispersion curves inFig. 9b) clearly show the four uncoupled electrical modes, in light blue, corresponding to the four degreesof freedom of the different charges in the diatomic unit cell, as well as the mechanical transversal wavemode in magenta.

Fig. 9 shows the dispersion curves for the diatomic configuration i, with L1=10 H and L2=1 H and viceversa. As seen in the dispersion curves of the 2D square lattice, monoatomic or diatomic, there are two veer-ing regions for the wavenumbers corresponding to Γ−χ andM−Γ. It is over these frequency/wavenumberranges that attenuation occurs due to the energy exchange between modes. The diatmoic configuration ihas a similar behavior to that of its monoatomic counterpart, as the two veering regions move along thefrequency axis in unison, regardless of the combination of inductance values. This is illustrated Fig. 9 a,b)where veering regions for the monoatomic occurs approximately over the same frequency range as those ofthe diatomic configuration. Furthermore, comparing the dispersion curves of the monoatomic and diatomic,Fig. 9b) suggest that for the diatomic configuration ,i, the value of the largest inductance (L2=10 H) isdominant in establishing the locat ion of the veering regions.

To further exploit the additional electrical degrees of freedom and achieve increased control over theattenuation along different wave propagation directions, the diatomic configuration ii was selected. As seenin Fig. 8, configuration ii also uses two different inductances, however contrary to configuration i, theinductances along each direction are invariant. As a result, configuration ii allows for more independent

ICAST2016: 27th International Conference on Adaptive Structures and TechnologiesOctober 3-5, 2016, Lake George, New York, USA

0

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Γ X M Γ−15 −10 −5 0 50

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Transmittance [dB]

Freq

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y [H

z] c)

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Γ X M Γ−15 −10 −5 0 50

2000

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Transmittance [dB]

Freq

uenc

y [H

z] d)

Figure 10. c) Dispersion and transmittance for diatomic unit cell configuration ii with L1=2 H andL2=10 H. d) Dispersion and transmittance for diatomic unit cell configuration ii wit h L1=10 H and

L2=2 H. The dashed green lines in the transmittance plots represent the purely structural response (circuitoff) while the solid red lines represent that of the hybrid medium (circuit on). The cyan lines in the

dispersion plots represent the uncouples electrical modes, the magenta represent the uncoupled mechanicalmodes and the blue represent the coupled mechanical and electrical modes.

control over the presence of mode veering along the different directions. Mode veering along Γ − χ willbe dominated by the inductance along the X direction. The latter can be seen in Fig. 10c) where L2=2 Hgives way to broad veering and attenuation around the vicinity of 1 kHz. Conversely, in Fig. 10d) forL2=10 H there is abrupt veering around 0.5 kHz and thus attenuation over a narrow frequency range. Lastly,in both Fig. 10c) and d) veering along M-Γ is around the same frequency range as it is dependent on thecombination of kx and ky.

5. CONCLUSION

In the presented 2D hybrid medium, we have introduced additional electrical modes that interact withthe mechanical modes of the medium to exploit eigenvalue veering. The frequency range at which strongattenuation of mechanical waves traversing the hybrid medium is observed, coincides with the range offrequencies at which veering of electrical and mechanical modes in the dispersion diagram occurs. Theeffect of these phenomena on the propagation of mechanical waves in a sub-wavelength regime, which canbe of special interest in a number of engineering applications, has been proven numerically. Furthermore,the 2D electrical network allows for anisotropic wave propagation by means of directional tuning throughthe selection of the inductance values. The latter allows has paramount effects on the mechanical waves,as the presence of mode veering between the two domains can be controlled, and thus the presence ofmechanical attenuation.

ICAST2016: 27th International Conference on Adaptive Structures and TechnologiesOctober 3-5, 2016, Lake George, New York, USA

The integration of a mechanical and electrical domain to create a hybrid medium allows for the adap-tation of both the mechanical and the electrical expression of its dispersion properties, by exploiting thetunability of the electric impedance. The electrical domain offers an ideal interface to digital devices (e.g.by integrating digital potentiometers in the circuit shown in the supplemental information section) such ascentral or distributed computing units. In the presented medium we can thus state, that an analog circuit isan integral part of the material. In this sense, we claim that the hybrid medium represents a demonstrationof the hardware element of a new class of smart materials comprised of mechanical, analog and -potentially-digital electric elements, with tunable and programmable properties.

ICAST2016: 27th International Conference on Adaptive Structures and TechnologiesOctober 3-5, 2016, Lake George, New York, USA

APPENDIX

˙Ix1 =V1 − V2eikxlc

L2(12)

˙Iy1 =−V1 + V4e

ikylc

L2(13)

˙Ix2 =−V2 + V1e

−ikxlc

L2(14)

˙Iy2 =V2 − V3eikylc

L2(15)

˙Ix3 =−V4 + V3e

ikxlc

L2(16)

˙Iy3 =V4 − V1e−ikylc

L2(17)

˙Ix4 =−V4 + V3e

ikxlc

L2(18)

˙Iy4 =V4 − V1e−ikylc

L2(19)

L1( ˙ID − ˙Ix4 + ˙Iy4 + q4)− V3 + V4 = 0 (20)

L1( ˙IB − ˙Ix2 + ˙Iy2 + q4)− V1 + V2 = 0 (21)

L1( ˙IC + ˙Ix3 − ˙Iy3 + q4)− V2 + V3 = 0 (22)

L1( ˙IA + ˙Ix1 − ˙Iy1 + q4) + V1 − V4 = 0 (23)

L1(−( ˙IB + ˙IC + ˙ID))− V1 + V2 (24)

L1(−( ˙IA + ˙IC + ˙ID))− V2 + V3 (25)

L1(−( ˙IA + ˙IB + ˙ID))− V3 + V4 (26)

L1(−( ˙IA + ˙IB + ˙IC)) + V1 − V4 (27)

A =

−−iL1 sin((lckx))+L1 cos((lckx))−iL1 sin((lcky))+L1 cos((lcky))+2L2L2Cp

iL1 sin((lcky))−L1 cos((lcky))+L1L2ω2Cp+2L1+L2L2Cp

..

iL1 sin((lcky))−L1 cos((lcky))+L1L2ω2Cp+2L1+L2L2Cp

iL1 sin((lcky))−L1 cos((lcky))+L1L2ω2Cp+2L1+L2L2Cp

..

L1((i sin((lckx))−cos((lckx))+i sin((lcky))−cos((lcky))+L2ω2Cp+2))

L2Cp

−iL1 sin((lckx))−L1 cos((lckx))+L1L2ω2Cp+2L1+L2L2Cp

..

iL1 sin((lckx))−L1 cos((lckx))+L1L2ω2Cp+2L1+L2L2Cp

L1((−2 cos((lckx))+i sin((lcky))−cos((lcky))+L2ω2Cp+2))

L2Cp..

Q1

Q2

Q3

Q4

ICAST2016: 27th International Conference on Adaptive Structures and TechnologiesOctober 3-5, 2016, Lake George, New York, USA

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