arXiv:1609.08404v1 [cond-mat.soft] 13 Sep 2016 · 1 Department of Civil, Environmental, and Geo- Engineering, ... ical properties of smart material inserts ... an electrostrictive

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Wave control through soft microstructural curling:bandgap shifting, reconfigurable anisotropy andswitchable chirality

Paolo Celli1, Stefano Gonella1, Vahid Tajeddini2, Anastasia

Muliana2, Saad Ahmed3, Zoubeida Ounaies3

1 Department of Civil, Environmental, and Geo- Engineering, University ofMinnesota, Minneapolis, MN 55455, USA2 Department of Mechanical Engineering, Texas A&M University, CollegeStation, TX 77840, USA3 Department of Mechanical and Nuclear Engineering, The Pennsylvania StateUniversity, University Park, PA 16802, USA

E-mail: [email protected]

Published article: Smart Materials and Structures 26 (3), 035001 (2017);http://dx.doi.org/10.1088/1361-665X/aa59ea

Abstract. In this work, we discuss and numerically validate a strategy to attainreversible macroscopic changes in the wave propagation characteristics of cellularmetamaterials with soft microstructures. The proposed cellular architectureis characterized by unit cells featuring auxiliary populations of symmetrically-distributed smart cantilevers stemming from the nodal locations. Throughan external stimulus (the application of an electric field), we induce extreme,localized, reversible curling deformation of the cantilevers—a shape modificationwhich does not affect the overall shape, stiffness and load bearing capability of thestructure. By carefully engineering the spatial pattern of straight (non activated)and curled (activated) cantilevers, we can induce several profound modificationsof the phononic characteristics of the structure: generation and/or shifting of totaland partial bandgaps, cell symmetry relaxation (which implies reconfigurable wavebeaming), and chirality switching. While in this work we discuss the specific caseof composite cantilevers with a PDMS core and active layers of electrostrictiveterpolymer P(VDF-TrFE-CTFE), the strategy can be extended to other smartmaterials (such as dielectric elastomers or shape-memory polymers).

Keywords: Metamaterials, Tunability, Soft active materials, Chirality, Symmetryrelaxation, Cellular solids

http://arxiv.org/abs/1609.08404v2

http://dx.doi.org/10.1088/1361-665X/aa59ea

Wave control through soft microstructural curling 2

1. Introduction

Metamaterials are media whose internal architectureis carefully designed to elicit properties that are notattainable by conventional materials available in na-ture. Some of the most promising opportunities formetamaterials engineering are found in the control ofacousto-elastic waves; acoustic metamaterials are typ-ically (but not necessarily) periodic structures whichdisplay exotic wave manipulation effects, such as sub-wavelength filtering [1] and waveguiding [2], negativerefraction [3] and cloaking [4]. One of the main draw-backs of metamaterials is their inherent passivity: ametamaterial is typically designed for a specific taskand for a prescribed operational condition. There-fore, its geometry is determined and “frozen” at thedesign stage. A modification of the excitation sce-nario (e.g., a variable frequency content) would mostlikely result in the need for a full re-design of themedium. To alleviate this issue, several strategies foractive or adaptive control have been introduced overthe past decade. The simplest approach involves amanual adjustment of the position of the constitu-tive elements of a periodic medium [5, 6]. More au-tomated strategies revolve around the application ofexternal non-mechanical stimuli to modify the mechan-ical properties of smart material inserts (e.g., shapememory alloys [7], shunted piezoelectric [8–15], mag-netoelastic [16–19], electrorheological [20] and magne-torheological [21] phases, thermally-stimulable poly-meric materials [22], aerodynamic-loading-sensitive el-ements [23]), eventually resulting in a reversible mod-ification of the wave propagation characteristics of themetamaterial. Tunable phononic characteristics arealso attainable using nonlinear periodic structures. Forexample, by inducing buckling in some structural ele-ments via the application of static loads, it is possibleto trigger large deformations that can result in dra-matic pattern reconfigurations [24–27]. Other avenuesleverage the onset of geometric and material nonlinear-ities. For example, changing the level of precompres-sion of a granular crystal results in a variation of itsnonlinear characteristics [28]; this concept has been ex-ploited to produce media with tunable bandgaps [29],wave directivity [30] and tunable wave focusing ca-pabilities [31]. Finally, nonlinearly-induced higher-harmonics have recently been used to produce wave-fields displaying modal mixing and augmented direc-tivity patterns [32].

Recent advances in the field of soft active materialshave opened new opportunities for tunability [33–39].Pioneering, in this sense, are the works of Yang andChen [33], who first proposed dielectric elastomers totune the wave characteristics of periodic structures,of Bayat and Gordaninejad [35], who studied magne-torheological shape transforming lattices, and of Nouh

et al. [36], who investigated periodic plates with softelectrically-stiffened PVDF inclusions. While theseworks introduce soft active materials in the context ofwave control devices, they only partially take advan-tage of the dramatic shape modifications enabled bysoft active materials, such as dielectric elastomers [40,41] and shape memory polymers [42,43]. In this work,in light of these recent advances, we discuss the effectsof localized and reversible shape modifications on thespectro-spatial wave control characteristics of soft cel-lular structures. The shape modifications are internaland localized, since they occur at the level of an aux-iliary microstructure—here comprising non-structuralcantilever elements. The reversibility comes from thefact that the cantilevers are equipped with smart actu-ators (here we consider layers of P(VDF-TrFE-CTFE)electrostrictive terpolymer [44] on a PDMS substrate).Excited with an external (electrical, in this case) stim-ulus, the actuators cause the auxiliary cantilevers tocurl (i.e., experience extreme—yet reversible—rollingdeformation), thus modifying their dynamic propertiesand their effect on the global wave manipulation capa-bilities of the medium. The idea introduced in thiswork is a logical continuation of a paradigm that hasrecently gained traction in the arena of phononic crys-tals/metamaterials [45–47], based on the idea of de-coupling the static (load-bearing) properties from itsdynamic (wave control) functionalities. Thanks to thedramatic shape modifications that can be achieved,the strategy presented herein is expected to producemore pronounced effects than previous implementa-tions, which merely relied on material property cor-rection. As a first step, we briefly discuss how globalmicrostructural curling produces reversible shifts inthe bandgaps landscape. We note that this initialpart of our analysis is conceptually similar to a re-cent investigation independently pursued by Zhang etal [48] in the context of 3D printed shape memorypolymers, in which the authors briefly touch on theappearance/disappearance of phononic bandgaps. Inthis respect, we hereby attempt to offer a complete ra-tionale of the physical mechanisms that govern the evo-lution of the bandgap behavior induced by microstruc-tural shape changes. We then proceed to show howselective curling can lead to symmetry relaxation ofthe unit cell, which in turn produces partial bandgapswhich give way to focused wave patterns (reconfig-urable anisotropy). Finally, we discuss how these inter-nal shape modifications can introduce non-trivial chiraleffects in the response of the periodic medium.

The paper is organized as follows. In Sec. 2 weintroduce our strategy for reconfigurable tunability ofthe lattice characteristics. In Sec. 3 we discuss howwe model the curling of a cantilever beam with smartmaterial inserts in response to an applied electric field.


In Sec. 4 we report the results of the wave analysis,in terms of bandgap shifting, reconfigurable anisotropyand switchable chirality. Finally, the conclusions of ourwork are drawn in Sec. 5.

2. Architecture and curling strategy

Given a cellular architecture, e.g. the regularhexagonal (RH) lattice sketched in Fig. 1a, weintroduce a symmetric population of microstructuralelements, consisting of cantilevers located at the nodallocations, as shown in Fig. 1b. These internal elements

Figure 1. Wave control strategy at a glance. (a) Regularhexagonal (RH) lattice frame. (b) RH frame with anauxiliary microstructure (straight cantilevers). (c) Example ofarchitecture attainable through a smart-material-induced shapetransformation (curling) of the microstructure.

constitute an auxiliary microstructure, in that theiraddition (or removal) does not affect the connectivityof the primary lattice (which is still regular hexagonal)and therefore does not affect its static propertiesnor its load bearing capabilities. The auxiliarycantilevers behave, to all intents and purposes,as resonators capable of enriching the dispersionrelation (wave response) of the cellular medium byopening locally resonant bandgaps. Expanding onthis passive architecture, we now let the cantileversbe instrumented with layers of active material andwe apply external stimuli to produce reversiblechanges in their local shape, as sketched in Fig. 1c.These shape transformations, which are localized atthe microstructural level, cause a modification ofthe resonant frequency of the cantilevers and, in

turn, a modification the global dynamic response(altering the frequency location of the locally resonantbandgaps). The net result is a medium with tunablewave characteristics where, remarkably, the tunabilitystrategy does not interfere with other importantstructural functionalities.

3. Nonlinear deformation of soft, smart

composite cantilevers

To obtain the levels of curling necessary to inducemacroscopic changes in the response of the cantilevers,we are required to work with a compliant materialsubstrate. To this end, we consider a cellularsolid skeleton (main lattice plus cantilevers) made ofPolydimethylsiloxane (PDMS), a soft polymer. Eachcantilever is instrumented with two thin patches of asoft active material, as shown in Fig. 2. We choose towork with P(VDF-TrFE- CTFE), an electrostrictiveterpolymer capable of inducing ∼ 4% electrostrictivestrain. This high electrostrictive strain coupled withits high Youngs modulus makes this polymer uniquelysuited for this internal cantilever application [44, 49].Note that, due to the nature of the electrostrictionphenomenon, the two patches are only capable ofcontracting in the thickness direction and expandingin the planar direction in response to a through-the-thickness electric field. For this reason, weonly activate one patch at a time: activating thebottom patch allows for counterclockwise curling, whileactivating the top one causes the cantilever to curlclockwise. The mechanical properties for the PDMSsubstrate are selected within the range of propertiesthat can be achieved with off-the-shelf PDMS kits:Young’s modulus E = 2MPa, Poisson’s ratio ν =0.5, density ρ = 965 kgm−3. The properties ofP(VDF-TrFE-CTFE) are: Young’s modulus Ep =200MPa, Poisson’s ratio νp = 0.48, density ρp =1300 kgm−3, coefficient of electrostriction β = β13 =3 · 10−18 m2V−2, breakdown electric field Ee

b =350MVm−1.

A single cantilever (before and after the applica-tion of an electric field to the bottom patch), with allits characteristic dimensions, is shown in Fig. 2. Thetwo lines oriented at ±60o and stemming from the ori-gin represent portions of the lattice links belongingto the hexagonal lattice structure. Throughout thisanalysis, we consider the following dimensions (whichhave been carefully selected to be compatible with fab-rication methods currently available): the length ofeach lattice link is L = 2.5 cm, the thickness of a lat-tice link is t = L/50 = 500µm, the length of a can-tilever is Lc = 0.8 · L = 2 cm, the thickness of a can-tilever is tc = Lc/65 = 308µm, the distance from thebase of the cantilever to the beginning of the patch


Figure 2. Counterclockwise curling of a terpolymer-PDMScomposite cantilever beam, obtained by activating the bottompatch with an electric field E

e = 150MVm−1.

is dp = 0.3 · Lc = 6mm, the length of a patch isLp = 0.6 · Lc = 12mm, the thickness of a patch istp = 10µm.

To predict the curling of the terpolymer-PDMSsandwich cantilevers, we resort to an analyticalmodel largely inspired by the work of Tajeddini andMuliana [50]—here adapted to the special case inwhich only one patch at a time is activated [51].The key aspects of the model are summarized in thefollowing (for a complete and detailed account of theformulation, refer to the Supplementary Data (SD)Section). The kinematics are based on Reissner’sfinite-strain beam theory [52], with the additionalassumptions of initially straight configuration andshear indeformability [53, 54]; the latter assumptionrestricts the model to the treatment of slender beams.The nonlinear strain-displacement relations are:

du

dx= (1 + ǫx0) cosφ− 1 , (1)

dv

dx= (1 + ǫx0) sinφ , (2)

where u and v are the beam’s axial and lateraldisplacements, ǫx0 is the axial strain at the beam’sneutral axis and φ is the rotational angle of the crosssection of the deformed beam.

The composite cantilever is equipped with thinpatches of a soft (compliant) active material, perfectlybonded to the substrate and symmetrically placed withrespect to the neutral axis of the beam. In responseto an external stimulus, a patch exerts axial forceson the substrate at their interface. Due to the factthat, for the considered geometry, the axial strain atthe centroidal axis (ǫx0) is typically negligible, theaction of the patch is effectively akin to a constantbending moment applied to the span of the substratesandwiched between the patches. In light of these

observations, the following relations can be written:

ǫx0 = 0 , (3)

dφ

dx=

Mac

EIc, (4)

where Ic is the second moment of area of the substrate(Ic = b t3c/12), and Mac is the moment due to theactuator’s action. Depending on whether we areactivating the bottom (B) or top (T) patch, themoment can assume different values. If we onlyactivate the bottom patch, the moment applied to thesubstrate is:

Mac =Ep E b tp t

2c

2(tcE + 6tpEp)ǫp ; (5)

if we only activate the top patch, the resulting momentis:

Mac = −Ep E b tp t

2c

2(tcE + 6tp Ep)ǫp , (6)

where ǫp is the free strain of the patch (strainundergone by an unconstrained patch under the actionof the electric field); for our specific choice of soft activematerial, i.e., for an electrostrictive terpolymer, thefree strain is:

ǫp = β (Ee)2 , (7)

where Ee is the applied electric field. Note that, toderive Eqs. 5-6, we also assumed the patches to bemuch thinner than the substrate; as a consequence, wecan safely assume the axial stress to be constant alongthe patch thickness. It is important to note that, asa result of the slenderness and of the pure-bendingactuation, the established deformation field featureslarge rotations and small strains.

The boundary value problem is solved in apiecewise fashion. Given the geometry considered inFig. 2, we can identify three intervals: 0 ≤ x ≤ dp,where Mac = 0 due to the absence of patches, dp ≤

x ≤ Lp + dp, where Mac is given by Eq. 5 or Eq. 6(depending on whether we are activating the bottomor top patch, respectively), and Lp+dp ≤ x ≤ L whereagain Mac = 0. The piecewise solution is given below.For 0 ≤ x ≤ dp:

{

u(x) = 0v(x) = 0

(8)

For dp ≤ x ≤ dp + Lp:

{

u(x) = EIcMac

sinMac(x−dp)

EIc− x+ dp

v(x) = EIcMac

[

1− cosMac(x−dp)

EIc

] (9)

For dp + Lp ≤ x ≤ L:

{

u(x)=cosMacLp

EIc(x−dp−Lp)−x+dp+

EIcMac

sinMacLp

EIc

v(x)=sinMacLp

EIc(x−dp−Lp)+

EIcMac

(

1−cosMacLp

EIc

)

(10)


Figure 3. Unit cell analysis perspective on bandgap shifting. (a) Regular hexagonal architecture. (b) Low-frequency region of theband diagram for the cell in (a). Note the steep P-mode running almost parallel to the vertical axis. (c) Microstructured architecturewith a symmetric population of straight cantilevers. (d) Band diagram for the cell in (c). (e) Architecture with all curled cantilevers,resulting from the application of a 150MVm−1 electric field to the bottom patch of each cantilever. (f) Band diagram for the cellin (e). (g) Hybrid architecture displaying three straight and three curled cantilevers. (h) Band diagram for the cell in (g).

The curled shape shown in Fig. 2 is obtained byimplementing Eqs. 8-10 with the selected geometricand material parameters, for an imposed electric fieldEe = 150MVm−1 applied to the bottom patch only.

4. Phononic analysis

The analysis of the phononic characteristics of the softhexagonal lattice with straight/curled microstructureis carried out using a unit cell discretized withbeam elements, and in-house MATLAB routines.Throughout this work, we refer to the cantilevers asstraight or curled, according to whether they are inthe undeformed or deformed configuration. The finiteelement model of the curled cantilevers is obtainedby discretizing the curved profile predicted by theanalytical model discussed in Sec. 3 (shown in Fig. 2).In addition to accounting for the effect of the patchesin the form of equivalent bending moment loads,we also explicitly incorporate the influence of theterpolymer patches on the mass and stiffness matricesby taking into account the multi-material structure

of the cantilevers’ layered cross section accordingto the relative thickness of the layers. While anonlinear model is employed to predict the curlingdeformation of the cantilevers, a linear small-on-largeanalysis suffices to describe the superimposed smalldeformations experienced during a transient waveevent, which are a perturbation about the equilibriumdeformed state reached after curling. For more detailson this, including a numerical justification for thevalidity of the small-on-large model assumption, referto the SD Section. It is of course implied that thestatic moments that cause the cantilevers to curl arepreserved throughout the dynamic analysis. Also notethat, throughout our analysis, we neglect viscoelasticdamping effects, which are outside the proof-of-concept scope of this work, although we recognizethat they may significantly affect the predicted wavemanipulation effects in future experimental settings.


4.1. Bandgap shifting

The most intuitive manifestation of microstructuralshape changes is the onset of bandgap tunability.This effect is here illustrated by comparing thewave response of several unit cell configurations—with straight cantilevers, curled cantilevers, or acombination of both—at different frequencies. Thisresult is achieved through a classical Bloch analysiswhich yields the dispersion relation for the lattice [55].Dispersion relations are here portrayed in the formof band diagrams (Fig. 3), obtained by plotting thefrequency values computed at points of the reciprocalwave vector space marked by a coordinate s runningalong the contour of the First Brillouin zone (BZ)of the lattice [56]. Details on the BZ, including thesignificance of points O, A, ..., G are reported in theSD Section. Note that, in order to guarantee thatthe adopted representation is automatically compatiblewith the entire spectrum of symmetries (and lackthereof) that are established by arbitrarily curling themicrostructural elements, we here work with the entireFirst Brillouin zone instead of the Irreducible Brillouinzone (IBZ) of the hexagonal lattice, which would onlybe applicable in selected highly-symmetric cases. Moreinformation on this point is also reported in the SDSection. This selection results in more complex (attimes redundant, yet always complete) band diagramscontaining a wealth of information on wave directivity.Details on how to navigate these plots and extract thenecessary phononic characteristics are provided below.

We begin our analysis by considering the referencecase of a regular hexagonal lattice without cantilevers.The unit cell and its band diagram are shown inFigs. 3a-b. In the frequency range of interest, weobserve two modes of wave propagation: a slowerS-mode, known to be dominated by beam bendingdeformation, and a very fast P-mode (note the almostvertical steep slope), dominated by longitudinal beamdeformation mechanisms. In Figs. 3c-d we can seethat adding a population of straight cantilevers yieldsa locally resonant bandgap (highlighted in red andlabeled BG1), along with the appearance of localizednew modes which originate from the “splitting” of theS-mode and cluster around the bandgap. Note that theonset of the bandgap (f1 ≈ 6.19Hz) coincides with thefirst natural frequency of one of the straight compositecantilevers. When computing the band diagrams forthe unit cells with cantilevers, we experienced somenumerical issues when s coincides with O (i.e., withthe origin of the k-plane); details on this issue andon the reason why it does not affect our results aregiven in the SD Section. Application of an electricfield to the bottom patch of each cantilever (Ee =150MVm−1) forces initially-straight cantilevers to curlcounterclockwise, resulting in the architecture shown in

Fig. 3e. Intuition suggests that the curled cantileversare stiffer than their straight counterparts, thusbehaving as frequency-upshifted resonators; indeed,the locally resonant bandgap (now highlighted in blueand labeled BG2) shifts upwards. Also, the onset ofthe bandgap (f2 ≈ 7.87Hz) coincides now with the firstnatural frequency of a curled cantilever. The shift canbe quantified by evaluating the relative change betweenthe onsets of the two bandgaps, and it amounts to27% for our choice of electric field. To understandhow the bandgap gradually evolves as a function of theelectric field, see Fig. 4. As we increase the electric field

Figure 4. Bandgap evolution as a function of the electric fieldapplied to the bottom patch of each cantilever (Ee). For acertain value of E

e, the bandgap spans the frequency rangeencompassed by the gray band. The inserts represent the unitcell at selected values of the electric field.

from 0MVm−1 up to 150MVm−1 and the cantileversare progressively curled and stiffened, the bandgapshifts towards higher frequencies. Finally, we explore ahybrid scenario where three cantilevers of the unit cellare curled while the others are undeformed, as shownin Fig. 3g. In this case, we observe bandgaps associatedwith both types of resonators, with the onsets of BG1and BG2 coinciding with those observed in Fig. 3d andFig. 3f.

4.2. Reconfigurable anisotropy

In this section, we discuss how, through a strategythat allows individual resonator tuning and asymmet-ric microstructural reconfiguration, we can achieve pro-found spatial wave manipulation effects. To this end,we revisit and enhance the idea of relaxed cell symme-try [45,47], which we had previously introduced in thecontext of piezo-shunt-controlled lattices. Recall thatthe anisotropic wave patterns that are established inperiodic structures [57] usually reflect closely the sym-metry (or lack thereof) of the unit cell. We can there-fore expect that, by curling selected subsets of can-tilevers, we would alter the symmetry landscape of thecell, thus inducing non-symmetric anisotropy patterns


Figure 5. Reconfigurable anisotropy, from a unit cell analysis perspective. (a) Band diagram for an architecture with relaxed cellsymmetry; the unit cell is also sketched as an insert in (a). (b) Cartesian iso-frequency contour of the 5th mode of the dispersionrelation, labeled “m.5” in (a). (c) Cartesian iso-frequency contour of the 7th mode of the dispersion relation, labeled “m.7” in (a).The hexagonal contour in (b) and (c) is the contour of the First Brillouin zone (BZ): characteristic points of the BZ are properlylabeled.

Figure 6. Reconfigurable anisotropy captured via full scale simulations. Both wavefields represent the response of a finite latticecharacterized by the same relaxed cell symmetry architecture shown in Fig. 5a and comprising 11 × 10 unit cells. The wavefields,excited by a 7-cycle burst signal, correspond to the total mechanical energy in the structure at a given time instant. The dashed linesprovide a guide to the eye, and are meant to loosely bound the high-energy regions. (a) Response to a burst with carrier frequency6.7Hz. (b) Response to a burst with carrier frequency 8.5Hz.

with pronounced wave beaming characteristics (in con-trast with the symmetric behavior of the host lattice).This scenario is explored in Fig. 5. We can see that twoof the cantilevers have been activated and curled, whilethe others are left in their undeformed state. From theband diagram in Fig. 5a, we can see that the loss ofsymmetry of the unit cell has drastic repercussions onits response. This is especially visible for the 5th and7th modes (highlighted in red and blue, respectively),and it can be best appreciated by looking at the cor-responding iso-frequency contours in Figs. 5b-c (i.e.,the dispersion surface of the selected mode is sliced at

different frequencies; an increase in frequency is asso-ciated with a transition from light to dark contours).The hexagon bounds the region of the Cartesian wavevector plane corresponding to the BZ, and the pointshighlighted on the contour are the same points indi-cated on the abscissa of Fig. 5a. We can see that boththe 5th and the 7th mode lose the typical six-foldedsymmetry of hexagonal lattices. Taking a closer look atthe 5th branch, we notice a partial bandgap manifestingas a pair of dips in the branch, spanning the BC and EFedges of the BZ. The presence of the partial bandgapand the morphology of the dispersion surface suggest


that the wave response should be attenuated along di-rections characterized by wavevectors pointing from Oto points on the BC and EF edges, and, conversely, befocused along directions OD and OG (and neighboringones). Similar considerations can be made for the 7th

mode, which features a partial bandgap spanning theCD and FG edges, with wave beaming expected alongdirections OE and OB.

To validate the predictions from the unit cellanalysis, we test the response of a finite latticecomprising 11 × 10 unit cells (Fig. 6) with thesame architecture shown in Fig. 5a (simulationsare performed with a Newmark-β time-integrationalgorithm). The bottom nodes of the lattice areclamped, and an in-plane excitation is applied tothe mid-point of the upper boundary, as indicatedby the arrow. Fig. 6a represents the response toa 7-cycle burst with carrier frequency f = 6.7Hz—belonging to the 5th mode. The wavefield depicts thetotal mechanical energy landscape in the lattice at acertain instant of propagation. We can see that thewave is mainly propagating along a direction whichcoincides with OD, while it is attenuated along thedirections corresponding to the partial bandgap (leftportion of the domain), in complete agreement withthe iso-frequency contour of Fig. 5b. Fig. 6b representsthe response of the same structure when the carrierfrequency of the burst is f = 8.5Hz—which belongsto the 7th mode. This wavefield displays an oppositepattern: the energy associated with the wave is nowmainly propagating in the left portion of the domain,while the right portion remains de-energized. Again,this result is consistent with the unit cell analysisprediction of Fig. 5c.

To summarize the results shown in this section,we can state that the selective curling of somecantilevers causes a profound modification of the waveanisotropy patterns. In particular, the availabilityof cantilevers resonating at different frequencies alongdifferent directions causes the appearance of partialbandgaps, which lead to spatially-selective and beamedwave patterns. It is also interesting to point outthat the same lattice presents opposite wave beamingcharacteristics at different frequencies. Due to thereversible nature of the curling, we can switch betweendifferent directivity patterns by simply curling othersets of cantilevers, ultimately enabling reconfigurablewave beaming.

4.3. Switchable chirality

In the previous section, we showed how relaxing theunit cell symmetry has drastic repercussions on themorphology of the wave patterns. To elucidate therich opportunities for symmetry relaxation attainablewith actively-curlable microstructures, we now proceed

to provide a mechanistic rationale to link symmetryin the structure to symmetry in the response. Forbrevity, we restrict our analysis to the S-mode. Thischoice is motivated by two considerations. First, thedeformation patterns associated with the S-mode areparticularly easy to interpret, due to the fact thatthe mode mostly involves flexural deformation of thelattice links [45]. Secondly, since an S-like mode isobserved at low frequencies for virtually every cellconfiguration, it offers a fair metric of comparisonbetween different architectures.

Let us consider two cell configurations in whichonly two cantilevers (located at opposite nodes of anhexagonal cell) are curled: in the case of Fig. 7a,both cantilevers are curled counterclockwise, whilein Fig. 7b one is curled clockwise and the othercounterclockwise. By inspecting the symmetry of thecell in Fig. 7a, we note that the cell is characterizedby geometric chirality—its shape cannot be recovered,after mirroring it about any axis, by resorting tosimple translations and rotations (simply put, thearchitecture does not possess mirror symmetries). Incontrast, the architecture in Fig. 7b is not chiral,since it features a mirror symmetry about the dashedaxis. It is important to realize that the chiralityis here introduced through shape modifications ofthe auxiliary microstructure, and can be switchedon/off through the application of an electric stimulus,without modifying the primary lattice network. In thisrespect, it is qualitatively different from the chiralitymost commonly observed in lattice materials, which isassociated with a special connectivity of the primarylattice [58–60]. To emphasize its sole dependencyon the microstructure, we refer to it as second-orderchirality.

We consider the iso-frequency contour of theS-mode evaluated at 0.49 fS

max (where fSmax is the

maximum frequency of the S-mode for any specificarchitecture). In Fig. 7c, we report the iso-frequencycontour for the case in Fig. 7a (thick black line),compared to that of a reference case with all straightcantilevers (thin red line), which is characterized bythe highest achievable degree of symmetry (6-foldrotational symmetry, 3 mirror axes and inversionsymmetry). We observe that the geometrically chiralpattern of Fig. 7a induces chirality in the response—as highlighted by the lack of mirror symmetries inthe black iso-frequency contour in Fig. 7c. On theother hand, the response of the non-chiral geometry inFig. 7b, represented by the black contour in Fig. 7d, is,as expected, non-chiral—as indicated by the existenceof two mirror axes. Alone, these observations wouldlead to the partial (and overly simplistic) conclusionthat geometric chirality in the unit cell implies chiralityin the wave response. In the following, we will


Figure 7. Effects of microstructural curling on the symmetry of the wave response: geometric chirality. Comparison betweentwo architectures characterized by two curled cantilevers and differing in terms of curling orientation. (a) Geometrically-chiralunit cell (both cantilevers are curled counterclockwise). (b) Non-chiral unit cell (one cantilever is curled clockwise and the othercounterclockwise; the dashed line represents an axis of mirror symmetry). (c, d) Responses for the architectures in (a) and (b),respectively (thick black contours; the thin red contours represent the response of an architecture featuring all straight cantilevers).The dashed lines represent mirror axes of symmetry of the response and the arrows indicate wave vectors corresponding to selecteddirections of propagation. (e, f) Analogous models for the cells in (a) and (b), respectively.

show that the connection between cell geometry andresponse, in terms of chirality, is significantly moresubtle.

To better elucidate the onset of chirality in theresponse, we analyze how a wave impinging on thehexagonal cell along three characteristic directionsinteracts with differently-oriented cantilevers. Toprovide a more intuitive rationale, we replace theoriginal cell with an analogous one in which the curledcantilevers are substituted with slanted (yet straight)ones, whose centers of mass, just like in the curledcase, are off-centered with respect to lines connectingthe hexagon’s vertices (Fig. 7e). Note that thisanalogous model, albeit structurally different, is, forall intents and purposes, identical to the original onein terms of symmetry and geometric chirality, thusproviding some useful qualitative information on thewave/cantilever interaction. The three directions of

wave propagation we consider are marked as OA, OA′

and OA′′ in Fig. 7e. These directions correspond towave vectors kOA, kOA′ and kOA′′ in Fig. 7c and to theinflection points of the iso-frequency contours (wherethe response chirality manifests the most). Whena shear wave impinges on the cell along a directionidentified by one of those wave vectors, the unit cell willlocally vibrate along a direction perpendicular to thewave vector. In Fig. 7e, this is schematically denotedby sliding clamp constraints which, individually, onlyallow translation perpendicularly to the direction ofthe incoming wave. While the unit cell features a fixedset of internal beam-like resonators, they naturallydisplay different vibrational characteristics accordingto the way in which they are excited. Our objectiveis to determine the landscape of effective internalresonating mechanisms that is available for wavestraveling along different directions. For example, a


Figure 8. Effects of microstructural curling on the symmetry of the wave response: geometric chirality versus functional chirality.Comparison between two architectures differing in terms of curling orientation of the entire cantilever microstructure. (a, b) Twogeometrically-chiral unit cells in which all cantilevers are curled counterclockwise and clockwise, respectively. (c, d) Responses forthe architectures in (a) and (b), respectively. The dashed lines represent mirror axes of symmetry of the response and the arrowsindicate wave vectors corresponding to selected directions of propagation. (e, f) Analogous models for the cells in (a) and (b),respectively.

shear wave impinging along OA (and shaking the cellalong the direction perpendicular to OA), engages acell characterized by two cantilevers parallel to OA(which are activated flexurally), two inclined by 30o

and two inclined by 60o with respect to OA (theselast four will undergo a blend of flexural and axialdeformation). A wave along OA′, on the other hand,effectively sees a cell in which two cantilevers areinclined by 90o with respect to OA′ (thus activatedaxially), two are parallel to OA′ (thus activatedflexurally) and two are inclined by 60o with respectto OA′ (mixed mode). Finally, with respect to a wavealong OA′′, two cantilevers are inclined by 30o and fourare inclined by 60o. In light of these considerations,we can conclude that the establishment of chiralityin the response is linked to the availability of threedistinct sets of resonating mechanisms along the threeconsidered directions. In Fig. 7f, we repeat the exercisefor the architecture in Fig. 7b. In this case, shear

waves along OA and OA′ see a cell characterized byan identical set of effective resonators, consisting oftwo cantilevers parallel and one perpendicular to OA(or OA′, respectively), two inclined by 60o and oneby 30o with respect to OA (or OA′, respectively).On the contrary, a wave along OA′′ engages a cellwith two cantilevers inclined by 30o and four inclinedby 60o with respect to OA′′. Consistently with thisadditional symmetry in the resonating mechanisms,the iso-frequency contour in Fig. 7d is identical alongOA and OA′ and does not display chirality.

Let us now dig deeper into the role of the mi-crostructural elements, to illustrate further implica-tions of the second order chirality. First, we con-sider the unit cell configuration in Fig. 8a, character-ized by six counterclockwise-curled cantilevers and dis-playing geometric chirality, as highlighted by the ab-sence of mirror symmetries. Interestingly, and counter-intuitively, its S-mode response, shown in Fig. 8c (thick


black contour), is not chiral (all the dashed lines areaxes of mirror symmetry). Identical considerations canbe made for the configuration in Fig. 8b, characterizedby six clockwise-curled cantilevers (whose response isshown in Figs. 8d). These examples suggest that ge-ometric chirality alone does not necessarily imply chi-rality of the response. To lift this apparent contradic-tion, we repeat the directional vibration analysis in-troduced above, here based on the analogous modelsof Figs. 8e-f. It is easy to recognize that, in both cases,we have the same identical availability of resonatingmechanisms along all directions. This explains whythe responses of Figs. 8c-d are identical. We can ar-gue that these architectures, despite being geometri-cally chiral, are functionally non-chiral—meaning thatmirroring the cell about any axis fully preserves the ef-fective functionality of the microstructure with respectto the S-mode.

Summarizing our findings, we can conclude that,for lattices with auxiliary microstructures that featuresecond order geometric chirality, functional chiralityimplies chirality in the response.

5. Conclusions

In this work, we have shown that we can resort tothe localized shape modification of a population ofsoft auxiliary microstructural elements to attain adramatic reconfiguration of the wave characteristicsof soft cellular structures. In our structures, themicrostructural elements are composite cantileverbeams with soft active material inserts, that cancurl upon the application of an electric field. Thisstrategy allows for tunable wave control, since thelocalized curling deformations can be reversed byremoving the electric fields. Another remarkableaspect of this strategy is that the wave controlcapabilities—enabled at the microstructural level—are completely independent from other functionalitiesand properties of the lattices (e.g., their load-bearingcapability). The independent controllability of eachcantilever allows considerable flexibility and allowsfor both spectral and spatial wave control. Bycurling all the cantilevers inside every unit cell inthe same fashion, we can shift the location of thelocally resonant bandgap. By curling selected sets ofcantilevers in each cell, on the other hand, we relaxthe symmetry of the architecture, we introduce partial(directional) bandgaps and achieve pronounced wavebeaming. Due to the peculiar symmetry landscapesintroduced by microstructural curling, we are also ableto observe chirality effects of the “second-order”—i.e.,independent from the lattice connectivity and onlyassociated with the mechanical functionality of themicrostructures.

Acknowledgements

S.G. and P.C. acknowledge the support of the NationalScience Foundation (grant CMMI-1266089). P.C.also acknowledges the support of the Universityof Minnesota through the Doctoral DissertationFellowship. A.M. and V.T. acknowledge the supportof the Air Force Office of Scientific Research(grant FA9550-14-1-0234). Z.O. and S.A. gratefullyacknowledge the support of the National ScienceFoundation (EFRI grant 1240459) and the Air ForceOffice of Scientific Research.

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Wave control through soft microstructural curling S1

Supplementary Data (SD)

S1. Analytical model for nonlinear curling of

an electro-actuated cantilever beam

In this section, we describe in detail the model adoptedto predict the large deformation of a cantilever beamequipped with patches made of soft active material (anelectrostrictive terpolymer, in our case). We beginfrom a general nonlinear beam formulation, we thenspecialize it to the case of a cantilever beam in purebending and, finally, we discuss a strategy to modelthe electro-actuation.

S1.1. Nonlinear deformation of a shear-indeformablebeam

Our starting point is the framework by Irschik andGerstmayr [H. Irschik and J. Gerstmayr, Acta Mech.206, 1–21, 2009], which provides a formulationfor shear-indeformable, nonlinear beams inspired byReissner’s beam theory [E. Reissner, Z. Angew.Math. Phys. 23 795–804, 1972]. The fundamentalassumptions of this formulation are the following:

• the beam is originally straight;

• all the fundamental assumptions of Euler-Bernoulli beam theory (cross sections remainundistorted, plane and perpendicular to the neu-tral axis) hold. These assumptions imply shear-indeformability; thus, implicitly, we are restrictingourselves to the case of very thin beams.

We consider an initially-straight beam undergoingnonlinear deformation, as sketched in Fig. S1. In

Figure S1. Nonlinear deformation of a thin beam.

this sketch, dx is the initial length of an infinitesimalbeam element, dl is the length of the same infinitesimalelement after deformation, φ is the rotation of the crosssection, u is the displacement along the x direction and

v is the displacement along the y direction. dl can berewritten as:

dl =dl

dxdx = Λx0 dx = (1 + ǫx0) dx , (S1)

where Λx0 = dl/dx is the axial stretch along thecentroidal axis and ǫx0 = (dl − dx)/dx = Λx0 − 1 isthe corresponding engineering strain. Note that theonly nonzero strain component in this formulation isthe axial one. From trigonometry, we can derive thefollowing kinematic relationships:

du

dx= (1 + ǫx0) cosφ− 1 , (S2)

dv

dx= (1 + ǫx0) sinφ . (S3)

We now shift our attention to the equilibriumequations, which will be initially written for a genericbeam and only later specialized to the case of ashear-indeformable beam. We consider a generic(deformed) beam element and we construct the freebody diagram shown in Fig. S2. Here N is the

Figure S2. Free body diagram of an infinitesimal beam element.

axial force resultant and n is a normal distributedforce, Q is the shear force resultant and q is ashear distributed force, M is the moment resultantand m is a distributed moment; the other quantitieshave been previously introduced. Equilibrium alongthe horizontal and vertical directions, together withthe moment equilibrium about point A, leads to thefollowing equations:

N ′−Qφ′ + n = 0 , (S4)

Q′ +N φ′ + q = 0 , (S5)

M ′ +Q (1 + ǫx0) +m = 0 , (S6)

where ( )′ stands for d( )/dx. Note that, to obtainthese formulae, we assumed dφ to be small. Next,we eliminate the shear force from the statement of


equilibrium. Manipulating Eq. S6, we obtain thefollowing expression for the shear force:

Q = −M ′ +m

1 + ǫx0; (S7)

substituting it into Eq. S4 and Eq. S5, we obtain thefollowing equations, which enforce the equilibrium of ashear-indeformable beam:

N ′ +

(

M ′ +m

1 + ǫx0

)

φ′ + n = 0 , (S8)

N φ′−

(

M ′ +m

1 + ǫx0

)

′

+ q = 0 . (S9)

Following [H. Irschik and J. Gerstmayr, ActaMech. 206, 1–21, 2009], we can define the constitutivebehavior in terms of nonlinear strain measures andtheir work conjugates. In particular, we can eitherintroduce a relationship between the Biot stress (Txx)and strain (Hxx), or between the second Piola-Kirchhoff stress (Sxx) and Green strain (Exx), knowingthat we can subsequently determine the axial force andmoment resultants as:

N =

∫

A0

Txx dA =

∫

A0

Λx Sxx dA , (S10)

M = −

∫

A0

Txx y dA = −

∫

A0

Λx Sxx y dA , (S11)

where A0 is the cross-sectional area in the undeformedconfiguration (which coincides with the deformed areaA due to one of the Euler-Bernoulli assumptions), Λx =Λx0−yφ′ = 1+ǫx0−yφ′ is the axial stretch ratio at anypoint of the beam’s cross section (a definition whichcan be derived using continuum mechanics arguments;note that φ′ = κ is the generalized curvature of thedeformed axis). Note that the expressions for the axialBiot and Green strains in our specific problem are:

Hxx = Λx − 1 = Λx0 − yφ′− 1 = ǫx0 − yφ′ , (S12)

Exx =1

2

(

Λ2x − 1

)

=1

2

[

(Λx0 − yφ′)2− 1

]

. (S13)

It is worth spending few words on the significance ofthe Biot strain (Hxx). Recalling from classical small-deformation Euler-Bernoulli theory that the axialengineering strain can be expressed as ǫx = ǫx0−yκ, wenotice that, as far as this specific problem is concerned,Hxx ≡ ǫx. As a consequence, the Biot and Cauchystresses coincide as well (Txx ≡ σx). The consequencesof this fact are significant and will affect the treatmentof the electro-actuation: we can base our argumentson engineering strains without falling in the pitfalls ofsmall-deformation analysis.

S1.2. The special case of a cantilever beamundergoing pure bending

The formulation reported in the following is inspiredby [A. Muliana, Int. J. Nonlinear Mech. 71, 152–164, 2015]. In order to model a pure bending scenario,we consider a cantilever beam subjected to a tipmoment M∗, as sketched in Fig. S3. Since the beam is

Figure S3. Sketch of a cantilever subjected to a tip moment.

undergoing pure bending, the established deformationfield features large rotations and small strains. Thus,we can assume a linear elastic constitutive model, i.e.,a linear relationship between Biot stress and strain:

Txx = EHxx = E (ǫx0 − yφ′) . (S14)

Substituting Eq. S14 into Eq. S10 and Eq. S11, weobtain:

N = E ǫx0

∫

A0

dA− E φ′

∫

A0

y dA , (S15)

M = E φ′

∫

A0

y2 dA− E ǫx0

∫

A0

y dA . (S16)

If we assume the cross section to be homogeneous,the area is

∫

A0

dA = A0, the first moment of area

is∫

A0

y dA = 0 and the second moment of area is∫

A0

y2 dA = I. Thus, we obtain:

N = E A0 ǫx0 , (S17)

M = EI φ′ . (S18)

Due to the specific loading and boundary conditions,we have that N = 0 and M = M∗; these equationsde facto represent an equilibrium statement andreplace Eq. S8 and Eq. S9 in this simpler scenario.Substituting N = 0 and M = M∗ into Eq. S17 andEq. S18, we obtain:

ǫx0 = 0 , (S19)

φ′ =M∗

EI. (S20)

As a consequence, we conclude that the neutral axisremains unstretched, while the curvature (κ = φ′) isconstant throughout the beam’s length.

Combining the equilibrium/material equations(Eq. S19 and Eq. S20) with the kinematic relationshipswritten in Eq. S2 and Eq. S3 (which, of course, also


hold for the cantilever problem), we can solve forthe displacement profile of the cantilever. IntegratingEq. S20, we obtain:

φ =

∫ x

0

M∗

EIdx =

M∗

EIx . (S21)

Substituting Eq. S21 and Eq. S19 into Eq. S2 andEq. S3, and integrating them, we obtain:

u(x) =EI

M∗sin

(

M∗

EIx

)

− x , (S22)

v(x) =EI

M∗

[

1− cos

(

M∗

EIx

)]

. (S23)

S1.3. Electro-actuated smart cantilever

Our goal is to leverage the shear-indeformablenonlinear beam formulation discussed in the previoussections to predict the curling of a cantilever beamequipped with patches made of a soft active material(in the following, an electro-actuated patch is alsoreferred to as “actuator”). The sketch of a genericbeam configuration is shown in Fig. S4. The

Figure S4. Sketch of a cantilever equipped with symmetrically-placed patches (actuator) made of a soft active material.

formulation we adopt is largely based on [V. Tajeddiniand A. Muliana, Composite Struct. 132, 1085–1093,2015], but is here adapted to the particular scenario inwhich only one patch at a time is activated. In additionto the assumptions introduced in Sec. S1, we assumethe following:

• the patches are symmetrically-placed with respectto the centroidal axis;

• the patches are perfectly bonded to the substrateand the thickness of the bonding layer is negligible;

• the patches are much thinner than the substrate(tp ≪ tc). In light of this, we can consider theaxial strain distribution to be constant along thepatch’s thickness;

• if the patches were made of a material which wascapable of both axial elongation and shrinking, itwould have been possible to apply specific electricfields to top and bottom patches in order toachieve a pure bending deformation. However, weare considering patches made of an electrostrictiveterpolymer, which can only elongate axially (andnot shrink) under the action of an electric field.Thus, we are required to only activate one patchat a time. While this loading configuration doesnot induce pure bending, we make a quasi-purebending assumption—as it can be shown that,for the selected parameters, the axial strain issufficiently small and can be neglected.

Due to the pure bending assumption, we considerthe action of a patch to be correctly modeled by anequivalent bending moment applied to the beam spanwhere the actuator is located (dp < x < dp + Lp

in Fig. S4). To evaluate this equivalent moment,we follow an approach developed within “laminatestheory”, in [B-T. Wang and C. Rogers, J. Intell. Mat.Sys. Struct. 2, 38–58, 1992]. This model, originallydeveloped for small deformations, is here extended toa large-deformation formulation. As a first step, weassume a linear elastic constitutive model for the patchand we express the axial stress in the actuator as:

σac = Ep (ǫp − ǫac) , (S24)

whereEp is the Young’s modulus of the patch, ǫac is theaxial strain in the actuator and ǫp is the actuator freestrain, i.e., the strain that an unconstrained actuatorwould undergo when subjected to an electric field.Note that this relationship written in terms of Cauchystress (σ) and engineering strain (ǫ), corresponds to anequivalent one involving Biot stress and strain, sincethese measures coincide for our specific problem (asdiscussed in Sec. S1). Intuitively, due to the constraintplaced by bonding, ǫac < ǫp. Thus, to have a positivestress when the residual strain ǫp − ǫac is positive, wechoose the sign convention as in Eq. S24.

We now argue that the thin electroactuatedpatches can be modeled by axial forces applied atthe top and bottom surfaces of the substrate. Theforces generated by the two patches are drawn inFig. S5a. Depending on their relative magnitude, thoseforces produce a bending moment, an axial force, ora combination of the two. In order to maximize thebending moment that is achieved, we only activate oneof the two patches at a time. This strategy reflectsthe fact that the terpolymer can only elongate axially.The activation of the bottom (B) patch results in theforce FB

ac = σac b tp applied at the bottom surface ofthe substrate (y = −tc/2), with FT

ac = 0. This, in


Figure S5. (a) Detail of an infinitesimal element of thecomposite cantilever beam, where the electroactuated patcheshave been replaced by axial forces applied at the top and bottomsurfaces. (b) Same infinitesimal beam element, showing theequivalent axial force and moment that mimick the behavior ofthe actuators.

turn, produces the following equivalent moment:

Mac = FBac

tc2

= σac btp tc2

. (S25)

On the other hand, if the top patch (T) is activated,we have FT

ac = σac b tp, FBac = 0 and, consequently, the

following equivalent moment:

Mac = −FTac

tc2

= −σac btp tc2

, (S26)

where the minus sign is due to the positive momentconvention. The equivalent axial force Fac, albeitnonzero, is neglected due to the quasi-pure bendingassumption.

We now need to determine the moment distribu-tion in the substrate and impose the compatibility ofmoments at the patch/substrate interface. For the sub-strate, due to the pure bending assumption, we assumea linear elastic constitutive behavior: σc = E ǫc, whereǫc is the “induced strain distribution” in the substratedue to the actuator. Here ǫc is to be interpreted as aninduced strain distribution which is compatible withthe actuation configuration of the two patches (e.g. ifwe activate the bottom patch but not the top one, theformer will be strained while the latter will not) andthat does not represent the actual strain profile in thebeam. Following [B-T. Wang and C. Rogers, J. Intell.Mat. Sys. Struct. 2, 38–58, 1992], we assume thisstrain to be linearly-distributed, as shown in Fig. S6.Depending on whether we are activating the B or Tpatch, we face two different scenarios. If we only acti-vate the B patch, we assume the induced strain distri-bution sketched in Fig. S6a; this results in ǫc being:

ǫc = ǫBc = ǫactc/2− y

tc. (S27)

On the other hand, if we only activate the T patch, theinduced strain in the beam is distributed as sketchedin Fig. S6b and ǫc can be written as:

ǫc = ǫTc = ǫactc/2 + y

tc. (S28)

Figure S6. Detail of the composite cantilever beam,highlighting the induced strain distribution due to the actuatoralong the cross section. (a) Case in which the bottom patch (B)is activated. (b) Case in which the top patch (T) is activated.

The moment resultant on the substrate due to ǫc(recalling Eq. S11) can be computed as:

Mc = −

∫ tc/2

tc/2

b y σc dy = −

∫ tc/2

tc/2

b y E ǫc dy . (S29)

If we consider the B patch, we substitute the straindistribution in Eq. S27 into Eq. S29 and, uponintegration, we obtain

MBc = E

b t2c12

ǫac ⇒ ǫac =12MB

c

E b t2c. (S30)

Substituting Eq. S30 into Eq. S24, plugging theresulting σac into Eq. S25 and setting MB

c = Mac toenforce compatibility, we obtain (after isolating Mac

on the left hand side of the equation):

Mac = MBc =

Ep E b tp t2c

2(tc E + 6tp Ep)ǫp . (S31)

Similarly, if the T patch is activated, we obtain:

Mac = MTc = −

Ep E b tp t2c

2(tcE + 6tp Ep)ǫp . (S32)

At this stage, we still have to model theelectromechanical coupling. Recalling that ǫp is thefree strain of the patch as a result of the applicationof an electric field, and invoking a quadratic relationbetween free strain and electric field (customary forelectrostrictive terpolymers), we set:

ǫp = β (Ee)2, (S33)

where Ee is the applied electric field and β isthe coefficient of electrostriction—which can be


experimentally measured. Note that, by substitutingEq. S33 into Eq. S32 and Eq. S31, we obtain consistentorientations for the beam deformation: if we activatethe B patch only, we produce a positive moment, whileif we activate the T patch we produce a negativemoment.

We now have all the necessary information toderive the displacement profile of a cantilever under theelectrostrictive action of one of the two symmetrically-placed actuators. Now recall Eq. S20, rewritten herefor this specific problem as:

φ′ =Mac

EIc, (S34)

where Mac is the moment modeling the actuator’saction and Ic is the second moment of area of thesubstrate (Ic = b t3c/12). Invoking the kinematicrelationships for nonlinear beam theory (Eqs. S2-S3)and recalling that ǫx0 = 0 for a cantilever in purebending, we obtain:

u′ = cosφ− 1 , (S35)

v′ = sinφ . (S36)

For each beam segment (0 < x < dp, dp < x < dp+Lp,dp + Lp < x < Lc), we need to integrate Eq. S34,substitute it into Eqs. S35-S36, and integrate again toobtain u(x) and v(x). The final result is the followingdisplacement profile. For 0 ≤ x ≤ dp:

{

u(x) = 0v(x) = 0

(S37)

For dp ≤ x ≤ dp + Lp:

{

u(x) = EIcMac

sinMac(x−dp)

EIc− x+ dp

v(x) = EIcMac

[

1− cosMac(x−dp)

EIc

] (S38)

For dp + Lp ≤ x ≤ L:

{

u(x)=cosMacLp

EIc(x−dp−Lp)−x+dp+

EIcMac

sinMacLp

EIc

v(x)=sinMacLp

EIc(x−dp−Lp)+

EIcMac

(

1−cosMacLp

EIc

)

(S39)

S2. Validation of the small-on-large wave

model assumption

In this section, we provide a validation of some of theassumptions invoked to simplify the treatment of wavepropagation in soft structures. In particular, our aimis to verify the following:

(i) Can we treat the propagating wave as a small(linear) dynamic perturbation of a structurenonlinearly pre-deformed through the applicationof a large static load (mimicking the applicationof an electric field)?

(ii) Does a fine mesh of straight beam elementsaccurately describe the behavior of curled beams?

To address these points, we compare some numericalresults for a simulation of a curled cantilever beamexcited by a time-evolving (burst-like) tip moment.Note that the dimensions of the cantilevers are thesame as those reported in Sec. 3, except for thefact that we only consider the PDMS substrate. Weapproach this simulation in a number of differentways, each invoking an additional layer of simplifyingassumptions, to quantify the net effects of theseassumptions on the numerical predictions.

First, we resort to a fully nonlinear finite elementcode to analyze the response of an initially-straightbeam, discretized by 6-nodes isoparametric triangularelements (see Fig. S7a), to the excitation profileshown in Fig. S7b. This first scenario is labeledcase A. The load profile comprises an initial slowramp, during which the beam is brought to its curledstate, followed by a burst, which represents a small-displacement perturbation about the curled state. Inessence, in this case, all phases of the deformationpath are explicitly modeled by the nonlinear code astransient events, although the curling phase, due to itslong time scale, is de facto a slow quasi-static event.The amplitude A0 selected for the simulations shownin this section is A0 = 1.54e-4 [Nm]. Note that,due to the fact that we are dealing with a slenderstructure subjected to pure moment loading, we expectto induce a small-strain, large-deformation (rotation)field; accordingly, we use a Saint-Venant/Kirchhoffmaterial model—which correctly implements a small-strain linear material law (compatible with the largedeformation mechanics framework). Moreover, toguarantee energy and momentum conservation in ourmodel, we use implicit time integration, employing thescheme discussed in the work of Simo and Tarnow[J.C. Simo and N. Tarnow, Z. Angew. Math. Phys.43, 757–792, 1992; R. Ganesh, Mechanisms of wavemanipulation in nonlinear periodic structures, PhDThesis, University of Minnesota, 2015]. The result ofcase A, in terms of x and y displacements of the mid-point of the beam’s tip, is shown by the solid lines inFig. S7f.

To address (i), we compare case A to case B—where we considered an already-curled geometry (seeFig. S7c, coinciding with the final state of the rampportion of the load profile in Fig. S7b) and we applythe load shown in Fig. S7e (which can be obtained fromthe perturbation part of the load shown in Fig. S7b, bycentering it at 0). The result of case B, in terms of xand y displacements of the mid-point of the beam’s tip,is shown by the dashed lines in Fig. S7f. We can seethat the response is almost identical to that obtainedin case A. In light of this, we can conclude that the


Figure S7. Case A: nonlinear transient analysis of a straightbeam discretized via isoparametric elements (a), and subjectedto a ramp+burst load (b). Case B: nonlinear analysis of a pre-curled beam discretized with isoparametric elements (c). Case C:linear analysis of a pre-curled beam discretized via Timoshenkobeam elements (d). (e) Load profile for cases B and C. (f) Resultsin terms of tip displacements of the beam for cases A, B, C.

propagating wave manifests indeed as a perturbation ofa pre-deformed state achieved through the applicationof a large ramp load which is kept constant.

To address (ii), we compare the previous resultsto case C—similar to case B, except for the fact thatwe consider a linear finite element model comprisingTimoshenko beam elements (see Fig. S7d). The resultof case C, in terms of x and y displacements ofthe beam’s tip, is shown by the dash-dotted linesin Fig. S7f. We can see that the response of thissimplified model agrees qualitatively with cases A andB. The detail of Fig. S7f, however, highlights howthe peaks of the response in case C are shifted withrespect to A and B. This discrepancy is due to thefact that the dispersion relation is affected by thefinite element discretization—since we are using only20 beam elements per cantilever, we are bound to facefrequency-shifts with respect to the true solution. Thisaspect, which is ininfluential for this proof-of-conceptanalysis, should be kept in mind when designingexperiments to test the strategy presented in this work.

S3. Proper selection of the Brillouin Zone

To access all the information on the wave propagationbehavior of a certain periodic structure, it is sufficientto consider the region of the reciprocal wave vectorspace delimited by the First Brillouin zone (BZ). TheBZ for an hexagonal lattice structure is shown inFig. S8 (the whole region bounded by the dashedcontour), where ξ1 and ξ2 are components of thewave vector in the reciprocal lattice coordinate system(see e.g. the work of Gonella and Ruzzene [S.Gonella and M. Ruzzene, J. Sound Vib. 312, 125–139, 2008]). Note that the BZ only depends on

Figure S8. First Brillouin zone (BZ, the region boundedby the whole dashed contour) of a regular hexagonal latticestructure and Irreducible Brillouin zone (IBZ, the shaded area)in a cantilever-free case or with cantilevers having all the sameproperties.


the primary lattice—i.e., it is not affected by thepresence of auxiliary microstructural elements, whichdo not change the lattice connectivity. Due to thesymmetries of the unit cell, it is sometimes possibleto define an Irreducible Brillouin zone (IBZ): it is thensufficient to investigate wave vectors belonging to thissmaller region of the wave vector plane to obtain afull phononic characterization the medium. Since theIBZ strongly depends on the symmetries of the waveresponse, its shape and size are affected by the presence(and by the curling deformation) of the auxiliarycantilevers: when considering architectures in whichthe symmetry has been relaxed due to the curlingof a subset of cantilevers, the IBZ shown in Fig. S8(the shaded area) is no longer sufficient. A differentIBZ could be identified for a given curling pattern,but it would change as soon as a different internalarchitecture is established. For these reasons, in orderto maintain consistency of the dispersion plots acrossall cases considered and compared in the manuscript,we decided to consider band diagrams calculated alongthe contour of the full BZ. Also note that, while inSec. 4.2 we simply refer to the hexagonal contour as aBZ, this is actually the equivalent representation of theBZ in a Cartesian wave vector space (sometimes knownas Wigner-Seitz cell); a Cartesian representation ispreferable in studying directivity, as directions inthe Cartesian wavevector plane correspond directly tophysical directions of wave propagation in the medium.

S4. Numerical error in the band diagram

calculation and more bandgap-related

information

In this section, we report on a numerical artifact weencountered when computing the dispersion relationof architectures featuring composite cantilevers. Theseissues occur only when s (the coordinate spanningthe contour of the BZ in the reciprocal wave vectorplane—the ξ-plane) coincides with the origin of the ξ-plane (point O in the BZ), and it manifests throughthe appearance of non-physical dispersion branches.This issue is illustrated in Fig. S9. In Fig. S9a wereport the band diagram for a cell configuration inwhich all cantilevers are straight (non-activated). Inorder to visualize the numerical artifact, we zoominto the region boxed in red, which extends froms = 0 to s = 0.4. The portion of the band diagramwithin this restricted range of s values—for variousdiscretizations of the contour of the BZ (i.e., fordifferent values of ∆s)—is shown in Figs. S9b-d. InFig. S9b, we can see that the branches behave oddlyfor s values ranging from 0 to ∆s, i.e., within the firstinterval of the discrete s array, in the frequency intervalcorresponding to the expected bandgap. These strange

Figure S9. Numerical artifacts in the computation of the banddiagrams of architectures featuring composite cantilevers. (a)Band diagram for a unit cell featuring all straight cantilevers,where the shaded region indicates the bandgap. The region ofinterest in this section is highlighted by the red box. (b), (c),(d) Detail of the red box region when ∆s = 0.05, ∆s = 0.02, and∆s = 0.01, respectively.

branches appear mainly around the bandgap region.In order to demonstrate that what we are observingis just a numerical artifact, we look at Figs. S9c-d and notice that this behavior strongly depends onthe discretization of the s vector. Specifically, these


Figure S10. (a) Band diagram for a unit cell with straight cantilevers, with detail of the mode shape of the unit cell for a s valuealong the OA branch and for a mode situated right before the bandgap. In the detail, the light gray, dark gray and black lines are theundeformed cell, the deformed main frame and the deformed cantilevers. (b) Transmissibility of a structure featuring 11 × 10 cellsand all straight cantilevers. (c) Band diagram for a case with curled cantilevers, with detail of a mode shape before the bandgap.(d) Transmissibility of a structure featuring 11 × 10 cells and all curled cantilevers.

branches only exist within the 0 to ∆s range for anyvalue of ∆s and they seem to change characteristics aswe change ∆s. The numerical nature of these branchesis also highlighted by the fact that the error manifestsas a high condition number when solving the eigenvalueproblem for s = 0, while the condition number isreasonably low for all other values of s.

Another piece of evidence that demonstrates thenon-physicality of the branches within the first ∆sinterval is given by the response of finite lattices. InFigs. S10a-b, we report the comparison between theband diagram of a unit cell configuration with straightcantilevers and the transmissibility φ of a finite latticecomprising 11× 10 unit cells and characterized by thesame architecture. Note that the transmissibility hasbeen obtained as φ = 20 log(uout

RMS/uinRMS), where u

outRMS

and uinRMS are the root mean squared displacements

recorded at a point away from the excitation andnear the excitation, respectively (considering the sameexcitation location and boundary conditions discussedin Sec. 4.2). The fact that the bandgap is “through”and that the branches are a numerical artifact isproven by the appearance of the bandgap in thetransmissibility plot. In particular, both onset andend-frequency of the bandgap are in good agreementin both representations of the lattice response. In

Fig. S10a, we also report the mode shape for a unitcell at a frequency belonging to a branch occurringimmediately before the bandgap. Since the deformedmode shape is superimposed to the undeformed one(light gray line), we can see that the main frame(dark gray line) is undeformed, while the cantileversare undergoing strong bending-like deformation—anaspect which confirms the locally-resonant nature ofthese bandgaps [S. Gonella, A.C. To and W.K. Liu,J. Mech. Phys. Solids 57, 621–633, 2009]. Notethat bending mainly occurs near the root of thecantilevers, where the cross section is weaker (featuringPDMS only, without terpolymer layers). The verticalcantilevers are not bent since the point we selectedfor the computation of the mode shape belongs tothe OA segment (this segment, for the P-like wavesassociated with the considered branch, correspondsto wave vectors associated with wave propagationalong the vertical direction, that do not engage thevertical cantilevers). Similar considerations can bemade for the architecture featuring curled cantilevers,whose band diagram and transmissibility plots arecompared in Figs. S10c-d. Also in this case, themode shape before the bandgap highlights how thecurled cantilevers are bending while the main frame isundeformed. Once again, the bandgap clearly appears


in the response of the finite lattice, confirming itspresence despite the numerical artifacts in the banddiagram.

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arXiv:1609.08404v1 [cond-mat.soft] 13 Sep 2016 · 1 Department of Civil, Environmental, and Geo- Engineering, ... ical properties of smart material inserts ... an electrostrictive