Multistable Metamaterial on Elastic Foundation Enables TunableMorphology for Elastic Wave Control

V. Ramakrishnan1 and M. J. Frazier1, a)

Department of Mechanical and Aerospace Engineering, University of California, San Diego, California 92093,USA

(Dated: 23 September 2020)

We present a strategy for continuously and reversibly tuning the propagation of elastic waves in one-dimensionalsystems without need for persistent external stimulation. The general approach places a bistable metamaterial onan elastic substrate which is subsequently deformed via prescribed boundary displacements. The internal substratedeformation, which is shaped by a prescribed spatial variation in elasticity, is reflected in the overlaying metamaterialand facilitates the reconfiguration of bistable elements over isolated regions. As each configuration is associated with aunique stiffness, these regions represent an adjustable, meso-scale morphology amenable to tuning elastic waves. Theessential bistability is characterized by an asymmetric, double-welled equipotential energy function and is developedby mechanical rather than phenomenological means. The asymmetry provides for the unique, configuration-specific(stable) equilibrium stiffnesses; the equipotential promotes reversibility (i.e., no one configuration is energeticallypreferred). From a uniform metamaterial-substrate system, we demonstrate the utility of our strategy by producinga waveguide with shifting pass band and a metamaterial with variable unit cell morphology.

I. INTRODUCTION

Architected metamaterials1–3 are remarkable for theirextraordinary thermal4–6, mechanical7–9, acoustic10–15, andoptical16,17 performance stemming from a cleverly designedinternal architecture, comprising both a characteristic (oftenperiodic) structural geometry and constituent materialmorphology – in a sense, representing the emancipation ofmaterial behavior from the limitations of chemistry. Inparticular, research pertaining to elastic wave propagationhas realized extreme/counterintuitive effective properties8,18

that bring once fantastic applications into practical reach19–22.In addition, illustrating a broad range of applicable scales,new strategies for, e.g., thermal management23,24, signalprocessing25, acoustic sensing26, energy harvesting27–29, andearthquake mitigation30,31 have also been proposed.

However, utilizing “stiff” material constituents in themetamaterial construction leaves little opportunity to adaptthe dynamic performance for scenarios where the targetfrequency range may fluctuate significantly (e.g., as inenvironmental vibration mitigation and energy harvestingapplications), thus limiting effectiveness outside a fixedservice range. One remedy is the class of active architectureswhich leverage shunted piezoelectric elements32–37 toadjust the effective stiffness via electromechanical coupling,thus tuning the native dispersion characteristics of themetamaterial. Nevertheless, it is conceivable that forcertain metamaterial applications, obviating the need for theaccompanying complex wiring and external controllers isdesirable. Serving to accommodate this requirement, a secondapproach toward inherently tunable performance utilizessoft material constituents for which deforming the internalarchitecture modifies the effective stiffness and, consequently,the characteristic wave dispersion. By exploiting instability

a)Electronic mail: mjfrazier@ucsd.edu.

and multistability in readily re-configurable soft architectures,efforts to imbue metamaterials with an inherent tuningcapability, generally, effect one of two outcomes: (i)the tuning is continuous and reversible, though the tunedconfigurations are reliant on the constant support of anexternal field or prescribed boundary condition38–45 or (ii)the tuning is among a finite number of discrete, stableconfigurations, though the process is energetically fatedto follow an energy-minimizing sequence46–52, prohibitingrestoration to an earlier configuration; thus, tuning isirreversible and, ultimately, terminated. Continuous tuningallows for small adjustments; reversible tuning facilitatesrepeated adjustments; and stable tuning eliminates theneed for an ever-present supporting field or boundarycondition. Xia et al.53 demonstrated these tuning abilitiesin a silicon-coated microlattice utilizing an electrochemicalreaction. A metamaterial system possessing a inherent tuningcapability independent of chemistry and external controllerswhich permits continuous, reversible, and stable adjustmentsto the dispersion characteristics [i.e., combining the desirableattributes of outcomes (i) and (ii)] is appealing and the aim ofthe present work.

In this article, we propose a class of architectedmetamaterial whose effective stiffness as well as itsdistribution is continuously, stably, and reversibly adjustablefor adaptive dynamic performance using the techniqueof strain engineering. The general strategy couples asoft, multistable metamaterial to a relatively stiff (thoughdeformable) foundation which, when strained, assists inreconfiguring the multistable architecture, modifying themetamaterial performance. A similar approach has beenexplored in the context of controllable electronic and photonicperformance in thin-film materials54,55. Following thisstrategy, the strain state in the elastic substrate is reflectedin the overlaying metamaterial, causing isolated regions toswitch to the opposing configuration characterized by aunique effective stiffness, i.e., one that is acoustically distinctto the propagating wave. Moreover, dependent upon the strain

2

magnitude, the size of the switched regions are customizable,enabling a post-fabrication redistribution of the stiffnessproperty. This redistribution of a mechanical property,achieved without an external controller or appeal to chemistry,adds a second dimension to the tuning operation and is uniqueto the relevant literature. As the non-convex potential energylandscape is not biased toward a particular configuration, thetuned stiffness patterning in the re-configured metamaterialis stable (permitting quasi-static removal of the instigatingboundary displacements) and potentially reversible (requiringthe same effort to elicit and reverse the tuned state). In onedimension, we introduce and analyze two such metamaterialsystems. While the multistability is localized in onesystem, it is interactive in another which affects reversibility.In addition, continuous tuning, i.e., that permitting smalladjustments in configuration, is demonstrated. Our approachwill be applied to the establishment of waveguides withshifting transparency and metamaterial unit cells with tunable(stiffness) morphology.

The remainder of the article is organized as follows:Section II presents simple realizations of our strategy inthe form of mass-spring systems, including a qualitativeand analytical description of the energetic and dispersioncharacteristics. In Section III, we analyze and discuss theresults of numerical simulations illustrating the tunable wavefiltering properties. We demonstrate the concept in the formof (1) a waveguide with an adjustable pass band and (2) ametamaterial with a variable unit cell (stiffness) morphology.Finally, in Sec. IV, we draw insights from the numericalresults and suggest avenues for further study.

II. MODEL DESCRIPTION

A. The Asymmetric, Equipotential Function

Figure 1a illustrates the basic structural element giving riseto the metamaterial multistability, representing an elaboratedconception of the bistable unit found in related works49,56.In part, the bistable element comprises two identical,pin-connected axial springs, kA, arranged symmetricallyabout and at an angle to the x-axis as illustrated. Thisconstruction ensures that, as the pinned end displaces, uA, thecombined reaction from the axial springs is always parallelto the displacement. A notable distinction with earlierworks, the remote ends of each spring are not fixed, butmay displace, uB, along the horizontal axis. Given therelative displacement, ∆ = (uA− uB)/dx, the correspondingnon-convex potential function, ψA(∆), is symmetric aboutthe unstable equilibrium configuration, ∆ = 1. In additionto ψA(∆), the total bistable energy has contributions from anelastic potential, ψT(∆), originating from a pair of torsionalsprings, kT, at the remote ends of each kA, and a magneticpotential, ψM(∆), stemming from the interaction between ahorizontally-aligned magnetic dipole moment, pm, affixed topinned end and a pair of dipole moments, ps, collocated withthe torsional springs and oriented an angle, θ , with respect tothe x-axis (see Fig. 1a). Together, these contributions foster

FIG. 1. (Color online). Bistable Metamaterial on Elastic Foundation.(a) The essential bistable element of the metamaterial-substratesystems. (b) The non-convex energy function is adjusted bythe geometric and material parameters to yield an asymmetric,equipotential profile. The asymmetry is reflected in thecorresponding stiffness function, resulting in a unique stiffness,k = {ks1,ks2}, for each of the two equilibrium configurations, ∆ ={∆s1,∆s2}. The shaded region identifies the negative stiffness region.(c) Two metamaterial-substrate (MM-SS) arrangements with latticeconstant, a: one in which the bistable element is local (LOC) toeach metamaterial degree-of-freedom; another in which the bistableelement couples two metamaterial degrees-of-freedom (INT). Forcompact display, the bistable element is represented as a nonlinearspring which is indicated by an arrow.

the crucial asymmetric equipotential necessary for stable,reversible tuning. Specifically, the total bistable potential

3

becomes

ψ(∆) = ψA(∆)+ψM(∆)+ψT(∆), (1)

where the full equations for each component is given in theAppendix. The corresponding stiffness function is given ask(∆) = ∂ 2ψ/∂∆2. The two stable equilibrium configurations,∆ = {∆s1,∆s2} (∆s1 < ∆s2), are characterized by stiffnessesk = {ks1,ks2}.

Figure 1b illustrates the effect of each component in Eq.(1) on the shape of the bistable energy landscape and stiffnessfunction. While kT = 0 and the dipole moments ps areoriented such that θ = 0, the bistable potential is symmetric[Fig. 1b.(i)]; consequently, ks1 = ks2. Changing either ofthese conditions alone, generally, biases the energy landscapetoward a specific equilibrium configuration [Fig. 1b.(ii)] andhinders reversible tuning; however, a suitable choice of kT andps will produce an asymmetric, equipotential landscape withks1 6= ks2 [Fig. 1b.(iii)].

B. The Governing Equations and Homogeneous DispersionRelations

Two metamaterial-substrate arrangements are considered,distinguished by the placement of the bistable element withinthe unit cell of an N-cell chain (schematically represented asa nonlinear spring in Fig. 1c). In one arrangement (LOC), thebistable element couples the local metamaterial and substratedegrees-of-freedom, i.e., {uA,uB} → {v j,Vj}; in another(INT), the bistable element links nearest neighbors in theoverlaying metamaterial, i.e., {uA,uB} → {v j−1,v j}. In thepreceding statement, the right arrow maps the displacementsof the bistable element, uA and uB (Fig. 1a), to those ofthe metamaterial layer, v, and substrate layer, V (Fig. 1c).The substrate features springs of variable stiffness, s j, joiningneighboring degrees-of-freedom. The variability enables aninhomogeneous strain. In addition, each arrangement featuressprings, kI, which are interactive in the LOC arrangement andlocal in the INT arrangement.

Dimensionless governing equations will enable theinvestigation of the dynamics without reference to explicitgeometric and material parameters. For this reason,displacements, position, and time are scaled by suitablequantities:

v̄ = v/dx, V̄ =V/dx, x̄ = x/a, t̄ = ω0t,

where ω20 = kI/m. In addition, kId2

x defines a convenient unitof energy. Thus, for a chain of N unit cells, the dimensionless

Lagrangian is

L LOC =N

∑j=1

[12( ˙̄v2

j +˙̄V 2

j )− ψ̄(V̄j, v̄ j)

]

− 12

N

∑j=2

[s̄ j(V̄j−V̄j−1)

2 +(v̄ j− v̄ j−1)2] , (2a)

L INT =12

N

∑j=1

[˙̄v2

j +˙̄V 2

j − (v̄ j−V̄j)2]

−N

∑j=2

[s̄ j

2(V̄j−V̄j−1)

2 + ψ̄(v̄ j, v̄ j−1)

], (2b)

where s̄ j = s j/kI is the normalized local substrate stiffnessand ψ̄ = ψ/kId2

x . Accordingly, for an arbitrary unit cell, j, theEuler-Lagrange governing equations stemming from Eqs. (2)are

¨̄v j +(2v̄ j− v̄ j−1− v̄ j+1)+∂ψ̄(V̄j, v̄ j)

∂ v̄ j= 0, (3a)

¨̄Vj + s̄ j(V̄j−V̄j−1)+ s̄ j+1(V̄j−V̄j+1)+∂ψ̄(V̄j, v̄ j)

∂V̄j= 0.

(3b)

for the LOC arrangement and

¨̄v j +(v̄ j−V̄j)+∂ψ̄(v̄ j, v̄ j+1)

∂ v̄ j+

∂ψ̄(v̄ j, v̄ j−1)

∂ v̄ j= 0, (4a)

¨̄Vj +(V̄j− v̄ j)+ s̄ j(V̄j−V̄j−1)+ s̄ j+1(V̄j−V̄j+1) = 0, (4b)

for the INT arrangement. In the simulations to follow, Eqs.(3) and (4) are used to model the response of the system toprescribed substrate boundary displacement.

To evaluate harmonic wave propagation in the overlayingmetamaterial, we assume s̄ j is such that the substrate iscomparatively rigid so that any disturbance propagatingwithin the metamaterial does not transfer a meaningful portionof its energy into the substrate. Thus, we need only considerEqs. (3a) and (4a) in the analysis. For the simplest caseof a homogeneous metamaterial (i.e., bistable elements inidentical configuration), the application of Bloch’s theoremprovides the characteristic dispersion relations for the LOCand INT arrangements, respectively, (see Appendix)

2+ k̄s(i)−1γ− γ = ω̄

2, (LOC) (5a)

1+ k̄s(i)

(2− 1

γ− γ

)= ω̄

2, (INT) (5b)

where ω̄ = ω/ω0 is the normalized wave frequency andγ = exp(iκa) the propagation constant with dimensionlesscomplex wavenumber, κa. In solving Eqs. (5) for γ , the real(propagating) and imaginary (attenuating) components of thecomplex wavenumber are extracted as: κRa = |Re(i lnγ)| andκIa = |Im(i lnγ)|.

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III. NUMERICAL RESULTS

For analysis, the material and geometric parameterscontained in Table I are utilized and are unique to each system.As detailed in the Appendix, the parameters define the bistablepotential function, ψ̄ . Consequently, for both the LOCand INT arrangements, k̄s2/k̄s1 = 10. Figure 2 displays thearrangement-specific dispersion curves for the metamaterialuniformly in each of the two stable configurations. Theconfiguration-specific frequency ranges over which themetamaterial supports propagating waves begins to illustratethe anticipated performance tuning capability.

In the following subsections, respectively, we utilizesubstrates with constant and periodic spatial variation instiffness, s̄ j, to re-configure the overlaying metamaterialand alter its dynamic performance in the context of wavepropagation. These will be applied to the establishment ofwaveguides with shifting transparency and metamaterials withtunable pass bands.

A. Waveguide with Shifting Transparency

To demonstrate the enhanced functionality acquired by abistable metamaterial coupled to a strained elastic substrate,we first study the case of a homogeneous foundation, i.e.,s̄ j = const. We consider a chain of N = 50 unit cells and,without loss of generality, initiate each bistable element

FIG. 2. Dispersion Diagram. For an asymmetric bistable potential,the equilibrium stiffnesses are, generally, distinct, producing uniquedispersion diagrams. In the above, the shaded region identifies theshared frequency range of the configuration-specific pass bands. Theinset shows the frequency region of the lower band gap.

in the first stable configuration, ∆s1; thus, the original,untuned metamaterial is characterized by the stiffness, k̄s1.Prescribed displacements are applied quasi-statically to thesubstrate boundaries, establishing a uniform internal strain,ε = (V̄N − V̄1)/N. As a consequence of substrate coupling,the metamaterial simultaneously deforms, effecting switchingin certain bistable elements. Subsequently, the boundarydisplacements are quasi-statically removed, leaving a stablepattern of original and re-configured bistable elements whichestablishes the tuned metamaterial.

For the LOC arrangement, the kI resist the metamaterialdeformation, leading to a relative displacement between thelocal metamaterial and substrate degrees-of-freedom, ∆ j =v̄ j − V̄j, which changes signs about the midpoint of themass-spring chain. Thus, switching is facilitated in bistableelements on one half of the metamaterial (especially closerto the boundary) and hindered in those elements on theopposite half. At most, substrate strain may re-configureonly half of the bistable elements in the chain. A singledomain wall separates the original and the switched elements,each characterized by acoustically distinct stiffnesses, k̄s1 andk̄s2, respectively. These effects are depicted in Fig. 3a forincreasing (compressive) substrate strain.

For the INT arrangement, the bistable elements oppose themetamaterial deformation, ∆ j = v̄ j− v̄ j+1, which, due to thefree metamaterial boundaries, decreases in magnitude fromthe chain midpoint but does not change sign. Therefore,all elements are susceptible to re-configuration, althoughelements nearer the metamaterial interior will switch first. Inaddition, as switching entails ∆ j 6= 0, in the absence of thesubstrate, the tuned metamaterial would exhibit a lingeringdeformation (i.e., a residual strain) – the more elementsthat switch, the greater the effect. However, the substratecounteracts this effect and, since it is rigid (though elastic)compared to the metamaterial, generally, ∆ j 6= ∆s(i), as forthe LOC arrangement. For the INT arrangement, two domainwalls form as a consequence of interior elements switchingbefore those nearer the boundaries; however, since ∆ j deviatesfrom the equilibrium values, ∆s(i), the local stiffness, k̄ j, alsodeviates from k̄s(i). These effects are depicted in Fig. 3b forincreasing (compressive) substrate strain.

To demonstrate the strain-dependent wave dynamics ofthe tuned metamaterial, Figs. 3c,d display the FFT ofthe time signal v̄50(t̄) following an impulse, ˙̄v1(t̄)δ (t̄ − t̄0),which instigates a propagating disturbance. An absorbingboundary consisting of an additional 200 cells with k̄ j = k̄50(51 ≤ j < 250) and linearly increasing damping is utilized.Dependent upon the specific tuned stiffness profile, k̄ j, thefrequency range over which the metamaterial is transparentto propagating waves shifts. For the LOC arrangementin Fig. 3c, following a substrate strain of ε = −6.75×

TABLE I. Material and Geometric Parameters

k̄A k̄T k̄M δ ω20

LOC 1/10 187/8791 71/565 2 100INT 100 1957/92 14200/113 2 1/10

5

FIG. 3. (Color online). Waveguide with Shifting Transparency. The relative displacement and stiffness profiles of bistable elements in the (a)LOC and (b) INT metamterial-substrate arrangements. The dashed lines identify the stable equilibrium displacements, ∆s(i), and stiffnesses,k̄s(i). The indicated strain, ε , is that induced in the substrate by a quasi-statically applied (and subsequently removed) boundary displacement.(c,d) The strain-dependent stiffness profile affects the propagation of waves generated by an impulse. This is apparent in the non-coincidentFourier transforms of v̄50(t̄) at different strains. The dashed lines represent the upper and lower frequency bounds of the pass bands for thehomogeneous ∆s1 and ∆s2 configurations. The shaded region identifies the shared frequency range of the configuration-specific pass bands

10−4, the metamaterial dynamics is essentially unchangedas few bistable elements are re-configured; consequently,components of the transmitted signal lie within the passband associated with k̄s1 in Fig. 2a. Following a substratestrain of ε = −0.47, the metamaterial exhibits two domainscharacterized by k̄s1 and k̄s2, respectively; therefore, thecomponents of the transmitted signal are restricted to thenarrower frequency range in which the dispersion curves inFig. 2a overlap. Outside this range, wave motion is hinderedby Bragg scattering and confined to the domain in which thedisturbance was generated. A similar effect is observed for theINT arrangement in Fig. 3d where sufficient substrate straincauses most of the bistable elements to switch. Note, however,that the residual strain causes the stiffness profile to deviatefrom k̄s2 in the switched region and so the frequency rangeof components within the transmitted wave deviates from thatindicated by Fig. 2b.

The results depicted in Fig. 3 demonstrate the utility ofan elastic substrate for tuning metamaterials with multistableelements: with access to only the substrate boundaries,the multistable elements at the boundary and within theinterior may be switched. In contrast, earlier works utilizingmultistable elements for tunable metamaterial performance

offer no means of switching individual or isolated groupsof elements in the interior. For both the LOC andINT arrangements, a strained substrate enables altering themetamaterial morphology (i.e., the stiffness distribution)57

and, as a result, the transmission of incoming waves.The manner in which the two arrangements facilitate

switching bistable elements impacts the reversibility of thetuned metamaterial configurations. Recall that, for the LOCarrangement, ∆ j exhibits a sign change about the systemmidpoint, promoting switching in one half of the mass-springchain and reinforcing the native state it in the other half.From the tuned metamaterial configuration, reversing thesubstrate strain inverts ∆ j and, thus, the bistable elements inre-configured half are directed back to the native state whilethose in the native state are switched. Therefore, generally,the LOC arrangement does not support reversible tuning.Alternatively, for the INT arrangement, ∆ j does not exhibita sign change but is maximum (in magnitude) near the systemmidpoint, causing switching to commence there. From thetuned metamaterial configuration, inverting ∆ j by reversingthe substrate strain returns the switched elements to theirnative configurations without re-configuring any elementspresently in the native configuration. Therefore, generally,

6

FIG. 4. Reversibility of LOC and INT Arrangements. Inresponse to substrate strain, the relative displacement, ∆ j , exhibitsa sign change in the LOC arrangement but not in the INTarrangement. Above, metamaterials in the (a) LOC and (b) INTarrangements are initialized in the tuned configurations in Figs.3a.(iii) and b.(iii), respectively, which were attained via substratecompression. Subsequently, submitting the substrate to tensionreveals the reversibility of tuning the overlaying metamaterial. TheINT metamaterial is able to return to its original, untuned state whilethe LOC metamaterial is not.

the INT arrangement supports reversibility. These effects areobserved in Fig. 4 and the Supplemental animations.

B. Unit Cell with Variable Morphology

In the preceding subsection, a homogeneous substrate(i.e., s̄ j = const.) supports a uniform strain distribution whichre-configures the bistable elements in the overlayingmetamaterial: for the LOC arrangement, switchingcommences at one of the metamaterial boundaries; forthe INT arrangement, switching initiates within the interior.Naturally, substrates with a variable stiffness profile (i.e.,s̄ j 6= const.) will alter these results and may deliver anenhanced tuning capability. A substrate with monotonicstiffness function, for example, (not presented here) wouldpermit switching elements in the LOC chain beyond (orshort of) the current limit at the chain midpoint and, in theINT chain, would allow switching to initiate at or nearera boundary rather than the interior. Nevertheless, in thefollowing, we consider an elastic foundation with periodicstiffness function in order the effect a metamaterial withvariable unit cell morphology for wave control.

For simplicity, we consider a bi-material substrate withstiffnesses α and β (α < β ) which, as before, are sufficientlylarge to ensure that the substrate is rigid compared to theoverlaying metamaterial. These stiffnesses are distributedperiodically within the substrate with a unit cell comprisingthirty springs; therefore, the overlaying metamaterial, once

tuned, is expected to exhibit the same periodicity. Inparticular, for the LOC arrangement, s̄ j = α for j ≤ 15 ands̄ j = β for 15 < j ≤ 30; for the INT arrangement, s̄1 = α ands̄ j = β for 1 < j≤ 30. The interface between the two stiffnessregions provides a nucleation site at which switching maycommence. Utilizing alternative bi-material distributions ormulti-material substrates, isolated regions of the metamaterialmay be designed to switch in a pre-determined sequence;nevertheless, this is not present interest. In the following, weconsider wave propagation within finite systems consisting oftwenty of the earlier defined unit cells.

Figures 5a,b illustrate the state of the bistable elementsin the metamaterial for the LOC and INT arrangements,respectively. In particular, the first row shows the relativedisplacement, ∆ j, for the twenty-cell systems correspondingto the indicated nominal strain induced by boundarydisplacement. Apparently, the periodic substrate supports aninternal strain field with the same periodicity which assists inswitching the bistable elements at the same spatial intervals.To the purpose of tunable wave propagation, the secondrow of each sub-figure, focuses on the variation of themorphology, k̄ j, over the unit cell at several levels of substratestrain. Substrate strain changes the distribution of the bistablestiffness within the unit cell which impacts the metamaterialdispersion characteristics.

We compute the dispersion curves utilizing the stable,strain-dependent metamaterial unit cells. In each stableconfiguration, following the definition in Sec. II. A., thelocal stiffness k̄ j = k(∆) is determined at each jth elementof the re-configured unit cell, and then a dispersion analysisis performed for the system of N = 30 elements by suitablyextending Eqs. (3) and (4) and apply Bloch’s theorem, v̄1 =v̄30γ (see Appendix). Figures 5c,d show how the frequencyrange of pass bands, respectively, in the overlaying LOCand INT metamaterials change as a function of substratestrain. We also determine the FFT of the time signalv̄600(t̄) following an impulse, ˙̄v1(t̄)δ (t̄ − t̄0). Again, anabsorbing boundary with k̄ j = k̄600 (601 ≤ j < 800) andlinearly increasing damping is utilized. For a particular strain,we compare the computed dispersion and the FFT from timesimulation. For both the LOC and the INT arrangement, thereis excellent agreement between the analytical and numericalresults. These results, together with those of the previoussection, support the notion of inherent and tunable wavecontrol via an adjustable material morphology. Comparing thecommon frequency regions in the FFTs in Figs. 3c,d and 5c,d,we notice that periodic distributions of k̄ j opens additionalband gaps, illustrating the impact of the present repeatingmorphology with respect to the relatively simple variationsin Sec. III A. Although, the overlaying metamaterial inthe LOC arrangement is capable of stable and continuoustuning, except for a narrow range of tuned configurations, itis irreversible. However, in the case of an INT-type system,the tuning processes is able to meet all three requirements –stable, reversible, and continuous.

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FIG. 5. (Color online). Metamaterial with Tunable Unit Cell Morphology. (a,b) Subject to boundary displacement, a substrate with periodicstiffness variation elicits a displacement profile, ∆ j , with similar periodicity, leading to periodic element re-configuration and stiffnessmodulation, k̄ j. The (analytical) dispersion pass bands and the (numerical) FFT of a simulated disturbance are compared, revealing roughagreement for the (c) LOC system away from flat bands and excellent agreement for the (d) INT arrangement at lower frequencies.

IV. CONCLUSIONS

Typically, metamaterial tuning schemes demonstratecontinuity, stability, or reversibility individually or inpair-wise but not triple combination. In this article,we propose a one-dimensional system comprised of abistable metamaterial on an elastic substrate which permitscontinuous, stable, and reversible adjustments to themorphology through strain engineering and, thus, tunabledynamic performance. This is demonstrated in the form

of a wave waveguide with variable pass band and ametamaterial unit cell with flexible morphology supportinga tunable dispersion. The strategy of strain engineering formorphological tuning can be extended to higher dimensions,however, the results are generally not stable58,59 in the absenceof persistent substrate strain. However, e.g., defects and theinteraction of nonlinear wave modes with the transition zonerepresent two promising methods to overcome this obstacle.In addition, as s j approaches k̄I, the substrate not onlybegins to participate in shaping the wave dynamics together

8

with the metamaterial but also begins to sustain an apparentdeformation profile (i.e., warping) in the face of non-uniformmetamaterial morphologies (see Supplementary Material).This effect suggests an alternative route to programmingshape in soft matter60.

SUPPLEMENTARY MATERIAL

See supplementary material for animations and furtherdetails of the tuning and potential recovery of themetamaterial in LOC and INT arrangements.

ACKNOWLEDGMENTS

This work is supported by start-up funds provided by theUniversity of California.

DATA AVAILABILITY

The data that supports the findings of this study areavailable within the article and its supplementary material.

APPENDIX

Potential Function

Consider each term in Eq. (1) in turn. From thegeometry illustrated in Fig. 1a, where δ = dy/dx, theinstantaneous length of an axial spring is `(∆) = dx ¯̀(∆) =dx√(1−∆)2 +δ 2 and, therefore, its natural length is `0 =

dx ¯̀(0). Thus, the contribution of axial springs to the on-sitepotential is given by

ψA(∆) = kAd2x[ ¯̀(∆)− ¯̀0

]2.

In general, for a dipole moment p in a magnetic field B, thepotential energy is given by ψM = −p ·B. The field B of anarbitrary, point-like magnetic dipole p = pp̂ is61,62

B(p,r) =µ0 p

4π|r|3

[3|r|2

(p̂ · r)r− p̂],

where r is the position vector of a point in space relativeto the position of the field-generating dipole and µ0 is thepermeability of free space. At present, the dipole momentunder consideration is that affixed to the mass, pm, and B isthat resulting from the superposition of fields generated bythe ps pair at the site of pm. To ensure a horizontal resultant,pm remains parallel to the x-axis while the orientation, θ , ofps is symmetric about the horizontal axis. For our system,the position vector can be written r = dxr̄ and the magneticpotential becomes

ψM = kMd2x

2

∑i=1

p̂m

|r̄i|3·[

3|r̄i|2

(p̂s(i) · r̄i)r̄i− p̂s(i)

],

where kM = µ0 ps pm/4πd5x . Notice that the magnetic potential

does not include contributions from dipole moments inneighboring bistable elements.

Finally, the torsional springs resist the angulardisplacement, ϕ(∆), of the axial springs relative to somearbitrary offset, ϕ0, and may be regarded as the bendingresistance from the ends of the axial springs fixed to thesubstrate. For simplicity, ϕ0 = tan−1[(1−∆s(i))/δ ], where thespecific value of i is that for which ∆s(i) is in the upper energywell of the total potential constructed from ψA and ψM alone;consequently,

ψT(∆) = kT

[tan−1 ( 1−∆

δ

)− tan−1

(1−∆s(i)

δ

)]2.

Following the normalization described in Sec. II B, we definek̄A = kA/kI, k̄M = kM/kI, and k̄T = kT/kId2

x .

Dispersion Relations

Following Eqs. (3) and (4), the matrix equations of motionfor an arbitrary unit cell are given by:[

1 −1−1 1+ k̄s(i)− ω̄2

][v̄ j−1v̄ j

]=

[f j−1f j

], (LOC)[

k̄s(i) −k̄s(i)−k̄s(i) 1+ k̄s(i)− ω̄2

][v̄ j−1v̄ j

]=

[f j−1f j

], (INT)

where f j and f j−1 are the forces applied by the adjacentunit cells. Following Sec. II. A., the local stiffness isgiven by k̄(∆) = ∂ 2ψ̄/∂∆2; therefore, k̄s(i) = k̄(∆s(i)) arethe stiffnesses associated with the two stable configurations,∆s(i), i = 1,2. Recall, for the LOC arrangement, ∆ =

v̄ j − V̄j and, for the INT arrangement, ∆ = v̄ j−1 − v̄ j.Following Bloch’s theorem, v̄ j−1 = v̄ jγ , where γ = e−iκa isthe complex propagation constant with wavenumber κ . Thus,the displacement transformation

[v̄ j−1, v̄ j

]T=Tv̄ j, where T=

[γ 1]T is the Bloch transformation matrix. Simultaneously,pre-multiplying the matrix equations of motion by theconjugate transpose, TH, and applying the displacementtransformation – recognizing that cell equilibrium requiresTHf = 0 – reduces the equation dimensions. The subsequentdeterminate establishes the metamaterial characteristic wavedispersion relations in Eqs. (5).

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