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New metamaterials with macroscopic behavior outside

that of continuum elastodynamics

Graeme W. MiltonDepartment of Mathematics, University of Utah, Salt Lake City UT 84112, USA

Abstract

Metamaterials are constructed such that, for a narrow range of frequencies,the momentum density depends on the local displacement gradient, and the stressdepends on the local velocity. In these models the momentum density generallydepends not only on the strain, but also on the local rotation, and the stress isgenerally not symmetric. A variant is constructed for which, at a fixed frequency,the momentum density is independent of the local rotation (but still depends onthe strain) and the stress is symmetric (but still depends on the velocity). Gen-eralizations of these metamaterials may be useful in the design of elastic cloakingdevices.

Keywords: cloaking, elastic metamaterials, novel elastodynamics

1 Introduction

The possibility of cloaking objects to make them invisible has attracted substantial recentattention. Alu and Engheta ([1], [2]) following earlier work of Kerker [3], found that thescattering from an object could be substantially reduced by surrounding it by a plasmonicor metamaterial coating. Milton and Nicorovici [4], expanding on the work of [5] and [6],proved that the dipole moments of a polarizable point dipole or clusters of polarizable linedipoles would vanish when placed within a cloaking region surrounding a superlens, andthis has been confirmed numerically [7]. Bruno And Lintner [8] show that only partialcloaking occurs when the polarizable object is not discrete. Ramm [9] and Miller [10]found that cloaking could be achieved by sensing and manipulating the fields near theboundary of the object.

Perhaps the greatest interest has been generated by transformation based cloaking.This type of cloaking was first discovered by Greenleaf Lassas and Uhlmann ([11], [12]) inthe context of electrical conductivity. Their idea was to apply the well known fact (see,for example, [13]) that the equations of electrical conductivity retain their form undercoordinate transformations, using a singular transformation which mapped a point to a

1

sphere (the cloaking region), and which was the identity outside a larger sphere. Perturb-ing the conductivity in the cloaking region corresponds to changing the conductivity ata point in the equivalent problem, and this has no effect on the fields outside the largersphere. This type of cloaking was extended to the full time harmonic Maxwell equa-tions by Pendry, Schurig and Smith [14], and at the same time Leonhardt [15] discovereda different transformation based, two-dimensional, geometric optics or high frequencyacoustic cloaking scheme which only utilized isotropic materials. The former cloakinghas been verified numerically ([16]; [17]; [18]), placed on a firm theoretical foundation([19]), and experimentally substantiated in the microwave regime in an approximate wayin two-dimensions by Schurig, Mock, Justice, Cummer, Pendry, Starr, and Smith [20] us-ing Schelkunoff and Friis’s idea ([21]) of utilizing metamaterials with split ring resonatorsto achieve artificial magnetism. Cai, Chettiar, Kildishev, and Shalaev [22] proposed an-other design which may experimentally achieve approximate two-dimensional cloaking inthe visible regime.

Milton, Briane and Willis [23] found that applying transformation based cloaking tocontinuum elastodynamics would require new materials with very unusual properties.This is because the continuum elastodynamic equations do not retain their form undercoordinate transformations, but rather take the form of the equations that Willis ([24],[25], [26]) found described the ensemble averaged behavior of composite materials. Oneinteresting feature is that the density at a given frequency is required to be matrix valued.Willis [27] had proved that his density operator had this property. It also follows fromthe work of Movchan and Guenneau [28] and Avila, Griso, and Miara [29], that thereexist high contrast materials with a local anisotropic effective density. Simple modelswith anisotropic and possibly complex density were obtained by Milton and Willis [30].These microstructures generalized a construction of Sheng, Zhang, Liu, and Chan [31]and Liu, Chan, and Sheng [32], who established that the density can be negative over arange of frequencies. The same conclusion was reached and rigorously proved by Avila,Griso, and Miara [29]. A comparison of Maxwell’s equations, which can be written inthe form

∂

∂xi

(

Cijkℓ

∂Eℓ

∂xk

)

= {ω2εE}j , (1.1)

whereCijkℓ = eijmekℓn{µ−1}mn, (1.2)

[in which E is the electric field, ε the electric permittivity tensor, µ the magnetic perme-ability tensor, and eijm = 1 (-1) if ijm is an even (odd) permutation of 123 and is zerootherwise] with the equations of continuum elastodynamics

∂

∂xi

(

Cijkℓ

∂uℓ

∂xk

)

= −{ω2ρu}j, (1.3)

[in which u is the displacement field, ρ is the density, and C is now the elasticity tensor]suggests that ρ as a function of the frequency ω could share many of the same proper-ties as ε(ω). This has been established, and in fact for every function ρ(ω) satisfying

2

these properties one can construct a model which has approximately that function as itseffective density as a function of frequency ([30]). Cummer and Schurig [33] investigatedtransformation based acoustic cloaking in two-dimensions and, due to a mathematicalequivalence with the electromagnetic problem, found that it could be achieved providedone could construct the necessary materials with anisotropic density.

Anisotropic density is not the only new property needed to achieve transformationbased elasticity cloaking. One needs materials where the constitutive law, like that inthe Willis equations, couples the stress with not only with the strain but also with thevelocity, and couples the momentum density not only with the velocity but also with thedisplacement gradient through the strain. Also, unlike in the Willis equations, one wantsthis dependence to be local in space and to apply to a single microgeometry rather thanto an ensemble average of microgeometries. Here we show that such unusual behaviorcan be realized in a model such that, for a narrow range of frequencies, the associatedwaves have wavelength much larger than the microstructure, and the momentum densitydepends on the displacement gradient, and the stress depends on the velocity. Curiouslythe momentum density depends not only on the strain, but also on the local rotation,and the stress is not symmetric. We also construct a variant of the model for which (ata fixed frequency) the momentum density is independent of the local rotation (but stilldepends on the strain) and the stress is symmetric (but still depends on the velocity).Both models rely heavily on the discovery Sheng, Zhang, Liu, and Chan [31] that onecan design structures which, at a fixed frequency, respond as if they have negative mass.Following their ideas, and the simplified constructions of Milton and Willis [30] whichincorporate these ideas, figure 1 illustrates a structure with negative mass. In the modelsconsidered in this paper both positive and negative masses are idealized as point masses.

This work on metamaterials with macroscopic behavior outside that of continuumelastodynamics, even though they are governed by continuum elastodynamics at the mi-croscale, is preceeded by work on metamaterials with macroscopic non-Ohmic, possiblynon-local, conducting behavior, even though they conform to Ohm’s law at the mi-croscale, ([34]; [35]; [36]; [37]; [38]; [39]) by work on metamaterials with non-Maxwellianmacroscopic electromagnetic behavior, even though they conform to Maxwell’s equationsat the microscale ([40]), and by work on metamaterials with a macroscopic higher or-der gradient or non-local elastic response even though they are governed by usual linearelasticity equations at the microscale ([41]; [42]; [43]).

For simplicity we consider a two dimensional model, although it can be generalizedto three dimensions. The model is illustrated in figure 2.

2 The momentum density in the model

Figure 3 shows the unit cell in the model. The masses are approximated as point masses,and the springs may be taken to have all the same spring constant hK, which scales inproportion to the size of the unit cell. This ensures that the spring network, by itself,responds as a elastic material in the limit h → 0.

3

Void

Rigid

Figure 1: This structure responds as if it had negative mass to oscillations at a fixedfrequency above resonance, such that the spherical core oscillates out of phase with therigid, but light, surrounding shell. The springs have the same spring constant to ensurethat the effective mass is isotropic.

Figure 2: The model with strange elastic behavior. The large solid black circles havepositive mass, while the neighbouring large white circles have negative mass at the givenfrequency. They are connected to the spring network by rods of fixed length, whichalternatively can be regarded as springs with infinite spring constants.

4

D

B

A E F C

2h

Figure 3: The unit cell of the model.

Without loss of generality let us place the origin at the center x0 of the unit cell underconsideration. Then when the material is a rest the points A,B,C,D,E and F in figure3 are at positions

xA = (−h, 0), xB = (0,−h), xC = (h, 0), xD = (0, h)

xE = (−ch, 0), xF = (ch, 0), (2.1)

where c is a parameter between 0 and 1 which controls the inclination of the rods joiningE and F to B and D: these rods have length h

√1 + c2. In the limit h → 0 we assume

the (infinitesimal) physical displacements at the points xA, xB, xC and xD derive fromsome smooth complex valued displacement field u(x), and in particular

uA = ℜe[u(xA)e−iωt] ≈ ℜe[(u0 − hq)e−iωt],

uB = ℜe[u(xB)e−iωt] ≈ ℜe[(u0 − hw)e−iωt],

uC = ℜe[u(xA)e−iωt] ≈ ℜe[(u0 + hq)e−iωt],

uD = ℜe[u(xD)e−iωt] ≈ ℜe[(u0 + hw)e−iωt],

(2.2)

where

u0 = u(x0), q =∂u

∂x1

∣

∣

∣

∣

x=x0

, w =∂u

∂x2

∣

∣

∣

∣

x=x0

, (2.3)

5

and x0 = 0 because the origin is at the center of the unit cell.At the macroscopic level we choose not keep track of the displacements uE and uF .

These are internal hidden variables which however can be recovered from uB and uD.They have the form

uE ≈ ℜe[(u0 + hs)e−iωt],

uF ≈ ℜe[(u0 − hs)e−iωt], (2.4)

where the complex vector s remains to be determined. Since there is a rigid rod connect-ing the points D and E we have, in the limit h → 0, the constraint that w − s must beperpendicular to the line joining D and E, i.e.

0 = (w − s) · (c, 1) = c(w1 − s1) + (w2 − s2). (2.5)

Similarly since there is a rigid rod connecting the points B and E we have, in the limith → 0, the constraint

0 = (w + s) · (−c, 1) = −c(w1 + s1) + (w2 + s2). (2.6)

These equations have the solution

s1 = w2/c =1

c

∂u2

∂x2

, s2 = cw1 = c∂u1

∂x2

. (2.7)

One can easily see that if c is replaced by −c then s gets replaced by −s, which justifiesthe formula (2.4) for uF .

Now suppose the masses at the points E and F , which we call a mass pair, arerespectively chosen to have the form

mE = hm, mF = −hm+ δh2, (2.8)

where m is a positive real constant, and δ is a possibly complex constant with a non-negative imaginary part. Then at any time t the physical momentum in the unit cell is,to second order in h,

ℜe{−iω[mE(u0 + hs) +mF (u0 − hs)]e−iωt} ≈ h2ℜe{−iω[2ms + δu0]e−iωt}. (2.9)

Since the area of the unit cell is 2h2 we see that the complex momentum density is

p = −iωms + (δ/2)(−iωu0), (2.10)

in which −iωu0 is the complex velocity of the unit cell, and s depends on the deformationgradient through (2.7).

6

3 The stress in the model

The acceleration of the material will also generate stress. To see this, note that to leadingorder in h, the masses at E and F must be accelerated by complex forces F = −ω2mhuand −F acting on the respective masses (the physical forces are obtained by multiplyingthese by e−iωt and taking the real part). Let FED and FFD be the forces which the rodsED and FD exert on the vertex D and let FEB and FFB be the forces which the rodsEB and FB exert on the vertex B. Balance of forces at E and F requires that

FED + FEB = −F, FFD + FFB = F. (3.1)

Since these forces are aligned with their respective rods we have

FED = −FFB = α(c, 1), FEB = −FFD = β(c,−1), (3.2)

where the constants α and β are such that (3.1) is satisfied:

(α + β)c = −F1, α− β = −F2. (3.3)

Forgetting for a moment the springs and surrounding material, to maintain the motionsa force

ht = −FED − FFD = −α(c, 1) + β(c,−1) = (cF2, F1/c) = −ω2mh(cu2, u1/c) (3.4)

needs to act on the vertex D to balance the forces which the rods ED and FD exerton this vertex. Similarly a force −ht needs to act on the vertex B to balance the forceswhich the rods EB and FB exert on this vertex. On the other hand, no additional forcesneed to be exerted on the vertices A and C.

Now consider a rectangular sample (with dimensions, say of order√h, small com-

pared to the wavelength) as in figure 2 with (complex) forces acting on the boundaryvertices to maintain the motions. These forces will have two components: an elastic (orpossibly viscoelastic) component to compensate for the (complex) stress σE caused bythe (complex) strain in the spring network, and an inertial component to compensate forthe inertial stress σI caused by the acceleration −ω2u of the material. The compensatinginertial component will be zero on the left and right sides of the network. At the top5 vertices in figure 2 the compensating inertial component will be approximately 2ht ateach vertex. The extra factor of 2 arises because of the extra load that the springs atthe top boundary carry to support the inertial component of the forces at the 4 verticesimmediately below the top. (At the other interior nodes these forces are balanced). Thusthe compensating inertial component per unit length, is t on the top, −t on the bottom,and zero on the sides. By the definition of stress this should be equated with σIn, wheren is the unit normal to the boundary. Thus we deduce that

σI =

(

0 t10 t2

)

=

(

0 −ω2mcu2

0 −ω2mu1/c

)

. (3.5)

7

Of course, in the limit h → 0, the stress in the spring network (excluding the masses andconnecting rods) will be governed by a normal relation

σE = C[∇u+ (∇u)T ], (3.6)

where C is the elasticity tensor of the network.Our choice to define the macroscopic stress through the forces at the vertices of the

unit cell (rather than as a volume average of the microscopic stress field, which wouldresult in a symmetric stress) makes good physical sense and is consistent with our choiceto define the macroscopic displacement field through the values of the displacement atthe vertices of the unit cell. These choices bear some resemblance to the way Pendry,Holden, Robbins and Willis [44] define macroscopic electromagnetic fields through fieldvalues at the boundary of the unit cell. In defining the stress in this way it is importantthat the unit cell be chosen so that each mass pair (of positive and negative masses) isnot split by the boundary of the unit cell: each mass pair and their four connecting rodsis regarded as an indivisible unit in the same way that in defining the electric polarizationfield one would not choose to place the boundary in a dielectric material so as to splitthe positive and negative charges of the constituent atoms.

4 Summary

In the limit h → 0 the constitutive law takes the form(

σ

p

)

=

(

C S

D ρ

)(

∇u

v

)

, (4.1)

where σ = σE + σI is the total stress, v = −iωu is the velocity, and ρ = (δ/2)I is thedensity. In the basis where σ, p, ∇u and v are represented by the vectors

σ =

σ11

σ21

σ12

σ22

, p =

(

p1p2

)

, ∇u =

∂u1/∂x1

∂u2/∂x1

∂u1/∂x2

∂u2/∂x2

, v =

(

v1v2

)

, (4.2)

the third-order tensors S and D, from (2.7), (2.10) and (3.5), are represented by thematrices

S =

0 00 00 −iωmc

−iωm/c 0

, D =

(

0 0 0 −iωm/c0 0 −iωmc 0

)

. (4.3)

Thus the matrix entering the constitutive law (4.1) is symmetric. Since the constitutivelaw and the equation of motion are asymptotically independent of h, the waves associated

8

D

B

A CFE

2h

G H

Figure 4: The unit cell of a generalized model.

with their solutions will have wavelength much larger that the microstructure. In thiscircumstance the usual continuum elastodynamic equations normally apply. Howeverthis model shows that this is not always the case.

The model has some unusual features. Although the net amount of mass in a regionof unit area is independent of h, the total amount of positive (or negative) mass in aregion of unit area scales like 1/h. However in any physical model, as usual, one never infact takes the limit h → 0, but instead one sets the microstructure as small as requiredto get a reasonable approximation to the limiting behavior for a desired set of appliedfields. Also the amount of positive (or negative) mass in the unit cell only representsthe amount of apparent mass, not gravitational mass, which could be quite different ifthe apparent mass arises from substructures which are close to resonance. Of coursethe analysis given here needs some rigorous justification to check, for example, that thedisplacements can be approximated by a smooth field u(x) in the limit h → 0. Also froma practical point of view it needs to be checked that the behavior is stable to sufficientlysmall variations in the masses or spring constants from cell to cell.

The model can easily be generalized. One simple generalization has additional massesand connecting rods as illustrated in figure 4, with masses

mG = −hm′ + δh2, mH = hm′, (4.4)

at the points G and H which, when the material is at rest, are located at

xG = (−c′h, 0), xH = (c′h, 0), (4.5)

9

where c′ andm′ are positive constants. By superposition the constitutive law will take theform (4.1) with a density ρ = δI and with the third-order tensors S and D representedby the matrices

S =

0 00 00 iω(m′c′ −mc)

iω(m′/c′ −m/c) 0

,

D =

(

0 0 0 iω(m′/c′ −m/c)0 0 iω(m′c′ −mc) 0

)

. (4.6)

In particular if, at a given frequency, m′c′ = mc we obtain a model in which the mo-mentum does not depend on the local rotation and the stress is symmetric. With thiscondition satisfied one has the relations

σ = C∇u+ S′u, ∇ · σ = −iωp = D′∇u− ω2ρu, (4.7)

whereS′ = −iωS, D′ = −iωD, (4.8)

and the only non-zero Cartesian elements of these two tensors are from (4.1), (4.2) and(4.6)

S ′

221= D′

122= ω2(m′/c′ −m/c). (4.9)

The general form of these tensors S′ and D′ corresponds with that required for elasticitycloaking: they are real and satisfy the identities S ′

ijk = S ′

jik = D′

kij. They may serveas a good starting point in the quest to find materials with a suitable combination ofproperties C, S′, D′ and ρ needed for elasticity cloaking. However, quite apart fromthis, the novel properties of this new class of metamaterials may have other importantapplications.

In the design of these models perhaps the key feature is that in the narrow frequencyrange under consideration the amount of negative mass in the unit cell (due to hiddeninternal masses moving out of phase with the motion) almost balances the amount ofpositive mass in the unit cell. This ensures that the momentum density approaches aconstant as the cell size approaches zero, while keeping non-trivial the stresses caused bythe moving masses in the unit cell. It will be interesting to see if other models with thiskey feature also have a local constitutive law of the form (4.1), with non-zero couplingterms S and D.

Acknowledgements

Graeme Milton is grateful to John Willis for helpful suggestions which improved themanuscript and for support from the National Science Foundation through grant DMS-0411035.

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