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Negative effective dynamic mass density and stiffness: Micro architectureand phononic transport in periodic compositesSia Nemat-Nasser and Ankit Srivastava Citation: AIP Advances 1, 041502 (2011); doi: 10.1063/1.3675939 View online: http://dx.doi.org/10.1063/1.3675939 View Table of Contents: http://aipadvances.aip.org/resource/1/AAIDBI/v1/i4 Published by the American Institute of Physics. Related Articles

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AIP ADVANCES 1, 041502 (2011)

Negative effective dynamic mass-density and stiffness:Micro-architecture and phononic transport in periodiccomposites

Sia Nemat-Nasser and Ankit Srivastavaa

Department of Mechanical and Aerospace Engineering University of California, San Diego La Jolla, California 92093-0416, USA

(Received 7 November 2011; accepted 16 December 2011; published online 29 December2011)

We report the results of the calculation of negative effective density and negative

effective compliance for a layered composite. We show that the frequency-dependent

effective properties remain positive for cases which lack the possibility of localized

resonances (a 2-phase composite) whereas they may become negative for cases

where there exists a possibility of local resonance below the length-scale of the

wavelength (a 3-phase composite). We also show that the introduction of damping in

the system considerably affects the effective properties in the frequency region close

to the resonance. It is envisaged that this demonstration of doubly negative materialcharacteristics for 1-D wave propagation would pave the way for the design and

synthesis of doubly negative material response for full 3-D elastic wave propagation.

Copyright 2011 Author(s). This article is distributed under a Creative Commons

Attribution 3.0 Unported License. [doi:10.1063/1.3675939]

I. INTRODUCTION

Rytov1 studied the Bloch-form2 or Floquet-type3 elastic waves propagating normal to layers

in a periodic layered composite and produced the expression for the dispersion relation that gives

the pass-bands and stop-bands in the frequency-wave number space. Because of the emergence of

structural composite materials with application to aerospace and other technologies, the 1960’s wit-nessed considerable scientific activity mostly focused on estimating the effective static properties of

composites, whereby elegant and rigorous bounds for their effective properties were established;4–11.

The early effort to study the dynamic response of elastic composite materials was mostly limited to

one-dimensional problems. To create a general numerical approach to solving elastic waves in com-

posites, Kohn et al.12 proposed using a modified version of the Rayleigh quotient in conjunction with

the Bloch-form waves to calculate the dispersion curves. To directly account for the strong disconti-

nuities that generally exist in the elastic properties of a composite’s constituents, Nemat-Nasser13–16

developed a mixed variational formulation to calculate the eigenfrequencies and modeshapes of

harmonic waves in 1-, 2-, and 3-dimensional periodic composites.

Recent research in the fields of metamaterials and phononic crystals has opened up intriguing

possibilities for the experimental realization of such exotic phenomena as negative refraction and

super-resolution. The realization of such phenomena, using crystal anisotropy, has met with recent

successes both in the fields of photonics17–20 and phononics.21–23 It is also possible to realizesuch anomalous wave propagation characteristics with the use of the so called doubly negative

materials.24 It has long been understood that this double negative behavior is a result of local

resonances existing below the length-scale of the wavelength. This physical intuition was used to

realize such materials for electromagnetic waves.25–28 Analogous arguments and results have also

been proposed for the elastodynamic case.29, 30 The central idea in this approach is that the traveling

aAuthor to whom correspondence should be addressed. Electronic mail: [email protected]

2158-3226/2011/1(4)/041502/10 C Author(s) 20111, 041502-1




041502-2 S. Nemat-Nasser and A. Srivastava AIP Advances 1, 041502 (2011)

wave experiences the averaged properties of the microstructure. Therefore, it becomes imperative

to define these averaged properties in a consistent manner in order to be able to explain and

predict wave propagation characteristics in such materials. There has been recent interest in the field

of dynamic homogenization which seeks to define the averaged material parameters which govern

electromagnetic/elastodynamic mean wave propagation.31–36 Subsequent efforts have led to effective

property definitions which satisfy both the averaged field equations and the dispersion relation of the composite.37–42 In the present paper, we show that the effective properties thus defined become

negative in the presence of local resonances, thereby capturing the dynamic effect first anticipated by

Veselago24 for the electromagnetic waves. Although the current treatment concerns 1-D composites,

it is expected that the physical intuition gained will help to design 3-D composites with extreme

material property profiles.

II. EFFECTIVE DYNAMIC PROPERTIES FOR LAYERED COMPOSITES

A brief overview of the effective property definitions is provided here for completeness (see39

for details). For harmonic waves traveling in a layered composite with a periodic unit cell ={ x : −a /2 ≤ x < a /2} the field variables (velocity, ˆu, stress, σ , strain, , and momentum, ˆ p) take the

following Bloch form:

F ( x , t ) = Re

F ( x ) exp[i (q x − ωt )]

(1)

Field equations are

∂ σ

∂ x + iω ˆ p = 0;

∂ ˆu

∂ x + iω = 0 (2)

We define the averaged field variable as

F ( x ) = F eiq x ; F = 1

a

+a/2

−a/2

F ( x )d x (3)

where F ( x ) is the periodic part of F ( x , t ). In general, the following constitutive relations hold41

= ¯ D

σ

+S 1

u

;

p

= S 2

σ

+ ρ

u

(4)

with nonlocal space and time parameters. For Bloch wave propagation, the above can be reduced

to39

= Deff σ ; p = ρeff u

Deff =¯ D

1 + v p S 1= i qu

σ ; ρeff = ρ

1 + v p S 2= p−iωu

(5)

where v p =ω / q= [ Deff ρeff ]−1/2 defines the phase velocity and the dispersion relation. These effective

properties satisfy the averaged field equations and the dispersion relation. These definitions have

been extended to the full 3-D case using micromechanics.42

III. EFFECTIVE PROPERTIES OF 2-PHASE COMPOSITES

A mixed-variational formulation is used to calculate the band-structure and modeshapes for thelayered composites (see appendix) which are required to calculate the effective properties.

Fig. 1 shows the effective properties calculated for the first two pass-bands of a 2-phase layered

composite. Each phase is 5mm thick and the material properties of the phases are

1. E P1 = 2 × 109 Pa; ρP1 = 1000 kg/m3

2. E P2 = 200 × 109 Pa; ρP2 = 3000 kg/m3

where E is the stiffness, ρ is the density, and P1 and P2 denote phases 1 and 2, respectively. It is seen

that the static values (0 frequency) of the effective parameters emerge as the domainant averages of

the material properties. The non-dispersive nature of the parameters in the low frequency regime is

reflected by the fact that the long wavelength waves are not affected by the microstructure. While





FIG. 1. Effective properties for a 2-phase composite. a. Schematic of 3 unit cells of the composite, b. Effective density, c.

Effective compliance.

both density and compliance are simultaneously real and positive in the pass-bands, one of these is

negative in the stop-bands (not plotted), thereby precluding the existence of propagating waves for

the associated frequency ranges. Even if we only consider the pass-bands, it is still possible to have

complex effective density and compliance for purely elastic but asymmetric unit cells. It doesn’t

seem possible, however, to obtain simultaneously negative properties without resorting to local

resonances even for the cases where a great impedance mismatch exists between the two phases.

IV. EFFECTIVE PROPERTY OF 3-PHASE COMPOSITES

Consider a unit cell, built on the physical intuition suggested in.29, 36, 43 Fig. 2 shows the unit cell

of a 3-phase composite. The central heavy and stiff phase can resonate locally if the second phase

is sufficiently compliant. The total thicknesses of phase 1, 2, and 3 in the calculations which follow,

are 2.9 mm, 1 mm, and 0.435 mm respectively.

A. Simultaneously negative density and compliance

Fig. 3 shows the effective properties calculated for the first two pass-bands of four different

3-phase unit cells, each with an increasingly compliant second phase. As the compliance of the

second phase is increased, the first two branches move to the lower frequency regime and the

effective properties become negative over a fraction of the second pass-band, which increases with

increasing second-phase compliance. Since the calculated effective properties are real and negative

in the pass-bands, they reflect the propagating nature of the bands, precisely satisfying the dispersion

relation, (ω / q)2 = 1/( Deff ρeff ).





FIG. 2. Schematic of a 3-phase unit cell.

FIG. 3. Effective properties for the 3-phase composite. Shaded area represents stop-band a. E P2 = 2.2 Gpa, b. E P2

= 1.0 Gpa, c. E P2 = 0.2 Gpa, d. E P2 = 0.02 Gpa.

As expected, the existence of simultaneously negative parameters is highly sensitive to the

density of P3 and the compliance (stiffness) of P2. The resonance is also contingent upon adequatestiffness mismatch between P2 and P3 whereas it is largely independent of the geometric and material

parameters of P1. These relations are more evident in the limiting regime where the definition of

effective mass is easily established.

B. Limiting case of a spring-mass system

Consider the one-dimensional model of Fig. 4 where a cylindrical cavity has been carved out

of a rigid bar of mass M 0. Another rigid body of mass m is placed within the cavity and connected

to the walls of the cavity by springs of a common stiffness K .





FIG. 4. Schematic of a 1-D material where the effective mass depends on frequency.

A harmonic force with frequency ω is applied to the bar. The macroscopic force F (t ) is related tothe macrosropic acceleration of the rigid bar with the effective mass of the system. This frequency-

dependent effective mass is given by,

M eff = M 0 +mω2

0

ω20 − ω2

(6)

where ω0 =√

2K /m is the resonant frequency of the system. M eff increases with increasing fre-

quency up to the resonant frequency, at which point it becomes infinite. Beyond the resonant

frequency M eff begins at negative infinity and approaches M 0. Now consider again the unit cell of

Fig. 2 and use the following material properties:

1. E P1 = 870 × 109 Pa; ρP1 = 2000 kg/m3

2. E P2 = 2 × 108 Pa; ρP2 = 5 kg/m3

3. E P3 = 320 × 109

Pa; ρP3 = 8000 kg/m3

P1 and P3 are rigid compared to P2, and P2 can be assumed to be massless. The equivalent

spring constant for P2 is E P2 / l2 where l2 = 1/2 = .5 mm is the thickness of one layer of the phase.

The mass of P3 is l3ρP3 where l3 = 0.435 mm is the thickness of the central layer. With these

values, the resonant frequency f = √ 2K /m/2π of the system is approximately 76 kHz. Fig. 5

shows the effective density for the corresponding first two pass-bands. It is seen that the variation of

ρeff is essentially congruent to the variation of M eff for the ideal case. At approximately 76 kHz, ρeff

increases towards infinity. Beyond this frequency, starting from negative infinity, it approaches zero.

There is a stop-band in Fig. 5 which is not present in Eq. (6) and reflects the fact that the system in

Fig. 4 is a finite system, while the example of Fig. 5 corresponds to an infinite periodic composite.

C. Inclusion of dissipation

We now consider the effect of including damping in the system on the negative characteristics

of the 3-phase composite. Consider the example of Fig. 3(d) where damping has been introduced

in the three phases. Damping is introduced as an additional imaginary part to the elastic modulus.

Although sophisticated damping models may be used, we use a constant imaginary part (as a small

percent of the real part) for the present purpose. Fig. 6 shows the effective property calculations for

the damped layered composite. For the damped case, both the effective density and the effective

compliance assume complex values, consistent with the fact that there are no clear pass-bands or

stopbands. Fig. 6 shows the real parts of the effective properties. Comparing the results with Fig. 3

it may be verified that the effective properties calculated for the small damping case (1 percent) are

very close to those of the undamped composite. Increasing the damping considerably affects the





−2 −1 0 1 2

x 104

0

2

4

6

8

10x 10

4

ρ effective (kg/m3)

F r e q u e n c y ( H z )

FIG. 5. Effective density for the limiting case.

FIG. 6. Effective properties (real parts) for the damped layered composite. a. Effective density, b. Effective compliance

properties in the region close to the resonance. Away from the resonance, however, the effect of

damping on the real parts of the properties are minimal.

V. DISCUSSION

Characterization of the effective dynamic properties of heterogeneous composites is more

complex than of their static properties. The average dynamic properties are non-local in space and

in time,37, 44, 45 and are generally non-unique.38 Still, for frequency-wavenumber pairs that satisfy

the dispersion relations of 1-D composites, the non-unique constitutive relation can be transformed

into a form with vanishing coupling parameters.41 The coefficients of this constitutive form ( Deff ,

ρeff in this paper) are uniquely determined from the microstructure and satisfy the averaged field

equations and the dispersion relations of the composite. For composites with symmetric unit cells,





these parameters in general are real-valued and positive on the pass-bands. They are complex-valued

for composites which have an asymmetric unit cell. For these unit cells, it is generally not easy to

achieve simultaneously real-valued and negative parameters. This paper explores the special cases

which are expected to give rise to simultaneously negative effective values of density and stiffness

(compliance), and shows that the constitutive form developed in39 adequately captures the physics

of internal resonances.Recent research has also led to the definition and computation of effective dynamic properties

for the 3-D case.42 The exact nature of these effective tensors for composites with internal resonances

remains to be studied. It is envisaged that the intuition gained from the present work would help to

design real3-D composites whichwould display extremematerial properties at predicted frequencies.

ACKNOWLEDGMENTS

This research has been conducted at the Center of Excellence for Advanced Materials (CEAM)

at the University of California, San Diego, under DARPA AFOSR Grants FA9550-09-1-0709 and

RDECOM W91CRB-10-1-0006 to the University of California, San Diego.

APPENDIX A: MIXED METHOD FOR CALCULATION OF EIGENMODES OF PERIODICCOMPOSITES

Consider harmonic waves in an unbounded periodic elastic composite consisting of a collection

of unit cells, . In view of periodicity, we have ρ(x) = ρ(x+ m I β), and C jk mn (x) = C j km n (x+m I β), where x is the position vector with components x j, j = 1, 2, 3, ρ(x) is the density and

C jk mn (x), ( j, k ,m, n = 1, 2, 3) are the components of the elasticity tensor in Cartesian coordinates.

m is any integer and I β , β = 1, 2, 3, denote the three vectors which form a parallelepiped enclosing

the periodic unit cell.

For time harmonic waves with frequency ω (λ = ω2), the field quantities are proportional to

e±iωt . The field equations become

σ jk ,k + λρu j = 0; σ j k = C j km num,n (A1)

For harmonic waves with wavevector q, the Bloch boundary conditions take the form

u j (x+ I β) = u j (x)eiq. I β ; t j (x+ I β ) = −t j (x)eiq. I β (A2)

for x on ∂, where t is the traction vector.

To find an approximate solution of the field equations (Eq. (A1)) subject to the boundary

conditions (Eq. (A2)), we consider the following expressions:

u j =+ M

α,β,γ =− M

U (αβγ ) j f (αβγ )(x) (A3)

σ jk

=

+ M

α,β,γ =− M

S (αβγ ) jk f (αβγ )(x) (A4)

where the approximating functions f (αβγ ) are continuous and continuously differentiable, satisfying

the Bloch periodicity conditions. The eigenvalues are obtained by rendering the following functional

stationary:

λ N = (σ j k , u j,k + u j,k , σ jk − D jk mnσ j k , σ mn )/ρu j , u j (A5)

where gu j , v j =

gu j v∗ j dV, with star denoting complex conjugate, and D jkmn are the components

of the elastic compliance tensor, the inverse of the elasticity tensor C jkmn.

Substituting Eq. (A3) and (A4) into Eq. (A5) and equating to zero the derivatives of λ N with

respect to the unknown coefficients U (αβγ ) j and S

(αβγ ) j k , we arrive at the following set of linear





FIG. 7. Schematic of a layered composite.

homogeneous equations:

σ jk ,k + λ N ρu j , f (αβγ ) = 0; 1 D jk mn σ mn − u j,k , f (αβγ ) = 0 2 (A6)

There are 6 M 3 p ( M p = 2 M + 1) equations in Eq. (A6)2 for a general 3-directionally periodiccomposite. They may be solved for S

(αβγ ) j k in terms of U

(αβγ ) j and the result substituted into Eq. (A6)1.

This leads to a system of 3 M 3 p linear equations. The roots of the determinant of these equations give

estimates of the first 3 M 3 p eigenvalue frequencies. The corresponding eigenvectors are U (αβγ ) j from

which the displacement field within the unit cell is reconstituted. The stress variation in the unit cell

is obtained from Eq. (A6)2.

1. Example: A 2-layered composite

To evaluate the effectiveness and accuracy of the mixed variational method, consider a layered

composite (Fig. 7) with harmonic longitudinal stress waves traveling perpendicular to the layers.

The displacement, u, and stress, σ , are approximated by

u =+ M

α=− M

U (α)ei (qx +2πα x /a); σ =+ M

α=− M

S (α)ei (qx +2πα x /a) (A7)

In the above equations, a is the periodicity length. Substituting these into Eq. (A6)2 we obtain

S α in terms of U α . The resulting equations are then substituted into Eq. (A6)1, providing a set

of M p linear homogeneous equations, the roots of whose determinant give the first M p eigenvalue

frequencies for a given wavenumber q.

The exact dispersion relation for 1-D longitudinal wave propagation in a periodic layered

composite has been given by Rytov:

cos(qa ) = cos(ωh1/c1) cos(ωh2/c2) − sin(ωh1/c1) sin(ωh2/c2) (A8)

= (1 + κ2)/(2κ); κ = ρ1c1/(ρ2c2) (A9)

where hi is the thickness, ρ i is the density, and c i is the longitudinal wave velocity of the i th layer

(i= 1,2) ina unitcell.In Fig. 8 we compare the frequency-wavenumber dispersion relations obtained

by this mixed variational method and the exact solution.

The first five modes calculated from the mixed variational formulation are shown in Fig. 8.

The accuracy of the results calculated from the mixed method depend upon the number of the

Fourier terms used in the approximation ( M p). For the case of Fig. 8, they are very close to the exact

solution. Since the exact dispersion relations are available for only fairly simple geometries like

layered composites, the mixed variational formulation provides an attractive and effective method





0 0.5 1 1.5 2 2.5 30

1

2

3

4

5

6x 10

5

Normalized Wavenumber (Q)

F r e q u e n c y ( H z )

FIG. 8. Frequency-wavenumber dispersion relations calculated from the mixed variational formulation. M p=21 terms are

used in Fourier expansion.

to calculate the eigenfrequencies and eigenvectors associated with three-dimensionally periodic

composites.

1 S. Rytov, Soviet Physics-Acoustics 2, 68 (1956).2 F. Bloch, Z. Phys 52, 555 (1928).3 G. Floquet, Ann De Lecole Normale Superieure 2, 47 (1883).4 Z. Hashin and S. Shtrikman, Journal of the Mechanics and Physics of Solids 10, 335 (1962).5 Z. Hashin and S. Shtrikman, Journal of the Mechanics and Physics of Solids 10, 343 (1962).6 R. Hill, Journal of the Mechanics and Physics of Solids 11, 357 (1963).7 S. Nemat-Nasser, T. Iwakuma, and M. Hejazi, Mechanics of materials 1, 239 (1982).8 J. Willis, Journal of applied mechanics 50, 1202 (1983).9 S. Nemat-Nasser, N. Yu, and M. Hori, Mechanics of materials 15, 163 (1993).

10 S. Nemat-Nasser and M. Hori, Journal of engineering materials and technology 117, 412 (1995).11 S. Nemat-Nasser and M. Hori, Micromechanics: overall properties of heterogeneous materials (Elsevier The Netherlands,

1999).12 W. Kohn, J. Krumhansl, and E. Lee, Journal of Applied Mechanics 39, 327 (1972).13 S. Nemat-Nasser, Journal of Applied Mechanics 39, 850 (1972).14 S. Nemat-Nasser, F. Fu, and S. Minagawa, International Journal of Solids and Structures 11, 617 (1975).15 S. Minagawa and S. Nemat-Nasser, International Journal of Solids and Structures 12, 769 (1976).16 I. Babuska and J. Osborn, Math. Comp 32, 991 (1978).17 E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. Soukoulis, Nature 423, 604 (2003).18 E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. Soukoulis, Physical review letters 91, 207401 (2003).19 P. Parimi, W. Lu, P. Vodo, J. Sokoloff, J. Derov, and S. Sridhar, Physical review letters 92, 127401 (2004).20 Z. Lu, J. Murakowski, C. Schuetz, S. Shi, G. Schneider, and D. Prather, Physical review letters 95, 153901 (2005).21 S. Yang, J. Page, Z. Liu, M. Cowan, C. Chan, and P. Sheng, Physical review letters 93, 24301 (2004).22 M. Ke, Z. Liu, C. Qiu, W. Wang, J. Shi, W. Wen, and P. Sheng, Physical Review B 72, 064306 (2005).23 A. Sukhovich, L. Jing, and J. Page, Physical Review B 77, 014301 (2008).24 V. Veselago, Physics-Uspekhi 10, 509 (1968).25 J. Pendry, A. Holden, D. Robbins, and W. Stewart, Microwave Theory and Techniques, IEEE Transactions on 47, 2075

(1999).26 J. Pendry, Physical Review Letters 85, 3966 (2000).





27 D. Smith, W. Padilla, D. Vier, S. Nemat-Nasser, and S. Schultz, Physical Review Letters 84, 4184 (2000).28 D. Smith, J. Pendry, and M. Wiltshire, Science 305, 788 (2004).29 Z. Liu, X. Zhang, Y. Mao, Y. Zhu, Z. Yang, C. Chan, and P. Sheng, Science 289, 1734 (2000).30 Z. Liu, C. Chan, and P. Sheng, Physical Review B 71, 014103 (2005).31 J. Willis, Journal of the Mechanics and Physics of Solids 28, 287 (1980).32 J. Willis, in Continuum micromechanics (Springer-Verlag New York, Inc., 1997) pp. 265–290.33

L. Wang and S. Rokhlin, The Journal of the Acoustical Society of America 112, 38 (2002).34 D. Smith and J. Pendry, JOSA B 23, 391 (2006).35 A. Amirkhizi and S. Nemat-Nasser, Comptes Rendus Mecanique 336, 24 (2008).36 G. Milton and J. Willis, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science 463, 855

(2007).37 J. Willis, Mechanics of Materials 41, 385 (2009).38 J. Willis, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science (2011).39 S. Nemat-Nasser, J. Willis, A. Srivastava, and A. Amirkhizi, Physical Review B 83, 104103 (2011).40 A. Shuvalov, A. Kutsenko, A. Norris, and O. Poncelet, Proceedings of the Royal Society A: Mathematical, Physical and

Engineering Science 467, 1749 (2011).41 S. Nemat-Nasser and A. Srivastava, “Overalldynamicconstitutive relationsof layered elastic composites,” (2011), accepted

in Journal of Mechanics and Physics of Solids, arXiv:1105.5173v1.42 A. Srivastava and S. Nemat-Nasser, “Overall dynamic properties of 3-d periodic elastic composites,” (2011), submitted to

Proceeding of the Royal Society.43 A. Avila, G. Griso, and B. Miara, Comptes Rendus Mathematique 340, 933 (2005).44 J. Willis, Advances in applied mechanics 21, 1 (1981).45

J. Willis, Wave Motion 3, 1 (1981).

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