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# Journal Club December 2007: Elastodynamic band gaps and metamaterials

Metals have a lustre because electromagnetic radiation in the visible range is almost completely absorbed at an exponential rate close to a metal surface and then radiated back. This effect and other properties of crystalline materials can be explained by their electronic band structure and the associated dispersion relations. In certain materials, notably semiconductors, distinctive band gaps are found indicating that electrons of certain forbidden energies which cannot propagate through the material. A concise description of the basics can be found here.

There is a close analogy between the motion of electrons in periodic crystals and electromagnetic waves in periodic dielectric structures. This analogy and the quest for materials with unusual electromagnetic properties has led to exploration of periodic materials with tunable electromagnetic band structures and band gaps. Such materials are called photonic crystals and photonic composites. A good review article on the subject is E. Yablonovitch, 1993, "Photonic band-gap crystals", *J. Phys.: Condens. Matter,* **5,** 2443-2460, doi:10.1088/0953-8984/5/16/004.

Researchers have long noted that the equations of linear elasticity have the same form as Maxwell's equations of electromagnetism. For example, at fixed frequency () Maxwell's equations can be written as .

This is very similar to the elasticity equation . Hence one can expect to see some of the same phenomena in elastodynamics as are observed in electrodynamics, including dispersion relations and band gaps. The holy grail of the field of phononic band gap materials is to design a material with such band gaps that can block certain frequencies of vibration. A more general term for such materials "Elastodynamic band gap materials". In the case where the shear component can be neglected, as in sound waves in fluids, the terms "sonic band gap" or "acoustic band gap" materials are also used. Note that most of the literature on elastodynamic band gap materials assumes some familiarity with classical electrodynamics and the literature on photonic band gap materials.

One of the first calculations of the acoustic band gaps of some simple periodic composites can be found in M. S. Kushwaha et al., 1993, "Acoustic band structure of periodic elastic composites," *Phys. Rev. Lett.*. 71, 2022 - 2025, doi:10.1103/PhysRevLett.71.2022 The calculations in that paper were carried out only for the case of antiplane shear. However, since 1993 there has been extensive research on the subject that has filled in many of the gaps. An extensive bibilography can be found in J. P. Dowling's page on Photonic and Acoustic Band Gaps and Metamaterials. A paper that I have found particularly useful for understanding the basics is by P. G. Martinsson and A. B. Movchan, 2003, "Vibrations of Lattice Structures and Phononic Band Gaps," *The Quarterly J. of Mechanics and Appl. Math.*, **56**(1), 45-64, doi:10.1093/qjmam/56.1.45.

Despite the promise of photonic and phononic band gap materials and the significant improvement in our understanding of such materials, economically viable applications have not been forthcoming. There are definitely some opportunities for experimental mechanicians here.

In the past five years, many of the workers in the area of photonic band gaps have moved on to explore electromagnetic metamaterials. The impetus was provided by a seminal paper by Pendry in April 2000 (J. B. Pendry, 2000, "Negative refraction makes a perfect lens," *Phys. Rev. Lett*. 85, 3966 - 3969, doi:10.1103/PhysRevLett.85.3966) who showed that negative refractive index materials could be realized and perfect lensing was a byproduct of such a behavior. The trick was to take advantage of the invariance of Maxwell's equations under certain transformations. Further exploration of these ideas have indicated that electromagnetic cloaking can also be achieved using such transformations (see Schurig et al., 2006, "Metamaterial Electromagnetic Cloak at Microwave Frequencies," *Science,* **314**(5801), 977 - 980, doi:10.1126/science.1133628.)

Extensions to elastodynamics have not been far behind. In 2000, Z. Liu and P. Sheng's group in Hong Kong showed that negative elastic moduli[1] could be achieved in certain resonant structures (Liu et al., 2000, "Locally Resonant Sonic Materials," *Science*, **289**(5485), 1734 - 1736, doi:10.1126/science.289.5485.1734.) These ideas have been given a sound theoretical footing and extended significantly by Graeme Milton and his coworkers. **Among the possibilities that this work has identified are materials with elastic cloaking, negative inertial mass, anisotropic density, and even materials which follow a modified Newton's second law.** These are exciting times for this field and we have barely scratched the surface for possible applications and further developments.

The papers that I recommend for this edition of the journal club deal with recent developments in elastodynamic metamaterials.

- The first paper shows that a material composed of periodic arrays of Helmholtz resonators can give rise to a negative bulk modulus.
Ultrasonic metamaterials with negative modulus by Nicholas Fang et al.,

*Nature Materials*,**5**, (2006), 452, doi:10.1038/nmat1644. - The second paper shows how ideas from electromagnetism can be applied to elastodynamic cloaking.
On cloaking for elasticity and physical equations with a transformation invariant form by Graeme Milton, Marc Briane and John Willis,

*New Journal of Physics*.,**8**, (2006), 248, doi:10.1088/1367-2630/8/10/248. - The third paper is an excellent collection of ideas on the unusual possibilities that arise when we homogenize resonant structures, i.e., negative mass, anisotropic density, and a stress that depends both on the velocity and the strain.
On modifications of Newton's second law and linear continuum elastodynamics by Graeme Milton and John Willis,

*Proc. Royal Soc. London, Series A*,**463**(2079), (2007), 855-880, doi:10.1098/rspa.2006.1795.

Please let me know if you cannot access any of the papers above and I will send you a copy. I have not attached any of them to this post for copyright reasons.

**Notes: **

[1] It was later discovered that Liu et al. had actually found a negative density material rather than a negative modulus material. See Liu, Chan, Sheng, 2005, "Analytic model of phononic crystals with local resonances ", Physical Review B 71, 014103.

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## Comments

## A great choice of a topic...

Biswajit,

Though I seldom post in Journal Clubs here, I do enjoy reading some of them. (If I remember it right, I have posted only one comment in a journal club so far, and the last time I checked, none had responded to it! ;-)) ... But yes, the last month's topic made for a very interesting reading, and just when I was wondering what might follow it, here comes another greatly interesting, and in fact, a very unusual kind of a topic... I can see three/four different subjects of my varied background come together in a very unusual manner here: materials (band theory), electromagnetic fields, elasticity (vibrations), composites.... A must read for those who (still) believe too much in "branches" and "disciplines." ...

Anyway, congratulations for a great choice of a topic!

Ajit

## Re: A great choice of a topic...

Ajit,

I hope it interests you and others enough for you start working on metamaterials :)

I am keen to know what questions might arise in readers' minds and what applications of these ideas are possible.

-- Biswajit

## effective medium theory always has something new to offer

Biswajit,

Thanks for the nice overview of this fast evolving topic - one that I have been trying to catch up on, so your list of article is timely and will be on my holiday reading list!

A quick search came up with other recent papers, including this by Ping Sheng's group: Effective dynamic mass density of composites J. Mei, Z. Liu, W. Wen and P. Sheng, Phys. Rev. B

76, 134205, 2007.(20 pages). This paper reminds me again why the area of composite materials can be such an interesting and rewarding area for research. Twenty years ago I was incredulous when I heard about acoustic bubble resonance. This is thelow frequencyphenomenon in which tiny air bubbles in water can have huge O(1) effects on sound propagation. How can a tiny bubble with diameter 1/1000th or less of a wavelength have any perceptible effect? The reason is that the effective medium comprises the inertia of the two phases (water and air) added in parallel (to use a simple metaphor) while the stiffnesses add in series. So at certain frequencies (or a narrow bandwidth) the effective medium has compressiblity on the order of air, with mass density of water. Put a lot of bubbles together and the bandwidth is broadened. You end up with an effective wave speed much smaller than that of water.The giant bubble resonance has a long history, see e.g.

The Acoustic Bubble, T. G. Leighton, Academic Press, 1994. However, it seems that it has been particularly well studied and understood in the last few decades with many nice experiments and applications (though I can't recall specifics). A related but obviously different and far more difficult phenomenon is sonoluminescence - but in a very real sense the success to date in understanding sonoluminescence has been due to prior work in bubble acoustics.The papers you cite and also the above mentioned one by Mei et al. use methods that are relatively standard in effective medium theory to come up with what seem to be strange results, e.g. negative inertia, or dynamic mass density to be more specific. The non-intuitive nature of these results make them all the more interesting, and there is plenty of room for theoretical investigation and even more for experiments. Maybe in twenty years all this will also be routine, with many applications and examples. And perhaps some entirely unexpected effect will materialize, or perhaps I should say metamaterialize!

## Re:Effective medium theories

Andy,

It is indeed amazing how many surprises linear elasticity still has to offer. Thanks for mentioning Mei et al. That paper hints at the possibility that ensemble averaging (as used by Willis in his early papers on the dynamics of composites and in the Milton-Willis paper) may not be a necessary part of the effective medium theory of metamaterials. However, more work is needed to show that that rigorously.

I should mention the pioneering work of Jim Berryman who discussed some similar ideas as early as 1980. You can find links to his papers at http://sepwww.stanford.edu/sep/berryman/rockphys.html.

-- Biswajit

## A timely theme

Dear all,

A nice introduction and a survey of literature on periodic (lattice) media

by Biswajit. This is, surprisingly, a growing area of interest - certainly on

the other side of the Atlantic. There is rich history

associated with the band-gap materials (rather micro-structures?) within the

aero structures community in the late 60s which diminished-only to be

rediscovered again.

As an act of 'self promotion', and to keep the solid mechanicians

informed of earlier literature in this area, I would like to draw your attention to a recent article that I wrote with

my colleagues at Cambridge on applying band-gap concepts to lattice materials

and metallic foams.

A. Srikantha Phani, J. Woodhouse and N. A. Fleck (2006), Wave propagation in

two-dimensional periodic lattices,

Journal of the Acoustical Society ofAmerica,Vol.119, issue 4, pp. 1995-2005

In the above paper, I ’applied’ the solid state physics concepts to

lattice materials (or metallic foams with a regular microstrucure) using the

Finite Element technique and Bloch's theorem. Hope this paper serves to keep

the works of earlier physicists/aero structures community in perspective.

The effective medium theory obtained via the method of longwaves can only

capture the dispersion upto a point. A cosserat-like model a la Mindlin-Toupin

couple stress theory might only extend the range of effective medium theories.

Essentially these are discrete media in their own right when dynamical

phenomena are to be investigated.

There is still a lot more engineering physics to be learnt and applied in

this area.

I am happy to learn from others, despite my committments being elsewhere at

the moment.

Best wishes,

Srikanth

Department of Mechanical Engineering

University of Bath, UK

http://www2.eng.cam.ac.uk/~skpa2

## Re: Wave propagation in two-dimensional periodic lattices

Srikanth,

Thanks for pointing out your paper. That certain structures can act as acoustic filters has indeed been known for a long while. I believe that M.S. Kushwaha actually demonstrated sonic band gaps in scultures in front of a building in the mid 1990s.

The recent buzz regarding elastic metamaterials is more to do with unusual properties such as negative density or unsymmetric stress tensors even in the long wavelength regime (long compared to the size of the unit cell). I will be interesting to see what happens when the wavelength of the elastic waves become smaller than that of the unit cell and how the theory develops so that it is able to deal with that problem.

-- Biswajit

## Biswajit, You are right

Biswajit,

You are right about the acoustic band-gap experiments (sonic crystals) of Kushwaha et al. What I was referring to is vibrational band-gaps of interest to structural mechanics. The ideas of band-gaps indeed go back to Lord Rayleigh's work in 19th century!

The non-symmetric stress tensor (as occurs in couple stress theories) is to be expected as these generalised continua are essentially of higher order nature. For example, in a latice network of beams, such as a garden trellis, any cross section would transmit axial, shear and bending moments. Kinematics can be non-affine too (a classic example being a regular hexagonal lattice as in a honeycomb) which makes even elastic strain-gradient like continuum models unable to explain experimental phenomena such as the correct scaling for fracture toughness of regular hexagonal honeycombs.

The notion of a unit cell is the key to understand the mechanics of the periodic media. I suspect that when the wavelength is shorter than the unit cell size, one could get away by defining a smaller unit cell (?) so that the machinery of solid state physics (Bloch's theorem) can still be used?

The concept of negative density is new to me. I must learn this.

Best wishes,

Srikanth

## Resonance

Srikanth,

Metamaterials are different from other composites in the sense that part of the unusual properties are due to some sort of resonance. The following paragraphs from "Three-component elastic wave band gap material " by Liu, Chan, and Sheng (Physical Review B, 2002, 65, 165116) give a lucid description of what's involved.

"In the past decade, there have been intense activities on photonic band-gap1 systems because of their novel physical properties and many potential applications in the photonics age. Elastic wave can also have forbidden gaps, but it has received less attention partly because it is a more complex mathematical problem. Another reason is that elastic wave band-gap material is usually conceived to have a length scale at least a few times that of the wavelength, implying gigantic structures for lower frequencies and making the application rather difficult. Previous considerations are mainly focused on two-component systems and absolute elastic wave band gaps are rather difficult to realize for two-component solid materials in three dimensions.

There are two mechanisms that can lead to a forbidden gap in classical waves.

. The existence of an absolute gap requires the overlap of the Bragg gaps in all directions. Symmetry, periodicity, and orderness of the periodic system are all important. The frequency of the spectral gap is of the order of c/a, where c is wave speed and a is the lattice constant. The photonic band gap, as it was originally perceived, is a Bragg gap that overlaps in all 4pi radians.One is Bragg scattering in periodic systems and the opening of a Bragg gap at the Brillouin zone (BZ) boundary. Examples are polariton gaps when light couples with optical phonons in ionic crystals. The frequency of a resonance gap is dictated by the frequency of the resonance, and is independent of orderness, periodicity and, symmetry unless there is a high concentration of resonating units so that they couple strongly with each other. Electromagnetic resonances derived from elementary excitations such as polariton or plasmon are usually associated with strong absorption, and are thus not very useful for creating photonic gaps.The other mechanism is derived from localized resonances. "The situation is different for elastic waves in at least two respects. First, the dissipation can be very small in certain mechanical excitations, making resonance a possible mechanism for the creation a elastic wave band gap. Second, we shall show in this paper that it is possible to tune the spectral gap ‘‘continuously’’ between a Bragg gap and a resonance gap in three-component elastic systems, while such manipulations would be very difficult for electromagnetic waves-- Biswajit

## Journal Club as a forum for learning

One of the great aspects of iMechanica from my point of view is the Journal Club. I have learned a considerable amount about some topics through the papers suggested and resulting discussions.

The December 07 JC by Biswajit Banerjee is a case in point. The papers he selected allowed me to quickly understand some, but by no means all, of the issues involved with dynamics of metamaterials. As a result I became interest in the topic of cloaking of waves.

This is the phenomenon that causes waves incident on the "cloak" to emerge on the other side as if the cloak and whatever it might happen to shroud, were not there. The difficulty in realizing the effect is that the cloak must exhibit this physical invariance for arbitrary wave incidence. One solution is the so-called transformation approach, in which the equations are viewed as being physically transformed due to a change of coordinates. The transformation method for acoustics can be understood in terms of finite deformation theory. I think this approach is one that people familiar with finite elasticity will find amenable.

The upshot of the initial interest sparked by Biswajit's Journal Club is a paper that recently appeared in Proc. R. Soc. A. You can find some movie simulations illustrating aspects of cloaking at this web page, which also contains links to other sites on cloaking.

## jClub

Andrew,

thanks so much for the feedback on the j-club, glad to hear that it is "working" in its educational mission. I think it goes both ways, too, a person can choose to lead a j-club topic to solidify their own knowledge on the topic, so the benefit goes both to the twice-monthly discussion leaders as well as to the general iMechanica group who find specific topics of interest and potentially learning.

Michelle