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# Journal Club Theme of April 2013: Mechanical Metamaterials

Discussion on metamaterials (particularly dynamics of phononic/acoustic metamaterials) was first initiated in Journal Club Dec. 2007 (by Dr. Banerjee) and then recently again in Journal Clubs May 2012 (by Dr. Spadoni), June 2012 (by Dr. Kochmann) and Dec. 2012 (by Dr. Ruzzene). For the last five years, the research area in mechanical metamaterials has broadened so much that it now covers various unconventional macroscopic characteristics, not only dynamic properties (e.g., band-gaps, dynamic modulus/density, etc.) but also static properties (e.g., negative modulus, ultra-low density, etc.).

For the recent development in this area, I would recommend a review paper by Lee et al. [1]. The paper summarizes several extraordinary mechanical characteristics and their fabrication techniques that have been recently reported. Unconventional functionalities can be classified into linear properties (e.g., elastic modulus-density, phononic band-gaps, Poisson’s ratio) and non-linear properties (e.g., specific energy absorption, high-strain-rate resistance). In this Journal Club, I would like to introduce a couple of related (and interesting!) topics which are not intensively covered in the review paper [1].

The so-called “extremal materials” are mechanical metamaterials characterized by the unconventional linear elastic properties. They were introduced by Milton and Cherkaev [2], and they are roughly defined as “materials for which the eigenvalues of the effective (elasticity) tensor only take (very) large or (very) small values.” The eigenmode corresponding to the very small eigenvalue is an easy mode of deformation, so the concept of nullmode, unimode, bimode, trimode, quadramode, and pentamode materials are introduced in 3-D elasticity. Using this terminology, an auxetic material having Poisson’s ratio of -1 can be viewed as a unimode material since it has substantially lower resistance for dilation than for shear deformations. So, the pentamode materials can only support a single type of stress component, and authors suggested that materials with arbitrary elasticity tensor can be developed using pentamode materials. Milton has recently published a series of papers on the characterization of these extremal materials [3, 4]. Moreover, the research group led by Wegener recently fabricated an “approximate” version of the conceptual pentamode material having negligible shear modulus [5]. In this line of research, lots of interesting results can be produced from these unconventional extremal materials in the future.

For the metamaterials characterized by their unconventional non-linear properties, it is worthwhile to mention materials having tunable properties though non-affine (i.e., non-homogeneous) transformation. In the recent paper by Li et al. [6], dramatic color-switching is achieved through non-affine mechanical pattern transformation in shape memory polymer (SMP). When membranes having hexagonal arrangement of micron-size circular holes were hot-pressed, the circular holes deformed to ellipses and eventually closed resulting in featureless surface of membranes (Fig1.A). During this procedure, thus the initial membranes with diffraction color become a transparent film. An extended version of this 2-D non-affine transformation to 3-D spherical shells was also reported by Shim et al. [7]. They introduce a class of continuum spherical shell structures patterned with a uniform arrangement of circular holes, and those structures undergo non-affine transformation induced by buckling under pressure, resulting in isotropic volume reduction (Fig1.B). Thanks to this unconventional feature, the proposed spherical structures can be used as building blocks to construct 3-D symmetric auxetic metamaterials.

Currently, investigation of phononic/acoustic metamaterials is mostly limited to small amplitudes of deformation, thus leading to linear analysis. In the future, nonlinear wave propagation analysis considering large deformation (particularly though non-affine transformation) could provide ample opportunity for practical applications (e.g. extreme loading conditions).

Fig. 1 (A) Top: Optical images of the original and deformed shape memory polymer (SMP) membranes on top of the “Penn” log. Bottom: Numerical simulation results for the SMP thermo-mechanical cycle. (B) The complete set of spherical shells patterned with uniform distribution of circular holes, leading to buckling-induced isotropic volume reduction.

**REFERENCES**

[1] J.-H. Lee, J.P. Singer and E.L. Thomas (2012) Micro-/Nanostructred mechanical metamaterials , *Advanced Materials*, **24**:4782.

[2] G.W. Milton and A.V. Cherkaev (1995) Which elasticity tensors are realizable? , *Journal of Engineering Materials and Technology*, **117**:483.

[3] G.W. Milton (2012) Complete characterization of the macroscopic deformations of periodic unimode metamaterials of rigid bars and pivots, *Journal of Mechanics and Physics of Solids*, http://dx.doi/org/10.1016/j.jmps.2012.08.011

[4] G.W. Milton (2012) Adaptable nonlinear bimode metamaterials using rigid bars, pivots, and actuators, *Journal of Mechanics and Physics of Solids*, http://dx.doi/org/10.1016/j.jmps.2012.08.012

[5] M. Kadic, T. Buckmann, N. Stenger, M. Thiel and M. Wegener (2012) On the practicability of pentamode mechanical metamaterials, *Applied Physics Letters*, **100**:191901.

[6] J. Li, J. Shim, J. Deng, J.T.B. Overvelde, X. Zhu, K. Bertoldi and S. Yang (2012) Switching photonic membranes via pattern transformation and shape memory effect , *Soft Matter*, **8**:10322.

[7] J. Shim, C. Perdigou, E.R. Chen, K. Bertoldi, P.M. Reis (2012) Buckling-induced encapsulation of structured elastic shells under pressure , *Proceedings of the National Academy of Sciences of the USA*, **109**:5978.

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## Comments

## nice post!

Jongmin - thanks for this nice review. Personally, I find the concept of "extremal materials" introduced by Milton fascinating. For sure it poses a number of questions: are we able to design robust materials characterized by such unual properties? do such material exist also in Nature? are there specfic applications that can take advantage of these properties?

## Extremal Materials

Thank you for your comment, Katia.

Do such materials exist in Nature?I think some of them exist. Like Milton suggested in his papers, auxetic materials (with ν=-1) can be viewed as one of extremal materials, so-called "unimode materials". As you pointed out in your previous post Journal Club April 2010, some ceramics and metals show auxetic behavior. But, for other types of extremal materials (e.g., pentamode materials), it would be very interesting if they can be found in Nature.For the realization of the extremal materials, the research group led by Wegener has been very active. Using dip-in direct-laser-writing optical lithography, they fabricated a practical (maybe, robust) version of pentamode materials [5], and they also recently reported how this practical variation affects the phonon band structure under particular conditions [PRB, 86:155116 (2012)]. But, I think that it is too early to say something definite for its practical applications although it has great potential.

## Re: Mechanical metamaterials

In my experience, most proposed extremal structures are unstable and have to be constrained appropriately. Unfortunately, these constraints spoil the interesting features of these structures - particularly for problems involving wave propagation. At alternative approach is to use layered media with a range of length scales (as proposed by Milton and others some time ago) to achieve extremal behavior. With the advent of cheap high resolution 3D printing, it has become easier to explore these microstructures. A couple of our attempts are shown below. These structures were printed by Hans Sutandie at the AUT University in Auckland and the designs were supervised by Prof. Sarat Sigamneni.

-- Biswajit

## interesting structures

Thank you very much sharing the nice pictures and your opinion. I have a quick question. Are these structures building blocks of the layered media, or something else? Thanks.

JS

## Re: Mechanical metamaterial structures

Sorry about the paucity of explanation re. the above structures. These are designed to correspond the structures proposed by Milton in his 2007 paper "New metamaterials with macroscopic behavior outside that of continuum elastodynamics",

Graeme W Milton 2007

New J. Phys.9359 doi:10.1088/1367-2630/9/10/359.These can be used as building blocks for hierarchical laminates and three-dimensional checkerboard structures (think phononic crystals).

I just found out that a paper has also been published on the matter (of which I am a co-author :)

"Elastic metamaterials: some initial experiences with the Milton-Willis structure based on a rapid prototyped model and a numerical analysis" by H.J. Sutandie, S. Singamneni, B. Banerji http://journalamme.polsl.pl/papers_vol55_2/58256.pdf

-- Biswajit

## Thanks for sharing your paper.

Thanks for sharing your paper. The attempt to fabricate the Milton-Willis materials was very interesting and meaningful although it did not seem to pan out nicely as you expected. Hopefully, in a few years we may get different outcome thanks to the advances in 3-D printing technology.

JS

## On specific applications of extremal metamaterials

Katia's question is important in the context of today's research funding scenario. Other than the cloaking-type applications proposed by Andy Norris and co-workers and the more recent dispersion studies (and band gap searches) by Wegener's group, few applications have been discovered for these structures.

It's a chicken and egg problem. To get funded to study these materials we need clear cut applications and to think up applications we need to study these materials first. My thought is that the onus is on unversity faculty to explore potential structures until we have a better understanding of the properties of a large class of metamaterials without thinking of appllications. Once those are known applications will follow and non-academic researchers like me are ideal for that because we interact with industry on a daily basis.

-- Biswajit

## Thanks

Very interesting topic. Thanks for sharing.

## Thanks for your interest.

Thanks for your interest.

## Thanks for sharing. I do not

Thanks for sharing. I do not understand it properly. Material is not my subject. I am working on cam follower mechanism and energy.