**The purpose of this website (META-MAT.ORG) is to disseminate research results in acoustic, thermal and mechanical metamaterials through the organization of online seminars, symposia, conferences, summer schools and workshops with industry and the academic world.**

**In 2001, the word “metamaterial” was coined by R. M. Walser [WALSER 01], who gave the following definition: macroscopic composite having a manmade, three-dimensional, periodic cellular architecture designed to produce an optimized combination, not available in nature, of two or more responses to specific excitation. Since then, the definition of metamaterials has been polished to encompass all kinds of waves and a generally accepted definition is that proposed by M. Wegener and coauthors for which metamaterials are rationally designed composites made of tailored building blocks, which are composed of one or more constituent bulk materials, leading to effective medium properties beyond those of their ingredients [KADIC 19]. The concept of metamaterial is rooted in the nano-scale world and electromagnetism, with the seminal works of V. Veselago [VESELAGO 68] and J. Pendry [PENDRY 99, PENDRY 00] that revolutionized the way researchers model light propagation in complex media, where the refraction index can take any value on the real line, or actually in the complex plane. Indeed metamaterials can have refraction index with a positive or negative real part, and same for their imaginary part, thanks to the concepts of passive and active media, in accordance with the Kramers-Kronig relations [GRALAK 10]. The refraction index need not be a complex number, it can also be a complex valued tensor describing anisotropic heterogeneous media with sign-shifting phases [MILTON 02, SMITH 04, CAI 10, ZHELUDEV 12]. The fine structure of negatively refracting media usually consists of periodic arrangements of elements with size much smaller than the considered wavelength (typically hundreds of nanometers) that acquire effective properties of materials with negative optical index [PEN 00], or highly anisotropic materials such as hyperbolic metamaterials [IOR 13, PODDUBNY 13] or invisibility cloaking devices [PEN 06], which are based on the form invariance of Maxwell’s equations that behave nicely under coordinate changes [NICOLET 94], unlike the Navier equations [MILTON 06a], which are transformed into so-called Willi’s equations [WILLIS 81], which go beyond Newton’s second law [MILTON 07]. Two other cloaking techniques, via scattering cancellation [ALU 05] and anomalous resonances [MILTON 06b] have also generated a huge interest in the applied mathematics community. The transition from the electromagnetic to acoustic metamaterials was made in particular possible thanks to phononic crystals, which are artificial handcrafted structures. They range from a few meters down to hundreds of nanometers or less. At this scale, matter appears as continuous and the laws of classical mechanics can be applied. The search for structures with complete phononic band gaps began in 1992 with work by Sigalas and Economou [ECONOMOU 93]. Interestingly, as unveiled in [OBRIEN 02] high-permittivity dielectric rods display stop bands induced by low frequency localized modes, which are associated with artificial magnetism. In a similar way, high-density rods and spheres display unique features upon resonance, such as a negative effective density [LIU 00]. Just like the refractive index, the effective density need not be a real number, but is, in general, a complex valued matrix, as can be seen in mechanical metamaterials [CHRISTENSEN 15, ACHAOUI 16, MINIACI 16, BERTOLDI 17, KADIC 19]**.

**Perhaps more surprisingly, the analogies drawn between electromagnetic, acoustic and mechanical waves propagating in complex media, have been further stretched to encompass heat, mass and light diffusion phenomena [GUENNEAU 13, SCHITTNY 13, SCHITTNY 14].**

References: https://en.wikipedia.org/wiki/Metamaterial

[ACHAOUI 16] Achaoui Y., Ungureanu B., Brûlé S., Enoch S. & Guenneau S., “Seismic wave damping with arrays of inertial resonators”, *Extreme Mechanics Letters*, **8**, 30-37, 2016 (http://dx.doi.org/10.1016/j.eml.2016.02.004).

[ALU 05] Alù A. & Engheta N., “Achieving transparency with plasmonic and metamaterial coatings”, *Physical Review E*, **72** (1), 016623, 2005.

[BERTOLDI 17] Bertoldi K., Vitelli V., Christensen J. & van Hecke M., “Flexible mechanical metamaterials”. *Nature Reviews Materials*, **2** (11), 1-11, 2017.

[CAI 10] Cai W. & Shalaev V. M., “Optical metamaterials”, *New York: Springer*, **10** (6011), 2010.

[CHRISTENSEN 15] Christensen J., Kadic M., Kraft O. & Wegener M., “Vibrant times for mechanical metamaterials”, *Mrs Communications*, **5** (3), 453-462, 2015.

[CRASTER 12] Craster R. V. & Guenneau S., “Acoustic metamaterials: Negative refraction, imaging, lensing and cloaking” (Vol. 166). *Springer Science & Business Media*, Eds. 2012.

[CUI 10] Cui T. J., Smith D. R. & Liu R., “Metamaterials” (p. 1). *Spring Street, NY: springer*, 2010.

[ECONOMOU 93] Economou E.N. & Sigalas M.M., “Classical wave propagation in periodic structures: Cermet versus network topology”, *Phys. Rev.* *B*, **48**, 13434-13438, 1993.

[GRALAK 10] Gralak B. & Tip A., “Macroscopic Maxwell’s equations and negative index media”, *Journal of mathematical physics*, **51** (5), 052902, 2010.

[GUENNEAU 13] Guenneau S. & Puvirajesinghe T.M., “Fick’s second law transformed: one path to cloaking in mass diffusion”, *Journal of The Royal Society Interface*, **10** (83), 20130106, 2013.

[IOR 13] Iorch I.V., Mukhin I.S., Shadrivov I.V., Belov P.A. & Kivshar Y.S., “Hyperbolic metamaterials based on multilayer graphene structures”, *Phys. Rev.* *B*,** 87**, 075416-6, 2013.

[KADIC 19] Kadic M., Milton G. W., van Hecke M. & Wegener M., “3D metamaterials”. *Nature Reviews Physics*, **1** (3), 198-210, 2019.

[LIU 00] Liu Z., Zhang X., Mao Y., Zhu Y., Yang Z., Chan C. T. & Sheng P. “Locally Resonant Sonic Materials”, *Science*, **289**, 1734, 2000.

[MILTON 02] Milton G.W., “The Theory of Composites” *(Cambridge Monographs on Applied and Computational Mathematics). Cambridge: Cambridge University Press*, 2002, doi:10.1017/CBO9780511613357

[MILTON 06a] Milton G.W., Briane M. & Willis J.R., “On cloaking for elasticity and physical equations with a transformation invariant form”, *New Journal of Physics*, **8** (10), 248, 2006.

[MILTON 06b] Milton G.W. & Nicorovici N.A.P., “On the cloaking effects associated with anomalous localized resonance”, *Proceedings of the Royal Society A, ***462**, 3027-3059, 2006.

[MILTON 07] Milton G.W. & Willis J.R., “On modifications of Newton’s second law and linear continuum elastodynamics”, *Proceedings of the Royal Society A*, **463**, 855-880, 2007.

[MINIACI 16] Miniaci M., Krushynska A., Movchan A. B., Bosia F. & Pugno N. M., “Spider web-inspired metamaterials”, *Appl. Phys. Lett.* 109, 071905, 2016.

[NICOLET 94] Nicolet A., Remacle J.F., Meys B., Genon A., Legros W., “Transformation methods in computational electromagnetics”,* Journal of Applied Physics*, **75**, 10, 6036-6038, 1994.

[OBRIEN 02] O’brien S. & Pendry J.B., “Photonic band-gap effects in dielecrtric composites”, *J. Phys. Cond. Matt.* **14**, 4035-4044, 2002.

[PENDRY 99] Pendry J.B., Holden A.J., Robbins D.J. & Stewart W.J., “Magnetism from conductors and enhanced nonlinear phenomena”, *IEEE Transactions on Microwave Theory and Techniques* **47** (11): 2075, 1999.

[PENDRY 00] Pendry J. B., “Negative refraction makes a perfect lens”, *Phys. Rev. Lett.* **85**, 3966-3969, 2000.

[PENDRY 06] Pendry J.B., Schurig D. & Smith D.R., “Controlling Electromagnetic Fields”, *Science* **312** (5781): 1789-1782, 2006.

[PODDUBNY 13] Poddubny A., Iorsh I., Belov P. & Kivshar, Y., “Hyperbolic metamaterials”. *Nature photonics*, **7** (12), 948, 2013.

[SCHITTNY 13] Schittny R., Kadic M., Guenneau S. & Wegener M., “Experiments on transformation thermodynamics: molding the flow of hea”, *Physical review letters*, **110** (19), 195901, 2013.

[SCHITTNY 14] Schittny R., Kadic M., Bückmann T. & Wegener M., “Invisibility cloaking in a diffusive light scattering medium”, *Science ***345** (6195), 427-429, 2014.

[SMITH 04] Smith D. R., Pendry J. B. & Wiltshire M. C., “Metamaterials and negative refractive index”. *Science*, **305** (5685), 788-792, 2004.

[VESELAGO 68] Veselago V.G., “The Electrodynamics of substances with simultaneously negative values of ε and μ”, *Soviet Physics Uspekhi* **10** (4), 509-514, 1968.

[WALSER 01] Walser R.M., “Electromagnetic metamaterials”, paper presented at the *International Society for Optical Engineering (SPIE)*, **4467**, pp. 1-165, 2001.

[WILLIS 81] Willis J.R., “Variational and related methods for the overall properties of composites”, *Advances in applied mechanics*, **21**, 1-78, 1981.

[ZHELUDEV 12] Zheludev N. I. & Kivshar Y. S., “From metamaterials to metadevices”. *Nature materials*, **11** (11), 917-924, 2012.