Engineering metamaterials showing unorthodox behaviors with respect to wave propagation are recently attracting growing attention for what concerns both modeling and experiments. They are conceived arranging small components into periodic or quasi-periodic patterns in such a way that the resulting structure possesses new incredible properties with respect to the original material.

Indeed, innumerable experimental evidences are today available which support the fact that engineering microstructured materials can inhibit wave propagation in particular frequency ranges. Such inhibition of wave propagation typically intervenes for precise frequency ranges which are known as “frequency band-gaps”.

Numerous modeling efforts are currently made trying to account for the observed band-gaps in a reliable manner. The most common models are intrinsically microscopic and are based on the use of Bloch's theorem for periodic microstructures (see [1] among many others) or on numerical homogeneization techniques (see [2]). Nevertheless, to the authors’ knowledge, a systematic treatment of band-gap modeling in the spirit of Generalized Continuum Mechanics is still lacking and deserves attention.

The idea of using generalized continuum theories to describe microstructured materials needs to be fully developed in order to achieve a relatively simplified modeling and more effective conception of large-scale engineering “metastructures” made up of metamaterials as building blocks. This would allow for the design of real, large-scale engineering structures which are able to resist to vibrations and shocks in a large range of frequencies

Theoretical models accounting for any single element of such metastructures rapidly show their limits both in terms of complexity and computational performances. For this reason, we propose to develop and use a new Generalized Continuum Model, called “Relaxed Multimorphic”, allowing to describe the behavior of metastructures in the simplified framework of continuum mechanics. The constitutive macroscopic parameters of our model will be identified on real metamaterials, opening the way to the efficient design and realization of fascinating engineering metastructures.


Based on the existing state of the art, the real challenge in the field of band gap metamaterials is of course not that of finding the microstructures which give rise to band-gap behaviors. Band-gap metamaterials already exist and different patents have been deposited on this subject. Everyday, specialized researchers find new microstructures which give rise to new metamaterials with performances which are each time more astonishing (see e.g. [3-5]).

Hence, deciding to focus our attention on this particular aspect of the problem would not make of us the only ones who are able to provide a unique expertise.

The real topical point is that of being able to set up and subsequently improve simplified models which allow to characterize band-gap metamaterials by means of the most limited possible number of averaged parameters. In this way, the conception of complex metastructures would become accessible with limited computational costs.

Based on our expertise, we believe to be the ones who can think to develop such new generalized continuum models which are sophisticated enough to characterize the desired band-gap metamaterials from a mechanical point of view.

Once such models will be developed and used to globally characterize the mechanical behavior of specific metamaterials, we will be capable of doing something that today is not conceivable: design morphologically complex structures composed of metamaterials which are globally able to inhibit wave propagation.

Of course, experimental tests will be needed during our journey to aid in the calibration of the new theoretical model that we propose to use to reach our cardinal goal. Although fundamental, this experimental assessment is not the original part of the work, since traditional experimental tools in dynamical material characterization are needed to perform the desired work. A Research Engineer who possesses the needed skills will be eventually hired in the framework of our ERC-STG project in order to proceed in this direction. Moreover, the experimental expertise of LGCIE SMS-ID at INSA-Lyon (Ali Daouadji [11]) can be an added value to accelerate the needed experimental campaigns and to eventually give a deeper insight into the possible micro-motions occurring in the considered metamaterials.

The seminal original ideas which are at the basis of this ERC-STG proposal can be found in [6-10]. The relaxed micromorphic model presented therein, if already capable to describe some aspects of band-gaps metamaterials, needs to be suitably generalized to be able to catch more realistic metamaterials behaviors. Such generalizations will bring us to the conception of the new "relaxed multimorphic model" which will be perfected in close collaboration with Patrizio Neff (University of Duisburg-Essen, Germany).


[1] A. Spadoni, M. Ruzzene, S. Gonella, F. Scarpa, (2009). “Phononic properties of hexagonal chiral lattices”. Wave Motion, Vol. 46, pp. 435-450. See the article

[2] K. Pham, V.G. Kouznetsova, M.G.D. Geers, (2013). “Transient computational homogenization for heterogeneous materials under dynamic excitation”. J. Mech. Phys. Solids, Vol. 61, pp. 2125-2146. See the article

[3] W. Man, M. Florescu, K. Matsuyama, P. Yadak, G. Nahal, S. Hashemizad, E. Williamson, P. Steinhardt, S. Torquato, P. Chaikin, (2013). “Photonic band gap in isotropic hyperuniform disordered solids with low dielectric contrast”. Optics Express, Vol. 21:17, pp. 19972-19981. See the article

[4] Z. Liu, X. Zhang, Y. Mao, Y. Zhu, Z. Yang, C. Chan, P. Sheng, (2000b). “Locally resonant sonic materials”. Science Vol. 289 (5485), pp. 1734-1736. See the article

[5] R. Lucklum, M. Ke, M. Zubtsov, (2012). “Two-dimensional phononic crystal sensor based on a cavity mode”. Sensors and Actuators B, Vol. 171–172, pp. 271–277. See the article

[6] A. Madeo, P. Neff, I.-D. Ghiba, L. Placidi and G. Rosi (2014). “Band gaps in the relaxed linear micromorphic continuum”. Z. Angew. Math. Mech. (ZAMM), Vol. 27, 551-570. See the article

[7] A. Madeo, P. Neff, I.D. Ghiba, L. Placidi, G. Rosi (2015). “Wave propagation in relaxed micromorphic continua: modeling metamaterials with frequency band gaps”. Continuum Mechanics and Thermodynamics, Vol. 27, pp. 551-570. See the article

[8] P. Neff, I.D. Ghiba, A. Madeo, L. Placidi, G. Rosi (2014). “A unifying perspective: the relaxed linear micromorphic continuum”. Continuum Mechanics and Thermodynamics, Vol. 26, pp. 639-681. See the article

[9] I.D. Ghiba, P. Neff, A. Madeo, L. Placidi, G. Rosi, (2015). “The relaxed linear micromorphic continuum: existence, uniqueness and continuous dependence in dynamics”. Mathematics and Mechanics of Solids, Vol. 20:10, pp. 1171-1197. See the article

[10] P. Neff, D. Pauly, K.J. Witsch, (2015). “Poincaré meets Korn via Maxwell: Extending Korn's First Inequality to Incompatible Tensor Fields”. J. Diff. Equations, Volume 258 (4), pp. 1267–1302. See the article

[11] G. Robin, M. Jrad, N. Mathieu, A. Daouadji, M. Daya, (2015). “Vibration analysis of corrugated beams: The effects of temperature and corrugation shape”. Mechanics Research Communications. Accepted.