The Role of Curl Terms in Micromorphic Models

Table of Links

Abstract and 1. Introduction

1.1 A Polyethylene-based metamaterial for acoustic control

2 Relaxed micromorphic modelling of finite-size metamaterials

2.1 Tetragonal Symmetry / Shape of elastic tensors (in Voigt notation)

3 Dispersion curves

4 New considerations on the relaxed micromorphic parameters

4.1 Consistency of the relaxed micromorphic model with respect to a change in the unit cell’s bulk material properties

4.2 Consistency of the relaxed micromorphic model with respect to a change in the unit cell’s size

4.3 Relaxed micromorphic cut-offs

5 Fitting of the relaxed micromorphic parameters: the particular case of vanishing curvature (without Curl P and Curl P˙)

5.1 Asymptotes

5.2 Fitting

5.3 Discussion

6 Fitting of the relaxed micromorphic parameters with curvature (with Curl P)

6.1 Asymptotes and 6.2 Fitting

6.3 Discussion

7 Fitting of the relaxed micromorphic parameters with enhanced kinetic energy (with Curl P˙) and 7.1 Asymptotes

7.2 Fitting

7.3 Discussion

8 Summary of the obtained results

9 Conclusion and perspectives, Acknowledgements, and References

A Most general 4th order tensor belonging to the tetragonal symmetry class

B Coefficients for the dispersion curves without Curl P

C Coefficients for the dispersion curves with P

D Coefficients for the dispersion curves with P◦

9 Conclusion and perspectives

In addition to the fitting comparison presented in the present paper, one main result that we present here is the fitting procedure itself that has been automatized to a big extent by asking only the cut-offs and asymptotes to be imposed a priori. This has been done by imposing the exact value of the cut-offs and minimizing the asymptotes’ mean square error compared to the exact numerical values issued via Boch-Floquet analysis. The rest of the curves’ fitting follows directly.

Based on the findings of this paper, we will briefly present some insight that will give directions to the follow-up research:

• Further enhance the relaxed micromorphic model via the addition of extra microscopic degrees of freedom to increase its precision at very small wavelengths (approaching the unit cell’s size);

• Design complex large-scale meta-structures that control elastic energy using the new labyrinthine metamaterial as a basic building block. This design would not be otherwise possible due to the huge number of degrees of freedom resulting from the meshing of all the tiny elements contained in the labyrinthine unit cells;

• Study negative refraction phenomena in meta-structures including the new labyrinthine metamaterial as a basic building block;

• Design complex structures for wave control simultaneously including the different metamaterials that were characterized via the relaxed micromorphic model until now.

Acknowledgements

Angela Madeo, Gianluca Rizzi and Jendrik Voss acknowledge support from the European Commission through the funding of the ERC Consolidator Grant META-LEGO, N◦ 101001759. Angela Madeo and Gianluca Rizzi acknowledge funding from the French Research Agency ANR, “METASMART” (ANR-17CE08-0006). Patrizio Neff acknowledges support in the framework of the DFG-Priority Programme 2256 “Variational Methods for Predicting Complex Phenomena in Engineering Structures and Materials”, Neff 902/10-1, Project-No. 440935806.

References

[1] A. Aivaliotis, A. Daouadji, G. Barbagallo, D. Tallarico, P. Neff, and A. Madeo. “Microstructure-related Stoneley waves and their effect on the scattering properties of a 2D Cauchy/relaxed-micromorphic interface”. en. Wave Motion 90 (Aug. 2019). Pp. 99–120. issn: 01652125. doi: 10.1016/j.wavemoti.2019.04.003. url: https://linkinghub.elsevier.com/ retrieve/pii/S0165212518304827 (visited on 04/23/2021).

[2] A. Aivaliotis, D. Tallarico, M.-V. d’Agostino, A. Daouadji, P. Neff, and A. Madeo. “Frequency- and angle-dependent scattering of a finite-sized meta-structure via the relaxed micromorphic model”. en. Archive of Applied Mechanics 90.5 (May 2020). Pp. 1073–1096. issn: 0939-1533, 1432-0681. doi: 10.1007/s00419-019-01651-9. url: http://link.springer. com/10.1007/s00419-019-01651-9 (visited on 04/23/2021).

[3] G. Allaire. “Homogenization and two-scale convergence”. SIAM Journal on Mathematical Analysis 23.6 (1992). Pp. 1482– 1518.

[4] I. Andrianov, V. Bolshakov, V. Danishevs’kyy, and D. Weichert. “Higher order asymptotic homogenization and wave propagation in periodic composite materials”. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 464.2093 (2008). Pp. 1181–1201.

[5] A. Bacigalupo and L. Gambarotta. “Second-gradient homogenized model for wave propagation in heterogeneous periodic media”. International Journal of Solids and Structures 51.5 (2014). Pp. 1052–1065.

[6] A. Bensoussan, J. Lions, and G. Papanicolaou. Asymptotic analysis for periodic structures. Vol. 374. American Mathematical Soc., 2011.

[7] D. S. Bernstein. Matrix Mathematics: Theory, Facts, and Formulas (Second Edition). Princeton reference. Princeton University Press, 2009.

[8] O. R. Bilal, D. Ballagi, and C. Daraio. “Architected lattices for simultaneous broadband attenuation of airborne sound and mechanical vibrations in all directions”. Physical Review Applied 10.5 (2018). P. 054060.

[9] G. Bouchitt´e and M. Bellieud. “Homogenization of a soft elastic material reinforced by fibers”. Asymptotic Analysis 32.2 (2002). Pp. 153–183.

[10] C. Boutin, A. Rallu, and S. Hans. “Large scale modulation of high frequency waves in periodic elastic composites”. Journal of the Mechanics and Physics of Solids 70 (2014). Pp. 362–381.

[11] T. B¨uckmann, M. Kadic, R. Schittny, and M. Wegener. “Mechanical cloak design by direct lattice transformation”. Proceedings of the National Academy of Sciences 112.16 (2015). Pp. 4930–4934.

[12] M. Camar-Eddine and P. Seppecher. “Determination of the closure of the set of elasticity functionals”. Archive for Rational Mechanics and Analysis 170.3 (2003). Pp. 211–245.

[13] P. Celli, B. Yousefzadeh, C. Daraio, and S. Gonella. “Bandgap widening by disorder in rainbow metamaterials”. Applied Physics Letters 114.9 (2019). P. 091903.

[14] W. Chen and J. Fish. “A dispersive model for wave propagation in periodic heterogeneous media based on homogenization with multiple spatial and temporal scales”. Journal of Applied Mechanics 68.2 (2001). Pp. 153–161.

[15] R. Craster, J. Kaplunov, and A. Pichugin. “High-frequency homogenization for periodic media”. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 466.2120 (2010). Pp. 2341–2362.

[16] S. A. Cummer, J. Christensen, and A. Al`u. “Controlling sound with acoustic metamaterials”. Nature Reviews Materials 1.3 (2016). Pp. 1–13.

[17] M. V. d’Agostino, G. Barbagallo, I.-D. Ghiba, B. Eidel, P. Neff, and A. Madeo. “Effective description of anisotropic wave dispersion in mechanical band-gap metamaterials via the relaxed micromorphic model”. en. Journal of Elasticity 139.2 (May 2020). Pp. 299–329. issn: 0374-3535, 1573-2681. doi: 10.1007/s10659-019-09753-9. url: http://link.springer. com/10.1007/s10659-019-09753-9 (visited on 04/23/2021).

[18] T. Frenzel, M. Kadic, and M. Wegener. “Three-dimensional mechanical metamaterials with a twist”. Science 358.6366 (2017). Pp. 1072–1074.

[19] M. Geers, V. Kouznetsova, and M. Brekelmans. “Multi-scale computational homogenization: Trends and challenges”. Journal of Computational and Applied Mathematics 234.7 (2010). Pp. 2175–2182.

[20] S. Guenneau, A. Movchan, G. P´etursson, and S. A. Ramakrishna. “Acoustic metamaterials for sound focusing and confinement”. New Journal of physics 9.11 (2007). P. 399.

[21] Z. Hashin and S. Shtrikman. “A variational approach to the theory of the elastic behaviour of multiphase materials”. Journal of the Mechanics and Physics of Solids 11.2 (1963). Pp. 127–140.

[22] R. Hill. “Elastic properties of reinforced solids: some theoretical principles”. Journal of the Mechanics and Physics of Solids 11.5 (1963). Pp. 357–372.

[23] R. Hu and C. Oskay. “Nonlocal homogenization model for wave dispersion and attenuation in elastic and viscoelastic periodic layered media”. Journal of Applied Mechanics 84.3 (2017).

[24] N. Kaina, A. Causier, Y. Bourlier, M. Fink, T. Berthelot, and G. Lerosey. “Slow waves in locally resonant metamaterials line defect waveguides”. Scientific reports 7.1 (2017). Pp. 1–11.

[25] N. Kaina, F. Lemoult, M. Fink, and G. Lerosey. “Negative refractive index and acoustic superlens from multiple scattering in single negative metamaterials”. en. Nature 525.7567 (Sept. 2015). Pp. 77–81. issn: 0028-0836, 1476-4687. doi: 10.1038/ nature14678. url: http://www.nature.com/articles/nature14678 (visited on 04/23/2021).

[26] A. O. Krushynska, M. Miniaci, F. Bosia, and N. M. Pugno. “Coupling local resonance with Bragg band gaps in single-phase mechanical metamaterials”. Extreme Mechanics Letters 12 (2017). Pp. 30–36.

[27] R. Lakes. “Foam structures with a negative Poisson’s ratio”. Science 235.4792 (1987). Pp. 1038–1040. [28] Z. Liu, X. Zhang, Y. Mao, Y. Zhu, Z. Yang, C. T. Chan, and P. Sheng. “Locally resonant sonic materials”. science 289.5485 (2000). Pp. 1734–1736.

[29] C. Miehe, J. Schr¨oder, and J. Schotte. “Computational homogenization analysis in finite plasticity simulation of texture development in polycrystalline materials”. Computer Methods in Applied Mechanics and Engineering 171.3-4 (1999). Pp. 387–418.

[30] G. Milton. “The Theory of Composites. 2002”. Cambridge Monographs on Applied and Computational Mathematics (2002).

[31] D. Misseroni, D. J. Colquitt, A. B. Movchan, N. V. Movchan, and I. S. Jones. “Cymatics for the cloaking of flexural vibrations in a structured plate”. Scientific reports 6.1 (2016). Pp. 1–11.

[32] P. Neff, I. D. Ghiba, M. Lazar, and A. Madeo. “The relaxed linear micromorphic continuum: well-posedness of the static problem and relations to the gauge theory of dislocations”. en. The Quarterly Journal of Mechanics and Applied Mathematics 68.1 (Feb. 2015). Pp. 53–84. issn: 0033-5614, 1464-3855. doi: 10 . 1093 / qjmam / hbu027. url: https : / / academic.oup.com/qjmam/article-lookup/doi/10.1093/qjmam/hbu027 (visited on 04/25/2021).

[33] P. Neff, B. Eidel, M. V. d’Agostino, and A. Madeo. “Identification of scale-independent material parameters in the relaxed micromorphic model through model-adapted first order homogenization”. en. Journal of Elasticity 139.2 (May 2020). Pp. 269–298. issn: 0374-3535, 1573-2681. doi: 10.1007/s10659-019-09752-w. url: http://link.springer.com/10.1007/ s10659-019-09752-w (visited on 04/23/2021).

[34] P. Neff, I.-D. Ghiba, A. Madeo, L. Placidi, and G. Rosi. “A unifying perspective: the relaxed linear micromorphic continuum”. en. Continuum Mechanics and Thermodynamics 26.5 (Sept. 2014). Pp. 639–681. issn: 0935-1175, 1432-0959. doi: 10.1007/s00161-013-0322-9. url: http://link.springer.com/10.1007/s00161-013-0322-9 (visited on 04/23/2021).

[35] C. Pideri and P. Seppecher. “A second gradient material resulting from the homogenization of an heterogeneous linear elastic medium”. Continuum Mechanics and Thermodynamics 9.5 (1997). Pp. 241–257.

[36] G Rizzi, F Dal Corso, D Veber, and D Bigoni. “Identification of second-gradient elastic materials from planar hexagonal lattices. Part II: Mechanical characteristics and model validation”. International Journal of Solids and Structures 176 (2019). Pp. 19–35.

[37] G. Rizzi, M. Collet, F. Demore, B. Eidel, P. Neff, and A. Madeo. “Exploring metamaterials’ structures through the relaxed micromorphic model: switching an acoustic screen into an acoustic absorber”. Frontiers in Materials 7 (Mar. 2021). P. 589701. issn: 2296-8016. doi: 10.3389/fmats.2020.589701. url: https://www.frontiersin.org/articles/10. 3389/fmats.2020.589701/full (visited on 04/23/2021).

[38] G. Rizzi, M. V. d’Agostino, P. Neff, and A. Madeo. “Boundary and interface conditions in the relaxed micromorphic model: exploring finite-size metastructures for elastic wave control”. (arXiv:2105.00963) to appear in Mathematics and Mechanics of Solids (2021).

[39] G. Rizzi, P. Neff, and A. Madeo. “Metamaterial shields for inner protection and outer tuning through a relaxed micromorphic approach”. arXiv preprint arXiv:2111.12001 (2021).

[40] E. S´anchez-Palencia. “Non-homogeneous media and vibration theory”. Lecture Notes in Physics 127 (1980).

[41] A. Sridhar, V. Kouznetsova, and M. Geers. “A general multiscale framework for the emergent effective elastodynamics of metamaterials”. Journal of the Mechanics and Physics of Solids 111 (2018). Pp. 414–433. [42] A. Srivastava and S. Nemat-Nasser. “On the limit and applicability of dynamic homogenization”. Wave Motion 51.7 (2014). Pp. 1045–1054.

[43] A. Srivastava and J. R. Willis. “Evanescent wave boundary layers in metamaterials and sidestepping them through a variational approach”. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473.2200 (2017). P. 20160765.

[44] P. Suquet. “Elements of Homogenization for Inelastic Solid Mechanics, Homogenization Techniques for Composite Media”. Lecture Notes in Physics 272 (1985). P. 193.

[45] D. Tallarico, A. Trevisan, N. V. Movchan, and A. B. Movchan. “Edge waves and localization in lattices containing tilted resonators”. Frontiers in Materials 4 (2017). P. 16.

[46] P. Wang, F. Casadei, S. Shan, J. C. Weaver, and K. Bertoldi. “Harnessing buckling to design tunable locally resonant acoustic metamaterials”. Physical review letters 113.1 (2014). P. 014301.

[47] J. R. Willis. “Negative refraction in a laminate”. en. Journal of the Mechanics and Physics of Solids 97 (Dec. 2016). Pp. 10–18. issn: 00225096. doi: 10.1016/j.jmps.2015.11.004. url: https://linkinghub.elsevier.com/retrieve/pii/ S0022509615302623 (visited on 04/23/2021).

[48] J. Willis. “Bounds and self-consistent estimates for the overall properties of anisotropic composites”. Journal of the Mechanics and Physics of Solids 25.3 (1977). Pp. 185–202.

[49] J. Willis. “Exact effective relations for dynamics of a laminated body”. Mechanics of Materials 41.4 (2009). Pp. 385–393.

[50] J. Willis. “Effective constitutive relations for waves in composites and metamaterials”. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 467.2131 (2011). Pp. 1865–1879.

[51] J. Willis. “The construction of effective relations for waves in a composite”. Comptes Rendus M´ecanique 340.4-5 (2012). Pp. 181–192.

[52] R. Zhu, X. N. Liu, G. K. Hu, C. T. Sun, and G. L. Huang. “Negative refraction of elastic waves at the deep-subwavelength scale in a single-phase metamaterial”. en. Nature Communications 5.1 (Dec. 2014). P. 5510. issn: 2041-1723. doi: 10. 1038/ncomms6510. url: http://www.nature.com/articles/ncomms6510 (visited on 04/23/2021).

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