Comptes Rendus
Introduction to mechanical metamaterials and their effective properties
Comptes Rendus. Physique, Volume 21 (2020) no. 7-8, pp. 751-765.

Metamaterials are rationally designed composites made of building blocks which are composed of one or more constituent materials. Metamaterial properties can go beyond those of the ingredient materials, both qualitatively and quantitatively. In addition, their properties can be mapped on some generalized continuum model. We present the general procedure of designing elastic metamaterials based on masses and springs. We show that using this simple approach we can design any set of effective properties including linear elastic metamaterials,—defined by bulk modulus, shear modulus, mass density—and nonlinear metamaterials,—with instabilities or programmable parts. We present designs and corresponding numerical calculations to illustrate their constitutive behavior. Finally, we discuss the addition of a thermal stimulus to mechanical metamaterials.

Les métamatériaux sont des composites de conception rationnelle constitués de briques élémentaires qui sont composées d’un ou plusieurs matériaux constitutifs. Les propriétés des métamatériaux peuvent aller au-delà de celles des matériaux constitutifs, à la fois qualitativement et quantitativement. En outre, leurs propriétés peuvent être mises en correspondance avec certains modèles de milieux continus généralisés. Nous présentons une procédure générale de conception de métamatériaux élastiques à base de systèmes de type masses et de ressorts. Nous montrons quavec cette approche simple, nous pouvons concevoir tout un ensemble de propriétés effectives, y compris celles de métamatériaux élastiques non linéaires avec instabilités ou parties programmables — définis par un module de masse, de cisaillement et une masse volumique. Nous présentons des designs et calculs numériques afin dillustrer les lois de comportement. Enfin, nous discutons de l’apport d’un stimulus thermique aux métamatériaux mécaniques.

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DOI: 10.5802/crphys.30
Keywords: Metamaterials, Effective parameters, Elasticity, Anisotropy, Waves, Cauchy elasticity, Navier equation
Mot clés : Métamatériaux, Paramètres effectifs, Élasticité, Anisotropie, Ondes, Élasticité de Cauchy, Équation de Navier

Xueyan Chen 1, 2; Nicolas Laforge 1; Qingxiang Ji 1, 2; Huifeng Tan 2; Jun Liang 3, 2; Gwenn Ulliac 1; Johnny Moughames 1; Samia Adrar 1; Vincent Laude 1; Muamer Kadic 1

1 Institut FEMTO-ST, UMR 6174, CNRS, Université de Bourgogne Franche-Comté, 25000 Besançon, France
2 National Key Laboratory of Science and Technology on Advanced Composites in Special Environments, Harbin Institute of Technology; 92 Xidazhi Street, Harbin, 150001, PR China
3 Institute of Advanced Structure Technology, Beijing Institute of Technology, No. 5 South Zhongguancun Street, Haidian District, Beijing, 100081, PR China
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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     title = {Introduction to mechanical metamaterials and their effective properties},
     journal = {Comptes Rendus. Physique},
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Xueyan Chen; Nicolas Laforge; Qingxiang Ji; Huifeng Tan; Jun Liang; Gwenn Ulliac; Johnny Moughames; Samia Adrar; Vincent Laude; Muamer Kadic. Introduction to mechanical metamaterials and their effective properties. Comptes Rendus. Physique, Volume 21 (2020) no. 7-8, pp. 751-765. doi : 10.5802/crphys.30. https://comptes-rendus.academie-sciences.fr/physique/articles/10.5802/crphys.30/

[1] M. J. Allen; V. C. Tung; R. B. Kaner Honeycomb carbon: a review of graphene, Chem. Rev., Volume 110 (2009) no. 1, pp. 132-145 | DOI

[2] Y. Shao; J. Wang; H. Wu; J. Liu; I. A. Aksay; Y. Lin Graphene based electrochemical sensors and biosensors: a review, Electroanalysis, Volume 22 (2010) no. 10, pp. 1027-1036 | DOI

[3] M. Yi; Z. Shen A review on mechanical exfoliation for the scalable production of graphene, J. Mater. Chem. A, Volume 3 (2015) no. 22, pp. 11700-11715 | DOI

[4] G. Mittal; V. Dhand; K. Y. Rhee; S.-J. Park; W. R. Lee A review on carbon nanotubes and graphene as fillers in reinforced polymer nanocomposites, J. Ind. Eng. Chem., Volume 21 (2015), pp. 11-25 | DOI

[5] A. D. Moghadam; E. Omrani; P. L. Menezes; P. K. Rohatgi Mechanical and tribological properties of self-lubricating metal matrix nanocomposites reinforced by carbon nanotubes (CNTs) and graphene–a review, Composites B, Volume 77 (2015), pp. 402-420 | DOI

[6] J. J. Carruthers; A. Kettle; A. Robinson Energy absorption capability and crashworthiness of composite material structures: a review, Appl. Mech. Rev., Volume 51 (1998) no. 10, pp. 635-649 | DOI

[7] D. Liu; Y. Tang; W. Cong A review of mechanical drilling for composite laminates, Compos. Struct., Volume 94 (2012) no. 4, pp. 1265-1279 | DOI

[8] R. F. Gibson A review of recent research on mechanics of multifunctional composite materials and structures, Compos. Struct., Volume 92 (2010) no. 12, pp. 2793-2810 | DOI

[9] X. Yu; J. Zhou; H. Liang; Z. Jiang; L. Wu Mechanical metamaterials associated with stiffness, rigidity and compressibility: A brief review, Prog. Mater. Sci., Volume 94 (2018), pp. 114-173 | DOI

[10] A. Srivastava Elastic metamaterials and dynamic homogenization: a review, Int. J. Smart Nano Mater., Volume 6 (2015) no. 1, pp. 41-60 | DOI

[11] K. Bertoldi; V. Vitelli; J. Christensen; M. van Hecke Flexible mechanical metamaterials, Nat. Rev. Mater., Volume 2 (2017) no. 11, p. 17066 | DOI

[12] J.-H. Lee; J. P. Singer; E. L. Thomas Micro-/nanostructured mechanical metamaterials, Adv. Mater., Volume 24 (2012) no. 36, pp. 4782-4810 | DOI

[13] E. Yablonovitch Photonic band-gap structures, J. Opt. Soc. Am. B, Volume 10 (1993) no. 2, pp. 283-295 | DOI

[14] E. Yablonovitch; T. Gmitter; K.-M. Leung Photonic band structure: The face-centered-cubic case employing nonspherical atoms, Phys. Rev. Lett., Volume 67 (1991) no. 17, p. 2295 | DOI

[15] H. J. Monkhorst; J. D. Pack Special points for Brillouin-zone integrations, Phys. Rev. B, Volume 13 (1976) no. 12, p. 5188 | DOI | MR

[16] P. E. Blöchl; O. Jepsen; O. K. Andersen Improved tetrahedron method for Brillouin-zone integrations, Phys. Rev. B, Volume 49 (1994) no. 23, p. 16223 | DOI

[17] A. C. Eringen; E. S. Suhubi; S. Cowin Elastodynamics (volume 1, finite motions), J. Appl. Mech., Volume 42 (1975), p. 748 | DOI

[18] G. A. Maugin Applications of an energy-momentum tensor in nonlinear elastodynamics: Pseudomomentum and eshelby stress in solitonic elastic systems, J. Mech. Phys. Solids, Volume 40 (1992) no. 7, pp. 1543-1558 | DOI | MR | Zbl

[19] J. Achenbach Wave Propagation in Elastic Solids, Vol. 16, Elsevier, Amsterdam, The Netherlands, 2012

[20] G. W. Milton; J. R. Willis On modifications of Newton’s second law and linear continuum elastodynamics, Proc. R. Soc. A, Volume 463 (2007) no. 2079, pp. 855-880 | DOI | MR | Zbl

[21] M. Kadic; G. W. Milton; M. van Hecke; M. Wegener 3D metamaterials, Nat. Rev. Phys., Volume 1 (2019) no. 3, pp. 198-210 | DOI

[22] P. Martinsson; A. Movchan Vibrations of lattice structures and phononic band gaps, Q. J. Mech. Appl. Math., Volume 56 (2003) no. 1, pp. 45-64 | DOI | MR

[23] D. Colquitt; I. Jones; N. Movchan; A. Movchan Dispersion and localization of elastic waves in materials with microstructure, Proc. R. Soc. A, Volume 467 (2011) no. 2134, pp. 2874-2895 | DOI | MR | Zbl

[24] A. Piccolroaz; A. Movchan Dispersion and localisation in structured Rayleigh beams, Int. J. Solids Struct., Volume 51 (2014) no. 25-26, pp. 4452-4461 | DOI

[25] A. N. Norris Low-frequency dispersion and attenuation in partially saturated rocks, J. Acoust. Soc. Am., Volume 94 (1993) no. 1, pp. 359-370 | DOI

[26] C. Findeisen; J. Hohe; M. Kadic; P. Gumbsch Characteristics of mechanical metamaterials based on buckling elements, J. Mech. Phys. Solids, Volume 102 (2017), pp. 151-164 | DOI | MR

[27] G. W. Milton; A. V. Cherkaev Which elasticity tensors are realizable?, J. Eng. Mater. Technol., Volume 117 (1995) no. 4, pp. 483-493 | DOI

[28] B. Banerjee An Introduction to Metamaterials and Waves in Composites, CRC Press, Boca Raton, Florida, USA, 2011 | DOI

[29] M. Kadic; T. Bückmann; N. Stenger; M. Thiel; M. Wegener On the practicability of pentamode mechanical metamaterials, Appl. Phys. Lett., Volume 100 (2012) no. 19, 191901 | DOI

[30] M. Kadic; T. Bückmann; R. Schittny; M. Wegener Metamaterials beyond electromagnetism, Rep. Prog. Phys., Volume 76 (2013) no. 12, 126501 | DOI

[31] T. Bückmann; R. Schittny; M. Thiel; M. Kadic; G. W. Milton; M. Wegener On three-dimensional dilational elastic metamaterials, New J. Phys., Volume 16 (2014) no. 3, 033032 | DOI

[32] T. Bückmann; N. Stenger; M. Kadic; J. Kaschke; A. Frölich; T. Kennerknecht; C. Eberl; M. Thiel; M. Wegener Tailored 3D mechanical metamaterials made by dip-in direct-laser-writing optical lithography, Adv. Mater., Volume 24 (2012) no. 20, pp. 2710-2714 | DOI

[33] T. Frenzel; M. Kadic; M. Wegener Three-dimensional mechanical metamaterials with a twist, Science, Volume 358 (2017) no. 6366, pp. 1072-1074 | DOI

[34] I. Fernandez-Corbaton; C. Rockstuhl; P. Ziemke; P. Gumbsch; A. Albiez; R. Schwaiger; T. Frenzel; M. Kadic; M. Wegener New twists of 3D chiral metamaterials, Adv. Mater., Volume 31 (2019) no. 26, 1807742 | DOI

[35] R. Gümrük; R. Mines Compressive behaviour of stainless steel micro-lattice structures, Int. J. Mech. Sci., Volume 68 (2013), pp. 125-139 | DOI

[36] T. Tancogne-Dejean; D. Mohr Elastically-isotropic truss lattice materials of reduced plastic anisotropy, Int. J. Solids Struct., Volume 138 (2018), pp. 24-39 | DOI

[37] T. Tancogne-Dejean; D. Mohr Stiffness and specific energy absorption of additively-manufactured metallic bcc metamaterials composed of tapered beams, Int. J. Mech. Sci., Volume 141 (2018), pp. 101-116 | DOI

[38] V. S. Deshpande; N. A. Fleck; M. F. Ashby Effective properties of the octet-truss lattice material, J. Mech. Phys. Solids, Volume 49 (2001) no. 8, pp. 1747-1769 | DOI | Zbl

[39] T. Frenzel; C. Findeisen; M. Kadic; P. Gumbsch; M. Wegener Tailored buckling microlattices as reusable light-weight shock absorbers, Adv. Mater., Volume 28 (2016) no. 28, pp. 5865-5870 | DOI

[40] T. Tancogne-Dejean; A. B. Spierings; D. Mohr Additively-manufactured metallic micro-lattice materials for high specific energy absorption under static and dynamic loading, Acta Mater., Volume 116 (2016), pp. 14-28 | DOI

[41] X. Cao; S. Duan; J. Liang; W. Wen; D. Fang Mechanical properties of an improved 3D-printed rhombic dodecahedron stainless steel lattice structure of variable cross section, Int. J. Mech. Sci., Volume 145 (2018), pp. 53-63 | DOI

[42] S. C. Han; J. W. Lee; K. Kang A new type of low density material: Shellular, Adv. Mater., Volume 27 (2015) no. 37, pp. 5506-5511 | DOI

[43] C. Bonatti; D. Mohr Smooth-shell metamaterials of cubic symmetry: Anisotropic elasticity, yield strength and specific energy absorption, Acta Mater., Volume 164 (2019), pp. 301-321 | DOI

[44] B. Florijn; C. Coulais; M. van Hecke Programmable mechanical metamaterials, Phys. Rev. Lett., Volume 113 (2014) no. 17, 175503 | DOI

[45] L. J. Gibson; M. F. Ashby Cellular Solids: Structure and Properties, Cambridge University Press, Cambridge, UK, 1999

[46] T. Tancogne-Dejean; M. Diamantopoulou; M. B. Gorji; C. Bonatti; D. Mohr 3D plate-lattices: An emerging class of low-density metamaterial exhibiting optimal isotropic stiffness, Adv. Mater., Volume 30 (2018) no. 45, 1803334

[47] J. Berger; H. Wadley; R. McMeeking Mechanical metamaterials at the theoretical limit of isotropic elastic stiffness, Nature, Volume 543 (2017) no. 7646, p. 533 | DOI

[48] V. Deshpande; M. Ashby; N. Fleck Foam topology: bending versus stretching dominated architectures, Acta Mater., Volume 49 (2001) no. 6, pp. 1035-1040 | DOI

[49] R. S. Lakes Viscoelastic Solids, Vol. 9, CRC Press, Boca Raton, Florida, USA, 1998

[50] R. Christensen Theory of Viscoelasticity: An Introduction, Academic Press Inc., New York, USA, 2012

[51] C. M. Zener; S. Siegel Elasticity and anelasticity of metals, J. Phys. Chem., Volume 53 (1949) no. 9, p. 1468-1468 | DOI

[52] X. Chen; Q. Ji; J. Wei; H. Tan; J. Yu; P. Zhang; V. Laude; M. Kadic Light-weight shell-lattice metamaterials for mechanical shock absorption, Int. J. Mech. Sci., Volume 169 (2020), 105288 | DOI

[53] G. Lu; T. Yu Energy Absorption of Structures and Materials, Woodhead Publishing Limited, Cambridge, UK, 2003

[54] L. Salari-Sharif; T. A. Schaedler; L. Valdevit Energy dissipation mechanisms in hollow metallic microlattices, J. Mater. Res., Volume 29 (2014) no. 16, pp. 1755-1770 | DOI

[55] L. R. Meza; S. Das; J. R. Greer Strong, lightweight, and recoverable three-dimensional ceramic nanolattices, Science, Volume 345 (2014) no. 6202, pp. 1322-1326 | DOI

[56] J. Ma; Z. You Energy absorption of thin-walled square tubes with a prefolded origami pattern part I: geometry and numerical simulation, J. Appl. Mech., Volume 81 (2014) no. 1, 011003

[57] S. Li; H. Fang; S. Sadeghi; P. Bhovad; K.-W. Wang Architected origami materials: How folding creates sophisticated mechanical properties, Adv. Mater., Volume 31 (2019) no. 5, 1805282

[58] T. A. Schaedler; A. J. Jacobsen; A. Torrents; A. E. Sorensen; J. Lian; J. R. Greer; L. Valdevit; W. B. Carter Ultralight metallic microlattices, Science, Volume 334 (2011) no. 6058, pp. 962-965 | DOI

[59] J. L. Silverberg; A. A. Evans; L. McLeod; R. C. Hayward; T. Hull; C. D. Santangelo; I. Cohen Using origami design principles to fold reprogrammable mechanical metamaterials, Science, Volume 345 (2014) no. 6197, pp. 647-650 | DOI

[60] S. Shan; S. H. Kang; J. R. Raney; P. Wang; L. Fang; F. Candido; J. A. Lewis; K. Bertoldi Multistable architected materials for trapping elastic strain energy, Adv. Mater., Volume 27 (2015) no. 29, pp. 4296-4301 | DOI

[61] K. Bertoldi Harnessing instabilities to design tunable architected cellular materials, Annu. Rev. Mater. Res., Volume 47 (2017), pp. 51-61 | DOI

[62] T.-C. Lim Negative thermal expansion in transversely isotropic space frame trusses, Phys. Status Solidi B, Volume 250 (2013) no. 10, pp. 2062-2069 | DOI

[63] D. G. Gilmore Spacecraft Thermal Control Handbook, Fundamental Technologies, vol. 1, American Institute of Aeronautics and Astronautics, Reston, Virginia, USA, 2002, pp. 373-403 (Online version available at: http://www.knovel.com/knovel2/Toc.jsp)

[64] Q. Zhang; J. Wommer; C. ORourke; J. Teitelman; Y. Tang; J. Robison; G. Lin; J. Yin Origami and kirigami inspired self-folding for programming three-dimensional shape shifting of polymer sheets with light, Extreme Mech. Lett., Volume 11 (2017), pp. 111-120 | DOI

[65] Y. Mao; Z. Ding; C. Yuan; S. Ai; M. Isakov; J. Wu; T. Wang; M. L. Dunn; H. J. Qi 3D printed reversible shape changing components with stimuli responsive materials, Sci. Rep., Volume 6 (2016), p. 24761 | DOI

[66] J. B. Hopkins; K. J. Lange; C. M. Spadaccini Designing microstructural architectures with thermally actuated properties using freedom, actuation, and constraint topologies, J. Mech. Design, Volume 135 (2013) no. 6, 061004

[67] S. Tibbits Design to self-assembly, Archit. Design, Volume 82 (2012) no. 2, pp. 68-73 | DOI

[68] J. C. Breger; C. Yoon; R. Xiao; H. R. Kwag; M. O. Wang; J. P. Fisher; T. D. Nguyen; D. H. Gracias Self-folding thermo-magnetically responsive soft microgrippers, ACS Appl. Mater. Interfaces, Volume 7 (2015) no. 5, pp. 3398-3405 | DOI

[69] G. Stoychev; N. Puretskiy; L. Ionov Self-folding all-polymer thermoresponsive microcapsules, Soft Matter, Volume 7 (2011) no. 7, pp. 3277-3279 | DOI

[70] R. Lakes Dense solid microstructures with unbounded thermal expansion, J. Mech. Behav. Mater., Volume 7 (1996) no. 2, pp. 85-92 | DOI

[71] J. Lehman; R. S. Lakes Stiff, strong, zero thermal expansion lattices via material hierarchy, Compos. Struct., Volume 107 (2014), pp. 654-663 | DOI

[72] Q. Wang; J. A. Jackson; Q. Ge; J. B. Hopkins; C. M. Spadaccini; N. X. Fang Lightweight mechanical metamaterials with tunable negative thermal expansion, Phys. Rev. Lett., Volume 117 (2016) no. 17, 175901 | DOI

[73] C. A. Steeves; S. L. d. S. e Lucato; M. He; E. Antinucci; J. W. Hutchinson; A. G. Evans Concepts for structurally robust materials that combine low thermal expansion with high stiffness, J. Mech. Phys. Solids, Volume 55 (2007) no. 9, pp. 1803-1822 | DOI | MR | Zbl

[74] G. Jefferson; T. A. Parthasarathy; R. J. Kerans Tailorable thermal expansion hybrid structures, Int. J. Solids Struct., Volume 46 (2009) no. 11-12, pp. 2372-2387 | DOI | Zbl

[75] O. Sigmund; S. Torquato Composites with extremal thermal expansion coefficients, Appl. Phys. Lett., Volume 69 (1996) no. 21, pp. 3203-3205 | DOI

[76] O. Sigmund; S. Torquato Design of materials with extreme thermal expansion using a three-phase topology optimization method, J. Mech. Phys. Solids, Volume 45 (1997) no. 6, pp. 1037-1067 | DOI | MR

[77] S. Watts; D. A. Tortorelli Optimality of thermal expansion bounds in three dimensions, Extreme Mech. Lett., Volume 12 (2017), pp. 97-100 | DOI

[78] J. Qu; M. Kadic; M. Wegener Poroelastic metamaterials with negative effective static compressibility, Appl. Phys. Lett., Volume 110 (2017) no. 17, 171901

[79] J. Qu; M. Kadic; A. Naber; M. Wegener Micro-structured two-component 3D metamaterials with negative thermal-expansion coefficient from positive constituents, Sci. Rep., Volume 7 (2017), p. 40643 | DOI

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