Flexible mechanical metamaterials


Mechanical metamaterials exhibit properties and functionalities that cannot be realized in conventional materials. Originally, the field focused on achieving unusual (zero or negative) values for familiar mechanical parameters, such as density, Poisson's ratio or compressibility, but more recently, new classes of metamaterials — including shape-morphing, topological and nonlinear metamaterials — have emerged. These materials exhibit exotic functionalities, such as pattern and shape transformations in response to mechanical forces, unidirectional guiding of motion and waves, and reprogrammable stiffness or dissipation. In this Review, we identify the design principles leading to these properties and discuss, in particular, linear and mechanism-based metamaterials (such as origami-based and kirigami-based metamaterials), metamaterials harnessing instabilities and frustration, and topological metamaterials. We conclude by outlining future challenges for the design, creation and conceptualization of advanced mechanical metamaterials.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Figure 1: Building blocks of mechanical metamaterials.
Figure 2: Mechanism-based metamaterials.
Figure 3: Origami-inspired metamaterials.
Figure 4: Mechanism-based, shape-morphing metamaterials.
Figure 5: Instability-based metamaterials.
Figure 6: Frustration and tunable metamaterials.
Figure 7: Prototypes of topological mechanical metamaterials.


  1. 1

    Soukoulis, C. & Wegener, M. Past achievements and future challenges in the development of three-dimensional photonic metamaterials. Nat. Photonics 5, 523–530 (2011).

    CAS  Article  Google Scholar 

  2. 2

    Cummer, S. A., Christensen, J. & Alù, A. Controlling sound with acoustic metamaterials. Nat. Rev. Mater. 1, 16001 (2016).

    Article  Google Scholar 

  3. 3

    Schittny, R., Kadic, M., Guenneau, S. & Wegener, M. Experiments on transformation thermodynamics: molding the flow of heat. Phys. Rev. Lett. 110, 195901 (2013).

    Article  CAS  Google Scholar 

  4. 4

    Pendry, J. B. Negative refraction makes a perfect lens. Phys. Rev. Lett. 85, 3966–3969 (2000).

    CAS  Article  Google Scholar 

  5. 5

    Kadic, M., Bückmann, T., Schittny, R. & Wegener, M. Metamaterials beyond electromagnetism. Rep. Prog. Phys. 76, 126501 (2013).

    Article  Google Scholar 

  6. 6

    Christensen, J., Kadic, M., Kraft, O. & Wegener, M. Vibrant times for mechanical metamaterials. MRS Commun. 5, 453–462 (2015).

    CAS  Article  Google Scholar 

  7. 7

    Lakes, R. Foam structures with a negative Poisson's ratio. Science 235, 1038–1040 (1987).

    CAS  Article  Google Scholar 

  8. 8

    Lakes, R. Deformation mechanisms of negative Poisson's ratio materials: structural aspects. J. Mater. Sci. 26, 2287–2292 (1991).

    Article  Google Scholar 

  9. 9

    Milton, G. Composite materials with poisson's ratios close to — 1. J. Mech. Phys. Solids 40, 1105–1137 (1992).

    Article  Google Scholar 

  10. 10

    Milton, G. W. & Cherkaev, A. V. Which elasticity tensors are realizable? J. Eng. Mater. Technol. 117, 483–493 (1995).

    Article  Google Scholar 

  11. 11

    Kadic, M., Bückmann, T., Stenger, N., Thiel, M. & Wegener, M. On the practicability of pentamode mechanical metamaterials. Appl. Phys. Lett. 100, 191901 (2012).

    Article  CAS  Google Scholar 

  12. 12

    Nicolaou, Z. G. & Motter, A. E. Mechanical metamaterials with negative compressibility transitions. Nat. Mater. 11, 608–613 (2012).

    CAS  Article  Google Scholar 

  13. 13

    Bückmann, T., Thiel, M., Kadic, M., Schittny, R. & Wegener, M. An elasto-mechanical unfeelability cloak made of pentamode metamaterials. Nat. Commun. 5, 4130 (2014).

    Article  CAS  Google Scholar 

  14. 14

    Schenk, M. & Guest, S. D. Geometry of miura-folded metamaterials. Proc. Natl Acad. Sci. USA 110, 3276–3281 (2013).

    CAS  Article  Google Scholar 

  15. 15

    Yasuda, H. & Yang, J. Reentrant origami-based metamaterials with negative Poisson's ratio and bistability. Phys. Rev. Lett. 114, 185502 (2015).

    CAS  Article  Google Scholar 

  16. 16

    Gatt, R. et al. Hierarchical auxetic mechanical metamaterials. Sci. Rep. 5, 8395 (2015).

    CAS  Article  Google Scholar 

  17. 17

    Babaee, S. et al. 3D soft metamaterials with negative Poisson's ratio. Adv. Mater. 25, 5044–5049 (2013).

    CAS  Article  Google Scholar 

  18. 18

    Liu, J. et al. Harnessing buckling to design architected materials that exhibit effective negative swelling. Adv. Mater. 28, 6619–6624 (2016).

    CAS  Article  Google Scholar 

  19. 19

    Miura, K. Method of packaging and deployment of large membranes in space. Inst. Space Astronaut. Sci. Rep. 618, 1–9 (1985).

    Google Scholar 

  20. 20

    Hawkes, E. et al. Programmable matter by folding. Proc. Natl Acad. Sci. USA 107, 12441–12445 (2010).

    CAS  Article  Google Scholar 

  21. 21

    Tachi, T. & Miura, K. Rigid-foldable cylinders and cells. J. Int. Shell Spat. Struct. 53, 217–226 (2012).

    Google Scholar 

  22. 22

    Wei, Z. Y., Guo, Z. V., Dudte, L., Liang, H. Y. & Mahadevan, L. Geometric mechanics of periodic pleated origami. Phys. Rev. Lett. 110, 215501 (2013).

    CAS  Article  Google Scholar 

  23. 23

    Lv, C., Krishnaraju, D., Konjevod, G., Yu, H. & Jiang, H. Origami based mechanical metamaterials. Sci. Rep. 4, 5979 (2014).

    CAS  Article  Google Scholar 

  24. 24

    Lechenault, F., Thiria, B. & Adda-Bedia, M. Mechanical response of a creased sheet. Phys. Rev. Lett. 112, 244301 (2014).

    CAS  Article  Google Scholar 

  25. 25

    Cheung, K. C., Tachi, T., Calisch, S. & Miura, K. Origami interleaved tube cellular materials. Smart Mater. Struct. 23, 094012 (2014).

    Article  CAS  Google Scholar 

  26. 26

    Cho, Y. et al. Engineering the shape and structure of materials by fractal cut. Proc. Natl Acad. Sci. USA 111, 17390 (2014).

    CAS  Article  Google Scholar 

  27. 27

    Waitukaitis, S., Menaut, R., Chen, B. G. & van Hecke, M. Origami multistability: from single vertices to metasheets. Phys. Rev. Lett. 114, 055503 (2015).

    Article  CAS  Google Scholar 

  28. 28

    Shyu, T. et al. A kirigami approach to engineering elasticity in nanocomposites through patterned defects. Nat. Mater. 14, 785–789 (2015).

    CAS  Article  Google Scholar 

  29. 29

    Filipov, E. T., Tachi, T. & Paulino, G. H. Origami tubes assembled into stiff, yet reconfigurable structures and metamaterials. Proc. Natl Acad. Sci. USA 112, 12321–12326 (2015).

    CAS  Article  Google Scholar 

  30. 30

    Coulais, C., Teomy, E., de Reus, K., Shokef, Y. & van Hecke, M. Combinatorial design of textured mechanical metamaterials. Nature 535, 529–532 (2016).

    CAS  Article  Google Scholar 

  31. 31

    Isobe, M. & Okumura, K. Initial rigid response and softening transition of highly stretchable kirigami sheet materials. Sci. Rep. 6, 24758 (2016).

    CAS  Article  Google Scholar 

  32. 32

    Dudte, L., Vouga, E., Tachi, T. & Mahadevan, L. Programming curvature using origami tessellations. Nat. Mater. 15, 583–588 (2016).

    CAS  Article  Google Scholar 

  33. 33

    Haghpanah, B. et al. Multistable shape-reconfigurable architected materials. Adv. Mater. 28, 7915–7920 (2016).

    CAS  Article  Google Scholar 

  34. 34

    Overvelde, J. T. et al. A three-dimensional actuated origami-inspired transformable metamaterial with multiple degrees of freedom. Nat. Commun. 7, 10929 (2016).

    CAS  Article  Google Scholar 

  35. 35

    Overvelde, J. T., Weaver, J., Hoberman, C. & Bertoldi, K. Rational design of reconfigurable prismatic architected materials. Nature 541, 347–352 (2017).

    CAS  Article  Google Scholar 

  36. 36

    Prodan, E. & Prodan, C. Topological phonon modes and their role in dynamic instability of microtubules. Phys. Rev. Lett. 103, 248101 (2009).

    Article  CAS  Google Scholar 

  37. 37

    Kane, C. L. & Lubensky, T. C. Topological boundary modes in isostatic lattices. Nat. Phys. 10, 39–45 (2014).

    CAS  Article  Google Scholar 

  38. 38

    Chen, B. G., Upadhyaya, N. & Vitelli, V. Nonlinear conduction via solitons in a topological mechanical insulator. Proc. Natl Acad. Sci. USA 111, 13004–13009 (2014).

    Article  CAS  Google Scholar 

  39. 39

    Vitelli, V., Upadhyaya, N. & Chen, B. G. Topological mechanisms as classical spinor fields. Preprint at http://arxiv.org/abs/1407.2890v2 (2014).

  40. 40

    Paulose, J., Chen, B. G. & Vitelli, V. Topological modes bound to dislocations in mechanical metamaterials. Nat. Phys. 11, 153–156 (2015).

    CAS  Article  Google Scholar 

  41. 41

    Paulose, J., Meeussen, A. S. & Vitelli, V. Selective buckling via states of self-stress in topological metamaterials. Proc. Natl Acad. Sci. USA 112, 7639–7644 (2015).

    CAS  Article  Google Scholar 

  42. 42

    Nash, L. M. et al. Topological mechanics of gyroscopic metamaterials. Proc. Natl Acad. Sci. USA 112, 14495–14500 (2015).

    CAS  Article  Google Scholar 

  43. 43

    Khanikaev, A. B., Fleury, R. & Mousavi, S. H. & Alù, A. Topologically robust sound propagation in an angular-momentum-biased graphene-like resonator lattice. Nat. Commun. 6, 8260 (2015).

    CAS  Article  Google Scholar 

  44. 44

    Susstrunk, R. & Huber, S. D. Observation of phononic helical edge states in a mechanical topological insulator. Science 349, 47–50 (2015).

    Article  CAS  Google Scholar 

  45. 45

    Meeussen, A. S., Paulose, J. & Vitelli, V. Geared topological metamaterials with tunable mechanical stability. Phys. Rev. X 6, 041029 (2016).

    Google Scholar 

  46. 46

    Rocklin, D. Z., Chen, B. G., Falk, M., Vitelli, V. & Lubensky, T. C. Mechanical Weyl modes in topological Maxwell lattices. Phys. Rev. Lett. 116, 135503 (2016).

    Article  CAS  Google Scholar 

  47. 47

    Kariyado, T. & Hatsugai, Y. Manipulation of Dirac cones in mechanical graphene. Sci. Rep. 5, 18107 (2015).

    CAS  Article  Google Scholar 

  48. 48

    Chen, B. G. et al. Topological mechanics of origami and kirigami. Phys. Rev. Lett. 116, 135501 (2016).

    Article  CAS  Google Scholar 

  49. 49

    Mousavi, S. H., Khanikaev, A. B. & Wang, Z. Topologically protected elastic waves in phononic metamaterials. Nat. Commun. 6, 8682 (2015).

    Article  CAS  Google Scholar 

  50. 50

    Xiao, M., Chen, W.-J., He, W.-Y. & Chan, C. T. Synthetic gauge flux and Weyl points in acoustic systems. Nat. Phys. 11, 920–924 (2015).

    CAS  Article  Google Scholar 

  51. 51

    Rocklin, D. Z., Zhou, S., Sun, K. & Mao, X. Transformable topological mechanical metamaterials. Nat. Commun. 8, 14201 (2017).

    CAS  Article  Google Scholar 

  52. 52

    Yang, Z. et al. Topological acoustics. Phys. Rev. Lett. 114, 1–4 (2015).

    CAS  Google Scholar 

  53. 53

    Süsstrunk, R. & Huber, S. D. Classification of topological phonons in linear mechanical metamaterials. Proc. Natl Acad. Sci. USA 113, E4767–E4775 (2016).

    Article  CAS  Google Scholar 

  54. 54

    Deymier, P. A., Runge, K., Swinteck, N. & Muralidharan, K. Torsional topology and fermion-like behavior of elastic waves in phononic structures. Comptes Rendus Mécanique 343, 700–711 (2015).

    Article  Google Scholar 

  55. 55

    Bi, R. & Wang, Z. Unidirectional transport in electronic and photonic Weyl materials by dirac mass engineering. Phys. Rev. B 92, 241109 (2015).

    Article  Google Scholar 

  56. 56

    Berg, N., Joel, K., Koolyk, M. & Prodan, E. Topological phonon modes in filamentary structures. Phys. Rev. E 83, 021913 (2011).

    Article  CAS  Google Scholar 

  57. 57

    Peano, V., Brendel, C., Schmidt, M. & Marquardt, F. Topological phases of sound and light. Phys. Rev. X 5, 031011 (2015).

    Google Scholar 

  58. 58

    Wang, Y.-T., Luan, P.-G. & Zhang, S. Coriolis force induced topological order for classical mechanical vibrations. New J. Phys. 17, 073031 (2015).

    Google Scholar 

  59. 59

    Wang, P., Lu, L. & Bertoldi, K. Topological phononic crystals with one-way elastic edge waves. Phys. Rev. Lett. 115, 104302 (2015).

    Article  CAS  Google Scholar 

  60. 60

    Po, H. C., Bahri, Y. & Vishwanath, A. Phonon analog of topological nodal semimetals. Phys. Rev. B 93, 205158 (2016).

    Article  CAS  Google Scholar 

  61. 61

    Coulais, C., Sounas, D. & Alù, A. Static non-reciprocity in mechanical metamaterials. Nature 542, 461–464 (2017).

    CAS  Article  Google Scholar 

  62. 62

    Mullin, T., Deschanel, S., Bertoldi, K. & Boyce, M. C. Pattern transformation triggered by deformation. Phys. Rev. Lett. 99, 084301 (2007).

    CAS  Article  Google Scholar 

  63. 63

    Bertoldi, K., Reis, P. M., Willshaw, S. & Mullin, T. Negative Poisson's ratio behavior induced by an elastic instability. Adv. Mater. 22, 361–366 (2010).

    CAS  Article  Google Scholar 

  64. 64

    Shim, J., Perdigou, C., Chen, E., Bertoldi, K. & Reis, P. Buckling-induced encapsulation of structured elastic shells under pressure. Proc. Natl Acad. Sci. USA 109, 5978–5983 (2012).

    CAS  Article  Google Scholar 

  65. 65

    Florijn, B., Coulais, C. & van Hecke, M. Programmable mechanical metamaterials. Phys. Rev. Lett. 113, 175503 (2014).

    Article  CAS  Google Scholar 

  66. 66

    Coulais, C., Overvelde, J. T., Lubbers, L. A., Bertoldi, K. & van Hecke, M. Discontinuous buckling of wide beams and metabeams. Phys. Rev. Lett. 115, 044301 (2015).

    Article  Google Scholar 

  67. 67

    Fargette, A., Neukirch, S. & Antkowiak, A. Elastocapillary snapping: capillarity induces snap-through instabilities in small elastic beams. Phys. Rev. Lett. 112, 137802 (2014).

    Article  CAS  Google Scholar 

  68. 68

    Silverberg, J. L. et al. Using origami design principles to fold reprogrammable mechanical metamaterials. Science 345, 647–650 (2014).

    CAS  Article  Google Scholar 

  69. 69

    Kang, S. H. et al. Complex ordered patterns in mechanical instability induced geometrically frustrated triangular cellular structures. Phys. Rev. Lett. 112, 098701 (2014).

    Article  CAS  Google Scholar 

  70. 70

    Shan, S. et al. Multistable architected materials for trapping elastic strain energy. Adv. Mater. 27, 4296 (2015).

    CAS  Article  Google Scholar 

  71. 71

    Silverberg, J. L. et al. Origami structures with a critical transition to bistability arising from hidden degrees of freedom. Nat. Mater. 14, 389–393 (2015).

    CAS  Article  Google Scholar 

  72. 72

    Zhang, Y. et al. A mechanically driven form of kirigami as a route to 3D mesostructures in micro/nanomembranes. Proc. Natl Acad. Sci. USA 112, 11757–11764 (2015).

    CAS  Article  Google Scholar 

  73. 73

    Raney, J. R. et al. Stable propagation of mechanical signals in soft media using stored elastic energy. Proc. Natl Acad. Sci. USA 113, 9722–9727 (2016).

    CAS  Article  Google Scholar 

  74. 74

    Rafsanjani, A., Akbarzadeh, A. & Pasini, D. Snapping mechanical metamaterials under tension. Adv. Mater. 55, 5931–5935 (2007).

    Google Scholar 

  75. 75

    Alexander, S. Amorphous solids: their structure, lattice dynamics and elasticity. Phys. Rep. 296, 65–236 (1998).

    CAS  Article  Google Scholar 

  76. 76

    van Hecke, M. Jamming of soft particles: geometry, mechanics, scaling and isostaticity. J. Phys. Condens. Matter 22, 033101 (2010).

    CAS  Article  Google Scholar 

  77. 77

    Katgert, G., Tighe, B. P. & van Hecke, M. The jamming perspective on wet foams. Soft Matter 9, 9739–9746 (2013).

    CAS  Article  Google Scholar 

  78. 78

    Jaeger, H. M., Nagel, S. R. & Behringer, R. P. Granular solids, liquids, and gases. Rev. Mod. Phys. 68, 1259 (2012).

    Article  Google Scholar 

  79. 79

    Broedersz, C. P. & MacKintosh, F. C. Modeling semiflexible polymer networks. Rev. Mod. Phys. 86, 995 (2014).

    CAS  Article  Google Scholar 

  80. 80

    Jacobs, D. J. & Thorpe, M. F. Generic rigidity percolation: the pebble game. Phys. Rev. Lett. 75, 4051–4054 (1995).

    CAS  Article  Google Scholar 

  81. 81

    Wyart, M., Liang, H., Kabla, A. & Mahadevan, L. Elasticity of floppy and stiff random networks. Phys. Rev. Lett. 101, 215501 (2008).

    CAS  Article  Google Scholar 

  82. 82

    Ellenbroek, W. G., Zeravcic, Z., van Saarloos, W. & van Hecke, M. Non-affine response: jammed packings versus spring networks. EPL 87, 34004 (2009).

    Article  CAS  Google Scholar 

  83. 83

    Ellenbroek, W. G., Hagh, V. F., Kumar, A., Thorpe, M. F. & van Hecke, M. Rigidity loss in disordered systems: three scenarios. Phys. Rev. Lett. 114, 135501 (2015).

    Article  CAS  Google Scholar 

  84. 84

    Goodrich, C. P., Liu, A. J. & Nagel, S. R. The principle of independent bond-level response: tuning by pruning to exploit disorder for global behavior. Phys. Rev. Lett. 114, 225501 (2015).

    Article  CAS  Google Scholar 

  85. 85

    O’Rourke, J. How to Fold It: The Mathematics of Linkages, Origami and Polyhedra (Cambridge Univ. Press, 2011).

    Google Scholar 

  86. 86

    Grima, J. N. & Evans, K. E. Auxetic behavior from rotating squares. J. Mater. Sci. Lett. 19, 1563–1565(2000).

    CAS  Article  Google Scholar 

  87. 87

    Overvelde, J. T., Shan, S. & Bertoldi, K. Compaction through buckling in 2D periodic, soft and porous structures: effect of pore shape. Adv. Mater. 24, 2337–2342 (2012).

    CAS  Article  Google Scholar 

  88. 88

    Overvelde, J. T. & Bertoldi, K. Relating pore shape to the non-linear response of periodic elastomeric structures. J. Mech. Phys. Solids 64, 351–366 (2014).

    Article  Google Scholar 

  89. 89

    Kuribayashi, K. et al. Self-deployable origami stent grafts as a biomedical application of Ni-rich TiNi shape memory alloy foil. Mater. Sci. Eng. A 419, 131–137 (2006).

    Article  CAS  Google Scholar 

  90. 90

    Song, Z. et al. Origami lithium-ion batteries. Nat. Commun. 5, 3140 (2014).

    Article  CAS  Google Scholar 

  91. 91

    Goldman, F. in Origami5: Fifth International Meeting of Origami Science, Mathematics, and Education (eds Wang-Iverson, P., Lang, R. J . & Yim, M. ) 99–110 (CRC Press, 2011).

    Google Scholar 

  92. 92

    Blees, K. et al. Graphene kirigami. Nature 524, 204–207 (2015).

    CAS  Article  Google Scholar 

  93. 93

    Song, Z. et al. Kirigami-based stretchable lithium-ion batteries. Sci. Rep. 5, 10988 (2015).

    CAS  Article  Google Scholar 

  94. 94

    Xu, L. et al. Kirigami nanocomposites as wide-angle diffraction gratings. ACS Nano 10, 6156–6162 (2016).

    CAS  Article  Google Scholar 

  95. 95

    Rafsanjani, A. & Bertoldi, K. Buckling-induced kirigami. Phys. Rev. Lett. 118, 084301 (2017).

    Article  Google Scholar 

  96. 96

    Lamoureux, A. et al. Dynamic kirigami structures for integrated solar tracking. Nat. Commun. 6, 8092 (2015).

    Article  Google Scholar 

  97. 97

    Eidini, M. & Paulino, G. H. Unraveling metamaterial properties in zigzag-base folded sheets. Sci. Adv. 1, e1500224 (2015).

    Article  Google Scholar 

  98. 98

    Sussman, D. et al. Algorithmic lattice kirigami: a route to pluripotent materials. Proc. Natl Acad. Sci. USA 112, 7449–7453 (2015).

    CAS  Article  Google Scholar 

  99. 99

    Bückmann, T. et al. Tailored 3D mechanical metamaterials made by dip-in direct-laser-writing optical lithography. Adv. Mater. 24, 2710–2714 (2012).

    Article  CAS  Google Scholar 

  100. 100

    Gibson, L. & Ashby, M. Cellular Solids: Structure and Properties 2 nd edn (Cambridge Univ. Press, 1999).

    Google Scholar 

  101. 101

    Wierzbicki, T. & Abramowicz, W. On the crushing mechanics of thin-walled structures. J. Appl. Mech. 50, 727–734 (1983).

    Article  Google Scholar 

  102. 102

    Papka, S. & Kyriakides, S. Biaxial crushing of honeycombs: — part1: experiments. Int. J. Solids Struct. 36, 4367–4396 (1999).

    Article  Google Scholar 

  103. 103

    Papka, S. & Kyriakides, S. In-plane biaxial crushing of honeycombs:— part II: analysis. Int. J. Solids Struct. 36, 4397–4423 (1999).

    Article  Google Scholar 

  104. 104

    Wu, E. & Jiang, W. Axial crush of metallic honeycombs. Int. J. Impact Eng. 19, 439–456 (1997).

    Article  Google Scholar 

  105. 105

    Hayes, A. M., Wang, A., Dempsey, B. M. & McDowell, D. L. Mechanics of linear cellular alloys. Mech. Mater. 36, 691–713 (2004).

    Article  Google Scholar 

  106. 106

    Zhang, Y. et al. One-step nanoscale assembly of complex structures via harnessing of an elastic instability. Nano Lett. 8, 1192–1196 (2008).

    CAS  Article  Google Scholar 

  107. 107

    Babaee, S., Shim, J., Weaver, J., Patel, N. & Bertoldi, K. 3D soft metamaterials with negative Poisson's ratio. Adv. Mater. 25, 5044–5049 (2013).

    CAS  Article  Google Scholar 

  108. 108

    Bertoldi, K. & Boyce, M. C. Mechanically-triggered transformations of phononic band gaps in periodic elastomeric structures. Phys. Rev. B 77, 052105 (2008).

    Article  CAS  Google Scholar 

  109. 109

    Wang, P., Casadei, F., Shan, S., Weaver, J. & Bertoldi, K. Harnessing buckling to design tunable locally resonant acoustic metamaterials. Phys. Rev. Lett. 113, 014301 (2014).

    Article  CAS  Google Scholar 

  110. 110

    Shan, S. et al. Harnessing multiple folding mechanisms in soft periodic structures for tunable control of elastic waves. Adv. Funct. Mater. 24, 4935–4942 (2014).

    CAS  Article  Google Scholar 

  111. 111

    Krishnan, D. & Johnson, H. Optical properties of two dimensional polymer photonic crystals after deformation induced pattern transformations. J. Mech. Phys. Solids 57, 1500–1513 (2009).

    CAS  Article  Google Scholar 

  112. 112

    Li, J. et al. Switching periodic membranes via pattern transformation and shape memory effect. Soft Matter 8, 10322–10328 (2012).

    CAS  Article  Google Scholar 

  113. 113

    Kang, S. et al. Buckling-induced reversible symmetry breaking and amplification of chirality using supported cellular structures. Adv. Mater. 25, 3380–3385 (2013).

    CAS  Article  Google Scholar 

  114. 114

    Bazant, Z. P. & Cedolin, L. Stability of Structures: Elastic, Inelastic, Fracture, and Damage Theories (Oxford Univ. Press, 1991).

    Google Scholar 

  115. 115

    Pandey, A., Moulton, D., Vella, D. & Holmes, D. Dynamics of snapping beams and jumping poppers. EPL 105, 24001 (2014).

    Article  CAS  Google Scholar 

  116. 116

    Restrepo, D., Mankame, N. D. & Zavattieri, P. D. Phase transforming cellular materials. EML 4, 52–60 (2015).

    Google Scholar 

  117. 117

    Hasan, M. Z. & Kane, C. L. Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).

    CAS  Article  Google Scholar 

  118. 118

    Lubensky, T. C., Kane, C. L., Mao, X., Souslov, A. & Sun, K. Phonons and elasticity in critically coordinated lattices. Rep. Prog. Phys. 78, 073901 (2015).

    CAS  Article  Google Scholar 

  119. 119

    Zhou, Y., Chen, B. G., Upadhyaya, N. & Vitelli, V. Kink-antikink asymmetry and impurity interactions in topological mechanical chains. Phys. Rev. E 95, 022202 (2017).

    Article  Google Scholar 

  120. 120

    Mitchell, N. P., Nash, L. M., Hexner, D., Turner, A. & Irvine, W. T. M. Amorphous gyroscopic topological metamaterials. Preprint at https://arxiv.org/abs/1612.09267 (2016).

  121. 121

    Souslov, A., van Zuiden, B. C., Bartolo, D. & Vitelli, V. Topological sound in active-liquid metamaterials. Nat. Phys.http://dx.doi.org/10.1038/nphys4193 (2017).

  122. 122

    Coulais, C., Kettenis, C. & van Hecke, M. A characteristic length scale causes anomalous size effects and boundary programmability in mechanical metamaterials. Nat. Phys.http://dx.doi.org/10.1038/nphys4269 (2017).

  123. 123

    Lang, R. J. in Proc. 12 th Ann. Symp. Comp. Geo. (eds Whitesides, S. H. ) 98–105 (ACM, 1996).

    Google Scholar 

  124. 124

    Felton, S., Tolley, M., Demaine, E., Rus, D. & Wood, R. A method for building self-folding machines. Science 345, 644–646 (2014).

    CAS  Article  Google Scholar 

  125. 125

    Zeravcic, Z., Manoharan, V. N. & Brenner, M. P. Size limits of self-assembled colloidal structures made using specific interactions. Proc. Natl Acad. Sci. USA 111, 15918–15923 (2014).

    CAS  Article  Google Scholar 

  126. 126

    Miskin, M. Z. & Jaeger, H. M. Adapting granular materials through artificial evolution. Nat. Mater. 12, 326–331 (2013).

    Article  CAS  Google Scholar 

Download references


J.C. acknowledges support from the European Research Council (ERC) through the Starting Grant No. 714577 PHONOMETA and from the Ministerio de Economía, Industria y Competitividad (MINECO) through a Ramon y Cajal grant (Grant No. RYC-2015-17156). M.vH. acknowledges funding from the Netherlands Organisation for Scientific Research through Grant VICI No. NWO-680-47-609. V.V. acknowledges support from the University of Chicago Materials Research Science and Engineering Center, which is funded by the National Science Foundation through Grant No. DMR-1420709.

Author information




All authors contributed equally to the preparation of this manuscript.

Corresponding author

Correspondence to Johan Christensen.

Ethics declarations

Competing interests

The authors declare no competing interests.

PowerPoint slides

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bertoldi, K., Vitelli, V., Christensen, J. et al. Flexible mechanical metamaterials. Nat Rev Mater 2, 17066 (2017). https://doi.org/10.1038/natrevmats.2017.66

Download citation

Further reading


Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing