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Topological kinematics of origami metamaterials


A variety of electronic phases in solid-state systems can be understood by abstracting away microscopic details and refocusing on how Fermi surface topology interacts with band structure to define available electron states1. In fact, topological concepts are broadly applicable to non-electronic materials and can be used to understand a variety of seemingly unrelated phenomena2,3,4,5,6. Here, we apply topological principles to origami-inspired mechanical metamaterials7,8,9,10,11,12, and demonstrate how to guide bulk kinematics by tailoring the crease configuration-space topology. Specifically, we show that by simply changing the crease angles, we modify the configuration-space topology, and drive origami structures to dramatically change their kinematics from being smoothly and continuously deformable to mechanically bistable and rigid. In addition, we examine how a topologically disjointed configuration space can be used to constrain the locally accessible deformations of a single folded sheet. While analyses of origami structures are typically dependent on the energetics of constitutive relations11,12,13,14, the topological abstractions introduced here are a separate and independent consideration that we use to analyse, understand and design these metamaterials.

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Fig. 1: Distinguishing the roles of topological and energetic considerations in origami mechanics.
Fig. 2: Configuration-space topology of origami-inspired mechanical metamaterials is determined by the underlying crease pattern.
Fig. 3: Coupling configuration-space topology with vertex–vertex interactions.
Fig. 4: Decoupling configuration-space topology with vertex–vertex interactions.

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  1. Qi, X.-L. & Zhang, S.-C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011).

    Article  ADS  Google Scholar 

  2. Bader, R. & Nguyen-Dang, T. T. A topological theory of molecular structure. Rep. Prog. Phys. 44, 893–948 (1981).

    Article  ADS  MathSciNet  Google Scholar 

  3. Avron, J. E., Osadchy, D. & Seiler, R. A topological look at the quantum Hall effect. Phys. Today 56, 38–42 (2003).

    Article  Google Scholar 

  4. Lu, L., Joannopoulos, J. D. & Soljačić, M. Topological photonics. Nat. Photon. 8, 821–829 (2014).

    Article  ADS  Google Scholar 

  5. Kane, C. L. & Lubensky, T. C. Topological boundary modes in isostatic lattices. Nat. Phys. 10, 39–45 (2013).

    Article  Google Scholar 

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

    Article  Google Scholar 

  7. Mahadevan, L. & Rica, S. Self-organized origami. Science 307, 1740–1740 (2005).

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

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

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  12. 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).

    Article  ADS  Google Scholar 

  13. Giampieri, A., Perego, U. & Borsari, R. A constitutive model for the mechanical response of the folding of creased paperboard. Int. J. Solids Struct. 48, 2275–2287 (2011).

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

  15. Hull, T. Project Origami: Activities for Exploring Mathematics. (CRC Press: Boca Raton, FL, 2006).

    MATH  Google Scholar 

  16. 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).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  18. Hanna, B. H., Lund, J. M., Lang, R. J., Magleby, S. P. & Howell, L. L. Waterbomb base: a symmetric single-vertex bistable origami mechanism. Smart Mater. Struct. 23, 094009 (2014).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  20. Yang, N. & Silverberg, J. L. Decoupling local mechanics from large-scale structure in modular metamaterials. Proc. Natl Acad. Sci. USA 114, 3590–3595 (2017).

    Article  ADS  Google Scholar 

  21. Brunck, V., Lechenault, F., Reid, A. & Adda-Bedia, M. Elastic theory of origami-based metamaterials. Phys. Rev. E 93, 033005 (2016).

    Article  ADS  Google Scholar 

  22. Waitukaitis, S. & van Hecke, M. Origami building blocks: Generic and special four-vertices. Phys. Rev. E 93, 023003 (2016).

    Article  ADS  MathSciNet  Google Scholar 

  23. Tachi, T. Rigid-foldable thick origami. In Origami 5: The 5th International Conference on Origami in Science Mathematics and Education (eds Wang-Iverson, P. et al.) 253–264 (Taylor & Francis, New York, NY, 2011).

  24. Evans, A. A., Silverberg, J. L. & Santangelo, C. D. Lattice mechanics of origami tessellations. Phys. Rev. E 92, 013205 (2015).

    Article  ADS  MathSciNet  Google Scholar 

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

    Google Scholar 

  26. Nakahara, M. Geometry, Topology and Physics 2nd edn (Taylor & Francis, Boca Raton, FL, 2003).

    MATH  Google Scholar 

  27. Nojima, T. Origami Modeling of Functional Structures Based on Organic Patterns. MSc thesis, Kyoto Univ. (2002).

  28. Tachi, T. Designing freeform origami tessellations by generalizing Resch’s patterns. J. Mech. Des. 135, 111006 (2013).

    Article  Google Scholar 

  29. Barreto, P. T. Lines meeting on a surface: the “Mars” paperfolding. In Proc. 2nd International Meeting of Origami Science and Scientific Origami 323–331 (ed. Miura, K.) (Sein Univ. Art and Design, Otsu, 1997).

  30. Nojima, T. Modelling of folding patterns in flat membranes and cylinders by origami. JSME Int. J. Ser. C 45, 364–370 (2002).

    Article  ADS  Google Scholar 

  31. Lang, R. J. The science of origami. Phys. World 20, 30 (2007).

    Article  Google Scholar 

  32. Kovac, M. & Sareh, P. Aerial devices capable of controlled flight. WO patent application PCT/GB2016/051,567 (2016);

  33. Francis, K. C. et al. From crease pattern to product: considerations to engineering origami-adapted designs. In Proc. ASME 2014 IEDTC & CIEC, Buffalo, NY, 17–20 August 2014, V05BT08A030 (ASME, New York, NY, 2014).

  34. Chen, Y., Peng, R. & You, Z. Origami of thick panels. Science 349, 396–400 (2015).

    Article  ADS  Google Scholar 

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The authors thank A. Ruina, T. Healy, J. Jenkins, U. Nguyen, L. Freni and the Cohen laboratory for useful discussions. We also thank F. Parish for assistance with the laser cutter, and S. Waitukaitis, P. Dieleman and M. van Hecke for providing the photo in Fig. 1b. This work was supported by the National Science Foundation grant no. EFRI ODISSEI-1240441. I.C. received continuing support from DMREF-1435829. B.L. acknowledges the support of the National Science Foundation grant no. NSF CBET-1706511. C.D.S. acknowledges the kind hospitality of the Kavli Institute of Theoretical Physics in Santa Barbara, CA, funded by the National Science Foundation under grant no. NSF PHY-1125915.

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B.L. and J.L.S. designed the research; B.L. conducted the research; B.L., J.L.S., A.A.E., R.J.L., T.C.H. and I.C. interpreted the results; C.D.S., R.J.L., T.C.H. and I.C. supervised the research; B.L., J.L.S., C.D.S., R.J.L., T.C.H. and I.C. prepared the manuscript.

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Correspondence to Bin Liu.

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Liu, B., Silverberg, J.L., Evans, A.A. et al. Topological kinematics of origami metamaterials. Nature Phys 14, 811–815 (2018).

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