Letter | Published:

Origami structures with a critical transition to bistability arising from hidden degrees of freedom

Nature Materials volume 14, pages 389393 (2015) | Download Citation

  • A Corrigendum to this article was published on 22 April 2015

This article has been updated

Abstract

Origami is used beyond purely aesthetic pursuits to design responsive and customizable mechanical metamaterials1,2,3,4,5,6,7,8. However, a generalized physical understanding of origami remains elusive, owing to the challenge of determining whether local kinematic constraints are globally compatible and to an incomplete understanding of how the folded sheet’s material properties contribute to the overall mechanical response9,10,11,12,13,14. Here, we show that the traditional square twist, whose crease pattern has zero degrees of freedom (DOF) and therefore should not be foldable, can nevertheless be folded by accessing bending deformations that are not explicit in the crease pattern. These hidden bending DOF are separated from the crease DOF by an energy gap that gives rise to a geometrically driven critical bifurcation between mono- and bistability. Noting its potential utility for fabricating mechanical switches, we use a temperature-responsive polymer-gel version of the square twist to demonstrate hysteretic folding dynamics at the sub-millimetre scale.

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Change history

  • 31 March 2015

    In the version of this Letter originally published, the authors Jesse L. Silverberg and Jun-Hee Na should have been denoted as having contributed equally to this work. This has now been corrected in the online versions of the Letter.

References

  1. 1.

    , , & Identifying links between origami and compliant mechanisms. Mech. Sci. 2, 217–225 (2011).

  2. 2.

    , & Axial crushing of thin-walled structures with origami patterns. Thin. Walled Struct. 54, 65–71 (2012).

  3. 3.

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

  4. 4.

    , , , & Geometric mechanics of periodic pleated origami. Phys. Rev. Lett. 110, 215501 (2013).

  5. 5.

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

  6. 6.

    , , & Origami multistability: From single vertices to metasheets. Phys. Rev. Lett. 114, 055503 (2015).

  7. 7.

    , , , & Origami based mechanical metamaterials. Sci. Rep. 4, 5979–5981 (2014).

  8. 8.

    , , , & Waterbomb base: A symmetric single-vertex bistable origami mechanism. Smart Mater. Struct. 23, 094009 (2014).

  9. 9.

    Curvature and creases: A primer on paper. IEEE Trans. Comput. 25, 1010–1019 (1976).

  10. 10.

    in Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium: Evolution and Trends in Design, Analysis and Construction of Shell and Spatial Structures (eds Domingo, A. & Lazaro, C.) 2287–2294 (Editorial Universitat Politècnica de València, 2009);

  11. 11.

    Project Origami: Activities for Exploring Mathematics (CRC Press, 2012).

  12. 12.

    & Relaxation mechanisms in the unfolding of thin sheets. Phys. Rev. Lett. 107, 025506 (2011).

  13. 13.

    , , & Geometric mechanics of curved crease origami. Phys. Rev. Lett. 109, 114301 (2012).

  14. 14.

    , & Mechanical response of a creased sheet. Phys. Rev. Lett. 112, 244301 (2014).

  15. 15.

    & Percolation on elastic networks: New exponent and threshold. Phys. Rev. Lett. 52, 216–219 (1984).

  16. 16.

    , , & Criticality and isostaticity in fibre networks. Nature Phys. 7, 983–988 (2011).

  17. 17.

    et al. Structure-function relations and rigidity percolation in the shear properties of articular cartilage. Biophys. J. 107, 1–10 (2014).

  18. 18.

    , , & Surface phonons, elastic response, and conformal invariance in twisted kagome lattices. Proc. Natl Acad. Sci. USA 109, 12369–12374 (2012).

  19. 19.

    & Topological boundary modes in isostatic lattices. Nature Phys. 10, 39–45 (2013).

  20. 20.

    , & Nonlinear conduction via solitons in a topological mechanical insulator. Proc. Natl Acad. Sci. USA 111, 13004–13009 (2014).

  21. 21.

    , & Topological modes bound to dislocations in mechanical metamaterials. Nature Phys. 11, 153–156 (2015).

  22. 22.

    & Nonlinear dynamics: Jamming is not just cool any more. Nature 396, 21–22 (1998).

  23. 23.

    , , & Measurement of growing dynamical length scales and prediction of the jamming transition in a granular material. Nature Phys. 3, 260–264 (2007).

  24. 24.

    , & Shock waves in weakly compressed granular media. Phys. Rev. Lett. 111, 218003 (2013).

  25. 25.

    Continuous deformations in random networks. J. Non-Cryst. Solids 57, 355–370 (1983).

  26. 26.

    The Science of Structural Engineering (World Scientific, 1999).

  27. 27.

    On the calculation of the equilibrium and stiffness of frames. Lond. Edinb. Dubl. Phil. Mag. J. Sci. 27, 294–299 (1864).

  28. 28.

    Buckminster Fuller’s “tensegrity” structures and Clerk Maxwell’s rules for the construction of stiff frames. Int. J. Solids Struct. 14, 161–172 (1978).

  29. 29.

    , , , & (Non) existence of pleated folds: How paper folds between creases. Graphs Combinator. 27, 377–397 (2011).

  30. 30.

    Origami3: Proceedings of the Third International Meeting of Origami Science, Mathematics, and Education 29–38 (A K Peters, 2002).

  31. 31.

    et al. Sacrificial bonds and hidden length: Unraveling molecular mesostructures in tough materials. Biophys. J. 90, 1411–1418 (2006).

  32. 32.

    et al. Geometrically controlled snapping transitions in shells with curved creases. Preprint at (2014)

  33. 33.

    et al. Programming reversibly self-folding origami with micropatterned photo-crosslinkable polymer trilayers. Adv. Mater. 27, 79–85 (2015).

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Acknowledgements

The authors thank J. Mosely, U. Nguyen, B. Johnson, B. Parker and M. Schneider for artistic inspiration, as well as O. Vincent, N. Bende, C-K. Tung, S. Waitukaitis and the Cohen lab for useful discussions. We also thank F. Parish for assistance with the laser cutter. This work was funded by the National Science Foundation through award EFRI ODISSEI-1240441.

Author information

Author notes

    • Jesse L. Silverberg
    •  & Jun-Hee Na

    These authors contributed equally to this work.

Affiliations

  1. Physics Department, Cornell University, Ithaca, New York 14853, USA

    • Jesse L. Silverberg
    • , Bin Liu
    •  & Itai Cohen
  2. Department of Polymer Science and Engineering, University of Massachusetts, Amherst, Massachusetts 01003, USA

    • Jun-Hee Na
    •  & Ryan C. Hayward
  3. Department of Physics, University of Massachusetts, Amherst, Massachusetts 01003, USA

    • Arthur A. Evans
    •  & Christian D. Santangelo
  4. Department of Mathematics, Western New England University, Springfield, Massachusetts 01119, USA

    • Thomas C. Hull
  5. Lang Origami, Alamo, California 94507, USA

    • Robert J. Lang

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Contributions

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

Competing interests

The authors declare no competing financial interests.

Corresponding author

Correspondence to Jesse L. Silverberg.

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DOI

https://doi.org/10.1038/nmat4232

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