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

Nature Materials volume 14, pages 389–393 (2015)Cite this article

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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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Figure 1: Schematics and photographs introducing the square twist’s essential geometric properties and mechanical characteristics.
Figure 2: Experimental strain-controlled mechanical data studying the transition between mono- and bistability in square twists.
Figure 3: Simulation results for the square twist with non-rigid facets.
Figure 4: A sub-millimetre-scale self-folding polymer-gel version of the square twist is used to verify the geometric nature of bistability in stress-controlled conditions.

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  • 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. Greenberg, H., Gong, M., Magleby, S. & Howell, L. Identifying links between origami and compliant mechanisms. Mech. Sci. 2, 217–225 (2011).

    Article  Google Scholar 

  2. Song, J., Chen, Y. & Lu, G. Axial crushing of thin-walled structures with origami patterns. Thin. Walled Struct. 54, 65–71 (2012).

    Article  Google Scholar 

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

    Article  CAS  Google Scholar 

  4. 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).

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

  6. 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  Google Scholar 

  7. Lv, C., Krishnaraju, D., Konjevod, G., Yu, H. & Jiang, H. Origami based mechanical metamaterials. Sci. Rep. 4, 5979–5981 (2014).

    Article  CAS  Google Scholar 

  8. 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  Google Scholar 

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

    Article  Google Scholar 

  10. Tachi, T. 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); http://go.nature.com/HbzSH1

    Google Scholar 

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

    Google Scholar 

  12. Thiria, B. & Adda-Bedia, M. Relaxation mechanisms in the unfolding of thin sheets. Phys. Rev. Lett. 107, 025506 (2011).

    Article  CAS  Google Scholar 

  13. Dias, M. A., Dudte, L. H., Mahadevan, L. & Santangelo, C. D. Geometric mechanics of curved crease origami. Phys. Rev. Lett. 109, 114301 (2012).

    Article  Google Scholar 

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

    Article  CAS  Google Scholar 

  15. Feng, S. & Sen, P. N. Percolation on elastic networks: New exponent and threshold. Phys. Rev. Lett. 52, 216–219 (1984).

    Article  Google Scholar 

  16. Broedersz, C. P., Mao, X., Lubensky, T. C. & MacKintosh, F. C. Criticality and isostaticity in fibre networks. Nature Phys. 7, 983–988 (2011).

    Article  CAS  Google Scholar 

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

    Article  Google Scholar 

  18. Sun, K., Souslov, A., Mao, X. & Lubensky, T. Surface phonons, elastic response, and conformal invariance in twisted kagome lattices. Proc. Natl Acad. Sci. USA 109, 12369–12374 (2012).

    Article  CAS  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  CAS  Google Scholar 

  22. Liu, A. J. & Nagel, S. R. Nonlinear dynamics: Jamming is not just cool any more. Nature 396, 21–22 (1998).

    Article  CAS  Google Scholar 

  23. Keys, A. S., Abate, A. R., Glotzer, S. C. & Durian, D. J. Measurement of growing dynamical length scales and prediction of the jamming transition in a granular material. Nature Phys. 3, 260–264 (2007).

    CAS  Google Scholar 

  24. Van den Wildenberg, S., van Loo, R. & van Hecke, M. Shock waves in weakly compressed granular media. Phys. Rev. Lett. 111, 218003 (2013).

    Article  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Book  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  29. Demaine, E. D., Demaine, M. L., Hart, V., Price, G. N. & Tachi, T. (Non) existence of pleated folds: How paper folds between creases. Graphs Combinator. 27, 377–397 (2011).

    Article  Google Scholar 

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

    Book  Google Scholar 

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

    Article  CAS  Google Scholar 

  32. Bende, N. P. et al. Geometrically controlled snapping transitions in shells with curved creases. Preprint at http://arxiv.org/abs/1410.7038 (2014)

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

    Article  CAS  Google Scholar 

Download references

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

  1. Jesse L. Silverberg and Jun-Hee Na: These authors contributed equally to this work.

Authors and 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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  1. Jesse L. Silverberg
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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.

Corresponding author

Correspondence to Jesse L. Silverberg.

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The authors declare no competing financial interests.

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Silverberg, J., Na, JH., Evans, A. et al. Origami structures with a critical transition to bistability arising from hidden degrees of freedom. Nature Mater 14, 389–393 (2015). https://doi.org/10.1038/nmat4232

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