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A non-linear rod model for folded elastic strips. (English) Zbl 1323.74044
Summary: We consider the equilibrium shapes of a thin, annular strip cut out in an elastic sheet. When a central fold is formed by creasing beyond the elastic limit, the strip has been observed to buckle out-of-plane. Starting from the theory of elastic plates, we derive a Kirchhoff rod model for the folded strip. A non-linear effective constitutive law incorporating the underlying geometrical constraints is derived, in which the angle the ridge appears as an internal degree of freedom. By contrast with traditional thin-walled beam models, this constitutive law captures large, non-rigid deformations of the cross-sections, including finite variations of the dihedral angle at the ridge. Using this effective rod theory, we identify a buckling instability that produces the out-of-plane configurations of the folded strip, and show that the strip behaves as an elastic ring having one frozen mode of curvature. In addition, we point out two novel buckling patterns: one where the centerline remains planar and the ridge angle is modulated; another one where the bending deformation is localized. These patterns are observed experimentally, explained based on stability analyses, and reproduced in simulations of the post-buckled configurations.

MSC:
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74G60 Bifurcation and buckling
74K20 Plates
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[1] Antman, S. S., Nonlinear problems of elasticity, (1995), Springer · Zbl 0820.73002
[2] Audoly, B.; Pomeau, Y., Elasticity and geometryfrom hair curls to the nonlinear response of shells, (2010), Oxford University Press · Zbl 1223.74001
[3] Chouaïeb, N., 2003. Kirchhoff’s Problem of Helical Solutions of Uniform Rods and Stability Properties. Ph.D. Thesis, École polytechnique fédérale de Lausanne, Lausanne, Switzerland.
[4] Demaine, E. D.; Demaine, M. L.; Hart, V.; Price, G. N.; Tachi, T., (non)existence of pleated foldshow paper folds between creases, Graphs Combinatorics, 27, 3, 377-397, (2011) · Zbl 1237.52012
[5] Dias, M. A.; Santangelo, C. D., The shape and mechanics of curved-fold origami structures, Europhys. Lett., 100, 5, 54005, (2012)
[6] Dias, M. A.; Dudte, L. H.; Mahadevan, L.; Santangelo, C. D., Geometric mechanics of curved crease origami, Phys. Rev. Lett., 109, 11, 114301, (2012)
[7] Dill, E. H., Kirchhoff’s theory of rods, Arch. Hist. Exact Sci., 44, 1-23, (1992) · Zbl 0762.01012
[8] do Carmo, M. P., Differential geometry of curves and surfaces, (1976), Prentice-Hall Englewood Cliffs · Zbl 0326.53001
[9] Doedel, E.J., Champneys, A.R., Fairgrieve, T.F., Kuznetsov, Y.A., Sandstede, B., Wang, X.J., 2007. AUTO-07p: Continuation and Bifurcation Software for Ordinary Differential Equations. 〈http://indy.cs.concordia.ca/auto/〉.
[10] Duncan, J. P.; Duncan, J. L., Folded developables, Proc. R. Soc. London Ser. A, 383, 1784, 191-205, (1982) · Zbl 0506.53003
[11] Engel, H., Structure systems, (1968), Praeger Westport, CT
[12] Fuchs, D.; Tabachnikov, S., More on paperfolding, Am. Math. Mon., 106, 1, 27-35, (1999) · Zbl 1037.53501
[13] Giomi, L.; Mahadevan, L., Multi-stability of free spontaneously curved anisotropic strips, Proc. R. Soc. AMath. Phys. Eng. Sci., 468, 2138, 511-530, (2012) · Zbl 1364.74064
[14] Goldstein, R. E.; Warren, P. B.; Ball, R. C., Shape of a ponytail and the statistical physics of hair fiber bundles, Phys. Rev. Lett., 108, 078101, (2012)
[15] Goriely, A., Twisted elastic rings and the rediscoveries of Michell’s instability, J. Elasticity, 84, 281-299, (2006) · Zbl 1098.74034
[16] Guinot, F.; Bourgeois, S.; Cochelin, B.; Blanchard, L., A planar rod model with flexible thin-walled cross-sections. application to the folding of tape springs, Int. J. Solids Struct., 49, 73-86, (2012)
[17] Hamdouni, A.; Millet, O., An asymptotic non-linear model for thin-walled rods with strongly curved open cross-section, Int. J. Non-Linear Mech., 41, 3, 396-416, (2006) · Zbl 1160.74374
[18] Hamdouni, A.; Millet, O., An asymptotic linear thin-walled rod model coupling twist and bending, Int. Appl. Mech., 46, 9, 1072-1092, (2011)
[19] Huffman, D. A., Curvature and creasesa primer on paper, IEEE Trans. Comput., C-25, 10, 1010-1019, (1976) · Zbl 0338.68074
[20] Jackson, P., Folding techniques for designersfrom sheet to form, (2011), Laurence King Publishers
[21] Kilian, M.; Flöry, S.; Chen, Z.; Mitra, N. J.; Sheffer, A.; Pottmann, H., Curved folding, SIGGRAPH Comput. Graph., 27, 3, (2008)
[22] Kuribayashi, K.; Tsuchiya, K.; You, Z.; Tomus, D.; Umemoto, M.; Ito, T.; Sasaki, M., Self-deployable origami stent grafts as a biomedical application of ni-rich tini shape memory alloy foil, Mater. Sci. Eng. A, 419, 1-2, 131-137, (2006)
[23] Lanczos, C., The variational principles of mechanics, (1970), University of Toronto Press · Zbl 0138.19706
[24] Love, A. E.H., A treatise on the mathematical theory of elasticity, (1944), Dover Publications USA · Zbl 0063.03651
[25] Mansfield, E. H., Large-deflexion torsion and flexure of initially curved strips, Proc. R. Soc. London A Math. Phys. Sci., 334, 1598, 279-298, (1973) · Zbl 0286.73039
[26] Miura, K., Method of packing and deployment of large membranes in space, (1980), The Institute of Space and Astronautical Science
[27] Moulton, D.; Lessinnes, T.; Goriely, A., Morphoelastic rods. part I. A single growing elastic rod, J. Mech. Phys. Solids, 61, 2, 398-427, (2013)
[28] Mouthuy, P.-O.; Coulombier, M.; Pardoen, T.; Raskin, J.-P.; Jonas, A. M., Overcurvature describes the buckling and folding of rings from curved origami to foldable tents, Nat. Commun., 3, 1290, (2012)
[29] Norman, A. D.; Seffen, K. A.; Guest, S. D., Morphing of curved corrugated shells, Int. J. Solids Struct., 46, 7-8, 1624-1633, (2009) · Zbl 1217.74080
[30] Pottmann, H.; Wallner, J., Computational line geometry, (2001), Springer-Verlag Berlin, Heidelberg · Zbl 1006.51015
[31] Powers, T. R., Dynamics of filaments and membranes in a viscous fluid, Rev. Mod. Phys., 82, 2, 1607-1631, (2010)
[32] Sadowsky, M., Ein elementarer beweis für die existenz eines abwickelbares Möbiusschen bandes und zurückfürung des geometrischen problems auf ein variationsproblem, Sitzungsber. Preuss. Akad. Wiss., 22, 412-415, (1930) · JFM 56.0601.02
[33] Schenk, M.; Guest, S. D., Origami foldinga structural engineering approach, (Wang-Iverson, P.; Lang, R. J.; Yim, M., Origami^5, Fifth International Meeting of Origami Science, Mathematics, and Education, (2011), A K Peters/CRC Press), 291-303
[34] Seffen, K. A., Compliant shell mechanisms, Philos. Trans. Ser. A Math. Phys. Eng. Sci., 370, 1965, 2010-2026, (2012)
[35] Seffen, K. A.; You, Z.; Pellegrino, S., Folding and deployment of curved tape springs, Int. J. Mech. Sci., 42, 2055-2073, (2000) · Zbl 0969.74502
[36] Shi, Y.; Hearst, J. E., The Kirchhoff elastic rod, the nonlinear Schrödinger equation, and DNA supercoiling, J. Chem. Phys., 101, 6, 5186-5200, (1994)
[37] Spivak, M., 1979. A Comprehensive Introduction to Differential Geometry, 2nd edition, vol. 3. Publish or Perish, Inc. · Zbl 0439.53002
[38] Starostin, E. L.; van der Heijden, G. H.M., The shape of a Möbius strip, Nat. Mater., 6, 11, 563-567, (2007)
[39] Starostin, E. L.; van der Heijden, G. H.M., Tension-induced multistability in inextensible helical ribbons, Phys. Rev. Lett., 101, 084301, (2008)
[40] Steigmann, D. J.; Faulkner, M. G., Variational theory for spatial rods, J. Elasticity, 33, 1, 1-26, (1993) · Zbl 0801.73039
[41] Wei, Z., Guo, Z., Dudte, L., Liang, H., Mahadevan, L., 2012. Geometric Mechanics of Periodic Pleated Origami. arXiv1211.6396, pp. 1-28.
[42] Wingler, H. M., Bauhausweimar, dessau, Berlin, Chicago, (1969), The MIT Press
[43] Wolfram Research, Inc., 2012. Mathematica Edition. Version 9.0. Champaign, IL, USA.
[44] Wolgemuth, C. W.; Goldstein, R. E.; Powers, T. R., Dynamic supercoiling bifurcations of growing elastic filaments, Phys. DNonlinear Phenomena, 190, 3-4, 266-289, (2004) · Zbl 1063.74021
[45] Wunderlich, W., Über ein abwickelbares möbiusband, Monatshefte für Math., 66, 3, 276-289, (1962) · Zbl 0105.14802
[46] Zajac, E. E., Stability of two planar loop elasticas, J. Appl. Mech., 29, 136-142, (1962) · Zbl 0106.38005
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