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One-dimensional von Kármán models for elastic ribbons

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Abstract

By means of a variational approach we rigorously deduce three one-dimensional models for elastic ribbons from the theory of von Kármán plates, passing to the limit as the width of the plate goes to zero. The one-dimensional model found starting from the “linearized” von Kármán energy corresponds to that of a linearly elastic beam that can twist but can deform in just one plane; while the model found from the von Kármán energy is a non-linear model that comprises stretching, bendings, and twisting. The “constrained” von Kármán energy, instead, leads to a new Sadowsky type of model.

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References

  1. Agostiniani V, DeSimone A, Koumatos K (2016) Shape programming for narrow ribbons of nematic elastomers. J Elast. doi:10.1007/s10659-016-9594-1

    MATH  Google Scholar 

  2. Armon S, Efrati E, Kupferman R, Sharon E (2011) Geometry and mechanics in the opening of chiral seed pods. Science 333:1726–1730

    Article  ADS  Google Scholar 

  3. Bartels S, Hornung P (2015) Bending paper and the Möbius strip. J Elast 119:113–136

    Article  MATH  Google Scholar 

  4. Chen Z (2014) Geometric nonlinearity and mechanical anisotropy in strained helical nanoribbons. Nanoscale 6:9443–9447

    Article  ADS  Google Scholar 

  5. Chopin J, Démery V, Davidovitch B (2015) Roadmap to the morphological instabilities of a stretched twisted ribbon. J Elast 119:137–189

    Article  MathSciNet  MATH  Google Scholar 

  6. Chung DS, Benedek GB, Konikoff FM, Donovan JM (1993) Elastic free energy of anisotropic helical ribbons as metastable intermediates in the crystallization of cholesterol. Proc Natl Acad Sci USA 90:11341–11345

    Article  ADS  Google Scholar 

  7. Dias MA, Audoly B (2015) “Wunderlich, Meet Kirchhoff”: a general and unified description of elastic ribbons and thin rods. J Elast 119:49–66

    Article  MathSciNet  MATH  Google Scholar 

  8. Efrati E (2015) Non-Euclidean ribbons: generalized Sadowsky functional for residually-stressed thin and narrow bodies. J Elast 119:251–261

    Article  MathSciNet  MATH  Google Scholar 

  9. Freddi L, Hornung P, Mora MG, Paroni R (2016) A corrected Sadowsky functional for inextensible elastic ribbons. J Elast 123:125–136

    Article  MathSciNet  MATH  Google Scholar 

  10. Freddi L, Hornung P, Mora MG, Paroni R (2016) A variational model for anisotropic and naturally twisted ribbons. SIAM J Math Anal 48:3883–3906

    Article  MathSciNet  MATH  Google Scholar 

  11. Freddi L, Mora MG, Paroni R (2012) Nonlinear thin-walled beams with a rectangular cross-section—part I. Math Models Methods Appl Sci 22:1150016

    Article  MathSciNet  MATH  Google Scholar 

  12. Freddi L, Mora MG, Paroni R (2013) Nonlinear thin-walled beams with a rectangular cross-section—part II. Math Models Methods Appl Sci 23:743–775

    Article  MathSciNet  MATH  Google Scholar 

  13. Freddi L, Morassi A, Paroni R (2007) Thin-walled beams: a derivation of Vlassov theory via \(\Gamma\)-convergence. J Elast 86:263–296

    Article  MathSciNet  MATH  Google Scholar 

  14. Friesecke G, James RD, Müller S (2006) A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence. Arch Ration Mech Anal 180:183–236

    Article  MathSciNet  MATH  Google Scholar 

  15. Fosdick R, Fried E (2016) The mechanics of ribbons and Möbius bands. Springer, Dordrecht

    Book  MATH  Google Scholar 

  16. Hinz DF, Fried E (2015) Translation of Michael Sadowsky’s paper “An elementary proof for the existence of a developable Möbius band and the attribution of the geometric problem to a variational problem”. J Elast 119:3–6

    Article  MATH  Google Scholar 

  17. Hornung P (2014) A remark on constrained von Kármán theories. Proc R Soc Lond Ser A Math Phys Eng Sci 470(2170):20140346

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Kirby NO, Fried E (2015) Gamma-limit of a model for the elastic energy of an inextensible ribbon. J Elast 119:35–47

    Article  MathSciNet  MATH  Google Scholar 

  19. Kit OO, Tallinen T, Mahadevan L, Timonen J, Koskinen P (2012) Twisting graphene nanoribbons into carbon nanotubes. Phys Rev B 85:085428

    Article  ADS  Google Scholar 

  20. Le Dret H (1991) Problèmes variationnels dans les multi-domaines. Modélisation des jonctions et applications. In: Recherches en Mathématiques Appliquées, vol 19. Masson, Paris

  21. Sadowsky M (1930) Ein elementarer Beweis für die Existenz eines abwickelbaren Möbiuschen Bandes und die Zurückführung des geometrischen Problems auf ein Variationsproblem. Sitzungsber. Preuss. Akad. Wiss. 1930 – Mitteilung vom 26. Juni, pp 412–415

  22. Sawa Y, Yeb F, Urayamaa K, Takigawaa T, Gimenez-Pintob T, Selingerb RLB, Selingerb JV (2011) Shape selection of twist-nematic-elastomer ribbons. Proc Natl Acad Sci USA 108:6364–6368

    Article  ADS  Google Scholar 

  23. Starostin EL, van der Heijden GHM (2015) Equilibrium shapes with stress localisation for inextensible elastic Möbius and other strips. J Elast 119:67–112

    Article  MATH  Google Scholar 

  24. Todres RE (2015) Translation of W. Wunderlich’s, “On a developable Möbius band”. J Elast 119:23–34

    Article  MathSciNet  MATH  Google Scholar 

  25. von Kármán T (1910) Festigkeitsproblem im Maschinenbau. Encyk. D. Math. Wiss. IV/4:311–385

    Google Scholar 

  26. Washizu K (1975) Variational methods in elasticity and plasticity, 2nd edn. Pergamon Press, New York

    MATH  Google Scholar 

  27. Wunderlich W (1962) Über ein abwickelbares Möbiusband. Monatsh Math 66:276–289

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

MGM acknowledges support by GNAMPA–INdAM under Project 2016 “Multiscale analysis of complex systems with variational methods” and by the ERC under Grant No. 290888 “Quasistatic and Dynamic Evolution Problems in Plasticity and Fracture”. PH acknowledges support by the DFG.

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Authors and Affiliations

  1. DMIF, Università degli Studi di Udine, via delle Scienze 206, 33100, Udine, Italy

    Lorenzo Freddi

  2. Fachrichtung Mathematik, TU Dresden, 01062, Dresden, Germany

    Peter Hornung

  3. Dipartimento di Matematica, Università di Pavia, via Ferrata 1, 27100, Pavia, Italy

    Maria Giovanna Mora

  4. DADU, Università degli Studi di Sassari, Palazzo del Pou Salit, 07041, Alghero, SS, Italy

    Roberto Paroni

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  1. Lorenzo Freddi
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  2. Peter Hornung
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  3. Maria Giovanna Mora
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  4. Roberto Paroni
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Correspondence to Roberto Paroni.

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Freddi, L., Hornung, P., Mora, M.G. et al. One-dimensional von Kármán models for elastic ribbons. Meccanica 53, 659–670 (2018). https://doi.org/10.1007/s11012-017-0666-5

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  • DOI: https://doi.org/10.1007/s11012-017-0666-5

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