Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Adjacent Equilibria in Highly Flexible Upright Loop on Rigid Foundation

Abstract

For very slender structural components, self-weight may compete with elastic flexural stiffness in determining equilibrium configurations. In cases where the inherent elastic stiffness is low (relative to self-weight) we observe a variety of types of highly nonlinear behavior in the equilibrium shapes, together with changes in the natural frequencies of small oscillations about these equilibrium configurations. This technical note describes a specific phenomenon observed in experiments on very slender polycarbonate loops. In addition to profound changes in equilibrium shapes as a function of weight-to-stiffness ratio, under some circumstances it is possible to have two adjacent, co-existing equilibrium configurations. This robust, highly nonlinear snap-through behavior is demonstrated by perturbing from one shape to the other.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

References

  1. 1.

    Wang CY, Watson LT (1981) Equilibrium of heavy elastic cylindrical shells. J Appl Mech 48:582–586

  2. 2.

    Virgin LN (2007) Vibration of Axially Loaded Structures. Cambridge University Press, Cambridge, UK

  3. 3.

    Hertel T, Walkup RE, Avouris P (1998) Deformation of carbon nanotubes by surface van der Waals forces. Phys Rev B 58:13870–13873

  4. 4.

    Pantano A, Parks DM, Boyce MC (2004) Mechanics of deformation of single- and multi-wall carbon nanotubes. J Mech Phys Solids 52:789–821

  5. 5.

    Raux PS, Reis PM, Bush JWM, Clanet C (2010) Rolling ribbons. Phys Rev Lett 105:044301:105

  6. 6.

    Zheng M, Ke C (2011) Mechanical deformation of carbon nanotube nano-rings on flat substrate. J Appl Phys 074304:109

  7. 7.

    Shi J, Muftu S, Wan KT (2012) Adhesion of a compliant cylindrical shell onto a rigid substrate. J Appl Mech 77:041013

  8. 8.

    Liu JL, Xia R (2013) A unified analysis of a micro-beam, droplet and CNT ring adhered on a substrate: Calculation of variation with movable boundaries. Acta Mechanica Sinica 29:62–72

  9. 9.

    Plaut RH, Virgin LN (2014) Deformation and vibration of upright loops on a foundation and of hanging loops. Int J Solids Struct 51:3067–3075

  10. 10.

    Zakrzhevskii AK, Tkachenko VF, Khoroshilov VS (2010) Natural modes and frequencies of in-plane vibrations of a fixed elastic ring. Int Appl Mech 46:1420–1427

  11. 11.

    Thompson JMT, Hunt GW (1973) A General Theory of Elastic Stability. Wiley, London

  12. 12.

    Bellow DG, Ford G, Kennedy JS (1965) Anticlastic behavior of flat plates. Exp Mech 5:227–232

  13. 13.

    Conway HD, Nickola WE (1965) Anticlastic action of flat sheets in bending. Exp Mech 5:115–119

  14. 14.

    Campanile LF, Jahne R, Hasse A (2011) Exact analysis of the bending of wide beams by a modified elastica approach. J Mech Eng Sci 225:2759–2764

  15. 15.

    Taylor M, Bertoldi K, Steigmann DJ (2014) Spatial resolution of wrinkle patterns in thin elastic sheets at finite strain. J Mech Phys Solids 62:163–180

Download references

Author information

Correspondence to L.N. Virgin.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Virgin, L., Plaut, R. & Cartee, E. Adjacent Equilibria in Highly Flexible Upright Loop on Rigid Foundation. Exp Mech 55, 1191–1197 (2015). https://doi.org/10.1007/s11340-015-0011-7

Download citation

Keywords

  • Weight effects
  • Adjacent equilibria
  • Loop
  • Vibration