Journal of Nonlinear Science

, Volume 26, Issue 4, pp 1097–1132 | Cite as

Instability Paths in the Kirchhoff–Plateau Problem

  • Giulio G. Giusteri
  • Paolo Franceschini
  • Eliot Fried


The Kirchhoff–Plateau problem concerns the equilibrium shapes of a system in which a flexible filament in the form of a closed loop is spanned by a soap film, with the filament being modeled as a Kirchhoff rod and the action of the spanning surface being solely due to surface tension. Adopting a variational approach, we define an energy associated with shape deformations of the system and then derive general equilibrium and (linear) stability conditions by considering the first and second variations of the energy functional. We analyze in detail the transition to instability of flat circular configurations, which are ground states for the system in the absence of surface tension, when the latter is progressively increased. Such a theoretical study is particularly useful here, since the many different perturbations that can lead to instability make it challenging to perform an exhaustive experimental investigation. We generalize previous results, since we allow the filament to possess a curved intrinsic shape and also to display anisotropic flexural properties (as happens when the cross section of the filament is noncircular). This is accomplished by using a rod energy which is familiar from the modeling of DNA filaments. We find that the presence of intrinsic curvature is necessary to obtain a first buckling mode which is not purely tangent to the spanning surface. We also elucidate the role of twisting buckling modes, which become relevant in the presence of flexural anisotropy.


Minimal surface Flexible boundary Kirchhoff rod  Buckling analysis Shape energy 

Mathematics Subject Classification

74G60 74K10 35B35 35Q74 



We gratefully acknowledges support from the Okinawa Institute of Science and Technology Graduate University with subsidy funding from the Cabinet Office, Government of Japan.


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Giulio G. Giusteri
    • 1
  • Paolo Franceschini
    • 1
    • 2
  • Eliot Fried
    • 1
  1. 1.Mathematical Soft Matter UnitOkinawa Institute of Science and Technology Graduate UniversityOnnaJapan
  2. 2.Dipartimento di Matematica e FisicaUniversità Cattolica del Sacro CuoreBresciaItaly

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