Eulerian formulation of constrained elastica

Vincent Denoël, Emmanuel Detournay

Research output: Contribution to journalArticlepeer-review

36 Scopus citations


The formulation of the constrained elastica problem proposed in this paper is predicated on two key concepts: first, the deformed elastica is described by means of the distance from the conduit axis; second, the problem is formulated in terms of the Eulerian curvilinear coordinate of the conduit rather than the natural curvilinear coordinate of the elastica. This approach is further implemented within a segmentation algorithm, which transforms the global constrained elastica problem into a sequence of analogous auxiliary problems that result from dividing the conduit and the elastica into segments limited by contacts. Each auxiliary segment entails solving a segment of elastica subject to isoperimetric constraints corresponding to the assumed positions of the segment ends along the conduit. This new formulation resolves in one stroke a series of issues that afflict the classical Lagrangian approach: (i) the contact detection is reduced to checking whether a threshold on the distance function is violated, (ii) the isoperimetric conditions are transformed into regular boundary conditions, instead of being treated as external integral constraints, (iii) the method yields a well-conditioned set of equations that does not degenerate with decreasing flexural rigidity of the elastica and/or decreasing clearance between the conduit and the elastica.

Original languageEnglish (US)
Pages (from-to)625-636
Number of pages12
JournalInternational Journal of Solids and Structures
Issue number3-4
StatePublished - Feb 2011


  • Constrained elastica
  • Eulerian coordinate
  • Hollow conduit
  • Unilateral contact

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