Eulerian Formulation for an Extensible Elastic Rod

This paper reformulates the governing equations of an extensible elastic rod by reference to a given spatial curve. This Eulerian formulation is motivated by the need to solve efficiently the constrained elastica problem encountered in many medical and engineering applications, in which a thin rod is inserted in a tortuous conduit. The Eulerian reformulation of the equations hinges on the restatement of the rod local equilibrium in terms of derivatives with respect to the curvilinear coordinate associated with the reference curve and the description of the rod deflection as a perturbation of this curve. The originality of the proposed formulation lays in the axially unconstrained character of the resulting system such that the determination of the rod configuration between two fixed points reduces to the resolution of a classical boundary value problem.