Classifications of ideal 3D elastica shapes at equilibrium

Abstract

We investigate the equilibrium configurations of the ideal 3D elastica, i.e., inextensible, unshearable, isotropic, uniform, and naturally straight and prismatic rods, with linear elastic constitutive relations. Infinite solution trajectories are expressed analytically and classified in terms of only three parameters related to physical quantities. Orientation of sections and mechanical loading are also well described analytically with these parameters. Detailed analysis of solution trajectories yields two main results. First, all particular trajectories are completely characterized and located in the space of these parameters. Second, a general geometric structure is exhibited for every ideal 3D elastic rod, where the trajectory winds around a core helix in a tube-shaped envelope. This remarkable structure leads to a classification of the general case according to three properties called chirality components. In addition, the geometry of the envelope provides another characterization of the ideal 3D elastica. For both results, the domains and the frontiers of every class are plotted in the space of the parameters.

Bifurcations of buckled, clamped anisotropic rods and thin bands under lateral end translations

“…The behavior of strips under our loading conditions is likely related to the phenomenon of lateral-torsional buckling, known to structural engineers [6].There is much prior work on the configurations of naturally straight rods. Work on the general behavior and classification of solutions includes that of Antman [7,8], Maddocks [9], Nizette and Goriely [10], and Cognet and co-workers [11]. Neukirch and Henderson made a detailed investigation into the connectivity of solutions for rods subject to end thrusts and coaxial twists [12,13].…”

Analytical expression of elastic rods at equilibrium under 3D strong anchoring boundary conditions

“…Then, the quantity µ = K 0 /F is used as unit of length and the ODEs are written using the cylindrical coordinates (ρ, φ, z) of axis (O, e k ). The resulting expressions are integrated analytically [16,19,30] from −∞ to +∞, leading to a full geometric and mechanical description of infinite rods : Although the definition domain parametrization has significant adv obeys to a hierarchy that leads to m fications. In particular, the dimens reduced from six to five by noting tha responds only to a solid rotation of t its tangent at S = 0.…”

“…where n is the normal vector of Frenet and F > 0 is the norm of the vector force F. Case F = 0 leads to helical rod trajectories and must be studied separately [19,30]. Three dimensionless constants (λ, t P , a) are identified, such that ∀ s ∈ R :…”

“…These constants are the three first integral of the problem, and parameter a is directly related to the Hamiltonian [30]. Then, the quantity µ = K 0 /F is used as unit of length and the ODEs are written using the cylindrical coordinates (ρ, φ, z) of axis (O, e k ).…”

“…We provide a general method to express analytically one equilibrium configuration of ideal 3D elastica without self-contact that fulfill any arbitrary strong anchoring boundary conditions. Our approach is based on a complete formalisation of the equilibrium of rods, which has led to classifications of all shapes in a three-dimensional space [30]. In section 2, our objective is transcribed into an inverse problem.…”