A theory of locally impenetrable elastic tubes

We present a reduced order theory of locally impenetrable elastic tubes. The constraint of local impenetrability – an inequality constraint on the determinant of the 3D deformation gradient – is transferred to the Frenet curvature of the centerline of the tube via reduced kinematics. The constraint is incorporated into a variational scheme, and a complete set of governing equations, jump conditions, and boundary conditions are derived. It is shown that with the local impenetrability actively enforced, configurations of an elastic tube comprise segments of solutions of the Kirchhoff rod theory appropriately connected to segments of constant Frenet curvature. The theory is illustrated by way of three examples: a fully flexible tube hanging under self-weight, an elastic tube hanging under self-weight, and a highly twisted elastic tube.

💡 Research Summary

This paper presents a novel reduced-order theory for modeling slender elastic tubes with uniform circular cross-sections, which explicitly enforces the fundamental continuum assumption of local material impenetrability. The local impenetrability condition, an inequality constraint on the determinant of the 3D deformation gradient (det F > 0), is translated via reduced kinematics into an upper bound on the Frenet curvature (κ) of the tube’s centerline: κ < 1/t, where t is the cross-sectional radius.

The core theoretical advancement lies in actively incorporating this inequality constraint into a variational framework for Kirchhoff rods. The constraint is converted to an equality using a slack function and a Lagrange multiplier (Λ). The ensuing analysis reveals that the equilibrium configuration of such a tube partitions into distinct regions: “inactive” segments where the curvature is below the limit (κ < 1/t, Λ=0) and which obey the standard Kirchhoff rod equations, and “active” segments where the curvature saturates the bound (κ = 1/t, g=0). The active segments are curves of constant Frenet curvature, known as Salkowski curves. The boundaries between these regions are not known a priori but are determined as part of the solution, with specific jump conditions derived from the variational principle.

Within active regions, the constitutive relation between internal moments and bending strains becomes nonlinear due to the presence of the Lagrange multiplier term. Crucially, the force and moment balance equations maintain their standard form throughout the rod, but their solutions in active and inactive regions are coupled through the continuity of position and tangent vectors, as well as the derived jump conditions.

The theory is demonstrated through three archetypal examples. First, a fully flexible (zero bending stiffness) tube hanging under its own weight develops an active region—a circular arc of constant curvature—at its bottom as its ends are brought horizontally closer together. Second, for an elastic tube with bending stiffness hanging under gravity, a dimensionless parameter comparing gravitational force to bending stiffness dictates whether the impenetrability constraint activates before the ends meet. Third, for a highly twisted elastic rod, the impenetrability constraint leads to the nucleation and spread of a helical active region at the rod’s center, preventing the bending energy concentration observed in the unconstrained solution.

In summary, this work provides a rigorous mathematical framework to embed a physical geometric limit, arising from finite thickness, into the classical elastic rod model. It bridges the gap between theories for perfectly flexible chains with hard curvature constraints and those for elastic rods, with implications for understanding phenomena in soft robotics, biomechanics, and material packing.