A theory of locally impenetrable elastic tubes

TL;DR

The paper develops a reduced-order theory for locally impenetrable elastic tubes by actively enforcing the local constraint through a slack function and a Lagrange multiplier within a variational framework based on Kirchhoff rod theory. It shows that deformed configurations decompose into inactive regions governed by the standard Kirchhoff response and active regions with constant curvature bounded by $=rac{1}{t}$, connected by jump conditions that ensure continuity of internal forces and moments. Three representative problems—fully flexible tubes, elastic tubes, and highly twisted 3D tubes—demonstrate how local impenetrability qualitatively changes curvature distribution, introduces nontrivial internal moments and shear, and prevents curvature concentration that would violate the cross-sectional radius. The approach provides insight into self-avoidance, knotting, and packing of slender tubes, with potential extensions to global impenetrability and self-avoidance in biomolecular and architectural contexts.

Abstract

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.

Figures (5)

• Figure 1: Reference configuration $\mathcal{R}_0$ of a circular tube of radius $t$ and centerline ${\bf R}(s)$ parametrised by the arc-length coordinate $s$ is shown on the left. A deformed configuration $\mathcal{R}$ of the tube with centerline ${\bm r} (s)$ is shown in the middle, where an orthonormal set of directors $\{{\bm d}_1,{\bm d}_2,{\bm d}_3\}$. The cross-sectional geometry is shown on the right.
• Figure 2: The solutions for a fully flexible tube of thickness $t=0.025L$ hanging under self-weight. Panels \ref{['fig:idealtube_config1']} and \ref{['fig:idealtube_config2']} are configurations of a fully flexible tube, and a locally impenetrable ideal tube, respectively. Variation of normalized curvature $u_2L$, bending moment $m_2L/K$, shear force $n_1L^2/K$ and axial force $n_3L^2/K$ along the length of the tube are respectively shown in panels \ref{['fig:idealtube_u2']}, \ref{['fig:idealtube_m2']}, \ref{['fig:idealtube_n1']} and \ref{['fig:idealtube_n3']}. The differences in the solutions are negligible away from the active region, and have therefore not been shown. The blue curves represent solutions without the local impenetrability constraint. The solutions for tube with impenetrability constraint is represented by green, with the active region depicted in red.
• Figure 3: Solutions of a pinned elastica of thickness $t/L=0.025$ hanging under self-weight with $\alpha=4.86 \times 10^4$ and varying values of $\Delta/L$. The configurations obtained from standard KRT and the present theory are shown in panels \ref{['fig:elastica_config1']} and \ref{['fig:elastica_config2']}. Panels \ref{['fig:elastica_u2']} and \ref{['fig:elastica_m2']}, respectively show the variation of normalized curvature $u_2L$, and normalized moment $m_2L/K$ and the parameter $\Lambda L/K$ along the length of the tube close to the active region (in red). Variation of shear force $n_1L^2/K$ and axial force $n_3L^2/K$ are shown in panels \ref{['fig:elastica_n1']} and \ref{['fig:elastica_n3']}. The blue curves represent solutions for the standard elastica. The solutions for elastica with local impenetrability constraint is represented by green with the active regions depicted in red.
• Figure 4: Twisted configurations of a Kirchhoff rod and a locally impenetrable elastic tube with radius $t/L=0.05$ and $K_3/K=1$ are shown in \ref{['fig:rod_config1']} and \ref{['fig:rod_config2']}. The position and orientation of the rod is fixed at $s=0$. At $s=1$ the position is fixed, director ${\bm d}_3$ is constrained to be along ${\bf E}_3$ and a twisting moment ${\bm m} \cdot {\bf E}_3$ is applied. Panels \ref{['fig:rod_kappa']} and \ref{['fig:rod_lambda']} show the variation along the length of the rod of normalized Frenet curvature $\kappa L$ and normalized parameter $\Lambda L/K$, respectively. The gray curves represent solutions for a Kirchhoff rod. The solutions for locally impenetrable elastic tube is denoted by blue curves, with the red curves indicating the active region. Solutions obtained from the standard KRT do not converge beyond $m_3\approx66.11$
• Figure 5: The top row shows the variation of the normalized bending strains (\ref{['fig:rod_u1']}) $u_1L$, \ref{['fig:rod_u2']}) $u_2L$ and (\ref{['fig:rod_u3']}) $u_3L$ in a Kirchhoff rod and locally impenetrable elastic tube with an applied moment ${\bm m} \cdot {\bf E}_3 = 65 K/L$. Panels \ref{['fig:rod_n1']} to \ref{['fig:rod_m3']} show the variations in director components of internal forces $\{n_1L^2/K,~n_2L^2/K,~n_3L^2/K\}$ and moments $\{m_1L/K,~m_2L/K,~m_3L/K\}$, respectively. The results for both Kirchhoff rod (gray curves) and locally impenetrable elastic tube (blue and red curves) are shown, with red indicating the active region.