A theory of locally impenetrable elastic tubes

TL;DR

This paper develops a reduced order elastic tube theory incorporating local impenetrability constraints, linking deformation to Frenet curvature, and derives governing equations with applications to various elastic tube configurations.

Contribution

It introduces a novel reduced kinematic framework that enforces local impenetrability via a variational approach, connecting elastic tube configurations with Kirchhoff rod segments.

Findings

Configurations involve Kirchhoff rod segments connected by constant curvature regions.

The theory successfully models flexible and elastic tubes under self-weight.

It handles highly twisted elastic tubes with the impenetrability constraint.

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.