On the static equilibria of branched elastic rods

Abstract

Structures featuring rod-like bodies connected in a branched ensemble are ubiquitous. They appear in antennae, replicating DNA strands, and, most visibly, in the plant kingdom. In spite of this, modeling these branched structures using rod theories have received little attention in the literature. With the help of a general rod theory, which was originally developed by Green and Naghdi, balance laws for these structures are discussed. The governing equations established are then shown to be equivalent to variational principles for tree-like structures. These principles are adopted from works by Ivanov and Tuzhilin and may lead to the development of nonlinear stability criteria for these structures. It is also shown how the conditions at a branching point in these structures can be understood using recent interpretations of Eshelby’s work on material (configurational) forces in adhesion problems. The static configurations are illuminated for several structures using Euler’s theory of the elastica. Instances of multiple possible configurations and novel extensions to the classical problem of the tallest column are also discussed.