General theory for plane extensible elastica with arbitrary undeformed shape

TL;DR

This paper derives a comprehensive strain energy expression for plane extensible elastica with arbitrary shapes, applicable to polymers and biological filaments, and establishes universal criteria for their conformational behavior.

Contribution

It introduces a general strain energy formula for extensible elastica with arbitrary undeformed shapes, advancing the modeling of biological and polymeric filaments.

Findings

Derived the strain energy expression for arbitrary undeformed shapes.

Established universal stress-strain relations and neutral curve criteria.

Highlighted the importance of natural curve existence in filament dynamics.

Abstract

A general expression for the strain energy of a homogeneous, isotropic, plane extensible elastica with an arbitrary undeformed configuration is derived. This energy constitutes the correct expression for one-dimensional models of polymers or vesicles, whose natural configuration is characterized by locally changing curvature. We derive the macroscopic stress-strain relations, providing an universal criterion for the neutral curve location. In this respect, we demonstrate that the natural curve existence constitutes the fundamental requirement for the conformational dynamics of any inextensbile biological filament.