General theory for plane extensible elastica with arbitrary undeformed shape

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.