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 expression appears to be suitable for one-dimensional models of polymers or vesicles, the natural configuration of which is characterized by locally changing curvature. In a linear setting, we derive the macroscopic stress–strain relations, providing an universal criterion for the neutral curve location. In this respect, we further demonstrate that the neutral curve existence constitutes the fundamental requirement for the conformational dynamics of any inextensbile biological filament.