The θ-formulation of the two-dimensional elastica: buckling and boundary layer theory

The equations of a planar elastica under pressure can be rewritten in useful form by parametrising the variables terms local orientation angle, $\theta$, instead arc length. This ``$\theta$-formulation'' lends itself to particularly easy boundary layer analysis limit weak bending stiffness. Within this parameterization, layers are located at inflexion points, where $\theta$ is extremum, and they connect regions low large curvature. A simple composite solution derived without resorting elliptic functions integrals. approximation used as an elementary building block describe complex shapes. Applying theory study elastic ring uniform subject set point forces, we discover snapping instability. instability confirmed numerical simulations. Finally, carry out experiments find good agreement with experimental shape deformed elastica.