Shaping Closed Elastic Rings Through a Jump in Natural Curvature

We investigate the shapes that a planar, inextensible closed rod can take in the presence of a piecewise inhomogeneity in its natural curvature. Since the associated mathematical problem may admit either unique or multiple solutions depending on the parameters, we reformulate the boundary value problem as a Cauchy problem with constrained initial conditions. This approach allows us to identify parameter regions corresponding to uniqueness or multiplicity of solutions; then, we exclude self-intersecting configurations. Using first integrals, geometrical arguments, perturbation analysis, and numerical simulations, we identify three distinct morphological regimes. Within a fully nonlinear framework, we determine the curves that partition the (κ,s0) plane, where κ denotes the jump in natural curvature located at the curvilinear coordinate s0, into regions associated with different shapes. The perturbation analysis, supported by geometrical arguments, shows good agreement with numerical results obtained from the fully nonlinear model. Our findings also apply to the equivalent case of a homogeneous rod subjected to two pointwise bending moments