  We investigate the equilibrium configurations of closed planar elastic curves
of fixed length, whose stiffness, also known as the bending rigidity, depends
on an additional density variable. The underlying variational model relies on
the minimization of a bending energy with respect to shape and density and can
be considered as a one-dimensional analogue of the Canham-Helfrich model for
heterogeneous biological membranes. We present a generalized Euler-Bernoulli
elastica functional featuring a density-dependent stiffness coefficient. In
order to treat the inherent nonconvexity of the problem we introduce an
additional length scale in the model by means of a density gradient term. We
derive the system of Euler-Lagrange equations and study the bifurcation
structure of solutions with respect to the model parameters. Both analytical
and numerical results are presented.
