Bifurcation of elastic curves with modulated stiffness

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