Bifurcation of elastic curves with modulated stiffness

Katharina Brazda (1) , Gaspard Jankowiak (2) , Christian Schmeiser (1) , Ulisse Stefanelli (1, 3, 4)

1 Fakultät für Mathematik [Wien]
2 RICAM - Johann Radon Institute for Computational and Applied Mathematics
3 ViRAPID - Vienna Research Platform on Accelerating Photoreaction Discovery
4 IMATI - Istituto di Matematica Applicata e Tecnologie Informatiche

We investigate the equilibrium configurations of closed planar elastic curves of fixed length, whose stiffness (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 onedimensional 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 regularization term. Then, we derive the system of Euler-Lagrange equations and study the bifurcation structure of solutions with respect to the model parameters. We present analytical results and numerical simulations.

Domaines: Equations aux dérivées partielles [math.AP]

Katharina Brazda, Gaspard Jankowiak, Christian Schmeiser, Ulisse Stefanelli. Bifurcation of elastic curves with modulated stiffness. 2020. ⟨hal-03025931v1⟩