Stability and bifurcation of piecewise-homogeneous closed elastic rods

This project is suitable for a mathematically oriented Master’s thesis or as the starting point of a PhD project in applied mathematics and continuum mechanics.

Context

Tubular epithelial ducts can be modeled, in planar cross-section, as closed elastic rods whose mechanical properties change across interfaces. This minimal model already produces several competing equilibrium shapes, including nearly circular, pear-shaped, and saddle-like configurations. Existing work has classified important equilibrium branches and computed morphological separatrices that divide their parameter regimes. The central open question is which of these equilibria are stable and how the branches reorganize when the interface position $s_0$ and the jump in natural curvature $\kappa$ vary.

The biological motivation comes from Ambrosi, Favata, Paroni, and Tomassetti (2026), Oncogenic transformation of tubular epithelial ducts: How mechanics affects morphology. The mathematical and computational starting point is the related five-author study by Ambrosi, Favata, Paroni, Tomassetti, and Turzi, Shaping closed elastic rings through a jump in natural curvature, together with its analytical and numerical code.

Main research questions

  1. How should the constrained second variation be formulated for a closed, piecewise-homogeneous elastica with interface and isoperimetric conditions?
  2. Which quasi-circular, pear-shaped, and saddle-like branches are stable, and where do they exchange stability?
  3. What is the complete two-parameter bifurcation diagram as $s_0$ and $\kappa$ vary?
  4. What happens at the triple point where the known separatrices meet?
  5. Can the singular limit in which one angular sector shrinks to zero be described by matched asymptotics or an appropriate renormalization?

Possible work packages

1. Variational and spectral formulation

Derive the constrained Hessian, including the correct transmission and closure (isoperimetric) constraints. Establish a practical criterion for local energetic stability from the spectrum of the constrained Hessian or Jacobi operator, and compare it with conjugate-point or index methods.

2. Numerical continuation

Implement a continuation and eigenvalue solver for equilibrium branches. Track genuine fold-bifurcation and stability-loss curves in the two-dimensional parameter space, and relate them to the known morphological separatrices and their meeting point without conflating the two kinds of boundary.

3. Singular and limiting regimes

Analyze the limit in which one part of the rod becomes vanishingly short. Determine the correct scaling, explain the divergence found by direct substitution, and compare the asymptotic prediction with computations.

4. Extensions

Depending on the student’s interests, the work can extend toward rigorous dimension reduction from a two-dimensional elastic model with surface energy, admissibility and self-contact, or non-planar rods with bending–torsion coupling.

Expected outcomes

Background

The project is best suited to a student with a solid foundation in calculus of variations, ordinary differential equations, and numerical analysis. Previous exposure to elasticity, bifurcation theory, or scientific Python is useful but not essential.

Code and manuscript sources

A private consolidated PRJ-Epitelium repository preserves the complete Git histories of the computational and manuscript repositories as subtrees. Access to the research package can be arranged with prospective students.

Students interested in the project are welcome to contact me to discuss scope, prerequisites, and supervision.