Plateau, the Cosserat, and the mechanics of shapes

Giulio Giuseppe Giusteri (Politecnico di Milano, Italy)

Thursday 7th February, 2019 14:00-15:00 Maths 311B


There are numerous situations in which the mechanics of slender or thin objects manifests itself through significant shape modifications, even in the presence of small stresses. This is because small local strains can have a significant cumulative effect on the shape of a continuum. By "shape" we indicate geometric features of an object that are invariant under isometries of the three-dimensional ambient space. We name "shape energies" of continua those elastic energy functionals that depend only on such features.

In this talk, recent results will be presented about the mechanics of filaments, modelled by special Cosserat rods, and of liquid membranes, modelled as area-minimizing surfaces. We will first discuss how the strains defining the shape of a rod can be successfully viewed as fundamental descriptors of the system. This entails encoding the shape not in a description of what can be seen, but of how the rod can be reconstructed starting from one of its cross sections. Moreover, it leads naturally to discretisation schemes that avoid the need for interpolation procedures that introduce some arbitrariness in popular numerical schemes.

Considering the coupled effects of the dynamics of rods and membranes leads to the statement of the Kirchhoff-Plateau problem, in which a liquid film spans a flexible filament (with arbitrary cross-sectional shape) in the form of a closed loop. Using techniques of the calculus of variations and geometric measure theory, the shape energy of the system is minimized under the constraint of non-interpenetration of matter, but allowing for points on the surface of the bounding loop to come into contact. This problem is rich in physical modelling and challenging for the presence of non-smooth and non-convex constraints. Finally, some research directions related to the modelling of structured surfaces will be introduced.

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