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Seminar in Mathematical physics and related subjects: “Plateau, the Cosserat, and the mechanics of shapes”

Tuesday, January 29, 2019, 14:30 - Room 2AB45 - Giulio Giusteri

ARGOMENTI: Seminars

Giulio Giusteri (Politecnico di Milano)
“Plateau, the Cosserat, and the mechanics of shapes”
Tuesday, January 29, 2019, 14:30-15:30
Room 2AB45

Abstract
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 due to the fact that small local strains can have a large 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, I will present recent results about the mechanics of filaments, modeled by special Cosserat rods, and of liquid membranes, modeled as area-minimizing surfaces. I will first discuss how the strains defining the shape of a rod can be successfully viewed as fundamental descriptors of the system both from the geometrical and the mechanical viewpoint.
I will then consider the coupled effects of the dynamics of rods and membranes that characterize the Kirchhoff-Plateau problem, in which a flexible filament (with arbitrary cross-sectional shape) in the form of a closed loop is spanned by a liquid film. 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 modeling and challenging for the presence of non-smooth and non-convex constraints.

References:
1. G. G. Giusteri, P. Franceschini, E. Fried (2016) “Instability paths in the Kirchhoff-Plateau problem.” J. Nonlinear Sci. 26, 1097-1132.
2. G. G. Giusteri, L. Lussardi, E. Fried (2017) “Solution of the Kirchhoff-Plateau problem.” J. Nonlinear Sci. 27, 1043-1063.
3. G. G. Giusteri, E. Fried (2018) “Importance and effectiveness of representing the shapes of Cosserat rods and framed curves as paths in the special Euclidean algebra.” J. Elast. 132, 43-65.