Matching properties of isometries and elasticity of thin shells

In this talk we will show how the mechanical response of an elastic film is affected by subtle geometric properties of its mid-surface. The crucial role here is played by spaces of weakly regular (Sobolev) isometries or infinitesimal isometries. These are the deformations of the mid-surface preserving its metric up to a certain prescribed order of magnitude, and hence contributing to the stretching energy of the film at a level corresponding to the magnitude of the given external force.

In this line, we will discuss results concerning the matching and density of infinitesimal isometries on convex, developable and axisymmetric surfaces. By a matching property, we refer to the possibility of modifying an infinitesimal isometry of a certain order to make it an infinitesimal isometry of a higher order. In particular, we will show that on a convex surface, any one parameter family of first order bendings generated by a first order isometry can be modified at a higher order of perturbation to a family of exact bendings (exact isometries).

In the second step, we will show how this analysis can be combined with the tools of calculus of variations towards the rigorous derivation of a hierarchy of thin shell theories. The validity of each theory depends on the scaling of the applied force in terms of the vanishing thickness of the reference shell. The obtained hierarchy extends the seminal result of Friesecke, James and Muller valid for flat (plate-like) films, to shells whose mid-surface may have arbitrary geometry. When a matching property is established, the above-mentioned infinte hierarchy effectively collapses to a finite number of theories.