Abstract

Since the beginning of the 1900s, linearized Cosserat elasticity is well known and often used in the engineering community for modeling micropolar elastic solids. But from a mathematician’s perspective a geometrically nonlinear version of the model is interesting. Existence of solutions for the latter has been known for about 20 years, while regularity questions were investigated only during the last couple of years.

In the first part of the talk, the Cosserat bulk model will be introduced and an overview of recently developed different regularity results will be given, for Cosserat energy minimizers as well as for critical points. Classical regularity theory for harmonic maps into manifolds is an essential tool in deriving those results. At the same time, the geometric nature of the model’s nonlinearities in some cases allows not only regular, but also quite singular solutions to exist, which will be the focus of the second part of the talk.

Finally, those singular solutions do not exist, when we go away from the (3d-) Cosserat bulk model towards a Cosserat model for shells, i.e. very thin materials, whose elastic behaviour is governed by their (2d-) midsurface.