Discretization of geometrically exact elasto-plastic Cosserat shells using geodesic finite elements

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Cosserat materials are a generalization of the classic continuum mechanics model. In addition to the usual displacement field, a field of rotations is considered. In the case of shell models, these rotations can be interpreted naturally as the orientation and torsion of the shell normal vector. They hence lead to a natural extension of the Mindlin-Reissner model for large strains.The discretization of such models is notoriously difficult. The usual finite elements cannot be used, since the configuration space has the structure of a nonlinear manifold. Existing approaches suffer from instabilities and outright crashes when dealing with large rotations, and in many cases they do not preserve the objectivity of continuous models.During the last few years, geodesic finite elements (GFE) have been developed by the applicant. These generalize normal Lagrange elements to the case of functions with values in a nonlinear manifold. As a special case, finite elements of arbitrary order for Cosserat materials are obtained. The discretization allows rotations of any size, and provably preserves the objectivity of continuous models.The goal of this project is to apply geodesic finite elements for the discretization of Cosserat shells. For this, a sequence of shell models of increasing complexity will be treated. Preliminary work covering the purely elastic case is already available.