Modelling and mathematical analysis of geometrically nonlinear Cosserat shells with higher order and residual effects

• Applicants: Professor Dr. Mircea Birsan; Professor Dr. Patrizio Neff
• Subject Area: Mathematics ; Mechanics
• Term: from 2018 to 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 415894848

We intend to investigate new geometrically nonlinear Cosserat shell models incorporating effects up to order h^5 in the thickness h. The isotropic model should combine membrane, bending and curvature effects at the same time. The Cosserat model naturally includes a frame of orthogonal directors, the last of which does not necessarily coincide with the normal of the surface. This rotation field is coupled to the shell-deformation and augments the well-known Reissner-Mindlin kinematic (one independet director) with so-called in-plane drill rotations.The aim is to formulate this higher order model which should be able to capture additional detailed geometric and topological effects of the initially curved shell. The model will also be extended to multiplicative plasticity, allowing for the consideration of residual stress effects. Other possible extensions concern the thermo-mechanical coupling and shells with residual stresses in applications to design-control problems of ultra-thin three-dimensional objects.At present, the mathematical well-posedness for such curved shell models is completely open. We intend to formulate the first overall existence proof. In our group we have obtained results for the simpler planar Cosserat shell modell with effects up to order h^3; note that in the planar case, no terms of order h^5 arise.The similarities with and differences to existing shell models, mainly based on the Kirchhoff-Love normality assumption, as well as the consistency with linear shell models will be discussed. The elastic and the elastic-viscoplastic shell models will also be investigated for well-posedness. The formulations will be given in matrix notation, which will simplify the FEM-implementation as well as the mathematical treatment, since the structure of the equations is closer to the 3D-formulation.Major challenges are the coupling of geometrical nonlinearities with the topology of the shell and the geometry of the group SO(3) for the additional orthogonal frame as well as the physical nonlinearity in the plastic coupling.The method we follow in the first period of the project is an educated ansatz for the three-dimensional shell deformation with analytical thickness integration, which leads us to obtain completely two-dimensional sets of equations in variational form. This programme has already been successfully applied to the plate (flat-shell) model.We expect major new insights into the deformation behaviour of thin structures. Furthermore, our mathematical inquiries will require novel mathematical tools, e.g. new Korn's inequalities for shells with residual stresses.

• DFG Programme: Research Grants