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

Subject Area Mathematics
Mechanics
Term from 2018 to 2023
Project identifier Deutsche Forschungsgemeinschaft (DFG) - Project number 415894848
 
Final Report Year 2024

Final Report Abstract

We have investigated new geometrically nonlinear Cosserat shell models incorporating effects up to order h5 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 independent director) with so-called in-plane drill rotations. The main goal is to formulate higher order models able to capture additional detailed geometrical and topological effects of the initially curved shell and incorporate more information about the original considered three-dimensional domain (which are omitted in a shell model with effects up to h3), to study the well-posedness and the accuracy of these models, and to compare them with the already existing approaches. The method that we follow in the envisaged first period of the project is an educated ansatz for the three-dimensional elastic shell deformation with analytical thickness integration, which leads us to obtain completely two-dimensional sets of equations in variational form. The step-by-step approach we follow allows us to give a transparent approximation of the three-dimensional parental problem. This programme has been successfully followed previously for the plate (flat-shell) model and it has been shown that, in this case, a quadratic ansatz in the thickness direction does not lead to terms of order O(h5 ). These higher-order effects become important only when a curved initial shell configuration is considered; it is plausible to assume a preference of the shell deformation for certain directions depending on the initially given curvature of the shell. The purely elastic nonlinear Cosserat shell model will also be extended to multiplicative plasticity. This has allowed us to address the problem of residual stress effects, which have applications in design control problems of nano three-dimensional objects, situations in which a model up to order O(h5 ) may be essential. Since the considered ansatz is quadratic with respect to the thickness, the solution of the resulting two-dimensional shell problem will be able to offer a very good approximation of the initial considered three-dimensional problem. The mathematical well-posedness for such curved shell models was completely open. We intend to formulate the first overall existence proof. 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 shell models is shown to be well-posed. The formulations was first be given in matrix notation in order to simplify the FEM implementation and to facilitate the mathematical treatment, since the structure of the equations is closer to the 3D-formulation. The major challenges was the coupling of geometrical nonlinearities with the initial 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. As planned we have found new substantial insights into the deformation behaviour of thin structures. The resulting well-posedness problems has requires new mathematical tools, e.g. new Korn-type inequalities.

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