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

我们打算研究新的几何非线性Cosserat壳模型，该模型考虑了厚度h的h^5阶效应。各向同性模型应同时考虑膜效应、弯曲效应和曲率效应。Cosserat模型自然包括一个正交指向标架，其中最后一个不一定与曲面的法线重合。这个旋转场与壳体变形相耦合，并用所谓的平面内钻探旋转来扩充众所周知的Reissner-Mindlin运动学(一个独立的导向器)，目的是建立这个高阶模型，它应该能够捕捉到最初弯曲的壳体的额外的详细的几何和拓扑效应。该模型还将扩展到乘性塑性，允许考虑残余应力效应。其他可能的扩展涉及热力耦合和残馀应力壳在超薄三维物体的设计控制问题中的应用。目前，这种曲壳模型的数学适定性是完全开放的。我们打算建立第一个整体存在的证明。在本课题组中，我们得到了影响达到h^3阶的更简单的平面Cosserat壳模型的结果；注意，在平面情况下，不存在h^5阶项。我们将讨论与现有壳模型的异同，主要基于Kirchhoff-Love正态假设，以及与线性壳模型的一致性。弹性壳模型和弹粘塑性壳模型的适定性也将被研究。公式将以矩阵形式给出，这将简化有限元的实现和数学处理，因为方程的结构更接近于三维公式。主要的挑战是几何非线性与壳的拓扑和群的几何耦合，SO(3)对于附加的正交框架以及塑性耦合中的物理非线性。我们在项目的第一阶段采用的方法是三维壳变形的解析厚度积分的有教益的分析方法，这使我们得到了完全二维的变分形式的方程组。这个程序已经成功地应用于板(扁壳)模型。我们期待着对薄结构的变形行为有重大的新见解。此外，我们的数学查询将需要新的数学工具，例如具有残余应力的壳的新Korn不等式。

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.