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

Cosserat材料是经典连续介质力学模型的推广。除了通常的位移场外，还考虑了一个旋转场。在壳模型的情况下，这些旋转可以很自然地解释为壳法向量的方向和扭转。因此，它们导致了对大菌株的Mindlin-Reissner模型的自然扩展。这种模型的离散化是出了名的困难。由于位形空间具有非线性流形的结构，因此不能使用通常的有限元。现有的方法在处理大旋转时存在不稳定性和彻底崩溃的问题，并且在许多情况下，它们不能保持连续模型的客观性。在过去的几年中，申请人已经开发了测地线有限元（GFE）。这些将正常的拉格朗日元素推广到非线性流形中有值的函数的情况。作为一种特殊情况，得到了任意阶的coserat材料的有限元。离散化允许任意大小的旋转，并且可以证明保持连续模型的客观性。本项目的目的是应用测地线有限元对Cosserat壳进行离散化。为此，将处理一系列越来越复杂的壳模型。关于纯弹性情况的初步工作已经有了。

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.