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

具有高阶和残差效应的几何非线性 Cosserat 壳的建模和数学分析

• 批准号: 415894848
• 负责人: Professor Dr. Mircea Birsan
• 金额: --
• 依托单位: Lehrstuhl für Analysis
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2018
• 资助国家: 德国
• 起止时间: 2017-12-31 至 2022-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/415894848?language=en
• 关键词: Modelling; mathematical; analysis; geometrically; nonlinear

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.

我们打算研究一种新的几何非线性Cosserat壳模型，该模型在厚度h上考虑了h^5阶效应。各向同性模型应同时结合联合收割机薄膜、弯曲和曲率效应。Cosserat模型自然包括正交指向标的框架，其中最后一个不一定与曲面的法线重合。这个旋转场耦合到壳变形和增强著名的Reissner-Mindlin运动学（一个independet导演）与所谓的面内drill rotations.The的目的是制定这个高阶模型，应该能够捕获额外的详细的几何和拓扑效果的最初弯曲的壳。该模型还将扩展到乘法塑性，允许考虑残余应力的影响。其他可能的扩展涉及热-机械耦合和具有残余应力的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。我们打算制定第一个全面的存在证明。在我们的小组中，我们已经得到了更简单的平面Cosserat壳模型的结果，其效应的阶数达到h^3;注意，在平面情况下，没有出现h^5阶的项。我们将讨论与现有壳模型的相似性和差异，主要是基于Kirchhoff-Love正态性假设，以及与线性壳模型的一致性。弹性和弹粘塑性壳模型也将被调查的适定性。公式将以矩阵符号给出，这将简化FEM实现以及数学处理，主要的挑战是几何非线性与壳的拓扑和SO（3）群的几何的耦合在第一期工程中，我们采用的方法是一种有根据的分析方法，它可以用来分析三种情况，三维壳变形与分析厚度积分，这使我们得到完全二维方程组的变分形式。该程序已成功地应用于板（扁壳）模型，我们期待着对薄结构的变形行为有新的认识。此外，我们的数学调查将需要新的数学工具，如新的科恩不等式壳与残余应力。