Finite Rotations and Finite Elements of Thin Elastic Shells

拟议的调查包括理论研究和计算研究。其基本概念已由作者早些时候提出。这种想法源于薄壳、板、拱和梁中经常发生的变形的几何性质；这些变形的特征通常是小应变，但往往伴随着有限的旋转。这些特点对当代用有限元方法逼近这类结构的过程产生了深远的影响。由于应变很小，元素内部的相对旋转也很小。因此，人们可以用线性方程来描述单个单元的变形，如果材料是钩形的，那么单元的整个描述就是线性方程。然后，有限旋转的出现和相关的非线性可以被引入到全局矩阵的形成中。上述观察结果表明，在用于描述薄结构并预测其响应的模型的应用和公式方面具有实际优势。然而，许多问题仍然没有得到回答，例如这种表述的准确性、趋同标准的适用性以及这种程序的实际利弊。该提议的目的是通过仔细研究理论基础、可选泛函、静止和/或极端标准来寻求答案。最近的互补官能团的配方提供了实现这一目标的手段。用于数值和符号调查的计算程序是探索和检验结果的实用性的其他手段。

The proposed investigation includes theoretical and computational studies. The underlying concept was presented earlier by the author. The idea stems from the geometrical nature of the deformations which often occur in thin shells, plates, arches and beams; these deformations are usually characterized by small strains but often accompanied by finite rotations. These features have far-reaching consequences in the contemporary procedures for approximating such structures by the method of finite elements. Since the strains are small, relative rotations are small within elements. Consequently, one can describe the deformations of the individual element by linear equations and, if the material is hookean, the entire description of the element is given by linear equations. Then the occurrence of finite rotations and the associated nonlinearities can be introduced in the formation of the global matrix. The foregoing observations suggest practical advantages in the application and formulation of the models which are used to describe the thin structures and to predict their response. However, many questions remain unanswered, questions about the accuracy of such formulation, applicability of convergence criteria, and practical advantages and disadvantages of such procedures. The purpose of the proposal is to seek the answers by scrutinizing the theoretical bases, the alternative functionals, stationary and/or extremal criteria. Recent formulations of the complementary functionals provide means to that end. Computational programs for numerical and symbolic investigations are other means for exploring and for testing the practicality of the results.