Kirchhoff Plates and Large Deformations - Modelling and C^1-continuous Discretization

In this thesis a theory for large deformation of plates is presented. Herein aspects of the common 3D-theory for large deformation with the Kirchhoff hypothesis for reducing the dimension from 3D to 2D is combined. Even though the Kirchhoff assumption was developed for small strain and linear material laws, the deformation of thin plates made of isotropic non-linear material was investigated in a numerical experiment. Finally a heavily deformed shell without any change in thickness arises. This way of modeling leads to a two-dimensional strain tensor essentially depending on the first two fundamental forms of the deformed mid surface. Minimizing the resulting deformation energy one ends up with a nonlinear equation system defining the unknown displacement vector U. The aim of this thesis was to apply the incremental Newton technique with a conformal, C^1-continuous finite element discretization. For this the computation of the second derivative of the energy functional is the key difficulty and the most time consuming part of the algorithm. The practicability and fast convergence are demonstrated by different numerical experiments.:1 Introduction 2 The deformation problem in the three-dimensional space 2.1 General differential geometry of deformation in the three-dimensional space 2.2 Equilibrium of forces 2.3 Material laws 2.4 The weak formulation 3 Newton’s method 3.1 The modified Newton algorithm 3.2 Second linearization of the energy functional 4 Differential geometry of shells 4.1 The initial mid surface 4.2 The initial shell 4.3 The plate as an exception of a shell 4.4 Kirchhoff assumption and the deformed shell 4.4.1 Differential geometry of the deformed shell 4.4.2 The Lagrangian strain tensor of the deformed plate 5 Shell energy and boundary conditions 5.1 The resulting Kirchhoff deformation energy 5.2 Boundary conditions 5.3 The resulting weak formulation 6 Newton’s method and implementation 6.1 Newton algorithm 6.2 Finite Element Method (FEM) 6.2.1 Bogner-Fox-Schmidt (BFS) elements 6.2.2 Hsiegh-Clough-Tocher (HCT) elements 6.3 Efficient solution of the linear systems of equation 6.4 Implementation 6.5 Newton’s method and mesh refinement 7 Numerical examples 7.1 Plate deflection 7.1.1 Approximation with FEM using BFS-elements 7.1.2 Approximation with FEM using reduced HCT-elements 7.2 Bending-dominated deformation 7.2.1 Approximation with FEM using BFS-elements 7.2.1.1 1st example: Cylinder 7.2.1.2 2nd example: Cylinder with further rotated edge normals 7.2.1.3 3rd example: Möbiusstrip 7.2.1.4 4th example: Plate with twisted edge 7.2.2 Approximation with FEM using reduced HCT-elements 7.2.2.1 1st example: Partly divided annular octagonal plate 7.2.2.2 2nd example: Divided annulus with rotated edge normals 8 Outlook and open questions Bibliography Notation Theses List of Figures List of Table