C1‐continuous FEM for Kirchhoff plates and large deformation

Abstract

In this article we present a theory for large deformation of plates. Herein we combine aspects of the common 3D‐theory for large deformation with the Kirchhoff hypothesis for reducing the dimension from 3D to 2D. Even though the Kirchhoff assumption was developed for small strain and linear material laws, we want to investigate the deformation of thin plates made of isotropic non‐linear material as 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 non‐linear equation system defining the unknown displacement vector U. The aim of this article is to apply the incremental Newton technique with a conformal, C1‐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. We demonstrate the practicability and fast convergence.