# The Hellan-Herrmann-Johnson Method for Nonlinear Shells

## TL;DR

This paper extends the Hellan-Herrmann-Johnson finite element method from linear Kirchhoff plates to nonlinear shells, enabling accurate modeling of large strains and rotations with improved discretization for complex geometries.

## Contribution

It introduces a generalized HHJ method for nonlinear shells, incorporating finite strains, large rotations, and geometric degrees of freedom for kinks.

## Key findings

- Demonstrates effectiveness through benchmark examples
- Enables straightforward discretization of structures with kinks
- Shows improved modeling of nonlinear shell behavior

## Abstract

In this paper we derive a new finite element method for nonlinear shells. The Hellan-Herrmann-Johnson (HHJ) method is a mixed finite element method for fourth order Kirchhoff plates. It uses convenient Lagrangian finite elements for the vertical deflection, and introduces sophisticated finite elements for the moment tensor. In this work we present a generalization of this method to nonlinear shells, where we allow finite strains and large rotations. The geometric interpretation of degrees of freedom allows a straight forward discretization of structures with kinks. The performance of the proposed elements is demonstrated by means of several established benchmark examples.