# Mathematical modeling of a Cosserat method in finite-strain holonomic plasticity

## TL;DR

This paper introduces a new numerical method for finite-strain Cosserat plasticity that improves efficiency and accuracy through quaternion-based micro-rotation parameterization and a novel two-pass preconditioning scheme, validated by benchmark tests.

## Contribution

The paper presents a novel two-pass preconditioning scheme and quaternion-based micro-rotation parameterization for finite-strain Cosserat plasticity, enhancing computational performance over previous methods.

## Key findings

- The new algorithm outperforms the old in efficiency and convergence.
- Numerical simulations validate the effectiveness of the proposed method.
- The model is suitable for complex materials with large size effects.

## Abstract

This article deals with the mathematical derivation and the validation over benchmark examples of a numerical method for the solution of a finite-strain holonomic (rate-independent) Cosserat plasticity problem for materials, possibly with microstructure. Two improvements are made in contrast to earlier approaches: First, the micro-rotations are parameterized with the help of an Euler-Rodrigues formula related to quaternions. Secondly, as main result, a novel two-pass preconditioning scheme for searching the energy-minimizing solutions based on the limited memory Broyden-Fletcher-Goldstein-Shanno quasi-Newton method is proposed that consists of a predictor step and a corrector-iteration. After outlining the necessary adaptations to the model, numerical simulations compare the performance and efficiency of the new and the old algorithm. The proposed numerical model can be effectively employed for studying the mechanical response of complicated materials featuring large size effects.