Dimension reduction for elastoplastic rods in the bending regime

TL;DR

This paper develops a rigorous dimension reduction model for elastoplastic rods in the bending regime, accounting for finite deformations with small strains and large rotations, using advanced mathematical techniques.

Contribution

It introduces a novel ansatz for the mutual recovery sequence in the bending regime, extending previous infinitesimal deformation models to finite deformations with small strains.

Findings

Derived an effective bending model from 3D finite plasticity.

Introduced a multiplicative decomposition for the Cosserat rod.

Implemented strain gradient regularization terms that vanish as thickness decreases.

Abstract

We rigorously derive an effective bending model for elastoplastic rods starting from three-dimensional finite plasticity. For the derivation we lean on a framework of evolutionary $\Gamma$-convergence for rate-independent systems. The main contribution of this paper is an ansatz for the mutual recovery sequence in the bending regime. In difference to previous works that deal with infinitesimal deformations in the limit, in the bending regime we are concerned with finite deformations that admit infinitesimally small strain but large rotations. To overcome these difficulties we provide new ideas based on a multiplicative decomposition for the Cosserat rod ansatz. As regularizing terms, we introduce strain gradient terms into the energy that vanish as the thickness of the rod tends to zero.