Dimension Reduction for Elastoplastic Rods in the Bending Regime

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 [Formula: see text]-convergence for rate-independent systems which establishes convergence of (approximate) energetic solutions to energetic solutions of the effective model. 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. For this, we consider a respective dependence of the strain gradient terms on the two parts of the polar decomposition.