Dimension reduction for elastoplastic rods in the bending regime

In our work we rigorously derive a limiting model for thin rods, starting from a full 3D model for finite plasticity. We are interested in a scaling of the elastic and plastic energy contributions like h^(-4), where h denotes the thickness of the rod. In the limit this results in a 1D bending theory.

For the derivation we lean on the framework of evolutionary Gamma-convergence for rate-independent systems, introduced by Mielke, Roubíček and Stefanelli in 2008. The main difficulty here is to establish a mutual recovery sequence for the stored energy and dissipation. Strategies have been developed by various authors in order to construct such a sequence, e.g. for linearization or in the von Kármán regime. However, these rely on considering infinitesimal deformations in the limit, which we cannot expect in the bending regime. Our approach relies on a construction based on a multiplicative decomposition of the rotation fields obtained via the rigidity estimate from Friesecke, James and Müller. In order to achieve enough regularity, we consider strain gradient terms in the energy, which act on the two parts of the polar decomposition individually. These terms vanish in the limit.

This is joined work with Stefan Neukamm.