Derivation of a bending model for rods from finite viscoelasticity

Abstract

Starting from a fully three dimensional model of finite viscoelasticity in the Kelvin-Voigt rheology, we derive a bending model for thin, viscoelastic rods. For this, we utilize a formulation of the model as a time-dependent metric gradient flow and pass to the limit by establishing convergence of curves of maximal slope using $Γ$-convergence. A posteriori, we relate the metric gradient flow limits to gradient flows in Hilbert spaces and discuss their properties. In contrast to previous results, where limit deformations are infinitesimal, in the bending regime, we deal with large rotations. This prevents us from using an additive ansatz for the construction of recovery sequences. Instead, we adapt from a previous result by the authors a multiplicative ansatz for elastoplastic Cosserat rods to the viscoelastic setting. For regularization purposes, we consider strain gradient terms in the three dimensional model that vanish in the limit.