Overdamped Stress Relaxation in Buckled Rods

TL;DR

This paper provides a detailed theoretical analysis of stress relaxation in weakly buckled rods in viscous environments, identifying distinct asymptotic regimes and boundary-layer behaviors, aligning with previous simulations and guiding future research.

Contribution

It introduces a comprehensive theoretical framework for stress relaxation in buckled rods, including new asymptotic solutions and boundary-layer analysis, enhancing understanding of their dynamics.

Findings

Identification of two self-similar intermediate asymptotic regimes.

Derivation of boundary-layer scenarios using multiple-scale perturbation.

Theoretical results align with previous numerical simulations.

Abstract

We present a comprehensive theoretical analysis of the stress relaxation in a multiply but weakly buckled incompressible rod in a viscous solvent. In the bulk two interesting regimes of generic self--similar intermediate asymptotics are distinguished, which give rise to two classes of approximate and exact power--law solutions, respectively. For the case of open boundary conditions the corresponding non--trivial boundary--layer scenarios are derived by a multiple--scale perturbation (``adiabatic'') method. Our results compare well with -- and provide the theoretical explanation for -- previous results from numerical simulations, and they suggest new directions for further fruitful numerical and experimental investigations.