On evolving natural curvature for an inextensible, unshearable, viscoelastic rod

TL;DR

This paper develops a mathematical model for an inextensible, unshearable, viscoelastic rod with evolving natural curvature, proving well-posedness and convergence of solutions to equilibrium states.

Contribution

It introduces a novel formulation for a viscoelastic rod with evolving natural configuration and analyzes the stability and long-term behavior of its quasistatic motion.

Findings

The dynamic equations are globally well-posed.

Existence of a smooth curve of static solutions for each terminal thrust.

Solutions converge to equilibrium points over time.

Abstract

We formulate and consider the problem of an inextensible, unshearable, viscoelastic rod, with evolving natural configuration, moving on a plane. We prove that the dynamic equations describing quasistatic motion of an Eulerian strut, an infinite dimensional dynamical system, are globally well-posed. For every value of the terminal thrust, these equations contain a smooth embedded curve of static solutions (equilibrium points). We characterize the spectrum of the linearized equations about an arbitrary equilibrium point, and using this information and a convergence result for dynamical systems due to Brunovsk\'y and Pol\'acik, we prove that every solution to the quasistatic equations of motion converges to an equilibrium point as time goes to infinity.