Continuum Dynamics on Manifolds: Application to Elasticity of Residually-Stressed Bodies

TL;DR

This paper develops a covariant framework for continuum dynamics on manifolds, focusing on elastic bodies with residual stress, and applies it to analyze vibrations of a curved elastic annulus.

Contribution

It introduces a covariant derivation of equations of motion for continua on manifolds, specifically addressing residual stresses in elastic bodies and their vibrational behavior.

Findings

Derived equations of motion for residually-stressed elastic bodies

Analyzed vibrational modes of a curved elastic annulus

Demonstrated geometric incompatibility causes residual stress

Abstract

This paper is concerned with the dynamics of continua on differentiable manifolds. We present a covariant derivation of equations of motion, viewing motion as a curve in an infinite-dimensional Banach space of embeddings of a body manifold in a space manifold. Our main application is the motion of residually-stressed elastic bodies; residual stress results from a geometric incompatibility between body and space manifolds. We then study a particular example of elastic vibrations of a two- dimensional curved annulus embedded in a sphere.