Covariant Linearization of elasticity

TL;DR

This paper develops a covariant linearization framework for continuum dynamics on manifolds, particularly addressing incompatible elasticity, by differentiating equations as 1-forms on configuration space with a specially constructed affine connection.

Contribution

It introduces a novel covariant linearization method for continuum mechanics on manifolds, expanding the theoretical tools for incompatible elasticity analysis.

Findings

Derived a general linearized theory for continuum dynamics on manifolds.

Provided detailed coordinate computations for linearized equations.

Applied the framework to problems in incompatible elasticity.

Abstract

In this paper we derive a general linearized theory for first-order continuum dynamics on manifolds with particular application to incompatible elasticity. We adopt a global approach viewing the equations of motion as a $1$-form on the configuration space which is the Banach manifold of $C^1$ time-dependent embeddings of a body manifold $\B$ into a space manifold $\S$. The linearization is done by differentiating the equations 1-form with respect to an affine connection which we construct and study extensively. We provide detailed coordinate computations for the linearized equations of a large class of problems in continuum dynamics on manifolds.