The Principle Bundle Structure of Continuum Mechanics

TL;DR

This paper demonstrates that the configuration space of continua has a principal bundle structure, enabling a coordinate-free separation of rigid body motions and deformations, and introduces a new approach to defining pure deformation spaces using screw theory.

Contribution

It reveals the principal bundle structure of continuum configuration spaces and applies screw theory to analyze deformations and rigid motions in a novel, coordinate-free manner.

Findings

Configuration space forms a principal bundle with fibers isomorphic to (3)

Allows separation of rigid body motion and deformation without coordinates

Defines a pure deformation space using screw theory

Abstract

In this paper it is shown that the structure of the configuration space of any continua is what is called in differential geometry a {\it principle bundle} \cite{Frankel2011ThePhysics}. A principal bundle is a structure in which all points of the manifold (each configuration in this case) can be naturally projected to a manifold called the {\it base manifold}, which in our case represents pure deformations. All configurations projecting to the same point on the base manifold (same deformation) are called fibers. Each of these fibers is then isomorphic to the Lie group $\mathfrak{se(3)}$ representing pure rigid body motions. Furthermore, it is possible to define what is called a connection and this allows to split any continua motion in a rigid body sub-motion and a deformable one in a completely coordinate free way. As a consequence of that it is then possible to properly define a pure deformation space on which an elastic energy can be defined. This will be shown using screw theory \cite{Ball:1900}, which is vastly used in the analysis of rigid body mechanisms but is not normally used to analyse continua. Beside the just mentioned result, screw theory will also be used to relate concepts like helicity and enstrophy to screw theory concepts.