The principal bundle structure of continuum mechanics

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 principle bundle. 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 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. In this paper, it is shown that each of these fibers is then naturally isomorphic to the Lie group SE(3) representing pure rigid body motions. Furthermore, it will be shown that it is possible to define intrinsically (coordinate free), 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 and this will be related to other literature on intrinsic strain which has not shown the geometric structure which will be presented in this work. This will be shown using screw theory and Lie groups. Screw theory is vastly used in the analysis of rigid body mechanisms but is not normally used to analyse continua. The paper will also relate known concepts of continua like helicity and enstrophy to screw theory concepts which will become evident by the presented construction.