Abstract
We discuss how the shape of a special Cosserat rod can be represented as a path in the special Euclidean algebra. By shape we mean all those geometric features that are invariant under isometries of the three-dimensional ambient space. The representation of the shape as a path in the special Euclidean algebra is intrinsic to the description of the mechanical properties of a rod, since it is given directly in terms of the strain fields that stimulate the elastic response of special Cosserat rods. Moreover, such a representation leads naturally to discretization schemes that avoid the need for the expensive reconstruction of the strains from the discretized placement and for interpolation procedures which introduce some arbitrariness in popular numerical schemes. Given the shape of a rod and the positioning of one of its cross sections, the full placement in the ambient space can be uniquely reconstructed and described by means of a base curve endowed with a material frame. By viewing a geometric curve as a rod with degenerate point-like cross sections, we highlight the essential difference between rods and framed curves, and clarify why the family of relatively parallel adapted frames is not suitable for describing the mechanics of rods but is the appropriate tool for dealing with the geometry of curves.
Highlights
Motivation and Main ResultsRod theory has undergone a systematic development and has provided a platform for endless applications
We show that the shape of a rod, namely those features that are invariant under direct isometries of the three-dimensional ambient space, can be identified with a square-integrable path in the special Euclidean algebra
We have described how the essential degrees of freedom that encode the shape of a special Cosserat rod, namely those geometric features that are invariant under isometries of the three-dimensional ambient space, correspond to a path traced in the special Euclidean algebra
Summary
Rod theory has undergone a systematic development and has provided a platform for endless applications. 5, the theory of framed curves as a limiting case of the Cosserat theory, we show that the former theory is not adequate to describe the mechanics of rods, since it is incapable of tracking twisting and shearing deformations and completely neglects any effect due to the actual shapes of the cross sections Even in those cases in which the frame along the curve is chosen to represent the material frame (and not merely determined by the curve geometry), the essential role accorded to the base curve makes it difficult to factor out global isometries. Our derivation of the theory of framed curves highlights the relevance of the results presented by Bishop [12] in 1975, results which are still surprisingly ignored in some recent publications We generalize his construction of relatively parallel adapted frames to the case of continuously differentiable regular curves. We conclude by remarking that, when treating purely geometric questions surrounding space curves, relatively parallel adapted frames are the appropriate tool, and any use of the Frenet frame should be abandoned
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