On in-plane drill rotations for Cosserat surfaces

TL;DR

This paper proves that under certain smoothness conditions, pure in-plane drill rotations cannot deform a shell surface without altering it, if boundary conditions are fixed, using differential geometry of surfaces.

Contribution

It establishes a mathematical impossibility result for in-plane drill rotations of Cosserat surfaces under boundary constraints.

Findings

Pure in-plane drill rotations cannot deform a shell surface with fixed boundary conditions.

Any isometry preserving normals and matching a surface at a boundary segment must be the identity.

The result is grounded in differential geometry and smoothness assumptions.

Abstract

We show under some natural smoothness assumptions that pure in-plane drill rotations as deformation mappings of a $C^2$-smooth regular shell surface to another one parametrized over the same domain are impossible provided that the rotations are fixed at a portion of the boundary. Put otherwise, if the tangent vectors of the new surface are obtained locally by only rotating the given tangent vectors, and if these rotations have a rotation axis which coincides everywhere with the normal of the initial surface, then the two surfaces are equal provided they coincide at a portion of the boundary. In the language of differential geometry of surfaces we show that any isometry which leaves normals invariant and which coincides with the given surface at a portion of the boundary, is the identity mapping.