Curvatures of a Surface and the Rotation of the Unit Normal Vector

Thanks to Weingarten’s formulae [1], which date to 1861, the bending deformation of a Kirchhoff-Love shell can be characterized by examining the variation of the unit normal vector to the surface of the shell. 

In our latest paper [2] we explore the variation of the unit normal vector by defining a rotation tensor. Although this tensor is not uniquely defined, the angular rates associated with the tensor can be used to establish new formulae for the component of the curvature tensor for the shell and the associated mean H and Gaussian K curvatures. We illustrate the formulae with application to a spherical shell.

[1] Julius Weingarten. Ueber eine Klasse auf einander abwickelbarer Fläachen. Journal für die Reine und Angewandte Mathematik. 59: 382–393, 1861.

[2] Nathaniel N. Goldberg and Oliver M. O’Reilly.  New Representations for the Curvature Tensor of a Surface with Application to Theories of Elastic Shells. Journal of Elasticity, Published Online March 17, 2022. [Open Access provided courtesy of the UC Berkeley Library].