Metric Implications in the Kinematics of Surfaces
Andre M. Sonnet, Epifanio G. Virga
TL;DR
This work develops a coordinate-free kinematic framework for soft thin shells, framing surface deformations through moving frames and the invariant rotation gradient $\boldsymbol{H}$ to distinguish three independent energy contents: stretching $w_s$, drilling $w_d$, and bending $w_b$. It analyzes how metric restrictions—conformal and isometric—affect these energies, deriving transformation laws for Gaussian and mean curvatures ($K^*$, $H^*$) and establishing conditions under which deformations are energy-free. Key contributions include explicit expressions linking $w_d$ and $w_b$ to $\boldsymbol{H}$, a comprehensive treatment of isometries (with results about uniform rotations and Bonnet transformations on minimal surfaces), and the demonstration that all surfaces of revolution admit pure bending eversion with energy-free configurations. The results illuminate how geometric constraints govern shell energetics, offering insights into soft elasticity, snap-through behavior, and the role of conformal factors in surface remodeling, using a framework that foregrounds intrinsic geometry over coordinate-based methods.
Abstract
In the direct approach to continua in reduced space dimensions, a thin shell is described as a mathematical surface in three-dimensional space. An exploratory kinematic study of such surfaces could be very valuable, especially if conducted with no use of coordinates. Three energy contents have been identified in a thin shell, which refer to three independent deformation modes: stretching, drilling, and bending. We analyze the consequences for the three energy contents produced by metric restrictions imposed on the admissible deformations. Would the latter stem from physical constraints, the elastic response of a shell could be hindered in ways that might not be readily expected.