Assorted remarks on bending measures and energies for plates and shells, and their invariance properties

In this note, we address several issues, including some raised in recent works and commentary, related to bending measures and energies for plates and shells, and certain of their invariance properties. We discuss the distinction between definitions and results in our and others' approaches, correct an error and citation oversights in our work, and provide additional brief observations regarding the relative size of energetic terms and the symmetrization of bending measures. Particular points of emphasis are a reiteration of some of the early history of dilation-invariant measures, the similarities between all such measures, and the non-dilation-invariance of our recently introduced bending measure for shells and curved rods. In the course of this discussion, we provide a simpler presentation of the elementary, but much overlooked, fact that the additional tangential stretch of material near the mid-surface of a thin body is the product of the mid-surface stretch and the change in curvature.