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

TL;DR

This paper discusses various issues related to bending measures and energies in plates and shells, focusing on invariance properties, historical context, and clarifying distinctions between different measures and definitions.

Contribution

It clarifies the differences between dilation-invariant and non-invariant bending measures, corrects previous errors, and offers a simplified presentation of key elementary facts.

Findings

Distinction between dilation-invariant and non-invariant measures emphasized.

Correction of errors and oversights in prior work provided.

Elementary relation between stretch, curvature, and derivatives clarified.

Abstract

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 overlaps and distinctions in results arising from two different definitions of stretching, correct an error and citation oversights in our prior work, reiterate some of the early history of dilation-invariant bending measures, and provide additional brief observations regarding the relative size of energetic terms and the symmetrization of bending measures. A particular point of emphasis is the distinction between dilation-invariant measures and a recently introduced non-dilation-invariant measure for shells and curved rods. In the course of this discussion, we provide a simpler presentation of the elementary, but much neglected, fact that the through-thickness derivative of tangential stretch of material near the mid-surface of a thin body is the product of the mid-surface stretch and change in curvature.