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

• Main Authors: Hanna, J. A, Vitral, E
• Format: Journal Article
• Language: English
• Published: 10-05-2024
• Subjects: Physics - Soft Condensed Matter
• DOI: 10.48550/arxiv.2405.06638

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.