Contrasting bending energies from bulk elastic theories

TL;DR

This paper compares four bulk elastic theories for thin plates and shells, revealing fundamental differences in their bending predictions and emphasizing the importance of kinematic measures for developing a covariant theory.

Contribution

It demonstrates that different elastic energy formulations lead to distinct bending behaviors and advocates for a covariant approach based on stretches instead of metrics.

Findings

Different elastic theories predict qualitatively different bending behaviors.

Not all theories predict an isometric ground state at finite thickness.

Kinematic measures from rod mechanics can unify definitions of stretching and bending.

Abstract

The choice of elastic energies for thin plates and shells is an unsettled issue with consequences for much recent modeling of soft matter. Through consideration of simple deformations of a thin body in the plane, we demonstrate that four bulk isotropic quadratic elastic theories have fundamentally different predictions with regard to bending behavior. At finite thickness, these qualitative effects persist near the limit of mid-surface isometry, and not all theories predict an isometric ground state. We discuss how certain kinematic measures that arose in early studies of rod mechanics lead to coherent definitions of stretching and bending, and promote the adoption of these quantities in the development of a covariant theory based on stretches rather than metrics.