Dilation-Invariant Bending of Elastic Plates, and Broken Symmetry in Shells

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 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.This note serves as an extended comment on our own and others' papers, particularly [1][2][3][4][5][6]. Therefore, some conceptual background is presumed on the part of the reader, and we forego introductory or concluding discussion. I. REMARKS ON DIFFERENT DEFINITIONS OF STRETCHING, AND RESULTING INVARIANT BENDING MEASURES

Pure measures of bending for soft plates

“…21 In particular, in the limiting case of plates, the one that concerns us here, two two-dimensional bending measures arose naturally from the method employed in ref. 20, one of these measures rmains invariant under arbitrary superimposed stretchings, and the other remains invariant only under superimposed dilations, both reducing to Antman's measure in the rod-limit.…”

“…The linear invariant of A, trA (which is quadratic in the principal curvatures of S), has already featured in the literature; it emerged as a strain energy density from a discretized model for membranes and plates 35 and, more recently, also from a dimension reduction to plates. 20 In the latter, the reduced strain energy density also involves the scalar invariants of the tensor P : ¼…”

“…Similarly, for a plate whose planar midsurface S in the reference configuration gets deformed into the surface embedded in three-dimensional space, the invariants of the curvature tensor ∇ s ν , where ν is the outer unit normal to and ∇ s denotes the surface gradient, cannot serve as pure measures of bending, as also neatly illustrated in ref. 12.…”

“…[16][17][18][19] was applied in ref. 20 to a three-dimensional isotropic quadratic strain energy formulated in terms of Biot's strain tensor to justify a model for shells. 21 In particular, in the limiting case of plates, the one that concerns us here, two two-dimensional bending measures arose naturally from the method employed in ref.…”

Stretch formulations and the Poynting effect in nonlinear elasticity

“…Therefore, particularly for small finite strains, we are interested in contrasting the mechanical response predicted by these two different energies. The quadratic-Biot material has been recently adopted to derive reduced plate and shell energies [18,19], which avoids the undesirable mixing between stretching and bending contents introduced when the reduction is performed for certain energies quartic in stretches [20][21][22], such as Saint Venant-Kirchhoff. It also leads to a complete two constant bending energy for an isotropic material, instead of the one constant bending energy derived from neo-Hookean.…”