Contrasting bending energies from bulk elastic theories

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. *

On the Kirchhoff-Love Hypothesis (Revised and Vindicated)

“…Geometric elasticity of plates (and shells) does blend together stretching and bending energies, which scale with different powers of h; the former, like w s in (59b), is of a pure metric nature, while in the latter, as in (59c), metric and curvature measures are combined together in an invariant way. Interesting remarks on this contamination of stretching and bending measures in w b are offered in [82].…”

Some observations on variational elasticity and its application to plates and membranes

“…A preference for blowup in compression rather than tension to describe real materials, something that happens naturally when using the present basis, led Magnusson and co-workers [78] to contrive a more complicated nonlinear constitutive relation than that corresponding to Biot energy. In another publication [15], my co-author and I confirm that a Swainger energy leads to the same ∂ 1 θ Λ 2 bending term as that obtained from the Almansi approach.…”

“…Different choices of linear constitutive relations between nonlinear stresses and strains lead to qualitatively different results in bulk elasticity at large strains [51,52]. It will be shown in Section III B and a separate publication [15] that constitutive choices have even more marked qualitative effects on derived bending elasticity. In particular, the Green form of the bulk energy favored in some recent works [12,14,50] appears to generate bending energies in qualitative contrast with intuitively sensible definitions even at moderate strains.…”

“…Using the Biot strain energy ( [79] and others) leads to a Reissner-like model in which expanding a circle is a pure stretch that preserves "bending energy", while extending an arc increases it. Using the Almansi or Swainger [15] strain energies (Section II C), expanding a circle decreases the "bending energy", while extending an arc is a pure stretch that preserves it. Of course, all of these are constitutive assumptions so, intuitive or not, cannot be right or wrong except as descriptions of a particular material's experimentally determined behavior.…”

Enforcing shaping of thin gel sheets by anisotropic swelling

“…Moreover, in our opinion it should also be better discussed the effectiveness of the reduced models of growing plates in predicting final steady shapes when three-dimensional elastic growth (swelling) is involved. As noted in [18,41], the choice of elastic energies for thin plates and shells is an unsettled issue with consequences for much recent modeling of soft matter. Figure B.6: Validation test for different number of parallelepiped elements n showed at the right of the figure.…”