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We propose bending energies for isotropic elastic plates and shells. For aplate, we define and employ a surface tensor that symmetrically couples stretchand curvature such that any elastic energy density constructed from itsinvariants is invariant under spatial dilations. This kinematic measure and itscorresponding isotropic quadratic energy resolve outstanding issues in thinstructure elasticity, including the natural extension of primitive bendingstrains for straight rods to plates, the assurance of a moment linear in thebending measure, and the avoidance of induced mid-plane strains in response topure moments as found in some commonly used analytical plate models. Ouranalysis also reveals that some other commonly used numerical models have theright invariance properties, although they lack full generality at quadraticorder in stretch. We further extend our result to naturally-curved rods andshells, for which the pure stretching of a curved rest configuration breaksdilation invariance; the new shell bending measure we provide contrasts withprevious \emph{ad hoc} postulated forms. The concept that unifies thesetheories is not dilation invariance, but rather through-thickness uniformity ofstrain as a definition of pure stretching deformations. Our results provide aclean basis for simple models of low-dimensional elastic systems, and shouldenable more accurate analytical probing of the structure of singularities insheets and membranes.

Dilation-invariant bending of elastic plates, and broken symmetry in shells