  We propose bending energies for isotropic elastic plates and shells. For a
plate, we define and employ a surface tensor that symmetrically couples stretch
and curvature such that any elastic energy density constructed from its
invariants is invariant under spatial dilations. This kinematic measure and its
corresponding isotropic quadratic energy resolve outstanding issues in thin
structure elasticity, including the natural extension of primitive bending
strains for straight rods to plates, the assurance of a moment linear in the
bending measure, and the avoidance of induced mid-plane strains in response to
pure moments as found in some commonly used analytical plate models. Our
analysis also reveals that some other commonly used numerical models have the
right invariance properties, although they lack full generality at quadratic
order in stretch. We further extend our result to naturally-curved rods and
shells, for which the pure stretching of a curved rest configuration breaks
dilation invariance; the new shell bending measure we provide contrasts with
previous \emph{ad hoc} postulated forms. The concept that unifies these
theories is not dilation invariance, but rather through-thickness uniformity of
strain as a definition of pure stretching deformations. Our results provide a
clean basis for simple models of low-dimensional elastic systems, and should
enable more accurate analytical probing of the structure of singularities in
sheets and membranes.
