  We derive stretching and bending energies for isotropic elastic plates and
shells. Through the dimensional reduction of a bulk elastic energy quadratic in
Biot strains, we obtain two-dimensional bending energies quadratic in bending
measures featuring a bilinear coupling of stretches and geometric curvatures.
For plates, the bending measure is invariant under spatial dilations and
naturally extends primitive bending strains for straight rods. For shells or
naturally-curved rods, the measure is not dilation invariant, and contrasts
with previous \emph{ad hoc} postulated forms. The corresponding field equations
and boundary conditions feature moments linear in the bending measures, and a
decoupling of stretching and bending such that application of a pure moment
results in isometric deformation of a unique neutral surface, primitive
behaviors in agreement with classical linear response but not displayed by
commonly used analytical models. We briefly comment on relations between our
energies, those derived from a neo-Hookean bulk energy, and a commonly used
discrete model for flat membranes. Although the derivation requires
consideration of stretch and rotation fields, the resulting energy and field
equations can be expressed entirely in terms of metric and curvature components
of deformed and reference surfaces.
