  A nonlinear small-strain elastic theory is constructed from a systematic
expansion in Biot strains, truncated at quadratic order. The primary motivation
is the need for a clean separation between stretching and bending energies for
shells, which appears to arise only from reduction of a bulk energy of this
type. An approximation of isotropic invariants, bypassing the solution of a
quartic equation or computation of tensor square roots, allows stretches,
rotations, stresses, and balance laws to be written in terms of derivatives of
position. Two-field formulations are also presented. Extensions to anisotropic
theories are briefly discussed.
