  The classical reduced model for a film/substrate bilayer is one in which the
film is governed by the Euler-Bernoulli beam equation and the substrate is
replaced by an array of springs (the so-called Winkler foundation assumption).
We derive a refined model in which the normal and shear tractions at the bottom
of the film are expressed in terms of the corresponding horizontal and vertical
displacements, and the response of the half-space is described by the exact
theory. The self-consistency of the refined model is confirmed by showing that
it yields a four-term (in the incompressible case) or six-term (in the
compressible case) expansion for the critical strain that agrees with the
expansion given by the exact theory.
