  The stressed state of flattened thin elastic sheet, as well as that of
translationally symmetric 3D solids, are effectively 2D problems. This paper
study equilibrium state-of-stress in metrically-incompatible 2D elastic
materials. The solution is represented by a scalar stress function,
generalizing the Airy stress function, which is determined by geometric
compatibility conditions. We develop a perturbative approximation method for
solving this stress function, valid for any constitutive relation. We apply the
method for the case of a Hookean solid to solve prototypical examples in which
the classical Airy approach is either inaccurate or inapplicable. Results are
shown to agree well with numerical results obtained in previous works.
