  The existing theory of incompatible elastic sheets uses the deviation of the
surface metric from a reference metric to define the strain tensor [Efrati et
al., J. Mech. Phys. Solids {\bf 57}, 762 (2009)]. For a class of simple
axisymmetric problems we examine an alternative formulation, defining the
strain based on deviations of distances (rather than distances squared) from
their rest values. While the two formulations converge in the limit of small
slopes and in the limit of an incompressible sheet, for other cases they are
found not to be equivalent. The alternative formulation offers several features
which are absent in the existing theory. (a) In the case of planar deformations
of flat incompatible sheets, it yields linear, exactly solvable, equations of
equilibrium. (b) When reduced to uniaxial (one-dimensional) deformations, it
coincides with the theory of extensible elastica; in particular, for a
uniaxially bent sheet it yields an unstrained cylindrical configuration. (c) It
gives a simple criterion determining whether an isometric immersion of an
incompatible sheet is at mechanical equilibrium with respect to normal forces.
For a reference metric of constant positive Gaussian curvature, a spherical cap
is found to satisfy this criterion except in an arbitrarily narrow boundary
layer.
