  We provide a fine description of the weak limit of sequences of regular
axisymmetric maps with equibounded neo-Hookean energy, under the assumption
that they have finite surface energy. We prove that these weak limits have a
dipole structure, showing that the singular map described by Conti-- De Lellis
is generic in some sense. On this map we provide the explicit relaxation of the
neo-Hookean energy. We also make a link with Cartesian currents showing that
the candidate for the relaxation we obtained presents strong similarities with
the relaxed energy in the context of \(\mathbb{S}^2\)-valued harmonic maps.
