  In a previous work, Henao \& Rodiac (2018) proved an existence result for the
neo-Hookean energy in 3D for the essentially 2D problem of axisymmetric domains
away from the symmetry axis. In this paper we pursue the investigation of this
problem, still in the axisymmetric setting, but without the assumption that the
domain is hollow and at a distance apart of the symmetry axis. We propose a
candidate for the associated relaxed energy defined in the space of
weak-$H^{1}$ limits of deformation maps without cavitation. More precisely, we
introduce an explicit energy functional (which coincides with the neo-Hookean
energy for regular maps and expresses the cost of a singularity in terms of the
jump and Cantor parts of the inverse) and an explicit admissible space (which
contains all weak $H^1$ limits of regular maps), for which we can prove the
existence of minimizers. Chances to succeed in establishing that minimizers do
not have dipoles singularities are higher in this explicit alternative
variational problem than when working with the abstract relaxation approach in
the abstract space of all weak limits. Our candidate for relaxed energy has
many similarities with the relaxed energy introduced by Bethuel-Br\' ezis-Coron
for a problem with lack of compactness in harmonic maps theory. The proof we
present in this paper for the lower semicontinuity of the augmented energy
functional, partly inspired by the prominent role played by conformality in
that context, further develops the connection between the minimization of the
3D neo-Hookean energy with the problem of finding a minimizing smooth harmonic
map from $\mathbb{B}^3$ into $\mathbb{S}^2$ with zero degree boundary data.
