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We consider the problem of minimizing the neo-Hookean energy in \(3D\). Thedifficulty of this problem is that the space of maps without cavitation is notcompact, as shown by Conti \& De Lellis with a pathological example involving adipole. In order to rule out this behaviour we consider the relaxation of theneo-Hookean energy in the space of axisymmetric maps without cavitation. Wepropose a minimization space and a new explicit energy penalizing the creationof dipoles. This new energy, which is a lower bound of the relaxation of theoriginal energy, bears strong similarities with the relaxed energy ofBethuel-Brezis-H\'elein in the context of harmonic maps into the sphere.

Harmonic dipoles and the relaxation of the neo-Hookean energy in 3D elasticity