Nonlinear elasticity with vanishing nonlocal self-repulsion

We prove that that for nonlinear elastic energies with strong enough energetic control of the outer distortion of admissible deformations, almost everywhere global invertibility as constraint can be obtained in the $\Gamma$-limit of the elastic energy with an added nonlocal self-repulsion term with asymptocially vanishing coefficient. The self-repulsion term considered here formally coincides with a Sobolev-Slobodecki\u{i} seminorm of the inverse deformation. Variants near the boundary or on the surface of the domain are also studied.