  We consider symmetric as well as non-symmetric coupling formulations of FEM
and BEM in the frame of nonlinear elasticity problems. In particular, the
Johnson-N\'ed\'elec coupling is analyzed. We prove that these coupling
formulations are well-posed and allow for unique Galerkin solutions if standard
discretizations by piecewise polynomials are employed. Unlike prior works, our
analysis does neither rely on an interior Dirichlet boundary to tackle the
rigid body motions nor on any assumption on the mesh-size of the discretization
used.
