The relation between incompressibility and divergence-free?

Hi all,

    I am learning continuum mechanics. One thing I am really confused
is the relation between incompressiblity and divergence-free. Since I am coding a finite element code that use mixed displacement and
pressure formulation. From what I got from Prof. Tom Hughes's book. Both incompressible solid and fluid can be characterized by divergence
free vector field (displacement for solid and velocity for fluid). But
is this only for small deformation in the case of solid? Since for large
deformation, the determinant of deformation gradient must be one, which
seems not equivalent to divergence free?

    What about for fluid, because it uses eularian frame, the
divergence-free should be equivalent to incomressbility for any case. Is
this correct?

Thanks for your clarification!

Mengda