  We introduce conformal mixed finite element methods for $2$D and $3$D
incompressible nonlinear elasticity in terms of displacement, displacement
gradient, the first Piola-Kirchhoff stress tensor, and pressure, where finite
elements for the $\mathrm{curl}$ and the $\mathrm{div}$ operators are used to
discretize strain and stress, respectively. These choices of elements follow
from the strain compatibility and the momentum balance law. Some inf-sup
conditions are derived to study the stability of methods. By considering $96$
choices of simplicial finite elements of degree less than or equal to $2$ in
$2$D and $3$D, we conclude that $28$ choices in $2$D and $6$ choices in $3$D
satisfy these inf-sup conditions. The performance of stable finite element
choices are numerically studied. Although the proposed methods are
computationally more expensive than the standard two-field methods for
incompressible elasticity, they are potentially useful for accurate
approximations of strain and stress as they are independently computed in the
solution process.
