  A new family of mixed finite element methods$-$compatible-strain mixed finite
element methods (CSFEMs)$-$are introduced for three-dimensional compressible
and incompressible nonlinear elasticity. A Hu-Washizu-type functional is
extremized in order to obtain a mixed formulation for nonlinear elasticity. The
independent fields of the mixed formulations are the displacement, the
displacement gradient, and the first Piola-Kirchhoff stress. A pressure-like
field is also introduced in the case of incompressible elasticity. We define
the displacement in $H^1$, the displacement gradient in $H(curl)$, the stress
in $H(div)$, and the pressure-like field in $L^2$. In this setting, for
improving the stability of the proposed finite element methods without
compromising their consistency, we consider some stabilizing terms in the
Hu-Washizu-type functional that vanish at its critical points. Using a
conforming interpolation, the solution and the test spaces are approximated
with some piecewise polynomial subspaces of them. In three dimensions, this
requires using the N\'ed\'elec edge elements for the displacement gradient and
the N\'ed\'elec face elements for the stress. This approach results in mixed
finite element methods that satisfy the Hadamard jump condition and the
continuity of traction on all internal faces of the mesh. This, in particular,
makes CSFEMs quite efficient for modeling heterogeneous solids. We assess the
performance of CSFEMs by solving several numerical examples, and demonstrate
their good performance for bending problems, for bodies with complex
geometries, and in the near-incompressible and the incompressible regimes.
Using CSFEMs, one can capture very large strains and accurately approximate
stresses and the pressure field. Moreover, in our numerical examples, we do not
observe any numerical artifacts such as checkerboarding of pressure, hourglass
instability, or locking.
