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A new family of mixed finite element methods$-$compatible-strain mixed finiteelement methods (CSFEMs)$-$are introduced for three-dimensional compressibleand incompressible nonlinear elasticity. A Hu-Washizu-type functional isextremized in order to obtain a mixed formulation for nonlinear elasticity. Theindependent fields of the mixed formulations are the displacement, thedisplacement gradient, and the first Piola-Kirchhoff stress. A pressure-likefield is also introduced in the case of incompressible elasticity. We definethe displacement in $H^1$, the displacement gradient in $H(curl)$, the stressin $H(div)$, and the pressure-like field in $L^2$. In this setting, forimproving the stability of the proposed finite element methods withoutcompromising their consistency, we consider some stabilizing terms in theHu-Washizu-type functional that vanish at its critical points. Using aconforming interpolation, the solution and the test spaces are approximatedwith some piecewise polynomial subspaces of them. In three dimensions, thisrequires using the N\'ed\'elec edge elements for the displacement gradient andthe N\'ed\'elec face elements for the stress. This approach results in mixedfinite element methods that satisfy the Hadamard jump condition and thecontinuity of traction on all internal faces of the mesh. This, in particular,makes CSFEMs quite efficient for modeling heterogeneous solids. We assess theperformance of CSFEMs by solving several numerical examples, and demonstratetheir good performance for bending problems, for bodies with complexgeometries, and in the near-incompressible and the incompressible regimes.Using CSFEMs, one can capture very large strains and accurately approximatestresses and the pressure field. Moreover, in our numerical examples, we do notobserve any numerical artifacts such as checkerboarding of pressure, hourglassinstability, or locking.

Compatible-Strain Mixed Finite Element Methods for 3D Compressible and Incompressible Nonlinear Elasticity