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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.

We introduce a new class of mixed finite element methods for 2D and 3D compressible nonlinear elasticity. The independent unknowns of these conformal methods are displacement, displacement gradient, and the first Piola-Kirchhoff stress tensor. The so-called edge finite elements of the curl operator is employed to discretize the trial space of displacement gradients. Motivated by the differential complex of nonlinear elasticity, this choice guarantees that discrete displacement gradients satisfy the Hadamard jump condition for the strain compatibility. We study the stability of the proposed mixed finite element methods by deriving some inf-sup conditions. By considering 32 choices of simplicial conformal finite elements of degrees 1 and 2, we show that 10 choices are not stable as they do not satisfy the inf-sup conditions. We numerically study the stable choices and conclude that they can achieve optimal convergence rates. By solving several 2D and 3D numerical examples, we show that the proposed methods are capable of providing accurate approximations of strain and stress.