We introduce conformal mixed finite element methods for 2D and 3D incompressible nonlinear elasticity in terms of displacement, displacement gradient, the first Piola-Kirchhoff stress tensor, and pressure, where finite elements for the curl and the 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 2D and 3D, we conclude that 28 choices in 2D and 6 choices in 3D 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. nonlinear elasticity. This inf-sup condition can be interpreted as the necessary and sufficient condition for the well-posedness of the linear problems associated to Newtons' iterations for solving nonlinear finite element methods. Based on this interpretation, we also write 6 other inf-sup conditions that are necessary for the stability of Newtons' iterations. Using these conditions, we derive some relations between the dimension of finite element spaces for different unknowns, which are necessary for the stability of Newtons' iterations.By considering 96 choices of simplicial finite elements of degree less than or equal to 2 for obtaining mixed finite element methods, we conclude that 68 choices in 2D and 90 choices in 3D do not satisfy all these inf-sup conditions, in general, and lead to unstable methods. The performance of stable choices in 2D and 3D are studied by solving numerical examples.This paper is organized as follows: The four-field mixed formulation and the associated conformal finite element methods are discussed in Sections 2 and 3. Newtons' iterations for solving nonlinear finite element methods are provided in Section 4. In Section 5, different inf-sup conditions are derived for studying the stability of finite element methods and it is shown that some choices of finite elements lead to unstable methods as they violate the inf-sup conditions. In Section 6, the inf-sup conditions are numerically studied. Moreover, by solving two numerical examples, the performance of stable finite element methods are investigated in 2D and 3D. Finally, some concluding remarks are given in Section 7.

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