Mixed finite elements; finite element exterior calculus; nonlinear elasticity; geometric mechanics; Hilbert complex

In this paper, using the Hilbert complexes of nonlinear elasticity, the approximation theory for Hilbert complexes, and the finite element exterior calculus, we introduce a new class of mixed finite element methods for 2D nonlinear elasticity -- compatible-strain mixed finite element methods (CSFEM). We consider a Hu-Washizu-type mixed formulation and choose the displacement, the displacement gradient, and the first Piola-Kirchhoff stress tensor as independent unknowns. We use the underlying spaces of the Hilbert complexes as the solution and test spaces.