Mixed finite element methods; finite element exterior calculus; nonlinear elasticity; incompressible elasticity; Hilbert complex

In this work, a new mixed finite element formulation is presented for the analysis of two-dimensional compressible solids in finite strain regime. A three-field Hu-Washizu functional, with displacement, displacement gradient and stress tensor considered as independent fields, is utilized to develop the formulation in spatial configuration. Certain constraints are imposed on displacement gradient and stress tensor so that they satisfy the required continuity conditions across the boundary of elements.

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 introduce a new family of mixed finite elements for incompressible nonlinear elasticity — compatible-strain mixed finite element methods (CSFEMs). Based on a Hu-Washizu-type functional, we write a four-field mixed formulation with the displacement, the displacement gradient, the first Piola-Kirchhoff stress, and a pressure-like field as the four independent unknowns. Using the Hilbert complexes of nonlinear elasticity, which describe the kinematics and the kinetics of motion, we identify the solution spaces of the independent unknown fields.