Second-order compatible-strain mixed finite elements for 2D compressible nonlinear elasticity

In recent years, a new class of mixed finite elements-compatible-strain mixed finite elements (CSMFEs)-has emerged that uses the differential complex of nonlinear elasticity. Their excellent performance in benchmark problems, such as numerical stability for modeling large deformations in near-incompressible solids, makes them a promising choice for solving engineering problems. Explicit forms exist for various shape functions of first-order CSMFEs. In contrast, existing second-order CSMFEs evaluate shape functions using numerical integration. In this paper, we formulate second-order CSMFEs with explicit shape functions for the displacement gradient and stress tensor. Concepts of vector calculus that stem from exterior calculus are presented and used to provide efficient forms for shape functions in the natural coordinate system. Covariant and contravariant Piola transformations are then applied to transform the shape functions to the physical space. Mid-nodes and pseudo-nodes are used to enforce the continuity constraints for the displacement gradient and stress tensor over the boundaries of elements. The formulation of the proposed second-order CSMFEs and technical aspects regarding their implementation are discussed in detail. Several benchmark problems are solved to compare the performance of CSMFEs with first-order CSMFEs and other second-order elements that rely on numerical integration. It is shown that the proposed CSMFEs are numerically stable for modeling near-incompressible solids in the finite strain regime.