Hybrid Equilibrium Element Formulation for Large Displacement Nonlinear Elasticity

The present paper proposes an equilibrium-based finite element formulation for the analysis of large displacement and large strain nonlinear elasticity problems. The proposed formulation is based on the Hybrid Equilibrium Element (HEE) and is developed in an updated Lagrangian framework. The solution is expressed in terms of a Cauchy stress field that strongly satisfies the domain and the boundary equilibrium equations in the deformed configuration, at the end of each loading step. The Cauchy stress field is independently modeled for each finite element, and the inter-element and boundary equilibrium conditions are imposed by the Lagrangian multiplier methods, for which the Lagrangian variable coincides with the element side displacement. The proposed formulation is developed in the theoretical framework of the Virtual Stress Principle, and the complementary weak form of the governing equations is derived in an incremental updated Lagrangian form. The proposed nonlinear HEE formulation is developed with two hyperelastic constitutive models, the Hencky model and the Neo-Hookean model. The proposed HEE formulations for large displacement and large deformation nonlinear elasticity have been implemented in a finite element code, using a triangular nine-node HEE with the quadratic stress field, and the results of three numerical simulations are presented, showing encouraging results for both compressible and quasi-incompressible materials.