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This paper presents stabilized mixed finite element formulations for tetrahedral elements at large deformations using volume and area bubble functions. To this end, the corresponding weak formulations are derived for the standard two-field method, the method of incompatible modes and the enhanced strain method. Then, the weak formulations will be linearized. Furthermore, the matrix formulations for the weak formulations and its linearizations are summarized. The numerical results for incompressible rubber-like materials using a Neo-Hookean material law show the locking-free performance and the drastic damping of the stresses for the new stabilized tetrahedral elements in finite deformation problems. This paper is an extension of the works published by the authors regarding small deformation problems for linear elasticity and plasticity.

SUMMARYIn order to overcome the oscillatory effects of the mixed bi-linear Galerkin formulation for tetrahedral elements, a stabilization approach is presented. To this end the mixed method of incompatible modes and the mixed method of enhanced strains are reformulated, thus giving both the interpretation of a mixed finite element method with stabilization terms. For non-linear problems, these are non-linearly dependent on the current deformation state and therefore are replaced by linearly dependent stabilization terms. The approach becomes most attractive for the numerical implementation, since the use of quantities related to the previous Newton iteration step, typically arising for mixed-enhanced elements, is completely avoided. The stabilization matrices for the mixed method of incompatible modes and the mixed method of enhanced strains are obtained with volume and area bubble functions. Various numerical examples are presented, which illustrate successfully the stabilization effect for bi-linear tetrahedral elements

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