Summary We summarize several previously published geometrically nonlinear EAS elements and compare their behavior. Various transformations for the compatible and enhanced deformation gradient are examined. Their effect on the patch test is one main concern of the work, and it is shown numerically and with a novel analytic proof that the improved EAS element proposed by Simo et al in 1993 does not fulfill the patch test. We propose a modification to overcome that drawback without losing the favorable locking‐free behavior of that element. Furthermore, a new transformation for the enhanced field is proposed and motivated in a curvilinear coordinate frame. It is shown in numerical tests that this novel approach outperforms all previously introduced transformations.

One of the most successful mixed finite element methods in solid mechanics is the enhanced assumed strain (EAS) method developed by Simo and Rifai in 1990. However, one major drawback of EAS elements is the highly mesh dependent accuracy. In fact, it can be shown that not only EAS elements, but every finite element with a symmetric stiffness matrix must either fail the patch test or be sensitive to mesh distortion in bending problems (higher order displacement modes) if the shape of the element is arbitrary. This theorem was established by MacNeal in 1992. In the present work we propose a novel Petrov-Galerkin approach for the EAS method, which is equivalent to the standard EAS method in case of regular meshes. However, in case of distorted meshes, it allows to overcome the mesh-distortion sensitivity without loosing other advantages of the EAS method. Three design conditions established in this work facilitate the construction of the element which does not only fulfill the patch test but is also exact in many bending problems regardless of mesh distortion and has an exceptionally high coarse mesh accuracy. Consequently, high quality demands on mesh topology might be relaxed.

Summary In this work, two well‐known approaches for mixed finite elements are combined to render three novel classes of elements. First, the widely used enhanced assumed strain (EAS) method is considered. Its key idea is to enhance a compatible kinematic field with an incompatible part. The second concept is a framework for mixed elements inspired by polyconvex strain‐energy functions, in which the deformation gradient, its cofactor and determinant are three principal kinematic fields. The key idea for the novel elements is to treat enhancement of those three fields separately. This approach leads to a plethora of novel enhancement strategies and promising mixed finite elements. Some key properties of the newly proposed mixed approaches are that they are based on a Hu‐Washizu type variational functional, fulfill the patch test, are frame‐invariant, can be constructed completely locking free and show no spurious hourglassing in elasticity. Furthermore, they give additional insight into the mechanisms of standard EAS elements. Extensive numerical investigations are performed to assess the elements' behavior in elastic and elasto‐plastic simulations.

The enhanced assumed strain (EAS) method is one of the most frequently used methods to avoid locking in solid and structural finite elements. One issue of EAS elements in the context of geometrically nonlinear analyses is their lack of robustness in the Newton-Raphson scheme, which is characterized by the necessity of small load increments and large number of iterations. In the present work we extend the recently proposed mixed integration point (MIP) method to EAS elements in order to overcome this drawback in numerous applications. Furthermore, the MIP method is generalized to generic material models, which makes this simple method easily applicable for a broad class of problems. In the numerical simulations in this work, we compare standard strain-based EAS elements and their MIP improved versions to elements based on the assumed stress method in order to explain when and why the MIP method allows to improve robustness. A further novelty in the present work is an inverse stress-strain relation for a Neo-Hookean material model. K E Y W O R D S enhanced assumed strain, inverse stress-strain relation, mixed finite elements, mixed integration point method, Newton-Raphson scheme, robustness This is an open access article under the terms of the Creative Commons Attribution-NonCommercial License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited and is not used for commercial purposes.

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