Keywords: finite elements, solid-shell, enhanced assumed strains, shell structures, transverse shear locking, volumetric locking.

Reliable numerical simulation of sheet metal forming processes is based on the correct reproduction of the high stress and strain gradients usually involved in those industrial processes. A crucial point in the computational modeling of elasto-plastic behavior, on the other hand, relies on the proper choice of the finite element formulation to be employed. During the last decades, shell elements represented the priority choice among end-users, mainly driven by the options offered by explicit finite element codes. Within this aspect, adoption of elements based on reduced integration was the trivial option, in order to fully benefit from the small CPU time. Nevertheless, as higher computational power has became increasingly cheaper and accessible, researchers' attention turned back to full integration-based elements, within a sounder theoretical framework and naturally free of hourglass instabilities, common in reduced schemes.

In this sense, the present paper aims to show one of the current research lines of the authors, focusing on the development of reliable and efficient shell and solid-shell elements. What distinguishes the present formulation from the majority of those previously presented in the literature is the fact that solely the Enhanced Assumed Strain (EAS) method is employed in order to improve the performance of conventional displacement-based elements. The EAS approach is responsible for the elimination of the transverse shear strain locking pathology, dominant in shell elements as thickness values tend to zero. In relation to solid-shell formulations, the EAS method provides an unified treatment of not only the transverse shear locking but also the volumetric locking, arising from the incompressibility constraints directly coming from the constant volume requirements within the plastic range, in elasto-plastic simulations. In both cases, the conventional formulation is enriched by a set of internal, element-wise, variables, consistent with the integration rules employed. These variables do not affect the overall dimensions of the equation system to be solved, once they are completely eliminated, at the element level, by static condensation procedures.

In the end of the paper, it is shown a numerical benchmark involving a small thickness shell structure, and including large displacements, large rotations and non-linear material constitutive laws. The example is treated by means of a coarse mesh, consisting of both regular and distorted meshes. Allied to the presence of a curved reference surface, elements pose themselves in warped configurations, which frequently introduce convergence and accuracy difficulties to EAS-based finite elements formulations. However, this does not seem to be the case with the present elements, which led to reliable results for all the situations analysed. The results obtained are compared with solutions coming from reduced numerical integration procedures, with the obtained results being equivalent. Nevertheless, the formulation presented appears to be more general, namely in the treatment of a general type constitutive laws and in the avoidance of parasitic (hourglass) numerical modes, usual in reduced integration schemes.

The present work is grounded on the work of the authors in the field, represented by the following references, which also contains the details about the theoretical and algorithmic aspects of the formulation, as well as an extensive set of linear and nonlinear benchmark tests.

1

César de Sá, J.M.A., Natal Jorge, R.M., Fontes Valente, R.A., Almeida Areias, P.M., "Development of shear locking-free shell elements using an enhanced assumed strain formulation", International Journal for Numerical Methods in Engineering, 53, 1721-1750, 2002. doi:10.1002/nme.360

2

Fontes Valente, R.A., Natal Jorge, R.M., Cardoso, R.P.R., César de Sá, J.M.A., Grácio, J.J., "On the use of an enhanced transverse shear strain shell element for problems involving large rotations", Computational Mechanics, 30, 286-296, 2003. doi:10.1007/s00466-002-0388-x

3

Fontes Valente, R.A., Natal Jorge, R.M., César de Sá, J.M.A., Grácio, J.J., "Enhanced transverse shear strain shell formulation applied to large elasto-plastic deformation problems", International Journal for Numerical Methods in Engineering, submitted, 2004.

4

Alves de Sousa, R.J., Natal Jorge, R.M., Fontes Valente, R.A., César de Sá, J.M.A., "A new volumetric and shear locking-free 3D enhanced strain element", Engineering Computations, 20, 896-925, 2003. doi:10.1108/02644400310502036

5

Fontes Valente, R.A., Alves de Sousa, R.J., Natal Jorge, R.M., "An enhanced strain 3D element for large deformation elasto-plastic thin-shell applications", Computational Mechanics, in press, 2004.

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