On the use of a reduced enhanced solid-shell (RESS) element for sheet forming simulations

Abstract

A recently proposed reduced enhanced solid-shell (RESS) element [Alves de Sousa, R.J., Cardoso, R.P.R., Fontes Valente, R.A., Yoon, J.W., Grácio, J.J., Natal Jorge, R.M., 2005. A new one-point quadrature enhanced assumed strain (EAS) solid-shell element with multiple integration points along thickness: Part I – Geometrically Linear Applications. International Journal for Numerical Methods in Engineering 62, 952–977; Alves de Sousa, R.J., Cardoso, R.P.R., Fontes Valente, R.A., Yoon, J.W., Grácio, J.J., Natal Jorge, R.M., 2006. A new one-point quadrature enhanced assumed strain (EAS) solid-shell element with multiple integration points along thickness: Part II – Nonlinear Applications. International Journal for Numerical Methods in Engineering, 67, 160–188.] is based on the enhanced assumed strain (EAS) method with a one-point quadrature numerical integration scheme. In this work, the RESS element is applied to large-deformation elasto-plastic thin-shell applications, including contact and plastic anisotropy. One of the main advantages of the RESS is its minimum number of enhancing parameters (only one), which when associated with an in-plane reduced integration scheme, circumvents efficiently well-known locking phenomena, leading to a computationally efficient performance when compared to conventional 3D solid elements. It is also worth noting that the element accounts for an arbitrary number of integration points through thickness direction within a single element layer. This capability has proven to be efficient, for instance, for accurately describing springback phenomenon in sheet forming simulations. A physical stabilization procedure is employed in order to correct the element’s rank deficiency. A general elasto-plastic model is also incorporated for the constitutive modelling of sheet forming operations with plastic anisotropy. Several examples including contact, anisotropic plasticity and springback effects are carried out and the results are compared with experimental data.

Introduction

The studies on the development and application of the low-order eight-node (brick) finite element for thin-walled applications have been increasing for the optimal balance between accuracy and efficiency in numerical simulations. Particularly, solid-shell concepts, based on conventional solid elements’ topologies but with improved bending performances, become popular. Solid-shell elements are kinematically similar to shell elements differing from the latter ones by the absence of rotational degrees of freedom, but keeping the three-dimensional structure of solid elements with eight physical nodes. A number of advantages encouraging the use of solid-shell elements can be listed by comparing these with conventional shell elements:

• Simpler procedure for configurations update, with no inclusion of rotational degrees of freedom;

• The use of full 3D constitutive laws without plane-stress assumptions on the constitutive equations;

• Natural calculation of thickness (strain) variations, as based on physical nodes;

• Accurate and automatic consideration of double sided contact problems.

However, such a class of low order finite elements is highly prone to the occurrence of the so-called locking phenomenon (Hughes, 2000). Overestimation of the stiffness matrix, due to the occurrence of locking, is the main source of poor results regarding this class of elements. Hauptmann et al. (2001) categorized several types of locking for solid-shell elements. Especially in the case of sheet metal forming simulations involving plasticity and thin-walled geometries, distinct cases of locking can be expected and should be avoided, namely:

• Volumetric locking, appearing due to the incompressible-type deformation in plasticity;

• Transverse shear locking, occurring as thickness to length ratio tends to zero in bending situations;

• Thickness locking, related to an inefficient reproduction of the strain field along the thickness direction.

The reduced (RI) and selective reduced (SRI) Integration techniques were the first successful numerical solutions to alleviate locking pathologies. For solid elements, both techniques correspond to the use of a lower quadrature rule (1 Gauss point, located in the center of the tri-unit cube) rather than the so-called complete quadrature rule (2 × 2 × 2 Gauss points). Such a reduced integrated element is especially attractive considering its computational efficiency. Nevertheless, due to the rank-deficiency, stabilization procedures are required in order to avoid the appearance of spurious deformation modes, also known as hourglass patterns. Starting from the works of Zienkiewicz et al., 1971, Hughes et al., 1978, the class of 3D reduced integrated elements has been continuously developed together with its many stabilization schemes through the last decades, being some examples Liu et al., 1998, Masud et al., 2000. Despite the accuracy and reliability of the results obtained from this kind of formulations, the fixed one-point integration rule gives rise to a critical drawback: when dealing with problems which require more than one integration point along the thickness direction (e.g. springback analysis), the accommodation of several elements layers causes a substantial decrease in computational efficiency.

The enhanced assumed strain (EAS) method was initially introduced by Simo and Rifai (1990). Within this formulation, the strain field is enlarged (under certain conditions) with the inclusion of an internal variables field, therefore resulting in additional deformation modes. The mathematical basis for such element formulation is given by the well-known Veubeke–Hu–Washizu three-field variational (Fraeijs de Veubeke, 1951). The EAS method has been widely applied for 2D, 3D, shell and solid-shell formulations. However, the computational inefficiency has proven to be the major disadvantage of EAS elements. In fact, the number of enhancing variables is seldom less than 10 (but reaching sometimes 30 or more), leading to hardly treatable stiffness matrices. Also, the use of a single layer of fully-integrated elements is commonly not enough to capture accurately bending effects due to the lack of integration points along the thickness direction.

In the scope of EAS and RI techniques, recent investigations proved that they can be combined to derive efficient and accurate solid-shell elements. Examples are the works of Puso, 2000, Reese, 2002, Reese, 2005, Reese et al., 2000, Legay and Combescure, 2003, with the common characteristic of using a fixed number of Gauss points in the thickness direction. Investigations carried out by Reese et al., 1999, Reese and Wriggers, 2000 have shown the importance of stabilization schemes in the EAS method: the hourglass instabilities can be avoided by a suitable form of stabilization.

In the present work, the recently proposed reduced enhanced solid-shell (RESS) element (Alves de Sousa et al., 2005, Alves de Sousa et al., 2006) is utilized for sheet metal forming applications. This solid-shell element is based on a one-point numerical quadrature scheme with eight physical nodes, but allowing for an arbitrary number of integration points through the thickness direction in a single element layer. This characteristic is distinctive within the class of solid-shell elements. In consequence, it avoids the use of several layers of elements in order to increase the number of thickness integration points in metal forming problems. The capabilities of the EAS and RI methods are combined together in order to eliminate locking problems. The computational efficiency is insured via the one-point quadrature integration scheme with just one enhancing parameter for the EAS method. The rank-deficiency caused by the in-plane reduced integration scheme demands an efficient and cost effective stabilization technique. In this work, the physical stabilization concept developed in the work of Cardoso et al. (2002) for shell elements is extended for the RESS element, efficiently eliminating three-dimensional hourglass modes without resorting to empirical parameters.

The main purpose of this paper is to assess the element’s performance for typical applications in sheet metal forming, including drawing, springback and hydroforming, which require multiple integration points through thickness direction. It is well known that in these cases, shell elements have difficulties in deal with double-sided contact and also conventional solid elements require several element layers to capture bending effects. Nevertheless, in the present work, simulations are conducted by the RESS solid-shell element using one element layer with multiple integration points through thickness. Anisotropic plasticity with recent yield functions (Barlat et al., 1991, Barlat et al., 2005) is also incorporated, accounting for full 3D stress states.