Enhanced assumed strain (EAS) and assumed natural strain (ANS) methods for one‐point quadrature solid‐shell elements

Abstract

. Developments on the 'RESS' element were motivated by the following reasons: first, solid-shell elements automatically incorporate the normal stress along the thickness direction, which makes them more suitable for the simulations with double-sided contact than their shell counterparts; second, they have only translational degrees of freedom, which alleviates some difficulties associated with formulating complex shell formulations using nodal rotations; third, general constitutive models can be used and a reformulation for plane-stress conditions is not necessary. was developed for one single-layer shell structure with reduced in-plane and multiple integration points along the thickness direction of the shell. The formulation consists of several combinations of well-known techniques to ameliorate locking problems as is the case of the enhanced assumed strain (EAS) method. However, the proposed 'RESS' solid shell did not consider the transverse shear components of hourglass (or physical stabilization) parts. This negligence produces a slightly flexible behavior of the element, but it can also cause the appearance of hourglass modes for several non-linear applications including large rigid body rotations. Sometimes, the negligence brings a non-positive-definite status on the stiffness matrix. In order to overcome such drawbacks, a modified assumed natural strain (ANS) method considering the top and bottom surfaces of the element was incorporated for the transverse shear components. At the same time, new hourglass strains for the membrane field were constructed based on the stabilization vectors of Liu et al. (Comput. Meth. Appl. Mech. Engng 1998; 154:69). With these modifications, the improved element (called 'M-RESS') passes both the membrane and bending patch tests and performs with remarkable stability and accuracy in sheet-forming simulations.

On the comparison of two solid-shell formulations based on in-plane reduced and full integration schemes in linear and non-linear applications

“…From these tests, we found out that the use of the second or the third ANS version (see Figure 3) leads almost to the same results. This observation agrees with the remark of Cardoso et al [32] and Schwarze and Reese [10] concerning the use of a full integration scheme. Therefore, the second ANS version is used for SSH3D solid-shell element.…”

“…This method was later extended to overcome the curvature thickness locking by modifying the interpolation of the transverse normal strain component E 33 for four-node shell elements, as proposed by Betsch and Stein [29] and also Bischoff and Ramm [31]. In the recent development of solid-shell elements, numerous formulations are developed based on the combination of the EAS and ANS methods such as the work of Hauptman et al [5], Miehe [16], Klinkel et al [18], Vu-Quoc and Tan [6], Cardoso et al [32], Schwarze and Reese [10] and the recent works of Rah et al [12] and Pramin et al [33]. Taking all these references and achievements in the literature, the main goal of the present paper is to present the formulations of the SSH3D (=Solid-SHell element in 3D) and RESS (=Reduced Enhanced Solid-Shell element) solid-shell elements.…”

Software Supported Implementation of Efficient Solid-Shell Finite Elements

“…[18,11,17]. More recent developments regarding especially the treatment of artificial stiffness effects by reduced integration techniques are given e. g. in [1,19,7,20] in the context of implicit finite element applications.…”

“…Though not as distinctive as in elements with linear displacement interpolation, locking also appears with quadratic shape functions and can be reduced -or even completely removed -by several corrections within the element formulation. Proposals for locking-free Solid-Shell elements, using reduced integration rules together with stabilization techniques against artificial kinematics can be found in [1,19,7,20]. In the current contribution, fully integrated element formulations are presented, where different locking phenomena are treated with the well-known methods of 'Assumed Natural Strains (ANS)' [2,6] and 'Enhanced Assumed Strains (EAS)' [23,22], which have already been applied to Solid-Shell elements for non-linear implicit analyses [11,9,10].…”

“…An application of the method to elements with quadratic shape functions was presented by BUCALEM and BATHE in [6]. CARDOSO ET.AL suggested for linear Solid-Shell elements in [7] an evaluation at two sampling points over the thickness (ζ = −1/ζ = 1), together with a linear interpolation in thickness direction. The implemented linear and quadratic Solid-Shell formulations contain interpolations of the transverse shear strains and -in order to cure membrane locking -the membrane strains of the quadratic elements are also interpolated.…”