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

A reduced enhanced solid-shell (RESS) finite element concept has been suggested recently by Alves de Sousa et al. (Int. J. Numer Meth. Engng 2005; 62:952; Int. J. Numer Meth. Engng 2006; 67:160; Int. J. Plasticity 2007; 23:490). 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. Furthermore, traditionally, solid elements have been developed based on fully integrated schemes with limited number of integration points per single element layers which renders them computationally expensive. On the contrary, the solid-shell element by Alves de Sousa et al. (Int. J. Numer. Meth. Engng 2005; 62:952; Int. J. Numer. Meth. Engng 2006; 67:160; Int. J. Plasticity 2007; 23:490) 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. Copyright © 2007 John Wiley & Sons, Ltd