Enhanced Allman quadrilateral for finite drilling rotations

The paper concerns a four-node quadrilateral element based on Allman shape functions undergoing finite (unrestricted) drilling rotations, and aims at improving its accuracy and facilitating its implementation.Firstly, the classical Allman shape functions are valid only for small in-plane rotations, and must be used with a co-rotational frame, which embeds finite rotations. We derive a new form of Allman shape functions, which is valid for finite drilling rotations, and allows to avoid the use of such a frame.Secondly, the classical Allman quadrilateral shows locking in the in-plane shear test. We study this problem, identify its source, and remove it by enhancing the element with two additional modes, via the Enhanced Assumed Displacement Gradient (EADG) method. To accomplish this, we extend the original version of the method to mixed functionals including rotations.Two variational formulations including the drilling rotation via the rotation constraint (RC) equation are considered; one based on the Green strain, and the other on the relaxed non-symmetric right stretch strain. Numerical tests of the corresponding finite elements show that the improved Allman elements are as exact in linear tests as the EADG4 element, and perform very well in a severe in-plane shear test for one layer of elements undergoing large rotations.