A Co-Rotational Formulation for 2-D Continua Including Incompatible Modes

The co-rotational finite element method is well known for decomposing the motion of an element into a rigid body motion and a strain-producing deformation. A key feature is the definition of the rotating frame and, partially for this reason, the method has almost exclusively been applied to beams and shells. A novel feature of the current paper is that the method is applied to continua—in the current case, two-dimensional. The main motivation is to allow the analyst to quickly introduce the latest and best linear elements into a non-linear context. To this end, in the current work, a set of ‘incompatible modes’ or ‘enhanced strains’ is added to the conventional four-noded elements. While the main body of the paper considers small strains, as a further novel aspect, it later applies the co-rotational method to problems with large strains (here via hyperelasticity) and, to this end, establishes a link between the co-rotational technique and a Biot stress formulation.