A robust non-linear mixed hybrid quadrilateral shell element

In the paper a non-linear quadrilateral shell element for the analysis of thin structures is presented. The variational formulation is based on a Hu–Washizu functional with independent displacement, stress and strain fields. The interpolation matrices for the mid-surface displacements and rotations as well as for the stress resultants and strains are specified. Restrictions on the interpolation functions concerning fulfillment of the patch test and stability are derived. The developed mixed hybrid shell element possesses the correct rank and fulfills the in-plane and bending patch test. Using Newton’s method the finite element approximation of the stationary condition is iteratively solved. Our formulation can accommodate arbitrary non-linear material models for finite deformations. In the examples we present results for isotropic plasticity at finite rotations and small strains as well as bifurcation problems and post-buckling response. The essential feature of the new element is the robustness in the equilibrium iterations. It allows very large load steps in comparison to other element formulations