The Hellan–Herrmann–Johnson and TDNNS methods for linear and nonlinear shells

In this paper we extend the recently introduced mixed Hellan–Herrmann–Johnson (HHJ) method for nonlinear Koiter shells to nonlinear Naghdi shells by means of a hierarchical approach. The additional shearing degrees of freedom are discretized by $H\left(\mathrm{curl}\right)$-conforming Nédélec finite elements entailing a shear locking free method. By linearizing the models we obtain in the small strain regime linear Kirchhoff–Love and Reissner–Mindlin shell formulations, which reduce for plates to the originally proposed HHJ and TDNNS methods for Kirchhoff–Love and Reissner–Mindlin plates, respectively. By interpolating the membrane strains into the so-called Regge finite element space we obtain locking-free arbitrary order shell methods. Additionally, the methods can be directly applied to structures with kinks and branched shells. Several numerical examples and experiments are performed validating the excellent performance of the proposed shell elements.