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(curl)-conforming N\'ed\'elec 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 method for Kirchhoff-Love and Reissner-Mindlin plates, respectively. By using the Regge interpolation operator 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 excellence performance of the proposed shell elements.

翻译：本文通过分层方法，将近期提出的用于非线性Koiter壳体的混合Hellan-Herrmann-Johnson（HHJ）方法推广至非线性Naghdi壳体。附加剪切自由度采用H(curl)相容的Nédélec有限元离散，从而得到无剪切锁定的方法。通过线性化模型，我们在小应变范围内得到了线性Kirchhoff-Love和Reissner-Mindlin壳体公式，这些公式分别退化为最初针对Kirchhoff-Love板和Reissner-Mindlin板提出的HHJ与TDNNS方法。利用Regge插值算子，我们获得了任意阶无锁定的壳体方法。此外，这些方法可直接应用于含尖角和分支壳体的结构。多个数值算例与实验验证了所提出的壳体单元具有优异性能。