A reduced constraint $hp$ finite element method for shell problems

We propose and analyze an $hp$ finite element method for the Nagdhi shell model, based on rectangular elements. We show that for the bending-dominated case, assuming sufficient smoothness on the solution, the method is locking free in terms of both $h$ and $p$, as the thickness of the shell tends to zero. Our results are established under the assumption that the geometrical coefficients appearing in the model are piecewise polynomial functions.