Reissner–Mindlin shell theory based on tangential differential calculus

Abstract

The linear Reissner–Mindlin shell theory is reformulated in the frame of the tangential differential calculus (TDC) using a global Cartesian coordinate system. The rotation of the normal vector is modelled with a difference vector approach. The resulting equations are applicable to both explicitly and implicitly defined shells, because the employed surface operators do not necessarily rely on a parametrization. Hence, shell analysis on surfaces implied by level-set functions is enabled, but also the classical case of parametrized surfaces is captured. As a consequence, the proposed TDC-based formulation is more general and may also be used in recent finite element approaches such as the TraceFEM and CutFEM where a parametrization of the middle surface is not required. Herein, the numerical results are obtained by isogeometric analysis using NURBS as trial and test functions for classical and new benchmark tests. In the residual errors, optimal higher-order convergence rates are confirmed when the involved physical fields are sufficiently smooth.