Intrinsic time integration scheme for directors in geometrically exact shell dynamics

This work presents an intrinsic time integration scheme for directors in shell dynamics. For shells without intersections, director rotations possess only two degrees of freedom. By revisiting the kinematics of directors, the two independent components of angular velocity and the corresponding rotational variations are rigorously identified. It is shown that, for sufficiently small relative rotations of directors between successive time steps, the time derivatives of the two non-vanishing rotation vector components are linearly related to the independent angular velocity components. Leveraging this relationship, and drawing inspiration from Lie-group integration and intrinsic integrator schemes, a generalized−α integration scheme specifically tailored for directors is developed. The shell formulation is based on a three-field Hu-Washizu variational principle, in which independent stress resultants and sectional strains are introduced. This leads to element-wise equations that allow the independent variables to be eliminated. Inertial forces are derived using Hamilton's variational principle, with the velocities of the mid-plane surface and the two independent angular velocity components of the directors being interpolated directly. The proposed scheme is validated through three numerical examples of increasing complexity. The results show lower errors compared with the time integration schemes reported for the numerical examples investigated.