Geometrically exact sandwich shells: The dynamic case Original Research Article

In this paper, we present a finite element formulation for the dynamic analysis of our geometrically exact multilayer shell model developed earlier. The dynamics of the motion of sandwich shells is referred directly to an inertial frame. This model accommodates large deformation and large overall motion. The layer directors at a point in the reference surface are connected to each other by universal joints, as in a chain of rigid links. Finite rotations of the directors in every layer are allowed, with shear deformation independently accounted for in each layer. The thickness and the length of each layer can be arbitrary, thus make it suitable to model shell structures with patches of constrained viscoelastic materials or of piezoelastic materials. The nonlinear dynamic weak form of the equations of motion of sandwich shells is constructed here. A time-stepping algorithm based on the generalized mid-point rule for the directors is then employed in the time discretization of the sandwich shell equations. This algorithm reduces exactly to an algorithm that conserves the total linear and angular momenta in single-layer shells. The exact linearization of the dynamic weak form and the associated configuration dynamic update involving the directors and the director rotations are obtained in closed form, leading to a configuration-dependent nonsymmetric tangent inertia matrix. As a result, asymptotically quadratic rate of convergence is attained in a Newton–Raphson iterative solution strategy. The applicability and generality of the proposed formulation are demonstrated through several numerical examples that include free vibration of a sandwich plate, and large overall motions of a free-flying sandwich plate.