A discontinuous Galerkin formulation of non-linear Kirchhoff–Love shells

Discontinuous Galerkin (DG) methods provide a means of weakly enforcing the continuity of the unknown-field derivatives and have particular appeal in problems involving high-order derivatives. This feature has previously been successfully exploited (Comput. Methods Appl. Mech. Eng. 2008; 197:2901-2929) to develop a formulation of linear Kirchhoff-Love shells considering only the membrane and bending responses. In this proposed one-field method - the displacements are the only unknowns, while the displacement field is continuous, the continuity in the displacement derivative between two elements is weakly enforced by recourse to a DG formulation. It is the purpose of the present paper to extend this formulation to finite deformations and non-linear elastic behaviors. While the initial linear formulation was relying on the direct linear computation of the effective membrane stress and effective bending couple-stress from the displacement field at the mid-surface of the shell, the non-linear formulation considered implies the evaluation of the general stress tensor across the shell thickness, leading to a reformulation of the internal forces of the shell. Nevertheless, since the interface terms resulting from the discontinuous Galerkin method involve only the resultant couple-stress at the edges of the shells, the extension to non-linear deformations is straightforward.