A mixed finite element for shell model with free edge boundary conditions Part 1. The mixed variational formulation Original Research Article

After a very brief recall on general shell theory, we construct a mixed variational formulation based on the introduction of a new unknown: the rotation of the normal to the medium surface. In Koiter shell theory (for instance), this rotation can be expressed with respect to the three components of the displacement field of the medium surface and their derivatives. The Lagrange multiplier corresponding to this relation (known as the Kirchhoff-Love kinematical assumption), is also introduced as an independent unknown. There are two main difficulties: one is due to the differential geometry of surfaces and is rather technical; the other is to define correctly the dual space for the Kirchhoff-Love relation. The difficulty is similar to the one met in the characterization of the dual space of the Sobolev space: H1(ω) (ω being the medium surface of the shell), for which a boundary component appears except for clamped shells which is a very restrictive situation.