Nonlinear Donati compatibility conditions on a surface-Application to the intrinsic approach for Koiter's model of a nonlinearly elastic shallow shell

An intrinsic approach to a mathematical model of a linearly or nonlinearly elastic body consists in considering the strain measures found in the energy of this model as the sole unknowns, instead of the displacement field in the classical approach. Such an approach thus provides a direct computation of the stresses by means of the constitutive equation. The main problem therefore consists in identifying specific compatibility conditions that these new unknowns, which are now matrix fields with components in L2, should satisfy in order that they correspond to an actual displacement field. Such compatibility conditions are either of Saint-Venant type, in which case they take the form of partial differential equations, or of Donati type, in which case they take the form of ortho-gonality relations against matrix fields that are divergence-free. The main objective of this paper consists in showing how an intrinsic approach can be successfully applied to the well-known Koiter's model of a nonlinearly elastic shallow shell, thus providing the first instance (at least to the authors' best knowledge) of a mathematical justification of this approach applied to a nonlinear shell model ("shallow" means that the absolute value of the Gaussian curvature of the middle surface of the shell is "uniformly small enough"). More specifically, we first identify and justify compatibility conditions of Donati type guaranteeing that the nonlinear strain measures found in Koiter's model correspond to an actual displacement field. Second, we show that the associated intrinsic energy attains its minimum over a set of matrix fields that satisfy these Donati compatibility conditions, thus providing an existence theorem for the intrinsic approach; the proof relies in particular on an interesting per se nonlinear Korn inequality on a surface. Incidentally, this existence result (once converted into an equivalent existence theorem for the classical displacement approach) constitutes a significant improvement over previously known existence theorems for Koiter's model of a nonlinearly elastic shallow shell.