Intrinsic nonlinear models of shells for Saint Venant-Kirchhoff materials

The first complete linear theory, based on Kirchhoff’s [1850] earlier works on plates, was given by Love [1888]. The modern theory (linear and nonlinear) comes from the works by Naghdi and Koiter in the sixties. In such models the displacement variables are functions defined on the mean surface and the resulting equations are differential equations on a smooth 2-dimensional submanifold of the three-dimensional Euclidean space. This is done by introducing local coordinates and local covariant and contravariant bases for the tangent plane and equations are written in terms of covariant and contravariant derivatives. In this framework expressions of constitutive laws become heavy and it is not always easy to separate purely mechanical approximations from approximations of the characteristics of the geometry of the mean surface. Research in Shell Theory has currently been stimulated by questions arising from a number of applications in control and design of large space structure, flexible robots, composite materials, etc, where intrinsic and mathematically more tractable models would be much prefered. Recently, Delfour and Zolsio (cf. [10] and [11]) introduced a new way to deal with the differential geometry, to express tangential differential operators, and to do the differential calculus on submanifolds of the n-dimensional Euclidean space. This was successfully used in the Natural Theory and the Love-Kirchhoff Theory (cf. for instance Germain [14], Dautray and Lions [7]) of static and dynamical linear thin/shallow shells. The new linear models are expressed in