Asymptotic Justification of the Kirchhoff–Love Assumptions for a Linearly Elastic Clamped Shell

The displacement vector of a linearly elastic shell can be computed by using the two-dimensional Koiter's model, based on the a priori Kirchhoff–Love assumptions. These hypotheses imply that the displacement of any point of the shell is an affine function of the transverse variable x3. The term independent of x3 of this approximation is equal to the displacement vector of the two-dimensional Koiter's model. The term linear in x3 depends on the infinitesimal rotation vector of the normal. After an appropriate scaling, we estimate here the difference between the three-dimensional displacement and this affine vector field in the case of shells clamped along their entire lateral face. Besides, in the case of shells with uniformly elliptic middle surface, taking into account the term depending of the rotation of the normal allows to improve the asymptotic estimate between the three-dimensionnal displacement and Koiter's bidimensional displacement.