Mathematical justification of the obstacle problem for elastic elliptic membrane shells

We consider a family of linearly elastic shells with thickness 2 epsilon ( where epsilon is a small parameter). The shells are clamped along a portion of their lateral face, all having the same middle surface S, and may enter in contact with a rigid foundation along the bottom face. We are interested in studying the limit behavior of both the three-dimensional problems, given in curvilinear coordinates, and their solutions (displacements u(epsilon) of covariant components ue u(i)(epsilon)) when epsilon tends to zero. To do that, we use asymptotic analysis methods. Guided by the insight of a previous work of the author where the formal asymptotical analysis was developed, in this paper it is shown that if the density of applied body forces is O(1) with respect to epsilon and the density of surface tractions is O(epsilon), the solution of the three-dimensional problem converges to the solution of the two-dimensional variational inequality which can be identified as the variational formulation of the obstacle problem of an elastic elliptic membrane shell.