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The obstacle problem for a shallow shells: a curvilinear approach

We start with a three-dimensional equilibrium problem involving a linearly elastic solid at small strains subjected to unilateral contact conditions. The reference configuration of the solid is assumed to be a thin shallow shell with a uniform thickness. We focus on the limit when the thickness tends to zero, i.e. when the three-dimensional domain tends to a two-dimensional one. In the generic case, this means that the initial Signorini problem, where the contact conditons hold on the boundary, tends to an obstacle problem, where the contact conditions hold in the domain. When the problem is stated in terms of curvilinear coordinates, the unilateral contact conditions involve a non penetrability inequality which couples the three covariant components of the displacement. We show that nevertheless we can uncouple these components and the contact conditions involve only the transverse covariant component of the displacement at the limit.