A Linearly Elastic Shell over an Obstacle: The Flexural Case

We study the equilibrium of a three-dimensional solid having a uniform thickness $2 \varepsilon $ along a middle surface which satisfies the usual assumptions of shell theory. The solid is linearly elastic at small strains and is submitted to unilateral contact conditions with an obstacle on a part of its boundary. When $\varepsilon $ tends to zero, the three-dimensional domain tends to a two-dimensional one, so that the contact conditions pass from a part of the boundary to the interior of the domain. We restrict our attention to the so-called bending case, that is when the shell undergoes only inextensional deformations. As a major difference with the case of a shallow shell, we get in general a coupling between the three components of the displacement in the contact conditions. The work is closed by explicit examples showing the corresponding variation of the non-penetrability condition along the surface of the shell and by comments about the model and the remaining difficulties.