An obstacle problem for flexural shells

Our objective is to identify two-dimensional equations that model an obstacle problem for a linearly elastic flexural shell subjected to a confinement condition expressing that all the points of the admissible deformed configurations remain in a given half-space. To this end, we embed the shell into a family of linearly elastic flexural shells, all sharing the same middle surface $\bm{\theta} (\overline{\omega})$, where $\omega$ is a domain in $\mathbb{R}^2$ and $\bm{\theta} : \overline{\omega} \to \mathbb{E}^3$ is a smooth enough immersion, all subjected to this confinement condition, and whose thickness $2 \varepsilon > 0$ is considered as a "small" parameter approaching zero. We then identify, and justify by means of a rigorous asymptotic analysis as $\varepsilon$ approaches zero, the corresponding "limit" two-dimensional variational problem. This problem takes the form of a set of variational inequalities posed over a non-empty closed and convex subset of the space $ H^1 (\omega) \times H^1 (\omega) \times H^2(\omega)$. The confinement condition considered here considerably departs from the Signorini condition usually considered in the existing literature, where only the "lower face" of the shell is required to remain above the "horizontal" plane. Such a confinement condition renders the asymptotic analysis considerably more difficult, however, as the constraint now bears on a vector field, the displacement vector field of the reference configuration, instead of on only a single component of this field.