An obstacle problem for elliptic membrane shells

Our objective is to identify two-dimensional equations that model an obstacle problem for a linearly elastic elliptic membrane 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 elliptic membrane shells, all sharing the same middle surface θ(ω¯), where ω is a domain in R2 and θ:ω¯→E3 is a smooth enough immersion, all subjected to this confinement condition, and whose thickness 2e>0 is considered as a “small” parameter approaching zero. We then identify, and justify by means of a rigorous asymptotic analysis as e approaches zero, the corresponding “limit” two-dimensional variational problem. This problem takes the form of a set of variational inequalities posed over a convex subset of the space H01(ω)×H01(ω)×L2(ω). The confinement condition considered here considerably departs from the Signorini condition usually considered in the existing literatu...