A time-dependent obstacle problem in linearised elasticity

In this paper we establish the existence of solutions to a time-dependent problem for a linearly elastic body subjected to a confinement condition, expressing that all the points of the deformed reference configuration remain confined in a prescribed half space. This problem takes the form of a set of hyperbolic variational inequalities posed over a nonempty, closed, ad convex subset of the space H 1 (Ω)×H 1 (Ω)×H 1 (Ω). The fact that any solution of the studied problem takes the form of a vector field, the generality of the confinement condition, and the choice of the function space where solutions are sought make the analysis substantially more complicated, thus requiring the adoption of new resolution strategies.