The system is the solid and an obstacle. At initial time, the solid occupies domain \(\mathcal {D}_{a}\) and its boundary \(\partial \mathcal {D}_{a}\), and at time t the solid occupies domain \(\mathcal {D}_{x}\) and its boundary \( \partial D_{x}\). A domain is a smooth, bounded, connected open set of \( \mathbb {R} ^{3}\). The solid is fixed to the obstacle which occupies domain \(\mathcal {D} _{obs}\) and its boundary \(\partial \mathcal {D}_{obs}\), on part \(\Gamma _{a}^{0}\subset \partial \mathcal {D}_{obs}\cap \partial \mathcal {D}_{a}\) of their boundaries. When moving, it can be in unilateral contact with the obstacle outside part \(\Gamma _{a}^{0}\), Fig. 30.1.

As usual, we consider the obstacle is part of the system which occupies domain \(\mathcal {D}_{a}\cup \mathcal {D}_{obs}\) and its boundary. For the sake of simplicity, we assume the obstacle is immobile. This is the case if it is very massive compared to the solid.