ABSTRACT
In this chapter we survey relevant available results for dynamic contact problems with a physically well posed contact condition in displacements. We confine ourselves to the contact of a body with a rigid undeformable obstacle. The reason is the lack of a satisfactory model for the contact of two bodies with dimension higher than one respecting the non-penetrability of both bodies, in particular the time dependence of the appropriate couples of boundary points which should describe this condition. In this respect the dynamic problem differs from the static one, where the non-penetrability can be modelled on the basis of the original configuration and from the case of strings, where the geometry is essentially simplified (cf. e.g. [33]). The main difficulty of the dynamic contact problems with friction arises
from the hyperbolic nature of the equilibrium of inner forces. In variational inequalities for hyperbolic problems it is much more comfortable to have the time derivatives of the solution in the test function instead of the solution itself. With the velocity as a test function, the signs at the velocity and at the space gradient will be the same after integration by parts in time and space. On the other hand, if the displacement is used as a test function, the space and time terms have opposite signs with many unpleasant consequences on estimates and convergence procedures. Hence, from a mathematical point of view the contact condition in velocities is much easier to handle. Before the first results on the original contact condition appeared there was a large literature on the solution to the contact condition in velocities, whose results were mostly mentioned in the monograph [42]. They included also the problem with a given time-independent friction force. The main tool in proofs of the existence of solutions to dynamic con-
tact problems is the penalization of the contact condition. This method may yield energy preserving solutions which is why we have focused our approach to results of this type. On the other hand, these results represent an essential part of the knowledge in that field. There are not many of them and even fewer of them are applicable to contacts of bodies. The results for purely elastic and therefore purely hyperbolic situations are limited to strings, to polyharmonic problems, or to special problems on halfspaces.
These results are surveyed in the first section. Including the viscous behaviour of the material, which is physically well
based, we can parabolize the problems. Such a parabolization allows solving wider classes of problems. There is a remarkable difference between domain and boundary obstacles. While the first require some direct estimate of the penalty term which does not give much information, the latter are much simpler to solve, because dual estimates of the acceleration can be employed and the penalty term is (up to the sign) the normal boundary traction, which can be controlled by the volume terms. We have studied two kinds of viscosity: the long and the short memory. The long timesingular memory yields slightly weaker results which do not enable us to consider the friction. For the short memory materials, satisfactory results for problems with given friction are derived. However, the original problem with Coulomb friction for the contact condition in displacements remains unsolved.
