ABSTRACT
When one or several solid deformable bodies move, they may come into contact and forces are transmitted through their common area of contact. These forces have usually two components: the normal component prevents the inter-penetration of the bodies, and the tangential component is created by friction. A contact problem is described by a system of partial differential equations which describe the motion and deformation of the bodies, together with the boundary conditions modeling contact and friction forces. Contact problems have applications to many fields of solid mechanics:
in machine dynamics and manufacturing, contact problems arise when two parts hit one another; in many areas of mechanical engineering, a contact problem arises when cracks open and close. During an earthquake, contacts may take place between different parts of a building or different buildings, and between buildings and the earth. Due to the lower friction the solitary buildings usually survive an earthquake remarkably better than the buildings being in some contact with other ones. Therefore, neglecting contact and friction in a computer simulation of the motion of large structures during an earthquake may lead to false predictions. An earthquake itself can be seen as a gigantic contact problem between different lithospheric plates of the earth’s surface, and indeed there are earthquake models which are precisely based on the build-up of tension in faults, followed by its release when an appropriate threshold is crossed, cf. e.g. [72]. Contact problems also occur in bio-mechanics; for instance a hip joint, be it natural or artificial, is subject to contact, and its wear depends on contact. The normal contact forces are basically prescribed by the so-called Si-
gnorini conditions, and there is not much room for anything else, since they are geometrical and they just account for the non-interpenetration condition via a variational-or virtual work-condition. Friction conditions are quite another story: the phenomenologically (not mathematically) simplest and most popular conditions are given by the Coulomb law of friction. This law introduced in 1781 (see [37]) is also in good harmony with practical experience.
