Contact problems are central to Solid Mechanics, because contact is the principal method of applying loads to a deformable body and the resulting stress concentration is often the most critical point in the body. Contact is characterized by unilateral inequalities, describing the physical impossibility of tensile contact tractions (except under special circumstances) and of material interpenetration. Additional inequalities and/or non-linearities are introduced when friction laws are taken into account. These complex boundary conditions can lead to problems with existence and uniqueness of quasi-static solution and to lack of convergence of numerical algorithms. In frictional problems, there can also be lack of stability, leading to stickยฑslip motion and frictional vibrations.
If the material is non-linear, the solution of contact problems is greatly complicated, but recent work has shown that indentation of a power-law material by a power law punch is self-similar, even in the presence of friction, so that the complete history of loading in such cases can be described by the (usually numerical) solution of a single problem.
Real contacting surfaces are rough, leading to the concentration of contact in a cluster of microscopic actual contact areas. This aects the conduction of heat and electricity across the interface as well as the mechanical contact process. Adequate description of such systems requires a random process or statistical treatment and recent results suggest that surfaces possess fractal properties that can be used to obtain a more ecient mathematical characterization.
Contact problems are very sensitive to minor proยฎle changes in the contacting bodies and hence are also aected by thermoelastic distortion. Important applications include cases where non-uniform temperatures are associated with frictional heating or the conduction of heat across a non-uniform interface. The resulting coupled thermomechanical problem can be unstable, leading to a rich range of physical phenomena.
Other recent developments are also brieยฏy surveyed, including examples of anisotropic materials, elastodynamic problems and fretting fatigue.