In the present review article general results concerning problems of contact interaction (contact problems) of pre-stressed bodies with rigid, and elastic punches (facings) are presented. The modern classification of investigations on contact problems for bodies with initial stresses is based on two approaches. The first approach is applicable to bodies with a specific type of elastic potential. The second approach is developed in parallel with the first one. It describes investigations on contact problems for pre-stressed bodies with an arbitrary structure of an elastic potential in a general form for compressible and incompressible materials. In the last case, investigations were carried out in a common form for the theory of large (finite) initial strains and various variants of the theory of small initial strains. The formulation of the problems is given, and methods of solving planar and spatial contact problems for bodies with initial stresses are developed. Procedures of the theory of complex variables and methods of Riemann-Hilbert are used. For planar problems, the authors introduce planar potentials of the linearized static and dynamic problems for elastic compressible and incompressible bodies with initial stresses. In the case of spatial contact problems, various procedures are used: general solutions of spatial static problems for elastic bodies with initial stresses, methods of harmonic potential theory, integral transformations, and integral equations using numerical methods of investigation. The essential numerical results in the form of graphs and tables are presented. These give the quantitative and qualitative analysis of the influence of initial stresses on the main characteristics of contact problems. This review contains 98 references.

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