In engineering applications, for example, pressure cylinders employed in paper industry, as a material of thin layer, polyurethane or natural and artificial rubbers in high elastic state is used. To define the stresses interval, where the material behavior can be considered as uncompressible and to specify the mechanical characteristics, the experiments on deep pressure and creep under the specimens made of polyurethane were carried out. The problem of thin layer made of incompressible viscous material subjected to compression by two parallel plates is considered. The analytical expressions for current lines, the rates of displacements, the deformations and stresses are derived and the corresponding diagrams are constructed. The dimensions of thin stratum near the contact planes are determined in which the stress condition can be considered as the state of hydrostatic pressure. This state may be responsible for degradation of the structure and the failure of thin layer and the construction as a whole. The res