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Point stationary lubricated contact of elastic bodies

Let's consider a case of external contact of two spherical bodies divided by a layer of lubricant (fig. 6.1).

Fig. 6.1

Considering that the size of zone of contact of bodies is small in comparison with radiuses of spheres, a gap between bodies in dimensional variables we write down as follows


where is co-ordinate of a point of breakage of a lubricant layer in section, , , is radiuses of sphere ,


is the elastic displacement of a surface of spheres (i=1,2). Displacement is considered positive if it is directed inside sphere.

As the sizes of contact zone are small in comparison with spheres radiuses then it is possible to replace spheres with semi spaces when calculating the displacement. As a result the elastic displacement can be defined under the formula


where are the module of elasticity and  Poisson’s ratio of body respectively.

Pressure in a lubricant layer at a known film thickness is defined from Reynolds's equation which in this case looks like


where are linear speeds of movement of surfaces.

To the equations (6.1) - (6.4) it is necessary to add dependences of viscosity and density of lubricant from pressure, an equilibrium condition of contacting bodies and boundary conditions for function.

The decision of the given problem possesses the same properties, as the decision of a problem for infinite cylinders. Namely, at high loadings the function has two maxima of pressure. At that the maximum located on an exit from a gap, is sharp. The pressure distribution in a lubricant layer is close to pressure distribution in dry contact, except for some vicinity of a point of the second maximum of pressure. Characteristic dependences and at the high loadings which are the solution of system of the equations (6.1) - (6.4) at the viscosity depending on pressure under the law (1.18), are presented on fig. 6.2 and 6.3.

Fig. 6.2


Fig. 6.3