Elastic Contact of Spheres Hertz, Model

The problem of the elastic contact of a flat cylindrical punch is less complex than that of a sphere due to the fact that for the flat punch, the contact radius is just given by the radius of the cylinder and thus known a priori. The problem of the elastic contact between a sphere and a planar surface and between two spheres was solved by Hertz in 1882 [846]. Under the assumption that the contact radius a is small compared to the sphere radii, that the contact is frictionless and no tensile stress exists within the area of contact. Hertz derived an equation for the contact radius a between the spheres  

Correspondingly, the force to achieve a certain penetration 8 is given as 

For a rigid sphere indenting an elastic half-space, the vertical displacement of the halfspace outside the contact area is given by 

The vertical stress distribution in the contact follows an elliptical shape  

The indentation force increases with a power of 3/2 with indentation depth. The contact area jta increases as and the mean contact pressure Fi/na increases with applied load as In contrast to the flat punch case, the contact does no longer act as a linear spring since Fl is proportional to due to the fact that the contact area 

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