Theories for Frictional Spheres in Contact

In the previous sections, only the normal contact of two elastic spheres with perfect smooth surfaces (i.e., no tangential force) is considered. However, for oblique contact between two frictional spheres, tangential forces are encountered, and, consequently, 

Assume that Amontons s law of sliding friction can be applied at each elementary area of the interface, such that 

Assume that the normal pressure and displacements are not affected by the existence of the tangential traction and resulting displacements. Hence, the normal pressure and contact area can be determined by the Hertzian theory. For the sliding contact of spheres, substituting Eq. (2.69) into Eq. (2.77) gives rise to the tangential traction as 

With the tangential traction parallel to the x-axis, the general displacements on the contact surface due to the tangential traction can be expressed by 

The detailed derivation of Eq. (2.80) and Eq. (2.81) is omitted here. Interested readers may refer to the work of Johnson (1985). 

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