Force-deflection relationship for spherical surfaces

For spheres of the same isotropic elastic material with plane strain modulus E and the same undeformed radius R, the deformed contact area is flat and its boundary is circular due to the symmetry of the configuration. According to the Hertz point of view, the normal stress distribution predicted for the contact area z = 0,0 r ais 

This displacement estimate must also be obtained from the elasticity solution for the half space problem it is exactly twice the normal displacement of the surface at the center of the contact area relative to the normal displacement at a remote point on the half space surface in the limit as r/a — oo. 

It is well established in the theory of elasticity that a pressure distribution (1 — applied to the surface of the elastic half-space z 0 

The corresponding relative displacement of the centers of the spheres , based on the solutions of the relevant elastic half-space problem, is 

Finally, the role of the parameter B in determining energy variations must be considered. 

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