Contact stress The Hertz formulas

By pressing a sphere upon a planar surface, a deformation occurs near the original point of contact, see Fig. F.8. This problem was first solved by Hertz in 1881 (see Landau and Lifshitz, 1986 Timoshenko and Goodier, 1970). The derivations are complicated. We state the results without proof. 

Consider a sphere of diameter D of material 1 in contact with a planar surface of material 2. The reaction of the load P is determined by an effective Young s modulus E, which is defined as 

The maximum normal stress at the point of contact, according to Hertz, is 1.5 times the average normal stress  

These formulas are important in the discussion of tip and sample deformation (see, for example, Pethica and Oliver, 1987), as well as some mechanical design problems of STM and AFM. 

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