Hi Bahman Daneshian . I just checked the PSG design databook that we commonly use in India for the hand calculations. It gives the formula for maximum contact stress (which I believe in this case is equal to the contact pressure) for two spheres contacting each other. I believe you just have to use young's modulus E = infinity for the rigid sphere. This will lead to one of the terms vanishing in the denominator. Hope this helps.

Hello Bahman,

I am not an expert in this field. However I know Prof. Peter Wriggers who is a great specialist on contact mechanics. He has many publications in this area, you can find most of them in Research Gate. Have a nice day and night.

see

http://multiscaleconsulting.com/publications/Contact_mechanics_for_randomly_rough_surfaces.pdf

http://home.ufam.edu.br/berti/nanomateriais/8403_PDF_CH03.pdf

http://www.dtic.mil/dtic/tr/fulltext/u2/a483610.pdf

For starters, in the case of low-velocity elastic impact of a perfectly smooth sphere everything can be found in the classical paper by Hertz (Über die Berührung fester elastischer Körper. Journal für die reine und angewandte Mathematik, 1882, 92, pp.156-171). For more complicated cases (with wave propagation, plasticity, surface roughness etc.) you should check textbooks on contact mechanics (Johnson, Barber, Wriggers) and impacts (Stronge).