The aim of this paper is to do an analytical research of internal stresses, which occur in contact of two rough surfaces. Research mainly has been done analytically and is based on the formulas described by von Mises yield criteria. As we can see from the obtained results, it’s possible to find a coefficient of friction by using von Mises yield criteria. To get more precise values of the coefficient of friction, it is necessary to take into account also surface geometry and adhesion forces between the contacting surfaces. This will be done in further research works.