In the paper numerical tests of the contact of a pair of flexible elements are presented. Non-linear phenomena occurring in the contact zone of two elements have been considered separately for the assembly and operational condition of the joint. Physical and mathematical models of the contact joint created using the finite element method are proposed. The contact joint is treated as a system composed of the subsystems: two of them are joined elements and the third one is the contact layer between the elements. The calculations results for the selected contact joint are given.