The description of the new contact mechanism between dissimilar materials during joint plastic deformation is proposed in this paper. To analyze the process of joint deformation of composite material layers, a multi-stage analytical model was developed based on the study of the contact interaction between the surfaces of the materials to be bonded using the slip line method. When mathematical simulation of the process of joint deformation of dissimilar materials, the influence of the geometrical surface profile of a harder layer of a composite, as a more significant factor, was estimated. For the entire range of influence of the investigated geometrical surface profile of a harder material of a composite, the final forming and stress state parameters in its intermediate zone were determined. To verify the analytical model, computer simulation of the process of joint deformation of composite material layers by the finite element method in two-dimensional formulation was carried out. The comparison of both solutions has confirmed the adequacy of the results obtained in the mathematical simulation. The theoretical model can be used in the development of bonding mechanisms between dissimilar materials, in the development of manufacturing technologies of new clad composite materials, as well as in the analysis and improvement of the existing manufacturing technologies of clad composite materials.