The stress-strain state of a plastic layer under compression by two rigid parallel rough plates under conditions of plane deformation is investigated. The hypothesis of flat sections is used. Tangential stresses are found in the part of the layer under load, as an approximate solution of the system of equations of the stress-strain state of the compressible layer. The solution is obtained in analytical form. On this basis the velocities of the points of the layer are calculated in the analytic form. The size and shape of the free surface are calculated depending on the thickness of the layer. The coefficients of this dependence are determined by recurrent relations. On this basis, an algorithm for finding the shape of a free surface is found. The computational experiments are provided and the results are compared with results obtained with FEM and ANSYS.

Mathematical models of the stress state of plastic layers (interlayers) under the tensile load under plane deformation are investigated. The layer of rectangular shape is included in the strip of more durable material. The method of characteristics (slip lines) is used. The reasons that contact hardening is not fully realized are investigated. The characteristic fields in the layer in the process of loading are analyzed. The classification of characteristic fields at the critical moment of loading is given. The criterion for the full realization of contact hardening of the layer material is obtained. It depends on the relative thickness of the layer and the coefficient of mechanical heterogeneity of the joint. Explicit analytical expressions for calculating the critical load in the case of the full implementation of contact hardening are obtained.

We investigate a critical state of the slope layer of less durable material in sheet sample by the loadings having opposite signs. It is shown that if there is no contact hardening of layer material, then a destruction occurs on the layer, independent of its slope angle. Using numerical experiments, we investigate a dependence of a sample strength from the slope angle of the layer for different relations between the external loads. We show that if there is a contact hardening of the layer, then a sample, containing less stable layer, may be equal in strength to a homogeneous sample for certain angles of layer slope.