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As is known, sheet metal undergoing cold reduction exerts large unit pressures on the rolls. Thus, the total pressures exerted on the rolls by the sheet are also large. With respect to the magnitudes of the stresses (working, mechanical) The problems involved in determining the mechanical stresses in the region of contact of rolling-mill rolls and sheet being rolled have been explored in many studies published in Russia and abroad. The investigations conducted by A. I. Tselikov, I. Ya. Tarnovskii, G. Hertz, I. Ya. Shtaerrnan, M. G. Ponomarev, and M. Ya. Vasilev are some examples of this research. Increasingly stringent requirements on the quality of cold-rolled sheet and the reliability of mechanical equipment are making it necessary to study the mechanisms which determine the dimensions and physico-mechanical properties of rolled products and the range of working loads. The well-known methods of determining the stress-strain state of the metal in the deformation zone give only approximate (with an error of 10-30%) values of these stresses. The results are only approximate because the procedure used to determine the normal contact stresses and the change in the friction coefficient along the deformation zone is flawed and does not account for elastic flattening of the sheet at the beginning of that zone and its elastic recovery at the end of the zone.We set out to construct a mathematical model that completely describes the features of the stress state of the metal of the sheet and the work roll in the region of their elastoplastic contact. We will use a fairly rigorous mathematical procedure to calculate the stress-strain state of the sheet metal [1, 2]. By analogy with the methods used in [3][4][5], our unidimensional mathematical model of cold reduction will be based on the recursive numerical solution of the finite-difference equation expressing the conditions of static equilibrium of an elementary volume of the metal in the region where it undergoes elastic and plastic deformation. Figure la shows the theoretical model being used to describe the integral detbrmation zone. The model includes regions of elastic/ell, le|2 and plastic lpl deformation. With allowance for the kinematic conditions, the plastic deformation region is further divided into regions of forward slip and backward slip. These regions have the lengths lbk and lfr, respectively (see Fig. la).The left AB and right CD bounding sections of the deformation zone are assumed to be vertical, due to the symmetry of the kinematic conditions and the geometric parameters. For the same reason, we further assumed that the section NT-the interface between the regions of forward and backward slip (Fig. la) -is also vertical. Several other assumptions were made, the most important of which are as follows: the deformation of the sheet is two-dimensional and steady over time; the mechanical properties of the sheet metal, the normal stresses 6x, and the kinematic and temperature-speed conditions of the rolling operation change only along the deformation zone ...