Contact problems are considered, taking account of friction forces for a three-layered elastic base, lying on a hard or on an elastic half-space. It is assumed that the layers are rigidly connected to each other and to the half-space. The base of the stamp is a parabola or a straight line. Normal and tangential stresses are related by the Coulomb law in the contact zone, normal and tangential forces act on the stamp, the system stamp - a three-layer base is in limiting equilibrium, stamp does not rotate in the deformation process of the layer. For the first time the exact integral equations of the first kind with kernels given in explicit analytic form are obtained for supplied problems with help of the analytical algorithm programs. The main properties of the kernels of integral equations are studied. It is shown that the numerator and denominator of the kernel symbols can be represented as an expansion of the powers products of the layers and half-space shear modules. It is shown that the expressions in these products contain hyperbolic and power functions of thickness and Poisson's ratio of the layers and half-space. Solution schemes of integral equations are built using asymptotic methods and direct collocation method. Asymptotic methods allow us to investigate the problem for relatively small or relatively large thicknesses of the layers. The proposed problem solution algorithm of the collocation method allows obtaining a solution for practically any value of initial parameters. With the help of the proposed methods the distribution of contact stresses, the sizes of the contact area, the relationship of stamp movement and acting on it forces, the stress-deformed state in the interior regions, particularly at the boundaries between layers with different mechanical parameters on depending of geometric and mechanical parameters of the layers and the coefficient friction to their optimal choice for provisioning of the required work resource of the simulated thus friction units are calculated. Comparison of the calculation results with the results obtained by finite element method is done.
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