Contact problems for a triple-layer strip in the presence of friction forces

Plane contact problems on the interaction of a punch and a triple-layer elastic strip, lying on a rigid or elastic half-space are considered, taking friction forces into account, on the assumption that the layers are rigidly connected with one another and with the half-space. It is also assumed that the punch base is plane or has the form of a parabola, the normal and shear stresses are related by Coulomb's law in the contact area, and normal and shear forces act on the punch. The punch – triple-layer base system is then in limit-equilibrium conditions and the punch does not turn when the layer is deformed. Exact integral equations of the first kind with kernels, represented in explicit analytical form, are first obtained using analytical computational programmes. The fundamental properties of the kernels of the integral equations are investigated and it is shown that the numerator and denominator of the kernel symbols can be represented in the form of an expansion in products of powers of the shear moduli of the layers and half-space. A scheme for solving the integral equations by the direct collocation method is described, which enables a solution of the problem to be obtained for practically any values of the initial parameters. The contact stress distributions, the dimensions of the contact area, the relation between the displacement of the punch and the forces acting on it are calculated as a function of the geometrical and mechanical parameters of the layers. The results of the calculations are compared in special cases with previously known results.