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The paper considers an axisymmetric stress state of a homogeneous elastic space with a circular disc-shaped crack, one of the edges of which is pressed into a cylindrical circular stamp with static friction. It is assumed that the contact zone is considered under the generalized law of dry friction, i.e. tangential contact stresses are proportional to normal contact pressure, while the proportionality coefficient depends on the radial coordinates of the points of the contacting surfaces and is directly proportional to them. Considering the fact that in this case the Abel images of contact stresses are also related in a similar way, the solution of the problem, with the help of rotation operators and theory of analytical functions, is reduced to an inhomogeneous Riemann problem for two functions and the closed solution in quadratures is constructed. A numerical analysis was carried out and regularities of changes in both normal and shear real contact stresses, as well as rigid displacement of the stamp depending on the physical and geometric parameters were revealed. [ABSTRACT FROM AUTHOR]

By the method of Hankel integral transformation, discontinuous solutions of equations of the axisymmetric elasticity theory are constructed for a piecewise homogeneous uniform layered space obtained by alternately joining two heterogeneous layers of equal thickness and whose junction contains a periodic system of parallel circular disk-shaped defects. On the basis of the solutions found, as examples, the governing systems of integral equations with Weber–Sonin kernels are presented for two cases: with defects in the form of absolutely rigid disk-shape inclusions and circular cracks. Using rotation operators, the governing systems of equations, in both cases, are reduced to a singular integral equation of the second kind, which is solved by the method of mechanical quadratures. Simple formulas for determining the rigid-body displacement of inclusions and crack opening are obtained. [ABSTRACT FROM AUTHOR]

Discontinuous solutions of equations of a plane elasticity problem for a piecewise homogeneous plane obtained by joining two dissimilar strips are found using the Airy stress function. In the general case, doubly periodic interphase defects are on their connection lines, on which displacements and stresses are discontinuous. Based on the solutions obtained, a system of governing singular integral equations is derived for a particular problem where the defect is a finite crack. This system is solved by the method of mechanical quadratures. Numerical calculations are also carried out. [ABSTRACT FROM AUTHOR]

An exact solution of a mixed boundary-value problem for an orthotropic plane with a slit on one of the principal directions of orthotropy is constructed in the case where an absolutely rigid stamp with a flat base is pressed into one face of the slit. Using the method of discontinuous solutions for the equations of elasticity theory, the problem stated is reduced to the Riemann problem for two functions, and a closed solution of this problem is constructed. Simple expressions for the basic mechanical characteristics, such as the opening of the slit, contact stresses under the die, and the intensity factor of breaking stresses at the end points of the slit are obtained. Solutions of similar problems for an orthotropic plane are found when the stamp reaches only one or both end points of the slit, as well as for a semi-infinite slit. A numerical analysis is carried out. [ABSTRACT FROM AUTHOR]