# singular integral equation

## A BRIEF OVERVIEW OF STATIONARY TWO-DIMENSIONAL THERMOELASTIC STATE MODELS IN HOMOGENEOUS AND PIECEWISE-HOMOGENEOUS BODIES WITH CRACKS

Purpose. A two-dimensional mathematical model of the problem of thermo-elasticity for piecewise-homogeneous component plate containing a crack has been built. The stress intensity coefficients in the vertices of the crack increase affecting strength of the body significantly. This leads to the growth of a crack and, as a result, to further local destruction of a material. Therefore, such a model reflects, to some extent, the destruction mechanism of the elements of engineering structures with cracks.

## Mathematical modeling of elastic state in a three-component plate containing a crack due to the action of unidirectional tension

Purpose. A two-dimensional mathematical model for the problem of elasticity theory in a three-component plate containing rectilinear crack due to the action of mechanical efforts is examined. As a consequence, the intensity of stresses in the vicinity of tops of the crack increases, which significantly affects strength of the body. This may lead to the growth of a crack and to the local destruction of a structure.

## Mathematical modeling of stationary thermoelastic state in a half plane containing a periodic system of cracks due to periodic local heating by a heat flux

Purpose. To determine the two-dimensional thermoelastic state in a semi-infinite solid (half-plane), weakened by a system of periodic internal cracks under conditions of local heating on the edge of the half plane. Heat flux due to frictional heating on the local area of the body, causes changes in temperature and stresses in the body, which significantly affects its strength, as it can lead to crack growth and local destruction. Therefore, the study of the problem of frictional heat is of a practical interest.

## Modeling of elastic state for two joined dissimilar semi-infinite plates with crack under the action of pressure on the shores of the crack

Purpose. A two-dimensional mathematical model for the problem of elasticity theory on joined dissimilar elastic half-planes containing rectilinear crack under the action of mechanical efforts on the shores of a crack is examined. As a consequence, the intensity of stresses in the vicinity of tops of the crack increases, which significantly affects strength of the body. This may lead to the growth of a crack and to the local destruction of a structure.

## Mathematical modeling of stationary thermoelastic state in a half plane containing an inclusion and a crack due to local heating by a heat flux

The two-dimensional stationary problems of heat conduction and  thermoelasticity for a semi-infinite elastic body containing an inclusion and a crack  are  considered.  For this purpose, mathematical models of these  two-dimensional problems in the form of a system of singular integral equations (SIEs) of the first and the second kinds are constructed.  The numerical solution of the system of integral equations in the case of a half plane  containing an inclusion and thermally insulated crack due to local heating by a heat flux is obtained using the method of mechanical quadratures.  We pre

## Mathematical modeling of the thermoelastic state in a circular disk with a crack due to the action of the heat source

Purpose. To determine the two-dimensional thermoelastic state in a circular plate, weakened by an edge or internal crack induced by a stationary heat sourse. This paper proposes using singular integral equation (SIE) to investigate thermostressed intensity in the vicinity of the crack tip, depending on the local heat source placement and identify typical mechanical effects.

## Determination of Stresses and Ultimate Loads for Composite Plates with Elastic Inclusions

In the article, the algorithm for determination of stresses in anisotropic plates with elastic inclusions of another anisotropic material was developed on the basis of complex singular integral equations. The solving of integral equations has been carried out numerically using the method of mechanical quadratures. The strength analysis (calculation of strength) of composite plates with inclusions has been performed using the Hoffman criterion.