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. This paper proposes to investigate the stress-deformed state in the vicinity of the crack tip, depending on the period of cracks placement. Methodology. The methods for studying two-dimensional thermoelastic state of a body with crack as stress concentrators are based on the method of complex variable function. Reducing the problem of stationary heat conduction and thermoelasticity to singular integral equations (SIE) of the first kind, the numerical solution by the method of mechanical quadrature was obtained. Findings. In this paper, we present graphical dependencies of stress intensity factors (SIF) at the crack tip on the angle of orientation and on the relative position of cracks. The obtained results will be used later to determine the critical value of the intensity of the local heat flux from equations of limit equilibrium at which crack growth and the local destruction of the body occur. Originality. The originality of our solution lies in the fact that the new two-dimensional problems of heat conduction and thermoelasticity for a half plane containing a periodic cracks due to local heating by a heat flux are obtained. Practical value. The practical value is the ability to extend our knowledge of the real situation in the thermoelastic elements of engineering structures with cracks that operate under conditions of heat stress (frictional heat) in various industries, particularly in mechanical engineering. The results of specific values of SIF at the crack tip in graphs may be useful in the development of sustainable modes of structural elements in terms of preventing the growth of cracks.

Determination of quasi-static thermoelastic state of layered thermosensitive plates

The technique of determining the quasistatic thermoelastic state of the layered thermosensitive plates free of load is illustrated. Much attention is paid to finding analytical-numerical solutions of one-dimensional non-stationary heat conduction problems taking into account the temperature dependences of the thermal and temperature conductivity coefficients. Their finding involves use of the Kirchhoff transformation, generalized functions, Green's functions of the corresponding linear heat conduction problem, exact sums of the series, in particular those for which the Gibbs effect takes place, linear splines and solving the received recurrent systems of nonlinear algebraic equations relative to the values in the nodes of the spline of the Kirchhoff variable on the layer division surfaces and the derivative in time on inner flat-parallel surfaces of layers. The results of numerical calculations of temperature fields in two-layer plates with different thicknesses of layers and the external surface heated by a constant heat flux are presented. The accuracy of the found solution is investigated. The comparison of the temperature fields, which are determined assuming simple nonlinearity, stable thermophysical characteristics with the ones based on the exact solution of the corresponding nonlinear stationary heat conduction problem is fulfilled.