Mathematical modeling of thermoelastic state in a tree-component piecewise-homogeneous plate containing a crack
Authors:   Department of Mathematics, Lviv Polytechnic National University, Zelenyak Volodymyr , Kolyasa Liubov ,   Department of Mathematics, Lviv Polytechnic National University, Klapchuk Myroslava ,   Department of Mathematics, Lviv Polytechnic National University
Published: June 2022

Purpose. A two-dimensional mathematical model of the problem of thermoelasticity for three-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. Graphic dependences of stress intensity factors (SIFs) at the tops of the crack have been built. This would make it possible to obtain the critical values of constant temperature in the two joined dissimilar elastic half-planes containing an inclusion and a crack in order to prevent crack growth, which would not allow the local destruction of the body. Methodology. Based on the method of the function of a complex variable we have studied the two-dimensional thermoelastic state for body with crack as stress concentrators. As result, the problem of thermoelasticity was reduced to a system of two singular integral equations (SIE) of the first and second kind, a numerical solution of which was found by the method of mechanical quadratures. The two-dimensional mathematical model of the thermoelastic state has been built in order to determine the stress intensity coefficients at the top of the crack and inclusion. The systems of singular integral equations of the first and second kinds of the specified problem on closed (contour of inclusion) and open (crack) contours are constructed. Numerical solution of the integral equations in the case of constant temperature in the two joined dissimilar elastic half-planes containing the crack and an inclusion was obtained by the mechanical quadrature method. Influence of thermophysical and mechanical properties of an inclusion on the SIF sat the crack types was investigated. Graphic dependences of the stress intensity factors which characterize distribution of the intensity of stresses at the vertices of a crack have been built, as well as on its elastic and thermoelastic characteristics of inclusion. This would make it possible to analyze the intensity of stresses in the neighborhood of a crack vertices depending on the geometrical and mechanical factors. As a result, this allow to determine the critical values of temperature in the three-component plate containing a crack in order to prevent the growth of the crack, as well as to prevent the local destruction of the body. It was found that that the appropriate selection of mechanical and thermophysical characteristics of the components of a three-component plate containing a crack can be useful to achieve an improvement in body strength in terms of the mechanics of destruction by reducing stress intensity factors at the crack’s vertices. Originality. The solutions of the new two-dimensional problem of thermoelasticity for a specified region (a two joined dissimilar elastic half-planes containing inclusion and a crack) due to the action of constant temperature is obtained. The studied model is the generalization of the previous models to determine the two-dimensional thermoelastic state in a piecewise-homogeneous plate weakened by internal cracks. Practical value. The practical application of this model is a more complete description of the stress-strain state in piecewise homogeneous structural elements with cracks operating under temperature loads. The results of numerical calculations obtained from the solution of systems of equations and presented in the form of graphs can be used in the design of rational modes of operation of structural elements. This takes into account the possibility of preventing the growth of cracks by appropriate selection of composite components with appropriate mechanical characteristics.