Abstract
A two-dimensional mathematical model of the thermoelastic state has been built for a circular plate containing a curvilinear inclusion and a crack, under the action of a uniformly distributed temperature across the entire piece-homogeneous plate. Using the apparatus of singular integral equations (SIEs), the problem was reduced to a system of two singular integral equations of the first and second kind on the contours of the crack and inclusion, respectively. Numerical solutions to the system of integral equations have been obtained for certain cases of the circular disk with an elliptical inclusion and a crack in the disk outside the inclusion, as well as within the inclusion. These solutions were applied to determine the stress intensity coefficients (SICs) at the tops of the crack. Stress intensity coefficients could later be used to determine the critical temperature values in the disk at which a crack begins to grow. Therefore, such a model reflects, to some extent, the destruction mechanism of the elements of those engineering structures with cracks that are operated in the thermal power industry and, therefore, is relevant. Graphic dependences of stress intensity coefficients on the shape of an inclusion have been built, as well as on its mechanical and thermal-physical characteristics, and a distance to the crack. This would make it possible to analyze the intensity of stresses in the neighborhood of the crack vertices, depending on geometric and mechanical factors. The study's specific results, given in the form of plots, could prove useful in the development of rational modes of operation of structural elements in the form of circular plates with an inclusion hosting a crack. The reported mathematical model builds on the earlier models of two-dimensional stationary problems of thermal conductivity and thermoelasticity for piece-homogeneous bodies with cracks.
Highlights
Elements of many modern structures are often designed to work under the conditions of thermal heating, which contribute to the emergence of temperature stresses in them
In work [10], the same method was applied to investigate the thermoelastic interaction between a two-component circular inclusion and a crack located in the plate under the conditions of a steady uniform temperature
We have theoretically studied the two-dimensional thermoelastic state in a circular plate containing a curvilinear inclusion and a crack based on the method of the function of a complex variable
Summary
Elements of many modern structures are often designed to work under the conditions of thermal heating, which contribute to the emergence of temperature stresses in them This is typical for tools and structures in the heat and power industry. The intensity of stresses at crack vertices is expressed through stress intensity coefficients (SICs) These parameters make it possible to determine the limit value of the heat load at which a crack begins to grow and the body locally collapses. The relevance of our research is predetermined by the importance of studying the thermoelastic state of piece-homogeneous bodies with cracks for practical applications in terms of assessing the strength and durability of structural elements, as well as in theoretical terms for devising new effective methods for determining temperature stresses
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