Задача термопружності для трансверсально ізотропного шару з коловими лініями розділу граничних умов на його поверхні

Abstract

Розроблено алгоритм дослідження напруженого стану, що виникає в трансверсально ізотропному шарі в результаті теплообміну за законом Ньютона між шаром та зовнішнім середовищем при наявності ліній розділу граничних умов третього роду для температури на граничних площинах шару.

In modern engineering many constructions, machine parts and device units operate under sufficient heat loads. Thermal stresses, which occur, can be of critical values and be crucial while structural designing. That is why the problems of thermoelasticity, which became of special importance in the middle of the last century [1, 2] are actual nowadays. The objective of the paper in question is the development of the method for the investigation of the stress state in the axis-symmetric problems of thermoelasticity, when the distribution lines of the boundary conditions of the 3-d order are available for the temperature on the boundary planes of the transversal isotropic layer. To demonstrate the developed method the problem of thermoelasticity for the transversal isotropic layer, when three circle lines of the boundary conditions distributions for the temperature are available, has been solved. Plane-parallel, transversal isotropic layer of the finite thickness 2h has been analyzed. The boundary plane of the layer are considered to be parallel to the isotropy and free from the external loads. Heat exchange according to the Newton law takes place between the points of the layer boundary planes and external environments, three lines of the boundary conditions distribution of the 3-d order for the temperature as the concentric circles being available on the upper layer boundary plane. The temperature of the external environment, which correspond different areas of the boundary planes, are different. Heat-exchange coefficients between the layer points and external environments are different too. To solve the problem the main equations and relations of the thermoelasticity theory for the transversal isotropic bodies obtained in the papers by W. Novatsky [3], has been used. When the boundary conditions of the problem are provided, the system of integral equations is obtained. To solve it the unknown function as the segment of the generalized Fourier’s series according to the Bessel’s function is introduced. The system of linear equations relatively unknown coefficients has been obtained. The system was built so, that the more the number of its equations is, the more accurate the solution of the task is. That is, taking advantage of the developed method, the problem of thermoelasticity can be solved with the preliminary defined accuracy. To demonstrate the developed method the numerical example has been analyzed. Expressions for the functions of temperature and stress distributions in the layer have been built. Using the obtained functions the effect of the environment temperature and conditions of contact with the layer on the temperature distribution in the layer, as well as on the value and nature of stresses inside the layer, has been analyzed.