Linear and nonlinear mathematical models for determining the temperature field, and later the analysis of temperature regimes in isotropic spatial inhomogeneous media exposed to internal and external thermal loads have been developed. To do this, the thermal conductivity for such structures is described as a whole using symmetric unit functions, which allows us to consider boundary thermal conductivity problems with one linear and nonlinear differential equation of thermal conductivity with discontinuous coefficients and linear and nonlinear boundary conditions on boundary surfaces. In the case of a nonlinear boundary value problem, the Kirchhoff transform is applied, which linearizes the initial nonlinear equation of thermal conductivity and nonlinear boundary conditions and results in a second-order linear differential equation with partial derivatives and singular coefficients with respect to the Kirchhoff function with linear conditions. To solve the obtained linear boundary value problem, the method of integral Fourier transform was used, as a result of which an analytical solution was obtained, which determines the Kirchhoff linearizing function. As an example, the linear and cubic dependences of the thermal conductivity of structural materials on the structure, which are often used in many practical problems, are chosen. As a result, analytical relations in the form of quadratic and biquadratic equations are obtained to determine the temperature distribution in the thermosensitive layer with foreign inclusion at external local heating. Numerical analysis of temperature behavior as a function of spatial coordinates for given values of geometric and thermophysical parameters is performed. The influence of foreign inclusion on the temperature distribution was studied if the material of the medium was selected ceramics VK94-I, and the inclusion – silver, aluminum and silicon. To determine the numerical values of temperature in these structures, as well as the analysis of heat transfer processes in the middle of these structures due to internal and external heat loads, developed software that uses a geometric representation of temperature distribution depending on spatial coordinates. The obtained numerical values of temperature testify to the correspondence of the developed mathematical models of the analysis of heat exchange processes in spatial inhomogeneous media with internal and external heating to the real physical process. Software also allows you to analyze this type of environment, which are exposed to internal and external heat loads, in terms of their heat resistance. As a result, it becomes possible to increase it and protect it from overheating, which can lead to the destruction of not only individual elements but also the entire structure.
[1] Azarenkov, V. I. (2012). Issledovanie i razrabotka teplovoi modeli i metodov analiza temperaturnikh polei konstruktcii radioelektronnoi apparaturi. Technology audit and production reserves, 3/1(5), 39–40. [In Russian].
[2] Carpinteri, A., & Paggi, M. (2008). Thermoelastic mismatch in nonhomogeneous beams. Journal of Engineering Mathematics, 61(2–4), 371–384. https://doi.org/10.1007/s10665-008-9212-8
[3] Dovbnia, K. M., & Dundar, O. D. (2016). Statsionarnyi teploobmin tonkykh polohykh izotropnykh obolonok, yaki znakhodiatsia pid diieiu dzherel tepla, zoseredzhenykh po dvovymirnii oblasti. Visnyk DonNU. Ser. A: Pryrodnychi nauky, 1–2, 107–112. [In Ukrainian].
[4] Havrysh, V. I., & Fedasjuk, D. V. (2012). Modelling of temperature regimes in piecewise-homogeneous structures. Lviv: Publishing house of Lviv Politechnic National University, 176 p.
[5] Havrysh, V. I., Baranetskiy, Ya. O., & Kolyasa, L. I. (2018). Investigation of temperature modes in thermosensitive non-uniform elements of radioelectronic devices. Radio electronics, computer science, management, 3(46), 7–15. https://doi.org/10.15588/1607-3274-2018-3-1
[6] Havrysh, V. I., Kolyasa, L. I., & Ukhanska, O. M. (2019). Determination of temperature field in thermally sensitive layered medium with inclusions. Naukovyi Visnyk NHU, 1, 94–100. https://doi.org/10.29202/nvngu/2019-1/5
[7] Hrytsiuk, Yu. I., & Andrushchakevych, O. T. (2018). Means for determining software quality by metric analysis methods. Scientific Bulletin of UNFU, 28(6), 159–171. https://doi.org/10.15421/40280631
[8] Hrytsiuk, Yu. I., & Buchkovska, A. Yu. (2018). Visualization of the results of expert evaluation of software quality using polar diagrams. Scientific Bulletin of UNFU, 27(10), 137–145. https://doi.org/10.15421/40271025
[9] Hrytsiuk, Yu. I., & Dalyavskyy, V. S. (2018). Using Petal Diagram for Visualizing the Results of Expert Evaluation of Software Quality. Scientific Bulletin of UNFU, 28(9), 97–106. https://doi.org/10.15421/411832
[10] Noda, N. (1991). Thermal stresses in materials with temperature-dependent properties. Applied Mechanics Reviews, 44, 383–397. https://doi.org/10.1115/1.3119511
[11] Otao, Y., Tanigawa, O., & Ishimaru, O. (2000). Optimization of material composition of functionality graded plate for thermal stress relaxation using a genetic algorithm. Journal of Thermal Stresses, 23, 257–271. https://doi.org/10.1080/014957300280434
[12] Podstrigach, Ia. S., Lomakin, V. A., & Koliano, Iu. M. (1984). Termouprugost tel neodnorodnoi struktury. Moscow: Nauka, 368 p. [In Russian].
[13] Tanigawa, Y., & Otao, Y. (2002). Transient thermoelastic analysis of functionally graded plate with temperature-dependent material properties taking into account the thermal radiation. Nihon Kikai Gakkai Nenji Taikai Koen Ronbunshu, 2, 133–134. https://doi.org/10.1299/jsmemecjo.2002.2.0_133
[14] Tanigawa, Y., Akai, T., & Kawamura, R. (1996). Transient heat conduction and thermal stress problems of a nonhomogeneous plate with temperature-dependent material properties. Journal of Thermal Stresses, 19(1), 77–102. https://doi.org/10.1080/01495739608946161
[15] Yangian, Xu, & Daihui, Tu. (2009). Analysis of steady thermal stress in a ZrO2/FGM/Ti-6Al-4V composite ECBF plate with temperature-dependent material properties by NFEM. 2009-WASE Int. Conf. on Informa. Eng, Vol. 2, 433–436. https://doi.org/10.1109/ICICTA.2009.842
[16] Koliano, Iu. M. (1992). Metody teploprovodnosti i termouprugosti neodnorodnogo tela. Kyiv: Naukova dumka, 280 p. https://doi.org/10.1192/bjp.161.2.280b
[17] Korn, G., & Korn, T. (1977). Spravochnik po matematike dlia nauchnykh rabotnikov i inzhenerov. Moscow: Nauka, 720 p. [In Russian].
[18] Kikoina, I. K. (1976). Tablitcy fizicheskikh velichin. Spravochnik. Moscow: Atomizdat, 1008 p. [In Russian].