Nonlinear mathematical models for the analysis of temperature regimes in a thermosensitive isotropic plate heated by locally concentrated heat sources have been developed. For this purpose, the heat-active zones of the plate are described using the theory of generalized functions. Given this, the equation of thermal conductivity and boundary conditions contain discontinuous and singular right parts. The original nonlinear equations of thermal conductivity and nonlinear boundary conditions are linearized by Kirchhoff transformation. To solve the obtained boundary value problems, the integral Fourier transform was used and, as a result, their analytical solutions in the images were determined. The inverse integral Fourier transform was applied to these solutions, which made it possible to obtain analytical expressions for determining the Kirchhoff variable. As an example, the linear dependence of the thermal conductivity on temperature is chosen, which is often used in many practical problems. As a result, analytical relations were obtained to determine the temperature in the heat-sensitive plate. The given analytical solutions are presented in the form of improper convergent integrals. According to Newtons method (three-eighths), numerical values of these integrals are obtained with a certain accuracy for given values of plate thickness, spatial coordinates, specific power of heat sources, the thermal conductivity of structural materials of the plate, and geometric parameters of the heat-active zone. The material of the plate is silicon and germanium. To determine the numerical values of temperature in the structure, as well as the analysis of heat transfer processes in the middle of the plate due to local heating, developed software, using which geometric mapping of temperature distribution depending on spatial coordinates, thermal conductivity, specific heat flux density. The obtained numerical values of temperature testify to the correspondence of the developed mathematical models of the analysis of heat exchange processes in the thermosensitive plate with local heating to the real physical process. The software also makes it possible to analyze such environments that are exposed to local heat loads in terms of their heat resistance. As a result, it becomes possible to increase it and to protect it from overheating, which can cause the destruction not only of individual elements but also of 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] Kikoina, I. K. (1976). Tablitcy fizicheskikh velichin. Spravochnik. Moscow: Atomizdat, 1008 p. [In Russian].
[8] Koliano, Iu. M. (1992). Metody teploprovodnosti i termouprugosti neodnorodnogo tela. Kyiv: Naukova dumka, 280 p.
[9] Korn, G., & Korn, T. (1977). Spravochnik po matematike dlia nauchnykh rabotnikov i inzhenerov. Moscow: Nauka, 720 p. [In Russian].
[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, 2, 433-436. https://doi.org/10.1109/ICICTA.2009.842