Thermal Reaction During Thermal Shock of a Massive Body with an Internal Cylindrical Cavity

Authors: Kartashov E.M. , Nenakhov E.V. Published: 14.12.2020
Published in issue: #6(93)/2020  
DOI: 10.18698/1812-3368-2020-6-60-79

Category: Physics | Chapter: Thermal Physics and Theoretical Heat Engineering  
Keywords: thermal shock, dynamic thermoelasticity, wave nature, stresses, accelerations

The paper examines mathematical models of thermal shock in terms of dynamic thermoelasticity and their application to specific cases during intense heating of a solid boundary. We introduce a stress compatibility equation for dynamical problems, generalizing the well-known Beltrami --- Mitchell relation for quasi-static cases. It is convenient to use this relation when considering numerous special cases in the theory of heat shock in Cartesian coordinates both for bounded bodies of a canonical form, i.e., an infinite plate, and for partially bounded bodies, i.e., space bounded from the inside by a flat surface. In the latter case, the obtained analytical solutions of dynamic problems of thermoelasticity lead to visual and convenient for physical analysis functional structures describing the kinetics of thermal stresses. For cylindrical and spherical coordinate systems, we propose a compatibility equation in displacements, which is convenient for studying the problem of thermal shock in bodies with a radial heat flux and under conditions of central symmetry. In the study, we singled out a class of problems in which the consideration of the geometric dimensions of a structure investigated for a thermomechanical reaction under conditions of intense heating concerns mainly the near-surface layers. According to the experimental results, it is these layers that absorb the main amount of heat during a time close to the beginning of heating, which corresponds to the time of the microsecond duration of the inertial effects. We investigated the thermal reaction of a massive body with an inner cylindrical cavity within the framework of dynamic thermoelasticity under various modes of intense heating of the cavity surface. Finally, we carried out numerical experiments and described the wave character of thermal stresses with the corresponding quasi-static values, and established the role of inertial effects in mathematical models of the theory of thermal shock


[1] Kartashov E.M., Kudinov V.A. Matematicheskie modeli teploprovodnosti i termouprugosti [Mathematical models of thermoelasticity]. Moscow, MIREA Publ., 2018.

[2] Parkus H. Instationare Warmespannugen. Vienna, Springer, 1959. DOI: https://doi.org/10.1007/978-3-7091-5710-7

[3] Boley B.A., Weiner J.H. Theory of thermal stresses. Weiner Chapman and Hall, 1960.

[4] Novatskiy V. Review of works on dynamical problems of thermoelasticity. Mekhanika, 1966, no. 6, pp. 101--142 (in Russ.).

[5] Kolyano Yu.M. Obobshchennaya termomekhanika (obzor) [Generalized thermal mechanics (review)]. Matematicheskie metody i fiziko-mekhanicheskie polya, 1975, no. 2, pp. 37--42 (in Russ.).

[6] Kartashov E.M., Nenakhov E.V. Dynamic thermoelasticity in the problem of heat shock based on the general energy equation. Teplovye protsessy v tekhnike [Thermal Processes in Engineering], 2018, vol. 10, no. 7-8, pp. 334--344 (in Russ.).

[7] Kartashov E.M. Originals of operating images for generalized problems of unsteady heat conductivity. Tonkie khimicheskie tekhnologii [Fine Chemical Technologies], 2019, vol. 14, no. 4, pp. 77--86 (in Russ.). DOI: https://doi.org/10.32362/2410-6593-2019-14-4-77-86

[8] Podstrigach Ya.S., Lomakin V.A., Kolyano Yu.M. Termouprugost’ tel neodnorodnoy struktury [Thermoelasticity of bodies with inhomogeneous structure]. Moscow, Nauka Publ., 1984.

[9] Kolyano Yu.M. Metody teploprovodnosti i termouprugosti neodnorodnogo tela [Thermal conductivity and thermoelasticity methods of inhomogeneous body]. Kiev, Naukova dumka Publ., 1992.

[10] Kolpashchikov V.L., Yanovskii S.Yu. Equation of dynamical thermoelasticity in a medium with thermal memory. J. Eng. Phys., 1984, vol. 47, no. 4, pp. 1241--1244. DOI: https://doi.org/10.1007/BF00869927

[11] Zarubin V.S., Kuvyrkin G.N. Matematicheskie modeli termomekhaniki [Mathematical models of thermal mechanics]. Moscow, FIZMATLIT Publ., 2002.

[12] Kartashov E.M., Bartenev G.M. Dinamicheskie effekty v tverdykh telakh v usloviyakh vzaimodeystviya s intensivnymi potokami energii [Dynamic effects in solids at the interaction with intense energy flows]. V: Itogi nauki i tekhniki. Ser. Khimiya i tekhnologiya vysokomolekulyarnykh soedineniy. T. 25 [In: Outcomes of Science and Technique. Ser. Chemistry and Technology of High-Molecular Compositions. Vol. 25]. Moscow, VINITI Publ., 1988, pp. 3--88 (in Russ.).

[13] Kartashov E.M., Parton V.Z. Dinamicheskaya termouprugost’ i problemy termicheskogo udara [Dynamic thermoelasticity in problem of thermal shock]. V: Itogi nauki i tekhniki. Ser. Mekhanika deformiruemogo tverdogo tela. T. 22 [In: Outcomes of Science and Technique. Ser. Mechanics of Deformable Solids. Vol. 22]. Moscow, VINITI Publ., 1991, pp. 55--127 (in Russ.).

[14] Kartashov E.M., Kudinov V.A. Analiticheskaya teoriya teploprovodnosti i prikladnoy termouprugosti [Analytical theory of heat conductivity and thermoelasticity]. Moscow, Librokom Publ., 2012.

[15] Carslaw H.S., Jaeger J.C. Conduction of heat in solids. Oxford Univ. Press, 1959.

[16] Lykov A.V. Teoriya teploprovodnosti [Thermal conductivity theory]. Moscow, Vysshaya shkola Publ., 1967.

[17] Kartashov E.M. Analiticheskie metody v teorii teploprovodnosti tverdykh tel [Analytical methods in thermal conductivity theory of solids]. Moscow, Vysshaya shkola Publ., 2001.

[18] Attetkov A.V., Volkov I.K. Formation of temperature fields in the region internally restricted by cylindrical hollow. Herald of the Bauman Moscow State Technical University, Series Mechanical Engineering, 1999, no. 1 (34), pp. 49--56 (in Russ.).

[19] Attetkov A.V., Belyakov N.S., Volkov I.K. Influence of boundary mobility on temperature field of solid body with cylindrical channel under non-stationary conditions of heat exchange with environment. Herald of the Bauman Moscow State Technical University, Series Natural Sciences, 2006, no. 1 (62), pp. 31--40 (in Russ.).

[20] Kartashov E.M., Nenakhov E.V. Model representations of heat shock. Izvestiya RAN. Energetika [Proceedings of the RAS. Power Engineering], 2019, no. 2, pp. 135--156 (in Russ.). DOI: https://doi.org/10.1134/S0002331019020158