On a method for approximate solution nonlinear heat conduction equation

V. B. Bednova

Čebyševskij sbornik, Tome 17 (2016) no. 4, pp. 11-22.

Résumé

In this paper a one-dimensional non-stationary heat conduction problem, modeling the process of rapid local heating of the sample beam type on the lateral surface is considered. The character of heating is such that it is possible to allocate only a certain directionof the heat propagation. Temperature fields are determined by an approximate method based on the idea of the thermal front. The solution is sought in the form of a power series in the coordinate with coefficients depending on time. The boundaries of the front heat distribution as a function of time are determined by the condition of the integral satisfaction of the heat conduction equation. Considered temperature fields arise in many industrial processes, such as laser material processing, when due to large temperature gradients can arise thermal stresses, leading to microcracking inner layers or the destruction of structural elements. Analytical view of the heat conduction problem’s solution allows to obtain analytical expressions for the thermal stresses and further facilitates the results analysis. The paper presents the solution of problems with boundary conditions of the first and second kinds for two monotonic and one non-monotonic dependencies of the thermal conductivity coefficient on temperature. The approximate solution and the exact solution of the non-stationary linear problem are compared and shows the suitability of the method for future use. Bibliography: 16 titles.

Détail

Keywords: Physical non-linearity of the heat conduction equation, approximate solution of the heat conduction equation, thermal front.

Bibliographie

[1] Barenblatt G. I., “On some approximate methods in the theory of one-dimensional non-stationary filtration in the elastic drive regime”, Izv. AN SSSR, OTN, 1954, no. 9, 35–49 | Zbl

[2] Bautin S. P., Analytical heat wave, Fizmatlit, M., 2003, 88 pp.

[3] Baharev M. S., Mirkin L. I., Shesterikov S. A., Yumasheva M. A., Structure and strength of materials with the laser action, Izd-vo Mosk. un-ta, M., 1988, 224 pp.

[4] Bednova V. B., “An approximate method for determining the temperature field with rapid local heating of the sample”, Trudy konferencii-konkursa molodyh uchenyh NII mekhaniki MGU imeni M. V. Lomonosova, 2013, 73–76

[5] Boley B. A., Weiner J. H., Theory of Thermal Stresses, Mir, M., 1964, 517 pp.

[6] Galaktionov V. A., Kurdyumov S. P., Mihajlov A. P., Samarskij A. A., “Heat localization in nonlinear media”, Differencial'nye uravneniya, XVII:10 (1981), 1826–1841

[7] Kalashnikov A. S., “On the equations of non-stationary filtration type with infinite speed of propagation of disturbances”, Vestnik Moskovskogo universiteta, 1972, no. 6, 45–49

[8] Carslaw H. S., Jaeger J. C., Conduction of Heat in Solids, Nauka, M., 1964, 488 pp.

[9] Lokoshchenko A. M., The creep and the long-term strength of metals in the aggressive media, Izd-vo Mosk. un-ta, M., 2000, 178 pp.

[10] Lykov A. V., Theory of heat conduction, Vysshaya shkola, M., 1967, 599 pp.

[11] Olejnik O. A., Kalashnikov A. S., Chzhou Yuj-Lin', “The Cauchy problem and boundary value problems for equations of non-stationary filtration”, Izv. AN SSSR, Seriya matematicheskaya, 22 (1958), 667–704 | Zbl

[12] Parcus H., Non-stationary thermal stresses, Gosudarstvennoe izd-vo fiziko-matematicheskoj literatury, M., 1963, 252 pp.

[13] Timoshenko S. P., Goodier J., Theory of Elasticity, Nauka, M., 1975, 576 pp.

[14] Shesterikov S. A., Yumasheva M. A., “An approximate method for estimating non-stationary temperature fields”, Deformirovanie i razrushenie tverdyh tel, Institut mekhaniki MGU. Nauchnye trudy, 23, Izd-vo Mosk. un-ta, M., 1973, 15–20

[15] Yumashev M. V., Bednova V. B., Vergazov M. M., Yumasheva M. A., “The destruction of brittle materials under local effect on the surface energy flow”, Mashinostroenie i inzhenernoe obrazovanie, 2014, no. 4, 52—58

[16] Yumashev M. V., Yumasheva M. A., Krasnova P. A., “Modeling of the body heating process with intensive exposure to heat to the surface”, Vestnik Moskovskogo universiteta, 2010, no. 4, 44—54