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- Category: Geotechnical and mining mechanical engineering, machine building
- Last Updated on 20 March 2019
- Published on 03 March 2019
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**Authors:**

**V.I.Havrysh**, Dr. Sc. (Tech.), Assoc. Prof., orcid.org/0000-0003-3092-2279, Lviv Polytechnic National University, Lviv, Ukraine, email: This email address is being protected from spambots. You need JavaScript enabled to view it.

**L.I.Kolyasa**, Cand. Sc. (Phys.-Math.), orcid.org/0000-0002-9690-8042, Lviv Polytechnic National University, Lviv, Ukraine, email: This email address is being protected from spambots. You need JavaScript enabled to view it.

**O.M.Ukhanska**, Cand. Sc. (Phys.-Math.), Assoc. Prof., orcid.org/0000-0003-4408-5491, Lviv Polytechnic National University, Lviv, Ukraine, email: This email address is being protected from spambots. You need JavaScript enabled to view it.

**V.B.Loik**, Cand. Sc. (Tech.), orcid.org/0000-0002-3772-1640, Lviv State University of Life Safety, Lviv, Ukraine

**Abstract:**

**Purpose.**Determination of the field of temperature which is caused by the presents of the heat flow thermally sensitive (thermo-physical parameters which depend on temperature) layered medium which contains through foreign inclusions.

**Methodology. **It is based on the use of generalized functions; this enables us to express the coefficient of heat conductivity for such a structure as something integral, provided the heat contact between conjugated surfaces of the layers and the inclusion is ideal. In this case, the boundary value problem is reduced to solving a single equation of heat conduction with discontinuous coefficients under the given boundary conditions at the boundary surfaces of the medium.

**Findings. **The heat flux is concentrated at one of the boundary surfaces of the medium, the other boundary surface is thermally insulated. There exists ideal heat contact at the surfaces of the conjugated layers. To determine temperature regimes in such a medium, a nonlinear equation of heat conduction with nonlinear boundary conditions is used. In order to solve the nonlinear boundary value problem of heat conduction, we introduce a linearizing function which enables us to obtain a partially linearized differential equation and linear boundary conditions to determine this function. After the piecewise-linear approximation of temperature with respect to spatial coordinates is carried out, a linear differential equation with discontinuous coefficients in the linearizing function is obtained. An analytical-numerical solution of the obtained linear boundary value problem is found with the use of Fourier integral transformation which determines the linearizing function and enables us to obtain calculation formulae for calculating temperature. For a two-layer medium with an inclusion, temperature distribution for a linear temperature dependence of the coefficient of heat conductivity of the material is found, and a comparative numerical analysis of the distribution of temperature is made with corresponding distribution for constant coefficients of heat conductivity of the materials of the layers (materials of the layers are Y12 and 08 steels). Calculation formulae for determining the distribution of temperature in a two-layer medium with a through inclusion are obtained in the work. Numerical calculations of the distribution in a layer and in a two-layer medium for constant and for linearly variable with respect to temperature coefficient of heat conductivity of materials of the layers are performed.

**Originality. **There has been carried out a partial linearization of nonlinear boundary value problem; due to this, we have obtained formulae for determination of the distribution of temperature in a thermo-sensitive piece-wise homogeneous medium.

**Practical value.**The practical value consists in the increase in accuracy calculation of temperature fields and in effectiveness of methods for investigation of thermo-sensitive piece-wise homogeneous media. The precision is achieved at the expanse of taking into account the piecewise homogeneous structure of the medium and that of the dependence of the coefficient of heat conductivity of the materials of the medium on temperature (nonlinear model).

**References.**

**1. **Vengerov, I.R., 2008. *Thermophysics of shafts and mines. Mathematical models. Vol.1. Paradigm analysis*. Donetsk: Nord-Press, 288 s. [pdf]. Available at: <https://www.e-reading.club/bookreader.php/137405/Vengerov_-_Teplofizika_shaht_i_rudnikov._Matematicheskie_modeli.pdf> [Accessed 11 December 2017].

**2. **Vengerov, I. R., 2007. Non-one-dimensional models of miming thermophysics. *Fiziko-tekhnicheskiye problemy gornogo proizvodstva. Sb. nauchn. tr.,* 10, pp. 60‒80. Available at: <http://www.ifgpdnr.ru/main/doc/%D0%A1%D0%B1%D0%BE%D1%80%D0%BD%D0%B8%D0%BA%20%D0%A4%D0%A2%D0%9F%D0% 93%D0%9F%202007.pdf > [Accessed 07 January 2018).

**3. **Smirnova, N. N., Nikolayeva, N. V., Brichkin, V. N. and Kuskov, V. B., 2014. Analytical solutions to certain problems of heat exchange in the processes of mining. *Fiziko-tekhnicheskiye problemy razrabotki poleznykh iskopayemykh / Gornaya teplofizika*, 1, pp. 80‒86.

**4. **Nemyrovskyi, Yu. V., 2007. Asymptotic analysis of problems of transient heat conduction of embedded aeolotropic nonhomogeneous plates under boundary conditions of the first and third types on the face surface. *Mathematical methods and physicomechanical fields* [online], 50(2), pp. 160‒175. Available at: <http://journals.iapmm.lviv.ua/ojs/index.php/MMPMF/article/view/2184/2398> [Accessed 25 December 2017].

**5. **Turii, O., 2008. Nonlinear contact boundary problem of thermomechanics for the irradiated two-layer plate connected by an intermediate layer. *Physico-mathematical modellingand informational technologies*, 8, pp. 118‒132. Available at: < http://dspace.nbuv.gov.ua/handle/123456789/22091> [Accessed 04 November 2017].

**6. **Havrysh, V. I., 2016. Research on temperature fields in a thermosensitive layer with a through inclusion. *Physicochemical Mechanics of materials*, 52(4), pp. 63‒68.

**7. **Bayat, A., Moosavi, H. and Bayat, Y., 2015. Thermo-mechanical analysis of functionally graded thick spheres with linearly time-dependent temperature. *Scientia Iranica*, 22(5), pp. 1801–1812.

**8. **Mohazzab, A.H. and Jabbari, M., 2011. Two-Dimensional Stresses in a Hollow FG Sphere with Heat Source. *Advanced Materials Research*, 264‒265, pp. 700–705. DOI: 10.4028/www.scientific.net/amr.264-265.700.

**9. **Ghannad, M. and Yaghoobi, M. P., 2015. A thermoelasticity solution for thick cylinders subjected to thermo-mechanical loads under various boundary conditions. *International Journal of Advanced Design & Manufacturing Technology*, 8(4), pp. 1–12.

**10.**Havrysh, V. Y., 2015. Modeling of temperature conditions in non-homogeneous elements of electronic devices with through foreign inclusions. *Electronic modeling*, 37(4), pp. 109‒118.

Tags: heat conduction • isotropic • temperature field • inclusion foreign • thermally sensitive layered medium • heat flux • heat contact

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