# Determination of the stationary thermal state of simple geometry layered structureswith themperature dependent heat conductivity factors

dc.citation.epage | 28 | |

dc.citation.issue | 3 | |

dc.citation.journalTitle | Econtechmod | |

dc.citation.spage | 23 | |

dc.citation.volume | 7 | |

dc.contributor.affiliation | Pidstryhach Institute for Applied Problems of Mechanics and Mathematics National Academy of Sciences of Ukraine | |

dc.contributor.affiliation | Lviv National Agrarian University | |

dc.contributor.author | Makhorkin, M. | |

dc.contributor.author | Makhorkin, I. | |

dc.contributor.author | Makhorkina, T. | |

dc.coverage.placename | Lublin | |

dc.date.accessioned | 2019-06-18T12:03:48Z | |

dc.date.available | 2019-06-18T12:03:48Z | |

dc.date.created | 2018-06-18 | |

dc.date.issued | 2018-06-18 | |

dc.description.abstract | An analytical-numerical method to determine the one-dimensional stationary thermal state of simple geometry multilayer structures for arbitrary dependences of heat-conductivity factors on temperature is proposed (the multilayer bodies of thermosensitive materials, referred to one of the classical orthogonal coordinate systems (a,b,g ) are considered, the thermal state caused by thermal load is characterized by a one-dimensional stationary temperature field t (a) ). The method is based on: · utilization of elements of generalized functions algebra; · approximation of temperature dependences of heatconductivity factors of materials by piecewise constant temperature functions –( ) ( ) ( ) ( ( ) ( ) ) ( ) 1 11() ( )mi i i i it j j ijl t t + S+ t t=» L = L +å L -L - ,where t0 = 0<t1 <t2 <...<tm <tk =tm+1; (i)Lj – with the given accuracy corresponds to the value of the heat-conductivity factor of the corresponding layer in the interval t j-1 < t < t j ,S+ (a ) = {0, a £ 0; 1, a > 0} ;· introduction into consideration the function of Kirchhoff function type –( ) ( )( ) ( )0 =1= L å òtniiiJ t x N a dx ,where ( ) ( ) ( ) Ni a = S+ a -ai-1 - S+ a -ai .Therefore, the temperature field is determined by therelation( ) ( ) ( ) ( ) ( )1 1n nii i ii it J F J N a J N a= =é ù é ù= ê + ú ê L úêë úû êë úûå å ,where ( ) ( ) ( )11 21ink i i iiCf C K Q S-+== + +å + - a J a J a a isthe solution of the partially degenerate equation derived from the heat equation in accordance with generalized functions algebra, taking into account the perfect thermal contact of the layers; С1 , С2 are the constants of integration, in the general case determined from the system of two nonlinear algebraic equations obtained from the boundary conditions; fk (a) , Ki , Qi are the functions and constants, determined by the recurrence relations obtained in the work. Approbation of the methodology by studying the stationary thermal state of a two-layer cylinder is realized. The cases of existence of a closed-form analytic solutions for the nonlinear heat conduction problem are considered. | |

dc.format.extent | 23-28 | |

dc.format.pages | 6 | |

dc.identifier.citation | Makhorkin M. Determination of the stationary thermal state of simple geometry layered structureswith themperature dependent heat conductivity factors / M. Makhorkin, I. Makhorkin, T. Makhorkina // Econtechmod. — Lublin, 2018. — Vol 7. — No 3. — P. 23–28. | |

dc.identifier.citationen | Makhorkin M. Determination of the stationary thermal state of simple geometry layered structureswith themperature dependent heat conductivity factors / M. Makhorkin, I. Makhorkin, T. Makhorkina // Econtechmod. — Lublin, 2018. — Vol 7. — No 3. — P. 23–28. | |

dc.identifier.issn | 2084-5715 | |

dc.identifier.uri | https://ena.lpnu.ua/handle/ntb/45134 | |

dc.language.iso | en | |

dc.relation.ispartof | Econtechmod, 3 (7), 2018 | |

dc.relation.references | 1. Rykalyn N. N. 1985. The effect of concentrated energy flows on materials. Moscow: Science, 246. (in Russian). | |

dc.relation.references | 2. Kushnir R. M., Popovych V. S. 2009. Thermoelasticity of thermosensitive solids, Lviv: SPOLOM, 412. (in Ukrainian) | |

dc.relation.references | 3. Popovych V. 2014. Methods for Determination of the Thermo-stressed State of Thermosensitive Solids Under Complex Heat Exchange Conditions. Encyclopedia of Thermal Stresses, No. 6, 2997–3008. | |

dc.relation.references | 4. Carpinteri A., Paggi M. 2008. Thermo-elastic mismatch in nonhomogeneous beams. J. Eng. Math.Vol. 61, No. 2–4, 371–384. | |

dc.relation.references | 5. Noda N. 1991. Thermal stresses in materials with temperature-dependent properties. Appl. Mech. Rev.Vol. 44, 383–397. | |

dc.relation.references | 6. Ootao Y., Tanigawa O., Ishimaru O. 2000. Optimization of material composition of functionality graded plate for thermal stress relaxation using a genetic algorithm. J. Therm. Stresses. Vol. 23, 257–271. | |

dc.relation.references | 7. Tanigawa Y., Akai T., Kawamura R. 1996. Transient heat conduction and thermal stress problems of a nonhomogeneous plate with temperaturedependent material properties. J. Therm. Stresses. Vol.19, No. 1, 77–102. | |

dc.relation.references | 8. Tanigawa Y., Ootao Y. 2002. Transient thermoelastic analysis of functionally graded plate with temperaturedependent material properties taking into account the thermal radiation. Nihon Kikai Gakkai Nenji Taikai Koen Ronbunshu, No. 2, 133–134. | |

dc.relation.references | 9. Yangiian 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 Inform. Eng. Vol. 02, 433–436. | |

dc.relation.references | 10. Kushnir R. M., Popovych V. S. 2013. On the determination of steady-state thermoelastic state of multilayer structures under high-temperature heating. Visnyk Shevchenko Kyiv. Nats. Univ. No. 3, 42–47.(in Ukrainian). | |

dc.relation.references | 11. Kushnir R. M., Protsiuk Yu. B. 2010. Thermoelastic state of layered thermosensitive bodies of revolution for the quadratic dependence of the heat-conduction coefficients. Materials Science, Vol.46, No. 1, 7–18. (in Ukrainian). | |

dc.relation.references | 12. Protsiuk Yu. B. 2010. Static thermoelasticity problems for thermosensitive plates with cubic dependence of heat conductivity coefficients on temperature. Mat. metody ta fiz.-mekh. polia, Vol. 53,No. 4, 151–162. (in Ukrainian). | |

dc.relation.references | 13. Makhorkin I. M., Mastykash L. V. 2015. Оn one analytical-numerical method of solution for the onedimensional quasi-static thermoelasticity problem for thermosensitive body of simple geometry. Mat. metody ta fiz.-mekh. polia, Vol. 58, No.4, 95–106. (inUkrainian). | |

dc.relation.references | 14. Makhorkin I.M., Makhorkin I.M., Mastykash L. V. 2016. Аnalytical-numerical determination of thermoelastic state of multilayer transtropic bodies with simple geometry. Prykl. probl. mekh. i mat. Vol. 14, 133–139. (in Ukrainian). | |

dc.relation.references | 15. Jaworski N. 2015. Effective-thermal-characteristicssynthesis- microlevel-models-in-the-problems-ofcomposite- materials-optimal-design. Econtechmod. an international quarterly journal, Vol. 04, No. 2, 3–12 | |

dc.relation.references | 16. Podstryhach Ya. S., Lomakyn V. A., Koliano Yu.M. 1984. Thermoelasticity of bodies of non-uniform structure. Moscow: Science, 368. (in Russian). | |

dc.relation.references | 17. Lomakyn V. A. 1976. Theory of elasticity of inhomogeneous bodies. Moscow: МGU, 376. (in Russian). | |

dc.relation.references | 18. MakhorkinM., SulymH. 2010. On determination of the stress-strain state of a multi-wedge system with thin radial defects under antiplane deformation. Civil and environmental engineering reports, Vol. 5, 235–251. | |

dc.relation.references | 19. MakhorkinM., Makhorkina T. 2017. Analytical determination of the order of stress field singularity in some configurations of multiwedge systems for the case of antiplane deformation. Econtechmod. An international quarterly journal, Vol. 6, No. 3, 45–52. | |

dc.relation.references | 20. Makhorkin I. 2018. Generalized functions in the stationary heat conduction problems for thermosensitive multilayer structures of simple geometry. Modern Problems of Mechanics and Mathematics, Vol. 1, 184–185. | |

dc.relation.referencesen | 1. Rykalyn N. N. 1985. The effect of concentrated energy flows on materials. Moscow: Science, 246. (in Russian). | |

dc.relation.referencesen | 2. Kushnir R. M., Popovych V. S. 2009. Thermoelasticity of thermosensitive solids, Lviv: SPOLOM, 412. (in Ukrainian) | |

dc.relation.referencesen | 3. Popovych V. 2014. Methods for Determination of the Thermo-stressed State of Thermosensitive Solids Under Complex Heat Exchange Conditions. Encyclopedia of Thermal Stresses, No. 6, 2997–3008. | |

dc.relation.referencesen | 4. Carpinteri A., Paggi M. 2008. Thermo-elastic mismatch in nonhomogeneous beams. J. Eng. Math.Vol. 61, No. 2–4, 371–384. | |

dc.relation.referencesen | 5. Noda N. 1991. Thermal stresses in materials with temperature-dependent properties. Appl. Mech. Rev.Vol. 44, 383–397. | |

dc.relation.referencesen | 6. Ootao Y., Tanigawa O., Ishimaru O. 2000. Optimization of material composition of functionality graded plate for thermal stress relaxation using a genetic algorithm. J. Therm. Stresses. Vol. 23, 257–271. | |

dc.relation.referencesen | 7. Tanigawa Y., Akai T., Kawamura R. 1996. Transient heat conduction and thermal stress problems of a nonhomogeneous plate with temperaturedependent material properties. J. Therm. Stresses. Vol.19, No. 1, 77–102. | |

dc.relation.referencesen | 8. Tanigawa Y., Ootao Y. 2002. Transient thermoelastic analysis of functionally graded plate with temperaturedependent material properties taking into account the thermal radiation. Nihon Kikai Gakkai Nenji Taikai Koen Ronbunshu, No. 2, 133–134. | |

dc.relation.referencesen | 9. Yangiian 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 Inform. Eng. Vol. 02, 433–436. | |

dc.relation.referencesen | 10. Kushnir R. M., Popovych V. S. 2013. On the determination of steady-state thermoelastic state of multilayer structures under high-temperature heating. Visnyk Shevchenko Kyiv. Nats. Univ. No. 3, 42–47.(in Ukrainian). | |

dc.relation.referencesen | 11. Kushnir R. M., Protsiuk Yu. B. 2010. Thermoelastic state of layered thermosensitive bodies of revolution for the quadratic dependence of the heat-conduction coefficients. Materials Science, Vol.46, No. 1, 7–18. (in Ukrainian). | |

dc.relation.referencesen | 12. Protsiuk Yu. B. 2010. Static thermoelasticity problems for thermosensitive plates with cubic dependence of heat conductivity coefficients on temperature. Mat. metody ta fiz.-mekh. polia, Vol. 53,No. 4, 151–162. (in Ukrainian). | |

dc.relation.referencesen | 13. Makhorkin I. M., Mastykash L. V. 2015. On one analytical-numerical method of solution for the onedimensional quasi-static thermoelasticity problem for thermosensitive body of simple geometry. Mat. metody ta fiz.-mekh. polia, Vol. 58, No.4, 95–106. (inUkrainian). | |

dc.relation.referencesen | 14. Makhorkin I.M., Makhorkin I.M., Mastykash L. V. 2016. Analytical-numerical determination of thermoelastic state of multilayer transtropic bodies with simple geometry. Prykl. probl. mekh. i mat. Vol. 14, 133–139. (in Ukrainian). | |

dc.relation.referencesen | 15. Jaworski N. 2015. Effective-thermal-characteristicssynthesis- microlevel-models-in-the-problems-ofcomposite- materials-optimal-design. Econtechmod. an international quarterly journal, Vol. 04, No. 2, 3–12 | |

dc.relation.referencesen | 16. Podstryhach Ya. S., Lomakyn V. A., Koliano Yu.M. 1984. Thermoelasticity of bodies of non-uniform structure. Moscow: Science, 368. (in Russian). | |

dc.relation.referencesen | 17. Lomakyn V. A. 1976. Theory of elasticity of inhomogeneous bodies. Moscow: MGU, 376. (in Russian). | |

dc.relation.referencesen | 18. MakhorkinM., SulymH. 2010. On determination of the stress-strain state of a multi-wedge system with thin radial defects under antiplane deformation. Civil and environmental engineering reports, Vol. 5, 235–251. | |

dc.relation.referencesen | 19. MakhorkinM., Makhorkina T. 2017. Analytical determination of the order of stress field singularity in some configurations of multiwedge systems for the case of antiplane deformation. Econtechmod. An international quarterly journal, Vol. 6, No. 3, 45–52. | |

dc.relation.referencesen | 20. Makhorkin I. 2018. Generalized functions in the stationary heat conduction problems for thermosensitive multilayer structures of simple geometry. Modern Problems of Mechanics and Mathematics, Vol. 1, 184–185. | |

dc.rights.holder | © Copyright by Lviv Polytechnic National University 2018 | |

dc.rights.holder | © Copyright by Polish Academy of Sciences 2018 | |

dc.rights.holder | © Copyright by University of Engineering and Economics in Rzeszów 2018 | |

dc.rights.holder | © Copyright by University of Life Sciences in Lublin 2018 | |

dc.subject | multilayer structures | |

dc.subject | solids of simple geometry | |

dc.subject | steady thermal state | |

dc.subject | temperature-dependent heat conduction factors | |

dc.subject | generalized functions | |

dc.title | Determination of the stationary thermal state of simple geometry layered structureswith themperature dependent heat conductivity factors | |

dc.type | Article |

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