cylindrical coordinate system, thermoelastic state of the body, the axisymmetric temperature state, temperature stresses and displacements.

The paper uses the system of Navier equations in the stationary case. A cylindrical coordinate system is considered, when the temperature does not depend on the angular variable. A partial solution of the system of Navier equations, which does not contain elastic displacements, is called a purely temperature solution. It was established that for purely temperature solutions the sum of normal stresses is zero and the volume deformation is equal T e  = 3 . An analytical expression of purely temperature displacements and stresses in the cylindrical coordinate system in the axisymmetric case was found. The solution of the boundary value problem of thermal conductivity, when the cylinder is heated on one end, cooled by liquid on the other with known heat losses on the side surface, is proposed. The solution of the boundary value problem of thermal conductivity for such a cylinder is given in the form of the sum of the basic temperature, which describes the heat balance, and the perturbed temperature. The basic temperature has a polynomial form and integrally satisfies the boundary conditions. The perturbed temperature has an exponential decrease with distance from the heated end and does not carry out integral heat transfer. The found dependencies were used and a new solution to the heat conduction equation was written in a cylindrical coordinate system. Simple formulas for expressing temperature changes have been obtained. A new temperature solution to the system of thermoelasticity equations in a cylindrical coordinate system has been written, when the temperature does not depend on the angular variable.

1. Noda N., Hetnarski R. B., Tanigawa Y. Thermal stresses, New York: Taylor&Francis, 2003, 502 p.

2. Nowacki W. Thermoelasticity, 2nd ed., Warsaw, Poland, Pergamon, 1986, 560 p.

3. Sadd M. H. Elasticity. Theory, applications, and numerics, Amsterdam: Academic Press; 2014, 600 p.

4. Melan E., Parkus H. Wärmespannungen: Infolge Stationärer Temperaturfelder Published by Springer, 2013, 121 p. ISBN 10: 3709139694.

5. Kovalenko A. D. Thermoelasticity: Basic Theory and Applications, Groningen, the Netherlands: Wolters Noordhoff, 1969, 302 p.

6. Timoshenko S. P., Goodier J. N. Theory of Elasticity, New York: McGraw-Hill, 1977, 567 p.

7. Rychahivskyy A. V. and Tokovyy Y. V. Correct analytical solutions to the thermoelasticity problems in a semi-plane, Thermal Stresses, 31, no. 11, 2008, рр. 1125–1145. Doi: 10.1080/01495730802250854.

8. Revenko V. P. Analytical solution of the problem of symmetric thermally stressed state of thick plates based on the 3d elasticity theory, Journal of Mechanical Engineering, vol. 24, no. 1. 2021, pp. 36–41.

9. Yuzvyak M., Tokovyy Y. and Yasinskyy A. Axisymmetric thermal stresses in an elastic hollow cylinder of finite length. J. Thermal Stresses, 44, no. 3, 2021, pp. 359–376. Doi: 10.1080/01495739.2020.1826376.

10. Revenko V. Construction of static solutions of the equations of elasticity and thermoelasticity theory. Scientific Journal of TNTU. Tern.: TNTU, 2022, vol. 108, no. 4, pp. 64–73. Available at: https://doi.org/ 10.33108/visnyk_tntu2022.04.064.

11. Revenko V. P. Solving the three-dimensional equations of the linear theory of elasticity, Int. Appl. Mech., vol. 45, no. 7, 2009, pp. 730–741. Available at: https://doi.org/10.1007/s10778-009-0225-4.

12. Revenko V. Finding physically justified partial solutions of the equations of the thermoelasticity theory in the cylindrical coordinate system, Scientific Journal of TNTU, 2023, vol. 112, no. 4, pp. 58–66. Available at: https://doi.org/10.33108/visnyk_tntu2023.04.058.

13. Korn G., Korn T. Mathematical Handbook for Scientists and Engineers. Definitions, Theorems, Formulas, New York: Dover Publications 2013, 1152 p. 14. Titchmarsh E. C. Eigenfunction expansions associated with second order differential equations. Part 1. L: Oxford University Press, 1962, 210 p.

1. Noda N., Hetnarski R. B., Tanigawa Y. Thermal stresses, New York: Taylor&Francis, 2003, 502 p.

2. Nowacki W. Thermoelasticity, 2nd ed., Warsaw, Poland, Pergamon, 1986, 560 p.

3. Sadd M. H. Elasticity. Theory, applications, and numerics, Amsterdam: Academic Press; 2014, 600 p.

4. Melan E., Parkus H. Wärmespannungen: Infolge Stationärer Temperaturfelder Published by Springer, 2013, 121 p. ISBN 10: 3709139694.

5. Kovalenko A. D. Thermoelasticity: Basic Theory and Applications, Groningen, the Netherlands: Wolters Noordhoff, 1969, 302 p.

6. Timoshenko S. P., Goodier J. N. Theory of Elasticity, New York: McGraw-Hill, 1977, 567 p.

7. Rychahivskyy A. V. and Tokovyy Y. V. Correct analytical solutions to the thermoelasticity problems in a semi-plane, Thermal Stresses, 31, no. 11, 2008, рр. 1125–1145. Doi: 10.1080/01495730802250854.

8. Revenko V. P. Analytical solution of the problem of symmetric thermally stressed state of thick plates based on the 3d elasticity theory, Journal of Mechanical Engineering, vol. 24, no. 1. 2021, pp. 36–41.

9. Yuzvyak M., Tokovyy Y. and Yasinskyy A. Axisymmetric thermal stresses in an elastic hollow cylinder of finite length. J. Thermal Stresses, 44, no. 3, 2021, pp. 359–376. Doi: 10.1080/01495739.2020.1826376.

10. Revenko V. Construction of static solutions of the equations of elasticity and thermoelasticity theory. Scientific Journal of TNTU. Tern.: TNTU, 2022, vol. 108, no. 4, pp. 64–73. Available at: https://doi.org/ 10.33108/visnyk_tntu2022.04.064.

11. Revenko V. P. Solving the three-dimensional equations of the linear theory of elasticity, Int. Appl. Mech., vol. 45, no. 7, 2009, pp. 730–741. Available at: https://doi.org/10.1007/s10778-009-0225-4.

12. Revenko V. Finding physically justified partial solutions of the equations of the thermoelasticity theory in the cylindrical coordinate system, Scientific Journal of TNTU, 2023, vol. 112, no. 4, pp. 58–66. Available at: https://doi.org/10.33108/visnyk_tntu2023.04.058.

13. Korn G., Korn T. Mathematical Handbook for Scientists and Engineers. Definitions, Theorems, Formulas, New York: Dover Publications 2013, 1152 p. 14. Titchmarsh E. C. Eigenfunction expansions associated with second order differential equations. Part 1. L: Oxford University Press, 1962, 210 p.