539.3

Navier’s equations, thermoelastic body, cylindrical coordinate system, characteristics of temperature state, stresses, displacements.

The paper considers the linear model of three-dimensional isotropic body of theories of thermoelasticity in the cylindrical coordinate system. We consider the case when the stationary temperature satisfies the Laplace equation. After substituting thermoelastic stresses into the equilibrium equation, the system of Navier’s differential equations were obtained. Its general solution is presented as the sum of homogeneous and partial solutions. The partial solution of the system of Navier’s equations, which is clearly determined by the stationary temperature and does not contain elastic displacements, is called the temperature solution. The physical and mathematical features of the thermoelastic stress state were taken into account and it was proved that in the temperature solution the sum of normal stresses is zero and the volume deformation is equal to . The found dependencies were used and the new temperature solution of the system of Navier’s equations were constructed in the cylindrical coordinate system, when the temperature does not depend on the axial variable. Simple formulas for expressing temperature stresses are given. The general solution of the equations of the theory of thermoelasticity by four harmonic functions is recorded.

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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. J. Thermal Stresses. 2008. Vol. 31. No. 11. P. 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. 2021. Vol. 24. No. 1. P. 36–41.
9. Yuzvyak M., Tokovyy Y. and Yasinskyy A. Axisymmetric thermal stresses in an elastic hollow cylinder of finite length. Journal of Thermal Stresses. 2021. Vol. 44. No. 3. P. 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. 2022. Vol. 108. No. 4. P. 64–73. Doi: 10.33108/visnyk_tntu2023.01.
11. Revenko V. P. Solving the three-dimensional equations of the linear theory of elasticity. Int. Appl. Mech. 2009. No. 7 (45). P. 730–741. URL: https://doi.org/10.1007/s10778-009-0225-4.
12. Korn G., Korn T. Mathematical Handbook for Scientists and Engineers. Definitions, Theorems, Formulas. New York: Dover Publications 2013. 1152 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: WoltersNoordhoff, 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. J. Thermal Stresses. 2008. Vol. 31. No. 11. P. 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. 2021. Vol. 24. No. 1. P. 36–41.
9. Yuzvyak M., Tokovyy Y. and Yasinskyy A. Axisymmetric thermal stresses in an elastic hollow cylinder of finite length. Journal of Thermal Stresses. 2021. Vol. 44. No. 3. P. 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. 2022. Vol. 108. No. 4. P. 64–73. Doi: 10.33108/visnyk_tntu2023.01.
11. Revenko V. P. Solving the three-dimensional equations of the linear theory of elasticity. Int. Appl. Mech. 2009. No. 7 (45). P. 730–741. URL: https://doi.org/10.1007/s10778-009-0225-4.
12. Korn G., Korn T. Mathematical Handbook for Scientists and Engineers. Definitions, Theorems, Formulas. New York: Dover Publications 2013. 1152 p.