Solving thermoelasticity problems for anisotropic solids of revolution

Deformable body mechanics


Ivanychev D. A.

Lipetsk State Technical University, 30, Moskovskaya str., Lipetsk, 398600, Russia



The goal of the work consists in determining the stress-strain state of anisotropic bodies of revolution in conditions of temperature exposure with no internal heat sources. The body boundary is free fr om external forces and kinematic dependencies.

The task setting is ensured by the inverse method development for the class of stationary axisymmetric thermoelasticity problems for transversely isotropic bodies. The author proposes the theory of the internal states’ spaces basis forming, including displacements, deformations, stresses and temperature. First, a basis of plane auxiliary states is formed for the case of plane deformation, based on the general solution of the plane thermoelasticity problem for a transversely isotropic body. Further, on its ground, a basis of internal spatial states is being induced by integral overlays method. This basis is being orthonormalized based on the Gram-Schmidt recursive-matrix orthogonalization algorithm, where integrals of the product of temperatures act as cross scalar products. After the basis orthogonalization the target state is being determined by the Fourier series, wh ere the coefficients represent definite integrals which kernels compose the functions of the temperatures in the basic elements multiplication by the specified temperature.

Verification of the solution is performed by comparing the specified temperature field with that obtained while the solving.

A strict solution of the test problem for a circular cylinder, and an approximate solution for a body in the form of a stepped cylinder, made of rock, with the corresponding conclusions on the series convergence are presented. A graphical visualization of the results is presented as well. The advantage of the presented approach consists in the fact that the most time-consuming calculations, namely an orthonormal basis construction, are performed once for a body of a certain configuration. Subsequently, this basis can be used for solving various thermoelastic problems for this body.


anisotropy, thermoelasticity, boundary state method, inverse method, axisymmetric problems, transversely isotropic bodies, Fourier series


  1. Ferrari M. Anisotropic layers with through-thickness thermal and material variations, Journal_of_Thermal_Stresses, 1992, vol. 15, no. 3, pp. 439 – 445.

  2. Ivanychev D.A. Trudy MAI, 2019, no. 105, available at:

  3. Novatskii V. Teoriya uprugosti (Theory of elasticity), Moscow, Mir, 1975, 872 p.

  4. Khan'zhov B.D. Izvestiya vuzov. Matematika, 1967, no. 12, pp. 84 – 89.

  5. Pen'kov V.B., Viktorov D.V., Satalkina L.V. Materialy mezhdunarodnoi nauchnoi konferentsii “Sovremennye problemy matematiki, mekhaniki, informatiki”, Tula, TulGU, 2008, pp. 274 – 277.

  6. Lur'e S.A., Dudchenko A.A., Nguen D.K. Trudy MAI, 2014, no. 75, available at:

  7. Lur'e S.A., Solyaev Yu.O., Nguen K., Medvedskii A.L., Rabinskii L.N. Trudy MAI, 2013, no. 71, available at:

  8. Nikolaev A.G., Orlov E.M. Problemi obchislyuval'noї mekhanіki mіtsnostі konstruktsіi, 2012, no. 20, available at:

  9. Levina L.V, Kuz'menko N.V. ХI Vserossiiskii s"ezd po fundamental'nym problemam teoreticheskoi i prikladnoi mekhaniki: sbornik dokladov Kazan', Izd-vo Kazanskogo universiteta, 2015, pp. 2276 – 2278.

  10. Belostochnyi G.N., Myl'tsina O.A. Trudy MAI, 2015, no. 82, available at:

  11. Gur'yanov N.G., Tyuleneva O.N. XV Mezhdunarodnaya nauchno-prakticheskaya konferentsiya “Teoriya i praktika sovremennoi nauki”: sbornik statei, Moscow, 2014, Izd-vo «Spetskniga», pp. 10 - 17.

  12. Gur'yanov N.G., Tyuleneva O.N. XI Vserossiiskii s"ezd po fundamental'nym problemam teoreticheskoi i prikladnoi mekhaniki: sbornik dokladov, Kazan', Kazanskii (Privolzhskii) federal'nyi universitet, 2015, pp. 1104 - 1106.

  13. Gur'yanov N.G., Tyuleneva O.N. Vestnik Permskogo natsional'nogo issledovatel'skogo politekhnicheskogo universiteta. Mekhanika, 2017, no. 2, pp. 60 - 77. DOI: 10.15593/perm.mech/2017.2.04

  14. Aleksandrov A.Ya., Solov'ev Yu.I. Prostranstvennye zadachi teorii uprugosti. Primenenie metodov teorii funktsii kompleksnogo peremennogo (Spatial problems of the theory of elasticity. Application of complex variable theory methods), Moscow, Nauka. Glavnaya redaktsiya fiziko-matematicheskoi literatury, 1978, 464 p.

  15. Pen'kov V.B., Pen'kov V.V. Dal'nevostochnyi matematicheskii zhurnal, 2001, vol. 2, no. 2, pp. 115 – 137.

  16. Satalkina L.V. Nauchnaya konferentsiya studentov i aspirantov Lipetskogo gosudarstvennogo tekhnicheskogo universiteta: sbornik tezisov dokladov, Lipetsk, LGTU, 2007, pp. 130 – 131.

  17. Ivanychev D.A. Vesti vysshikh uchebnykh zavedenii Chernozem'ya, 2014, no. 1, pp. 19 – 26.

  18. Lekhnitskii S.G. Teoriya uprugosti anizotropnogo tela (Theory of anisotropic body elasticity), Moscow, Nauka, 1977, 416 p.

  19. Levina L.V., Novikova O.S., Pen'kov V.B. Vestnik Lipetskogo gosudarstvennogo tekhnicheskogo universiteta, 2016, no. 2 (28), pp. 16 – 24.

  20. Yudin V.A., Korolev A.V., Afanaskin I.V., Vol'pin S.G. Teploemkost' i teploprovodnost' porod i flyuidov bazhenovskoi svity – iskhodnye dannye dlya chislennogo modelirovaniya teplovykh sposobov razrabotki (Heat capacity and thermal conductivity of rocks and fluids of the Bazhenov formation are the initial data for the numerical simulation of thermal methods of development), Moscow, NIISI RAN, 2015, 22 p.

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