Unsteady one-dimensional problem of thermoelastic diffusion for homogeneous multicomponent medium with plane boundaries

A.V. Vestyak^a*, S.A. Davydov^a**, A.V. Zemskov^a,b***, D.V. Tarlakovskii^b,a****

^a Moscow Aviation Institute (National Research University), Moscow, 125993 Russia

^b Research Institute of Mechanics, Moscow State University, Moscow, 119192 Russia

E-mail:  *v.a.vestyak@mail.ru,  **xenon_93@inbox.ru, ***azemskov1975@mail.ru, ****tdvhome@mail.ru

Received December 14, 2017

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Abstract

The paper deals with the problem of determining the stress-strain state of a thermoelastic multicomponent medium with plane boundaries (layer and half-space) taking into account the presence of diffusion fluxes in each medium component. The effect of changes in the concentration and temperature on the stress-strain state of the medium has been studied with the help of a locally equilibrium model of thermoelastic diffusion, which includes the coupled system of equations of motion, heat transfer, and mass transfer. The solution has been found using the Laplace transform, as well as using the Fourier expansion for the layer and the sine-cosine transform for the half-space. The surface Green's functions have been expressed and analyzed. Test calculation has been performed.

Keywords: mechanical diffusion, multicomponent media, thermoelastic diffusion, integral transforms, Fourier series, Green's functions

Figure Captions

Fig. 1. Change u in time τ and by layer depth x.

Fig. 2. Change υ in time τ and by layer depth x.

Fig. 3. Change η₁ in time τ and by layer depth x.

Fig. 4. Change η₂ in time τ and by layer depth x.

References

1. Knyazeva A.G. Vvedenie v termodinamiku neobratimykh protsessov. Lektsii o modelyakh [Introduction to Thermodynamics of Irreversible Processes. Lectures on Models]. Tomsk, Izd. "Ivan Fedorov", 2014. 172 p. (In Russian)

2. Atwa S.Y. Generalized thermoelastic diffusion with effect of fractional parameter on plane waves temperature-dependent elastic medium. J. Mater. Chem. Eng., 2013, vol. 1, no. 2, pp. 55–74.

3. Kumar R., Chawla V. A study of Green's functions for three-dimensional problem in thermoelastic diffusion media. Afr. J. Math. Comput. Sci. Res., 2014, vol. 7, no. 7, pp. 68–78. doi: 10.5897/AJMCSR2014.0564.

4. Othman M.I.A., Elmaklizi Y.D. 2-D problem of generalized magneto- thermoelastic diffusion, with temperature-dependent elastic moduli. J. Phys., 2013, vol. 2, no. 3, pp. 4–11.

5. Deswal S., Kalkal K.K., Sheoran S.S. Axi-symmetric generalized thermoelastic diffusion problem with two-temperature and initial stress under fractional order heat conduction. Phys. B, 2016, vol. 496, pp. 57–68. doi: 10.1016/j.physb.2016.05.008.

6. Aouadi M. A generalized thermoelastic diffusion problem for an infinitely long solid cylinder. Int. J. Math. Math. Sci., 2006, vol. 2006, art. 25976, pp. 1–15. doi: 10.1155/IJMMS/2006/25976.

7. Elhagary M.A. A two-dimensional generalized thermoelastic diffusion problem for a half-space subjected to harmonically varying heating. Acta Mech., 2013, vol. 224, no. 12, pp. 3057–3069. doi: 10.1007/s00707-013-0902-6.

8. El-Sayed A.M. A two-dimensional generalized thermoelastic diffusion problem for a half-space. Math. Mech. Solids., 2016, vol. 21, no. 9, pp. 1045–1060. doi: 10.1177/1081286514549877.

9. Davydov S.A., Zemskov A.V., Tarlakovskii D.V. Surface Green's functions in non-stationary problems of thermomechanical diffusion. Probl. Prochn. Plast., 2017, vol. 79, no. 1, pp. 38–47. (In Russian)

10. Tarlakovskii D.V., Vestyak V.A., Zemskov A.V. Dynamic processes in thermoelectromagnetoelastic and thermoelastodiffusive media. In: Encyclopedia of Thermal Stress. Vol. 2. Dordrecht, Heidelberg, New York, London, Springer, 2014, pp. 1064–1071.

11. Davydov S.A., Zemskov A.V., Tarlakovskii D.V. An elastic half-space under the action of one-dimensional time-dependent diffusion perturbations. Lobachevskii J. Math., 2015, vol. 36, no. 4, pp. 503–509. doi: 10.1134/S199508021504023X.

12. Zemskov A.V., Tarlakovskiy D.V. Two-dimensional nonstationary problem elastic for diffusion an isotropic one-component layer. J. Appl. Mech. Tech. Phys., 2015, vol. 56, no. 6, pp. 1023–1030. doi: 10.1134/S0021894415060127.

13. Davydov S.A., Zemskov A.V., Tarlakovskii D.V. Two-component elastic diffusion half-space under the influence of time-dependent perturbations. Ekol. Vestn. Nauchn. Tsentrov Chernomorsk. Ekon. Sotr., 2014, no. 2, pp. 31–38. (In Russian)

14. Prudnikov A.P., Brychkov Yu.A., Marichev O.I. Integraly i ryady [Integrals and Series]. Vol. 1: Elementary Functions. Moscow, Nauka, 1981. 797 p. (In Russian)

15. Zhuravskii A.M. Spravochnik po ellipticheskim funktsiyam [Handbook on Elliptic Functions]. Moscow, Leningrad, Izd. Akad. Nauk SSSR, 1941. 236 p. (In Russian)

For citation: Vestyak A.V., Davydov S.A., Zemskov A.V., Tarlakovskii D.V. Unsteady one-dimensional problem of thermoelastic diffusion for homogeneous multicomponent medium with plane boundaries. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2018, vol. 160, no. 1, pp. 183–195. (In Russian)

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