The coupled non-stationary thermo-electro-elasticity problem for a~long hollow cylinder

D. A. Shlyakhin ; M. A. Kal'mova

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 24 (2020) no. 4, pp. 677-691.

Résumé

A new closed solution of the coupled non-stationary thermo-electro-elasticity problem for a long piezoelectric ceramic radially polarized cylinder is constructed while satisfying the boundary conditions of thermal conductivity of the 1st and 3rd kind on its front surfaces. The case when the rate of change of the temperature field does not affect the inertia characteristics of the elastic system is considered. This makes it possible to include linear equations of equilibrium, electrostatics, and thermal conductivity with respect to the radial component of the displacement vector, electric potential, and also the function of changing the temperature field in the initial calculated relations of the problem under consideration. In the calculations, the classical Fourier law of thermal conductivity is used. To solve the problem, the mathematical apparatus of incomplete separation of variables is used in the form of a generalized biorthogonal finite integral transformation based on the multicomponent relation of the eigenvector functions of two homogeneous boundary value problems. An important point in the procedure of the structural algorithm of this method is the selection of the adjoint operator, without which it is impossible to solve non-self-adjoint linear problems of mathematical physics. The constructed calculated relationships make it possible to determine the stress-strain state, temperature and electric fields induced in a piezoelectric ceramic element under an arbitrary temperature external influence. An analysis of the numerical results makes it possible to determine the cylinder wall thickness at which the electric field leads to a redistribution of the temperature field. It is established that the rate of change in the volume of a piezoceramic body under external temperature influence does not significantly affect the temperature field. The developed calculation algorithm finds its application in the design of non-resonant piezoelectric temperature sensors.

Détail

Keywords: long piezoceramic cylinder, coupled thermo-electro-elasticity problem, unsteady temperature effect, finite integral transformations.

Bibliographie

[1] Kozlov V. L., Optoelektronnye datchiki [Optoelectronic Sensors], Belarus. State Univ., Minsk, 2005, 116 pp. (In Russian)

[2] Kulchin Yu. N., Raspredelennye volokonno-opticheskie izmeritel'nye sistemy [Distributed Fiberoptic Measuring Systems], Fizmatlit, Moscow, 2001, 272 pp. (In Russian)

[3] Dmitriev S. A., Slepov N. N., Volokonno-opticheskaia tekhnika: sovremennoe sostoianie i novye perspektivy [Fiber-Optic Technology: Current Status and New Perspectives], Tekhnosfera, Moscow, 2010, 608 pp. (In Russian)

[4] Pan'kov A. A., “Resonant diagnostics of temperature distribution by the piezo-electro-luminescent fiber-optical sensor according to the solution of the Fredholm integral equation”, PNRPU Mechanics Bulletin, 2018, no. 2, 72–82 (In Russian) | DOI

[5] Abbas I. A., Youssef H. M., “Finite element analysis of two-temperature generalized magneto-thermoelasticity”, Arch. Appl. Mech., 79:10 (2009), 917–925 | DOI | Zbl

[6] He T., Tian X., Shen Y., “A generalized electromagneto-thermoelastic problem for an infinitely long solid cylinder”, Eur. J. Mech. a/Solids, 24:2 (2005), 349–359 | DOI | Zbl

[7] Abbas I. A., Zenkour A. M., “LS model on electro-magneto-thermoelastic response of an infinite functionally graded cylinder”, Compos. Struct., 96 (2013), 89–96 | DOI

[8] Vatulyan A. O., Kiryutenko A. Yu., Nasedkin A. V., “Plane waves and fundamental solutions in linear thermoelectroelasticity”, J. Appl. Mech. Tech. Phys., 37:5 (1996), 727–733 | DOI | MR | Zbl

[9] Vatulyan A. O., Nesterov S. A., “The dynamic problem of thermoelectroelasticity for functionally graded layer”, Computational Continuum Mechanics, 10:2 (2017), 117–126 (In Russian) | DOI

[10] Kovalenko A. D., Vvedenie v termouprugost' [Introduction to Thermoelasticity], Nauk. dumka, Kiev, 1965, 204 pp. (In Russian) | MR

[11] Shlyakhin D. A., Dauletmuratova Zh. M., “Nonstationary axisymmetric thermoelasticity problem for a rigidly fixed circular plate”, Engineering Journal: Science and Innovation, 2018, no. 78, 1–18 (In Russian) | DOI

[12] Shlyakhin D. A., Dauletmuratova Zh. M., “Non-stationary coupled axisymmetric thermoelasticity problem for a rigidly fixed round plate”, PNRPU Mechanics Bulletin, 2019, no. 4, 191–200 (In Russian) | DOI

[13] Lychev S. A., Manzhirov A. V., Joubert S. V., “Closed solutions of boundary value problems of coupled thermoelasticity”, Mech. Solids, 45:4 (2010), 610–623 | DOI

[14] Radayev Yu. N., Taranova M. V., “Wavenumbers of type III thermoelastic waves in a long waveguide under sidewall heat interchanging”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2011, no. 2(23), 53–61 (In Russian) | DOI | Zbl

[15] Grinchenko V. T., Ulitko A. F., Shul'ga N. A., Mekhanika sviazannykh polei v elementakh konstruktsii [Mechanics of Coupled Fields in Structural Elements], Nauk. dumka, Kiev, 1989, 279 pp. (In Russian)

[16] Senitskij Yu. E., “A biorthogonal multicomponent finite integral transformation and its application to boundary value problems in mechanics”, Russian Math. (Iz. VUZ), 40:8 (1996), 69–79 | MR | Zbl

[17] Kobzar' V. N., Fil'shtinskii L. A., “The plane dynamic problem of coupled thermoelasticity”, J. Appl. Math. Mech., 72:5 (2008), 611–618 | DOI | Zbl