### Non-axisymmetric non-stationary problem of thermoelectroelasticity for a long piezo-ceramic cylinder

**Published:**20.07.2023

**Authors:** Shlyakhin D.A., Yurin V.A.

**Published in issue: **#7(139)/2023

**DOI: **10.18698/2308-6033-2023-7-2288

**Category:** Mechanics | **Chapter:** Mechanics of Deformable Solid Body

A new closed solution to the uncoupled non-axisymmetric problem of thermoelectroelasticity was constructed for a long hollow piezo-ceramic cylinder for the case of non-stationary temperature alteration on its inner surface, taking into account the convective heat transfer between the outer surface and the environment. Cylindrical surfaces were electroded and connected to a measurement device with large input resistance (idle mode), while the internal surface was grounded. The Fourier—Kirchhoff non-stationary heat conduction equation was considered without taking into account the influence of alteration in the body dimensions and the electric field on the temperature field. The closed solution to the heat conduction problem was constructed by the finite integral transformations (FIT) method. The quasi-static coupled electroelasticity problem at a certain temperature field was solved without taking into account the cylinder inertial properties by the FIT method. The calculated relations obtained made it possible to determine the temperature field, the stress-strain state, as well as the electric field in the long piezo-ceramic cylinder under non-stationary non-axisymmetric action in the form of a function of the temperature alteration. Numerical analysis of the results made it possible to determine the cylinder wall thickness and the region of the temperature effect alteration, where deformation could most efficiently transform into the electrical pulse.

**References**

**[1]**Ionov B.P., Ionov A.B. Spektralno-statisticheskiy podkhod k beskontaktnomu izmereniyu temperatury [Spectral statistical approach to contactless measurement of temperature]. Datchiki i sistemy — Sensors & Systems, 2009, no. 2, pp. 9–11.

**[2]**Kazaryan A.A. Tonkoplenochnyi datchik davleniya i temperatury [Thin-film sensor of pressure and temperature]. Datchiki i sistemy — Sensors & Systems, 2016, no. 3, pp. 50–56.

**[3]**Pankov A.A. Rezonansnaya diagnostika raspredeleniya temperatury pyezoelectrolyuministsentnym optovolokonnym datchikom po resheniyu integralnogo uravneniya Fredgolma [Resonant diagnostics of temperature distribution by the piezo-electro-luminescent fiber-optical sensor according to the solution of the Fredholm integral equation]. Vestnik Permskogo natsionalnogo issledovatelskogo politekhnicheskogo universiteta. Mekhanika — PNRPU Mechanics Bulletin, 2018, no. 2, pp. 72–82.

**[4]**Kalmova M. The scope of application of devices whose operation is based on taking into account the connectivity of thermoelectroelastic fields. Austrian J. of Technical and Natural Sciences, 2022, vol. 3 (4), pp. 14–16. https://doi.org/10.29013/AJT-22-3.4-14-16

**[5]**Mindlin R.D. Equations of high frequency vibrations of thermopiezoelectric crystal plates. Int. J. of Solids and Structures, 1974, vol. 10 (6), pp. 625–637.

**[6]**Lord H.W., Shulman Y. A generalized dynamical theory of thermoelasticity. J. of the Mechanics and Physics of Solids, 1967, vol. 15 (5), pp. 299–309.

**[7]**Green A.E., Naghdi P.M. Thermoelasticity without energy dissipation. J. of Elasticity, 1993, vol. 31, pp. 189–208.

**[8]**Vatulyan А.О., Nesterov S.А. Dinamicheskaya zadacha termoelektrouprugosti dlya funktsionalno-gradientnogo sloya [The dynamic problem of thermoelectroelasticity for functionally graded layer]. Vychislitelnaya mekhanika sploshnykh sred — Computational Continuum Mechanics, 2017, vol. 10, no. 2, pp. 117–126.

**[9]**Babeshko V.A., Ratner S.V., Syromyatnikov P.V. On mixed problems for thermoelectroelastic media with discontinuous boundary conditions. Doklady Physics, 2007, vol. 52, pp. 90–95.

**[10]**Saadatfar М., Razavi A.S. Piezoelectric hollow cylinder with thermal gradient. J. of Mechanical Science and Technology, 2009, vol. 23, pp. 45–53.

**[11]**Akbarzadeh A.H., Babaei M.H., Chen Z.T. The thermoelectromagnetoelastic behavior of a rotating functionally graded piezoelectric cylinder. Smart Materials and Structures, 2011, vol. 20 (6). https://doi.org/10.1088/0964-1726/20/6/065008

**[12]**Rahimi G.H., Arefi M., Khoshgoftar M.J. Application and analysis of functionally graded piezoelectrical rotating cylinder as mechanical sensor subjected to pressure and thermal loads. Applied Mathematics and Mechanics, 2011, vol. 32 (8), pp. 997–1008.

**[13]**Shlyakhin D.A., Kalmova M.A. Svyazannaya nestatsionarnaya zadacha termoelektrouprugosti dlya dlinnogo pologo tsilindra [The coupled non-stationary thermo-electro-elasticity problem for a long hollow cylinder]. Vestnik Samarskogo gosudarstvennogo tekhnicheskogo universiteta. Seriya Fiziko-matematicheskie nauki — Journal of Samara State Technical University. Ser. Physical and Mathematical Sciences, 2020, vol. 14, no. 4, pp. 677–691.

**[14]**Shlyakhin D.A., Kalmova M.A. Nestatsionarnaya zadacha termoelekt-rouprugosti dlya dlinnogo pyezokeramicheskogo tsilindra [The nonstationary thermoelectric elasticity problem for a long piezoceramic cylinder]. Vestnik Permskogo natsionalnogo issledovatelskogo politekhnicheskogo universiteta. Mekhanika — PNRPU Mechanics Bulletin, 2021, no. 2, pp. 181–190.

**[15]**Shlyakhin D.A., Kalmova M.A. Svyazannaya dinamicheskaya osesim-metrichnaya zadacha termoelektrouprugosti dlya dlinnogo pologo pyezokeramicheskogo tsilindra [Related dynamic axisymmetric thermo-electroelasticity problem for a long hollow piezoceramic cylinder]. Advanced Engineering Research, 2022, vol. 22, no. 2, pp. 81–90.

**[16]**Dai H.L., Wang X, Dai Q.H. Thermoelectroelastic responses in orthotropic piezoelectric hollow cylinders subjected to thermal shock and electric excitation. J. of Reinforced Plastics and Composites, 2005, vol. 24 (10), pp. 1085–1103. https://doi.org/10.1177/0731684405048834

**[17]**Dai H.L., Luo W.F., Dai T., Luo W.F. Exact solution of thermoelectroelastic behavior of a fluid-filled FGPM cylindrical thin-shell. Composite Structures, 2017, vol. (162), pp. 411–423. https://doi.org/10.1016/j.compstruct.2016.12.002

**[18]**Dai H.L., Wang X. Thermo-electro-elastic transient responses in piezoelectric hollow structures. International J. of Solids and Structures, 2005, vol. 42 (3–4), pp. 1151–1171. https://doi.org/10.1016/j.ijsolstr.2004.06.061

**[19]**Jabbari M., Sohrabpour S., Eslami M.R. General solution for mechanical and thermal stresses in a functionally graded hollow cylinder due to non-axisymmetric steady-state loads. J. of Applied Mechanics, 2003, vol. 70 (1), pp. 111–118.

**[20]**Atrian A., Fesharaki J.J., Majzoobi G.H., Sheidaee M. Effects of electric potential on thermo-mechanical behavior of functionally graded piezoelectric hollow cylinder under non-axisymmetric loads. International J. of Mechanical, Aerospace, Industrial, Mechatronic and Manufacturing Engineering, 2011, vol. 5 (11), pp. 2441–2444.

**[21]**Obata Y., Noda N. Steady thermal stresses in a hollow circular cylinder and a hollow sphere of a functionally gradient material. J. of Thermal Stresses, 1994, vol. 17 (3), pp. 471–487.

**[22]**Ishihara M., Ootao Y., Kameo Y. A general solution technique for electroelastic fields in piezoelectric bodies with D∞ symmetry in cylindrical coordinates. J. of Wood Science, 2016, vol. 62, pp. 29–41.

**[23]**Ishihara M., Ootao Y., Kameo Y. Analytical technique for thermoelectroelastic field in piezoelectric bodies with D ∞ symmetry in cylindrical coordinates. J. of Thermal Stresses, 2017, vol. 41 (6), pp. 1–20.

**[24]**Chen W.Q., Shioya T. Piezothermoelastic behavior of a pyroelectric spherical shell. J. of Thermal Stresses, 2001, vol. 24, pp. 105–120.

**[25]**Kovalenko A.D. Osnovy termouprugosti [Fundamentals of thermoelasticity]. Kiev, Naukova Dumka Publ., 1970, 307 p.

**[26]**Parton V.Z., Kudryavtsev B.A. Elektromagnitouprugost pyezoelektricheskikh i elektroprovodnykh tel [Electromagnetoelasticity of piezoelectric and electrically conductive solids]. Moscow, Nauka Publ., 1988, 470 p.

**[27]**Sneddon I.N. Fourier Transforms. McGraw-Hill Book Company, Inc., 1951 [In Russ.: Sneddon I.N. Preobrazovaniya Furye. Moscow, Inostrannaya Literatura Publ., 1955, 668 p.].

**[28]**Senitsky Y.E. Issledovanie uprugogo deformirovaniya elementov konstruktsii pri dinamicheskikh vozdeystviyakh metodom konechnykh integralnykh preobrazovaniy [Study of the elastic deformations of structural elements under dynamic influences by the method of finite integral transformations]. Saratov, Saratov University Publ., 1985, 174 p.

**[29]**Janke E., Emde F., Losch F. Tafeln Höherer Funktionen. B.G. Teubner Verlagsgesellschaft, 1960 [In Russ.: Janke E., Emde F., Losch F. Spetsialnye funktsii. Moscow, Nauka Publ., 1977, 342 p.].

**[30]**Hong C.H., Kim H.P., Choi B.Y., Han H.S., Son J.S., Ahn C.W., Jo W. Lead-free piezoceramics – Where to move on? J. of Materiomics, 2016, vol. 2 (1), pp. 1–24. https://doi.org/10.1016/j.jmat.2015.12.002

**[31]**Shlyakhin D.A., Kalmova M.A. Uncoupled problem of thermoelectroelasticity for a cylindrical shell. In: XXX Russian—Polish—Slovak Seminar Theoretical Foundation of Civil Engineering (RSP 2021), pp. 263–271. https://doi.org/10.1007/978-3-030-86001-1_31