Finite element analysis on generalized piezothermoelastic interactions in an unbounded piezoelectric medium containing a spherical cavity
-
Ibrahim Abbas
,
Areej Almuneef
Abstract
This paper addresses the theoretical analysis of a piezothermoelastic problem involving an unbounded thermopiezoelectric medium with a spherical cavity subjected to pulse heating flux. The generalized piezo-thermo-elastic formulations of Lord and Shulman with thermal relaxation effects are used in this work. Unlike previous studies, which often consider simplified boundary conditions or steady-state thermal loading, our work incorporates generalized piezo-thermo-elastic formulations based on the Lord and Shulman model, accounting for thermal relaxation effects under dynamic thermal loading conditions. The numerical solution of the governing equations is done using the finite element approach, and temporal evolution is solved using the implicit scheme. New numerical results provide insight into the dynamic behavior of the piezoelectric medium subjected to thermal conditions. Thermal relaxation time and pulse heating flux are analyzed in their influence on the coupled thermal, mechanical and electrical fields and, thus, on the response of the system.
Funding source: Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia
Award Identifier / Grant number: PNURSP2025R742
-
Research ethics: Not applicable.
-
Informed consent: Not applicable.
-
Author contributions: The authors have accepted responsibility for the entire content of this manuscript and approved its submission.
-
Use of Large Language Models, AI and Machine Learning Tools: None declared.
-
Conflict of interest: The authors state no conflict of interest.
-
Research funding: This research work was funded by Institutional Fund Projects under grant no. (PNURSP2025R742).
-
Data availability: Not applicable.
References
[1] A. Safari and E. K. Akdoğan, “Piezoelectric and acoustic materials for transducer applications,” in Piezoelectric and Acoustic Materials for Transducer Applications, 2008, pp. 1–481.10.1007/978-0-387-76540-2Suche in Google Scholar
[2] M. A. Biot, “Thermoelasticity and irreversible thermodynamics,” J. Appl. Phys., vol. 27, no. 3, pp. 240–253, 1956. https://doi.org/10.1063/1.1722351.Suche in Google Scholar
[3] H. W. Lord and Y. Shulman, “A generalized dynamical theory of thermoelasticity,” J. Mech. Phys. Solids, vol. 15, no. 5, pp. 299–309, 1967. https://doi.org/10.1016/0022-5096(67)90024-5.Suche in Google Scholar
[4] D. S. Chandrasekharaiah, “A generalized linear thermoelasticity theory for piezoelectric media,” Acta Mech., vol. 71, nos. 1–4, pp. 39–49, 1988. https://doi.org/10.1007/bf01173936.Suche in Google Scholar
[5] N. Cheng and C. Sun, “Wave propagation in two− layered piezoelectric plates,” J. Acoust. Soc. Am., vol. 57, no. 3, pp. 632–638, 1975.10.1121/1.380479Suche in Google Scholar
[6] S. Biswas, “Surface waves in piezothermoelastic transversely isotropic layer lying over piezothermoelastic transversely isotropic half-space,” Acta Mech., vol. 232, no. 2, pp. 373–387, 2021. https://doi.org/10.1007/s00707-020-02848-8.Suche in Google Scholar
[7] S. Guha and A. K. Singh, “Plane wave reflection/transmission in imperfectly bonded initially stressed rotating piezothermoelastic fiber-reinforced composite half-spaces,” Eur. J. Mech. A/Solids, vol. 88, 2021, Art. no. 104242. https://doi.org/10.1016/j.euromechsol.2021.104242.Suche in Google Scholar
[8] M. Ragab, S. M. Abo-Dahab, A. E. Abouelregal, and A. A. Kilany, “A thermoelastic piezoelectric fixed rod exposed to an axial moving heat source via a dual‐phase‐lag model,” Complexity, vol. 2021, no. 1, pp. 1–11, 2021. https://doi.org/10.1155/2021/5547566.Suche in Google Scholar
[9] Y. Ma and T. He, “Dynamic response of a generalized piezoelectric-thermoelastic problem under fractional order theory of thermoelasticity,” Mech. Adv. Mater. Struct., vol. 23, no. 10, pp. 1173–1180, 2016. https://doi.org/10.1080/15376494.2015.1068397.Suche in Google Scholar
[10] A. H. Akbarzadeh, M. H. Babaei, and Z. T. Chen, “Thermopiezoelectric analysis of a functionally graded piezoelectric medium,” Int. J. Appl. Mech., vol. 03, no. 01, pp. 47–68, 2012. https://doi.org/10.1142/s1758825111000865.Suche in Google Scholar
[11] T. He, L. Cao, and S. Li, “Dynamic response of a piezoelectric rod with thermal relaxation,” J. Sound Vib., vol. 306, nos. 3–5, pp. 897–907, 2007, https://doi.org/10.1016/j.jsv.2007.06.018.Suche in Google Scholar
[12] G. O. Putra, D.-H. Hwang, and J.-H. Han, “Effective elastic, thermoelastic, and piezoelectric properties of braided composites using equivalent laminate modeling,” Adv. Compos. Mater., vol. 30, no. 4, pp. 338–364, 2020. https://doi.org/10.1080/09243046.2020.1815133.Suche in Google Scholar
[13] C. Hwu, “Piezoelectric and magneto-electro-elastic materials,” in Anisotropic Elasticity with Matlab, C. Hwu, Ed., Cham, Springer International Publishing, 2021, pp. 265–287.10.1007/978-3-030-66676-7_11Suche in Google Scholar
[14] R. Tiwari, R. Kumar, and A. E. Abouelregal, “Analysis of a magneto-thermoelastic problem in a piezoelastic medium using the non-local memory-dependent heat conduction theory involving three phase lags,” Mech. Time-Depend. Mater., vol. 26, no. 2, pp. 271–287, 2022. https://doi.org/10.1007/s11043-021-09487-z.Suche in Google Scholar
[15] Y. J. Yu and Z. C. Deng, “Fractional order thermoelasticity for piezoelectric materials,” Fractals, vol. 29, no. 04, p. 2150082, 2021. https://doi.org/10.1142/s0218348x21500821.Suche in Google Scholar
[16] T. Saeed, “Hybrid finite element method to thermo-elastic interactions in a piezo-thermo-elastic medium under a fractional time derivative model,” Mathematics, vol. 10, no. 4, p. 650, 2022. https://doi.org/10.3390/math10040650.Suche in Google Scholar
[17] M. Marin, A. Öchsner, S. Vlase, D. O. Grigorescu, and I. Tuns, “Some results on eigenvalue problems in the theory of piezoelectric porous dipolar bodies,” Continuum Mech. Thermodyn., vol. 35, no. 5, pp. 1969–1979, 2023. https://doi.org/10.1007/s00161-023-01220-0.Suche in Google Scholar
[18] I. A. Abbas and R. Kumar, “Deformation due to thermal source in micropolar generalized thermoelastic half-space by finite element method,” J. Comput. Theor. Nanosci., vol. 11, no. 1, pp. 185–190, 2014. https://doi.org/10.1166/jctn.2014.3335.Suche in Google Scholar
[19] I. A. Abbas, R. Kumar, and V. Chawla, “Response of thermal source in a transversely isotropic thermoelastic half-space with mass diffusion by using a finite element method,” Chin. Phys. B, vol. 21, no. 8, p. 084601, 2012. https://doi.org/10.1088/1674-1056/21/8/084601.Suche in Google Scholar
[20] V. Stelmashchuk and H. Shynkarenko, “Stability of hybrid time integration scheme for Lord–Shulman thermopiezoelectricity,” Results Appl. Math., vol. 23, 2024, Art. no. 100467. https://doi.org/10.1016/j.rinam.2024.100467.Suche in Google Scholar
[21] A. M. Zenkour, “Piezoelectric behavior of an inhomogeneous hollow cylinder with thermal gradient,” Int. J. Thermophys., vol. 33, no. 7, pp. 1288–1301, 2012. https://doi.org/10.1007/s10765-012-1248-3.Suche in Google Scholar
[22] A. M. Zenkour, “Exact solution of thermal stress problem of an inhomogeneous hygrothermal piezoelectric hollow cylinder,” Appl. Math. Model., vol. 38, no. 24, pp. 6133–6143, 2014. https://doi.org/10.1016/j.apm.2014.05.028.Suche in Google Scholar
[23] F. Ebrahimi and M. R. Barati, “Vibration analysis of smart piezoelectrically actuated nanobeams subjected to magneto-electrical field in thermal environment,” J. Vib. Control, vol. 24, no. 3, pp. 549–564, 2016. https://doi.org/10.1177/1077546316646239.Suche in Google Scholar
[24] M. I. A. Othman and E. A. A. Ahmed, “Effect of gravity field on piezothermoelastic medium with three theories,” J. Therm. Stresses, vol. 39, no. 4, pp. 474–486, 2016. https://doi.org/10.1080/01495739.2016.1152136.Suche in Google Scholar
[25] M. Pakdaman and Y. Tadi Beni, “Size-dependent generalized piezothermoelasticity of microlayer,” J. Appl. Comput. Mech., vol. 11, no. 1, pp. 223–238, 2025. https://doi.org/10.22055/jacm.2024.46393.4510.Suche in Google Scholar
[26] A. E. Abouelregal, S. S. Alsaeed, H. M. Sedighi, M. E. Elzayady, and A. R. Hafezi, “A comprehensive study on heterogeneous media with spherical cavity: the higher-order fractional triple-phase-lag thermoelasticity with local kernels,” Int. J. Appl. Mech., vol. 16, no. 10, 2024, Art. no. 2450120. https://doi.org/10.1142/S1758825124501205.Suche in Google Scholar
[27] I. A. Abbas, “Generalized thermoelastic interaction in functional graded material with fractional order three-phase lag heat transfer,” J. Cent. South Univ., vol. 22, no. 5, pp. 1606–1613, 2015. https://doi.org/10.1007/s11771-015-2677-5.Suche in Google Scholar
[28] E. Carrera, A. E. Abouelregal, I. A. Abbas, and A. M. Zenkour, “Vibrational analysis for an axially moving microbeam with two temperatures,” J. Therm. Stresses, vol. 38, no. 6, pp. 569–590, 2015. https://doi.org/10.1080/01495739.2015.1015837.Suche in Google Scholar
[29] I. Abbas, T. Saeed, and M. Alhothuali, “Hyperbolic two-temperature photo-thermal interaction in a semiconductor medium with a cylindrical cavity,” Silicon, vol. 13, no. 6, pp. 1871–1878, 2021. https://doi.org/10.1007/s12633-020-00570-7.Suche in Google Scholar
[30] R. Tiwari, A. E. Abouelregal, O. N. Shivay, and S. F. Megahid, “Thermoelastic vibrations in electro-mechanical resonators based on rotating microbeams exposed to laser heat under generalized thermoelasticity with three relaxation times,” Mech. Time-Depend. Mater., vol. 28, no. 2, pp. 423–447, 2022. https://doi.org/10.1007/s11043-022-09578-5.Suche in Google Scholar
[31] F. S. Alzahrani and I. A. Abbas, “Photo-thermal interactions in a semiconducting media with a spherical cavity under hyperbolic two-temperature model,” Mathematics, vol. 8, no. 4, p. 585, 2020. https://doi.org/10.3390/math8040585.Suche in Google Scholar
[32] K. Lotfy and R. S. Tantawi, “Photo-thermal-elastic interaction in a functionally graded material (FGM) and magnetic field,” Silicon, vol. 12, no. 2, pp. 295–303, 2019. https://doi.org/10.1007/s12633-019-00125-5.Suche in Google Scholar
[33] I. A. Abbas, “A GN model for thermoelastic interaction in a microscale beam subjected to a moving heat source,” Acta Mech., vol. 226, no. 8, pp. 2527–2536, 2015. https://doi.org/10.1007/s00707-015-1340-4.Suche in Google Scholar
[34] T. He and L. Cao, “Generalized thermoelastic responses of a piezoelectric rod subjected to a moving heat source,” Key Eng. Mater., vols. 353–358, pp. 1149–1152, 2007, https://doi.org/10.4028/www.scientific.net/kem.353-358.1149.Suche in Google Scholar
[35] I. A. Abbas and H. M. Youssef, “Two-dimensional fractional order generalized thermoelastic porous material,” Lat. Am. J. Solids Struct., vol. 12, no. 7, pp. 1415–1431, 2015. https://doi.org/10.1590/1679-78251584.Suche in Google Scholar
[36] I. Abbas, A. Hobiny, and A. El-Bary, “The effects of fractional time derivatives in bioheat conduction technique on tumor thermal therapy,” J. Non-Equilib. Thermodyn., vol. 49, no. 1, pp. 61–72, 2024. https://doi.org/10.1515/jnet-2023-0065.Suche in Google Scholar
[37] Z. Alqahtani and I. Abbas, “Analytical solutions for nonequilibrium bioheat transfer in tumor during magnetic nanoparticles hyperthermia,” J. Non-Equilib. Thermodyn., vol. 49, no. 4, pp. 529–542, 2024. https://doi.org/10.1515/jnet-2024-0035.Suche in Google Scholar
[38] Z. Alqahtani, I. Abbas, and A. A. El-Bary, “Generalized piezothermoelastic interactions in a piezoelectric rod subjected to pulse heat flux,” J. Non-Equilib. Thermodyn., vol. 50, no. 1, pp. 173–184, 2025. https://doi.org/10.1515/jnet-2024-0077.Suche in Google Scholar
[39] P. Lata, “Thermomechanical interactions in a transversely isotropic thermoelastic media with diffusion due to inclined load,” Struct. Eng. Mech., vol. 90, no. 3, pp. 263–272, 2024. https://doi.org/10.12989/sem.2024.90.3.263.Suche in Google Scholar
[40] S. Singh and P. Lata, “Time harmonic interactions in a nonlocal isotropic thermoelastic thick circular plate without energy dissipation,” Adv. Mater. Res. (South Korea), vol. 13, no. 5, pp. 417–429, 2024. https://doi.org/10.12989/amr.2024.13.5.417.Suche in Google Scholar
[41] P. Lata and H. Kaur, “Stoneley wave propagation in transversely isotropic thermoelastic media using new modified couple stress theory and two-temperature theory,” Coupled Syst. Mech., vol. 13, no. 5, pp. 395–409, 2024. https://doi.org/10.12989/csm.2024.13.5.395.Suche in Google Scholar
[42] P. Lata and H. Kaur, “Interactions in transversely isotropic new modified couple stress solid due to Hall current, rotation, inclined load with energy dissipation,” Coupled Syst. Mech., vol. 13, no. 1, pp. 21–41, 2024. https://doi.org/10.12989/csm.2024.13.1.021.Suche in Google Scholar
[43] H. M. Youssef and A. A. El-Bary, “Generalized thermoelastic infinite layer subjected to ramp-type thermal and mechanical loading under three theories—state space approach,” J. Therm. Stresses, vol. 32, no. 12, pp. 1293–1309, 2009. https://doi.org/10.1080/01495730903249276.Suche in Google Scholar
[44] M. A. Ezzat and A. A. El-Bary, “Two-temperature theory of magneto-thermo-viscoelasticity with fractional derivative and integral orders heat transfer,” J. Electromagn. Waves Appl., vol. 28, no. 16, pp. 1985–2004, 2014. https://doi.org/10.1080/09205071.2014.953639.Suche in Google Scholar
[45] K. Lotfy, A. El-Bary, W. Hassan, and M. Ahmed, “Hall current influence of microtemperature magneto-elastic semiconductor material,” Superlattices Microstruct., vol. 139, 2020, Art. no. 106428. https://doi.org/10.1016/j.spmi.2020.106428.Suche in Google Scholar
[46] K. Lotfy, A. El-Bary, W. Hassan, A. Alharbi, and M. Almatrafi, “Electromagnetic and Thomson effects during photothermal transport process of a rotator semiconductor medium under hydrostatic initial stress,” Results Phys., vol. 16, 2020, Art. no. 102983. https://doi.org/10.1016/j.rinp.2020.102983.Suche in Google Scholar
[47] H. M. Youssef and A. A. El-Bary, “Two-temperature generalized thermoelasticity with variable thermal conductivity,” J. Therm. Stresses, vol. 33, no. 3, pp. 187–201, 2010. https://doi.org/10.1080/01495730903454793.Suche in Google Scholar
[48] S. M. H. Jani, Y. Kiani, and Y. Tadi Beni, “Generalized piezothermoelasticity of hollow spheres under thermal shock using Lord–Shulman theory,” J. Therm. Stresses, vol. 47, no. 3, pp. 347–362, 2024. https://doi.org/10.1080/01495739.2023.2277215.Suche in Google Scholar
[49] I. A. Abbas and M. I. A. Othman, “Plane waves in generalized thermo-microstretch elastic solid with thermal relaxation using finite element method,” Int. J. Thermophys., vol. 33, no. 12, pp. 2407–2423, 2012. https://doi.org/10.1007/s10765-012-1340-8.Suche in Google Scholar
[50] S. M. N. Mehrian, A. Nazari, and M. H. Naei, “Coupled thermoelasticity analysis of annular laminate disk using laplace transform and galerkin finite element method,” Appl. Mech. Mater., vol. 656, pp. 298–304, 2014, https://doi.org/10.4028/www.scientific.net/amm.656.298.Suche in Google Scholar
[51] J. P. Carter and J. R. Booker, “Finite element analysis of coupled thermoelasticity,” Comput. Struct., vol. 31, no. 1, pp. 73–80, 1989. https://doi.org/10.1016/0045-7949(89)90169-7.Suche in Google Scholar
[52] T. Saeed and I. Abbas, “Finite element analyses of nonlinear DPL bioheat model in spherical tissues using experimental data,” Mech. Based Des. Struct. Mach., vol. 50, no. 4, pp. 1287–1297, 2022. https://doi.org/10.1080/15397734.2020.1749068.Suche in Google Scholar
[53] M. Marin, A. Hobiny, and I. Abbas, “Finite element analysis of nonlinear bioheat model in skin tissue due to external thermal sources,” Mathematics, vol. 9, no. 13, p. 1459, 2021. https://doi.org/10.3390/math9131459.Suche in Google Scholar
[54] X. Tian, Y. Shen, and C. Chen, “A direct finite element method study of generalized thermoelastic problems,” Int. J. Solids Struct., vol. 43, nos. 7–8, pp. 2050–2063, 2006. https://doi.org/10.1016/j.ijsolstr.2005.06.071.Suche in Google Scholar
[55] X. Tian, J. Zhang, Y. Shen, and T. J. Lu, “Finite element method for generalized piezothermoelastic problems,” Int. J. Solids Struct., vol. 44, nos. 18–19, pp. 6330–6339, 2007. https://doi.org/10.1016/j.ijsolstr.2007.02.035.Suche in Google Scholar
[56] I. A. Abbas and M. I. A. Othman, “Generalized thermoelsticity of the thermal shock problem in an isotropic hollow cylinder and temperature dependent elastic moduli,” Chin. Phys. B, vol. 21, no. 1, p. 014601, 2012. https://doi.org/10.1088/1674-1056/21/1/014601.Suche in Google Scholar
[57] F. Kheibari, Y. Tadi Beni, and Y. Kiani, “Lord-Shulman based generalized thermoelasticity of piezoelectric layer using finite element method,” Struct. Eng. Mech., vol. 92, no. 1, pp. 81–88, 2024. https://doi.org/10.12989/sem.2024.92.1.081.Suche in Google Scholar
[58] A. H. Akbarzadeh, M. H. Babaei, and Z. T. Chen, “Coupled thermopiezoelectric behaviour of a one-dimensional functionally graded piezoelectric medium based on C–T theory,” Proc. Inst. Mech. Eng., Part C, vol. 225, no. 11, pp. 2537–2551, 2011. https://doi.org/10.1177/0954406211406954.Suche in Google Scholar
[59] M. H. Babaei and Z. T. Chen, “The transient coupled thermo-piezoelectric response of a functionally graded piezoelectric hollow cylinder to dynamic loadings,” Proc. R. Soc. A, vol. 466, no. 2116, pp. 1077–1091, 2010. https://doi.org/10.1098/rspa.2009.0543.Suche in Google Scholar
© 2025 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Original Research Articles
- Heat transfer at nano-scale and boundary conditions: a comparison between the Guyer-Krumhansl model and the Thermomass theory
- Exergy-based efficient ecological-function optimization for endoreversible Carnot refrigerators
- Effect of depositional nanoparticles on heat transfer at the solid–liquid interface using molecular dynamics simulations
- Optimization of injection parameters, and ethanol shares for cottonseed biodiesel fuel in diesel engine utilizing artificial neural network (ANN) and taguchi grey relation analysis (GRA)
- A general relativistic kinetic theory approach to linear transport in generic hydrodynamic frame
- Asymmetric quantum harmonic Otto engine under hot squeezed thermal reservoir
- Approaches of finite-time thermodynamics in conceptual design of heat exchange systems
- Thermal transport in a silicon/diamond micro-flake with quantum dots inserts
- Finite element analysis on generalized piezothermoelastic interactions in an unbounded piezoelectric medium containing a spherical cavity
Artikel in diesem Heft
- Frontmatter
- Original Research Articles
- Heat transfer at nano-scale and boundary conditions: a comparison between the Guyer-Krumhansl model and the Thermomass theory
- Exergy-based efficient ecological-function optimization for endoreversible Carnot refrigerators
- Effect of depositional nanoparticles on heat transfer at the solid–liquid interface using molecular dynamics simulations
- Optimization of injection parameters, and ethanol shares for cottonseed biodiesel fuel in diesel engine utilizing artificial neural network (ANN) and taguchi grey relation analysis (GRA)
- A general relativistic kinetic theory approach to linear transport in generic hydrodynamic frame
- Asymmetric quantum harmonic Otto engine under hot squeezed thermal reservoir
- Approaches of finite-time thermodynamics in conceptual design of heat exchange systems
- Thermal transport in a silicon/diamond micro-flake with quantum dots inserts
- Finite element analysis on generalized piezothermoelastic interactions in an unbounded piezoelectric medium containing a spherical cavity
