Two-Temperature Thermoelastic Interactions in Modified Couple Stress Beam in Context of GN Theories of Type II and III
Keywords:
Modified Couple Stress Theory, Isotropic, Euler- Bernoulli Nanobeam, Laplace Transform, Energy Dissipation, Green–Naghdi (GN) Theory, Two-TemperatureAbstract
In this study, a thermoelastic beam is analyzed using the modified couple stress theory in the framework of Green–Naghdi (GN) thermoelectricity theories Types II and III, incorporating the two-temperature model. The combined use of MCST, two-temperature theory, and GN models aims to provide a comprehensive understanding of micro-scale beam responses, linking the gap between classical mechanics and modern thermal theories The mathematical model is developed based on the Euler–Bernoulli beam assumptions and solved using the Laplace transform technique. Physical quantities such as lateral deflection, thermal moment, and average axial stress induced by heat flux are derived and computed numerically. The combined influence of the two-temperature parameter and GN theories (types II and III) on these responses is illustrated through graphical representations. Additionally, several particular cases of interest are discussed to highlight the distinctive effects under various boundary conditions.
How to cite this article:
K Harpreet, Two-Temperature Thermoelastic Interactions in Modified Couple Stress Beam in Context of GN Theories of Type II and III, J Adv Res Appl Math Stat, Vol 11, Issue 3&4 2026: Pg. No. 55-73.
DOI: https://doi.org/10.24321/2455.7021.202610
References
Abbas, I.A. (2018), “Free vibrations of nanoscale beam under two-temperature Green and Naghdi model”, Int. J. Acoust. Vib., 23(3), 289–293. https://doi.org/10.20855/ijav.2018.23.31051
Abbas, I.A., Hobiny, A., Alshehri, H., Vlase, S. and Marin, M. (2022), “Analysis of thermoelastic interaction in a polymeric orthotropic medium using the finite element method”, Polymers (Basel), 14(10), 2112. https://doi.org/10.3390/polym14102112
Abdelrahman, A.A., Abd-El-Mottaleb, H.E. and Eltaher, M.A. (2020), “On bending analysis of perforated microbeams including the microstructure effects”, Struct. Eng. Mech., 76(6), 765–779. https://doi.org/10.12989/sem.2020.76.6.765
Abouelregal, A.E. and Marin, M. (2020), “The size-dependent thermoelastic vibrations of nanobeams subjected to harmonic excitation and rectified sine wave heating”, Mathematics, 8, 1128. https://doi.org/10.3390/math8071128
Abouelregal, A.E., Rabih, M.N.A. and Megahid, S.F. (2024), “A modified couple stress model to analyze the effect of size-dependent thermal interactions in rotating nanobeams whose properties change with temperature”, Math. Mech. Solids, 29(8). https://orcid.org/0000-0003-4187-839X
Arbind, A., Reddy, J.N. and Srinivasa, A.R. (2014), “Modified couple stress-based third-order theory for nonlinear analysis of functionally graded beams”, Lat. Am. J. Solids Struct., 11(3), 459–487.
Awrejcewicz, J., Krysko, V.A., Pavlov, S.P., Zhigalov, M.V., Kalutsky, L.A. and Krysko, A.V. (2019), “Thermoelastic vibrations of a Timoshenko microbeam based on the modified couple stress theory”, Nonlinear Dyn., 99(2), 919–943. https://doi.org/10.1007/s11071-019-04976-w
Chabekh, M., Messaoudi, S.A. and Chougui, N. (2025), “Analysis of a laminated beam with dual-phase-lag thermoelasticity”, Discrete Contin. Dyn. Syst. B, 30(12), 5006–5047. https://doi.org/10.3934/dcdsb.2025095
Dai, H.L., Wang, Y.K. and Wang, L. (2015), “Nonlinear dynamics of cantilevered microbeams based on modified couple stress theory”, Int. J. Eng. Sci., 94, 103–112.
Dehrouyeh-Semnani, A.M. and Bahrami, A. (2016), “On size-dependent Timoshenko beam element based on modified couple stress theory”, Int. J. Eng. Sci., 107, 134–148.
Esen, A.A.A. and Eltaher, M.A. (2020), “Dynamics analysis of Timoshenko perforated microbeams under moving loads”, Eng. Comput., https://doi.org/10.1007/s00366-020-01212-7
Ezzat, M.A. (2011), “Magneto-thermoelasticity with thermoelectric properties and fractional derivative heat transfer”, Phys. B Condens. Matter, 406(1), 30–35.
Ezzat, M.A., Ezzat, S.M. and Alduraibi, N.S. (2021), “On size-dependent thermo-viscoelasticity theory for piezoelectric materials”, Waves Random Complex Media, 35(2), 2366–2388.
Hadian, M., Torabi, K. and Jazi, S.H. (2018), “Nonlinear vibration analysis of an elastically connected double-non-classical Timoshenko microbeam subject to moving particle based on the modified couple stress theory”, J. Braz. Soc. Mech. Sci. Eng., 42(5), 1–12.
Jalaei, M.H., Thai, H.T. and Civalek, Ö. (2022), “On viscoelastic transient response of magnetically imperfect functionally graded nanobeams”, Int. J. Eng. Sci., 172, 103629.
Jorge Silva, M.A. and Ma, T.F. (2023), “Fundamentals of thermoelasticity for curved beams”, Anal. PDE, arXiv preprint arXiv:2310.07496 [math.AP], 29 pages. https://doi.org/10.48550/arXiv.2310.07496
Kaur, I., Lata, P. and Singh, K. (2021), “Study of transversely isotropic nonlocal thermoelastic thin nano-beam resonators with multi-dual-phase-lag theory”, Arch. Appl. Mech., 91, 317–341. https://doi.org/10.1007/s00419-020-01771-7
Kaur, I., Lata, P. and Singh, K. (2022), “Thermoelastic damping in generalized simply supported piezo-thermo-elastic nanobeam”, Struct. Eng. Mech., 81, 29–37. https://doi.org/10.12989/sem.2022.81.1.0
Krysko, A.V., Awrejcewicz, J., Kutepov, I.E. and Krysko, V.A. (2015), “On a contact problem of two-layer beams coupled by boundary conditions in a temperature field”, Mech. Adv. Mater. Struct., 22(5), 468–484. https://doi.org/10.1080/01495739.2015.1015848
Kumar, R. and Devi, S. (2017) “Interactions of Thermoelastic Beam in Modified Couple Stress Theory”, Appl. Appl. Math. 12(2) ,910 – 923.
Kumar, R. and Kumar, R. (2021), “Effect of two-temperature parameter on thermoelastic vibration in micro and nano beam resonator”, Eur. J. Mech. A Solids, 89, 104310.
Lagnese, J.E., Leugering, G. and Schmidt, E.J.P.G. (1993), “Modelling of dynamic networks of thin thermoelastic beams”, Math. Methods Appl. Sci., 16(5), 327–358. https://doi.org/10.1002/mma.1670160503
Lata, P. and Singh, S. (2020), “Time harmonic interactions in nonlocal thermoelastic solid with two temperatures”, Struct. Eng. Mech., 74, 341–350.
Madjid, B.H. and Mohamed, C. (2018), “Influence of temperature on the deformation of a two-layer beam functionally graded under electrostatic loading”, J. Adv. Res. Des., 42(1).
Sahrawat, R.K., Duhan, A. and Kumar, K. (2023), “Study of vibrations in micro-scale piezothermoelastic beam resonator utilising modified couple stress theory”, Acta Mech., 234(8), 1–17. https://doi.org/10.1007/s00707-023-03575-6
Saw, G.R., Singh, S. and Chakraborty, G. (2025), “Analysis of pull-in parameters of a microelectromechanical beam with van der Waals force using modified couple stress theory”, J. Appl. Mech., 92(6), 061007.
Singh, A.K., Guha, S. and Maji, A. (2025), “Analysis of thermoelastic damping and frequency shift of nano-scale piezoelectric fiber-reinforced thermoelastic composite beam under single, dual, and three phase-lag models: A comparative approach”, Thin-Walled Struct., 211, 113134. https://doi.org/10.1016/j.tws.2025.113134
Sobhy, M. and Zenkour, A.M. (2020), “The modified couple stress model for bending of normal deformable viscoelastic nanobeams resting on visco-Pasternak foundations”, Mech. Adv. Mater. Struct., 27(7), 525–538. https://doi.org/10.1080/15376494.2018.1482579
Togun, N. and Bağdatli, S.M. (2018), “The vibration of nanobeam resting on elastic foundation using modified couple stress theory”, J. Appl. Comput. Mech., 12(4), 221–225.
Togun, N. and Bağdatli, S.M. (2024), “Application of modified couple-stress theory to nonlinear vibration analysis of nanobeam with different boundary conditions”, J. Vib. Eng. Technol., 12(10). https://doi.org/10.1007/s42417-024-01294-3
Uzun, B. and Yayli, M. (2023), “Bending analysis of a perforated microbeam with Laplace transform”, Proc. 23rd Natl. Mech. Congr. TUMTMK, 04–08 September 2023, Konya, Türkiye.
Wang, Y.-W., Zhang, X.-Y. and Li, X.-F. (2020), “Thermoelastic damping in a micro-beam based on the memory-dependent generalized thermoelasticity”, Waves Random Complex Media, 32(1), 1–18. https://doi.org/10.1080/17455030.2020.1865590
Wen, S.R., Ting, X.H., Chang, H. and Sun, J.Y. (2021), “A two-dimensional thermoelasticity solution for bimodular material beams under the combination action of thermal and mechanical loads”, Mathematics, 9, 1556. http://dx.doi.org/10.3390/math9131556
Zenkour, A.M. (2020), “Magneto-thermal shock for a fiber-reinforced anisotropic half-space due to a refined multi-dual-phase-lag model”, J. Phys. Chem. Solids, 137, 109213.
Zenkour, A.M. and Abouelregal, A.E. (2014), “Effect of harmonically varying heat on FG nanobeams in the context of a nonlocal two-temperature thermoelasticity theory”, J. Vibroeng., 16(8), 3665–3678.
Zenkour, A.M. and Abouelregal, A.E. (2014), “State-space approach for an infinite medium with a spherical cavity based upon two-temperature generalized thermoelasticity theory and fractional heat conduction”, Z. Angew. Math. Phys., 65, 149–164.
Zenkour, A.M. and Abouelregal, A.E. (2015), “Thermoelastic vibration of an axially moving microbeam subjected to sinusoidal pulse heating”, Int. J. Struct. Stab. Dyn., 15(6), 1450081.
