Abstract
This study provides an analytical investigation of the thermomechanical behavior of biological skin tissue subjected to harmonic thermal loading within the framework of four thermoelastic theories. The four employed thermoelastic theories, namely the classical dynamic coupled theory (CDC), the Lord–Shulman (LS) theory, the dual-phase-lag (DPL) theory, and the nonlocal dual-phase-lag (NLDPL) theory, are utilised to represent various heat conduction mechanisms. The governing equations are derived for skin tissues and solved using the normal mode technique in conjunction with an eigenvalue approach. Numerical simulations are conducted to analyze the distributions of temperature, displacement, and stress fields, with the results illustrated through two- and three-dimensional graphical representations. The effects of angular frequency and the nonlocal parameter on the thermomechanical response are examined in detail. A comparative evaluation of the four thermoelastic theories (CDC, LS, DPL, and NLDPL) highlights their respective capabilities under harmonic heating conditions. The findings offer valuable insights into the behavior of skin tissues under varying conditions. These results may significantly contribute to the advancement of treatments such as hyperthermia therapy and laser surgery, thereby potentially improving patient care.
Similar content being viewed by others
Data availability
All data generated or analysed in this study are included in the article. The computational work presented has been carried out using MATLAB (R2021a).
References
Pennes, H. H. Analysis of tissue and arterial blood temperatures in the resting human forearm. J. Appl. Physiol. 1(2), 93–122. https://doi.org/10.1152/jappl.1948.1.2.93 (1948).
Biot, M. A. Thermoelasticity and irreversible thermodynamics. J. Appl. Phys. 27(3), 240–253. https://doi.org/10.1063/1.1722351 (1956).
Lord, H. W. & Shulman, Y. A generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids 15(5), 299–309. https://doi.org/10.1016/0022-5096(67)90024-5 (1967).
Choudhuri, S. R. On a thermoelastic three-phase-lag model. J. Therm. Stress. 30(3), 231–238. https://doi.org/10.1080/01495730601130919 (2007).
Tzou, D. Y. A unified field approach for heat conduction from macro-to micro-scales. J. Heat Transfer 117(1), 8–16. https://doi.org/10.1115/1.2822329 (1995).
Abouelregal, A. E., Civalek, Ö., Akgöz, B., Foul, A. & Askar, S. S. Analysis of thermoelastic behavior of porous cylinders with voids via a nonlocal space-time elastic approach and Caputo-tempered fractional heat conduction. Mech. Time-Depend. Mater. 29(2), 1–32. https://doi.org/10.1007/s11043-025-09770-3 (2025).
Gupta, M. & Mukhopadhyay, S. A study on generalized thermoelasticity theory based on non-local heat conduction model with dual-phase-lag. J. Therm. Stress. 42(9), 1123–1135. https://doi.org/10.1080/01495739.2019.1614503 (2019).
Eringen, A. C. Theory of nonlocal thermoelasticity. Int. J. Eng. Sci. 12(12), 1063–1077. https://doi.org/10.1016/0020-7225(74)90033-0 (1974).
Abouelregal, A. E., Akgöz, B. & Civalek, Ö. Nonlocal thermoelastic vibration of a solid medium subjected to a pulsed heat flux via Caputo-Fabrizio fractional derivative heat conduction. Appl. Phys. A 128(8), 660. https://doi.org/10.1007/s00339-022-05786-5 (2022).
Kumar, D., Singh, S. & Rai, K. N. Analysis of classical Fourier, SPL and DPL heat transfer model in biological tissues in the presence of metabolic and external heat source. Heat Mass Transf. 52(6), 1089–1107. https://doi.org/10.1007/s00231-015-1617-0 (2016).
Hobiny, A. & Abbas, I. The effect of fractional derivatives on thermo-mechanical interaction in biological tissues during hyperthermia treatment using eigenvalues approach. Fractal Fract. 7(6), 432. https://doi.org/10.3390/fractalfract7060432 (2023).
Zenkour, A. M., Saeed, T. & Al-Raezah, A. A. A 1D thermoelastic response of skin tissue due to ramp-type heating via a fractional-order Lord-Shulman model. J. Comput. Appl. Mech. 54(3), 365–377. https://doi.org/10.22059/jcamech.2023.364796.871 (2023).
El-Sapa, S., El-Bary, A. A., Albalawi, W. & Atef, H. M. Modelling Pennes’ bioheat transfer equation in thermoelasticity with one relaxation time. J. Electromagn. Waves Appl. 38(1), 105–121. https://doi.org/10.1080/09205071.2023.2272612 (2024).
Bajaj, S., Shrivastav, A. K., Somvanshi, A. & Saini, G. L. Effect of gravity and variable thermal conductivity in a thermoelastic half space with dual phase lag model. Sci. Rep. 15, 44362. https://doi.org/10.1038/s41598-025-28050-1 (2025).
Das, B., Islam, N. & Lahiri, A. Study of non-local thermoelasticity of a rectangular plate. J. Therm. Stresses 48(2), 186–208. https://doi.org/10.1080/01495739.2025.2466099 (2025a).
Megahid, S. F. Two-dimensional biothermomechanical effects in a layer of skin tissue exposed to variable thermal loading using a fourth-order MGT model. Sci. Rep. 15(1), 1–21. https://doi.org/10.1038/s41598-025-01745-1 (2025).
Hu, Y., Zhang, X. Y. & Li, X. F. Thermoelastic response of skin using time-fractional dual-phase-lag bioheat heat transfer equation. J. Therm. Stresses 45(7), 597–615. https://doi.org/10.1080/01495739.2022.2078452 (2022).
Marin, M., Hobiny, A. & Abbas, I. Finite element analysis of nonlinear bioheat model in skin tissue due to external thermal sources. Mathematics 9(13), 1459. https://doi.org/10.3390/math9131459 (2021).
Alqahtani, Z., Abbas, I. A., El-Bary, A. A. & Almuneef, A. Analytical solutions of thermomechanical interaction in living tissues under dual phase-lag model. Indian J. Phys. 98(14), 4663–4669. https://doi.org/10.1007/s12648-024-03245-w (2024).
Abouelregal, A.E., Civalek, O. & Akgoz, B. A size-dependent non-fourier heat conduction model for magneto-thermoelastic vibration response of nanosystems, 11(2), 344–357 (2025) https://doi.org/10.22055/jacm.2024.46746.4584.
Abbas, I. A., El-Bary, A. A. & Mohamed, A. O. Generalized thermomechanical interaction in two-dimensional skin tissue using eigenvalues approach. J. Therm. Biol. 119, 103777. https://doi.org/10.1016/j.jtherbio.2023.103777 (2024).
Zhang, Q., Sun, Y. & Yang, J. Bio-heat response of skin tissue based on three-phase-lag model. Sci. Rep. 10(1), 16421. https://doi.org/10.1038/s41598-020-73590-3 (2020).
Zakria, A. et al. Response of nonlocal thermoelastic nanobeams supported by Pasternak foundations to the effect of generalized fractional theory with three-phase lags. Sci. Rep. 15(1), 22317. https://doi.org/10.1038/s41598-025-00987-3 (2025).
Parmar, P., Karmakar, S., Lahiri, A. & Sarkar, S. P. Study of generalized two-dimensional bioheat problem in the context of memory-dependent derivative. J. Therm. Biol. 129, 104107. https://doi.org/10.1016/j.jtherbio.2025.104107 (2025).
Bera, A., Dutta, S., Misra, J. C. & Shit, G. C. Computational modeling of the effect of blood flow and dual phase lag on tissue temperature during tumor treatment by magnetic hyperthermia. Math. Comput. Simul. 188, 389–403. https://doi.org/10.1016/j.matcom.2021.04.020 (2021).
Abbas, I., Hobiny, A. & El-Bary, A. Numerical solutions of nonlocal heat conduction technique in tumor thermal therapy. Acta Mech. 235(4), 1865–1875. https://doi.org/10.1007/s00707-023-03803-z (2024).
Islam, N., Das, B., Shit, G. C. & Lahiri, A. Thermoelastic and electromagnetic effects in a semiconducting medium. Acta Mech. 236, 2171–2191. https://doi.org/10.1007/s00707-025-04267-z (2025).
Das, B., Islam, N. & Lahiri, A. Analytical study of micropolar thermoelastic rectangular plate under three theories. J. Therm. Stresses 47(4), 521–536. https://doi.org/10.1080/01495739.2024.2314062 (2024).
Das, B. & Lahiri, A. Generalized magnetothermoelasticity for isotropic media. J. Therm. Stresses 38(2), 210–228. https://doi.org/10.1080/01495739.2014.985564 (2015).
Das, B., Islam, N. & Lahiri, A. Comparative thermoelastic analysis of semiconductors with an external heat source under three theories. Sci. Rep. 15(1), 40120. https://doi.org/10.1038/s41598-025-23984-y (2025).
Zenkour, A. M., Saeed, T. & Alnefaie, K. M. Analysis of the bio-thermoelasticity response of biological tissues subjected to harmonic heating using a refined Green-Lindsay model. J. Comput. Appl. Mech. 54(4), 588–606. https://doi.org/10.22059/jcamech.2023.366508.889 (2023).
Acknowledgements
The authors express gratitude to the reviewers for their suggestions, which have improved the current work.
Author information
Authors and Affiliations
Contributions
All authors reviewed the manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
About this article
Cite this article
Islam, N., Das, B. & Lahiri, A. Theoretical analysis of thermomechanical response for biological skin tissues. Sci Rep (2026). https://doi.org/10.1038/s41598-026-41406-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s41598-026-41406-5
