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New insights on fractional thermoelasticity from anomalous heat conduction

基于反常热传导的分数阶热弹耦合理论

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Abstract

Anomalous heat transport of low-dimensional nanomaterial, e.g., divergent effective thermal conductivity, has been observed from both atomistic simulations and experimental studies. It is greatly urgent to establish the phenomenological anomalous thermoelastic model, and study the thermoelastic coupling due to strong heat transfer. The aim of this work is to revisit fractional wave-type thermoelastic models from the anomalous heat conductive viewpoint. Firstly, it has been suggested that anomalous heat conduction is due to the second sound, hence wave-type heat conduction is considered: the analogy between wave-type heat conduction and viscoelastic model is given, and thus the connection between Cattaneo-Vernotte and Green-Naghdi heat conductive models is clarified. Secondly, it has been recognized that the divergent thermal conductivity of one-dimensional systems satisfies the fractional order power law, therefore fractional derivative should be incorporated: Fractional order thermoelastic models based on Cattaneo-Vernotte and Green-Naghdi theories are summarized and compared, theoretically. Numerical investigations are conducted by using Laplace transform method, and the plot of thermoelastic responses vs. fractional order parameter shows: for all fractional order range [0, 1], fractional Cattaneo-Vernotte (FCV) I model and fractional Green-Naghdi (FGN) I-III models can predict anomalous thermoelastic responses, i.e., higher temperature and compressive stress than classical thermoelasticity. Furthermore, the history of temperature or stress indicates: FGN II model can predict anomalous responses for all time range. Further systematical studies are expected for Green-Naghdi model and its fractional versions to shed light on anomalous heat conduction and thermoelastic coupling, and to facilitate the applications of nanomaterials due to such anomalous behaviors.

摘要

原子模拟和实验研究表明低维纳米材料具有反常热传导特性, 建立唯象反常热弹耦合模型, 研究强热传导导致的热力耦合响应, 对微纳器件的安全稳定运行至关重要. 本文旨在从反常热传导的视角重新审视分数阶热弹耦合模型. 反常热传导由第二声效应引起, 考虑波动热传导理论, 建立了波动热传导与粘弹性理论的类比, 阐明了Cattaneo-Vernotte模型和Green-Naghdi模型的关联. 通过引入分数阶导数, 构建了基于Cattaneo-Vernotte和Green-Naghdi理论的分数阶热弹耦合模型. 数值结果表明: 对分数阶次区间[0,1], 分数阶Cattaneo-Vernotte (FCV) I模型和分数阶Green-Naghdi (FGN) I-III模型都能预测反常热弹耦合响应: 比经典热弹性理论更高的温度和应力; 温度和应力随时间的变化显示: FGN II模型在所有时间范围内都能得到反常响应. 对Green-Naghdi模型及其分数阶模型的进一步系统研究, 有助于揭示反常热传导和热弹耦合机理, 并促进纳米材料的广泛应用.

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References

  1. Z. Zhang, Y. Ouyang, Y. Cheng, J. Chen, N. Li, and G. Zhang, Size-dependent phononic thermal transport in low-dimensional nanomaterials, Phys. Rep. 860, 1 (2020).

    Article  MathSciNet  Google Scholar 

  2. S. N. Li, and B. Y. Cao, Fractional Boltzmann transport equation for anomalous heat transport and divergent thermal conductivity, Int. J. Heat Mass Transfer 137, 84 (2019).

    Article  Google Scholar 

  3. X. Gu, Y. Wei, X. Yin, B. Li, and R. Yang, Colloquium: Phononic thermal properties of two-dimensional materials, Rev. Mod. Phys. 90, 041002 (2018).

    Article  Google Scholar 

  4. R. Livi, Anomalous transport in low-dimensional systems: A pedagogical overview, Phys. A-Statist. Mech. Appl. 631, 127779 (2023).

    Article  MathSciNet  Google Scholar 

  5. A. Dhar, Heat transport in low-dimensional systems, Adv. Phys. 57, 457 (2008).

    Article  Google Scholar 

  6. B. Li, and J. Wang, Anomalous heat conduction and anomalous diffusion in one-dimensional systems, Phys. Rev. Lett. 91, 044301 (2003).

    Article  Google Scholar 

  7. S. Liu, P. Hänggi, N. Li, J. Ren, and B. Li, Anomalous heat diffusion, Phys. Rev. Lett. 112, 040601 (2014).

    Article  Google Scholar 

  8. C. W. Chang, D. Okawa, H. Garcia, A. Majumdar, and A. Zettl, Breakdown of Fourier’s law in nanotube thermal conductors, Phys. Rev. Lett. 101, 075903 (2008).

    Article  Google Scholar 

  9. V. Lee, C. H. Wu, Z. X. Lou, W. L. Lee, and C. W. Chang, Divergent and ultrahigh thermal conductivity in millimeter-long nanotubes, Phys. Rev. Lett. 118, 135901 (2017).

    Article  Google Scholar 

  10. N. Yang, G. Zhang, and B. Li, Violation of Fourier’s law and anomalous heat diffusion in silicon nanowires, Nano Today 5, 85 (2010).

    Article  Google Scholar 

  11. X. Xu, L. F. C. Pereira, Y. Wang, J. Wu, K. Zhang, X. Zhao, S. Bae, C. Tinh Bui, R. Xie, J. T. L. Thong, B. H. Hong, K. P. Loh, D. Donadio, B. Li, and B. Özyilmaz, Length-dependent thermal conductivity in suspended single-layer graphene, Nat. Commun. 5, 3689 (2014).

    Article  Google Scholar 

  12. S. N. Li, and B. Y. Cao, Anomalous heat equations based on non-Brownian descriptions, Phys. A-Statist. Mech. Appl. 526, 121141 (2019).

    Article  MathSciNet  Google Scholar 

  13. Y. J. Yu, W. Hu, and X. G. Tian, A novel generalized thermoelasticity model based on memory-dependent derivative, Int. J. Eng. Sci. 81, 123 (2014).

    Article  MathSciNet  Google Scholar 

  14. A. Cepellotti, G. Fugallo, L. Paulatto, M. Lazzeri, F. Mauri, and N. Marzari, Phonon hydrodynamics in two-dimensional materials, Nat. Commun. 6, 6400 (2015).

    Article  Google Scholar 

  15. S. Huberman, R. A. Duncan, K. Chen, B. Song, V. Chiloyan, Z. Ding, A. A. Maznev, G. Chen, and K. A. Nelson, Observation of second sound in graphite at temperatures above 100 K, Science 364, 375 (2019).

    Article  Google Scholar 

  16. V. Peshkov, Second sound in helium II, J. Phys. USSR. 8, 381 (1944).

    Google Scholar 

  17. R. H. Xia, X. G. Tian, and Y. P. Shen, MD simulation of a copper rod under thermal shock, Acta Mech. Sin. 26, 599 (2010).

    Article  Google Scholar 

  18. X. Qian, Y. Luo, Q. Chai, Y. Zhang, and L. Zhao, A thermomechanical investigation on laser ablation of aluminum alloy, Acta Mech. Solid Sin. 36, 658 (2023).

    Article  Google Scholar 

  19. X. Yunsheng, G. Yingkui, and G. Zengyuan, Experimental research on transient heat transfer in sand, Acta Mech. Sin. 12, 39 (1996).

    Article  Google Scholar 

  20. C. Catteneo, A form of heat conduction equation which eliminates the paradox of instantaneous propagation, 247, 431 (1958).

  21. P. Vernotte, Paradoxes in the continuous theory of the heat conduction, Comptes Rendus 246, 3154 (1958).

  22. A. E. Green, and P. M. Naghdi, On undamped heat waves in an elastic solid, J. Therm. Stresses 15, 253 (1992).

    Article  MathSciNet  Google Scholar 

  23. A. E. Green, and P. M. Naghdi, Thermoelasticity without energy dissipation, J. Elasticity 31, 189 (1993).

    Article  MathSciNet  Google Scholar 

  24. A. Compte, and R. Metzler, The generalized Cattaneo equation for the description of anomalous transport processes, J. Phys. A-Math. Gen. 30, 7277 (1997).

    Article  MathSciNet  Google Scholar 

  25. H. H. Sherief, A. M. A. El-Sayed, and A. M. Abd El-Latief, Fractional order theory of thermoelasticity, Int. J. Solids Struct. 47, 269 (2010).

    Article  Google Scholar 

  26. M. A. Ezzat, Magneto-thermoelasticity with thermoelectric properties and fractional derivative heat transfer, Physica B-Condensed Matter 406, 30 (2011).

    Article  Google Scholar 

  27. M. A. Ezzat, and A. S. El Karamany, Theory of fractional order in electro-thermoelasticity, Eur. J. Mech.-A Solids 30, 491 (2011).

    Article  Google Scholar 

  28. Y. J. Yu, and Z. C. Deng, Fractional order theory of Cattaneo-type thermoelasticity using new fractional derivatives, Appl. Math. Model. 87, 731 (2020).

    Article  MathSciNet  Google Scholar 

  29. H. M. Youssef, Theory of fractional order generalized thermoelasticity, J. Heat Transfer 132, 061301 (2010).

    Article  Google Scholar 

  30. A. S. El-Karamany, and M. A. Ezzat, Fractional phase-lag Green-Naghdi thermoelasticity theories, J. Therm. Stresses 40, 1063 (2017).

    Article  Google Scholar 

  31. M. A. Ezzat, and A. A. El-Bary, Unified GN model of electrothermoelasticity theories with fractional order of heat transfer, Microsyst. Technol. 24, 4965 (2018).

    Article  Google Scholar 

  32. A. Hobiny, and I. Abbas, Fractional order GN model on photothermal interaction in a semiconductor plane, Silicon 12, 1957 (2020).

    Article  Google Scholar 

  33. M. Islam, and M. Kanoria, Short-time analysis of magnetothermoelastic wave under fractional order heat conduction law, J. Therm. Stresses 38, 1217 (2015).

    Article  Google Scholar 

  34. A. Sur, and M. Kanoria, Fractional order two-temperature thermoelasticity with finite wave speed, Acta Mech. 223, 2685 (2012).

    Article  MathSciNet  Google Scholar 

  35. X. Li, Z. Xue, and X. Tian, A modified fractional order generalized bio-thermoelastic theory with temperature-dependent thermal material properties, Int. J. Therm. Sci. 132, 249 (2018).

    Article  Google Scholar 

  36. Y. Qiao, X. Wang, H. Qi, and H. Xu, Numerical simulation and parameters estimation of the time fractional dual-phase-lag heat conduction in femtosecond laser heating, Int. Commun. Heat Mass Transfer 125, 105355 (2021).

    Article  Google Scholar 

  37. Y. Li, M. Peng, T. He, X. Tian, and K. Liao, A fractional dual-phase-lag generalized thermoelastic model of ultrashort pulse laser ablation with variable thermal material properties, vaporization and plasma shielding, Int. J. Therm. Sci. 177, 107556 (2022).

    Article  Google Scholar 

  38. A. E. Abouelregal, Three-phase-lag thermoelastic heat conduction model with higher-order time-fractional derivatives, Ind. J. Phys. 94, 1949 (2020).

    Article  Google Scholar 

  39. A. E. Abouelregal, Ö. Civalek, and H. F. Oztop, Higher-order time-differential heat transfer model with three-phase lag including memory-dependent derivatives, Int. Commun. Heat Mass Transfer 128, 105649 (2021).

    Article  Google Scholar 

  40. B. Gu, T. He, and Y. Ma, Thermoelastic damping analysis in microbeam resonators considering nonlocal strain gradient based on dual-phase-lag model, Int. J. Heat Mass Transfer 180, 121771 (2021).

    Article  Google Scholar 

  41. S. Shi, T. He, and F. Jin, Thermoelastic damping analysis of size-dependent nano-resonators considering dual-phase-lag heat conduction model and surface effect, Int. J. Heat Mass Transfer 170, 120977 (2021).

    Article  Google Scholar 

  42. N. Bazarra, J. R. Fernández, and R. Quintanilla, Analysis of a Moore-Gibson-Thompson thermoelastic problem, J. Comput. Appl. Math. 382, 113058 (2021).

    Article  MathSciNet  Google Scholar 

  43. A. E. Abouelregal, I. E. Ahmed, M. E. Nasr, K. M. Khalil, A. Zakria, and F. A. Mohammed, Thermoelastic processes by a continuous heat source line in an infinite solid via Moore-Gibson-Thompson thermoelasticity, Materials 13, 4463 (2020).

    Article  Google Scholar 

  44. R. Quintanilla, Moore-Gibson-Thompson thermoelasticity with two temperatures, Appl. Eng. Sci. 1, 100006 (2020).

    Google Scholar 

  45. Y. J. Yu, X. G. Tian, and Q. L. Xiong, Nonlocal thermoelasticity based on nonlocal heat conduction and nonlocal elasticity, Eur. J. Mech.-A Solids 60, 238 (2016).

    Article  MathSciNet  Google Scholar 

  46. L. Brancik, in Programs for fast numerical inversion of Laplace transforms in MATLAB language environment: Proceedings of the 7th Conference MATLAB’99, 1999. pp. 27–39.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 11802242), and the Fundamental Research Funds for the Central Universities (Grant No. D5000230066).

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Authors and Affiliations

  1. School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi’an, 710072, China

    Ya-Jun Yu  (尉亚军), Hua Wu  (吴华) & Zi-Chen Deng  (邓子辰)

  2. MIIT Key Laboratory of Dynamics and Control of Complex Systems, Northwestern Polytechnical University, Xi’an, 710072, China

    Ya-Jun Yu  (尉亚军), Hua Wu  (吴华) & Zi-Chen Deng  (邓子辰)

  3. State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong University, Xi’an, 710049, China

    Ya-Jun Yu  (尉亚军)

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  1. Ya-Jun Yu  (尉亚军)
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Contributions

Author contributions Yajun Yu: Conceptualization, Investigation, Methodology, Validation, Writing–original draft. Hua Wu: Investigation, Methodology, Validation, Writing–review & editing. Zichen Deng: Project administration, Supervision, Writing–review & editing.

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Correspondence to Ya-Jun Yu  (尉亚军).

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Yu, YJ., Wu, H. & Deng, ZC. New insights on fractional thermoelasticity from anomalous heat conduction. Acta Mech. Sin. 40, 423419 (2024). https://doi.org/10.1007/s10409-023-23419-x

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