Temperature-Rate Dependence Thermoelasticity Theory with Memory-Dependent Derivative: Stability and Uniqueness
Publication: Journal of Engineering Mechanics
Volume 147, Issue 3
Abstract
This article discusses the stability analysis of thermal signals of a thermodynamic consistent model including temperature-rate dependence thermoelasticity theory (Green-Lindsay) with a memory-dependent derivative (MDD). A unifying approach (an extension of Lyapunov’s original method to the stability theory developed by Zubov together with Korn’s inequality in elasticity under homogeneous boundary condition on displacement components) is employed to characterize the stability of the present thermoelastic system. In continuation, a uniqueness of the solutions of the present thermoelastic system is presented as a corollary of the stability theorem, and corresponding results in the absence of MDD are also mentioned as special cases. Finally, based on theoretical importance and understanding, with the help of an analogy between the homogeneous and nonhomogeneous boundary conditions on the displacement components, an open mathematical problem as alternate to the employed unifying approach is proposed.
Get full access to this article
View all available purchase options and get full access to this article.
Data Availability Statement
No data, models, or code were generated or used during the research.
Acknowledgments
The author expresses sincere thanks and gratitude to CSIR New Delhi for financial support of the research work.
References
Abeyaratne, R., and J. K. Knowles. 1999. “On the stability of thermoelastic materials.” J. Elast. 53 (3): 199–213. https://doi.org/10.1023/A:1007513631783.
Alfutov, N. A. 2000. Stability of elastic structures. Berlin: Springer.
Bachher, M. 2019. “Plane harmonic waves in thermoelastic materials with a memory-dependentderivative.” J. Appl. Mech. Tech. Phys. 60 (1): 123–131. https://doi.org/10.1134/S0021894419010152.
Biot, M. A. 1956. “Thermoelasticity and irreversible thermodynamics.” J. Appl. Phys. 27 (3): 240–253. https://doi.org/10.1063/1.1722351.
Bolotin, V. V. 1963. Nonconservative problems of the theory of elastic stability. London: Pergamon.
Chandrasekharaiah, D. S. 1998. “Hyperbolic thermoelasticity: A review of recent literature.” Appl. Mech. Rev. 51 (12): 705–729. https://doi.org/10.1115/1.3098984.
Choudhuri, S. K. R. 2007. “On a thermoelastic three-phase-lag model.” J. Therm. Stresses 30 (3): 231–238. https://doi.org/10.1080/01495730601130919.
Dafermos, C. M. 1968. “On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity.” Arch. Rational Mech. Anal. 29 (4): 241–271. https://doi.org/10.1007/BF00276727.
Dym, C. L. 1974. Stability theory and its application to structural mechanics. Leyden, IL: Noordhoff International.
El-Karamany, A. S., and M. A. Ezzat. 2015. “Modified Fourier’s law with time-delay and kernel function: Application in thermoelasticity.” J. Therm. Stresses 38 (7): 811–834. https://doi.org/10.1080/01495739.2015.1040309.
El-Karamany, A. S., and M. A. Ezzat. 2016. “Thermoelastic diffusion with memory-dependent derivative.” J. Therm. Stresses 39 (9): 1035–1050. https://doi.org/10.1080/01495739.2016.1192847.
Ericksen, J. L. 1966. “A thermo-kinetic view of elastic stability theory.” Int. J. Solids Struct. 2 (4): 573–580. https://doi.org/10.1016/0020-7683(66)90039-4.
Ezzat, M. A., A. S. El-Karamany, and A. A. El-Bary. 2014. “Generalized thermo-viscoelasticity with memory-dependent derivatives.” Int. J. Mech. Sci. 89 (Dec): 470–475. https://doi.org/10.1016/j.ijmecsci.2014.10.006.
Ezzat, M. A., A. S. El-Karamany, and A. A. El-Bary. 2016. “Electro-thermoelasticity theory with memory-dependent derivative heat transfer.” Int. J. Eng. Sci. 99 (Feb): 22–38. https://doi.org/10.1016/j.ijengsci.2015.10.011.
Ezzat, M. A., A. S. El-Karamany, and A. A. El-Bary. 2017. “On dual-phase-lag thermoelasticity theory with memory-dependent derivative.” Mech. Adv. Mater. Struct. 24 (11): 908–916. https://doi.org/10.1080/15376494.2016.1196793.
Green, A. E., and K. A. Lindsay. 1972. “Thermoelasticity.” J. Elast. 2 (1): 1–7. https://doi.org/10.1007/BF00045689.
Green, A. E., and P. M. Naghdi. 1991. “A re-examination of the basic postulates of thermomechanics.” Proc. R. Soc. London, Ser. A 432 (1885): 171–194. https://doi.org/10.1098/rspa.1991.0012.
Green, A. E., and P. M. Naghdi. 1992. “On undamped heat waves in an elastic solid.” J. Therm. Stresses 15 (2): 253–264. https://doi.org/10.1080/01495739208946136.
Green, A. E., and P. M. Naghdi. 1993. “Thermoelasticity without energy dissipation.” J. Elast. 31 (3): 189–208. https://doi.org/10.1007/BF00044969.
Horgan, C. O. 1995. “Korn’s inequalities and their applications in continuum mechanics.” SIAM Rev. 37 (4): 491–511. https://doi.org/10.1137/1037123.
Knops, R. J., and E. W. Wilkes. 1966. “On Movchan’s theorems for stability of continuous system.” Int. J. Eng. Sci. 4 (4): 303–329. https://doi.org/10.1016/0020-7225(66)90034-6.
Leipholz, H. 1980. Stability of elastic system. Alphen aan den Rinn, Netherlands: Sijthoff and Noordhoff.
Li, Y., and T. He. 2019. “A generalized thermoelastic diffusion problem with memory-dependent derivative.” Math. Mech. Solids 24 (5): 1438–1462. https://doi.org/10.1177/1081286518797988.
Li, Y., P. Zhang, C. Li, and T. He. 2019. “Fractional order and memory-dependent analysis to the dynamic response of a bi- layered structure due to laser pulse heating.” Int. J. Heat Mass Transfer 144 (Dec): 118664. https://doi.org/10.1016/j.ijheatmasstransfer.2019.118664.
Lord, H. W., and Y. Shulman. 1967. “A generalized dynamical theory of thermoelasticity.” J. Mech. Phys. Solids 15 (5): 299–309. https://doi.org/10.1016/0022-5096(67)90024-5.
Lyapunov, A. M. 1992. “The general problem of the stability of motion.” Int. J. Control 55 (3): 531–534. https://doi.org/10.1080/00207179208934253.
Rahimi, Z., G. Rezazadah, W. Sumelka, and X. J. Yang. 2017. “A study of critical point instability of micro and nano beams under a distributed variable-pressure force in the framework of the inhomogeneous non-linear nonlocal theory.” Arch. Mech. 69 (6): 413–433.
Sarkar, I., and B. Mukhopadhyay. 2019. “A domain of influence theorem for generalized thermoelasticity with memory-dependent derivative.” J. Therm. Stresses 42 (11): 1447–1457. https://doi.org/10.1080/01495739.2019.1642169.
Sarkar, I., and B. Mukhopadhyay. 2020a. “Generalized thermo-viscoelasticity with memory-dependent derivative: Uniqueness and reciprocity.” Arch. Appl. Mech. 1–13. https://doi.org/10.1007/s00419-020-01799-9.
Sarkar, I., and B. Mukhopadhyay. 2020b. “On energy, uniqueness theorems and variational principle for generalized thermoelasticity with memory-dependent derivative.” Int. J. Heat Mass Transfer 149 (Mar): 119112. https://doi.org/10.1016/j.ijheatmasstransfer.2019.119112.
Sarkar, I., and B. Mukhopadhyay. 2020c. “On the spatial behavior of thermal signals in generalized thermoelasticity with memory-dependent derivative.” Acta Mech. 231 (7): 2989–3001. https://doi.org/10.1007/s00707-020-02687-7.
Sarkar, I., and B. Mukhopadhyay. 2020d. “Thermo-viscoelastic interaction under dual-phase-lag model with memory-dependent derivative.” Waves Random Complex Medium 1–24. https://doi.org/10.1080/17455030.2020.1736733.
Shaw, S. 2019. “Theory of generalized thermoelasticity with memory- derivatives.” J. Eng. Mech. 145 (3): 04019003. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001569.
Shaw, S., and B. Mukhopadhyay. 2017. “A discontinuity analysis of generalized thermoelasticity theory with memory-dependent derivatives.” Acta Mech. 228 (7): 2675–2689. https://doi.org/10.1007/s00707-017-1853-0.
Singh, B., and S. Pal. 2020. “Magneto-thermoelastic interaction with memory response due to laser pulse under green-naghdi theory in an orthotropic medium.” Mech. Based Des. Struct. Mach. 1–18. https://doi.org/10.1080/15397734.2020.1798780.
Singh, B., S. Pal, and K. Barman. 2020. “Eigenfunction approach to generalized thermo-viscoelasticity with memory dependent derivative due to three-phase-lag heat transfer.” J. Therm. Stresses 1–20. https://doi.org/10.1080/01495739.2020.1770642.
Singh, B., and I. Sarkar. 2020. “Temperature-rate-dependent thermoelasticity theory with memory-dependent derivative: Energy, uniqueness theorems, and variational principle.” J. Heat Transfer 142 (10): 102103. https://doi.org/10.1115/1.4047510.
Sumelka, W., and G. Z. Voyiadjis. 2017. “A hyperelastic fractional damage material model with memory.” Int. J. Solids Struct. 124 (Oct): 151–160. https://doi.org/10.1016/j.ijsolstr.2017.06.024.
Thompson, J. M. T., and G. W. Hunt. 1973. A general theory of elastic stability. New York: Wiley.
Timoshenko, S. P., and J. M. Gere. 1963. Theory of elastic stability. New York: McGraw-Hill.
Tzou, D. Y. 1995. “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.
van der Heijden, A. M. A. 2009. W. T. Koiter’s elastic stability of solids and structures. Cambridge, UK: Cambridge University Press.
Wang, J.-L., and H.-F. Li. 2011. “Surpassing the fractional derivative: Concept of the memory-dependent derivative.” Comput. Math. Appl. 62 (3): 1562–1567. https://doi.org/10.1016/j.camwa.2011.04.028.
Xue, Z., X. Tian, and J. Liu. 2020. “Thermal shock fracture of a crack in a functionally gradient half-space based on the memory-dependent heat conduction model.” Appl. Math. Model. 80 (Apr): 840–858. https://doi.org/10.1016/j.apm.2019.11.021.
Yang, W., and Z. Chen. 2019. “Fractional single-phase lag heat conduction and transient thermal fracture in cracked viscoelastic materials.” Acta Mech. 230 (10): 3723–3740. https://doi.org/10.1007/s00707-019-02474-z.
Yu, Y.-J., W. Hu, and X. Tian. 2014. “A novel generalized thermoelasticity model based on memory-dependent derivative.” Int. J. Eng. Sci. 81 (Aug): 123–134. https://doi.org/10.1016/j.ijengsci.2014.04.014.
Zhang, X.-Y., Z.-T. Chen, and X.-F. Li. 2018. “Thermal shock fracture of an elastic half-space with a subsurface penny-shaped crack via fractional thermoelasticity.” Acta Mech. 229 (12): 4875–4893. https://doi.org/10.1007/s00707-018-2252-x.
Zhang, X.-Y., and X.-F. Li. 2017. “Transient thermal stress intensity factors for a circumferential crack in a hollow cylinder based on generalized fractional heat conduction.” Int. J. Therm. Sci. 121 (Nov): 336–347. https://doi.org/10.1016/j.ijthermalsci.2017.07.015.
Ziegler, H. 1968. Principles of structural stability. Waltham, MA: Blaisdell.
Zubov, V. I. 1964. Methods of A. M. Lyapunov and their application. Groningen, Netherlands: P. Noordhoff.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 147 • Issue 3 • March 2021
Copyright
© 2021 American Society of Civil Engineers.
History
Received: Aug 25, 2020
Accepted: Nov 29, 2020
Published online: Jan 6, 2021
Published in print: Mar 1, 2021
Discussion open until: Jun 6, 2021
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.