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Meccanica

, Volume 53, Issue 10, pp 2543–2554 | Cite as

A modified Green–Lindsay thermoelasticity with strain rate to eliminate the discontinuity

  • Y. Jun Yu
  • Zhang-Na Xue
  • Xiao-Geng Tian
Article

Abstract

Probing the mechanism of ultrafast thermoelastic processes is becoming increasingly important with the development of laser-assisted micro/nano machining. Although thermoelastic models containing temperature rate have been historically proposed, the strain rate has not been considered yet. In this work, a generalized thermoelastic model is theoretically established by introducing the strain rate in Green–Lindsay (GL) thermoelastic model with the aid of extended thermodynamics. Numerically, a semi-infinite one-dimensional problem is considered with traction free at one end and subjected to a temperature rise. The problem is solved using the Laplace transform method, and the transient responses, i.e. displacement, temperature and stresses are graphically depicted. Interestingly, it is found that the strain rate may eliminate the discontinuity of the displacement at the elastic and thermal wave front. Also, the present model is compared with Green–Naghdi (GN) models. It is found that the thermal wave speed of the present model is faster than GN model without energy dissipation, and slower than GN model with energy dissipation. In addition, the thermoelastic responses from the present model are the largest. The present model based upon GL model is free of the jump of GL model in the displacement distribution, and is safer in engineering practices than GN model. The present work will benefit the theoretical modeling and numerical prediction of thermoelastic process, especially for those under extreme fast heating.

Keywords

Green–Lindsay model Relaxation times Strain rate Transient responses Green–Naghdi model 

Notes

Funding

This study was funded by National Key R&D Problem of China (No. 2017YFB1102801), National Natural Science Foundation of China (No. 11572237), and Fundamental Research Funds for the Central Universities (3102017OQD072).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Afrin N, Zhang Y, Chen JK (2014) Dual-phase lag behavior of a gas-saturated porous-medium heated by a short-pulsed laser. Int J Therm Sci 75:21–27CrossRefGoogle Scholar
  2. 2.
    Chen JK, Tzou DY, Beraun JE (2006) A semiclassical two-temperature model for ultrafast laser heating. Int J Heat Mass Transf 49(1–2):307–316CrossRefzbMATHGoogle Scholar
  3. 3.
    Hosoya N, Kajiwara I, Inoue T, Umenai K (2014) Non-contact acoustic tests based on nanosecond laser ablation: generation of a pulse sound source with a small amplitude. J Sound Vib 333(18):4254–4264ADSCrossRefGoogle Scholar
  4. 4.
    Abd-alla AN, Giorgio I, Galantucci L, Hamdan AM, Del Vescovo D (2016) Wave reflection at a free interface in an anisotropic pyroelectric medium with nonclassical thermoelasticity. Continuum Mech Thermodyn 28(1–2):67–84ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cattaneo C (1948) Sulla conduzione del calore. Atti Sem Mat Fis Univ Modena 3:83–101MathSciNetzbMATHGoogle Scholar
  6. 6.
    Partap G, Chugh N (2017) Thermoelastic damping in microstretch thermoelastic rectangular Plate. Microsyst Technol 23(12):5875–5886CrossRefGoogle Scholar
  7. 7.
    Hosseini SM (2017) Shock-induced nonlocal coupled thermoelasticity analysis (with energy dissipation) in a MEMS/NEMS beam resonator based on Green–Naghdi theory: a meshless implementation considering small scale effects. J Therm Stresses 40(9):1134–1151CrossRefGoogle Scholar
  8. 8.
    Hosseini SM (2018) Analytical solution for nonlocal coupled thermoelasticity analysis in a heat-affected MEMS/NEMS beam resonator based on Green–Naghdi theory. Appl Math Model 57:21–36MathSciNetCrossRefGoogle Scholar
  9. 9.
    Liu S, Sun Y, Ma J, Yang J (2018) Theoretical analysis of thermoelastic damping in bilayered circular plate resonators with two-dimensional heat conduction. Int J Mech Sci 135:114–123CrossRefGoogle Scholar
  10. 10.
    Peshkov V (1944) Second sound in helium II. J. Phys. 8:381–382Google Scholar
  11. 11.
    Singh B (2012) Wave propagation in dual-phase-lag anisotropic thermoelasticity. Continuum Mech Thermodyn 25(5):675–683ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Cattaneo C (1958) A form of heat equation which eliminates the paradox of instantaneous propagation. Comptes Rendus 247:431–433zbMATHGoogle Scholar
  13. 13.
    Vernotte P (1958) Paradoxes in the continuous theory of the heat conduction. Comptes Rendus 246:3154–3155zbMATHGoogle Scholar
  14. 14.
    Tzou DY (1995) A unified field approach for heat conduction from macro to micro scales. ASME J Heat Transf 117:8–16CrossRefGoogle Scholar
  15. 15.
    Guo ZY, Hou QW (2010) Thermal wave based on the thermomass model. J Heat Transf 132(7):072403CrossRefGoogle Scholar
  16. 16.
    Kuang ZB (2014) Discussions on the temperature wave equation. Int J Heat Mass Transf 71:424–430CrossRefGoogle Scholar
  17. 17.
    Lord H, Shulman Y (1967) A generalized dynamic theory of thermoelasticity. J Mech Phys Solids 15:299–309ADSCrossRefzbMATHGoogle Scholar
  18. 18.
    Green AE, Lindsay K (1972) Thermoelasticity. J Elast 2:1–7CrossRefzbMATHGoogle Scholar
  19. 19.
    Green AE, Naghdi PM (1992) On undamped heat waves in an elastic solid. J Therm Stresses 15:252–264ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Green AE, Naghdi PM (1993) Thermoelasticity without energy dissipation. J Elast 31:189–208MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Quintanilla R (2001) Instability and non-existence in the nonlinear theory of thermoelasticity without energy dissipation. Continuum Mech Thermodyn 13(2):121–129ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kuang ZB (2008) Variational principles for generalized dynamical theory of thermopiezoelectricity. Acta Mech 203(1–2):1–11zbMATHGoogle Scholar
  23. 23.
    Wang YZ, Zhang XB, Song XN (2013) A generalized theory of thermoelasticity based on thermomass and its uniqueness theorem. Acta Mech 225(3):797–808MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Povstenko YZ (2004) Fractional heat conduction equation and associated thermal stress. J Therm Stresses 28(1):83–102MathSciNetCrossRefGoogle Scholar
  25. 25.
    Youssef HM (2010) Theory of fractional order generalized thermoelasticity. J Heat Transfer 132(6):061301CrossRefGoogle Scholar
  26. 26.
    Sherief HH, El-Sayed AMA, El-Latief AM (2010) Fractional order theory of thermoelasticity. Int J Solids Struct 47(2):269–275CrossRefzbMATHGoogle Scholar
  27. 27.
    Ezzat MA (2011) Magneto-thermoelasticity with thermoelectric properties and fractional derivative heat transfer. Physica B 406(1):30–35ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    Yu YJ, Hu W, Tian XG (2014) A novel generalized thermoelasticity model based on memory-dependent derivative. Int J Eng Sci 81:123–134MathSciNetCrossRefGoogle Scholar
  29. 29.
    Yu YJ, Tian XG, Lu TJ (2013) Fractional order generalized electro-magneto-thermo-elasticity. Eur J Mech A Solids 42:188–202MathSciNetCrossRefGoogle Scholar
  30. 30.
    Yu YJ, Tian XG, Liu XR (2015) Size-dependent generalized thermoelasticity using Eringen’s nonlocal model. Eur J Mech A Solids 51:96–106ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    Ignaczak J, Ostoja-Starzewski M (2009) Thermoelasticity with finite wave speeds. Oxford University Press, OxfordCrossRefzbMATHGoogle Scholar
  32. 32.
    Di Paola M et al (2014) On the influence of the initial ramp for a correct definition of the parameters of fractional viscoelastic materials. Mech Mater 69(1):63–70CrossRefGoogle Scholar
  33. 33.
    Di Lorenzo S et al (2016) Non-linear viscoelastic behavior of polymer melts interpreted by fractional viscoelastic model. Meccanica 52(8):1843–1850MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Povstenko Y (2015) Fractional thermoelasticity. Springer, New YorkCrossRefzbMATHGoogle Scholar
  35. 35.
    Alotta G et al (2016) On the behavior of a three-dimensional fractional viscoelastic constitutive model. Meccanica 52(9):2127–2142MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Xiao R, Sun H, Chen W (2016) An equivalence between generalized Maxwell model and fractional Zener model. Mech Mater 100:148–153CrossRefGoogle Scholar
  37. 37.
    Brancik L (1999) Programs for fast numerical inversion of Laplace transforms in MATLAB language environment. In: Proceedings of the 7th conference MATLAB’99, Czech Republic, Prague, pp 27–39Google Scholar
  38. 38.
    Yu YJ, Tian X-G, Xiong Q-L (2016) Nonlocal thermoelasticity based on nonlocal heat conduction and nonlocal elasticity. Eur J Mech A Solids 60:238–253MathSciNetCrossRefGoogle Scholar
  39. 39.
    Yu YJ et al (2016) The dilemma of hyperbolic heat conduction and its settlement by incorporating spatially nonlocal effect at nanoscale. Phys Lett A 380(1–2):255–261ADSCrossRefGoogle Scholar
  40. 40.
    Othman MIA, Zidan MEM, Hilal MIM (2015) The effect of initial stress on thermoelastic rotation medium with voids due to laser pulse heating with energy dissipation. J Therm Stresses 38(8):835–853CrossRefGoogle Scholar
  41. 41.
    Othman MIA, Zidan MEM, Hilal MIM (2014) Effect of magnetic field on a rotating thermoelastic medium with voids under thermal loading due to laser pulse with energy dissipation. Can J Phys 92(11):1359–1371ADSCrossRefGoogle Scholar

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Engineering Mechanics, School of Mechanics, Civil Engineering and ArchitectureNorthwestern Polytechnical UniversityXi’anPeople’s Republic of China
  2. 2.State Key Laboratory for Strength and Vibration of Mechanical StructureXi’an Jiaotong UniversityXi’anPeople’s Republic of China

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