Skip to main content
SpringerLink
Log in

Generalized thermoelastic problem of an infinite body with a spherical cavity under dual-phase-lags

  • Published:
Journal of Applied Mechanics and Technical Physics Aims and scope

Abstract

The aim of the present contribution is the determination of the thermoelastic temperatures, stress, displacement, and strain in an infinite isotropic elastic body with a spherical cavity in the context of the mechanism of the two-temperature generalized thermoelasticity theory (2TT). The two-temperature Lord–Shulman (2TLS) model and two-temperature dual-phase-lag (2TDP) model of thermoelasticity are combined into a unified formulation with unified parameters. The medium is assumed to be initially quiescent. The basic equations are written in the form of a vector matrix differential equation in the Laplace transform domain, which is then solved by the state-space approach. The expressions for the conductive temperature and elongation are obtained at small times. The numerical inversion of the transformed solutions is carried out by using the Fourier-series expansion technique. A comparative study is performed for the thermoelastic stresses, conductive temperature, thermodynamic temperature, displacement, and elongation computed by using the Lord–Shulman and dual-phase-lag models.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. E. Gurtin and W. O. Williams, “On the Clausius–Duhem Inequality,” Z. Angew. Math. Phys. 7, 626–633 (1966).

    Article  Google Scholar 

  2. M. E. Gurtin and W. O.Williams, “An Axiomatic Foundation for Continuum Thermodynamics,” Arch. Rational Mech. Anal. 26, 83–117 (1967).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. P. J. Chen and M. E. Gurtin, “On a Theory of Heat Conduction Involving Two Temperatures,” Z. Angew. Math. Phys. 19, 614–627 (1968).

    Article  MATH  Google Scholar 

  4. P. J. Chen, M. E. Gurtin, and W. O. Williams, “A Note on Non Simple Heat Conduction,” Z. Angew. Math. Phys. 19, 969–970 (1968).

    Article  Google Scholar 

  5. P. J. Chen, M. E. Gurtin, and W. O. Williams, “On the Thermodynamics of Non-Simple Elastic Materials with Two Temperatures,” Z. Angew. Math. Phys. 20, 107–112 (1969).

    Article  MATH  Google Scholar 

  6. W. E. Warren and P. J. Chen, “Wave Propagation in Two Temperatures Theory of Thermoelasticity,” Acta Mech. 16, 83–117 (1973).

    Article  Google Scholar 

  7. D. Lesan, “On the Linear Coupled Thermoelasticity with Two Temperatures,” Z. Angew. Math. Phys. 21, 583–591 (1970).

    Article  MathSciNet  Google Scholar 

  8. P. Puri and P. M. Jordan, “On the Propagation of Harmonic PlaneWaves under the Two-Temperature Theory,” Int. J. Eng. Sci. 44, 1113–1126 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  9. R. Quintanilla, “On Existence, Structural Stability, Convergence and Spatial Behavior in Thermoelasticity with Two Temperatures,” Acta Mech. 168, 61–73 (2004).

    Article  MATH  Google Scholar 

  10. H. Lord and Y. Shulman, “A Generalized Dynamical Theory of Thermoelasticity,” J. Mech. Phys. Solids 15, 299 (1967).

    Article  ADS  MATH  Google Scholar 

  11. J. Ignaczak, “Uniqueness in Generalized Thermoelasticity,” J. Thermal Stresses 2, 171–175 (1979).

    Article  MathSciNet  Google Scholar 

  12. J. Ignaczak, “A Note on Uniqueness in Thermoelasticity with One Relaxation Time,” J. Thermal Stresses 5, 257–263 (1982).

    Article  MathSciNet  Google Scholar 

  13. R. S. Dhaliwal and H. Sherief, “Generalized Thermoelasticity for Anisotropic Media,” Quart. Appl. Math. 33, 1–8 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  14. H. H. Sherief, “On Uniqueness and Stability in Generalized Thermoelasticity,” Quart. Appl. Math. 45, 773–778 (1987).

    MathSciNet  MATH  Google Scholar 

  15. A. E. Green and K. A. Lindsay, “Thermoelasticity,” J. Elasticity 2, 1–7 (1972).

    Article  MATH  Google Scholar 

  16. A. E. Green and P. M. Naghdi, “A Re-Examination of the Basic Results of Thermomechanics,” Proc. Roy. Soc. London, Ser. A 432, 171–194 (1992).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. A. E. Green and P.M. Naghdi, “Thermoelasticity without Energy Dissipation,” J. Elasticity 31, 189–208 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  18. A. E. Green and P. M. Naghdi, “On Undamped Heat Waves in an Elastic Solid,” J. Thermal Stresses 15, 252–264 (1992).

    Article  ADS  MathSciNet  Google Scholar 

  19. S. H. Mallik and M. Kanoria, “Two Dimensional Problem in Generalized Thermoelasticity for a Rotating Orthotropic Infinite Medium with Heat Sources,” Indian J. Math. 49 (1), 47–70 (2007).

    MathSciNet  MATH  Google Scholar 

  20. S. H. Mallik and M. Kanoria, “A Unified Generalized Thermoelastic Formulation: Application to an Penny Shaped Crack Analysis,” J. Thermal Stresses 32, 945–965 (2009).

    Article  Google Scholar 

  21. D. Y. Tzou, “A Unified Approach for Heat Conduction from Macro to Micro-Scales,” Trans. ASME, J. Heat Transfer 117, 8–16 (1995).

    Article  Google Scholar 

  22. S. K. Roychoudhury, “One-Dimensional Thermoelastic Waves in Elastic Half-Space with Dual-Phase-Lag Effect,” J. Mech. Mater. Struct. 2, 489–502 (2007).

    Article  Google Scholar 

  23. R. Quintanilla and C. O. Horgan, “Spatial Behaviour of Solutions of the Dual-Phase-Lag Heat Equation,” Math. Methods Appl. Sci. 28, 43–57 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  24. R. Quintanilla and R. Racke, “A Note on Stability in Dual-Phase-Lag Heat Conduction,” Int. J. Heat Mass Transfer 49, 1209–1213 (2006).

    Article  MATH  Google Scholar 

  25. R. Quintanilla, “A Well-Posed Problem for the Three-Dual-Phase-Lag Heat Conduction,” J. Thermal Stresses 32, 1270–1278 (2009).

    Article  Google Scholar 

  26. R. Quintanilla, “Exponential Stability in the Dual-Phase-Lag Heat Conduction Theory,” J. Non-Equib. Thermodyn. 27, 217–227 (2002).

    ADS  MATH  Google Scholar 

  27. R. Quintanilla, “A Condition on the Delay Parameters in the One-Dimensional Dual-Phase-Lag Thermoelastic Theory,” J. Thermal Stresses 26, 713–721 (2003).

    Article  MathSciNet  Google Scholar 

  28. R. Prasad, R. Kumar, and S. Mukhopadhyay, “Propagation of Harmonic Plane Waves under Thermoelasticity with Dual-Phase-Lags,” Int. J. Eng. Sci. 48, 2028–2043 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  29. H. M. Youssef, “Theory of Two-Temperature Generalized Thermoelasticity,” IMA J. Appl. Math. 71, 1–8 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  30. S. Mukhopadhyay and R. Kumar, “Thermoelastic Interactions on Two Temperature Generalized Thermoelasticity in an Infinite Medium with a Cylindrical Cavity,” J. Thermal Stresses 32, 341–360 (2009).

    Article  Google Scholar 

  31. R. Kumar, R. Prasad, and S. Mukhopadhyay, “Variational and Reciprocal Principles in Two-Temperature Generalized Thermoelasticity,” J. Thermal Stresses 33, 161–171 (2010).

    Article  Google Scholar 

  32. A. Magane and R. Quintanilla, “Uniqueness and Growth of Solutions in Two-Temperature Generalized Thermoelastic Theories,” J. Math. Mech. Solids 14, 622–634 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  33. S. Banik and M. Kanoria, “Two Temperature Generalized Thermoelastic Interactions in an Infinite Body with a Spherical Cavity,” Int. J. Thermophys. 32, 1247–1270 (2011).

    Article  ADS  Google Scholar 

  34. A. Sur and M. Kanoria, “Fractional Order Two-Temperature Thermoelasticity with Finite Wave Speed,” Acta Mech. 223, (12), 2685–2701 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  35. S. Mondal, S. H. Mallik, and M. Kanoria, “Fractional Order Two-Temperature Dual-Phase-Lag Thermoelasticity with Variable Thermal Conductivity,” Int. Scholarly Res. Notices 2014, 646049 (2014).

    Google Scholar 

  36. L. Y. Bahar and R. B. Hetnarski, “State Space Approach to Thermoelasticity,” in Proc. of the 6th Canad. Congress of Applied Mechanics, Vancouver (Canada), May 30 to June 3, 1977 (Univ. of British Columbia, 1977), pp. 17–18.

    Google Scholar 

  37. G. Honig and U. Hirdes, “A Method for the Numerical Inversion of Laplace Transform,” J. Comput. Appl. Math. 10, 113–132 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  38. S. Banik and M. Kanoria, “Two-Temperature Generalized Thermoelastic Interactions in an Infinite Body with a Spherical Cavity,” Int. J. Thermophys. 32 (6), 1247–1270 (2011); DOI: 10.1007/s10765-011-1002-2.

    Article  ADS  Google Scholar 

  39. L. Y. Bahar and R. B. Hetnarski, “State Space Approach to Thermoelasticity,” J. Thermal Stresses 1 (1), 135–145 (1978).

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Calcutta, 92 A.P.C. Road, 700009, Kolkata, West Bengal, India

    R. Karmakar, A. Sur & M. Kanoria

Authors
  1. R. Karmakar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Sur
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Kanoria
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. Karmakar.

Additional information

Translated from PrikladnayaMekhanika i Tekhnicheskaya Fizika, Vol. 57, No. 4, pp. 91–106, July–August, 2016.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Karmakar, R., Sur, A. & Kanoria, M. Generalized thermoelastic problem of an infinite body with a spherical cavity under dual-phase-lags. J Appl Mech Tech Phy 57, 652–665 (2016). https://doi.org/10.1134/S002189441604009X

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S002189441604009X

Keywords

Search

Navigation