Skip to main content
SpringerLink
Log in

Eigenvalue approach on a two-dimensional thermal shock problem with weak, normal and strong conductivity

  • Regular Article
  • Published:
The European Physical Journal Plus Aims and scope Submit manuscript

Abstract.

The present paper is devoted to the study of a two-dimensional thermal shock problem with weak, normal and strong conductivity using the eigenvalue approach. The governing equations are taken in the context of the new consideration of heat conduction with fractional order generalized thermoelasticity with the Lord-Shulman model (LS model). The bounding surface of the half-space is taken to be traction free and subjected to a time-dependent thermal shock. The Laplace and the exponential Fourier transform techniques are used to obtain the analytical solutions in the transformed domain by the eigenvalue approach. Numerical computations have been done for copper-like material for weak, normal and strong conductivity and the results are presented graphically to estimate the effects of the fractional order parameter.

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.A. Biot, J. Appl. Phys. 27, 240 (1956)

    Article  ADS  MathSciNet  Google Scholar 

  2. H.W. Lord, Y. Shulman, J. Mech. Phys. Solids 15, 299 (1967)

    Article  ADS  Google Scholar 

  3. A.E. Green, K.A. Lindsay, J. Elasticity 2, 1 (1972)

    Article  Google Scholar 

  4. R.S. Dhaliwal, H.H. Sherief, Quart. Appl. Math. 38, 1 (1980)

    MathSciNet  Google Scholar 

  5. H.M. Youssef, J. Heat Transf. 132, 061301 (2010)

    Article  Google Scholar 

  6. H.M. Youssef, E.A. Al-Lehaibi, Appl. Math. Lett. 23, 1183 (2010)

    Article  MathSciNet  Google Scholar 

  7. M.A. Ezzat, A.S. El-Karamany, Can. J. Phys. 89, 311 (2011)

    Article  ADS  Google Scholar 

  8. M.A. Ezzat, A.S. El Karamany, Eur. J. Mech. A/Solids 30, 491 (2011)

    Article  ADS  Google Scholar 

  9. M.A. Ezzat, Appl. Math. Model. 35, 4965 (2011)

    Article  MathSciNet  Google Scholar 

  10. H.H. Sherief, A.M.A. El-Sayed, A.M. Abd El-Latief, Int. J. Solids Struct. 47, 269 (2010)

    Article  Google Scholar 

  11. S. Santra et al., J. Therm. Stresses 38, 309 (2015)

    Article  Google Scholar 

  12. D.S. Mashat, A.M. Zenkour, A.E. Abouelregal, Mech. Adv. Mater. Struct. 22, 925 (2015)

    Article  Google Scholar 

  13. E.M. Hussein, J. Therm. Stresses 38, 133 (2015)

    Article  Google Scholar 

  14. A. Sur, M. Kanoria, J. Solid Mech. 6, 54 (2014)

    Google Scholar 

  15. M.A. Ezzat et al., J. Electromagn. Waves Appl. 28, 64 (2014)

    Article  Google Scholar 

  16. S. Deswal, K.K. Kalkal, Wave Motion 51, 100 (2014)

    Article  MathSciNet  Google Scholar 

  17. I.A. Abbas, J. Comput. Theor. Nanosci. 11, 380 (2014)

    Article  Google Scholar 

  18. N. Sarkar, A. Lahiri, Meccanica 48, 231 (2013)

    Article  MathSciNet  Google Scholar 

  19. H.M. Youssef, Acta Mech. 223, 1219 (2012)

    Article  MathSciNet  Google Scholar 

  20. M.A. Ezzat, A.S. El-Karamany, M.A. Fayik, J. Therm. Stresses 35, 637 (2012)

    Article  Google Scholar 

  21. M.A. Ezzat, A.S. El-Karamany, J. Appl. Polym. Sci. 124, 2187 (2012)

    Article  Google Scholar 

  22. A.M. Zenkour, A.E. Abouelregal, Arch. Mech. 67, 53 (2015)

    MathSciNet  Google Scholar 

  23. H.M. Youssef, Theory of generalized thermoelasticity with fractional order strain, J. Vibrat. Control (2015) DOI:1177/1077546314566837

  24. Y. Wang, D. Liu, Q. Wang, Acta Mech. Solida Sin. 28, 285 (2015)

    Article  Google Scholar 

  25. H.H. Sherief, A.M. Abd El-Latief, Math. Mech. Solids 20, 512 (2015)

    Article  MathSciNet  Google Scholar 

  26. R. Kumar, V. Gupta, I.A. Abbas, J. Comput. Theor. Nanosci. 10, 2520 (2013)

    Article  Google Scholar 

  27. I.A. Abbas, J. Magn. & Magn. Mater. 377, 452 (2015)

    Article  ADS  Google Scholar 

  28. I.A. Abbas, H.M. Youssef, Latin Am. J. Solids Struct. 12, 1415 (2015)

    Article  Google Scholar 

  29. H.M. Youssef, I.A. Abbas, J. Vibroeng. 16, 4077 (2014)

    Google Scholar 

  30. I.A. Abbas, Theor. Appl. Fract. Mech. 74, 18 (2014)

    Article  Google Scholar 

  31. A.M. Zenkour, A.E. Abouelregal, Z. Angew. Math. Phys. 65, 149 (2014)

    Article  MathSciNet  Google Scholar 

  32. I.A. Abbas, Appl. Math. Model. 39, 6196 (2015)

    Article  MathSciNet  Google Scholar 

  33. R. Kumar et al., Multidiscipline Model. Mater. Struct. 12, 133 (2016)

    Article  Google Scholar 

  34. R. Kumar, P. Sharma, Curved Layered Struct. 3, 96 (2016)

    Google Scholar 

  35. N.C. Das, A. Lahiri, R.R. Giri, Indian J. Pure Appl. Math. 28, 1573 (1997)

    MathSciNet  Google Scholar 

  36. I.A. Abbas, Comp. Math. Appl. 68, 2036 (2014)

    Article  Google Scholar 

  37. I.A. Abbas, J. Mech. Sci. Technol. 28, 4193 (2014)

    Article  Google Scholar 

  38. I.A. Abbas, Mech. Based Design Struct. Machines 43, 501 (2015)

    Article  Google Scholar 

  39. I.A. Abbas, Can. J. Phys. 93, 585 (2015)

    Article  ADS  Google Scholar 

  40. H. Stehfest, Commun. ACM 13, 47 (1970)

    Article  Google Scholar 

  41. S. Lee, M. Chien, W. Culham, Vertical single-well pulse testing of a three-layer stratified reservoir, in SPE Annual Technical Conference and Exhibition (Society of Petroleum Engineers, 1984)

  42. H.H. Sherief, N.M. El-Maghraby, J. Therm. Stresses 28, 465 (2005)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box. 80203, 21589, Jeddah, Saudi Arabia

    Faris S. Alzahrani

  2. Department of Mathematics, Faculty of Science and Arts - Khulais, University Of Jeddah, Jeddah, Saudi Arabia

    Ibrahim A. Abbas

  3. Nonlinear Analysis and Applied Mathematics Research Group (NAAM), Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia

    Faris S. Alzahrani & Ibrahim A. Abbas

  4. Department of mathematics, Faculty of Science, Sohag University, Sohag, Egypt

    Ibrahim A. Abbas

Authors
  1. Faris S. Alzahrani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ibrahim A. Abbas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Faris S. Alzahrani.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Alzahrani, F., Abbas, I. Eigenvalue approach on a two-dimensional thermal shock problem with weak, normal and strong conductivity. Eur. Phys. J. Plus 131, 289 (2016). https://doi.org/10.1140/epjp/i2016-16289-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjp/i2016-16289-9

Search

Navigation