Skip to main content
SpringerLink
Log in

Modeling of Memory Dependent Derivative Under Three-Phase Lag in Generalized Thermo-Viscoelasticity

  • Original Paper
  • Published:
International Journal of Applied and Computational Mathematics Aims and scope Submit manuscript

Abstract

In this study, a new generalized model of thermo-viscoelasticity with three phase-lag (TPL) theory concerning memory-dependent derivative (MDD) theory is emphasized. The governing combined equations of the novel model associated with kernel function and time delay are considered in a two-dimensional semi-infinite space. The bounding surface of the medium is assumed to be free of traction and subjected to time-dependent thermal loading. Using Laplace and Fourier Transform techniques, the problem is transformed from the space–time domain and then solved numerically. Suitable numerical technique is used to find the inversion of Fourier and Laplace transforms. In the simulation, the effects of the time-delay parameter and kernel function on the distributions of the displacement components, stresses and temperature field are studied and represented graphically. The results shows that the presence of TPL, the time-delay and kernel function extensively affect all the distributions.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17

Similar content being viewed by others

References

  1. Gutierrez-Lemini, D.: Engineering Viscoelasticity. Springer, Berlin (2014)

    MATH  Google Scholar 

  2. Gross, B.: Mathematical Structure of the Theories of Viscoelasticity. Hemann, Paris (1953)

    MATH  Google Scholar 

  3. Stratonova, M.M.: A method of solving dynamic problems of viscoelasticity. Polym. Mech. 7(4), 646–648 (1971)

    Google Scholar 

  4. Pobedrya, B.E.: Coupled problems of thermoviscoelasticity. Polym. Mech. 5(3), 353–358 (1969)

    Google Scholar 

  5. Biot, M.: Variational principle in irreversible thermodynamics with application to viscoelasticity. Phys. Rev. 97, 1463–1469 (1955)

    MathSciNet  MATH  Google Scholar 

  6. Huilgol, R., Phan-Thien, N.: Fluid Mechanics of Viscoelasticity. Elsevier, Amsterdam (1997)

    Google Scholar 

  7. Tanner, R.: Engineering Rheology. Oxford Univ. Press, Oxford (1988)

    MATH  Google Scholar 

  8. Tschoegl, N.: Time dependence in material properties: an overview. Mech. Time-Depend. Mater. 1, 3–31 (1997)

    Google Scholar 

  9. Morland, L., Lee, E.: Stress analysis for linear viscoelastic materials with temperature variation. Trans. Soc. Rheol. 4, 233–263 (1960)

    MathSciNet  Google Scholar 

  10. Lord, H.W., Shulman, Y.: A generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids 15(5), 299–309 (1967)

    MATH  Google Scholar 

  11. Green, A.E., Lindsay, K.A.: Thermoelasticity. J. Elast. 2, 1–7 (1972)

    MATH  Google Scholar 

  12. Sherief, H.H., Allam, M.N., El-Hagary, M.A.: Generalized theory of thermoviscoelasticity and a half-space problem. Int. J. Thphys. 32, 1271–1275 (2011)

    Google Scholar 

  13. Chandrasekharaiah, D.S.: Thermoelasticity with second sound: a review. Appl. Mech. Rev. 39, 355–376 (1986)

    MATH  Google Scholar 

  14. Chandrasekharaiah, D.S.: Hyperbolic thermoelasticity: a review of recent literature. Appl. Mech. Rev. 51, 705–729 (1998)

    Google Scholar 

  15. Tzou, D.Y.: A unified field approach for heat conduction from macro-to micro-scales. J. Heat Transf. 117(1), 8–16 (1995)

    Google Scholar 

  16. Choudhuri, S.R.: On a thermoelastic three-phase-lag model. J. Therm. Stress. 30(3), 231–238 (2007)

    Google Scholar 

  17. Caputo, M.: Vibrations of an infinite viscoelastic layer with a dissipative memory. J. Acoust. Soc. Am. 56, 897–904 (1974)

    MATH  Google Scholar 

  18. Povstenko, Y.: Fractional heat conduction equation and associated thermal stress. J. Therm. Stress. 28(1), 83–102 (2004)

    MathSciNet  Google Scholar 

  19. Mondal, S., Mallik, S.H., Kanoria, M.: Fractional order two-temperature dual-phase-lag thermoelasticity with variable thermal conductivity. Int. Sch. Res. Notices. (2014)

  20. Sur, A., Kanoria, M.: Fractional order two-temperature thermoelasticity with finite wave speed. Acta Mech. 223(12), 2685–2701 (2012)

    MathSciNet  MATH  Google Scholar 

  21. Bhattacharya, D., Kanoria, M.: The influence of two-temperature fractional order generalized thermoelastic diffusion inside a spherical shell. Int. J. Appl. Innov. Eng. Manag. 3, 96–108 (2014)

    Google Scholar 

  22. Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Elsevier, Amsterdam (1998)

    MATH  Google Scholar 

  23. Sherief, H.H., El-Sayed, A.M.A., El-Latief, A.M.A.: Fractional order theory of thermoelasticity. Int. J. Solids Struct. 47(2), 269–275 (2010)

    MATH  Google Scholar 

  24. Povstenko, Y.: Fractional Thermoelasticity. Springer, Berlin (2015)

    MATH  Google Scholar 

  25. Wang, J.L., Li, H.F.: Surpassing the fractional derivative: concept of the memory-dependent derivative. Comput. Math. Appl. 62(3), 1562–1567 (2011)

    MathSciNet  MATH  Google Scholar 

  26. Yu, Y.J., Hu, W., Tian, X.G.: A novel generalized thermoelasticity model based on memory-dependent derivative. Int. J. Eng. Sci. 81, 123–134 (2014)

    MathSciNet  MATH  Google Scholar 

  27. Ezzat, M.A., El-Karamany, A.S., El-Bary, A.A.: Generalized thermo-viscoelasticity with memory-dependent derivatives. Int. J. Mech. Sci. 89, 470–475 (2014)

    Google Scholar 

  28. Ezzat, M.A., El-Karamany, A.S., El-Bary, A.A.: Generalized thermoelasticity with memory-dependent derivatives involving two temperatures. Mech. Adv. Mater. Struct. 23(5), 545–553 (2016)

    Google Scholar 

  29. Kant, S., Mukhopadhyay, S.: An investigation on responses of thermoelastic interactions in a generalized thermoelasticity with memory-dependent derivatives inside a thick plate. Math. Mech. Solids. 24(8), 2392–2409 (2019)

    MathSciNet  MATH  Google Scholar 

  30. Banerjee, S., Shaw, S., Mukhopadhyay, B.: Memory response on thermoelastic deformation in a solid half-space with a cylindrical hole. Mech Based Des Struct Mach. (2020). https://doi.org/10.1080/15397734.2019.1686989

    Article  Google Scholar 

  31. Banerjee, S., Shaw, S., Mukhopadhyay, B.: Memory response on thermal wave propagation emanating from a cavity in an unbounded elastic solid. J. Therm. Stress. 42(2), 294–311 (2019)

    Google Scholar 

  32. Sarkar, I., Mukhopadhyay, B.: On energy, uniqueness theorems and variational principle for generalized thermoelasticity with memory dependent derivative. Int. J. Heat Mass Transf. 149, 119112 (2020)

    Google Scholar 

  33. Shaw, S., Mukhopadhyay, B.: A discontinuity analysis of generalized thermoelasticity theory with memory-dependent derivatives. Act. Mech. 228(7), 2675–2689 (2017)

    MathSciNet  MATH  Google Scholar 

  34. Mondal, S., Othman, M.I.: Memory dependent derivative effect on generalized piezo-thermoelastic medium under three theories. Wav. Rand. Comp. Med. (2020)

  35. Shaw, S.: Theory of generalized thermoelasticity with memory-dependent derivatives. J. Eng. Mech. 145(3), 04019003 (2019)

    MathSciNet  Google Scholar 

  36. Ezzat, M.A., El-Bary, A.A.: Magneto-thermoelectric viscoelastic materials with memory-dependent derivative involving two-temperature. Int. J. Appl. Electromagn. Mech. 50(4), 549–567 (2016)

    Google Scholar 

  37. Ezzat, M.A., El-Karamany, A.S., El-Bary, A.A.: On dual-phase-lag thermoelasticity theory with memory-dependent derivative. Mech. Adv. Mater. Struct. 24(11), 908–916 (2017)

    Google Scholar 

  38. Singh, B., Pal, S., Barman, K.: Eigenfunction approach to generalized thermo-viscoelasticity with memory dependent derivative due to three-phase-lag heat transfer. J. Therm. Stress. 43(9), 1100–1119 (2020)

    Google Scholar 

  39. Sarkar, I., Mukhopadhyay, B.: On the spatial behavior of thermal signals in generalized thermoelasticity with memory dependent derivative. Act. Mech. 231, 2989–3001 (2020)

    MathSciNet  MATH  Google Scholar 

  40. Ezzat, M.A., El-Karamany, A.S., El-Bary, A.A.: Thermoelectric viscoelastic materials with memory-dependent derivative. Struct. Sys. 19, 539–551 (2017)

    Google Scholar 

  41. Singh, B., Pal, S.: Magneto thermoelastic interaction with memory response due to laser pulse under Green–Naghdi theory in an orthotropic medium. Mech. Based Des. Struct. Mach. (2020). https://doi.org/10.1080/15397734.2020.1798780

    Article  Google Scholar 

  42. Singh, B., Sarkar, I., Pal, S.: Temperature-rate-dependent thermoelasticity theory twith memory-dependent derivative: energy, uniqueness theorems, and variational principle. J. Heat Transf. 142(10), 102103 (2020)

    Google Scholar 

  43. Lotfy, K., Sarkar, N.: Memory-dependent derivatives for photo thermal semiconducting medium in generalized thermoelasticity with two-temperature. Mech. Time-Depend. Mater. 21(4), 519–534 (2017)

    Google Scholar 

  44. Singh, B., Pal, S., Barman, K.: Thermoelastic interaction in the semi-infinite solid medium due to three-phase-lag effect involving memory-dependent derivative. J. Therm. Stress. 42(7), 874–889 (2019)

    Google Scholar 

  45. Das, N.C., Lahiri, A., Giri, R.R.: Eigenvalue approach to generalized thermoelasticity. Indian J. Pure Appl. Math. 28, 1573–1594 (1997)

    MathSciNet  MATH  Google Scholar 

  46. Green, A.E., Naghdi, P.M.: Thermoelasticity without energy dissipation. J. Elastic. 31, 189–208 (1993)

    MathSciNet  MATH  Google Scholar 

  47. Bellman, R., Kalaba, R.E., Lockett, J.: Numerical Inversion of the Laplace Transform. American Elsevier, New York (1966)

    MATH  Google Scholar 

  48. Baksi, A., Bera, R.K., Debnath, L.: A study of magneto-thermoelastic problems with thermal relaxation and heat sources in a three-dimensional infinite rotating elastic medium. Int. J. Eng. Sci. 43, 1419–1434 (2005)

    MathSciNet  MATH  Google Scholar 

  49. Othman, M.I., Ezzat, M.A., Zaki, S.A., El-Karamany, A.S.: Generalized thermo-viscoelastic plane waves with two relaxation times. Int. J. Eng. Sci. 40(12), 1329–1347 (2002)

    MathSciNet  MATH  Google Scholar 

  50. Ezzat, M.A., Othman, M.I., El-Karamany, A.S.: State space approach to generalized thermo-viscoelasticity with two relaxation times. Int. J. Eng. Sci. 40(3), 283–302 (2002)

    MathSciNet  MATH  Google Scholar 

  51. Ezzat, M.A., El-Karamany, A.S., Samaan, A.A.: State-space formulation to generalized thermoviscoelasticity with thermal relaxation. J. Therm. Stress. 24(9), 823–846 (2001)

    Google Scholar 

  52. Ezzat, M.A., El-Karamany, A.S.: Fractional thermoelectric viscoelastic materials. J. Appl. Polym. Sci. 124(3), 2187–2199 (2012)

    Google Scholar 

  53. Ezzat, M.A., El-Karamany, A.S.: On uniqueness and reciprocity theorems for generalized thermo-viscoelasticity with thermal relaxation. Can. J. Phys. 81(6), 823–833 (2003)

    Google Scholar 

  54. Ezzat, M.A., El-Karamany, A.S.: The uniqueness and reciprocity theorems for generalized thermo-viscoelasticity with two relaxation times. Int. J. Eng. Sci. 40(11), 1275–1284 (2002)

    MathSciNet  MATH  Google Scholar 

  55. Sherief, H.H., Allam, M.N., El-Hagary, M.A.: Generalized theory of thermoviscoelasticity and a half-space problem. Int. J. Thermophys. 32(6), 1271–1295 (2011)

    Google Scholar 

  56. Sarkar, I., Mukhopadhyay, B.: Generalized thermo-viscoelasticity with memory-dependent derivative: uniqueness and reciprocity. Arch. Appl. Mech. 91(3), 965–977 (2021)

    Google Scholar 

  57. Christensen, R.M.: Theory of Viscoelasticity. Academic Press, New York (1982)

    Google Scholar 

  58. Othman, M.I.: Effect of rotation in case of 2-D problem of the generalized thermo-viscoelasticity with two relaxation times. Mech. Mecha. Eng. 13(2), 105–127 (2009)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah, West Bengal, 711103, India

    Biswajit Singh & Smita Pal Sarkar

  2. Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Paschim Medinipur, West Bengal, 721102, India

    Krishnendu Barman

Authors
  1. Biswajit Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Smita Pal Sarkar
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Krishnendu Barman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Krishnendu Barman.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Singh, B., Sarkar, S.P. & Barman, K. Modeling of Memory Dependent Derivative Under Three-Phase Lag in Generalized Thermo-Viscoelasticity. Int. J. Appl. Comput. Math 7, 243 (2021). https://doi.org/10.1007/s40819-021-01174-4

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40819-021-01174-4

Keywords

Search

Navigation