Acta Mechanica

, Volume 230, Issue 7, pp 2325–2338 | Cite as

Transient response in a piezoelastic medium due to the influence of magnetic field with memory-dependent derivative

  • Sudip Mondal
  • Abhik Sur
  • M. KanoriaEmail author
Original Paper


Enlightened by the Caputo fractional derivative, the present study deals with a novel mathematical model of generalized thermoelasticity to investigate the transient phenomena for a piezoelastic half-space due to the influence of a magnetic field in the context of the dual-phase-lag model of generalized thermoelasticity, which is defined in an integral form of a common derivative on a slipping interval by incorporating the memory-dependent heat transfer. The bounding plane of the medium is assumed to be stress free and subjected to a thermal shock. Employing the Laplace transform as a tool, the problem is transformed to the space domain, where the solution in the space–time domain is achieved by applying a suitable numerical technique based on Fourier series expansion technique. According to the graphical representations corresponding to the numerical results, conclusions about the new theory are constructed. Excellent predictive capability is demonstrated due to the presence of electric field, memory-dependent derivative and magnetic field.


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We are grateful to the Editor and reviewers for their valuable suggestions for the improvement of the paper.


  1. 1.
    Sur, A., Kanoria, M.: Propagation of thermal waves in a functionally graded thick plate. Math. Mech. Solids 22(4), 718–736 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Sur, A., Kanoria, M.: Thermoelastic interaction in a viscoelastic functionally graded half-space under three phase lag model. Eur. J. Comput. Mech. 23, 179–198 (2014)CrossRefGoogle Scholar
  3. 3.
    Das, P., Kanoria, M.: Study of finite thermal waves in a magneto-thermo-elastic rotating medium. J. Therm. Stress. 37, 405–428 (2014)CrossRefGoogle Scholar
  4. 4.
    Das, P., Kar, A., Kanoria, M.: Analysis of magneto-thermoelastic response in a transversely isotropic hollow cylinder under thermal shock with three-phase-lag effect. J. Therm. Stress. 36, 239–258 (2013)CrossRefGoogle Scholar
  5. 5.
    Sur, A., Pal, P., Kanoria, M.: Modeling of memory-dependent derivative in a fibre-reinforced plate under gravitational effect. J. Therm. Stress. 41(8), 973–992 (2018). CrossRefGoogle Scholar
  6. 6.
    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. Phys. 57(4), 652–665 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Sur, A., Kanoria, M.: Finite thermal wave propagation in a half-space due to variable thermal loading. Appl. Appl. Math. 9(1), 94–120 (2014)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Tzou, D.Y.: A unified approach for heat conduction from macro to micro-scales. ASME J. Heat Transf. 117, 8–16 (1995)CrossRefGoogle Scholar
  9. 9.
    Ignaczak, J., Ostoja-Starzewski, M.: Thermoelasticity with Finite Wave Speeds. Oxford University Press, New York. ISBN: 978-0-19-954164-5 (2010)Google Scholar
  10. 10.
    Quintanilla, R., Racke, R.: A note on stability in dual-phase-lag heat conduction. Int. J. Heat Mass Transf. 49, 1209–1213 (2006)CrossRefzbMATHGoogle Scholar
  11. 11.
    Chakravorty, S., Ghosh, S., Sur, A.: Thermo-viscoelastic interaction in a three-dimensional problem subjected to fractional heat conduction. Procedia Engrng. 173, 851–858 (2017)CrossRefGoogle Scholar
  12. 12.
    Sur, A., Kanoria, M.: Fractional order generalized thermoelastic functionally graded solid with variable material properties. J. Solid Mech. 6, 54–69 (2014)Google Scholar
  13. 13.
    Sur, A., Kanoria, M.: Three dimensional thermoelastic problem under two-temperature theory. Int. J. Comput. Methods. 14(3), 1750030 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, Berlin (2010)CrossRefzbMATHGoogle Scholar
  15. 15.
    Caputo, M.: Linear models of dissipation whose Q is almost frequency independent II. Geophys. J. R. Astron. Soc. 3, 529–539 (1967)CrossRefGoogle Scholar
  16. 16.
    Caputo, M., Mainardi, F.: Linear model of dissipation in anelastic solids. Rivis. ta. el. Nuovo. cimento. 1, 161–198 (1971)CrossRefGoogle Scholar
  17. 17.
    Sur, A., Kanoria, M.: Fractional order two-temperature thermoelasticity with finite wave speed. Acta Mech. 223, 2685–2701 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Purkait, P., Sur, A., Kanoria, M.: Thermoelastic interaction in a two dimensional infinite space due to memory dependent heat transfer. Int. J. Adv. Appl. Math. Mech. 5(1), 28–39 (2017)MathSciNetGoogle Scholar
  19. 19.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Ezzat, M.A., El-Karamany, A.S., El-Bary, A.A.: Electro-thermoelasticity theory with memory-dependent derivative heat transfer. Int. J. Eng. Sci. 99, 22–38 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Ezzat, M.A., El-Bary, A.A.: Memory-dependent derivatives theory of thermo-viscoelasticity involving two-temperature. J. Mech. Sci. Technol. 29, 4273–4279 (2015)CrossRefGoogle Scholar
  22. 22.
    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, 545–553 (2016)CrossRefGoogle Scholar
  23. 23.
    Ezzat, M.A., El-Karamany, A.S., El-Bary, A.A.: Modeling of memory-dependent derivative in generalized thermoelasticity. Eur. Phys. J. Plus 131, 372 (2016)CrossRefGoogle Scholar
  24. 24.
    Lotfy, K., Sarkar, N.: Memory-dependent derivatives for photothermal semiconducting medium in generalized thermoelasticity with two-temperature. Mech. Time-Dep. Mater. 21(4), 519–534 (2017). CrossRefGoogle Scholar
  25. 25.
    Wang, J.L., Li, H.F.: Surpassing the fractional derivative: concept of the memory-dependent derivative. Comput. Math. Appl. 62, 1562–1567 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Sur, A., Pal, P., Mondal, S., Kanoria, M.: Finite element analysis in a fiber-reinforced cylinder due to memory-dependent heat transfer. Acta Mech. (2019). Google Scholar
  27. 27.
    El-Karamany, A.S., Ezzat, M.A.: Thermoelastic diffusion with memory-dependent derivative. J. Therm. Stress. 39, 1035–1050 (2016)CrossRefGoogle Scholar
  28. 28.
    Sur, A., Kanoria, M.: Modeling of memory-dependent derivative in a fibre-reinforced plate. Thin Walled Struct. (2017). zbMATHGoogle Scholar
  29. 29.
    Sherief, H.H., Helmy, A.K.: A two dimensional problem for a half-space in magneto-thermoelasticity with thermal relaxation. Int. J. Eng. Sci. 40, 587–604 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Othman, M.I.A.: Generalized electromagneto-thermoelastic plane waves by thermal shock problem in a finite conductivity half-space with one relaxation time. Multidiscip. Model. Mater. Struct. 1, 231–250 (2005)CrossRefGoogle Scholar
  31. 31.
    Othman, M.I.A., Zidan, M.E.M., Hilal, M.I.M.: The effect of magnetic field on a rotating thermoelastic medium with voids under thermal loading due to laser pulse with energy dissipation. Canad. J. Phys. 92, 1359–1371 (2014)CrossRefGoogle Scholar
  32. 32.
    Sur, A., Kanoria, M.: Fibre-reinforced magneto-thermoelastic rotating medium with fractional heat conduction. Proc. Eng. 127, 605–612 (2015)CrossRefGoogle Scholar
  33. 33.
    Sur, A., Kanoria, M.: Modeling of fibre-reinforced magneto-thermoelastic plate with heat sources. Procedia Eng. 173, 875–882 (2017)CrossRefGoogle Scholar
  34. 34.
    Kalamkarov, A.L.: Modeling of anisotropic magneto-piezoelastic materials. In: Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory (DIPED), XXIst International Seminar/Workshop, IEEE, pp. 113–117 (2016)Google Scholar
  35. 35.
    Nixdorf, T.A., Pan, E.: Static plane-strain deformation of transversely isotropic magneto-electro-elastic and layered cylinders to general surface loads. Appl. Math. Model. 60, 208–219 (2018)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Nedin, R.D., Dudarev, V.V., Vatulyan, A.O.: Vibrations of inhomogeneous piezoelectric bodies in conditions of residual stress–strain state. Appl. Math. Model. 63, 219–242 (2018)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Mindlin, R.D.: On the equations of motion of piezoelectric crystals. In: Muskilishivili, N.I. (ed.) Problems of Continuum Mechanics, 70th Birthday Volume, pp. 282–290. SIAM, Philadelphia (1961)Google Scholar
  38. 38.
    Tiersten, H.F.: On the nonlinear equations of thermoelectroelasticity. Int. J. Eng. Sci. 9, 587–604 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Mindlin, R.D.: Equations of high frequency vibrations of thermo-piezoelectric plate. Int. J. Solids Struct. 10, 625–637 (1974)CrossRefzbMATHGoogle Scholar
  40. 40.
    Nowacki, W.: Foundations of linear piezoelectricity, In H. Parkus (ed.), Electromagnetic Interactions in Elastic Solids. Springer, Wein, Chapter 1 (1979)Google Scholar
  41. 41.
    Aouadi, M.: Generalized thermo-piezoelectric problems with temperature-dependent properties. Int. J. Solids Struct. 43, 6347–6358 (2006)CrossRefzbMATHGoogle Scholar
  42. 42.
    Banik, S., Kanoria, M.: Study of two-temperature generalized thermo-piezoelastic problem. J. Therm. Stress. 36, 71–93 (2013)CrossRefGoogle Scholar
  43. 43.
    Ezzat, M.A., El-Karamany, A.S., El-Bary, A.A.: A novel magneto-thermoelasticity theory with memory-dependent derivative. J. Electromagn. Waves Appl. 29(8), 1018–1031 (2015). CrossRefGoogle Scholar
  44. 44.
    Honig, G., Hirdes, U.: A method for the numerical inversion of the Laplace transform. J. Comput. Appl. Math. 10, 113–132 (1984)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsBasirhat CollegeKolkataIndia
  2. 2.Department of MathematicsTechno IndiaKolkataIndia
  3. 3.Department of Applied MathematicsUniversity of CalcuttaKolkataIndia
  4. 4.Department of MathematicsSister Nivedita UniversityNewtownIndia

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