On thermoelectric materials with memory-dependent derivative and subjected to a moving heat source

  • Mohamed H. Hendy
  • Sayed I. El-Attar
  • Magdy A. EzzatEmail author
Technical paper


We develop a model of generalized thermoelasticity with memory-dependent derivative (MDD) heat conduction law for a thermoelectric half-space. Some urgent theories take after as most remote point cases. The Laplace transform and state-space procedures are utilized to urge the overall account for any arrangement of limit conditions. The general solution acquired is connected to the particular issue of a half-space exposed to a uniform magnetic field, a moving heat source with consistent speed and ramp-type heating. The inverse Laplace transforms are registered numerically. The impacts of various estimations of the figure-of-merit quantity, heat source speed, MDD parameters, the magnetic number and the ramping time parameter are thought about.

List of symbols

λ, μ

Lame’s constants






Specific heat at constant strain


Components of magnetic field strength


Components of electric field vector


Conduction electric density vector


Magnetic field intensity


Components of heat flux vector


Constant component of magnetic field


Magnetic permeability


Electric conductivity


Permutation symbol


Components of stress tensor


Components of strain tensor


Components of displacement vector


\(= T - T_{o}\)


Reference temperature chosen so that \(\left| {T \, - \, T_{o} } \right|/T_{o}\) ≪ 1


= ui,i dilatation


Thermal conductivity


Coefficient of linear thermal expansion


= \((3\lambda + 2\mu )\alpha_{T}\)


Peltier coefficient at To


Seebeck coefficient at To


\(= \, \frac{{\delta_{o} \,\gamma }}{{\rho \,C_{E} }}\) thermoelastic parameter


\(= \, \frac{{\sigma_{o} \,B_{o}^{2} }}{{\eta_{o} \rho \,c_{o}^{2} }}\) magnetic number


\(= \frac{1}{{\sigma_{o} \mu_{o} }}\) magnetic diffusivity


\(= \frac{{\rho \,C_{E} }}{k}\)


\(= \;(\lambda \, + \,\;2\,\mu )/\rho\)



The authors gratefully acknowledge the approval and the support of this research study by the Grant no. SCI-2018-3-9-F-7583 from the Deanship of Scientific Research in Northern Border University, Arar, KSA.


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Copyright information

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

Authors and Affiliations

  • Mohamed H. Hendy
    • 1
    • 2
  • Sayed I. El-Attar
    • 1
  • Magdy A. Ezzat
    • 3
    • 4
    Email author
  1. 1.Department of Mathematics, Faculty of ScienceNorthern Border UniversityArarKingdom of Saudi Arabia
  2. 2.Department of Mathematics, Faculty of ScienceAl Arish UniversityArishEgypt
  3. 3.Department of Mathematics, Faculty of EducationAlexandria UniversityAlexandriaEgypt
  4. 4.Faculty of Science and ArtsAl-Qassim UniversityAl-BukairyahSaudi Arabia

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