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

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A Correction to this article was published on 12 January 2021

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Abstract

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.

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Change history

  • 12 January 2021

    The University of the corresponding author should be Department of Mathematics, College of Science and Arts, Qassim University, Al Bukairiyah, Saudi Arabia.

Abbreviations

λ, μ :

Lame’s constants

ρ :

Density

t :

Time

C E :

Specific heat at constant strain

\(B_{i}\) :

Components of magnetic field strength

E i :

Components of electric field vector

J i :

Conduction electric density vector

H i :

Magnetic field intensity

q i :

Components of heat flux vector

H o :

Constant component of magnetic field

μ o :

Magnetic permeability

σ o :

Electric conductivity

ε ijk :

Permutation symbol

σ ij :

Components of stress tensor

\(e_{ij}\) :

Components of strain tensor

u i :

Components of displacement vector

\(\theta\) :

\(= T - T_{o}\)

T o :

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

e :

= ui,i dilatation

\(k\) :

Thermal conductivity

\(\alpha_{T}\) :

Coefficient of linear thermal expansion

\(\gamma\) :

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

πo :

Peltier coefficient at To

k o :

Seebeck coefficient at To

\(\varepsilon\) :

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

\(M\) :

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

\(\eta\) :

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

\(\eta_{o}\) :

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

\(c_{o}^{2}\) :

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

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Acknowledgements

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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Author notes

  1. Magdy A. Ezzat

    Present address: Faculty of Science and Arts, Al-Qassim University, Al-Bukairyah, Saudi Arabia

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Northern Border University, Arar, Kingdom of Saudi Arabia

    Mohamed H. Hendy & Sayed I. El-Attar

  2. Department of Mathematics, Faculty of Science, Al Arish University, Arish, Egypt

    Mohamed H. Hendy

  3. Department of Mathematics, Faculty of Education, Alexandria University, Alexandria, Egypt

    Magdy A. Ezzat

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  1. Mohamed H. Hendy
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Correspondence to Magdy A. Ezzat.

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Hendy, M.H., El-Attar, S.I. & Ezzat, M.A. On thermoelectric materials with memory-dependent derivative and subjected to a moving heat source. Microsyst Technol 26, 595–608 (2020). https://doi.org/10.1007/s00542-019-04519-8

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