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On dual-phase-lag magneto-thermo-viscoelasticity theory with memory-dependent derivative

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Abstract

A new mathematical model of generalized magneto-thermo-viscoelasticity theories with memory-dependent derivatives (MDD) of dual-phase-lag heat conduction law is developed. The equations for one-dimensional problems including heat sources are cast into matrix form using the state space and Laplace transform techniques. The resulting formulation is applied to a problem for the whole space with a plane distribution of heat sources. It is also applied to a perfect conducting semi-space problem with a traction-free surface and plane distribution of heat sources located inside the medium. The inversion of the Laplace transforms is carried out using a numerical approach. Numerical results for the temperature, displacement, stress and heat flux distributions as well as the induced magnetic and electric fields are given and illustrated graphically. A comparison is made with the results obtained in the coupled theory. The impacts of the MDD heat transfer parameter and Alfven velocity on a viscoelastic material, for example, poly (methyl methacrylate) (Perspex) are discussed.

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Abbreviations

t :

Time

\( \varvec{x} \) :

Space coordinates

\( u_{i} \) :

Components of displacement vector

\( q_{i} \) :

Components of heat flux

\( e_{ij} \) :

Components of strain tensor

C o :

\( \sqrt {\frac{\lambda \, + \,\;2\,\mu }{\rho }} \), Speed of propagation of isothermal elastic waves

\( K_{o} \) :

\( \lambda \; + (2/3)\,\mu \), bulk modulus

\( C_{E} \) :

Specific heat at constant strain

\( \sigma_{o} \) :

The electrical conductivity

\( \varvec{H} \) :

Magnetic intensity vector

\( \varvec{E} \) :

Electric intensity vector

\( \mu_{o} \) :

Magnetic permeability

\( \varepsilon_{o} \) :

Electric permeability

\( \varvec{B} \) :

\( \mu_{o} \varvec{H}, \), magnetic induction vector

J :

Conduction current density vector

\( e \) :

Dilatation

\( k \) :

Thermal conductivity

\( T \) :

Absolute thermodynamic temperature

\( T_{ \circ } \) :

Reference temperature

\( \vartheta \) :

\( T - T_{o} \); \( \left| {\,\frac{\vartheta }{{T_{o} }}} \right| \ll 1 \)

\( \gamma \) :

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

\( \delta (.) \) :

Dirac delta function

\( \delta_{ij} \) :

Kronecker’s delta

\( \varepsilon \) :

Thermal coupling parameter

\( \varepsilon_{ij} \) :

Components of strain tensor

\( \sigma_{ij} \) :

Components of stress tensor

\( S_{ij} \) :

Components of stress deviator tensor

\( \lambda ,\;\mu \) :

Lame’ constants

\( \rho \) :

Mass Density

\( \eta_{o} \) :

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

\( \varepsilon \) :

\( \frac{{ \, \gamma^{2} T_{o} }}{{k\eta_{o} \rho C_{o}^{2} }} \), Thermal coupling parameter

\( \alpha_{o} \) :

\( \sqrt {\frac{{\mu_{o} \,H_{o}^{2} }}{\rho }} \), Alfven velocity

\( \tau \) :

Relaxation time

\( \tau_{q} ,\tau_{\theta } \) :

Phase-lags

\( {\varGamma (}\text{.)} \) :

Gamma function

Q :

Strength of applied heat source per unit mass

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Acknowledgements

The authors gratefully acknowledge the approval and the support of this research study by the Grant No. SCI-2017-1-8-F-7322 from the Deanship of Scientific Research in Northern Border University, Arar, KSA.

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Authors and Affiliations

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

    Dalia A. Aldawody & Mohamed H. Hendy

  2. Department of Mathematics, Faculty of Science, Port Said University, Port Said, Egypt

    Dalia A. Aldawody

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

    Mohamed H. Hendy

  4. Department of Mathematic, Faculty of Education, Alexandria University, Alexandria, Egypt

    Magdy A. Ezzat

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  1. Dalia A. Aldawody
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  2. Mohamed H. Hendy
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  3. Magdy A. Ezzat
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Correspondence to Magdy A. Ezzat.

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Aldawody, D.A., Hendy, M.H. & Ezzat, M.A. On dual-phase-lag magneto-thermo-viscoelasticity theory with memory-dependent derivative. Microsyst Technol 25, 2915–2929 (2019). https://doi.org/10.1007/s00542-018-4194-6

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