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Generalized thermoviscoelastic novel model with different fractional derivatives and multi-phase-lags

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Abstract

In the current investigation, we introduce a generalized modified model of thermoviscoelasticity with different fractional orders. Based on the Kelvin–Voigt model and generalized thermoelasticity theory with multi-phase-lags, the governing system equations are derived. In limited cases, the proposed model is reduced to several previous models in the presence and absence of fractional derivatives. The model is then adopted to investigate a problem of an isotropic spherical cavity, the inner surface of which is exposed to a time-dependent varying heat and constrained. The system of governing differential equations has been solved analytically by applying the technique of Laplace transform. To clarify the effects of the fractional-order and viscoelastic parameters, we depicted our numerical calculations in tables and figures. Finally, the results obtained are discussed in detail and also confirmed with those in the previous literature.

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Data availability

This manuscript has associated data in a data repository. [Authors’ comment: All data generated or analysed during this study are included in this published article [and its supplementary information files].]

Abbreviations

μ 0, λ 0 :

Lame’s constants

μ v, λ v :

Thermoviscoelastic relaxation times

α t :

Thermal expansion coefficient

C e :

Specific heat

γ 0 = (3 λ 0 + 2 μ 0) α t :

Thermal coupling parameter

T 0 :

Environmental temperature

θ = T – T 0 :

Temperature increment

T:

Absolute temperature

\( \vec{u} \) :

Displacement vector

\( e = {\text{div}}\;\vec{u} \) :

Cubical dilatation

\( \sigma_{ij} \) :

Stress tensor

K :

Thermal conductivity

ρ :

Material density

Q :

Heat source

t :

The time

\( \delta_{ij} \) :

Kronecker’s delta function

\( \vec{E} \) :

Induced electric field

τ q :

Phase lag of heat flux

τ θ :

Phase lag of temperature

e ij :

Strain tensor

q :

Heat flux vector

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Authors and Affiliations

  1. Department of Mathematics, College of Science and Arts, Jouf University, Gurayat, Saudi Arabia

    A. Soleiman, Ahmed E. Abouelregal, K. M. Khalil & M. E. Nasr

  2. Department of Mathematics, Faculty of Science, Benha University, Benha, 13518, Egypt

    A. Soleiman, K. M. Khalil & M. E. Nasr

  3. Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, 35516, Egypt

    Ahmed E. Abouelregal

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  1. A. Soleiman
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  2. Ahmed E. Abouelregal
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  3. K. M. Khalil
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  4. M. E. Nasr
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Correspondence to A. Soleiman.

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Soleiman, A., Abouelregal, A.E., Khalil, K.M. et al. Generalized thermoviscoelastic novel model with different fractional derivatives and multi-phase-lags. Eur. Phys. J. Plus 135, 851 (2020). https://doi.org/10.1140/epjp/s13360-020-00842-6

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