Generalized thermoviscoelastic novel model with different fractional derivatives and multi-phase-lags
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Generalized thermoviscoelastic novel model with different fractional derivatives and multi-phase-lags A. Soleiman1,2,a M. E. Nasr1,2,d
, Ahmed E. Abouelregal1,3,b
, K. M. Khalil1,2,c
,
1 Department of Mathematics, College of Science and Arts, Jouf University, Gurayat, Saudi Arabia 2 Department of Mathematics, Faculty of Science, Benha University, Benha 13518, Egypt 3 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
Received: 12 July 2020 / Accepted: 6 October 2020 © Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2020
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 timedependent 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.
List of symbols μ0, λ0 μv, λv αt Ce γ 0 (3 λ0 + 2 μ0) α t T0 θ T – T0 T u e div u σi j
Lame’s constants Thermoviscoelastic relaxation times Thermal expansion coefficient Specific heat Thermal coupling parameter Environmental temperature Temperature increment Absolute temperature Displacement vector Cubical dilatation Stress tensor
a e-mail: [email protected] (corresponding author) b e-mail: [email protected] c e-mail: [email protected] d e-mail: [email protected]
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K ρ Q t δi j E τq τθ eij q
Eur. Phys. J. Plus
(2020) 135:851
Thermal conductivity Material density Heat source The time Kronecker’s delta function Induced electric field Phase lag of heat flux Phase lag of temperature Strain tensor Heat flux vector
1 Introduction Generalized thermoelastic models have been progressed to eliminate the contradiction in the infinite velocity of heat propagation concealed in the classical dynamical coupled thermoelasticity theory [1]. In these generalized models, the basic equations contain thermal relaxation times of hyperbolic type [2–5]. Furthermore, Tzou [6–8] investigated the dual-phase-lag heat conduction theory by including two different phase delays correlating with the heat flow and temperature gradient. Chandrasekharaiah [9] introduced a generalized model improved from the heat conduction model established by Tzou [7, 8]. In the recent past, fractional calculus has been effectively applied in many
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