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Two-temperature theory in Green–Naghdi thermoelasticity with fractional phase-lag heat transfer

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Abstract

A mathematical model of two-temperature phase-lag Green–Naghdi thermoelasticty theories based on fractional derivative heat transfer is given. The GN theories as well as the theories of coupled and of generalized thermoelasticity with thermal relaxation follow as limit cases. The resulting non dimensional coupled equations together with the Laplace transforms techniques are applied to a specific problem of a half space subjected to arbitrary heating which is taken as a function of time and is traction free. The inverse transforms are obtained by using a numerical method based on Fourier expansion techniques. The predictions of the theory are discussed and compared with those for the generalized theory of thermoelasticity with one relaxation time. The effects of temperature discrepancy and fractional order parameters on copper-like material are discussed in different types of GN theories.

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Abbreviations

\(a_{1} ,a_{2} ,a_{3}\) :

Key numbers, each equals to 0 or 1

x :

(x 1, x 2, x 3), position

t :

Time

\(C_{E}\) :

Specific heat at constant strain

\(k_{ij}\) :

Thermal conductivity tensor

\(k_{ij}^{ * }\) :

Conductivity rate tensor

T :

Temperature

\(T_{o}\) :

Reference temperature

\(u_{i}\) :

Components of displacement vector

\(\text{v}\) :

\(\left[ {(\lambda + 2{\kern 1pt} \mu \,){\kern 1pt} /\,\rho } \right]{\kern 1pt}^{1/2}\), speed of propagation of longitudinal waves

q i :

Components of heat flux vector

e :

\(u_{i,i} ,\) dilatation

a :

Temperature discrepancy

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

Lame’s constants

\(\rho\) :

Density

\(\alpha_{T}\) :

Coefficient of linear thermal expansion

\(\upsilon\) :

Thermal displacement \(\dot{\upsilon } = \theta\)

\(\varepsilon\) :

Thermoelastic coupling parameter

\(\gamma\) :

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

\(\delta_{ij}\) :

Kronecker delta function

\(\tau_{o}\) :

Relaxation time

\(\tau_{q}\) :

Phase-lag of the heat flux

\(\tau_{\theta }\) :

Phase-lag of the temperature gradient

\(\tau_{\upsilon }\) :

Phase-lag of the thermal displacement gradient

\(\alpha\) :

Fractional derivative order

\(\sigma_{ij}\) :

Components of stress tensor

\(\eta\) :

\(\rho C_{E} /k\)

\(\theta\) :

\(T - T_{0}\), such that \(\left| {\theta /T_{0} } \right| < < 1\)

\(\phi\) :

Conductive temperature

\(\varphi\) :

\(\phi - \varphi_{0}\), such that \(\left| {\phi /\varphi_{0} } \right| < < 1\)

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Authors and Affiliations

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

    Magdy A. Ezzat

  2. Department of Mathematical and Physical Sciences, Nizwa University, Nizwa-611, P.O. Box 1357, Nizwa, Oman

    Ahmed S. El-Karamany

  3. Arab Academy for Science and Technology, P.O. Box 1029, Alexandria, Egypt

    Alaa A. El-Bary

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  1. Magdy A. Ezzat
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  2. Ahmed S. El-Karamany
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  3. Alaa A. El-Bary
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Correspondence to Magdy A. Ezzat.

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Ezzat, M.A., El-Karamany, A.S. & El-Bary, A.A. Two-temperature theory in Green–Naghdi thermoelasticity with fractional phase-lag heat transfer. Microsyst Technol 24, 951–961 (2018). https://doi.org/10.1007/s00542-017-3425-6

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