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On Thermo-viscoelasticity with Variable Thermal Conductivity and Fractional-Order Heat Transfer

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Abstract

The equations of generalized thermo-viscoelasticity for an isotropic medium with variable thermal conductivity and fractional-order heat transfer are given. The resulting formulation is applied to a half-space subjected to arbitrary heating which is taken as a function of time and is traction free. The Laplace transform technique is used. A numerical method is employed for the inversion of the Laplace transforms. Numerical results for temperature, displacement, and stress distributions are given and illustrated graphically for the problem. The effects of the fractional order and the variable thermal conductivity for heat transfer on a viscoelastic material such as poly(methyl methacrylate) (Perspex) are discussed.

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Abbreviations

\(\lambda , \mu \) :

Lamé constants

\(\rho \) :

Mass density

\(t\) :

Time

\(T\) :

Temperature

\(e\) :

Dilatation

\(u_i\) :

Components of displacement vector

\(\sigma _{ij}\) :

Components of stress tensor

\(e_{ij}\) :

Components of strain deviator tensor

\(S_{ij}\) :

Components of stress deviator tensor

\(\varepsilon _{ij}\) :

Components of strain tensor

\(T_\circ \) :

Reference temperature

\(\varepsilon \) :

\( = \frac{\gamma T_\mathrm{o} k_\mathrm{o} }{(\lambda +2\mu )\kappa _\mathrm{o} },\) thermal coupling parameter

\(\eta _\mathrm{o}\) :

\(=\frac{\rho C_E }{k}\)

\(k_\mathrm{o}\) :

Thermal diffusivity

\(C_E \) :

Specific heat at constant strain

\(k\) :

Thermal conductivity

\(\alpha \) :

Fractional order

\(\tau _\mathrm{o}\) :

Relaxation time

\(\alpha _T\) :

Coefficient of linear thermal expansion

K :

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

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

\(= \frac{K}{\rho },\) longitudinal wave speed

\(\gamma \) :

\(= (3\lambda +2\mu )\alpha _T\)

\(\delta (\cdot )\) :

Dirac delta function

\(\delta _{ij} \) :

Kronecker delta

\(\varGamma (\cdot )\) :

Gamma function

\(H(\cdot )\) :

Heaviside unit step function

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Author notes

  1. M. A. Ezzat

    Present address: Department of Mathematics, Faculty of Science and Letter in Al Bukayriyyah, Al-Qassim University, Al-Qassim, Saudi Arabia

Authors and Affiliations

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

    M. A. Ezzat

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

    A. S. El-Karamany

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

    A. A. El-Bary

Authors
  1. M. A. Ezzat
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  2. A. S. El-Karamany
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  3. A. A. El-Bary
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Ezzat, M.A., El-Karamany, A.S. & El-Bary, A.A. On Thermo-viscoelasticity with Variable Thermal Conductivity and Fractional-Order Heat Transfer. Int J Thermophys 36, 1684–1697 (2015). https://doi.org/10.1007/s10765-015-1873-8

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