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Analysis of thermoelectric viscoelastic wave characteristics in the presence of a continuous line heat source with memory dependent derivatives

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Abstract

In the present work, the definition of memory dependent derivative (MDD) heat transfer in an infinite solid body was used to investigate the problem of wave characteristics in an unbounded thermoelectric viscoelastic solid caused by a continuous line heat source in the presence of a uniform magnetic field. Both Laplace and Hankel transform strategies are used to acquire the widespread answer in a closed form. Analytical findings were obtained for the distribution within the medium of various fields such as temperature, displacement, and stresses. For the inversion of the Laplace transformations, a computational approach is used. The distributions of the numerical consequences of the non-dimensional considered bodily variables are represented graphically. Detailed comparative evaluation is represented thru the numerical outcomes to estimate the results of the kernels, time-delay and magnetic number on the behavior of all variable. The effect offers a concept to research main thermoelectric viscoelastic materials as any other type of pertinent materials.

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Abbreviations

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

Lame' constants

\(\rho\) :

Density

t :

Time

T :

Absolute temperature

T o :

Reference temperature

\(u_{i}\) :

Components of displacement tensor

\(\sigma_{ij}\) :

Components of stress tensor

e :

Dilatation

\(e_{ij}\) :

Components of strain deviator tensor

\(S_{ij}\) :

Components of stress deviator tensor

\(\varepsilon_{ij}\) :

Components of strain tensor

E i :

Components of electric field vector

J i :

Components of electric density vector

H i :

Components of magnetic field intensity

\(\mu_{{\text{o}}}\) :

Magnetic permeability

\(\sigma_{{\text{o}}}\) :

Electric conductivity

\(s_{{\text{o}}}\) :

Seebeck coefficient

\(\pi_{{\text{o}}}\) :

Peltier coefficient

M :

Magnetic parameter

\(\varepsilon\) :

\(\frac{{\gamma T_{{\text{o}}} {\kern 1pt} k}}{{(\lambda + 2\mu )\kappa_{{\text{o}}} }},\) Thermoelastic coupling parameter

Q :

Strength of the heat source

\(k_{\text{o}}\) :

Thermal diffusivity

\(k\) :

Thermal conductivity

q i :

Components of heat flux vector

\(\alpha_{{\text{T}}}\) :

Coefficient of linear thermal expansion

\(K_{{\text{o}}}\) :

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

\({\text{c}}_{{\text{o}}}^{{2}}\) :

\(K_{{\text{o}}} /\rho ,\) Longitudinal wave speed

\(C_{E}\) :

Specific heat at constant strain

\(\eta\) :

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

\(\gamma\) :

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

\(\delta (.)\) :

Dirac delta function

\(\delta_{ij}\) :

Kronecker’s delta

\(\Gamma (.)\) :

Gamma function

\(H(.)\) :

Heaviside unit step function

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Authors and Affiliations

  1. Department of Mathematics, College of Science and Arts, Qassim University, Al Bukairyah, Saudi Arabia

    Magdy A. Ezzat

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

    Magdy A. Ezzat

  3. Basic and Applied Science Institute, Arab Academy for Science, Technology and Maritime Transport, P.O. Box 1029, Alexandria, Egypt

    Alaa A. El-Bary

  4. National Committee for Mathematics, Academy of Scientific Research and Technology, Cairo, Egypt

    Alaa A. El-Bary

  5. Council of Future Studies and Risk Management, Academy of Scientific Research and Technology, Cairo, Egypt

    Alaa A. El-Bary

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Correspondence to Magdy A. Ezzat.

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Ezzat, M.A., El-Bary, A.A. Analysis of thermoelectric viscoelastic wave characteristics in the presence of a continuous line heat source with memory dependent derivatives. Arch Appl Mech 93, 605–619 (2023). https://doi.org/10.1007/s00419-022-02287-y

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