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Fractional order heat conduction law in magneto-thermoelasticity involving two temperatures

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Abstract

A new mathematical model of two-temperature magneto-thermoelasticity is constructed where the fractional order heat conduction law is considered. The state space approach is adopted for the solution of one-dimensional application for a perfect conducting half-space of elastic material with heat sources distribution in the presence of a transverse magnetic field. The Laplace-transform technique is used. A numerical method is employed for the inversion of the Laplace transforms. According to the numerical results and its graphs, conclusions about the new theory are given. Some comparisons are shown in figures to estimate the effects of the temperature discrepancy and the fractional order parameter on all the studied fields.

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Abbreviations

a :

temperature discrepancy

B i :

components of magnetic field strength

c:

the speed of light

c o :

\({\sqrt{\frac{\lambda + 2 \mu}{\rho}}}\), speed of propagation of isothermal elastic waves

C E :

specific heat at constant strain

e :

dilatation

e ij :

components of strain tensor

E i :

components of electric field vector

H(.):

Heaviside unit step function

H i :

magnetic field intensity

J i :

components electric density vector

k :

thermal conductivity

q :

heat flux vector

Q :

the intensity of applied heat source per unit volume

t :

time

T :

absolute thermodynamic temperature

T o :

\({\varphi_0}\) reference temperature

u i :

components of displacement vector

α, υ:

fractional orders

α 0 :

\({\sqrt{\mu_0 H_0^2 /\rho}}\), Alfven velocity

α T :

coefficient of linear thermal expansion

α*:

\({1+(\alpha _0 /c)^{2}}\)

β :

\({(\alpha_0 /c_0)^{2}}\)

β 0 :

dimensionless temperature discrepancy

\({\beta_0^c}\) :

the critical value of β 0

γ :

(3λ + 2μ)α T

δ(.):

Dirac delta function

δ ij :

Kronecker’s delta

\({\varepsilon_{ij}}\) :

components of strain tensor

\({\varepsilon_0}\) :

electric permittivity

\({\varepsilon}\) :

thermal coupling parameter

η0 :

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

θ :

\({T-\varphi_0 ;\, \left|{\frac{\theta}{\varphi_0}}\right| << 1}\)

λ, μ:

Lame’ constants

μ 0 :

magnetic permeability

ρ :

mass Density

σ ij :

components of stress tensor

τ 0 :

relaxation time

\({\phi}\) :

conductive absolute temperature

\({\varphi}\) :

\({\phi -\varphi _0 ;\, \left|{\frac{\varphi}{\varphi_0}}\right| << 1}\)

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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, P. O. Box 1357, Nizwa, 611, Oman

    Ahmed S. El Karamany

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  1. Magdy A. Ezzat
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  2. Ahmed S. El Karamany
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Correspondence to Magdy A. Ezzat.

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Ezzat, M.A., El Karamany, A.S. Fractional order heat conduction law in magneto-thermoelasticity involving two temperatures. Z. Angew. Math. Phys. 62, 937–952 (2011). https://doi.org/10.1007/s00033-011-0126-3

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