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On thermo-viscoelastic infinitely long hollow cylinder with variable thermal conductivity

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Abstract

Present work deals with the problem of an infinite long hollow cylinder with variable thermal conductivity in the context of generalized thermo-viscoelasticity theory with thermal relaxation. A mapping of Kirchhoff’s transformation was used to solve a problem with variable thermal conductivity. The Laplace transform is used. The inversion process is carried out using a numerical method based on Fourier series expansions. Numerical computations for the temperature, the displacement and the stress distributions are carried out and represented graphically. The results indicate that the thermal conductivity play a major role in all considered distributions.

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Abbreviations

λ, μ :

Lame’ constants

ρ :

Mass density

t :

Time

C E :

Specific heat at constant strains

K :

=λ + (2/3)μ, bulk modulus

ɛ ij :

Components of strain tensor

e ij :

Components of strain deviator tensor

σ ij :

Components of stress tensor

S ij :

Components of stress deviator tensor

e :

=ɛ ii , Dilatation

k :

Thermal conductivity

T :

Absolute temperature

u i :

Components of displacement vector

α T :

Coefficient of linear thermal expansion

γ:

=3 K α T

δ ij :

Kronecker’s delta

T o :

Reference temperature

c 1 :

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

η o :

=ρC E /k

ɛ :

\({ = }\frac{{ \, \gamma^{2} T_{0} }}{{k_{o} \eta_{o} \rho c_{1}^{2} }}\), Thermal coupling parameter

θ :

=T − T 0, such that |θ/T 0| < < 1

τ o :

Relaxation time

\(\Gamma (.)\) :

Gamma function

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Authors and Affiliations

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

    M. A. Ezzat

  2. 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. A. El-Bary
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Correspondence to M. A. Ezzat.

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Ezzat, M.A., El-Bary, A.A. On thermo-viscoelastic infinitely long hollow cylinder with variable thermal conductivity. Microsyst Technol 23, 3263–3270 (2017). https://doi.org/10.1007/s00542-016-3101-2

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