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Two-temperature generalized thermoelastic infinite medium with cylindrical cavity subjected to moving heat source

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Abstract

In this work, a problem of thermoelastic interactions in an elastic infinite medium with cylindrical cavity thermally shocked at its bounding surface and subjected to moving heat source with constant velocity has been solved. The governing equations are taken in the context of two-temperature generalized thermoelasticity theory (Youssef model). The analytical solution with direct approach in the Laplace transforms domain has been obtained. The derived analytical expressions have been computed for specific situations. Numerical results for the dynamical and conductive temperatures, stress, strain, and displacement are represented graphically with comparisons by one-temperature generalized thermoelasticity (Lord–Shulman model).

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Abbreviations

λ, μ:

Lamé’s constants

ρ :

Density

C E :

Specific heat at constant strain

α T :

Coefficient of linear thermal expansion

γ :

 = (3λ + 2μ)α T

t :

Time

T :

Temperature

T 0 :

Reference temperature

θ :

= (T − T 0) Temperature increment suchthat \({\frac{|\theta|}{T_{0}}< <1 }\)

σ ij :

Components of stress tensor

e ij :

Components of strain tensor

u i :

Components of displacement vector

K :

Thermal conductivity

τ o :

Relaxation times

c o :

\({=\sqrt {\frac{\lambda +2\mu}{\rho}}}\) Longitudinal wave speed

η :

\({=\frac{\rho\,{C}_{E}}{K}}\) The thermal viscosity

\({\varepsilon}\) :

\({=\frac{\gamma}{\rho \,C_{E}}}\) Dimensionless mechanical coupling constant

α :

\({=\frac{\gamma T_{0}}{\mu}}\) Dimensionless thermoelastic coupling constant

ω :

 = a c 2 o η 2 Dimensionless two-temperature parameter

β :

\({=\left({\frac{\lambda +2\mu}{\mu}} \right)^{{1}\mathord{\left/ {\vphantom {12}}\right.\kern-\nulldelimiterspace}2}}\)

Q :

Heat source

v :

The velocity of the heat source

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Author notes

  1. Hamdy M. Youssef

    Present address: Faculty of Engineering, Umm Al-Qura University, PO. 5555, Makkah, Saudi Arabia

Authors and Affiliations

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

    Hamdy M. Youssef

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  1. Hamdy M. Youssef
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Correspondence to Hamdy M. Youssef.

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Youssef, H.M. Two-temperature generalized thermoelastic infinite medium with cylindrical cavity subjected to moving heat source. Arch Appl Mech 80, 1213–1224 (2010). https://doi.org/10.1007/s00419-009-0359-1

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