Acta Mechanica

, Volume 229, Issue 6, pp 2431–2444 | Cite as

Effect of the porosity on a porous plate saturated with a liquid and subjected to a sudden change in temperature

Original Paper
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Abstract

This manuscript investigates the thermal stresses and temperatures in a porous plate hydrated with a liquid. The upper surface is taken to be impermeable, traction free and subjected to a thermal shock. The lower surface is laid on a rigid foundation. The effect of the porosity is analyzed through graphs. It is noticed that all functions for the two phases increase with the increasing porosity except for the stress and the displacement. The effect of time is analyzed through graphs. It is observed that the heat and elastic effects propagate with finite speeds. Comparison is made with a problem with the same configuration in the absence of fluid when the medium is not porous. It was found that the existence of the fluid decreases the temperature and the displacement, whereas opposite behavior is observed for the stress.

List of symbols

NAGR

Elastic moduli

\(\alpha _{{s},} \, \alpha _{{ sf}},\,\alpha _{{ fs}},\,\alpha _{{f}}\)

Coefficients of thermal expansion

\({k}_{{s}},\, {k}_{{f}}\)

Thermal conductivity

\({c}_{{ es},}\, {c}_{{ ef}}\)

Specific heats

\(\kappa \)

Coefficients of interphase heat transfer

\(\rho _{{s},} \, \rho _{{f},}\rho _{12}\)

Densities of the phases and interphase coupling

\(\beta \)

Porosity

\(\tau _{{s}},\, \tau _{{f}}\)

Relaxation times

\({e}_{{s},}\, {e}_{{f}}\)

Strain of the two phases,

\(\theta _{{s}},\, \theta _{{f}}\)

Temperatures of the phases

\({u}_{{ is}}\) and \({u}_{{ if}}\)

Displacements of the phases

\(v_{f}\)

Velocity of the fluid

\(\sigma _{{ ij}}, \sigma \)

Stresses of the phases

r1(t), r2(t)

Magnitude of surface heating

r0

A constant

H(t)

Heaviside unit step function

\({R}_{11}\)

\(\alpha _{{s}}\,{P}+ \alpha _{{ fs}}\,{G}\)

\({R}_{12}\)

\(\alpha _{{f}}\,{G} + \alpha _{{ sf}}{P}\)

\({R}_{22}\)

\(\alpha _{{f}}\,{R} +3 \alpha _{{ sf}}\,{G}\)

\({R}_{21}\)

\( 3\alpha _{{s}}\,{G} + \alpha _{{ fs}}\,{R}\)

P

\(2{N}+3{A}\)

\({F}_{11 }\)

\(\rho \,{c}_{{ es}}\)

\({F}_{22 }\)

\(\rho \,{c}_{{ ef}}\)

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Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceUniversity of DamanhourDamanhourEgypt

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