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

  • Eman M. Hussein
Original Paper


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


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

\(\beta \)


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

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


Velocity of the fluid

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

Stresses of the phases

r1(t), r2(t)

Magnitude of surface heating


A constant


Heaviside unit step function


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


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


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


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



\({F}_{11 }\)

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

\({F}_{22 }\)

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


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  1. 1.
    George, F., William, G.: Essentials of Multiphase Flow and Transport in Porous Media. Wiley, London (2008)Google Scholar
  2. 2.
    Donald, A., Bejan, A.: Convection in Porous Media, Theory, 3rd edn. Springer, New-York (2006)zbMATHGoogle Scholar
  3. 3.
    Nowinski, J.: Theory of Thermoelasticity with Applications. Sijthoff Noordhoff Int. Pub, Alphen Aan Den Rijn (1978)CrossRefzbMATHGoogle Scholar
  4. 4.
    Coussy, O., Dormieux, L., Detourna, E.: From mixture theory to Biot’s approach for porous media. Int. J. Solids Struct. 24, 3508–3524 (1998)Google Scholar
  5. 5.
    Biot, M.: Theory of elasticity and consolidation for a porous anisotropic solid. J. Appl. Phys. 26, 182–198 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Biot, M.: Theory of propagation of elastic waves in fluid–saturated porous solid. J. Acoust. Soc. Am. 28, 168–171 (1956)CrossRefGoogle Scholar
  7. 7.
    Mercer, G., Barry, S.: Flow and deformation in poroelasticity—II. Numerical method. Math. Comput. Model. 30, 31–38 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Reddy, P., Tajuddin, M.: Exact analysis of the plane–strain vibrations of thick-walled hollow poroelastic cylinders. Int. J. Solids Struct. 37, 3439–3456 (2000)CrossRefzbMATHGoogle Scholar
  9. 9.
    Tajuddin, M., Shah, S.: Longitudinal shear vibrations of composite poroelastic cylinders. Int. J. Eng. Sci. Technol. 3, 22–33 (2011)CrossRefGoogle Scholar
  10. 10.
    Tajuddin, M., Nageswara, C., Manoj, J.: Axial–shear vibrations of an infinitely long poroelastic composite circular cylinder. Spec. Top. Rev. Porous Media Int. J. 2, 133–143 (2011)CrossRefGoogle Scholar
  11. 11.
    Shanker, B., Manoj, J., Shah, S., Nath, N.: Radial vibrations of an infinitely long poroelastic composite hollow circular cylinder. Int. J. Eng. Sci. Technol. 4, 17–33 (2012)Google Scholar
  12. 12.
    Singh, B., Kumar, R.: Reflection and refraction of micropolar elastic waves at an interface between liquid–saturated porous solid and micropolar elastic solid. Proc. Natl. Acad. Sci. 70, 397–410 (2000)zbMATHGoogle Scholar
  13. 13.
    Sherief, H., Hussein, E.: A mathematical model for short-time filtration in poroelastic media with thermal relaxation and two temperatures. Transp. Porous Media 91, 199–223 (2012)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Hussein, E.: Problem in poroelastic media for an infinitely long solid circular cylinder with thermal relaxation. Transp. Porous Media 106, 145–161 (2015)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Xiong, X., Xiao, R., Tian, O.: Effect of initial stress on a fiber-reinforced thermoelastic porous media without energy dissipation. Transp. Porous Media 111, 81–95 (2016)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Xiong, Q., Tian, X.: Transient magneto–thermo–elasto-diffusive responses of rotating porous media without energy dissipation under thermal shock. Meccanica 51, 2435–2447 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Xiong, C., Guo, Y., Diao, Y.: Normal mode analysis to a poroelastic half-space problem under generalized thermoelasticity. Latin Am. J. Solids Struct. 14, 930–949 (2017)CrossRefGoogle Scholar
  18. 18.
    Ezzat, M., Ezzat, S.: Fractional thermoelasticity applications for porous asphaltic materials. Petrol. Sci. 13, 550–560 (2016)CrossRefGoogle Scholar
  19. 19.
    Ramagiri, M., Perati, M.: Three dimensional vibrations of thermoporoelastic solids with two temperatures. Proc. Eng. 127, 824–829 (2015)CrossRefGoogle Scholar
  20. 20.
    Honig, G., Hirdes, U.: A method for the numerical inversion of the laplace transform. J. Comput. Appl. Math. 10, 113–132 (1984)MathSciNetCrossRefzbMATHGoogle Scholar

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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