# Thermomechanical Response in Thermoelastic Medium with Double Porosity

Document Type: Research Paper

Authors

1 Department of Mathematics, Kurukshetra University, Kurukshetra, Haryana,India

2 Department of Mathematics& Statistics, H.P.University, Shimla, HP, India

Abstract

A dynamic two dimensional problem of thermoelasticity with double porous structure has been considered to investigate the disturbance due to normal force and thermal source. Laplace and Fourier transform technique is applied to the governing equations to solve the problem. The transformed components of stress and temperature distribution are obtained .The resulting expressions are obtained in the physical domain by using numerical inversion technique. Numerically computed results for these quantities are depicted graphically to study the effect of porosity. Results of Kumar & Rani [42] and Kumar & Ailawalia [43] have also been deduced as special cases from the present investigation.

Keywords

### References

[1] De Boer R., 2000, Theory of Porous Media , Springer-Verleg, New York.
[2] De Boer R., EhlersW.,1988, A historical review of the foundation of porous media theories, Acta Mechanica 74: 1-8.
[3] Biot M. A., 1941, General theory of three-dimensional consolidation, Journal of Applied Physics 12: 155-164.
[4] Bowen R.M.,1980, Incompressible porous media models by use of the theory of mixtures, International Journal of Engineering Science 18: 1129-1148.
[5] De Boer R., Ehlers W., 1990, Uplift, friction and capillarity-three fundamental effects for liquid saturated porous solids, International Journal of Solids and Structures 26: 43-57.
[6] Barenblatt G.I., Zheltov I.P., Kochina I.N., 1960, Basic concept in the theory of seepage of homogeneous liquids in fissured rocks (strata), Journal of Applied Mathematics and Mechanics 24: 1286-1303.
[7] Wilson R. K., Aifantis E. C.,1982, On the theory of consolidation with double porosity, International Journal of Engineering Science 20(9): 1009-1035.
[8] Khaled M.Y., Beskos D. E., Aifantis E. C.,1984, On the theory of consolidation with double porosity-III, International Journal for Numerical and Analytical Methods in Geomechanics 8: 101-123.
[9] Wilson R. K., Aifantis E. C., 1984, A double porosity model for acoustic wave propagation in fractured porous rock, International Journal of Engineering Science 22(8-10): 1209-1227.
[10] Beskos D. E., Aifantis E. C.,1986., On the theory of consolidation with double porosity-II, International Journal of Engineering Science 24(111): 1697-1716.
[11] Khalili N., Valliappan S., 1996, Unified theory of flow and deformation in double porous media, European Journal of Mechanics - A/Solids 15: 321-336.
[12] Aifantis E. C., 1977, Introducing a multi –porous medium, Developments in Mechanics 8: 209-211.
[13] Aifantis E. C.,1979, On the response of fissured rocks, Developments in Mechanics 10: 249-253.
[14] Aifantis E.C., 1980, On the problem of diffusion in solids, Acta Mechanica 37: 265-296.
[15] Aifantis E.C., 1980, The Mechanics of Diffusion in Solids, T.A.M. Report No. 440, Department of Theoretical and Applied Mechanics, University of Illinois, Urbana, Illinois.
[16] Moutsopoulos K. N., Eleftheriadis I. E., Aifantis E. C.,1996, Numerical simulation of transport phenomena by using the double porosity/ diffusivity Continuum model, Mechanics Research Communications 23(6): 577-582.
[17] Khalili N., Selvadurai A. P. S., 2003, A fully coupled constitutive model for thermo-hydro –mechanical analysis in elastic media with double porosity, Geophysical Research Letters 30: 2268-2271.
[18] Pride S. R., Berryman J. G., 2003, Linear dynamics of double –porosity dual-permeability materials-I, Physical Review E 68: 036603.
[19] Straughan B., 2013, Stability and uniqueness in double porosity elasticity, International Journal of Engineering Science 65:1-8.
[20] Svanadze M., 2005, Fundamental solution in the theory of consolidation with double porosity, Journal of the Mechanical Behavior of Materials 16:123-130.
[21] Svanadze M., 2010, Dynamical problems on the theory of elasticity for solids with double porosity, Applied Mathematics and Mechanics 10: 209-310.
[22] Svanadze M., 2012, Plane waves and boundary value problems in the theory of elasticity for solids with double porosity, Acta Applicandae Mathematicae 122: 461-470.
[23] Svanadze M., 2014, On the theory of viscoelasticity for materials with double porosity, Discrete and Continuous Dynamical Systems - Series B 19(7): 2335-2352.
[24] Svanadze M., 2014, Uniqueness theorems in the theory of thermoelasticity for solids with double porosity, Meccanica 49: 2099-2108.
[25] Scarpetta E., Svanadze M., Zampoli V., 2014, Fundamental solutions in the theory of thermoelasticity for solids with double porosity, Journal of Thermal Stresses 37(6): 727-748.
[26] Scarpetta E., Svanadze M., 2014 ,Uniqueness theorems in the quasi-static theory of thermo elasticity for solids with double porosity, Journal of Elasticity 120: 67-86.
[27] Kumar R., Ailawalia P., 2005, Elastodynamics of inclined loads in an micropolar cubic crystal, Mechanics and Mechanical Engineering 9(2): 57-75.
[28] Kumar R., Kaushal S., Miglani A., 2010, Analysis of deformation due to various sources in micropolar thermodiffusive elastic medium, International Journal for Computational Methods in Engineering Science and Mechanics 11:196-210.
[29] Kumar R., Singh D., Kumar A.,2014, A problem in microtstetch thermoelastic diffusion medium, International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering 8(1): 24-27.
[30] Iesan D., Quintanilla R., 2014, On a theory of thermoelastic materials with a double porosity structure, Journal of Thermal Stresses 37: 1017-1036.
[31] Sherief H., Saleh H., 2005, A half space problem in the theory of generalized thermoelastic diffusion, International Journal of Solids and Structures 42: 4484-4493.
[32] Khalili N., 2003, Coupling effects in double porosity media with deformable matrix, Geophysical Research Letters 30(22): 2153-2155.
[33] Nowacki W., 1967, On the completeness of stress functions in thermoelasticity, Bulletin De L’academie Polonaise des Sciences 15(9): 583-591.
[34] Wang W., Wang M.Z., 1992, Constructivity and completeness of the general solutions in elastodynamics, Acta Mechanica 91: 209-214.
[35] Eskandar-Ghadi M., 2005, A complete solution of the wave equations for transversely isotropic media, Journal of Elasticity 81:1-19.
[36] Eskandari-Ghadi M., Pak Ronald Y.S., 2008, Elastodynamics and elastostatic by a unified method of potentials for convex domains, Journal of Elasticity 92:187-194.
[37] Hayati Y., Eskandari-Ghadi M., Raoofian M., Rahimian M., Ardalan A.A., 2013, Frequency domain analysis of an axisymmetric thermoelastic transversely isotropic half-space, Journal of Engineering Mechanics 139:1407-1418.
[38] Hayati Y., Eskandari-Ghadi M., Raoofian M., Rahimian M., Ardalan A.A., 2013, Domain Green’s functions of an axisymmetric thermoelastic half-space by a method of potentials, Journal of Engineering Mechanics 139:1166-1177.
[39] Eskandari-Ghadi M., Rahimian M., Sture S., Forati M., 2014, Thermoelastodynamics in transversely isotropic media with scalar potential functions, Journal of Applied Mechanics 81: 021013.
[40] Raoofian Naeeni M., Eskandri-Ghadi M., Ardalan A.A., Strure S., Rahimian M., 2015, Transient response of a thermoelastic half-space to mechanical and thermal buried source, Zeitschrift für Angewandte Mathematik und Mechanik 95(4): 354-376.
[41] Kumar R., Rani L., 2004, Response of Generalized thermoelastic half-spcae with voids dut to mechanical and thermal sources, Meccanica 39: 563-584.
[42] Kumar R., Rani L., 2005, Interaction due to mechanical and thermal sources in thermoelastic half-space with voids, Journal of Vibration and Control 11: 499-517.
[43] Kumar R., Ailawalia P, 2006, Deformations due to mechanical sources in elastic solid with voids, International Journal of Applied Mechanics and Engineering 11(4): 865-880.
[44] Unger D.J., Aifantis E.C.,1988, Completeness of solutions in the double porosity theory, Acta Mechanica 75:269-274.
[45] Honig G., Hirdes U., 1984, A method for the numerical inversion of the Laplace transform, Journal of Computational and Applied Mathematics 10:113-132.
[46] Press W.H., Teukolsky S.A., Vellerlig W.T., Flannery B.P., 1986, Numerical Recipes in Fortran , Cambridge University Press, Cambridge.
[47] Raoofian Naeeni M., Campagna R., Eskandri-Ghadi M., Ardalan A.A., 2015,Performance comparison of numerical inversion methods for Laplace and Hankel integral transforms in engineering problems, Applied Mathematics and Computation 250: 759-775.
[48] Sharma J.N., Chauhan R.S., 2001, Mechanical and thermal sources in a generalized thermoelastic half-space, Journal of Thermal Stresses 24: 651-675.
[49] Bradie B., 2007, A Friendly Introduction to Numerical Analysis, Pearson Education/Prentice Hall, New Delhi, India.