Mathematical Modeling for Thermoelastic Double Porous Micro-Beam Resonators

Document Type: Research Paper


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

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


In the present work, the mathematical model of a homogeneous, isotropic thermoelastic double porous micro-beam, based on the Euler-Bernoulli theory is developed in the context of Lord-Shulman [1] theory of thermoelasticity. Laplace transform technique has been used to obtain the expressions for lateral deflection, axial stress, axial displacement, volume fraction field and temperature distribution. A numerical inversion technique has been applied to recover the resulting quantities in the physical domain. Variations of axial displacement, axial stress, lateral deflection, volume fraction field and temperature distribution with axial distance are depicted graphically to show the effects of porosity and thermal relaxation time. Some particular cases are also deduced.                         


[1] Lord H., Shulman Y., 1967, A generalized dynamical theory of thermoelasticity, Journal of the Mechanics and Physics of Solids 15: 299-309.
[2] Biot M. A., 1941, General theory of three-dimensional consolidation, Journal of Applied Physics 12: 155-164.
[3] 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.
[4] Aifantis E. C., 1977, Introducing a multi –porous medium, Developments in Mechanics 8: 209-211.
[5] Aifantis E. C., 1979, On the response of fissured rocks, Developments in Mechanics 10: 249-253.
[6] Aifantis E. C., 1980, On the problem of diffusion in solids, Acta Mechanica 37: 265-296.
[7] Wilson R.K., Aifantis E.C., 1984, 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] Beskos D.E., Aifantis E.C., 1986, On the theory of consolidation with double porosity-II, International Journal of Engineering Science 24: 1697-1716.
[10] Khalili N., Salvadorian 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.
[11] Svanadze M., 2005, Fundamental solution in the theory of consolidation with double porosity, Journal of the Mechanical Behavior of Materials 16: 123-130.
[12] 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.
[13] Straughan B., 2013, Stability and uniqueness in double porosity elasticity, International Journal of Engineering Science 65: 1-8.
[14] Nunziato J.W., Cowin S.C., 1979, A nonlinear theory of elastic materials with voids, Archive for Rational Mechanics and Analysis 72: 175-201.
[15] Cowin S.C., Nunziato J.W., 1983, Linear elastic materials with voids, Journal of Elasticity 13: 125-147.
[16] Iesan D., Quintanilla R., 2014, On a theory of thermoelastic materials with a double porosity structure, Journal of Thermal Stresses 37: 1017-1036.
[17] Fritz J., Baller M.K., Lang H.P., Rothuizen H., Vettiger P., Meyer E., Gntherodt H.J., Gerber C., Gimzewski J.K., 2001,Translating bio-molecular recognition into nanomechanics , Science 288: 316-318.
[18] Sidles J. A., 1991, Noninductive detection of single proton-magnetic resonance, Applied Physics Letters 58: 2854-2856.
[19] Nabian A., Rezazadeh G., Haddad-derafshi M., Tahmasebi A., 2008, Mechanical behavior of a circular micro plate subjected to uniform hydrostatic and non-uniform electrostatic pressure, Microsystem Technologies 14: 235-240.
[20] Fathalilou M., Motallebi A., Rezazadeh G., Yagubizade H., Shirazi K., Alizadeh Y., 2009, Mechanical behavior of an electrostatically-actuated microbeam under mechanical shock, Journal of Solid Mechanics 1: 45-57.
[21] Dimarogonas A., 1996, Vibration for Engineers, Prentice-Hall, Inc.
[22] Meirovitch L., 2001, Fundamentals of Vibrations, McGraw-Hill, International Edition.
[23] Boley B.A., 1972, Approximate analyses of thermally induced vibrations of beams and plates, Journal of Applied Mechanics 39: 212-216.
[24] Manolis G.D., Beskos D.E., 1980, Thermally induced vibrations of beam structures, Computer Methods in Applied Mechanics and Engineering 21: 337-355.
[25] Al-Huniti N.S., Al-Nimr M.A., Naij M., 2001, Dynamic response of a rod due to a moving heat Source under the hyperbolic heat conduction model, Journal of Sound and Vibration 242: 629-640.
[26] Biondi B., Caddemi S., 2005, Closed form solutions of Euler-Bernoulli beams with singularities, International Journal of Solids and Structures 42: 3027-3044.
[27] Fang D.N., Sun Y.X., Soh A.K., 2006, Analysis of frequency spectrum of laser-induced vibration of micro beam resonators, Chinese Physics Letters 23: 1554-1557.
[28] Sharma J.N., Grover D., 2011. Thermoelastic vibrations in micro-/nano-scale beam resonators with voids, Journal of Sound and Vibration 330: 2964-2977.
[29] Esen I., 2015, A new FEM procedure for transverse and longitudinal vibration analysis of thin rectangular plates subjected to a variable velocity moving load along an arbitrary trajectory, Latin American Journal of Solids and Structures 12: 808-830.
[30] Kumar R., 2016, Response of thermoelastic beam due to thermal source in modified couple stress theory, Computational Methods in Science and Technology 22(2): 95-101.
[31] Ghadiri M., Shafiei N., 2016, Vibration analysis of rotating functionally graded Timoshenko micro beam based on modified couple stress theory under different temperature distributions, Acta Astronautica 121: 221-240.
[32] Zenkour A. M. , 2016, Free vibration of a microbeam resting on Pasternak's foundation via the GN thermoelasticity theory without energy dissipation, Journal of Low Frequency Noise, Vibration and Active Control 35(4): 303-311.
[33] Kaghazian A., Hajnayeb A., Foruzande H., 2017, Free vibration analysis of a piezoelectric nanobeam using nonlocal elasticity theory , Structural Engineering and Mechanics 61(5): 617-624.
[34] Ebrahimi F., Barati M.R., 2017, Vibration analysis of embedded size dependent FG nanobeams based on third-order shear deformation beam theory , Structural Engineering and Mechanics 61(6): 721-736.
[35] Zenkour A. M., 2017, Thermoelastic response of a micro beam embedded in Visco-Pasternak’s medium based on GN-III model, Journal of Thermal Stresses 40(2): 198-210.
[36] Arefi M., Zenkour A.M., 2017, Vibration and bending analysis of a sandwich micro beam with two integrated piezo-magnetic face-sheet, Composite Structures 159: 479-490.
[37] Honig G., Hirdes U., 1984, A method for the numerical inversion of the Laplace transforms, Journal of Computational and Applied Mathematics 10: 113-132.
[38] Tzou D., 1996, Macro-to-Micro Heat transfer, Taylor& Francis, Washington DC.
[39] 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.
[40] Khalili N., 2003, Coupling effects in double porosity media with deformable matrix, Geophysical Research Letters 30(22): 2153-2155.