Abstract
This paper deals with the problem of magneto-thermo-elastic interactions in an unbounded, perfectly conducting elastic medium due to the presence of periodically varying heat sources in the context of linear theory of generalized thermo-elasticity with energy dissipation (TEWED or GN-III model), without energy dissipation (TEWOED or GN-II model) and three-phase-lag model (3P model). The governing equations of generalized thermo-elasticity of the above models under the influence of a magnetic field are established. The Laplace-Fourier double transform technique has been used to get the solution. The inversion of the Fourier transform has been done by using residual calculus, where poles of the integrand are obtained numerically in a complex domain by using Laguerre’s method, and the inversion of the Laplace transformation is done numerically using a method based on Fourier series expansion technique. Displacement, temperature, stress and strain distributions have been computed numerically and presented graphically in numbers of figures. A comparison of the results for different theories (GN-II, GN-III and 3P model) and the effect of magnetic field and damping coefficient on the physical quantities has been discussed.
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Abbreviations
- u :
-
Displacement vector
- λ, μ :
-
Lamé’s constants
- ρ :
-
Constant mass density of the medium
- γ :
-
Thermal modulus
- α t :
-
Coefficient of linear thermal expansion
- T 0 :
-
Uniform reference temperature
- T :
-
Small temperature increase above the reference temperature T 0
- J :
-
Electric current density vector
- B :
-
Magnetic induction vector
- c v :
-
Specific heat of the medium at constant strain
- K*:
-
A material constant characteristic for the GN theory
- H :
-
Total magnetic field vector at any time
- E :
-
Electric field vector
- μ e :
-
Magnetic permeability of the medium
- σ :
-
Electric conductivity of the medium
- c T :
-
Non-dimensional finite thermal wave speed of GN theory of thermo-elasticity II
- \({{\epsilon _{\rm T}}}\) :
-
Thermo-elastic coupling constant
- K :
-
Thermal conductivity
- κ :
-
Thermal diffusivity
References
Lord H.W., Shulman Y.: A generalized dynamical theory of thermo-elasticity. J. Mech. Phys. Solids 15, 299–309 (1967)
Green A.E., Lindsay K.A.: Thermo-elasticity. J. Elasticity 2, 1–7 (1972)
Paria G.: On magneto-thermo-elastic plane waves. Proc. Camb. Philos. Soc. 58, 527–531 (1962)
Nayfeh A., Nemat-Nasser S.: Thermo-elastic waves in solids with thermal relaxation. Acta Mech. 12, 43–69 (1971)
Nayfeh A., Nemat-Nasser S.: Electro-magneto-thermo-elastic plane waves in solid with thermal relaxation. J. Appl. Mech. 39, 108–113 (1972)
Roychoudhuri S.K., Chatterjee G.: A coupled magneto-thermo-elastic problem in a perfectly conducting elastic half-space with thermal relaxation. Int. J. Math. Mech. Sci. 13, 567–578 (1990)
Hsieh, R.K.T.: Mechanical modelling of new electromagnetic materials. In: Proceedings of IUTAM symposium, Stockholm, Sweden, 2–6 April (1990)
Ezzat M.A.: State space approach to generalized magneto-thermoelasticity with two relaxation times in a medium of perfect conductivity. J. Eng. Sci. 35, 741–752 (1997)
Ezzat M.A., Othman M.I., El-Karamany A.S.: Electro-magneto-thermo-elastic plane waves with thermal relaxation in a medium of perfect conductivity. J. Therm. Stresses 24, 411–432 (2001)
Sherief H.H., Yossef H.M.: Short time solution for a problem in magneto thermoelasticity with thermal relaxation. J. Therm. Stresses 27, 537–559 (2004)
Baksi A., Bera R.K.: Eigen function method for the solution of magneto-thermoelastic problems with thermal relaxation and heat source in three dimensions. Sci. Direct Math. Comput. Model. 42, 533–552 (2005)
Ezzat, A., EI-Karamany, A.S.: Fractional order heat conduction law in magneto-thermoelasticity involving two temperatures. ZAMP. (2011). doi:10.1007/s00033-011-0126-3
Hetnarski R.B., Ignaczak J.: Generalized thermoelasticity: closed form solutions. J. Therm. Stresses 16, 473–498 (1993)
Hetnarski R.B., Ignaczak J.: Generalized thermoelasticity: response of semi-space to a short laser pulse. J. Therm. Stresses 17, 377–396 (1994)
Green A.E., Naghdi P.M.: A re-examination of the basic postulate of thermo-mechanics. Proc. R. Soc. Lond. Ser. 432, 171–194 (1991)
Green A.E., Naghdi P.M.: On undamped heat waves in an elastic solid. J. Therm. Stresses 15, 252–264 (1992)
Green A.E., Naghdi P.M.: Thermoelasticity without energy dissipation. J. Elasticity 31, 189–208 (1993)
Roychoudhuri S.K.: Magneto-thermo-elastic waves in an infinite perfectly conducting solid without energy dissipation. J. Tech. Phys. 47, 63–72 (2006)
Chandrasekhariah D.S.: A note on the uniqueness of solution in the linear theory of thermoelasticity without energy dissipation. J. Elasticity 43, 279–283 (1996)
Chandrasekhariah D.S.: A uniqueness theorem in the theory of thermoelasticity without energy dissipation. J. Therm. Stresses 19, 267–272 (1996)
Chandrasekhariah D.S.: One dimensional wave propagation in the linear theory of thermoelasticity. J. Therm. Stresses 19, 695–710 (1996)
Chandrasekhariah D.S., Srinath K.S.: Thermoelastic interaction without energy dissipation due to a point heat source. J. Elasticity 50, 97–108 (1998)
Banik S., Mallik S.H., Kanoria M.: Thermoelastic Interaction with energy dissipation in an infinite solid with distributed periodically varying heat sources. J. Pure Appl. Math. 34, 231–246 (2007)
Mallik S.H., Kanoria M.: Generalized thermoelastic functionally graded infinite solids with a periodically varying heat source. Int. J. Solids Struct. 44, 7633–7645 (2007)
Das P., Kanoria M.: Magneto-thermo-elastic response in a functionally graded isotropic medium under a periodically varying heat source. Int. J. Thermophys. 30, 2098–2121 (2009)
Kar A., Kanoria M.: Themoelastic interaction with energy dissipation in a transversely isotropic thin circular disc. Eur. J. Mech. A Solids 26, 969–981 (2007)
Kar A., Kanoria M.: Themoelastic interaction with energy dissipation in an unbounded body with a spherical hole. Int. J. Solids Struct. 44, 2961–2971 (2007)
Mallik S.H., Kanoria M.: A two dimensional problem for a transversely isotropic generalized thermoelastic thick plate with spatially varying heat source. Eur. J. Mech. A Solids 27, 607–621 (2008)
Das P., Kanoria M.: Magneto-thermo-elastic waves in an infinite perfectly conducting elastic solid with energy dissipation. Appl. Math. Mech. 30, 221–228 (2009)
Islam M., Kanoria M.: Study of dynamical response in a two-dimensional transversely isotropic thick plate due to heat source. J. Therm. Stresses 34, 702–723 (2011)
Islam M., Mallik S.H., Kanoria M.: Study of dynamical response in a two-dimensional transversely isotropic thick plate with spatially varying heat sources and body forces. Appl. Math. Mech. 32, 1315–1332 (2011)
Kar A., Kanoria M.: Generalized thermoelastic problem of a spherical shell under thermal shock. Eur. J. Pure Appl. Math. 2, 125–146 (2009)
Tzou D.Y.: A unified field approach for heat conduction from macro to micro scales. ASME J. Heat Transfer 117, 8–16 (1995)
Chandrasekharaiah D.S.: Hyperbolic thermoelasticity: a review of recent literature. Appl. Mech. Rev. 51, 8–16 (1998)
Roychoudhuri S.K.: One-dimensional thermoelastic waves in elastic half-space with dual-phase-lag effects. J. Mech. Mater. Struct. 2, 489–503 (2007)
Kumar R., Mukhopadhyay S.: Effect of three-phase-lags on generalized thermoelasticity for an infinite medium with a cylindrical cavity. J. Therm. Stresses 32, 1149–1165 (2009)
Roychoudhuri S.K.: On a thermoelastic three-phase-lag model. J. Therm. Stresses 30, 231–238 (2007)
Quintanilla R., Racke R.: A note on stability in three-phase-lag heat conduction. Int. J. Heat Mass Transfer 51, 24–29 (2008)
Kar A., Kanoria M.: Generalized thermoelastic functionally graded orthotropic hollow sphere under thermal shock with three-phase-lag effect. Eur. J. Mech. A Solids 1, 1–11 (2009)
Honig G., Hirdes U.: A method for the numerical inversion of Laplace transforms. J. Comput. Appl. Math. 10, 113–132 (1984)
Roychoudhuri S.K., Dutta P.S.: Thermoelastic interaction without energy dissipation in an infinite solid with distributed periodically varying heat sources. Int. J. Solids Struct. 42, 4192–4203 (2005)
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Das, P., Kanoria, M. Magneto-thermo-elastic response in a perfectly conducting medium with three-phase-lag effect. Acta Mech 223, 811–828 (2012). https://doi.org/10.1007/s00707-011-0591-y
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DOI: https://doi.org/10.1007/s00707-011-0591-y