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Two-temperature theory in generalized magneto-thermoelasticity with two relaxation times

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Abstract

The model of one-dimensional equations of the two-temperature generalized magneto-thermoelasticity theory with two relaxation times in a perfect electric conducting medium is established. The state space approach developed in Ezzat (Can J. Phys. Rev. 86(11):1241–1250, 2008) is adopted for the solution of one-dimensional problems. The resulting formulation together with the Laplace transform techniques are applied to a specific problem of a half-space subjected to thermal shock and traction-free surface. The inversion of the Laplace transforms is carried out using a numerical approach. Some comparisons have been shown in figures to estimate the effects of the temperature discrepancy and the applied magnetic field.

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Abbreviations

λ,μ :

Lamè constants

t :

Time

ρ :

Density

C E :

Specific heat at constant strain

H :

Magnetic field intensity vector

E :

Electric field intensity vector

H o :

Constant component of magnetic field

J :

Conduction current density vector

T :

Thermodynamic temperature

φ :

Conductive temperature

T o :

Reference temperature

α T :

Coefficient of linear thermal expansion

σ ij :

Components of stress tensor

e ij :

Components of strain tensor

u i :

Components of displacement vector

e :

=u i,i , dilatation

k :

Thermal conductivity

κ :

Diffusivity

μ o :

Magnetic permeability

ε o :

Electric permeability

τ,ν:

Two relaxation times

β o :

The dimensionless temperature discrepancy

ε :

\({=}\frac{\varphi _{o}\gamma ^{2}}{\rho ^{2}c_{o}^{2}C_{E}}\), the thermal coupling parameter

δ ij :

Kronecker’s delta

γ :

=(3λ+2μ)α T

η o :

\({=}\frac{\rho C_{E}}{K}\)

c o :

\({=}(\frac{\lambda +2\mu }{\rho })^{\frac{1}{2}}\), speed of propagation of isothermal elastic waves

α o :

\({=}(\frac{\mu _{o}H_{o}^{2}}{\rho })^{\frac{1}{2}}\), the Alfven velocity

c :

\({=}\frac{1}{\sqrt{\mu _{0}\varepsilon _{0}}}\), speed of light.

References

  1. Ezzat M (2008) State space approach to solids and fluids. Can J Phys Rev 86(11):1241–1250

    Article  ADS  Google Scholar 

  2. Gurtin ME, Williams WO (1966) On the Clausius-Duhem inequality. Z Angew Math Phys 17:626–633

    Article  Google Scholar 

  3. Gurtin ME, Williams WO (1967) An axiom foundation for continuum thermodynamics. Arch Rat Mech Anal 26:83–117

    Article  MATH  MathSciNet  Google Scholar 

  4. Chen PJ, Gurtin ME (1968) On a theory of heat conduction involving two temperatures. Z Angew Math Phys 19:614–627

    Article  MATH  Google Scholar 

  5. Chen PJ, Gurtin ME, Williams WO (1968) A note on non-simple heat conduction. Z Angew Math Phys 19:969–970

    Article  Google Scholar 

  6. Chen PJ, Gurtin ME, Williams WO (1969) On the thermodynamics of non-simple elastic materials with two temperatures. Z Angew Math Phys 20:107–112

    Article  MATH  Google Scholar 

  7. Ieşan D (1970) On the linear coupled thermoelasticity with two temperatures. Z Angew Math Phys 21:583–591

    Article  MATH  MathSciNet  Google Scholar 

  8. Quintanilla R (2004) On existence, structural stability, convergence and spatial behavior in thermoelasticity with two temperatures. Acta Mech 168:61–73

    Article  MATH  Google Scholar 

  9. Youssef HM (2006) Theory of two-temperature generalized thermoelasticity. IMA J Appl Math 71:383–390

    Article  MATH  MathSciNet  Google Scholar 

  10. Puri P, Jordan P (2006) On the propagation of harmonic plane waves under the two-temperature theory. Int J Eng Sci 44:1113–1126

    Article  MATH  MathSciNet  Google Scholar 

  11. Magaña A, Quintanilla R (2009) Uniqueness and growth of solutions in two-temperature generalized thermoelastic theories. Math Mech Solids 14:622–634

    Article  MATH  MathSciNet  Google Scholar 

  12. Cattaneo C (1948) Sullacondizione del calore. Atti Sem Mat Fis Univ Modena

  13. Truesdell C, Muncaster RG (1980) Fundamentals of Maxwell’s kinetic theory of a simple monatonic gas. Academic Press, New York

    Google Scholar 

  14. Glass DE, Brian V (1985) Hyperbolic heat conduction with surface radiation. Int J Heat Mass Transf 28:1823-1830

    Article  ADS  Google Scholar 

  15. Joseph DD, Preziosi L (1989) Heat waves. Rev Mod Phys 61:41–73

    Article  ADS  MATH  MathSciNet  Google Scholar 

  16. Puri P, Kythe PK (1995) Non-classical thermal effects in Stoke’s second problem. Acta Mech 112:1–9

    Article  MATH  Google Scholar 

  17. Chandrasekharaiah DS (1998) Hyperbolic thermoelasticity, a review of recent literature. Appl Mech Rev 51:705–729

    Article  ADS  Google Scholar 

  18. Lord H, Shulman YA (1967) Generalized dynamical theory of thermoelasticity. Mech Phys Solid 15:299–309

    Article  ADS  MATH  Google Scholar 

  19. Müller I (1971) The coldness, a universal function in thermo-elastic solids. Arch Rat Mech Anal 41:319–332

    Article  MATH  Google Scholar 

  20. Green A, Laws A (1972) On the entropy production inequality. Arch Rat Anal 54:7–23

    Google Scholar 

  21. Green A, Lindsay K (1972) A generalized dynamical theory of thermoelasticity. J Elast 2:1–7

    Article  MATH  Google Scholar 

  22. Şuhubi E (1975) Thermoelastic solids. In: Eringen AC (ed) Cont Phys II. Academic Press, New York. Chap. 2

    Google Scholar 

  23. Ezzat MA (2004) Fundamental solution in generalized magneto-thermoelasticity with two relaxation times for perfect conductor cylindrical region. Int J Eng Sci 42:1503–1519

    Article  MATH  MathSciNet  Google Scholar 

  24. Dhaliwal RS, Rokne JG (1989) One dimensional thermal shook problem with two relaxation times. J Therm Stresses 12:259–279

    Article  MathSciNet  Google Scholar 

  25. Tzou DY (1995) A unified approach for heat conduction from macro to micro-scales. J Heat Transf 117:8–16

    Article  Google Scholar 

  26. Nayfeh A, Nemat-Nasser S (1972) Eelectromagneto-thermoelastic plane waves in solids with thermal relaxation. J Appl Mech 39:108–113

    Article  MATH  MathSciNet  Google Scholar 

  27. Choudhuri S (1984) Electro-magneto-thermo-elastic plane waves in rotating media with thermal relaxation. Int J Eng Sci 22:519–530

    Article  MATH  Google Scholar 

  28. Sharma JN, Chand D (1988) Transient generalized magnetothermoelastic waves in a half space. Int J Eng Sci 26:951–958

    Article  MATH  Google Scholar 

  29. Sherief HH, Ezzat MA (1998) A problem in generalized magneto-thermoelasticity for an infinitely long annular cylinder. J Eng Math 34:387–402

    Article  MATH  MathSciNet  Google Scholar 

  30. Ezzat MA, Youssef HM (2005) Generalized magneto-thermoelasticity in a perfectly conducting medium. Int J Solid Struct 42:6319–6334

    Article  MATH  Google Scholar 

  31. Ezzat MA Othman MI, Samaan AA (2001) State space approach to two-dimensional electromagneto-thermoelastic problem with two relaxation times. Int J Eng Sci 39:1383–1404

    Article  Google Scholar 

  32. Bahar LY, Hetnarski RB (1978) State space approach to thermoelasticity. J Therm Stresses 1:135–145

    Article  Google Scholar 

  33. Ezzat MA (1997) State space approach to generalized magneto-thermoelasticity with two relaxation times in a medium of perfect conductivity. Int J Eng Sci 35:741–752

    Article  MATH  Google Scholar 

  34. Honig G, Hirdes U (1984) A method for the numerical inversion of the Laplace transform. J Comput Appl Math 10:113–132

    Article  MATH  MathSciNet  Google Scholar 

  35. Paria G (1967) Magneto-elasticity and magneto-thermoelasticity. Adv Appl Meeh 10:73–112

    Article  Google Scholar 

  36. El-Karamany AS, Ezzat MA (2004) Discontinuities in generalized thermo-viscoelasticity under four theories. J Therm Stresses 27:1187–1212

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Education, Alexandria University, Alexandria, Egypt

    Magdy A. Ezzat

  2. Department of Mathematical and Physical Sciences, Nizwa University, P.O. Box 1357, Nizwa, 611, Oman

    Ahmed S. El-Karamany

Authors
  1. Magdy A. Ezzat
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  2. Ahmed S. El-Karamany
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Correspondence to Magdy A. Ezzat.

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Ezzat, M.A., El-Karamany, A.S. Two-temperature theory in generalized magneto-thermoelasticity with two relaxation times. Meccanica 46, 785–794 (2011). https://doi.org/10.1007/s11012-010-9337-5

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  • DOI: https://doi.org/10.1007/s11012-010-9337-5

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