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
Ezzat M (2008) State space approach to solids and fluids. Can J Phys Rev 86(11):1241–1250
Gurtin ME, Williams WO (1966) On the Clausius-Duhem inequality. Z Angew Math Phys 17:626–633
Gurtin ME, Williams WO (1967) An axiom foundation for continuum thermodynamics. Arch Rat Mech Anal 26:83–117
Chen PJ, Gurtin ME (1968) On a theory of heat conduction involving two temperatures. Z Angew Math Phys 19:614–627
Chen PJ, Gurtin ME, Williams WO (1968) A note on non-simple heat conduction. Z Angew Math Phys 19:969–970
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
Ieşan D (1970) On the linear coupled thermoelasticity with two temperatures. Z Angew Math Phys 21:583–591
Quintanilla R (2004) On existence, structural stability, convergence and spatial behavior in thermoelasticity with two temperatures. Acta Mech 168:61–73
Youssef HM (2006) Theory of two-temperature generalized thermoelasticity. IMA J Appl Math 71:383–390
Puri P, Jordan P (2006) On the propagation of harmonic plane waves under the two-temperature theory. Int J Eng Sci 44:1113–1126
Magaña A, Quintanilla R (2009) Uniqueness and growth of solutions in two-temperature generalized thermoelastic theories. Math Mech Solids 14:622–634
Cattaneo C (1948) Sullacondizione del calore. Atti Sem Mat Fis Univ Modena
Truesdell C, Muncaster RG (1980) Fundamentals of Maxwell’s kinetic theory of a simple monatonic gas. Academic Press, New York
Glass DE, Brian V (1985) Hyperbolic heat conduction with surface radiation. Int J Heat Mass Transf 28:1823-1830
Joseph DD, Preziosi L (1989) Heat waves. Rev Mod Phys 61:41–73
Puri P, Kythe PK (1995) Non-classical thermal effects in Stoke’s second problem. Acta Mech 112:1–9
Chandrasekharaiah DS (1998) Hyperbolic thermoelasticity, a review of recent literature. Appl Mech Rev 51:705–729
Lord H, Shulman YA (1967) Generalized dynamical theory of thermoelasticity. Mech Phys Solid 15:299–309
Müller I (1971) The coldness, a universal function in thermo-elastic solids. Arch Rat Mech Anal 41:319–332
Green A, Laws A (1972) On the entropy production inequality. Arch Rat Anal 54:7–23
Green A, Lindsay K (1972) A generalized dynamical theory of thermoelasticity. J Elast 2:1–7
Şuhubi E (1975) Thermoelastic solids. In: Eringen AC (ed) Cont Phys II. Academic Press, New York. Chap. 2
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
Dhaliwal RS, Rokne JG (1989) One dimensional thermal shook problem with two relaxation times. J Therm Stresses 12:259–279
Tzou DY (1995) A unified approach for heat conduction from macro to micro-scales. J Heat Transf 117:8–16
Nayfeh A, Nemat-Nasser S (1972) Eelectromagneto-thermoelastic plane waves in solids with thermal relaxation. J Appl Mech 39:108–113
Choudhuri S (1984) Electro-magneto-thermo-elastic plane waves in rotating media with thermal relaxation. Int J Eng Sci 22:519–530
Sharma JN, Chand D (1988) Transient generalized magnetothermoelastic waves in a half space. Int J Eng Sci 26:951–958
Sherief HH, Ezzat MA (1998) A problem in generalized magneto-thermoelasticity for an infinitely long annular cylinder. J Eng Math 34:387–402
Ezzat MA, Youssef HM (2005) Generalized magneto-thermoelasticity in a perfectly conducting medium. Int J Solid Struct 42:6319–6334
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
Bahar LY, Hetnarski RB (1978) State space approach to thermoelasticity. J Therm Stresses 1:135–145
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
Honig G, Hirdes U (1984) A method for the numerical inversion of the Laplace transform. J Comput Appl Math 10:113–132
Paria G (1967) Magneto-elasticity and magneto-thermoelasticity. Adv Appl Meeh 10:73–112
El-Karamany AS, Ezzat MA (2004) Discontinuities in generalized thermo-viscoelasticity under four theories. J Therm Stresses 27:1187–1212
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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