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Two-dimensional problem of generalized thermoelastic half-space subjected to moving heat source

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Abstract

In this work, we will consider a half-space filled with an elastic material, which has constant elastic parameters. The governing equations are taken in the context of the theory generalized thermoelasticity. A linear temperature ramping function is used to more realistically model thermal loading of the half-space surface. The medium is assumed initially quiescent and subjected to moving heat source with constant velocity in one direction. Laplace and Fourier transform techniques are used to obtaining the general solution for any set of boundary conditions. The inverse Laplace and Fourier transforms are computed numerically using a method based on Fourier expansion techniques. Some comparisons have been shown in figures to estimate the effect of the ramping parameter of heating and the velocity of the heat source.

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Abbreviations

λ, μ:

Lame’s constants

ρ:

Density

CE :

Specific heat at constant strain

t:

Time

T:

Temperature

To :

Reference temperature

θ:

Dynamical temperature

αT :

Coefficient of linear thermal expansion

γ:

α T (3λ + 2μ)

σij :

Components of stress tensor

eij :

Components of strain tensor

ui = (u, v, w):

Displacement vector

Fi :

Body force vector

K:

Thermal conductivity

τo :

Relaxation time

t 0 :

Ramping parameter

i, j = 1, 2, 3:

Indicate the reference axis

\( c_{o} = \sqrt {\frac{\lambda + 2\,\mu }{\rho }} \) :

Longitudinal waves speed

\( \eta = \frac{{\rho \,C_{E} }}{K} \) :

The thermal viscosity

\( \varepsilon = \frac{{\gamma^{2} {\text{T}}_{\text{o}} \,}}{{\rho \,{\text{C}}_{\text{E}} \,(\lambda + 2\mu )}} \) :

Dimensionless thermoelastic coupling constant

β:

\( \left( {\frac{\lambda + 2\mu }{\mu }} \right)^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}} \)

References

  • Bargmann H (1974) Recent developments in the field of thermally induced waves and vibrations. Nucl Eng Des 27:372–398

    Article  Google Scholar 

  • Boit M (1956) Thermoelasticity and irreversible thermo-dynamics. J Appl Phys 27:240–253

    Article  MathSciNet  Google Scholar 

  • Boley B (1980) In: Hasselman DPH, Heller RA (eds) Thermal stresses, vol 1-II. Plenum Press, New York

  • Chandrasekharaiah DS (1986) Thermoelasticity with second sound: A review. Appl Mech Rev 39(3):355–372

    Article  MATH  Google Scholar 

  • Ezzat Magdy A, El-Bary Alaa A (2014) Two-temperature theory of magneto-thermo-viscoelasticity with fractional derivative and integral orders heat transfer. J Electromagn Waves Appl 28:1985–2004

    Article  Google Scholar 

  • Ezzat MA, El-Bary AA (2015a) A novel magneto-thermoelasticity theory with memory-dependent derivative. J Electromagn Waves Appl 29:1018–1031

    Article  Google Scholar 

  • Ezzat MA, El-Bary AA (2015b) Memory-dependent derivatives theory of thermo-viscoelasticity involving two-temperature. J Mech Sci Technol 29:4273–4279

    Article  Google Scholar 

  • Ezzat MA, El-Bary AA (2016) Effects of variable thermal conductivity and fractional order of heat transfer on a perfect conducting infinitely long hollow cylinder. Int J Therm Sci 108:62–69

    Article  Google Scholar 

  • Ezzat M, El-Bary AA, Ezzat S (2011) Combined heat and mass transfer for unsteady Mhd flow of perfect conducting. Energy Convers Manag 52:934–945

    Article  Google Scholar 

  • Ezzat MA, El-Bary AA, Fayik MA (2013a) Fractional Fourier law with three-phase lag of thermoelasticity. Mech Adv Mater Struct 20:593–602

    Article  Google Scholar 

  • Ezzat MA, El-Karamany AS, El-Bary AA (2014b) Generalized thermal-viscoelasticity with memory-dependent derivatives. Int J Mech Sci 89:470–475

    Article  Google Scholar 

  • Ezzat Magdy A, El-Karamany Ahmed S, El-Bary Alaa A (2015a) Electro-magnetic waves in generalized thermo-viscoelasticity for different theories. J Appl Electromagn Mech 47(1):95–111

    Google Scholar 

  • Ezzat MA, El-Karamany AS, El-Bary AA (2015b) Thermo-viscoelastic materials with fractional relaxation operators. Appl Math Model 39(23):7499–7512

    Article  MathSciNet  Google Scholar 

  • Ezzat Magdy A, El-Karamany Ahmed S, El-Bary Alaa A (2016a) Generalized thermoelasticity with memory- dependent derivatives involving two temperatures. Mech Adv Mater Struct 23:545–553

    Article  Google Scholar 

  • Ezzat MA, El-Karamany AS, El-Bary AA, Fayik MA (2013b) Fractional calculus in one-dimensional isotropic thermo-viscoelasticity. C R Mec 341:553–566

    Article  Google Scholar 

  • Ezzat MA, El-Karamany AS, El-Bary AA, Fayik MA (2014a) Fractional ultrafast laser-induced magneto thermoelastic behavior in perfect conducting metal films. J Electromagn Waves Appl 28(1):64–82

    Article  Google Scholar 

  • Ezzat MA, El Karamany AS, El-Bary AA (2016b) Electro-thermoelasticity theory with memory-dependent derivative heat transfer. Int J Eng Sci 99:22–38

    Article  MathSciNet  Google Scholar 

  • Ezzat MA, Youssef HM (2009) Stat-space approach of conducting magneto-thermoelastic medium with variable thermal conductivity subjected to ramp-type heating. J Therm Stress 32:414–427

    Article  Google Scholar 

  • Ezzat MA, Zakaria M, El-Bary AA (2010) Thermo-electric-visco-elastic material. J Appl Polym Sci 117:1934–1944

    Article  Google Scholar 

  • Ezzat MA, Zakaria M, El-Bary AA (2012) Two-temperature theory in thermo-electric viscoelastic material subjected to modified Ohm’s and Fourier’s laws. Mech Adv Mater Struct 19:453–464

    Article  Google Scholar 

  • Lord H, Shulman Y (1967) A generalized dynamical theory of thermoelasticity. J Mech Phys Sol 15:299–309

    Article  MATH  Google Scholar 

  • Misra JC, Samanta SC, Chakrabarti AK (1991b) Magneto-thermoelastic interaction in an aeolotropic solid cylinder subjected to a ramp-type heating. Int J Eng Sci 29:1065–1075

    Article  MATH  Google Scholar 

  • Misra JC, Samanta SC, Chakrabarti AK (1992) Thermoviscoelastic waves in an infinite aeolotropic body with a cylindrical cavity—a study under the review of generalized theory of thermoelasticity. Comput Struct 43:951–965

    Article  Google Scholar 

  • Misra JC, Samanta SC, Chakrabarti AK, Misra SC (1991a) Magneto-thermoelastic interaction in an infinite elastic continuum with a cylindrical hole subjected to ramp-type heating. Int J Eng Sci 29:1505–1514

    Article  MATH  Google Scholar 

  • Mukhopadhyay S (2002) Thermoelastic interactions in a transversely isotropic elastic medium with a cylindrical hole subjected to a ramp-type increase in boundary temperature or load. J Therm Stress 25(4):341–362

    Article  Google Scholar 

  • Youssef M (2005a) State-space on generalized thermoelasticity for an infinite material with a spherical cavity and variable thermal conductivity subjected to ramp-type heating. J CAMQ Appl Math Inst 13(4):369–390

    MathSciNet  MATH  Google Scholar 

  • Youssef HM (2005b) Dependence of modulus of elasticity and thermal conductivity on reference temperature in generalized thermoelasticity for an infinite material with a spherical cavity. Appl Math Mech 26(4):470–475

    Article  MATH  Google Scholar 

  • Youssef HM (2005c) Generalized thermoelasticity of an infinite body with a cylindrical cavity and variable material properties. J Therm Stress 28(5):521–532

    Article  Google Scholar 

  • Youssef HM (2006a) Problem of generalized thermoelastic infinite medium with cylindrical cavity subjected to a ramp-type heating and loading. J Arch Appl Mech 75:553–565

    Article  MATH  Google Scholar 

  • Youssef HM (2006b) Two-dimensional generalized thermoelasticity problem for a half-space subjected to ramp-type heating. Eur J Mech A Solids 25:745–763

    Article  MathSciNet  MATH  Google Scholar 

  • Youssef HM, El-Bary AA (2006a) Thermal shock problem of a generalized thermoelastic layered composite material with variable thermal conductivity. Math Probl Eng 2006:87940

    Article  MathSciNet  MATH  Google Scholar 

  • Youssef HM, El-Bary AA (2006b) Mathematical model for thermal shock problem of a generalized thermoelastic layered composite material with variable thermal conductivity. Comput Methods Sci Technol 12:165–171

    Article  MATH  Google Scholar 

  • Youssef HM (2009a) Generalized thermoelastic infinite medium with cylindrical cavity subjected to moving heat source. Mech Res Commun 36:487–496

    Article  MathSciNet  MATH  Google Scholar 

  • Youssef HM (2009b) State-space approach of two-temperature generalized thermoelastic medium subjected to moving heat source and ramp-type heating. J Mech Mater Struct 4(9):1637–1649

    Article  Google Scholar 

  • Youssef HM, El-Bary AA (2009) Generalized thermoelastic infinite layer subjected to ramp-type thermal and mechanical loading under three theories—state space approach. J Therm Stress 32:1293–1310

    Article  Google Scholar 

  • Youssef HM (2010) Generalized thermoelastic infinite medium with spherical cavity subjected to moving heat source. J Comput Math Model 21(2):212–227

    Article  MATH  Google Scholar 

  • Youssef HM, El-Bary AA (2012) Generalized magneto-thermoelastic medium with variable material properties subjected to ramp-type heating. J Appl Polym Sci 124:5209–5219

    Google Scholar 

  • Youssef HM, El-Maghraby NM (2004) State space approach to generalized thermoelastic problem with thermo-mechanical shock. J Appl Math Comput 156(2):577–586

    Article  MATH  Google Scholar 

  • Youssef HM, Elsibai KA, El-Bary AA (2014) Vibration of gold nanobeam in context of two—temperature generalized thermoelasticity subjected to laser pulse. Lat Am J Solids Struct 11:2460–2482

    Article  Google Scholar 

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Acknowledgement

The authors wish to acknowledge the approval and the support of this research study by the grant No. 5–38–1436–5 from the Deanship of Scientific Research in Northern Border University, Arar, KSA.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Northern Border University, Arar, Kingdom of Saudi Arabia

    M. M. Amin

  2. Mathematics Department, Faculty of Science, Alarish University, Alarish, Egypt

    M. M. Amin

  3. Basic and Applied Science Department, Arab Academy for Science and Technology, P.O. Box 1029, Alexandria, Egypt

    A. A. El-Bary

  4. Mechanics Department, Faculty of Engineering and Islamic Architecture, Umm Al-Qura University, Mecca, Kingdom of Saudi Arabia

    Hamdy M. Youssef

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

    Hamdy M. Youssef

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  1. M. M. Amin
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  2. A. A. El-Bary
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  3. Hamdy M. Youssef
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Correspondence to A. A. El-Bary.

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Amin, M.M., El-Bary, A.A. & Youssef, H.M. Two-dimensional problem of generalized thermoelastic half-space subjected to moving heat source. Microsyst Technol 23, 4611–4617 (2017). https://doi.org/10.1007/s00542-017-3281-4

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  • DOI: https://doi.org/10.1007/s00542-017-3281-4

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