Skip to main content
SpringerLink
Log in

Eigenvalue Approach to Fractional-Order Dual-Phase-Lag Thermoviscoelastic Problem of a Thick Plate

  • Research Paper
  • Published:
Iranian Journal of Science and Technology, Transactions of Mechanical Engineering Aims and scope Submit manuscript

Abstract

The present paper deals with the problem of thermoviscoelastic interactions in a homogeneous isotropic thick plate whose upper surface is stress-free and is subjected to a known temperature distribution, while the lower surface rests on a rigid foundation and is thermally insulated. The problem is treated on the basis of fractional-ordered dual-phase-lag model of thermoelasticity. To study the viscoelastic nature of the material, Kelvin–Voigt model of linear viscoelasticity is employed. The governing equations are transformed into a vector-matrix differential equation with the use of joint Laplace and Fourier transforms, which is then solved by the eigenvalue approach. Numerical estimates of displacements, stresses and temperature are computed for copper material by using a numerical inversion technique. Finally, all the physical fields are represented graphically to estimate and highlight the effects of the fractional parameter, viscosity and time.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

References

  • Abbas IA (2015) Eigen value approach to fractional order generalized magneto-thermoelastic medium subjected to moving heat source. J Magn Magn Mater 377:452–459

    Article  Google Scholar 

  • Abouelregal A, Zenkour AM (2014) Effect of phase lags on thermoelastic functionally graded microbeams subjected to ramp-type heating. IJST Trans Mech Eng 38:321–335

    Google Scholar 

  • Al-Nimr M, Al-Huniti NS (2000) Transient thermal stresses in a thin elastic plate due to a rapid dual-phase-lag heating. J Therm Stress 23:731–746

    Article  Google Scholar 

  • Bachher M, Sarkar N, Lahiri A (2015) Fractional order thermoelastic interactions in an infinite porous material due to distributed time-dependent heat sources. Meccanica 50:2167–2178

    Article  MathSciNet  MATH  Google Scholar 

  • Biot MA (1956) Thermoelasticity and irreversible thermodynamics. J Appl Phys 27:243–253

    MathSciNet  MATH  Google Scholar 

  • Caputo M (1967) Linear model of dissipation whose q is almost frequency independent-ii. Geophys J R Astron Soc 13:529–539

    Article  Google Scholar 

  • Cattaneo C (1958) Sur Une forme de l’equation de la chaleur elinant le paradoxes d’une propagation instance. C R Acad Sci 247:431–432

    MATH  Google Scholar 

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

    Article  Google Scholar 

  • Deswal S, Kalkal KK (2014) Plane waves in a fractional order micropolar magneto-thermoelastic half-space. Wave Motion 51:100–113

    Article  MathSciNet  MATH  Google Scholar 

  • Dhaliwal RS, Sherief HH (1980) Generalized thermoelasticity for anisotropic media. Q Appl Math 33:1–8

    Article  MathSciNet  MATH  Google Scholar 

  • El-Karamany AS, Ezzat MA (2014) On the dual-phase-lag thermoelasticity theory. Meccanica 49:79–89

    Article  MathSciNet  MATH  Google Scholar 

  • El-Maghraby NM (2005) A two-dimensional problem for a thick plate with heat sources in generalized thermoelasticity. J Therm Stress 28:1227–1241

    Article  Google Scholar 

  • El-Maghraby NM (2009) Two-dimensional thermoelasticity problem for a thick plate under the action of a body force in two relaxation times. J Therm Stress 32:863–876

    Article  Google Scholar 

  • Elhagary MA (2014) A two-dimensional generalized thermoelastic diffusion problem for a thick plate subjected to thermal loading due to laser pulse. J Therm Stress 37:1416–1432

    Article  Google Scholar 

  • 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  MathSciNet  MATH  Google Scholar 

  • Ezzat MA (2010) Thermoelectric MHD non-Newtonian fluid with fractional derivative heat transfer. Physica B 405:4188–4194

    Article  Google Scholar 

  • Ezzat MA, El-Karamany AS, Ezzat SM (2012) Two-temperature theory in magneto-thermoelasticity with fractional order dual-phase-lag heat transfer. Nucl Eng Des 252:267–277

    Article  Google Scholar 

  • Green AE, Lindsay KA (1972) Thermoelasticity. J Elast 2:1–7

    Article  MATH  Google Scholar 

  • Green AE, Naghdi P (1991) A re-examination of the basic postulate of thermo-mechanics. Proc R Soc Lond Ser A 432:171–194

    Article  MATH  Google Scholar 

  • Green AE, Naghdi P (1992) On undamped heat waves in an elastic solid. J Therm Stress 15:252–264

    Article  MathSciNet  Google Scholar 

  • Green AE, Naghdi P (1993) Thermoelasticity without energy dissipation. J Elast 31:189–208

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Jumarie G (2010) Derivation and solutions of some fractional Black–Scholes equations in coarse-grained space and time. Application to Mertons optimal portfolio. Comput Math Appl 59:1142–1164

    Article  MathSciNet  MATH  Google Scholar 

  • Lord HW, Shulman Y (1967) A generalized dynamical theory of thermoelasticity. J Mech Phys Solid 15:299–306

    Article  MATH  Google Scholar 

  • Miller KS, Ross B (1993) An introduction to the fractional calculus and fractional differential equations. Wiley, New York

    MATH  Google Scholar 

  • Podlubny I (1999) Fractional differential equations. Academic Press, New York

    MATH  Google Scholar 

  • Povstenko YZ (2005) Fractional heat conduction equation and associated thermal stress. J Therm Stress 28:83–102

    Article  MathSciNet  Google Scholar 

  • Povstenko YZ (2010) Fractional radial heat conduction equation in an infinite medium with a cylindrical cavity and associated thermal stresses. Mech Res Commun 37:436–440

    Article  MATH  Google Scholar 

  • Quintanilla R, Racke R (2006) A note on stability of dual-phase-lag heat conduction. Int J Heat Mass Transf 49:1209–1213

    Article  MATH  Google Scholar 

  • Quintanilla R, Racke R (2007) Qualitative aspects in dual-phase-lag heat conduction. Proc R Soc Lond A 463:659–674

    Article  MathSciNet  MATH  Google Scholar 

  • Ross B (1977) The development of fractional calculus. Hist Math 4:75–89

    Article  MathSciNet  MATH  Google Scholar 

  • Roychoudhuri SK (2007) One-dimensional thermoelastic waves in elastic half space with dual-phase-lag effects. J Mech Mater Struct 2:489–503

    Article  Google Scholar 

  • Said SM, Othman MIA (2016) Gravitational effect on a fiber-reinforced thermoelastic medium with temperature-dependent properties for two different theories. Iran J Sci Technol Trans Mech Eng 40:223–232

    Article  Google Scholar 

  • Sarkar N, Lahiri A (2013) The effect of fractional parameter on a perfect conducting elastic half-space in generalized magneto-thermoelasticity. Meccanica 48:231–245

    Article  MathSciNet  MATH  Google Scholar 

  • Sherief HH, El-Sayed AM, El-Latief AM (2010) Fractional order theory of thermoelasticity. Int J Solids Struct 47:269–275

    Article  MATH  Google Scholar 

  • Thomas L (1980) Fundamentals of Heat Transfer. Prentice-Hall Inc., Englewood Cliffs

    Google Scholar 

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

    Article  Google Scholar 

  • Verma KL, Hasebe N (2001) Wave propagation in plates of general anisotropic media in generalized thermoelasticity. Int J Eng Sci 39:1739–1763

    Article  Google Scholar 

  • Vernotte P (1958) Les paradoxes de la theorie continue de l’equation de la chaleur. C R Acad Sci 246:3154–3155

    MATH  Google Scholar 

  • Youssef HM (2010) Theory of fractional order generalized thermoelasticity. J Heat Transf 132:1–7

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Guru Jambheshwar University of Science and Technology, Hisar, Haryana, 125001, India

    Kapil Kumar Kalkal & Sunita Deswal

  2. Department of Mathematics, Govt. College, Narnaund, Hisar, Haryana, India

    Renu Yadav

Authors
  1. Kapil Kumar Kalkal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sunita Deswal
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Renu Yadav
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Renu Yadav.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kalkal, K.K., Deswal, S. & Yadav, R. Eigenvalue Approach to Fractional-Order Dual-Phase-Lag Thermoviscoelastic Problem of a Thick Plate. Iran J Sci Technol Trans Mech Eng 43 (Suppl 1), 917–927 (2019). https://doi.org/10.1007/s40997-018-0202-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40997-018-0202-9

Keywords

Mathematics Subject Classification

Search

Navigation