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

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## 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.

## Keywords

Dual-phase-lag Fractional-order thermoelasticity Viscosity Eigenvalue approach## Mathematics Subject Classification

74Fxx 74Dxx 74Jxx## References

- Abbas IA (2015) Eigen value approach to fractional order generalized magneto-thermoelastic medium subjected to moving heat source. J Magn Magn Mater 377:452–459CrossRefGoogle Scholar
- Abouelregal A, Zenkour AM (2014) Effect of phase lags on thermoelastic functionally graded microbeams subjected to ramp-type heating. IJST Trans Mech Eng 38:321–335Google 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 23:731–746CrossRefGoogle 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 50:2167–2178MathSciNetCrossRefzbMATHGoogle Scholar
- Biot MA (1956) Thermoelasticity and irreversible thermodynamics. J Appl Phys 27:243–253MathSciNetzbMATHGoogle Scholar
- Caputo M (1967) Linear model of dissipation whose q is almost frequency independent-ii. Geophys J R Astron Soc 13:529–539CrossRefGoogle Scholar
- Cattaneo C (1958) Sur Une forme de l’equation de la chaleur elinant le paradoxes d’une propagation instance. C R Acad Sci 247:431–432zbMATHGoogle Scholar
- Chandrasekharaiah DS (1998) Hyperbolic thermoelasticity: a review of recent literature. Appl Mech Rev 51:705–729CrossRefGoogle Scholar
- Deswal S, Kalkal KK (2014) Plane waves in a fractional order micropolar magneto-thermoelastic half-space. Wave Motion 51:100–113MathSciNetCrossRefzbMATHGoogle Scholar
- Dhaliwal RS, Sherief HH (1980) Generalized thermoelasticity for anisotropic media. Q Appl Math 33:1–8MathSciNetCrossRefzbMATHGoogle Scholar
- El-Karamany AS, Ezzat MA (2014) On the dual-phase-lag thermoelasticity theory. Meccanica 49:79–89MathSciNetCrossRefzbMATHGoogle Scholar
- El-Maghraby NM (2005) A two-dimensional problem for a thick plate with heat sources in generalized thermoelasticity. J Therm Stress 28:1227–1241CrossRefGoogle 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 32:863–876CrossRefGoogle 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 37:1416–1432CrossRefGoogle Scholar
- Ezzat MA (2004) Fundamental solution in generalized magneto-thermoelasticity with two relaxation times for perfect conductor cylindrical region. Int J Eng Sci 42:1503–1519MathSciNetCrossRefzbMATHGoogle Scholar
- Ezzat MA (2010) Thermoelectric MHD non-Newtonian fluid with fractional derivative heat transfer. Physica B 405:4188–4194CrossRefGoogle 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 252:267–277CrossRefGoogle Scholar
- Green AE, Lindsay KA (1972) Thermoelasticity. J Elast 2:1–7CrossRefzbMATHGoogle Scholar
- Green AE, Naghdi P (1991) A re-examination of the basic postulate of thermo-mechanics. Proc R Soc Lond Ser A 432:171–194CrossRefzbMATHGoogle Scholar
- Green AE, Naghdi P (1992) On undamped heat waves in an elastic solid. J Therm Stress 15:252–264MathSciNetCrossRefGoogle Scholar
- Green AE, Naghdi P (1993) Thermoelasticity without energy dissipation. J Elast 31:189–208MathSciNetCrossRefzbMATHGoogle Scholar
- Honig G, Hirdes U (1984) A method for the numerical inversion of Laplace transforms. J Comput Appl Math 10:113–132MathSciNetCrossRefzbMATHGoogle 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 59:1142–1164MathSciNetCrossRefzbMATHGoogle Scholar
- Lord HW, Shulman Y (1967) A generalized dynamical theory of thermoelasticity. J Mech Phys Solid 15:299–306CrossRefzbMATHGoogle Scholar
- Miller KS, Ross B (1993) An introduction to the fractional calculus and fractional differential equations. Wiley, New YorkzbMATHGoogle Scholar
- Podlubny I (1999) Fractional differential equations. Academic Press, New YorkzbMATHGoogle Scholar
- Povstenko YZ (2005) Fractional heat conduction equation and associated thermal stress. J Therm Stress 28:83–102MathSciNetCrossRefGoogle Scholar
- Povstenko YZ (2010) Fractional radial heat conduction equation in an infinite medium with a cylindrical cavity and associated thermal stresses. Mech Res Commun 37:436–440CrossRefzbMATHGoogle Scholar
- Quintanilla R, Racke R (2006) A note on stability of dual-phase-lag heat conduction. Int J Heat Mass Transf 49:1209–1213CrossRefzbMATHGoogle Scholar
- Quintanilla R, Racke R (2007) Qualitative aspects in dual-phase-lag heat conduction. Proc R Soc Lond A 463:659–674MathSciNetCrossRefzbMATHGoogle Scholar
- Ross B (1977) The development of fractional calculus. Hist Math 4:75–89MathSciNetCrossRefzbMATHGoogle Scholar
- Roychoudhuri SK (2007) One-dimensional thermoelastic waves in elastic half space with dual-phase-lag effects. J Mech Mater Struct 2:489–503CrossRefGoogle 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 40:223–232CrossRefGoogle Scholar
- Sarkar N, Lahiri A (2013) The effect of fractional parameter on a perfect conducting elastic half-space in generalized magneto-thermoelasticity. Meccanica 48:231–245MathSciNetCrossRefzbMATHGoogle Scholar
- Sherief HH, El-Sayed AM, El-Latief AM (2010) Fractional order theory of thermoelasticity. Int J Solids Struct 47:269–275CrossRefzbMATHGoogle Scholar
- Thomas L (1980) Fundamentals of Heat Transfer. Prentice-Hall Inc., Englewood CliffsGoogle Scholar
- Tzou DY (1995) A unified field approach for heat conduction from macro to micro-scales. J Heat Transf 117:8–16CrossRefGoogle Scholar
- Verma KL, Hasebe N (2001) Wave propagation in plates of general anisotropic media in generalized thermoelasticity. Int J Eng Sci 39:1739–1763CrossRefGoogle Scholar
- Vernotte P (1958) Les paradoxes de la theorie continue de l’equation de la chaleur. C R Acad Sci 246:3154–3155zbMATHGoogle Scholar
- Youssef HM (2010) Theory of fractional order generalized thermoelasticity. J Heat Transf 132:1–7CrossRefGoogle Scholar