On harmonic plane wave propagation under fractional order thermoelasticity: an analysis of fractional order heat conduction equation

Abstract

In the present paper, we investigate the propagation of an harmonic plane wave propagating with assigned frequency by implementing the thermoelasticity theory based on a fractional order heat conduction law where the fractional order parameter $α$ satisfies $0 < α < 1$ . After formulating the problem, the exact dispersion relation solutions for the plane wave are determined analytically and asymptotic expressions of different characterization of the wave are analyzed in two special cases, namely for a high-frequency field and low-frequency field. We consider the case of longitudinal wave which is coupled with the thermal field and we ignore the transverse wave as it is observed to be independent to the thermal field. Two different modes: predominantly thermal and predominantly elastic mode longitudinal waves are found. Finally we compute wave characterizations for the intermediate values of frequency and verify our analytical results for the limiting cases of wave frequency. A detailed analysis is presented to highlight the effects of the fractional order parameter, $α$ , on the wave fields. Several important points are highlighted and the most important point which we have found is that $α$ plays a very important role in the behavior of the wave. As $α \leq 0.5$ , the thermal mode wave behaves similarly as in the case of a classical thermoelastic model and the nature of the wave changes significantly as $α$ gets nearer to the value 1, behaving more similarly to the case of a generalized thermoelastic model (Lord–Shulman model).

Keywords

Fractional order heat conduction harmonic plane waves phase velocity specific loss thermoelasticity generalized thermoelasticity Lord–Shulman model

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References

Lord

Shulman

. A generalized dynamical theory of thermoelasticity. J Mech Phys Solids 1967; 15: 299–309.

Green

Lindsay

. Thermoelasticity. J Elasticity 1972; 2: 1–7.

Green

Naghdi

. A re-examination of the basic postulates of thermomechanics. Proc Roy Soc London 1991; A432: 171–194.

Green

Naghdi

. On undamped heat waves in an elastic solid. J Therm Stresses 1992; 15: 253–264.

Green

Naghdi

. Thermoelasticity without energy dissipation. J Elasticity 1993; 31: 189–209.

Puri

. Plane thermoelastic waves in rotating media. Bull Acad Polon Sci Tech 1976; 24: 137–144.

Roychoudhuri

. Effects of rotation and relaxation times on plane waves in generalized thermoelasticity. J Elasticity 1985; 15: 59–68.

Chadwick

Sneddon

. Plane waves in an elastic solid conducting heat. J Mech Phys Solids 1958; 6: 223–230.

Chadwick

. Thermoelasticity: The dynamic theory. In: Hill

Sneddon

(eds) Progress in solid mechanics, vol. 1. Amsterdam: North-Holland, 1960, 263–328.

10.

Nayfeh

Nemat-Nasser

. Thermoelastic waves in solids with thermal Relaxation. Acta Mech 1971; 12: 53–69.

11.

Puri

. Plane waves in generalized thermoelasticity. Int J Eng Sci 1973; 11: 735–744.

12.

Haddow

Wegner

. Plane harmonic waves for three thermoelastic theories. Math Mech Solids 1996; 1: 111–127.

13.

Suh

Burger

. Effects of thermomechanical coupling and relaxation times on wave spectrum in dynamic theory of generalized thermoelasticity. J Appl Mech (Trans ASME) 1998; 65: 605–613.

14.

Agarwal

. On plane waves in generalized thermoelasticity. Acta Mech 1979; 31: 185–189.

15.

Chandrasekharaiah

. Thermoelastic plane waves without energy dissipation. Mech Res Commun 1996; 23: 549–555.

16.

Puri

Jordan

. On the propagation of plane waves in type-III thermoelastic media. Proc R Soc 2004; A460: 3203–3221.

17.

Povstenko

. Fractional heat conduction equation and associated thermal stress. J Thermal Stresses 2005; 28: 83–102.

18.

Caputo

. Linear model of dissipation whose Q is almost frequency independent II. Geophys J Roy Astron Soc 1967; 13: 529–539.

19.

Youssef

. Theory of fractional order generalized thermoelasticity. J Heat Transfer 2010; 132. DOI: 10.1115/1.4000705.

20.

Youssef

Al-Lehaibi

. Fractional order generalized thermoelastic half-space subjected to ramp type heating. Mech Res Commun 2010; 37: 448–452.

21.

Sherief

El Sayed

El Latief

. Fractional order theory of thermoelasticity. Int J Solids Structures 2010; 47: 269–275.

22.

Tiwari

Mukhopadhyay

. Boundary integral equations formulation for fractional order thermoelasticity. Computat Meth Sci Technol 2014; 20(2): 49–58.

23.

Kumar

Gupta

. Wave propagation at the boundary surface of an elastic and thermoelastic diffusion media with fractional order derivative. Appl Math Mod 2015; 5–6: 1674–1688.

24.

Othman

Said

. Effect of hydrostatic initial stress on a fiber-reinforced thermoelastic medium with fractional derivative heat transfer. Multidisc Mod Mater Structures 2013; 9: 410–426.

25.

Atwa

. Generalized thermoelastic diffusion with effect of fractional parameter on plane waves temperature-dependent elastic medium. J Therm Stresses 2011; 34: 851–872.

26.

Ezzat

. Magneto-thermoelasticity with thermoelectric properties and fractional derivative heat transfer. Physica B 2011; 406: 30–35.

27.

Ezzat

El-Karamany

. Fractional order theory of a perfect conducting thermoelastic medium. Can J Phys 2011; 89: 311–318.

28.

Punnusamy

. Foundation of Complex Analysis. Narosa Publishing House, 2001.

29.

Kumar

Kant

Mukhopadhyay

. An in-depth investigation on plane harmonic waves under two-temperature thermoelasticity with two relaxation parameters. Math Mech Solids 2015. DOI: 10.1177/1081286515578495.