# On thermoelectric materials with memory-dependent derivative and subjected to a moving heat source

- 16 Downloads

## Abstract

We develop a model of generalized thermoelasticity with memory-dependent derivative (MDD) heat conduction law for a thermoelectric half-space. Some urgent theories take after as most remote point cases. The Laplace transform and state-space procedures are utilized to urge the overall account for any arrangement of limit conditions. The general solution acquired is connected to the particular issue of a half-space exposed to a uniform magnetic field, a moving heat source with consistent speed and ramp-type heating. The inverse Laplace transforms are registered numerically. The impacts of various estimations of the figure-of-merit quantity, heat source speed, MDD parameters, the magnetic number and the ramping time parameter are thought about.

## List of symbols

*λ, μ*Lame’s constants

*ρ*Density

*t*Time

*C*_{E}Specific heat at constant strain

- \(B_{i}\)
Components of magnetic field strength

*E*_{i}Components of electric field vector

*J*_{i}Conduction electric density vector

*H*_{i}Magnetic field intensity

*q*_{i}Components of heat flux vector

*H*_{o}Constant component of magnetic field

*μ*_{o}Magnetic permeability

*σ*_{o}Electric conductivity

*ε*_{ijk}Permutation symbol

*σ*_{ij}Components of stress tensor

- \(e_{ij}\)
Components of strain tensor

*u*_{i}Components of displacement vector

- \(\theta\)
\(= T - T_{o}\)

*T*_{o}Reference temperature chosen so that \(\left| {T \, - \, T_{o} } \right|/T_{o}\) ≪ 1

*e*=

*u*_{i,i}dilatation- \(k\)
Thermal conductivity

- \(\alpha_{T}\)
Coefficient of linear thermal expansion

- \(\gamma\)
= \((3\lambda + 2\mu )\alpha_{T}\)

- π
_{o} Peltier coefficient at

*T*_{o}*k*_{o}Seebeck coefficient at

*T*_{o}- \(\varepsilon\)
\(= \, \frac{{\delta_{o} \,\gamma }}{{\rho \,C_{E} }}\) thermoelastic parameter

- \(M\)
\(= \, \frac{{\sigma_{o} \,B_{o}^{2} }}{{\eta_{o} \rho \,c_{o}^{2} }}\) magnetic number

- \(\eta\)
\(= \frac{1}{{\sigma_{o} \mu_{o} }}\) magnetic diffusivity

- \(\eta_{o}\)
\(= \frac{{\rho \,C_{E} }}{k}\)

- \(c_{o}^{2}\)
\(= \;(\lambda \, + \,\;2\,\mu )/\rho\)

## Notes

### Acknowledgements

The authors gratefully acknowledge the approval and the support of this research study by the Grant no. SCI-2018-3-9-F-7583 from the Deanship of Scientific Research in Northern Border University, Arar, KSA.

## References

- Biot M (1956) Thermoelasticity and irreversible thermodynamics. J Appl Phys 27:240–253MathSciNetCrossRefzbMATHGoogle Scholar
- Chandrasekharaiah DS (1986) Thermoelasticity with second sound: a review. Appl Mech Rev 39:355–376CrossRefzbMATHGoogle Scholar
- Chandrasekharaiah DS (1998) Hyperbolic thermoelasticity, a review of recent literature. Appl Mech Rev 51:705–729CrossRefGoogle Scholar
- El-Karamany AS, Ezzat MA (2002) On the boundary integral formulation of thermo-viscoelasticity theory. Int J Eng Sci 40:1943–1956MathSciNetCrossRefzbMATHGoogle Scholar
- Ezzat MA (1994) State space approach to unsteady two-dimensional free convection flow through a porous medium. Can J Phys 72:311–317MathSciNetCrossRefGoogle Scholar
- Ezzat MA (2006) The relaxation effects of the volume properties of electrically conducting viscoelastic material. Math Sci Eng B 130:11–23CrossRefGoogle Scholar
- Ezzat MA (2011) Thermoelectric MHD with modified Fourier’s law. Int J Therm Sci 50:449–455CrossRefGoogle Scholar
- Ezzat MA, Abd-Elaal MZ (1997a) State space approach to viscoelastic fluid flow of hydromagnetic fluctuating boundary-layer through a porous medium. ZAMM 77:197–207MathSciNetCrossRefzbMATHGoogle Scholar
- Ezzat MA, Abd-Elaal MZ (1997b) Free convection effects on a viscoelastic boundary layer flow with one relaxation time through a porous medium. J Frank Inst 334:685–706MathSciNetCrossRefzbMATHGoogle Scholar
- Ezzat MA, Awad ES (2010) Constitutive relations, uniqueness of solution, and thermal shock application in the linear theory of micropolar generalized thermoelasticity involving two temperatures. J Therm Stress 33:226–250CrossRefGoogle Scholar
- Ezzat MA, El-Bary AA (2015) Memory-dependent derivatives theory of thermo-viscoelasticity involving two-temperature. J Mech Sci Tech 29:4273–4279CrossRefGoogle Scholar
- Ezzat MA, El-Bary AA (2016) Thermoelectric MHD with memory-dependent derivative heat transfer. Int Commun Heat Mass Transfer 75:270–281CrossRefGoogle Scholar
- Ezzat MA, El-Bary AA (2017) A functionally graded magneto-thermoelastic half space with memory-dependent derivatives heat transfer. Steel Compos Struct Int J 25:177–186Google Scholar
- Ezzat MA, El-Bary AA (2018) Unified GN model of electro-thermoelasticity theories with fractional order of heat transfer. Micro Syst 24:4965–4979Google Scholar
- Ezzat MA, El-Karamany AS (2003) On uniqueness and reciprocity theorems for generalized thermoviscoelasticity with thermal relaxation. Can J Phys 81:823–833CrossRefGoogle Scholar
- Ezzat MA, Youssef HM (2010) Stokes’ first problem for an electro-conducting micropolar fluid with thermoelectric properties. Can J Phys 88:35–48CrossRefGoogle Scholar
- Ezzat MA, Othman MI, El-Karamany AS (2001) State space approach to generalized thermo-viscoelasticity with two relaxation times. Int J Eng Sci 40:283–302MathSciNetCrossRefGoogle Scholar
- Ezzat MA, El-Karamany AS, El-Bary AA (2014) Generalized thermo-viscoelasticity with memory-dependent derivatives. Int J Mech Sci 89:470–475CrossRefGoogle Scholar
- Fox N (1969) Generalized thermoelasticity. Int J Eng Sci 7:437–445CrossRefzbMATHGoogle Scholar
- Green A, Lindsay K (1972) Thermoelasticity. J Elast 2:1–7CrossRefzbMATHGoogle Scholar
- Hendy MH, Amin MM, Ezzat MA (2018) Magneto-electric interactions without energy dissipation for a fractional thermoelastic spherical cavity. Micro Sys 24:2895–2903Google Scholar
- Hetnarski RB, Ignaczak J (1999) Generalized thermoelasticity. J Therm Stress 22:451–476MathSciNetCrossRefzbMATHGoogle Scholar
- Hiroshige Y, Makoto O, Toshima N (2007) Thermoelectric figure-of-merit of iodine- doped copolymer of phenylenevinylene with dialkoxyphenylenevinylene. Synth Metals 157:467–474CrossRefGoogle Scholar
- Honig G, Hirdes U (1984) A method for the numerical inversion of Laplace transforms. J Comput Appl Math 10:113–132MathSciNetCrossRefzbMATHGoogle Scholar
- Ignaczak J, Ostoja-starzeweski M (2009) Thermoelasticity with finite wave speeds. Oxford University Press, OxfordCrossRefzbMATHGoogle Scholar
- Kaliski S, Nowacki W (1963) Combined elastic and electro-magnetic waves produced by thermal shock in the case of a medium of finite electric conductivity. Int J Eng Sci 1:163–175CrossRefGoogle Scholar
- Lata P, Kumar R, Sharma N (2016) Plane waves in an anisotropic thermoelastic. Steel Compos Struct Int J 22:567–587CrossRefGoogle Scholar
- Li D, He T (2018) Investigation of generalized piezoelectric-thermoelastic problem with nonlocal effect and temperature-dependent properties. Heliyon 4:e00860CrossRefGoogle Scholar
- Lord H, Shulman Y (1967) A generalized dynamical theory of thermoelasticity. J Mech Phys Solids 15:299–309CrossRefzbMATHGoogle Scholar
- Lotfy Kh (2017) A novel solution of fractional order heat equation for photothermal waves in a semiconductor medium with a spherical cavity. Chaos Solis Fract 99:233–242MathSciNetCrossRefzbMATHGoogle Scholar
- Lotfy K, Sarkar N (2017) Memory-dependent derivatives for photothermal semiconducting medium in generalized thermoelasticity with two-temperature. Mech Time Depend Mat 21:519–534CrossRefGoogle Scholar
- Othman MI, Lotfy K (2013) The effect of magnetic field and rotation of the 2-D problem of a fiber-reinforced thermoelastic under three theories with influence of gravity. Mech Mater 60:129–143CrossRefGoogle Scholar
- Othman MI, Ezzat MA, Zaki SA, El-Karamany AS (2002) Generalized thermo- viscoelastic plane waves with two relaxation times. Int J Eng Sci 40:1329–1347MathSciNetCrossRefzbMATHGoogle Scholar
- Rowe DM (1995) Handbook of thermoelectrics. CRC Press, Boca RatonCrossRefGoogle Scholar
- Sarkar N, Lotfy K (2018) A 2D problem of time-fractional heat order for two-temperature thermoelasticity under hydrostatic initial stress. Mech Adv Math Struct 25:279–285CrossRefGoogle Scholar
- Shaw S (2019) Theory of generalized thermoelasticity with memory-dependent derivatives. J Eng Mech 145:04019003CrossRefGoogle Scholar
- Shercliff JA (1979) Thermoelectric magnetohydrodynamics. J Fluid Mech 19:231–251CrossRefzbMATHGoogle Scholar
- Shereif HH (1986) Fundamental solution of generalized thermoelastic problem for short times. J Therm Stress 9:151–164CrossRefGoogle Scholar
- Sherief HH, Abd El-Latief AM (2013) Effect of variable thermal conductivity on a half-space under the fractional order theory of thermoelasticity. Int J Mech Sci 74:185–189CrossRefGoogle Scholar
- Sherief HH, Abd El-Latief AM (2014) Application of fractional order theory of thermoelasticity to a 1D problem for a half-space. J Appl Math Mech 94:509–515MathSciNetzbMATHGoogle Scholar
- Sherief HH, El-Said A, Abd El-Latief A (2010) Fractional order theory of thermoelasticity. Int J Solids Struct 47:269–275CrossRefzbMATHGoogle Scholar
- Tiwari R, Mukhopadhyay S (2018) Analysis of wave propagation in the presence of a continuous line heat source under heat transfer with memory dependent derivatives. Math Mech Solids 23:820–834MathSciNetCrossRefzbMATHGoogle Scholar
- Tritt TM (1999) Thermoelectric materials new directions and approaches. Math Res Soc Syma Proc 545:233–246Google Scholar
- Tritt TM (2000) Semiconductors and semimetals, recent trends in thermoelectric materials research. Academic Press, San DiegoGoogle Scholar
- Xiong C, Niu Y (2017) Fractional-order generalized thermoelastic diffusion theory. Appl Math Mech 38:1091–1108MathSciNetCrossRefzbMATHGoogle Scholar
- Xue Z-N, Chen Z-T, Tian X-G (2018) Thermoelastic analysis of a cracked strip under thermal impact based on memory-dependent heat conduction model. Eng Fract Mech 200:479–498CrossRefGoogle Scholar
- Yu Y-J, Hu W, Tian X-G (2014) A novel generalized thermoelasticity model based on memory-dependent derivative. Int J Eng Sci 81:123–134MathSciNetCrossRefzbMATHGoogle Scholar