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Analysis of a magneto-thermoelastic problem in a piezoelastic medium using the non-local memory-dependent heat conduction theory involving three phase lags

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Abstract

One of the most important features of the non-local heat conduction theory is that it is able to analyze and investigate the effect of all points of the body on a material point. Studying this non-local phenomenon helps researchers to capture microscopic effects at the macroscopic level with greater accuracy. This paper is devoted to understanding the non-local behavior of a piezoelastic half-space permeated with a magnetic field in the context of three-phase lag heat conduction with a memory-dependent derivative. The current study, which includes the presence of the propagation of thermal waves at limited speeds and non-locality in space and delay in time, has great significance in its application in many different physical fields. A thermal shock has been applied to the non-traction boundary. The problem has been solved by employing the Laplace transform technique. Physical variables such as temperature, displacement as well as stress have been determined in the time-domain by an accurate numerical method. Computational results have been provided to show the influence of effective parameters such as non-local correlation length parameter, kernel functions as well as time delay on the physical variables. By discussing the numerical results that have been graphically represented, significant differences have been seen to be attributable to the studied fields due to the non-locality effect, memory effect and time delay.

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Abbreviations

\(\sigma _{ij}\) :

stress tensor.

\(e_{ij}\) :

strain tensor.

\(c_{ijkl}\) :

isothermal elastic parameters.

\(\lambda , \mu \) :

Lamé constants.

\(\gamma\,=\) :

\(\left ( 3\lambda + 2\mu \right ) \alpha _{T}\).

\(\alpha _{T}\) :

coefficient of volume expansion.

\(\theta \) :

temperature change.

\(T_{0}\) :

initial reference temperature.

\(\rho \) :

mass density.

\(K^{*}\) :

thermal conductivity rate.

\(K\) :

thermal conductivity.

\(C_{E}\) :

specific heat at constant strain.

\(\tau _{q}\) :

phase lag of heat flux.

\(\tau _{T} \) :

phase lag of the temperature gradient.

\(\tau _{\nu } \) :

phase lag of thermal gradient.

\(\boldsymbol{\lambda }_{\boldsymbol{q}}\) :

non-local correlating length vector.

\(\omega \) :

time delay.

\(\boldsymbol{H}\) :

magnetic field vector.

\(\boldsymbol{E}\) :

electric field vector.

\(\boldsymbol{F}\) :

Lorentz force vector.

\(\mu _{0}\) :

magnetic permeability.

\(\epsilon _{0}\) :

electric permittivity.

\(\boldsymbol{J}\) :

current density vector.

\(h_{ijk}\) :

piezoelectric moduli.

\(\zeta _{e}\) :

free charge of the medium.

\(D_{i}\) :

electric displacement.

\(\varepsilon _{ij}\) :

dielectric moduli.

\(p_{i}\) :

pyroelectric moduli.

\(\gamma _{ij}\) :

thermal elastic coupling tensor.

\(\zeta \) :

free charge of the medium.

\(\boldsymbol{u}\) :

displacement vector.

References

  • Al-Huniti, N.S., Al-Nimr, M.A.: Thermoelastic behavior of a composite slab under a rapid dual-phase-lag heating. J. Therm. Stresses 27, 607–623 (2004)

    Article  Google Scholar 

  • Banik, S., Kanoria, M.: Effects of three-phase-lag on two-temperature generalized thermoelasticity for infinite medium with spherical cavity. Appl. Math. Mech. 33, 483–498 (2012)

    Article  MathSciNet  Google Scholar 

  • Bellmen, R., Kolaba, R.E., Lockette, J.A.: Numerical Inversion of the Laplace Transform. American Elsevier Pub. Co., New York (1966)

    Google Scholar 

  • Biot, M.A.: Thermoelasticity and irreversible thermodynamics. J. Appl. Phys. 27, 240 (1956)

    Article  MathSciNet  Google Scholar 

  • Cattaneo, C.: A form of heat conduction equation which eliminates the paradox of instantaneous propagation. C. R. Acad. Sci. 247, 431–433 (1958)

    MATH  Google Scholar 

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

    Article  Google Scholar 

  • Chiriţă, S., D’Apice, C., Zampoli, V.: The time differential three-phase-lag heat conduction model: thermodynamic compatibility and continuous dependence. Int. J. Heat Mass Transf. 102, 226–232 (2016)

    Article  Google Scholar 

  • Dai, T.M.: Restudy of coupled field theories for micropolar continua (II) – Thermopiezoelectricity and magnetothermoelasticity. Appl. Math. Mech. 23, 249–258 (2002)

    Article  Google Scholar 

  • Eringen, A.C.: Nonlocal Continuum Field Theories. Springer, New York (2002)

    MATH  Google Scholar 

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

    Article  Google Scholar 

  • Ezzat, M.A., El-Karamany, A.S., El-Bary, A.A.: On thermo-viscoelasticity with variable thermal conductivity and fractional-order heat transfer. Int. J. Thermophys. 36, 1684–1697 (2015)

    Article  Google Scholar 

  • Green, A.E., Lindsay, K.A.: Thermoelasticity. J. Elast. 2, 1–7 (1972)

    Article  Google Scholar 

  • Green, A.E., Naghdi, P.M.: A re-examination of the basic postulates of thermomechanics. Proc. R. Soc. A, Math. Phys. Eng. Sci. 432, 171–194 (1991)

    MathSciNet  MATH  Google Scholar 

  • Green, A.E., Naghdi, P.M.: On undamped heat waves in an elastic solid. J. Therm. Stresses 15, 253–264 (1992)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Ho, J.R., Kuo, C.P., Jiaung, W.S.: Study of heat transfer in multilayered structure within the framework of dual-phase-lag heat conduction model using lattice Boltzmann method. Int. J. Heat Mass Transf. 48, 55–69 (2003)

    Article  Google Scholar 

  • Itu, C., Öchsner, A., Vlase, S., Marin, M.I.: Improved rigidity of composite circular plates through radial ribs. Proc. Inst. Mech. Eng. L: J. Mat.: Design Appl. 233, 1585–1593 (2019)

    Google Scholar 

  • Khamis, A.K., Lotfy, K., El-Bary, A.A., Mahdy, A.M., Ahmed, M.H.: Thermal-piezoelectric problem of a semiconductor medium during photo-thermal excitation. Waves Random Complex Media, 1–15 (2020)

  • Kumar, R.: Effect of phase-lag on thermoelastic vibration of Timoshenko beam. J. Therm. Stresses 43, 1337–1354 (2020)

    Article  Google Scholar 

  • Kumar, R., Mukhopadhyay, S.: Effects of three phase lags on generalized thermoelasticity for an infinite medium with a cylindrical cavity. J. Therm. Stresses 32, 1149–1165 (2009)

    Article  Google Scholar 

  • Kumar, R., Tiwari, R., Kumar, R.: Significance of memory-dependent derivative approach for the analysis of thermoelastic damping in micromechanical resonators. Mech. Time-Depend. Mater., 1–18 (2020). https://doi.org/10.1007/s11043-020-09477-7

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

    Article  Google Scholar 

  • Lotfy, K., Sarkar, N.: Memory-dependent derivatives for photothermal semiconducting medium in generalized thermoelasticity with two-temperature. Mech. Time-Depend. Mater. 21, 519–534 (2017)

    Article  Google Scholar 

  • Mindlin, R.D.: Electromagnetic radiation from a vibrating quartz plate. Int. J. Sol. Struct. 9, 697–702 (1973)

    Article  Google Scholar 

  • Mindlin, R.D.: Equations of high frequency vibrations of thermopiezoelectric crystal plates. Int. J. Solids Struct. 10, 625–637 (1974)

    Article  Google Scholar 

  • Nowacki, W.: Foundations of linear piezoelectricity. Electromagn. Interact. Elast. Solids 257, 105–157 (1979)

    Google Scholar 

  • Quintanilla, R., Racke, R.: A note on stability in three-phase-lag heat conduction. Int. J. Heat Mass Transf. 51, 24–29 (2008)

    Article  Google Scholar 

  • Ramadan, K.: Semi-analytical solutions for the dual phase lag heat conduction in multilayered media. Int. J. Therm. Sci. 48, 14–25 (2009)

    Article  Google Scholar 

  • Roy Choudhuri, S.K.: On a thermoelastic three-phase-lag model. J. Therm. Stresses 30, 231–238 (2007)

    Article  Google Scholar 

  • Roy Choudhuri, S.K., Banerjee, M.: Magneto-viscoelastic plane waves in rotating media in the generalized thermoelasticity II. Int. J. Math. Math. Sci. 11, 1819–1834 (2005). https://doi.org/10.1155/IJMMS.2005.1819

    Article  MathSciNet  MATH  Google Scholar 

  • Sharifi, Z., Khordad, R., Gharaati, A., Forozani, G.: An analytical study of vibration in functionally graded piezoelectric nanoplates: nonlocal strain gradient theory. Appl. Math. Mech. 40, 1723–1740 (2019)

    Article  MathSciNet  Google Scholar 

  • Sur, A.: Non-local memory-dependent heat conduction in a magneto-thermoelastic problem. Waves Random Complex Media, 1–21 (2020). https://doi.org/10.1080/17455030.2020.1770369

  • Tiersten, H.F.: On the nonlinear equations of thermo-electroelasticity. Int. J. Eng. Sci. 9, 587–604 (1971)

    Article  MathSciNet  Google Scholar 

  • Tiwari, R., Misra, J.C.: Magneto-thermoelastic excitation induced by a thermal shock: a study under the purview of three phase lag theory. Waves Random Complex Media (2020). https://doi.org/10.1080/17455030.2020.1800861

    Article  Google Scholar 

  • Tiwari, R., Mukhopadhyay, S.: On electromagneto-thermoelastic plane waves under Green–Naghdi theory of thermoelasticity-II. J. Therm. Stresses 40, 1040–1062 (2017)

    Article  Google Scholar 

  • Tiwari, R., Kumar, R., Kumar, A.: Investigation of thermal excitation induced by laser pulses and thermal shock in the half space medium with variable thermal conductivity. Waves Random Complex Media (2020). https://doi.org/10.1080/17455030.2020.1851067

    Article  Google Scholar 

  • Tzou, D.Y.: Thermal shock phenomena under high rate response in solids. Annu. Rev. Heat Transf. 4, 111–185 (1992). https://doi.org/10.1615/annualrevheattransfer.v4.50

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Tzou, D.Y.: Macro-to Microscale Heat Transfer: The Lagging Behavior. Wiley, New York (1997)

    Google Scholar 

  • Tzou, D.Y., Guo, Z.Y.: Nonlocal behavior in thermal lagging. Int. J. Therm. Sci. 49, 1133–1137 (2010)

    Article  Google Scholar 

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

    MATH  Google Scholar 

  • Vernotte, P.: Some possible complications in the phenomena of thermal conduction. C. R. Acad. Sci. 252, 2190–2191 (1961)

    Google Scholar 

  • Vlase, S., Marin, M., Öchsner, A., Scutaru, M.L.: Motion equation for a flexible one-dimensional element used in the dynamical analysis of a multibody system. Contin. Mech. Thermodyn. 31, 715–724 (2019)

    Article  MathSciNet  Google Scholar 

  • Wang, J.L., Li, H.F.: Surpassing the fractional derivative: concept of the memory-dependent derivative. Comput. Math. Appl. 62, 1562–1567 (2011)

    Article  MathSciNet  Google Scholar 

  • Youssef, H.M., El-Bary, A.A.: Generalized thermoelastic infinite layer subjected to ramp-type thermal and mechanical loading under three theories-state space approach. J. Therm. Stresses 32, 1293–1309 (2009)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Nitishwar College, A constituent unit of Babasaheb Bhimrao Ambedkar Bihar University, Muzaffarpur, Bihar, India

    Rakhi Tiwari

  2. Department of Mathematics, Central University of South Bihar, Gaya, Bihar, India

    Ravi Kumar

  3. Department of Mathematics, College of Science and Arts, Jouf University, Al-Qurayyat, Saudi Arabia

    Ahmed E. Abouelregal

  4. Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, 35516, Egypt

    Ahmed E. Abouelregal

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  1. Rakhi Tiwari
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Correspondence to Rakhi Tiwari.

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R. Tiwari’s former address: Department of Mathematical Sciences, Indian Institute of Technology (BHU), Varanasi, India.

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Tiwari, R., Kumar, R. & Abouelregal, A.E. Analysis of a magneto-thermoelastic problem in a piezoelastic medium using the non-local memory-dependent heat conduction theory involving three phase lags. Mech Time-Depend Mater 26, 271–287 (2022). https://doi.org/10.1007/s11043-021-09487-z

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