A thermo-elastic model with a single relaxation time of an unbounded medium heated by a heat supply and moving vertically

Document Type : Research Paper

Authors

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

Abstract

The current paper presents a thermoelastic model with a single relaxation time to examine the thermoelastic interaction in an isotropic infinite medium. The unbounded medium is exposed to a thermal shock with varying temperatures due to a vertically moving heat source in a planar region. The basic partial differential equations were solved using the Laplace transform method. Physical fields are studied and compared in terms of how the speed of the heat source, the relaxation time parameter, and the time parameters affect their behavior. Graphical presentations are used to analyze physical field variables like temperature changes, thermal stress and deformation.

Keywords

[1]          H. W. Lord, Y. Shulman, A generalized dynamical theory of thermoelasticity, Journal of the Mechanics and Physics of Solids, Vol. 15, No. 5, pp. 299-309, 1967.
[2]          A. E. Green, K. Lindsay, Thermoelasticity, Journal of elasticity, Vol. 2, No. 1, pp. 1-7, 1972.
[3]          A. Green, P. Naghdi, On undamped heat waves in an elastic solid, Journal of Thermal Stresses, Vol. 15, No. 2, pp. 253-264, 1992.
[4]          A. E. Green, P. Naghdi, A re-examination of the basic postulates of thermomechanics, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences, Vol. 432, No. 1885, pp. 171-194, 1991.
[5]          A. Green, P. Naghdi, Thermoelasticity without energy dissipation, Journal of elasticity, Vol. 31, No. 3, pp. 189-208, 1993.
[6]          D. Y. Tzou, A unified field approach for heat conduction from macro-to micro-scales, 1995.
[7]          D. Y. Tzou, The generalized lagging response in small-scale and high-rate heating, International Journal of Heat and Mass Transfer, Vol. 38, No. 17, pp. 3231-3240, 1995.
[8]          D. T. M.-t. M. Heat, Transfer: The Lagging Behavior, Taylor and Francis, New York, 1997.
[9]          A. E. Abouelregal, Two-temperature thermoelastic model without energy dissipation including higher order time-derivatives and two phase-lags, Materials Research Express, Vol. 6, No. 11, pp. 116535, 2019.
[10]        A. E. Abouelregal, A novel model of nonlocal thermoelasticity with time derivatives of higher order, Mathematical Methods in the Applied Sciences, Vol. 43, No. 11, pp. 6746-6760, 2020.
[11]        M. Mohammadi, A. Farajpour, M. Goodarzi, H. Mohammadi, Temperature Effect on Vibration Analysis of Annular Graphene Sheet Embedded on Visco-Pasternak Foundation Journal of Solid Mechanics, Vol. 5, No. 3, pp. 305-323, 2013.
[12]        M. Mohammadi, A. Farajpour, A. Moradi, M. Hosseini, Vibration analysis of the rotating multilayer piezoelectric Timoshenko nanobeam, Engineering Analysis with Boundary Elements, Vol. 145, pp. 117-131, 2022/12/01/, 2022.
[13]        A. E. Abouelregal, A novel generalized thermoelasticity with higher-order time-derivatives and three-phase lags, Multidiscipline Modeling in Materials and Structures, 2019.
[14]        A. Abouelregal, On Green and Naghdi thermoelasticity model without energy dissipation with higher order time differential and phase-lags, Journal of Applied and Computational Mechanics, Vol. 6, No. 3, pp. 445-456, 2020.
[15]        Z. Hu, Z. Liu, Heat Conduction Simulation of 2D Moving Heat Source Problems Using a Moving Mesh Method, Advances in Mathematical Physics, Vol. ID 6067854, pp. 1-16, 2020.
[16]        D. W. Hahn, M. N. Özisik, 2012, Heat conduction, John Wiley & Sons,
[17]        S. B. Powar, P. M. Patane, S. L. Deshmukh, A review paper on numerical simulation of moving heat source, International Journal of Current Engineering and Technology, Vol. 4, pp. 63-66, 2016.
[18]        M. T. Pamuk, A. Savaş, Ö. Seçgin, E. Arda, Numerical simulation of transient heat transfer in friction-stir welding, International Journal on Heat and Technology, 2018.
[19]        Y. Sun, S. Liu, Z. Rao, Y. Li, J. Yang, Thermodynamic response of beams on Winkler foundation irradiated by moving laser pulses, Symmetry, Vol. 10, No. 8, pp. 328, 2018.
[20]        E. Mirkoohi, D. E. Seivers, H. Garmestani, S. Y. Liang, Heat source modeling in selective laser melting, Materials, Vol. 12, No. 13, pp. 2052, 2019.
[21]        K. He, Q. Yang, D. Xiao, X. Li, Analysis of thermo-elastic fracture problem during aluminium alloy MIG welding using the extended finite element method, Applied Sciences, Vol. 7, No. 1, pp. 69, 2017.
[22]        M. Akbari, D. Sinton, M. Bahrami, Geometrical effects on the temperature distribution in a half-space due to a moving heat source, Journal of Heat Transfer, Vol. 133, No. 6, 2011.
[23]        J. Winczek, The influence of the heat source model selection on mapping of heat affected zones during surfacing by welding, Journal of Applied Mathematics and Computational Mechanics, Vol. 15, No. 3, 2016.
[24]        W. Huang, R. D. Russell, 2010, Adaptive moving mesh methods, Springer Science & Business Media,
[25]        T. Flint, J. Francis, M. Smith, A. Vasileiou, Semi-analytical solutions for the transient temperature fields induced by a moving heat source in an orthogonal domain, International Journal of Thermal Sciences, Vol. 123, pp. 140-150, 2018.
[26]        S. Mondal, Interactions of a heat source moving over a visco-thermoelastic rod kept in a magnetic field in the Lord–Shulman model under a memory dependent derivative, Computational Mathematics and Modeling, Vol. 31, No. 2, pp. 256-276, 2020.
[27]        R. Tiwari, Analysis of phase lag effect in generalized magneto thermoelasticity with moving heat source, Waves in Random and Complex Media, pp. 1-18, 2021.
[28]        W. W. Mohammed, A. E. Abouelregal, D. Atta, F. Khelifi, Thermoelastic responses in a nonlocal infinite solid with a circular cylindrical cavity due to a moving heat supply under the MGT model of thermal conductivity, Physica Scripta, Vol. 97, No. 3, pp. 035705, 2022.
[29]        A. Abouelregal, Rotating magneto-thermoelastic rod with finite length due to moving heat sources via Eringen’s nonlocal model, Journal of Computational Applied Mechanics, Vol. 50, No. 1, pp. 118-126, 2019.
[30]        H. Asemi, S. Asemi, A. Farajpour, M. Mohammadi, Nanoscale mass detection based on vibrating piezoelectric ultrathin films under thermo-electro-mechanical loads, Physica E: Low-dimensional Systems and Nanostructures, Vol. 68, pp. 112-122, 2015.
[31]        S. R. Asemi, A. Farajpour, M. Mohammadi, Nonlinear vibration analysis of piezoelectric nanoelectromechanical resonators based on nonlocal elasticity theory, Composite Structures, Vol. 116, pp. 703-712, 2014.
[32]        M. Baghani, M. Mohammadi, A. Farajpour, Dynamic and Stability Analysis of the Rotating Nanobeam in a Nonuniform Magnetic Field Considering the Surface Energy, International Journal of Applied Mechanics, Vol. 08, No. 04, pp. 1650048, 2016.
[33]        M. Mohammadi, A. Farajpour, M. Goodarzi, R. Heydarshenas, Levy Type Solution for Nonlocal Thermo-Mechanical Vibration of Orthotropic Mono-Layer Graphene Sheet Embedded in an Elastic Medium, Journal of Solid Mechanics, Vol. 5, No. 2, pp. 116-132, 2013.
[34]        H. Moosavi, M. Mohammadi, A. Farajpour, S. H. Shahidi, Vibration analysis of nanorings using nonlocal continuum mechanics and shear deformable ring theory, Physica E: Low-dimensional Systems and Nanostructures, Vol. 44, No. 1, pp. 135-140, 2011/10/01/, 2011.
[35]        M. Mohammadi, A. Moradi, M. Ghayour, A. Farajpour, Exact solution for thermo-mechanical vibration of orthotropic mono-layer graphene sheet embedded in an elastic medium, Latin American Journal of Solids and Structures, Vol. 11, No. 3, pp. 437-458, 2014.
[36]        H. Mohammadi, M. Ghayour, A. Farajpour, Analysis of free vibration sector plate based on elastic medium by using new version of differential quadrature method, Journal of Simulation and Analysis of Novel Technologies in Mechanical Engineering, Vol. 3, No. 2, pp. 47-56, 2010.
[37]        A. Farajpour, A. Shahidi, M. Mohammadi, M. Mahzoon, Buckling of orthotropic micro/nanoscale plates under linearly varying in-plane load via nonlocal continuum mechanics, Composite Structures, Vol. 94, No. 5, pp. 1605-1615, 2012.
[38]        M. A. Biot, Thermoelasticity and irreversible thermodynamics, Journal of applied physics, Vol. 27, No. 3, pp. 240-253, 1956.
[39]        C. Cattaneo, A form of heat-conduction equations which eliminates the paradox of instantaneous propagation, Comptes Rendus, Vol. 247, pp. 431, 1958.
[40]        G. Honig, U. Hirdes, A method for the numerical inversion of Laplace transforms, Journal of Computational and Applied Mathematics, Vol. 10, No. 1, pp. 113-132, 1984.
[41]        M. Bachher, N. Sarkar, A. Lahiri, Generalized thermoelastic infinite medium with voids subjected to a instantaneous heat sources with fractional derivative heat transfer, International Journal of Mechanical Sciences, Vol. 89, pp. 84-91, 2014.
[42]        R. V. Singh, S. Mukhopadhyay, Relaxation effects on thermoelastic interactions for time-dependent moving heat source under a recent model of thermoelasticity, Zeitschrift für angewandte Mathematik und Physik, Vol. 72, No. 1, pp. 1-13, 2021.
Volume 53, Issue 3
September 2022
Pages 431-443
  • Receive Date: 13 August 2022
  • Revise Date: 26 August 2022
  • Accept Date: 26 August 2022