Journal of Computational Applied Mechanics

Study of a Half-Space Via a Generalized Dual Phase-lag Model with Variable Thermal Material Properties and Memory-Dependent Derivative

Document Type : Research Paper

Authors

1 Department of Mathematics, College of Science and Arts, Jouf University, Gurayat, Saudi Arabia

2 Department of Mathematics, Faculty of Science, Benha University, Benha 13518, Egypt

3 Department of Mathematics, Faculty of Science, South Valley University, Qena, Egypt

Abstract

The aim of this work is to investigate a generalized dual phase-lag model with variable thermal material parameters and memory dependent derivatives (VMDPL). In view of this model, the thermoelastic behavior on a half-space under an external body force and subjected to exponentially varying heat is analytically investigated. The governing differential equations are numerically solved using the Laplace transform approach. The effects of the variable thermal material properties and memory dependent derivative on all the physical quantities of a half-space are discussed. The obtained results demonstrate that the physical fields of a half-space depend not only on the distance, but also on the memory time delay and the variable thermal parameter. Furthermore, the variable thermal parameter and the variable thermal parameter has a clear effect on the temperature and the stress but has a negligible effect on the displacement. Finally, the validity of results is acceptable by comparing the displacement, stress and temperature according to the present generalized model (VMDPL) with those due to other thermoelasticity theories

Keywords

[1]          M. Biot, Thermoelasticity and Irreversible Thermodynamics, Journal of Applied Physics Vol. 27, No. 3, pp. 240-253, 1956.
[2]          C. Cattaneo, Sur une forme de l'equation de la chaleur eliminant la paradoxe d'une propagation instantantee, Compt. Rendu, Vol. 247, pp. 431-433, 1958.
[3]          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.
[4]          A. E. Green, K. Lindsay, Thermoelasticity, Journal of elasticity, Vol. 2, No. 1, pp. 1-7, 1972.
[5]          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.
[6]          A. Green, P. Naghdi, On undamped heat waves in an elastic solid, Journal of Thermal Stresses, Vol. 15, No. 2, pp. 253-264, 1992.
[7]          A. Green, P. Naghdi, Thermoelasticity without energy dissipation, Journal of elasticity, Vol. 31, No. 3, pp. 189-208, 1993.
[8]          D. Y. Tzou, 2014, Macro-to microscale heat transfer: the lagging behavior, John Wiley & Sons,
[9]          D. Y. Tzou, A unified field approach for heat conduction from macro-to micro-scales, Journal of Heat Transfer, Vol. 117, No. 1, pp. 8-16, 1995.
[10]        M. Caputo, F. Mainardi, Linear models of dissipation in anelastic solids, La Rivista del Nuovo Cimento (1971-1977), Vol. 1, No. 2, pp. 161-198, 1971.
[11]        M. Caputo, Vibrations of an infinite viscoelastic layer with a dissipative memory, The Journal of the Acoustical Society of America, Vol. 56, No. 3, pp. 897-904, 1974.
[12]        H. M. Youssef, Theory of fractional order generalized thermoelasticity, Journal of Heat Transfer, Vol. 132, No. 6, 2010.
[13]        Y. Povstenko, Fractional Cattaneo-type equations and generalized thermoelasticity, Journal of thermal Stresses, Vol. 34, No. 2, pp. 97-114, 2011.
[14]        M. A. Ezzat, Magneto-thermoelasticity with thermoelectric properties and fractional derivative heat transfer, Physica B: Condensed Matter, Vol. 406, No. 1, pp. 30-35, 2011.
[15]        M. Ezzat, A. El-Bary, Unified fractional derivative models of magneto-thermo-viscoelasticity theory, Archives of Mechanics, Vol. 68, No. 4, pp. 285-308, 2016.
[16]        H. H. Sherief, E. M. Hussein, Fundamental solution of thermoelasticity with two relaxation times for an infinite spherically symmetric space, Zeitschrift für angewandte Mathematik und Physik, Vol. 68, No. 2, pp. 1-14, 2017.
[17]        A. E. Abouelregal, Modified fractional thermoelasticity model with multi-relaxation times of higher order: application to spherical cavity exposed to a harmonic varying heat, Waves in Random and Complex Media, Vol. 31, No. 5, pp. 812-832, 2021.
[18]        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.
[19]        A. E. Abouelregal, M. A. Elhagary, A. Soleiman, K. M. Khalil, Generalized thermoelastic-diffusion model with higher-order fractional time-derivatives and four-phase-lags, Mechanics Based Design of Structures and Machines, pp. 1-18, 2020.
[20]        S. 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.
[21]        S. R. Asemi, M. Mohammadi, A. Farajpour, A study on the nonlinear stability of orthotropic single-layered graphene sheet based on nonlocal elasticity theory, Latin American Journal of Solids and Structures, Vol. 11, No. 9, pp. 1515-1540, 2014.
[22]        M. Danesh, A. Farajpour, M. Mohammadi, Axial vibration analysis of a tapered nanorod based on nonlocal elasticity theory and differential quadrature method, Mechanics Research Communications, Vol. 39, No. 1, pp. 23-27, 2012.
[23]        A. Farajpour, M. H. Yazdi, A. Rastgoo, M. Loghmani, M. Mohammadi, Nonlocal nonlinear plate model for large amplitude vibration of magneto-electro-elastic nanoplates, Composite Structures, Vol. 140, pp. 323-336, 2016.
[24]        A. Farajpour, M. Yazdi, A. Rastgoo, M. Mohammadi, A higher-order nonlocal strain gradient plate model for buckling of orthotropic nanoplates in thermal environment, Acta Mechanica, Vol. 227, No. 7, pp. 1849-1867, 2016.
[25]        M. Mohammadi, A. Farajpour, M. Goodarzi, F. Dinari, Thermo-mechanical vibration analysis of annular and circular graphene sheet embedded in an elastic medium, Latin American Journal of Solids and Structures, Vol. 11, pp. 659-682, 2014.
[26]        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.
[27]        M. Mohammadi, A. Farajpour, M. Goodarzi, Numerical study of the effect of shear in-plane load on the vibration analysis of graphene sheet embedded in an elastic medium, Computational Materials Science, Vol. 82, pp. 510-520, 2014.
[28]        M. Mohammadi, A. Farajpour, A. Moradi, M. Ghayour, Shear buckling of orthotropic rectangular graphene sheet embedded in an elastic medium in thermal environment, Composites Part B: Engineering, Vol. 56, pp. 629-637, 2014.
[29]        M. Mohammadi, M. Safarabadi, A. Rastgoo, A. Farajpour, Hygro-mechanical vibration analysis of a rotating viscoelastic nanobeam embedded in a visco-Pasternak elastic medium and in a nonlinear thermal environment, Acta Mechanica, Vol. 227, No. 8, pp. 2207-2232, 2016.
[30]        H. Asemi, S. Asemi, A. Farajpour, M. J. P. E. L.-d. S. Mohammadi, Nanostructures, Nanoscale mass detection based on vibrating piezoelectric ultrathin films under thermo-electro-mechanical loads, Vol. 68, pp. 112-122, 2015.
[31]        S. Asemi, A. Farajpour, H. Asemi, M. J. P. E. L.-d. S. Mohammadi, Nanostructures, Influence of initial stress on the vibration of double-piezoelectric-nanoplate systems with various boundary conditions using DQM, Vol. 63, pp. 169-179, 2014.
[32]        M. Baghani, M. Mohammadi, A. J. I. J. o. A. M. Farajpour, Dynamic and stability analysis of the rotating nanobeam in a nonuniform magnetic field considering the surface energy, Vol. 8, No. 04, pp. 1650048, 2016.
[33]        A. Farajpour, M. Danesh, M. J. P. E. L.-d. S. Mohammadi, Nanostructures, Buckling analysis of variable thickness nanoplates using nonlocal continuum mechanics, Vol. 44, No. 3, pp. 719-727, 2011.
[34]        A. Farajpour, M. H. Yazdi, A. Rastgoo, M. Loghmani, M. J. C. S. Mohammadi, Nonlocal nonlinear plate model for large amplitude vibration of magneto-electro-elastic nanoplates, Vol. 140, pp. 323-336, 2016.
[35]        A. Farajpour, M. Mohammadi, A. Shahidi, M. J. P. E. L.-d. S. Mahzoon, Nanostructures, Axisymmetric buckling of the circular graphene sheets with the nonlocal continuum plate model, Vol. 43, No. 10, pp. 1820-1825, 2011.
[36]        A. Farajpour, A. Rastgoo, M. J. M. R. C. Mohammadi, Surface effects on the mechanical characteristics of microtubule networks in living cells, Vol. 57, pp. 18-26, 2014.
[37]        A. Farajpour, A. Rastgoo, M. J. P. B. C. M. Mohammadi, Vibration, buckling and smart control of microtubules using piezoelectric nanoshells under electric voltage in thermal environment, Vol. 509, pp. 100-114, 2017.
[38]        A. Farajpour, A. Shahidi, M. Mohammadi, M. J. C. S. Mahzoon, Buckling of orthotropic micro/nanoscale plates under linearly varying in-plane load via nonlocal continuum mechanics, Vol. 94, No. 5, pp. 1605-1615, 2012.
[39]        M. R. Farajpour, A. Rastgoo, A. Farajpour, M. J. M. Mohammadi, N. Letters, Vibration of piezoelectric nanofilm-based electromechanical sensors via higher-order non-local strain gradient theory, Vol. 11, No. 6, pp. 302-307, 2016.
[40]        N. Ghayour, A. Sedaghat, M. Mohammadi, Wave propagation approach to fluid filled submerged visco-elastic finite cylindrical shells, 2011.
[41]        M. Goodarzi, M. Mohammadi, A. Farajpour, M. Khooran, Investigation of the effect of pre-stressed on vibration frequency of rectangular nanoplate based on a visco-Pasternak foundation, 2014.
[42]        M. Mohammadi, A. Farajpour, M. Goodarzi, H. J. J. o. S. M. Mohammadi, Temperature Effect on Vibration Analysis of Annular Graphene Sheet Embedded on Visco-Pasternak Foundati, Vol. 5, No. 3, pp. 305-323, 2013.
[43]        M. Mohammadi, M. Ghayour, A. J. C. P. B. E. Farajpour, Free transverse vibration analysis of circular and annular graphene sheets with various boundary conditions using the nonlocal continuum plate model, Vol. 45, No. 1, pp. 32-42, 2013.
[44]        M. Mohammadi, M. Ghayour, A. J. J. o. S. Farajpour, A. o. N. T. i. M. Engineering, Analysis of free vibration sector plate based on elastic medium by using new version of differential quadrature method, Vol. 3, No. 2, pp. 47-56, 2010.
[45]        M. Mohammadi, M. Goodarzi, M. Ghayour, S. Alivand, Small scale effect on the vibration of orthotropic plates embedded in an elastic medium and under biaxial in-plane pre-load via nonlocal elasticity theory, 2012.
[46]        M. Mohammadi, M. Goodarzi, M. Ghayour, A. J. C. P. B. E. Farajpour, Influence of in-plane pre-load on the vibration frequency of circular graphene sheet via nonlocal continuum theory, Vol. 51, pp. 121-129, 2013.
[47]        M. Mohammadi, M. Hosseini, M. Shishesaz, A. Hadi, A. J. E. J. o. M.-A. S. Rastgoo, Primary and secondary resonance analysis of porous functionally graded nanobeam resting on a nonlinear foundation subjected to mechanical and electrical loads, Vol. 77, pp. 103793, 2019.
[48]        M. Mohammadi, A. Moradi, M. Ghayour, A. J. L. A. J. o. S. Farajpour, Structures, Exact solution for thermo-mechanical vibration of orthotropic mono-layer graphene sheet embedded in an elastic medium, Vol. 11, No. 3, pp. 437-458, 2014.
[49]        M. Mohammadi, A. J. S. E. Rastgoo, A. I. l. J. Mechanics, Nonlinear vibration analysis of the viscoelastic composite nanoplate with three directionally imperfect porous FG core, Vol. 69, No. 2, pp. 131-143, 2019.
[50]        M. Mohammadi, A. J. M. o. A. M. Rastgoo, Structures, Primary and secondary resonance analysis of FG/lipid nanoplate with considering porosity distribution based on a nonlinear elastic medium, Vol. 27, No. 20, pp. 1709-1730, 2020.
[51]        H. Moosavi, M. Mohammadi, A. Farajpour, S. J. P. E. L.-d. S. Shahidi, Nanostructures, Vibration analysis of nanorings using nonlocal continuum mechanics and shear deformable ring theory, Vol. 44, No. 1, pp. 135-140, 2011.
[52]        M. Safarabadi, M. Mohammadi, A. Farajpour, M. Goodarzi, Effect of surface energy on the vibration analysis of rotating nanobeam, 2015.
[53]        J.-L. Wang, H.-F. Li, Surpassing the fractional derivative: Concept of the memory-dependent derivative, Computers & Mathematics with Applications, Vol. 62, No. 3, pp. 1562-1567, 2011.
[54]        Y.-J. Yu, W. Hu, X.-G. Tian, A novel generalized thermoelasticity model based on memory-dependent derivative, International Journal of Engineering Science, Vol. 81, pp. 123-134, 2014.
[55]        S. Shaw, A note on the generalized thermoelasticity theory with memory-dependent derivatives, Journal of Heat Transfer, Vol. 139, No. 9, 2017.
[56]        M. A. Ezzat, A. S. El-Karamany, A. A. El-Bary, On dual-phase-lag thermoelasticity theory with memory-dependent derivative, Mechanics of Advanced Materials and Structures, Vol. 24, No. 11, pp. 908-916, 2017.
[57]        S. Mondal, P. Pal, M. Kanoria, Transient response in a thermoelastic half-space solid due to a laser pulse under three theories with memory-dependent derivative, Acta Mechanica, Vol. 230, No. 1, pp. 179-199, 2019.
[58]        P. Lata, S. Singh, Thermomechanical interactions in a non local thermoelastic model with two temperature and memory dependent derivatives, Coupled systems mechanics, Vol. 9, No. 5, pp. 397-410, 2020.
[59]        I. Kaur, P. Lata, K. Singh, Memory-dependent derivative approach on magneto-thermoelastic transversely isotropic medium with two temperatures, International Journal of Mechanical and Materials Engineering, Vol. 15, No. 1, pp. 1-13, 2020.
[60]        A. Soleiman, A. E. Abouelregal, H. Ahmad, P. Thounthong, Generalized thermoviscoelastic model with memory dependent derivatives and multi-phase delay for an excited spherical cavity, Physica Scripta, Vol. 95, No. 11, pp. 115708, 2020
 
[61]        I. Sarkar, B. Mukhopadhyay, On energy, uniqueness theorems and variational principle for generalized thermoelasticity with memory-dependent derivative, International Journal of Heat and Mass Transfer, Vol. 149, No. 5, pp. 119112, 2020.
[62]        E. Awwad, A. E. Abouelregal, A. Soleiman Thermoelastic Memory-dependent Responses to an Infinite Medium‎ with a Cylindrical Hole and Temperature-dependent Properties, Journal of Applied and Computational Mechanics, Vol. 7, No. 2, pp. 870-882, 2021.
[63]        T. H. He, S. H. Shi, Effect of temperature-dependent problems on thermoelastic problem with thermal relaxations, Acta Mech. Solida Sin., Vol. 27, pp. 412–419, 2014.
[64]        H. Sherief, A. M. Abd El-Latief, Effect of variable thermal conductivity on a half-space under the fractional order theory of thermoelasticity, International Journal of Mechanical Sciences, Vol. 74, pp. 185-189, 2013.
[65]        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.
[66]        R. Tiwaria, A. Singhalb, A. Kumarc, Effects of variable thermal properties on thermoelastic waves induced by sinusoidal heat source in half space medium, Materials Today: Proceedings, 2022.
Journal of Computational Applied Mechanics
Volume 53, Issue 2
June 2022
Pages 264-281
  • Receive Date: 23 March 2022
  • Revise Date: 29 May 2022
  • Accept Date: 31 May 2022