Rotating magneto-thermoelastic rod with finite length due to moving heat sources via Eringen’s nonlocal model

Document Type : Research Paper


1 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

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


The article is concerned with a new nonlocal model based on Eringen’s nonlocal elasticity and generalized thermoelasticity. A study is made of the magneto-thermoelastic waves in an isotropic conducting thermoelastic finite rod subjected to moving heat sources permeated by a primary uniform magnetic field and rotating with a uniform angular velocity. The Laplace transform technique with respect to time is utilized. The inverse transforms to the physical domain are obtained in a numerical manner for the nonlocal thermal stress, temperature, and displacement distributions. Finally, some graphical presentations have been made to assess the effects of various parameters; nonlocal parameter, rotating, applied magnetic field as well as the speed of the heat source on the field variables. The results obtained in this work should be useful for researchers in nonlocal material science, low-temperature physicists, new materials designers, as well as to those who are working on the development of the theory of nonlocal thermoelasticity.


Main Subjects

[1]     Sparagen W., Claussen G.E., 1937, Temperature distribution during welding, The Welding Journal 16: 4-10.
[2]     Knopoff L., 1955, The interaction between elastic wave motion and a magnetic field in electrical conductors, Journal of Geophysical Research 60: 441-456.
[3]     Kaliski S., Petykiewicz J., 1959, Equation of motion coupled with the field of temperature in a magnetic field involving mechanical and electrical relaxation for anisotropic bodies, Proceedings of Vibration Problems 4: 1-12.
[4]     Chadwick P., 1957, Elastic wave propagation in a magnetic field, in Proceedings of the International Congress of Applied Mechanics, Brussels, Belgium 7: 143-153.
[5]     Nayfeh A.H., S. Nemat-Nasser, 1972, Electromagneto-thermoelastic plane waves in solids with thermal relaxation, Journal of Applied Mechanics, Transactions ASME 39(1): 108-113.
[6]     Allam M.N., Elsibai K.A., Abouelregal A.E., 2010, Magnetothermoelasticity for an infinite body with a spherical cavity and variable material properties without energy dissipation, International Journal of Solids and Structures 47(20); 2631-2638.
[7]     Abouelregal A.E., Abo-Dahab S.M., 2012, Dual phase lag model on magneto-thermoelasticity infinite non-homogeneous solid having a spherical cavity, Journal of Thermal Stresses 35(9): 820-841.
[8]     Abouelregal A.E., Abo-Dahab S.M., 2014, Dual-phase-lag diffusion model for Thomson’s phenomenon on electromagneto-thermoelastic an infinitely long solid cylinder, Journal of Computational and Theoretical Nanoscience 11(4) 1031-1039.
[9]       Zenkour A.M., Abouelregal A.E., 2016, Non-simple magnetothermoelastic solid cylinder with variable thermal conductivity due to harmonically varying heat, Earthquakes and Structures 10(3): 681-697.Eringen, A.C., 1972, Nonlocal polar elastic continua, International Journal of Engineering Science 10: 1-16.
[10]    Eringen A.C., Edelen, D.G.B., 1972, On nonlocal elasticity, International Journal of Engineering Science 10: 233-248.
[11]    Eringen A.C., 1983, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics 54: 4703-4710.
[12]    Inan E., Eringen A.C., 1991, Nonlocal theory of wave propagation in thermoelastic plates, International Journal of Engineering Science 29: 831-843.
[13]    Wang J., Dhaliwal, R.S., 1993, Uniqueness in generalized nonlocal thermoelasticity, Journal of Thermal Stresses 16: 71-77.
[14]    Zenkour, A.M., Abouelregal, A.E., 2014, Nonlocal thermoelastic vibrations for variable thermal conductivity nanobeams due to harmonically varying heat, Journal of Vibroengineering 16: 3665-3678.
[15]    Koutsoumaris C., Eptaimeros K.G., Tsamasphyros G.J., 2017, A different approach to Eringen.s nonlocal integral stress model with applications for beams, International Journal of Solid and Structures 112: 222-238.
[16]    Liew K.M., Zhang Y., Zhang, L.W., 2017, .Nonlocal elasticity theory for grapheme modeling and simulation : prospects and challenges, Journal of Modeling in Mechanics and Materials doi:10.1515/jmmm-2016-0159.
[17]    Rajneesh K., Aseem M. Rekha R., 2018, Transient analysis of nonolocal microstretch thermoelastic thick circular plate with phase lags, Mediterranean Journal of Modeling & Simulation 9: 025-042.
[18]    Abouelregal A.E., Mohamed B.O., 2018, Fractional order thermoelasticity for a functionally graded thermoelastic nanobeam induced by a sinusoidal pulse heating, Journal of Computational and Theoretical Nanoscience 15: 1233-1242.
[19]    Khisaeva Z., Ostoja-Starzewski M., 2006, Thermoelastic damping in nanomechanical resonators with finite wave speeds", Journal of Thermal Stresses 29(3): 201-216.
[20]    Abouelregal A.E., Zenkour A.M., 2017, Thermoelastic response of nanobeam resonators subjected to exponential decaying time varying load, Journal of Theoretical and Applied Mechanics 55(3): 937-948.
[21]    Afzali, J., Alemipour Z. and Hesam, M., 2013, High resolution image with multi-wall carbon nanotube atomic force microscopy tip, International Journal of Engineering Science 26(6): 671-676.
[22]    Abouelregal A.E., Zenkour A.M., 2018, Nonlocal thermoelastic model for temperature-dependent thermal conductivity nanobeams due to dynamic varying loads, Microsystem Technologies 24(2): 1189-1199.
[23]    Zenkour A.M., Abouelregal A.E., 2016, Nonlinear effects of thermo-sensitive nanobeams via a nonlocal thermoelasticity model with relaxation time, Microsystem Technologies 22(10): 2407-2415.
[24]    Ribeiro P., 2016, Non-local effects on the non-linear modes of vibration of carbon nanotubes under electrostatic actuation, International Journal of Non-Linear Mechanics 87: 1–20.
[25]    Zenkour A.M., Abouelregal A.E., 2015, Nonlocal thermoelastic nanobeam subjected to a sinusoidal pulse heating and temperature-dependent physical properties, Microsystem Technologies 21(8): 1767-1776.
[26]    Lord H.W., Shulman Y., 1967, A generalized dynamical theory of thermoelasticity, Journal of Mech. Phys. Solid 15: 299-309.
[27]    Mohammadi M., Ghayour M., Farajpour A., 2013, Free transverse vibration analysis of circular and annular graphene sheets with various boundary conditions using the nonlocal continuum plate model, Composites Part B: Engineering 45(1): 32-42.
[28]  Danesh M., Farajpour A., Mohammadi M., 2012, Axial vibration analysis of a tapered nanorod based on nonlocal elasticity theory and differential quadrature method, Mechanics Research Communications 39(1): 23-27.
[29]  Farajpour A., Yazdi M.R.H., Rastgoo A., Loghmani M., Mohammadi M., 2016, Nonlocal nonlinear plate model for large amplitude vibration of magneto-electro-elastic nanoplates, Composite Structures 140: 323-336.
[30]  Mohammadi M., Safarabadi M., Rastgoo A., Farajpour A., 2016, Hygro-mechanical vibration analysis of a rotating viscoelastic nanobeam embedded in a visco-Pasternak elastic medium and in a nonlinear thermal environment, Acta Mechanica, 227(8): 2207-2232.
[31]  Mohammadi M., Farajpour A., Moradi A., Ghayour M., 2014, Shear buckling of orthotropic rectangular graphene sheet embedded in an elastic medium in thermal environment, Composites Part B: Engineering 56: 629-637.
[32]  Moosavi H., Mohammadi M., Farajpour A., Shahidi S.H., 2011, Vibration analysis of nanorings using nonlocal continuum mechanics and shear deformable ring theory, Physica E: Low-dimensional Systems and Nanostructures 44(1): 135-140.
[33]  Goodarzi M., Mohammadi M., Farajpour A., Khooran M., 2014, Investigation of the effect of pre-stressed on vibration frequency of rectangular nanoplate based on a visco-Pasternak foundation, Journal of Solid Mechanics 6(1): 98-121.
[34]  Asemi S.R., Mohammadi M., Farajpour A., 2014, Study on the nonlinear stability of orthotropic single-layered graphene sheet based on nonlocal elasticity theory, Latin American Journal of Solids and Structures 11(9): 1515-1540.
[35]  Mohammadi M., Farajpour A., Goodarzi M., Mohammadi H., 2013, Temperature effect on vibration analysis of annular graphene sheet embedded on visco-Pasternak foundation, Journal of Solid Mechanics 5(3): 305-323.
[36]  Mohammadi M., Goodarzi M., Ghayour M., Alivand S., 2012, 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, Journal of Solid Mechanics 4(2): 128-143.
[37]  Mohammadi M., Farajpour A., Goodarzi M., 2014, 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 82: 510-520.
[38]    Wang H., Dong K., Men F., Yan Y.J., Wang X., 2010, Influences of longitudinal magnetic field on wave propagation in carbon nanotubes embedded in elastic matrix, Applied Mathematical Modelling 34: 878-889.
[39]  Mashat D.S., Zenkour A.M., Abouelregal A.E., 2017, Thermoelastic interactions in a rotating infinite orthotropic elastic body with a cylindrical hole and variable thermal conductivity, Archive of Mechanical Engineering 64(4): 481-498.
[40]  Schoenberg M. Censor D., 1973, Elastic waves in rotating media, Quarterly of Applied Mathematics 31: 115-125.
[41]  Abouelregal A.E., Abo-Dahab S.M., 2018, A two-dimensional problem of a mode-I crack in a rotating fibre-reinforced isotropic thermoelastic medium under dual-phase-lag model, Sådhanå 43:13,
[42]  Roychoudhuri S.K., Mukhopadhyay S., 2000, Effect of rotation and relaxation times on plane waves in generalized thermo-viscoelasticity; International Journal of Mathematics and Mathematical Sciences 23: 497-505.
[43]  He T., Cao L., 2009, A problem of generalized magnetothermoelastic thin slim strip subjected to a moving heat source, Mathematical and Computer Modelling 49(7-8), 1710-1720.
[44]  Honig G., and Hirdes U., 1984, A method for the numerical inversion of Laplace transforms, Journal of Computational and Applied Mathematics 10(1): 113-132.
[45]  Bayones F.S., Abd-Alla A.M., 2018, Eigenvalue approach to coupled thermoelasticity in a rotating isotropic medium, Results in Physics 8: 7-15.
Volume 50, Issue 1
June 2019
Pages 118-126
  • Receive Date: 12 January 2019
  • Revise Date: 25 February 2019
  • Accept Date: 25 February 2019