Nonlocal thermoelastic semi-infinite medium with variable thermal conductivity due to a laser short-pulse

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

3 Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, SAUDI ARABIA

4 Department of Mathematics, Faculty of Science, Kafrelsheikh University, Kafrelsheikh 33516, EGYPT


In this article, the thermoelastic interactions in an isotropic and homogeneous semi-infinite medium with variable thermal conductivity caused by an ultra-short pulsed laser heating based on the linear nonlocal theory of elasticity has been considered. We consider that the thermal conductivity of the material is dependent on the temperature. The surface of the surrounding plane of the medium is heated by an ultra-short pulse laser. Basic equations are solved along with the corresponding boundary conditions numerically by means of the Laplace transform technique. The influences of the rise time of the laser pulse, as well as the nonlocal parameter on thermoelastic wave propagation in the medium, have also been investigated in detail. Presented numerical results, graphs and discussions in this work lead to some important deductions. The results obtained here will 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. Wang X., Xu X., 2001, Thermoelastic wave induced by pulsed laser heating, Applied Physics A 73: 107-114.
  2. Tzou D.Y., 1996, Macro- to Microscale Heat Transfer-The Lagging Behavior, Taylor and Francis, Washington.
  3. Wang, X., Xu X., 2002, Thermoelastic wave in metal induced by ultrafast laser pulses, Journal of Thermal Stresses 25: 457-473.
  4. Scruby C.B., Drain L.E., 1990, Laser Ultrasonics Techniques and Applications, Adam Hilger, Bristol, UK.
  5. Qiu T.Q., Tien C.L., 1993, Heat transfer mechanisms during short-pulse laser heating of metals, ASME Journal of Heat Transfer 115: 835-841.
  6. Miao L., and Massoudi M., 2015, Heat transfer analysis and flow of a slag-type fluid: Effects of variable thermal conductivity and viscosity, International Journal of Nonlinear Mechanics 76: 8-19.
  7. Wang Y.Z., Liu D., Wang Q., Shu C., 2015, Thermoelastic response of thin plate with variable material properties under transient thermal shock, International Journal of Mechanical Sciences 104: 200-206.
  8. Abouelregal A.E., 2011, Fractional order generalized thermo-piezoelectric semi-infinite medium with temperature-dependent properties subjected to a ramp-type heating, Journal of Thermal Stresses 11: 1139-1155.
  9. Li C.L., Guo H.L., Tian X., Tian X.G., 2017, Transient response for a half-space with variable thermal conductivity and diffusivity under thermal and chemical shock, Journal of Thermal Stresses 40: 389–401.
  10. Zenkour A.M., Abouelregal A.E., 2015, Nonlocal thermoelastic nanobeam subjected to a sinusoidal pulse heating and temperature-dependent physical properties, Microsystem Technologies 21: 1767-1776.
  11. Zenkour A.M., Abouelregal A.E., Alnefaie K.A., Abu-Hamdeh N.H., 2017, Seebeck effect on a magneto-thermoelastic long solid cylinder with temperature-dependent thermal conductivity, European Journal of Pure and Applied Mathematics 10(4): 786-808.
  12. A. M. Zenkour, A. E. Abouelregal, 2015, Effects of phase-lags in a thermoviscoelastic orthotropic continuum with a cylindrical hole and variable thermal conductivity, Archive of Mechanics 67(6): 457-475.
  13. Dogonchi A.S., Ganji D.D., 2016, Convection-radiation heat transfer study of moving fin with temperature-dependent thermal conductivity, heat transfer and heat generation, Applied Thermal Engineering 103: 705-712.
  14. Nowacki W., 1974, Dynamical problems of thermodiffusion in elastic solids, Proc. Vib. Probl. 15: 105-128.
  15. Zenkour A.M., 2016, Effects of phase-lags and variable thermal conductivity in a thermoviscoelastic solid with a cylindrical cavity, Honam Mathematical Journal 38(3): 435-454.
  16. Zenkour A.M., 2016, Effect of a temperature-dependent thermal conductivity on a fixed unbounded solid with a cylindrical cavity, U.P.B. Sci. Bull., Series A 78(4): 231-242.
  17. Abouelregal A.E., Zenkour A.M., 2017, Thermoviscoelastic orthotropic solid cylinder with variable thermal conductivity subjected to temperature pulse heating, Earthquakes and Structures 13(2): 201-209.
  18. 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.
  19. 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.
  20. Eringen A.C., 1972, Nonlocal polar elastic continua, International Journal of Engineering Science 10: 1-16.
  21. Eringen A.C., Edelen, D.G.B., 1972, On nonlocal elasticity, International Journal of Engineering Science 10: 233-248.
  22. Eringen A.C., 1983, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, J. Appl. Phys. 54: 4703-4710.
  23. Inan E., Eringen A.C., 1991, Nonlocal theory of wave propagation in thermoelastic plates, International Journal of Engineering Science 29, 831-843.
  24. Zenkour A.M., Abouelregal A.E., 2014, Nonlocal thermoelastic vibrations for variable thermal conductivity nanobeams due to harmonically varying heat, Journal of Vibroengineering 16(8): 3665-3678.
  25. Zenkour A.M., Abouelregal A.E., 2014, Vibration of FG nanobeams induced by sinusoidal pulse-heating via a nonlocal thermoelastic model, Acta Mechanica 225(12): 3409-3421.
  26. 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.
  27. Zenkour A.M., Abouelregal A.E., Alnefaie K.A., Abu-Hamdeh N.H., Aljinaidi A.A., Aifantis E.C., 2015, State space approach for the vibration of nanobeams based on the nonlocal thermoelasticity theory without energy dissipation, Journal of Mechanical Science and Technology 29 (7): 2921-2931.
  28. 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(4): 1233-1242.
  29. Nejad M.Z., Hadi A., Rastgoo A., 2016, Buckling analysis of arbitrary two-directional functionally graded Euler-Bernoulli nano-beams based on nonlocal elasticity theory, International Journal of Engineering Science 103: 1-10.
  30. A. Daneshmehr, A. Rajabpoor, and A. Hadi, Size dependent free vibration analysis of nanoplates made of functionally graded materials based on nonlocal elasticity theory with high order theories, International Journal of Engineering Science, 95, 23-35, 2015.
  31. Zamani N.M., Hadi A., 2016, Non-local analysis of free vibration of bi-directional functionally graded Euler-Bernoulli nano-beams, International Journal of Engineering Science 105: 1-11.
  32. Nejad M.Z., Hadi A., 2016, Eringen's non-local elasticity theory for bending analysis of bi-directional functionally graded Euler-Bernoulli nano-beams, International Journal of Engineering Science 106: 1-9.
  33. Lord H.W., Shulman Y., 1967, A generalized dynamical theory of thermoelasticity, Journal of Mechanics and Physics of Solids 15: 299-309.
  34. Green A.E., Lindsay K.A., 1972, Thermoelasticity, Journal of Elasticity 2: 1-7.
  35. Nowacki W., 1975, Dynamic Problems of Thermoelasticity, Noordhoff, Leyden, The Netherlands.
  36. Noda N., Hetnarski, R.B., 1986, Thermal stresses in materials with temperature-dependent properties, thermal stresses I, North-Holland, Amsterdam.
  37. Barletta A., Pulvirenti B., 1998, Hyperbolic thermal waves in a solid cylinder with a non-stationary boundary heat flux, International Journal of Heat and Mass Transfer 41: 107-116.
  38. Honig G., Hirdes U., 1984, A method for the numerical inversion of Laplace transforms, Journal of Computational and Applied Mathematics 10(1): 113-132.
  39. Sherief H.H., Hamza F.A., 2016, Modeling of variable thermal conductivity in a generalized thermoelastic infinitely long hollow cylinder, Meccanica 51: 551-558.
  40. Wang Y., Liu D., Wang Q., Zhou J., 2015, Effect of fractional order parameter on thermoelastic behaviors of elastic medium with variable properties, Acta Mechanica Solida Sinica 28(6): 682-692.
  41. Arias I., Achenbach J.D., 2003, Thermoelastic generation of ultrasound by line-focused laser irradiation, International Journal of Solids and Structures 40: 6917-6935.
  42. McDonald, F.A., 1990, On the precursor in laser-generated ultrasound waveforms in metals, Applied Physics Letters 56(3): 230-232.
  43. Qi H-T., Xua H-Y., Guo X-W., 2013, The Cattaneo-type time fractional heat conduction equation for laser heating, Computers and Mathematics with Applications 66: 824-831.
  44. Li Y., Li L., Wei P., Wang C., 2018, Reflection and refraction of thermoelastic waves at an interface of two couple-stress solids based on Lord-Shulman thermoelastic theory, Applied Mathematical Modelling 55: 536-550.
  45. Wang B.L., Li J.E., 2013, Hyperbolic heat conduction and associated transient thermal fracture for a piezoelectric material layer, International Journal of Solids and Structures 50: 1415-1424.