Advertisement

Fractional Order Thermoelastic Wave Assessment in a Nanoscale Beam Using the Eigenvalue Technique

  • I. Abbas
  • F. Alzahrani
  • A. N. Abdalla
  • F. BertoEmail author
Article
  • 3 Downloads

This paper presents an analytical approach associated with Laplace transforms and a sequential concept over time to obtain the increment of temperature in nanoscale beam with fractional order heat conduction clamped from both ends. The governing equations are written in the forms of differential equations of matrix-vector in the domain of the Laplace transforms and are then solved by the eigenvalue technique. The analytical solutions are obtained for the increment of temperature, displacement, lateral deflection, and stresses in the Laplace domain. Numerical simulations are provided for silicon-like nanoscale beam material, with graphical display of calculated results. The physical implications of distributions of physical variables considered in this article are discussed.

Keywords

Laplace transforms nanoscale beam fractional order eigenvalues approach 

References

  1. 1.
    M. A. Biot, “Thermoelasticity and irreversible thermodynamics,” J. Appl. Phys., 27, No. 3, 240–253 (1956).Google Scholar
  2. 2.
    H. W. Lord and Y. Shulman, “A generalized dynamical theory of thermoelasticity,” J. Mech. Phys. Solids, 15, No. 5, 299–309 (1967).Google Scholar
  3. 3.
    A. E. Green and K. A. Lindsay, “Thermoelasticity,” J. Elasticity, 2, No. 1, 1–7 (1972).Google Scholar
  4. 4.
    M. A. Ezzat and M. A. Fayik, “Modeling for fractional ultra-laser two-step thermoelasticity with thermal relaxation,” Arch. Appl. Mech., 83, No. 11, 1679–1679 (2013),  https://doi.org/10.1007/s00419-013-0765-2.Google Scholar
  5. 5.
    I. A. Abbas, “Three-phase lag model on thermoelastic interaction in an unbounded fiber-reinforced anisotropic medium with a cylindrical cavity,” J. Comput. Theor. Nanos., 11, No. 4, 987–992 (2014).Google Scholar
  6. 6.
    I. A. Abbas and H. M. Youssef, “A nonlinear generalized thermoelasticity model of temperature-dependent materials using finite element method,” Int. J. Thermophys., 33, No. 7, 1302–1313 (2012).Google Scholar
  7. 7.
    H. H. Sherief and F. A. Megahed, “Two-dimensional problems for thermoelasticity, with two relaxation times in spherical regions under axisymmetric distributions,” Int. J. Eng. Sci., 37, No. 3, 299–314 (1999).Google Scholar
  8. 8.
    I. A. Abbas, “A GN model based upon two-temperature generalized thermoelastic theory in an unbounded medium with a spherical cavity,” Appl. Math. Comput., 245, 108–115 (2014).Google Scholar
  9. 9.
    I. A. Abbas, “Fractional order GN model on thermoelastic interaction in an infinite fibre-reinforced anisotropic plate containing a circular hole,” J. Comput. Theor. Nanos., 11, No. 2, 380–384 (2014).Google Scholar
  10. 10.
    M. Marin and A. Ochsner, “The effect of a dipolar structure on the Hölder stability in Green–Naghdi thermoelasticity,” Continuum Mech. Thermodyn., 29, No. 6, 1365– 1374 (2017).Google Scholar
  11. 11.
    M. Hassan, M. Marin, Abdullah Alsharif, and R. Ellahi, “Convective heat transfer flow of nanofluid in a porous medium over wavy surface,” Phys. Lett. A, 382, No. 38, 2749–2753 (2018).Google Scholar
  12. 12.
    R. Kumar and I. A. Abbas, “Deformation due to thermal source in micropolar thermoelastic media with thermal and conductive temperatures,” J. Comput. Theor. Nanos., 10, No. 9, 2241–2247 (2013).Google Scholar
  13. 13.
    I. A. Abbas, “A problem on functional graded material under fractional order theory of thermoelasticity,” Theor. Appl. Fract. Mech., 74, 18–22 (2014).Google Scholar
  14. 14.
    A.-E.-N. N. Abd-Alla and I. Abbas, “A problem of generalized magnetothermoelasticity for an infinitely long, perfectly conducting cylinder,” J. Therm. Stresses, 25, No. 11, 1009–1025 (2002).Google Scholar
  15. 15.
    I. A. Abbas and H. M. Youssef, “Two-temperature generalized thermoelasticity under ramp-type heating by finite element method,” Meccanica, 48, No. 2, 331–339 (2013).Google Scholar
  16. 16.
    I. A. Abbas, “Finite element analysis of the thermoelastic interactions in an unbounded body with a cavity,” Forsch. Ingenieurwes., 71, Nos. 3–4, 215–222 (2007).Google Scholar
  17. 17.
    M. A. Ezzat and A. S. El-Karamany, “Fractional order theory of a perfect conducting thermoelastic medium,” Can. J. Phys., 89, No. 3, 311–318 (2011).Google Scholar
  18. 18.
    M. A. Ezzat and A. S. El Karamany, “Theory of fractional order in electrothermoelasticity,” Eur. J. Mech. A-Solid., 30, No. 4, 491–500 (2011).Google Scholar
  19. 19.
    M. A. Ezzat, “Theory of fractional order in generalized thermoelectric MHD,” Appl. Math. Model., 35, No. 10, 4965–4978 (2011).Google Scholar
  20. 20.
    H. M. Youssef, “Theory of fractional order generalized thermoelasticity,” J. Heat Transfer, 132, No. 6, 061301 (2010),  https://doi.org/10.1115/1.4000705.Google Scholar
  21. 21.
    H. H. Sherief, A. M. A. El-Sayed, and A. M. Abd El-Latief, “Fractional order theory of thermoelasticity,” Int. J. Solids Struct., 47, No. 2, 269–275 (2010).Google Scholar
  22. 22.
    H. Sherief and A. M. Abd El-Latief, “Effect of variable thermal conductivity on a half-space under the fractional order theory of thermoelasticity,” Int. J. Mech. Sci., 74, 185–189 (2013).Google Scholar
  23. 23.
    R. Kumar, V. Gupta, and I. A. Abbas, “Plane deformation due to thermal source in fractional order thermoelastic media,” J. Comput. Theor. Nanos., 10, No. 10, 2520– 2525 (2013).Google Scholar
  24. 24.
    K. Y. Yasumura, T. D. Stowe, T. W. Kenny, and D. Rugar, “Thermoelastic energy dissipation in silicon nitride microcantilever structures,” Bull. Am. Phys. Soc., 44, 540 (1999).Google Scholar
  25. 25.
    G. Rezazadeh, A. S. Vahdat, S. Tayefeh-rezaei, and C. Cetinkaya, “Thermoelastic damping in a micro-beam resonator using modified couple stress theory,” Acta Mech., 223, No. 6, 1137–1152 (2012).Google Scholar
  26. 26.
    Y. Sun, D. Fang, and A. K. Soh, “Thermoelastic damping in micro-beam resonators,” Int. J. Solids Struct., 43, No. 10, 3213–3229 (2006).Google Scholar
  27. 27.
    J. N. Sharma and D. Grover, “Thermoelastic vibration analysis of Mems/Nems plate resonators with voids,” Acta Mech., 223, No. 1, 167–187 (2012).Google Scholar
  28. 28.
    J. N. Sharma and D. Grover, “Thermoelastic vibrations in micro-/nano-scale beam resonators with voids,” J. Sound Vib., 330, No. 12, 2964–2977 (2011).Google Scholar
  29. 29.
    A. M. Zenkour, A. E. Abouelregal, and I. A. Abbas, “Generalized thermoelastic vibration of an axially moving clamped microbeam subjected to ramp-type thermal loading,” J. Therm. Stresses, 37, No. 11, 1302–1323 (2014).Google Scholar
  30. 30.
    I. A. Abbas, “A GN model for thermoelastic interaction in a microscale beam subjected to a moving heat source,” Acta Mech., 226, No. 8, 2527–2536 (2015).Google Scholar
  31. 31.
    J. N. Sharma, “Thermoelastic damping and frequency shift in micro/nanoscale anisotropic beams,” J. Therm. Stresses, 34, No. 7, 650–666 (2011).Google Scholar
  32. 32.
    Y. X. Sun, Y. Jiang, and J. L. Yang, “Thermoelastic damping of the axisymmetric vibration of laminated trilayered circular plate resonators,” Appl. Mech. Mater., 313–314, 600–603 (2013).Google Scholar
  33. 33.
    A. S. El-Karamany and M. A. Ezzat, “On fractional thermoelasticity,” Math. Mech. Solids, 16, No. 3, 334–346 (2011).Google Scholar
  34. 34.
    M. A. Ezzat, “Magneto-thermoelasticity with thermoelectric properties and fractional derivative heat transfer,” Physica B, 406, No. 1, 30–35 (2011).Google Scholar
  35. 35.
    N. C. Das, A. Lahiri, and R. R. Giri, “Eigenvalue approach to generalized thermoelasticity,” Indian J. Pure Appl. Math., 28, No. 12, 1573–1594 (1997).Google Scholar
  36. 36.
    H. M. Youssef and I. A. Abbas, “Fractional order generalized thermoelasticity with variable thermal conductivity,” J. Vibroeng., 16, No. 8, 4077–4087 (2014).Google Scholar
  37. 37.
    D. Y. Tzou, Macro- to Micro-Scale Heat Transfer: The Lagging Behavior, CRC Press (1996).Google Scholar
  38. 38.
    D. Grover, “Transverse vibrations in micro-scale viscothermoelastic beam resonators,” Arch. Appl. Mech., 83, No. 2, 303–314 (2013).Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • I. Abbas
    • 1
    • 2
  • F. Alzahrani
    • 2
  • A. N. Abdalla
    • 1
  • F. Berto
    • 3
    Email author
  1. 1.Department of Mathematics, Faculty of ScienceSohag UniversitySohagEgypt
  2. 2.Nonlinear Analysis and Applied Mathematics Research Group (NAAM), Department of MathematicsKing Abdulaziz UniversityJeddahSaudi Arabia
  3. 3.Department of Engineering Design and MaterialsNorwegian University of Science and Technology (NTNU)TrondheimNorway

Personalised recommendations