Skip to main content
SpringerLink
Log in

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

  • Published:
Strength of Materials Aims and scope

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.
Fig. 6.
Fig. 7.
Fig. 8.

Similar content being viewed by others

References

  1. M. A. Biot, “Thermoelasticity and irreversible thermodynamics,” J. Appl. Phys., 27, No. 3, 240–253 (1956).

    Article  Google Scholar 

  2. H. W. Lord and Y. Shulman, “A generalized dynamical theory of thermoelasticity,” J. Mech. Phys. Solids, 15, No. 5, 299–309 (1967).

    Article  Google Scholar 

  3. A. E. Green and K. A. Lindsay, “Thermoelasticity,” J. Elasticity, 2, No. 1, 1–7 (1972).

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  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. 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).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  13. I. A. Abbas, “A problem on functional graded material under fractional order theory of thermoelasticity,” Theor. Appl. Fract. Mech., 74, 18–22 (2014).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  19. M. A. Ezzat, “Theory of fractional order in generalized thermoelectric MHD,” Appl. Math. Model., 35, No. 10, 4965–4978 (2011).

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  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. 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).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  31. J. N. Sharma, “Thermoelastic damping and frequency shift in micro/nanoscale anisotropic beams,” J. Therm. Stresses, 34, No. 7, 650–666 (2011).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  33. A. S. El-Karamany and M. A. Ezzat, “On fractional thermoelasticity,” Math. Mech. Solids, 16, No. 3, 334–346 (2011).

    Article  Google Scholar 

  34. M. A. Ezzat, “Magneto-thermoelasticity with thermoelectric properties and fractional derivative heat transfer,” Physica B, 406, No. 1, 30–35 (2011).

    Article  Google Scholar 

  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. 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. D. Y. Tzou, Macro- to Micro-Scale Heat Transfer: The Lagging Behavior, CRC Press (1996).

  38. D. Grover, “Transverse vibrations in micro-scale viscothermoelastic beam resonators,” Arch. Appl. Mech., 83, No. 2, 303–314 (2013).

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Sohag University, Sohag, Egypt

    I. Abbas & A. N. Abdalla

  2. Nonlinear Analysis and Applied Mathematics Research Group (NAAM), Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia

    I. Abbas & F. Alzahrani

  3. Department of Engineering Design and Materials, Norwegian University of Science and Technology (NTNU), Trondheim, Norway

    F. Berto

Authors
  1. I. Abbas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. F. Alzahrani
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. N. Abdalla
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. F. Berto
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to F. Berto.

Additional information

Translated from Problemy Prochnosti, No. 3, pp. 126 – 139, May – June, 2019.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Abbas, I., Alzahrani, F., Abdalla, A.N. et al. Fractional Order Thermoelastic Wave Assessment in a Nanoscale Beam Using the Eigenvalue Technique. Strength Mater 51, 427–438 (2019). https://doi.org/10.1007/s11223-019-00089-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11223-019-00089-2

Keywords

Search

Navigation