Skip to main content
SpringerLink
Log in

State space approach for the vibration of nanobeams based on the nonlocal thermoelasticity theory without energy dissipation

  • Published:
Journal of Mechanical Science and Technology Aims and scope Submit manuscript

Abstract

In this article, an Euler-Bernoulli beam model based upon nonlocal thermoelasticity theory without energy dissipation is used to study the vibration of a nanobeam subjected to ramp-type heating. Classical continuum theory is inherently size independent, while nonlocal elasticity exhibits size dependence. Among other things, this leads to a new expression for the effective nonlocal bending moment as contrasted to its classical counterpart. The thermal problem is addressed in the context of the Green-Naghdi (GN) theory of heat transport without energy dissipation. The governing partial differential equations are solved in the Laplace transform domain by the state space approach of modern control theory. Inverse of Laplace transforms are computed numerically using Fourier expansion techniques. The effects of nonlocality and ramping time parameters on the lateral vibration, temperature, displacement and bending moment 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

Similar content being viewed by others

References

  1. M. A. Biot, Thermoelasticity and irreversible thermodynamics, Journal of Applied Physics, 27 1996 240–253.

    Article  MathSciNet  Google Scholar 

  2. H. W. Lord and Y. Shulman, A generalized dynamical theory of thermoelasticity, Journal of the Mechanics and Physics of Solids, 15 1967 299–309.

    Article  Google Scholar 

  3. A. E. Green and K. A. Lindsay, Thermoelasticity, Journal of Elasticity, 2 1972 1–7.

    Article  Google Scholar 

  4. A. E. Green and P. M. Naghdi, Thermoelasticity without energy dissipation, Journal of Elasticity, 31 1993 189–209.

    Article  MathSciNet  Google Scholar 

  5. D. N. Fang, Y. X. Sun and A. K. Soh, Analysis of frequency spectrum of laser-induced vibration of microbeam resonators, Chinese Physics Letters, 23 2006 1554–1557.

    Article  Google Scholar 

  6. N. S. Al-Huniti, M. A. Al-Nimr and M. Naij, Dynamic response of a rod due to a moving heat source under the hyperbolic heat conduction model, Journal of Sound and Vibration, 242 2001 629–640.

    Article  Google Scholar 

  7. J. Kidawa-Kukla, Application of the Green functions to the problem of the thermally induced vibration of a beam, Journal of Sound and Vibration, 262 2003 865–876.

    Article  MATH  Google Scholar 

  8. B. A. Boley, Approximate analyses of thermally induced vibrations of beams and plates, Journal of Applied Mechanics, 39 1972 212–216.

    Article  Google Scholar 

  9. G. D. Manolis and D. E. Beskos, Thermally induced vibrations of beam structures, Computer Methods in Applied Mechanics and Engineering, 21 1980 337–355.

    Article  MathSciNet  Google Scholar 

  10. A. C. Eringen, Nonlocal continuum field theories, New York: Springer Verlag (2002).

    MATH  Google Scholar 

  11. A. C. Eringen, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics, 54 1983 4703–4710.

    Article  Google Scholar 

  12. J. Peddieson, R. Buchanan and R. P. McNitt, Application of nonlocal continuum models to nanotechnology, International Journal of Engineering Science, 41 2003 305–312.

    Article  MATH  Google Scholar 

  13. L. J. Sudak, Column buckling of multi-walled carbon nanotubes using nonlocal continuum mechanics, Journal of Applied Physics, 94 2003 7281–7287.

    Article  Google Scholar 

  14. L. Y. Bahar and R. B. Hetnarski, State space approach to thermoelasticity, Journal of Thermal Stress, 1 1978 135–145.

    Article  Google Scholar 

  15. H. H. Sherief, State space formulation for generalized thermoelasticity with one relaxation time including heat sources, Journal of Thermal Stress, 16 1993 163–180.

    Article  MathSciNet  Google Scholar 

  16. H. H. Sherief and K. Helmy, A two dimensional generalized thermoelasticity problem for a halfspace, Journal of Thermal Stress, 22 1999 897–910.

    Article  MathSciNet  Google Scholar 

  17. E. C. Aifantis, On the role of gradients in the localization of deformation and fracture, International Journal of Engineering Science, 30 1992 1279–1299.

    Article  Google Scholar 

  18. B. S. Altan and E. C. Aifantis, On some aspects in the special theory of gradient elasticity, Journal of the Mechanical Behavior of Materials, 8 1997 231–282.

    Article  Google Scholar 

  19. M. Y. Gutkin and E. C. Aifantis, Dislocations and disclinations in the gradient theory of elasticity, Physica Status Solidi, 41 1999 1980–1988.

    Google Scholar 

  20. E. C. Aifantis, Update on a class of gradient theories, Mechanics of Materials, 35 2003 259–280.

    Article  Google Scholar 

  21. H. Askes and E. C. Aifantis, Gradient elasticity theories in statics and dynamics-a unification of approaches, International Journal of Fracture, 139 2006 297–304.

    Article  Google Scholar 

  22. K. E. Aifantis and H. Askes, Gradient elasticity with interfacial effects, Journal of the Mechanical Behavior of Materials, 18 2007 283–306.

    Article  Google Scholar 

  23. S. Forest and E. C. Aifantis, Some links between recent gradient thermo-elasto-plasticity theories and the thermomechanics of generalized continua, International Journal of Solids and Structures, 47 2010 3367–3376.

    Article  MATH  Google Scholar 

  24. H. Askes and E. C. Aifantis, Gradient elasticity in statics and dynamics: an overview of formulations, length scale identification procedures, finite element implementations and new results, International Journal of Solids and Structures, 48 2011 1962–1990.

    Article  MATH  Google Scholar 

  25. E. C. Aifantis, On the gradient approach-relation to Eringen’s nonlocal theory, International Journal of Engineering Science, 49 2011 1357–1367.

    Article  MathSciNet  Google Scholar 

  26. A. C. Eringen, Nonlocal polar elastic continua, International Journal of Engineering Science, 10 1972 1–16.

    Article  MathSciNet  Google Scholar 

  27. A. C. Eringen and D. G. B. Edelen, On nonlocal elasticity, International Journal of Engineering Science, 10 1972 233–248.

    Article  MATH  MathSciNet  Google Scholar 

  28. D. Y. Tzou, Macro- to micro heat transfer: The lagging behavior, Taylor&Francis, Washington DC (1996).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

    A. M. Zenkour

  2. Department of Mathematics, Faculty of Science, Kafrelsheikh University, Kafr El-Sheikh, 33516, Egypt

    A. M. Zenkour

  3. Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, 35516, Egypt

    A. E. Abouelregal

  4. Department of Mechanical Engineering, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

    K. A. Alnefaie, N. H. Abu-Hamdeh, A. A. Aljinaidi & E. C. Aifantis

  5. Laboratory of Mechanics and Materials, Polytechnic School, Aristotle University of Thessaloniki, Thessaloniki, GR-54124, Greece

    E. C. Aifantis

  6. College of Engineering, Michigan Technological University, Houghton, MI, 49931, USA

    E. C. Aifantis

  7. International Laboratory of Modern Functional Materials, ITMO University, St. Petersburg, 197101, Russian

    E. C. Aifantis

Authors
  1. A. M. Zenkour
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. E. Abouelregal
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. K. A. Alnefaie
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. N. H. Abu-Hamdeh
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. A. A. Aljinaidi
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. E. C. Aifantis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. C. Aifantis.

Additional information

Recommended by Associate Editor Moon Ki Kim

Elias C. Aifantis is a Professor of Mechanics at Aristotle University of Thessaloniki/ GR, Emeritus Professor at Michigan Technological University/USA, Distinguished Adjunct Professor at King Abdulaziz University/SA and (formerly) Distinguished Foreign Expert at ITMO University/RU. He has promoted highly interdisciplinary work in mechanics of materials by bringing into the field of solids mechanics ideas from diffusion theory, chemical reactions, and nonlinear physics. He published about 550 articles and received > 7400 citations with 43 h-factor (ISI Web of Science). He has also included in the ISI Web of knowledge list of the world’s most highly cited authors in engineering.

A. M. Zenkour is a Professor of Applied Mathematics at King Abdulaziz University (Saudi Arabia). His research interests are in structural stability, vibration, plated structures and shells. He is the author or co-author of over 150 scientific publications, reviewer of many international journals in solid mechanics and applied mathematics, and an editorial member of some Journals. In addition, he delivered various lectures at national and international conferences. His research papers have been cited in many articles and textbooks. He has contributed to the development and application of continuum mechanics for predicting the bending, vibration and buckling behaviour of hygrothermal structures.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zenkour, A.M., Abouelregal, A.E., Alnefaie, K.A. et al. State space approach for the vibration of nanobeams based on the nonlocal thermoelasticity theory without energy dissipation. J Mech Sci Technol 29, 2921–2931 (2015). https://doi.org/10.1007/s12206-015-0623-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12206-015-0623-y

Keywords

Search

Navigation