Skip to main content
SpringerLink
Log in

Dynamic modeling of preloaded size-dependent nano-crystalline nano-structures

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

The vibration behavior of size-dependent nano-crystalline nano-beams is investigated based on nonlocal, couple stress and surface elasticity theories. A nanocrystalline nano-beam is composed of three phases which are nano-grains, nano-voids, and interface. Nano-voids or porosities inside the material have a stiffness-softening impact on the nano-beam. A Eringen’s nonlocal elasticity theory is applied in the analysis of nano-crystalline nano-beams for the first time. Residual surface stresses which are usually neglected in modeling nano-crystalline nano-beams are incorporated into nonlocal elasticity to better understand the physics of the problem. Also, a modified couple stress theory is used to capture rigid rotations of grains. Applying a differential transform method (DTM) satisfying various boundary conditions, the governing equations obtained from the Hamilton’s principle are solved. Reliability of the proposed approach is verified by comparing the obtained results with those provided in the literature. The effects of the nonlocal parameter, surface effect, couple stress, grain size, porosities, and interface thickness on the vibration characteristics of nano-crystalline nano-beams are explored.

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. Shaat, M. and Abdelkefi, A. Modeling of mechanical resonators used for nanocrystalline materials characterization and disease diagnosis of HIVs. Microsystem Technologies, 22, 305–318 (2016)

    Article  Google Scholar 

  2. Shaat, M. Effects of grain size and microstructure rigid rotations on the bending behavior of nanocrystalline material beams. International Journal of Mechanical Sciences, 94, 27–35 (2015)

    Article  Google Scholar 

  3. Wang, G. F., Feng, X. Q., Yu, S. W., and Nan, C. W. Interface effects on effective elastic moduli of nanocrystalline materials. Materials Science and Engineering A, 363, 1–8 (2003)

    Article  Google Scholar 

  4. Gleiter, H. Nanostructured materials: basic concepts and microstructure. Acta Materialia, 48, 1–29 (2000)

    Article  Google Scholar 

  5. Kim, H. S. and Bush, M. B. The effects of grain size and porosity on the elastic modulus of nanocrystalline materials. Nanostructured Materials, 11, 361–367 (1999)

    Article  Google Scholar 

  6. Yang, F., Chong, A. C. M., Lam, D. C. C., and Tong, P. Couple stress based strain gradient theory for elasticity. International Journal of Solids and Structures, 39, 2731–2743 (2002)

    Article  MATH  Google Scholar 

  7. Shaat, M. and Abdelkefi, A. Modeling the material structure and couple stress effects of nanocrystalline silicon beams for pull-in and bio-mass sensing applications. International Journal of Mechanical Sciences, 101, 280–291 (2015)

    Article  Google Scholar 

  8. Shaat, M., Khorshidi, M. A., Abdelkefi, A., and Shariati, M. Modeling and vibration characteristics of cracked nano-beams made of nanocrystalline materials. International Journal of Mechanical Sciences, 115, 574–585 (2016)

    Article  Google Scholar 

  9. Shaat, M. and Abdelkefi, A. Pull-in instability of multi-phase nanocrystalline silicon beams under distributed electrostatic force. International Journal of Engineering Science, 90, 58–75 (2015)

    Article  MathSciNet  Google Scholar 

  10. Gurtin, M. E. and Murdoch, A. I. A continuum theory of elastic material surfaces. Archive for Rational Mechanics and Analysis, 57, 291–323 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  11. Guo, J. G. and Zhao, Y. P. The size-dependent bending elastic properties of nanobeams with surface effects. Nanotechnology, 18, 295701 (2007)

    Article  Google Scholar 

  12. Ebrahimi, F., Shaghaghi, G. R., and Boreiry, M. An investigation into the influence of thermal loading and surface effects on mechanical characteristics of nanotubes. Structural Engineering and Mechanics, 57, 179–200 (2016)

    Article  MATH  Google Scholar 

  13. Gheshlaghi, B. and Hasheminejad, S. M. Surface effects on nonlinear free vibration of nanobeams. Composites Part B: Engineering, 42, 934–937 (2011)

    Article  Google Scholar 

  14. Ansari, R., Mohammadi, V., Shojaei, M. F., Gholami, R., and Rouhi, H. Nonlinear vibration analysis of Timoshenko nanobeams based on surface stress elasticity theory. European Journal of Mechanics-A/Solids, 45, 143–152 (2014)

    Article  MathSciNet  Google Scholar 

  15. Ansari, R., Mohammadi, V., Shojaei, M. F., Gholami, R., and Sahmani, S. Postbuckling analysis of Timoshenko nanobeams including surface stress effect. International Journal of Engineering Science, 75, 1–10 (2014)

    Article  Google Scholar 

  16. Sahmani, S., Bahrami, M., and Ansari, R. Surface energy effects on the free vibration characteristics of postbuckled third-order shear deformable nanobeams. Composite Structures, 116, 552–561 (2014)

    Article  Google Scholar 

  17. Ebrahimi, F. and Boreiry, M. Investigating various surface effects on nonlocal vibrational behavior of nanobeams. Applied Physics A, 121, 1305–1316 (2015)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  20. Ebrahimi, F. and Barati, M. R. A nonlocal higher-order shear deformation beam theory for vibration analysis of size-dependent functionally graded nanobeams. Arabian Journal for Science and Engineering, 41, 1679–1690 (2016)

    Article  MathSciNet  Google Scholar 

  21. Ebrahimi, F. and Barati, M. R. A nonlocal higher-order refined magneto-electro-viscoelastic beam model for dynamic analysis of smart nanostructures. International Journal of Engineering Science, 107, 183–196 (2016)

    Article  Google Scholar 

  22. Aydogdu, M. A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration. Physica E: Low-Dimensional Systems and Nanostructures, 41, 1651–1655 (2009)

    Article  Google Scholar 

  23. Ebrahimi, F., Barati, M. R., and Dabbagh, A. A nonlocal strain gradient theory for wave propagation analysis in temperature-dependent inhomogeneous nanoplates. International Journal of Engineering Science, 107, 169–182 (2016)

    Article  Google Scholar 

  24. Reddy, J. N. Nonlocal theories for bending, buckling and vibration of beams. International Journal of Engineering Science, 45, 288–307 (2007)

    Article  MATH  Google Scholar 

  25. Eltaher, M. A., Mahmoud, F. F., Assie, A. E., and Meletis, E. I. Coupling effects of nonlocal and surface energy on vibration analysis of nanobeams. Applied Mathematics and Computation, 224, 760–774 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. Şimşek, M. Large amplitude free vibration of nanobeams with various boundary conditions based on the nonlocal elasticity theory. Composites Part B: Engineering, 56, 621–628 (2014)

    Article  Google Scholar 

  27. Eltaher, M. A., Alshorbagy, A. E., and Mahmoud, F. F. Vibration analysis of Euler-Bernoulli nanobeams by using finite element method. Applied Mathematical Modelling, 37, 4787–4797 (2013)

    Article  MathSciNet  Google Scholar 

  28. Berrabah, H. M., Tounsi, A., Semmah, A., and Adda, B. Comparison of various refined nonlocal beam theories for bending, vibration and buckling analysis of nanobeams. Structural Engineering and Mechanics, 48, 351–365 (2013)

    Article  Google Scholar 

  29. Tounsi, A., Semmah, A., and Bousahla, A. A. Thermal buckling behavior of nanobeams using an efficient higher-order nonlocal beam theory. Journal of Nanomechanics and Micromechanics, 3, 37–42 (2013)

    Article  Google Scholar 

  30. Murmu, T. and Adhikari, S. Nonlocal elasticity based vibration of initially pre-stressed coupled nanobeam systems. European Journal of Mechanics-A/Solids, 34, 52–62 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  31. Zenkour, A. M., Abouelregal, A. E., Alnefaie, K. A., Abu-Hamdeh, N. H., Aljinaidi, A. A., and Aifantis, E. C. State space approach for the vibration of nanobeams based on the nonlocal thermoelasticity theory without energy dissipation. Journal of Mechanical Science and Technology, 29, 2921–2931 (2015)

    Article  Google Scholar 

  32. Ebrahimi, F., Ghadiri, M., Salari, E., Hoseini, S. A. H., and Shaghaghi, G. R. Application of the differential transformation method for nonlocal vibration analysis of functionally graded nanobeams. Journal of Mechanical Science and Technology, 29, 1207–1215 (2015)

    Article  Google Scholar 

  33. Ansari, R., Oskouie, M. F., and Gholami, R. Size-dependent geometrically nonlinear free vibration analysis of fractional viscoelastic nanobeams based on the nonlocal elasticity theory. Physica E: Low-Dimensional Systems and Nanostructures, 75, 266–271 (2016)

    Article  Google Scholar 

  34. Ebrahimi, F. and Barati, M. R. Dynamic modeling of a thermo-piezo-electrically actuated nanosize beam subjected to a magnetic field. Applied Physics A, 122, 1–18 (2016)

    Google Scholar 

  35. Ebrahimi, F. and Barati, M. R. Vibration analysis of smart piezoelectrically actuated nanobeams subjected to magneto-electrical field in thermal environment. Journal of Vibration and Control (2016) DOI 10.1177/1077546316646239

    Google Scholar 

  36. Ebrahimi, F. and Barati, M. R. Buckling analysis of nonlocal third-order shear deformable functionally graded piezoelectric nanobeams embedded in elastic medium. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 39, 1–16 (2016)

    Google Scholar 

  37. Ebrahimi, F. and Barati, M. R. Magnetic field effects on buckling behavior of smart size-dependent graded nanoscale beams. The European Physical Journal Plus, 131, 1–14 (2016)

    Article  Google Scholar 

  38. Ebrahimi, F. and Barati, M. R. Vibration analysis of nonlocal beams made of functionally graded material in thermal environment. The European Physical Journal Plus, 131, 279–300 (2016)

    Article  Google Scholar 

  39. Ebrahimi, F. and Barati, M. R. Small scale effects on hygro-thermo-mechanical vibration of temperature dependent nonhomogeneous nanoscale beams. Mechanics of Advanced Materials and Structures, 24, 924–936 (2017)

    Article  Google Scholar 

  40. Ebrahimi, F. and Barati, M. R. A unified formulation for dynamic analysis of nonlocal heterogeneous nanobeams in hygro-thermal environment. Applied Physics A, 122, 792–805 (2016)

    Article  Google Scholar 

  41. Attia, M. A. and Mahmoud, F. F. Modeling and analysis of nanobeams based on nonlocal-couple stress elasticity and surface energy theories. International Journal of Mechanical Sciences, 105, 126–134 (2016)

    Article  Google Scholar 

  42. Huang, Y., Hu, K. X., Wei, X., and Chandra, A. A generalized self-consistent mechanics method for composite materials with multiphase inclusions. Journal of the Mechanics and Physics of Solids, 42, 491–504 (1994)

    Article  MATH  Google Scholar 

  43. Ke, L. L., Wang, Y. S., Yang, J., and Kitipornchai, S. Nonlinear free vibration of sizedependent functionally graded microbeams. International Journal of Engineering Science, 50, 256–267 (2012)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Faculty of Engineering, Imam Khomeini International University, Qazvin, 34148-96818, Iran

    F. Ebrahimi

  2. Aerospace Engineering Department and Center of Excellence in Computational Aerospace, Amirkabir University of Technology, Tehran, 39518-79611, Iran

    M. R. Barati

Authors
  1. F. Ebrahimi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. R. Barati
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to F. Ebrahimi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ebrahimi, F., Barati, M.R. Dynamic modeling of preloaded size-dependent nano-crystalline nano-structures. Appl. Math. Mech.-Engl. Ed. 38, 1753–1772 (2017). https://doi.org/10.1007/s10483-017-2291-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-017-2291-8

Keywords

Chinese Library Classification

2010 Mathematics Subject Classification

Search

Navigation