Skip to main content
SpringerLink
Log in

In-plane thermal loading effects on vibrational characteristics of functionally graded nanobeams

  • Published:
Meccanica Aims and scope Submit manuscript

Abstract

In the present study, the effect of in-plane thermal loading on vibrational behaviour of functionally graded (FG) nanobeams are carried out by presenting both Navier type solution and differential transform method. Classical and first order shear deformation beam theories are adopted to count for the effect of shear deformations. Thermo-mechanical properties of FG nanobeam are supposed to vary smoothly throughout the thickness based on power-law model and material properties are assumed to be temperature-dependent. Eringen’s nonlocal elasticity theory is exploited to describe the size dependency of FG nanobeam. Using Hamilton’s principle, the nonlocal equations of motion together with corresponding boundary conditions are obtained for the free vibration analysis of FG nanobeams based on Euler–Bernoulli and Timoshenko beam theories. According to the numerical results, it is revealed that the proposed modeling and semi analytical approach can provide accurate frequency results of the FG nanobeams as compared to analytical results and also some cases in the literature. In following a parametric study is accompanied to examine the effects of the several parameters such as temperature change, gradient indexes, small scale parameter, mode number and boundary conditions on the natural frequencies of the temperature-dependent FG nanobeams in detail. It is explicitly shown that the vibration behaviour of a FG nanobeam are significantly influenced by these effects. Numerical results are presented to serve as benchmarks for future analyses of FG nanobeams.

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
Fig. 9

Similar content being viewed by others

References

  1. Ebrahimi F, Rastgoo A (2008) Free vibration analysis of smart annular FGM plates integrated with piezoelectric layers. Smart Mater Struct 17:015044. doi:10.1088/0964-1726/17/1/015044

    Article  ADS  Google Scholar 

  2. Ebrahimi F, Rastgoo A (2008) An analytical study on the free vibration of smart circular thin FGM plate based on classical plate theory. Thin-Walled Struct 46:1402–1408. doi:10.1016/j.tws.2008.03.008

    Article  Google Scholar 

  3. Ebrahimi F, Rastgoo A, Atai AA (2009) A theoretical analysis of smart moderately thick shear deformable annular functionally graded plate. Eur J Mech A Solids 28:962–973. doi:10.1016/j.euromechsol.2008.12.008

    Article  MATH  Google Scholar 

  4. Ebrahimi F, Naei MH, Rastgoo A (2009) Geometrically nonlinear vibration analysis of piezoelectrically actuated FGM plate with an initial large deformation. J Mech Sci Technol 23:2107–2124. doi:10.1007/s12206-009-0358-8

    Article  Google Scholar 

  5. Iijima S (1991) Helical microtubules of graphitic carbon. Nature 354:56–58

    Article  ADS  Google Scholar 

  6. Pradhan SC, Mandal U (2013) Finite element analysis of CNTs based on nonlocal elasticity and Timoshenko beam theory including thermal effect. Phys E 53:223–232. doi:10.1016/j.physe.2013.04.029

    Article  Google Scholar 

  7. Wang Q (2005) Wave propagation in carbon nanotubes via nonlocal continuum mechanics. J Appl Phys 98:124301. doi:10.1063/1.2141648

    Article  ADS  Google Scholar 

  8. Wang Q, Varadan VK (2006) Vibration of carbon nanotubes studied using nonlocal continuum mechanics. Smart Mater Struct 15:659. doi:10.1088/0964-1726/15/2/050

    Article  ADS  Google Scholar 

  9. Eringen AC (1972) Nonlocal polar elastic continua. Int J Eng Sci 10:1–16. doi:10.1016/0020-7225(72)90070-5

    Article  MathSciNet  MATH  Google Scholar 

  10. Eringen AC (1983) On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 54:4703–4710. doi:10.1063/1.332803

    Article  ADS  Google Scholar 

  11. Fu Y, Du H, Zhang S (2003) Functionally graded TiN/TiNi shape memory alloy films. Mater Lett 57:2995–2999. doi:10.1016/S0167-577X(02)01419-2

    Article  Google Scholar 

  12. Lü CF, Lim CW, Chen WQ (2009) Size-dependent elastic behavior of FGM ultra-thin films based on generalized refined theory. Int J Solids Struct 46:1176–1185. doi:10.1016/j.ijsolstr.2008.10.012

    Article  MATH  Google Scholar 

  13. Rahaeifard M, Kahrobaiyan MH, Ahmadian MT (2009) Sensitivity analysis of atomic force microscope cantilever made of functionally graded materials. Paper presented at the 3rd international conference on micro- and nanosystems

  14. Carbonari RC, Silva EC, Paulino GH (2009) Multi-actuated functionally graded piezoelectric micro-tools design: a multiphysics topology optimization approach. Int J Numer Methods Eng 77:301–336. doi:10.1002/nme.2403

    Article  MATH  Google Scholar 

  15. Lun FY, Zhang P, Gao FB, Jia HG (2006) Design and fabrication of micro optomechanical vibration sensor. Microfab Technol 120:61–64

    Google Scholar 

  16. Witvrouw A, Mehta A (2005) The use of functionally graded poly-SiGe layers for MEMS applications. Mater Sci Forum 492:255–260. doi:10.4028/www.scientific.net/MSF.492-493.255

    Article  Google Scholar 

  17. Lee Z, Ophus C, Fischer LM, Nelson-Fitzpatrick N, Westra KL, Evoy S, Radmilovic V, Dahmen U, Mitlin D (2006) Metallic NEMS components fabricated from nanocomposite Al–Mo films. Nanotechnology 17:3063. doi:10.1088/0957-4484/17/12/042

    Article  Google Scholar 

  18. Kim HS, Yang Y, Koh JT, Lee KK, Lee DJ, Lee KM, Park SW (2009) Fabrication and characterization of functionally graded nano-micro porous titanium surface by anodizing. J Biomed Mater Res B Appl Biomater 88:427–435. doi:10.1002/jbm.b.31124

    Article  Google Scholar 

  19. Kerman K, Lai BK, Ramanathan S (2012) Nanoscale compositionally graded thin-film electrolyte membranes for low-temperature solid oxide fuel cells. Adv Energy Mater 2:656–661. doi:10.1002/aenm.201100751

    Article  Google Scholar 

  20. Bafekrpour E, Simon GP, Habsuda J, Naebe M, Yang C, Fox B (2012) Fabrication and characterization of functionally graded synthetic graphite/phenolic nanocomposites. Mater Sci Eng A 545:123–131. doi:10.1016/j.msea.2012.02.097

    Article  Google Scholar 

  21. Wang Y, Ni QQ, Zhu Y, Natsuki T (2014) Fabrication of functionally graded nano-TiO2-reinforced epoxy matrix composites. Polym Compos 35:557–563. doi:10.1002/pc.22695

    Article  Google Scholar 

  22. Akgöz B, Civalek Ö (2014) Thermo-mechanical buckling behavior of functionally graded microbeams embedded in elastic medium. Int J Eng Sci 85:90–104. doi:10.1016/j.ijengsci.2014.08.011

    Article  Google Scholar 

  23. Şimşek M, Reddy JN (2013) A unified higher order beam theory for buckling of a functionally graded microbeam embedded in elastic medium using modified couple stress theory. Compos Struct 101:47–58. doi:10.1016/j.compstruct.2013.01.017

    Article  Google Scholar 

  24. Ansari R, Gholami R, Sahmani S (2011) Free vibration analysis of size-dependent functionally graded microbeams based on the strain gradient Timoshenko beam theory. Compos Struct 94:221–228. doi:10.1016/j.compstruct.2011.06.024

    Article  MATH  Google Scholar 

  25. Nateghi A, Salamat-talab M (2013) Thermal effect on size dependent behavior of functionally graded microbeams based on modified couple stress theory. Compos Struct 96:97–110. doi:10.1016/j.compstruct.2012.08.048

    Article  Google Scholar 

  26. Ke LL, Wang YS, Yang J, Kitipornchai S (2012) Nonlinear free vibration of size-dependent functionally graded microbeams. Int J Eng Sci 50:256–267. doi:10.1016/j.ijengsci.2010.12.008

    Article  MathSciNet  Google Scholar 

  27. Eltaher MA, Emam SA, Mahmoud FF (2012) Free vibration analysis of functionally graded size-dependent nanobeams. Appl Math Comput 218:7406–7420. doi:10.1016/j.amc.2011.12.090

    MathSciNet  MATH  Google Scholar 

  28. Eltaher MA, Alshorbagy AE, Mahmoud FF (2013) Determination of neutral axis position and its effect on natural frequencies of functionally graded macro/nanobeams. Compos Struct 99:193–201. doi:10.1016/j.compstruct.2012.11.039

    Article  Google Scholar 

  29. Eltaher MA, Emam SA, Mahmoud FF (2013) Static and stability analysis of nonlocal functionally graded nanobeams. Compos Struct 96:82–88. doi:10.1016/j.compstruct.2012.09.030

    Article  Google Scholar 

  30. Hosseini-Hashemi S, Nazemnezhad R (2013) An analytical study on the nonlinear free vibration of functionally graded nanobeams incorporating surface effects. Compos B Eng 52:199–206. doi:10.1016/j.compositesb.2013.04.023

    Article  Google Scholar 

  31. Asgharifard Sharabiani P, Yazdi MRH (2013) Nonlinear free vibrations of functionally graded nanobeams with surface effects. Compos B Eng 45:581–586. doi:10.1016/j.compositesb.2012.04.064

    Article  Google Scholar 

  32. Şimşek M, Yurtcu HH (2013) Analytical solutions for bending and buckling of functionally graded nanobeams based on the nonlocal Timoshenko beam theory. Compos Struct 97:378–386. doi:10.1016/j.compstruct.2012.10.038

    Article  Google Scholar 

  33. Kiani K (2014) Longitudinal and transverse instabilities of moving nanoscale beam-like structures made of functionally graded materials. Compos Struct 107:610–619. doi:10.1016/j.compstruct.2013.07.035

    Article  Google Scholar 

  34. Niknam H, Aghdam MM (2015) A semi analytical approach for large amplitude free vibration and buckling of nonlocal FG beams resting on elastic foundation. Compos Struct 119:452–462. doi:10.1016/j.compstruct.2014.09.023

    Article  Google Scholar 

  35. Thermophysical Properties Research Center (1967) Thermophysical properties of high temperature solid materials. In: Touloukian YS (ed) Macmillan

  36. Eringen AC, Edelen DGB (1972) On nonlocal elasticity. Int J Eng Sci 10:233–248. doi:10.1063/1.332803

    Article  MathSciNet  MATH  Google Scholar 

  37. Tauchert TR (1974) Energy principles in structural mechanics. McGraw-Hill, London

    Google Scholar 

  38. Hassan IAH (2002) On solving some eigenvalue problems by using a differential transformation. Appl Math Comput 127:1–22. doi:10.1016/S0096-3003(00)00123-5

    MathSciNet  MATH  Google Scholar 

  39. Rahmani O, Pedram O (2014) Analysis and modeling the size effect on vibration of functionally graded nanobeams based on nonlocal Timoshenko beam theory. Int J Eng Sci 77:55–70. doi:10.1016/j.ijengsci.2013.12.003

    Article  MathSciNet  Google Scholar 

  40. Ju SP (2004) Application of differential transformation to transient advective–dispersive transport equation. Appl Math Comput 155:25–38. doi:10.1016/S0096-3003(03)00755-0

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Farzad Ebrahimi & Erfan Salari

  2. Department of Mechanical Engineering, University of Zanjan, Zanjan, Iran

    Seyed Amir Hosein Hosseini

Authors
  1. Farzad Ebrahimi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Erfan Salari
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Seyed Amir Hosein Hosseini
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Farzad Ebrahimi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ebrahimi, F., Salari, E. & Hosseini, S.A.H. In-plane thermal loading effects on vibrational characteristics of functionally graded nanobeams. Meccanica 51, 951–977 (2016). https://doi.org/10.1007/s11012-015-0248-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11012-015-0248-3

Keywords

Search

Navigation