Skip to main content
SpringerLink
Log in

Vibration analysis of FG nanobeams on the basis of fractional nonlocal model: a variational approach

  • Technical Paper
  • Published:
Microsystem Technologies Aims and scope Submit manuscript

Abstract

The nonlocal vibrations of Euler–Bernoulli nanobeams are studied in this paper within the framework of fractional calculus. It is assumed that the material properties are functionally graded in the thickness direction and are estimated using the power-law function. Hamilton’s principle is applied to drive the fractional equation of motion which is then solved based on a new numerical approach named as variational finite difference method (VFDM). VFDM is formulated by the finite difference method (FDM) and matrix differential/integral operators. Since the method is directly applied to the variational form of governing equation, it is advantageous over existing approaches used for the fractional nonlocal models. The effects of nonlocality, fractional parameters and gradient of material on the fundamental frequencies of nanobeams subject to fully clamped, fully simply supported and clamped-simply supported boundary conditions are analyzed through illustrative examples.

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

Similar content being viewed by others

References

  • Agrawal OP (2002) Formulation of Euler-Lagrange equations for fractional variational problems. J Math Anal Appl 272:368–379

    Article  MathSciNet  MATH  Google Scholar 

  • Agrawal OP (2006) Fractional variational calculus and the transversality conditions. J Phys A 39:10375–10384

    Article  MathSciNet  MATH  Google Scholar 

  • Ansari R, Rouhi H, Sahmani S (2014a) Free vibration analysis of single- and double-walled carbon nanotubes based on nonlocal elastic shell models. J Vib Control 20:670–678

    Article  MathSciNet  MATH  Google Scholar 

  • Ansari R, Rouhi H, Mirnezhad M (2014b) A hybrid continuum and molecular mechanics model for the axial buckling of chiral single-walled carbon nanotubes. Curr Appl Phys 14:1360–1368

    Article  Google Scholar 

  • Ansari R, Rouhi H, Nasiri Rad A (2014c) Vibrational analysis of carbon nanocones under different boundary conditions: an analytical approach. Mech Res Commun 56:130–135

    Article  Google Scholar 

  • Ansari R, Shahabodini A, Rouhi H (2015) A nonlocal plate model incorporating interatomic potentials for vibrations of graphene with arbitrary edge conditions. Curr Appl Phys 15:1062–1069

    Article  Google Scholar 

  • Atanackovic T, Pilipovic S, Stankovic B, Zorica D (2014a) Fractional calculus with applications in mechanics: vibrations and diffusion processes. ISTE-Wiley,

  • Atanackovic T, Pilipovic S, Stankovic B, Zorica D (2014b) Fractional calculus with applications in mechanics: wave propagation, impact and variational principles. ISTE-Wiley, New York

  • Carpinteri A, Cornetti P, Sapora A, Di Paola M, Zingales M (2009) Fractional calculus in solid mechanics: local versus non-local approach. Phys Scr T136:014003

  • Carpinteri A, Cornetti P, Sapora A (2011) A fractional calculus approach to nonlocal elasticity. Eur Phys J Special Topics 193:193–204

    Article  MATH  Google Scholar 

  • Carpinteri A, Cornetti P, Sapora A (2014) Nonlocal elasticity: an approach based on fractional calculus. Meccanica 49:2551–2569

    Article  MathSciNet  MATH  Google Scholar 

  • Challamel N, Zorica D, Atanackovic TM, Spasic DT (2013) On the fractional generalization of Eringen’s nonlocal elasticity for wave propagation. C R Mec 341:298–303

    Article  Google Scholar 

  • Demir Ç, Civalek Ö (2017) A new nonlocal FEM via Hermitian cubic shape functions for thermal vibration of nano beams surrounded by an elastic matrix. Compos Struct 168:872–884

    Article  Google Scholar 

  • Di Paola M, Zingales M (2008) Long-range cohesive interactions of non-local continuum faced by fractional calculus. Int J Solids Struct 45:5642–5659

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

  • Eringen AC (2002) Nonlocal continuum field theories. Springer, New York

    MATH  Google Scholar 

  • Gurtin ME, Murdoch AI (1975) A continuum theory of elastic material surface. Arch Rat Mech Anal 57:291–323

    Article  MathSciNet  MATH  Google Scholar 

  • Gurtin ME, Murdoch AI (1978) Surface stress in solids. Int J Solids Struct 14:431–440

    Article  MATH  Google Scholar 

  • He L, Li H, Li M (2016) Optomechanical measurement of photon spin angular momentum and optical torque in integrated photonic devices. Adv Sci. https://doi.org/10.1126/sciadv.1600485

    Google Scholar 

  • Hu B, Ding Y, Chen W, Kulkarni D, Shen Y, Tsukruk VV, Wang ZL (2010) External-strain induced insulating phase transition in VO2 nanobeam and its application as flexible strain sensor. Adv Mater 22:5134–5139

    Article  Google Scholar 

  • Ice GE, Budai JD, Pang JWL (2011) The race to X-ray microbeam and nanobeam science. Science 334:1234–1239

    Article  Google Scholar 

  • Kirkby KJ, Grime GW, Webb RP, Kirkby NF, Folkard M, Prise K, Vojnovic B (2007) A scanning focussed vertical ion nanobeam: a new UK facility for cell irradiation and analysis. Nucl Instrum Meth Phys Res B 260:97–100

    Article  Google Scholar 

  • Koiter WT (1964) Couple stresses in the theory of elasticity. Proc K Ned Akad Wet (B) 67:17–44

    MathSciNet  MATH  Google Scholar 

  • Lam DCC, Yang F, Chong ACM, Wang J, Tong P (2003) Experiments and theory in strain gradient elasticity. J Mech Phys Solids 51:1477–1508

    Article  MATH  Google Scholar 

  • Lazopoulos KA (2006) Non-local continuum mechanics and fractional calculus. Mech Res Commun 33:753–757

    Article  MathSciNet  MATH  Google Scholar 

  • Li R, Kardomateas GA (2007) Thermal buckling of multi-walled carbon nanotubes by nonlocal elasticity. ASME J Appl Mech 74:399–405

    Article  MATH  Google Scholar 

  • Liu F, Meerschaert MM, McGough RJ, Zhuang P, Liu Q (2013) Numerical methods for solving the multi-term time-fractional wave-diffusion equation. Fract Calc Appl Anal 16:9–25

    Article  MathSciNet  MATH  Google Scholar 

  • Miandoab EM, Yousefi-Koma A, Nejat Pishkenari H (2015) Nonlocal and strain gradient based model for electrostatically actuated silicon nano-beams. Microsyst Technol 21:457–464

    Article  MATH  Google Scholar 

  • Mindlin RD (1964) Micro-structure in linear elasticity. Arch Ration Mech Anal 6:51–78

    Article  MathSciNet  MATH  Google Scholar 

  • Mindlin RD (1965) Second gradient of strain and surface tension in linear elasticity. Int J Solids Struct 1:417–438

    Article  Google Scholar 

  • Mindlin RD, Tiersten HF (1962) Effects of couple-stresses in linear elasticity. Arch Ration Mech Anal 11:415–448

    Article  MathSciNet  MATH  Google Scholar 

  • Peddieson J, Buchanan GR, McNitt RP (2003) Application of nonlocal continuum models to nanotechnology. Int J Eng Sci 41:305–312

    Article  Google Scholar 

  • Rahmani O, Hosseini SAH, Parhizkari M (2017) Buckling of double functionally-graded nanobeam system under axial load based on nonlocal theory: an analytical approach. Microsyst Technol 23:2739–2751

    Article  Google Scholar 

  • Rouhi H, Ansari R, Darvizeh M (2016a) Size-dependent free vibration analysis of nanoshells based on the surface stress elasticity. Appl Math Model 40:3128–3140

    Article  MathSciNet  MATH  Google Scholar 

  • Rouhi H, Ansari R, Darvizeh M (2016b) Analytical treatment of the nonlinear free vibration of cylindrical nanoshells based on a first-order shear deformable continuum model including surface influences. Acta Mech 227:1767–1781

    Article  MathSciNet  MATH  Google Scholar 

  • Sapora A, Cornetti P, Carpinteri A (2013) Wave propagation in nonlocal elastic continua modelled by a fractional calculus approach. Commun Nonlinear Sci Numer Simul 18:63–74

    Article  MathSciNet  MATH  Google Scholar 

  • Shambat G, Ellis B, Petykiewicz J, Mayer MA, Sarmiento T, Harris J, Haller EE, Vuckovic J (2011) Nanobeam photonic crystal cavity light-emitting diodes. Appl Phys Lett 99:071105

    Article  Google Scholar 

  • Sudak LJ (2003) Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics. J Appl Phys 94:7281–7287

    Article  Google Scholar 

  • Sumelka W (2014) Application of fractional continuum mechanics to rate independent plasticity. Acta Mech 225:3247–3264

    Article  MathSciNet  MATH  Google Scholar 

  • Sumelka W (2015) Non-local Kirchhoff-Love plates in terms of fractional calculus. Arch Civil Mech Eng 15:231–242

    Article  Google Scholar 

  • Sumelka W, Blaszczyk T, Liebold C (2015) Fractional Euler-Bernoulli beams: theory, numerical study and experimental validation. Eur J Mech A/Solids 54:243–251

    Article  MathSciNet  Google Scholar 

  • Tarasov VE, Aifantis EC (2015) Non-standard extensions of gradient elasticity: fractional non-locality, memory and fractality. Commun Nonlinear Sci Numer Simul 22:197–227

    Article  MathSciNet  MATH  Google Scholar 

  • Yang F, Chong ACM, Lam DCC, Tong P (2002) couple stress based strain gradient theory for elasticity. Int J Solids Struct 39:2731–2743

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, University of Guilan, P.O. Box 3756, Rasht, Iran

    M. Faraji Oskouie & R. Ansari

  2. Department of Engineering Science, Faculty of Technology and Engineering, East of Guilan, University of Guilan, Rudsar-Vajargah, 44891-63157, Iran

    H. Rouhi

Authors
  1. M. Faraji Oskouie
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. Ansari
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. H. Rouhi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. Ansari.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Faraji Oskouie, M., Ansari, R. & Rouhi, H. Vibration analysis of FG nanobeams on the basis of fractional nonlocal model: a variational approach. Microsyst Technol 24, 2775–2782 (2018). https://doi.org/10.1007/s00542-018-3776-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00542-018-3776-7

Search

Navigation