Skip to main content
SpringerLink
Log in

Analytical mathematical solution for vibrational response of postbuckled laminated FG-GPLRC nonlocal strain gradient micro-/nanobeams

  • Original Article
  • Published:
Engineering with Computers Aims and scope Submit manuscript

Abstract

An inhomogeneous beam model incorporating size effects is formulated based on the nonlocal strain gradient theory of elasticity within the framework of a third-order shear deformation beam theory to analyze the size dependency in vibration behavior of postbuckled laminated functionally graded (FG) micro-/nanobeams made from graphene platelet-reinforced composite (GPLRC). The graphene platelets are randomly dispersed in each individual layer in such a way that the weight fraction of the nanofiller varies layerwise on the basis of different patterns of FG dispersion. Based upon the Halpin–Tsai micromechanical scheme, the effective material properties of laminated FG-GPLRC micro-/nanobeams are extracted. With the aid of the Hamilton’s principle, the nonlocal strain gradient equations of motion are constructed. After that, an improved perturbation technique is put to use to capture the small-scale effects on the fundamental frequencies of laminated FG-GPLRC nonlocal strain gradient micro-/nanobeams. The size-dependent natural frequencies of laminated third-order shear deformable FG-GPLRC micro-/nanobeams are obtained as function of applied axial compressive load within both the prebuckling and postbuckling domains. It is seen that in the prebuckling regime, the softening-stiffness effect of the nonlocality causes to reduce the natural frequencies of laminated FG-GPLRC micro-/nanobeam, but the hardening-stiffness influence of strain gradient size effect leads to increase the natural frequencies. However, by moving to the postbuckling regime, this pattern becomes vice versa, as the nonlocal size dependency leads to increase the natural frequencies of micro-/nanobeam while the strain gradient size effect causes to reduce them.

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. Koizumi M (1997) FGM activities in Japan. Compos Part B Eng 28:1–4

    Article  Google Scholar 

  2. Yasuna K, Terauchi M, Otsuki A, Ishihara KN, Shingu PH (2000) Formation of nanoscale Fe:Ag multi-layer by repeated press-rolling and its layer thickness dependence of magneto resistance. Mater Sci Eng A 85:412–417

    Article  Google Scholar 

  3. Nobuhiro TJ, Yoshihiro S, Lee SH, Yoritoshi M (2003) ARB and other new techniques to produce bulk ultrafine grained materials. Adv Eng Mater 5:338–344

    Article  Google Scholar 

  4. Luo JG, Acoff VL (2004) Using cold roll bonding and annealing to process Ti/Al multilayered composites from elemental foils. Mater Sci Eng A 379:164–172

    Article  Google Scholar 

  5. Kundu S, Ghosh M, Laik A, Bhanumurthy K, Kale GB, Chatterjee S (2005) Diffusion bonding of commercially pure titanium to 304 stainless steel using copper interlayer. Mater Sci Eng A 407:154–160

    Article  Google Scholar 

  6. Yang J, Wu H, Kitipornchai S (2017) Buckling and postbuckling of functionally graded multilayer graphene platelet-reinforced composite beams. Compos Struct 161:111–118

    Article  Google Scholar 

  7. Song M, Kitipornchai S, Yang J (2017) Free and forced vibrations of functionally graded polymer composite plates reinforced with graphene nanoplatelets. Compos Struct 159:579–588

    Article  Google Scholar 

  8. Feng C, Kitipornchai S, Yang J (2017) Nonlinear bending of polymer nanocomposite beams reinforced with non-uniformly distributed graphene platelets (GPLs). Compos Part B Eng 110:132–140

    Article  Google Scholar 

  9. Wu H, Yang J, Kitipornchai S (2017) Dynamic instability of functionally graded multilayer graphene nanocomposite beams in thermal environment. Compos Struct 162:244–254

    Article  Google Scholar 

  10. Fu Y, Du H, Zhang S (2003) Functionally graded TiN/TiNi shape memory alloy films. Mater Lett 57:2995–2999

    Article  Google Scholar 

  11. Fu Y, Du H, Huang W, Zhang S, Hu M (2004) TiNi-based thin films in MEMS applications: a review. Sens Actuators A Phys 112:395–408

    Article  Google Scholar 

  12. Kahrobaiyan MH, Asghari M, Rahaeifard M, Ahmadian MT (2010) Investigation of the size-dependent dynamic characteristics of atomic force microscope microcantilevers based on the modified couple stress theory. Int J Eng Sci 48:1985–1994

    Article  Google Scholar 

  13. 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

    Article  Google Scholar 

  14. Lei J, He Y, Zhang B, Gan Z, Zeng P (2013) Bending and vibration of functionally graded sinusoidal microbeams based on the strain gradient elasticity theory. Int J Eng Sci 72:36–52

    Article  MATH  Google Scholar 

  15. Liu D, He Y, Dunstan DJ, Zhang B, Gan Z, Hu P, Ding H (2013) Toward a further understanding of size effects in the torsion of thin metal wires: an experimental and theoretical assessment. Int J Plast 41:30–52

    Article  Google Scholar 

  16. Reddy JN, El-Borgi S, Romanoff J (2014) Non-linear analysis of functionally graded microbeams using Eringen׳s non-local differential model. Int J Non-Linear Mech 67:308–318

    Article  Google Scholar 

  17. Jung W-Y, Han S-C, Park W-T (2014) A modified couple stress theory for buckling analysis of S-FGM nanoplates embedded in Pasternak elastic medium. Compos Part B Eng 60:746–756

    Article  Google Scholar 

  18. Sahmani S, Bahrami M, Ansari R (2014) Nonlinear free vibration analysis of functionally graded third-order shear deformable microbeams based on the modified strain gradient elasticity theory. Compos Struct 110:219–230

    Article  Google Scholar 

  19. Shojaeian M, Tadi Beni Y (2015) Size-dependent electromechanical buckling of functionally graded electrostatic nano-bridges. Sens Actuators A Phys 232:49–62

    Article  Google Scholar 

  20. Zhang B, He Y, Liu D, Shen L, Lei J (2015) Free vibration analysis of four-unknown shear deformable functionally graded cylindrical microshells based on the strain gradient elasticity theory. Compos Struct 119:578–597

    Article  Google Scholar 

  21. Jung W-Y, Han S-C (2015) Static and eigenvalue problems of Sigmoid functionally graded materials (S-FGM) micro-scale plates using the modified couple stress theory. Appl Math Model 39:3506–3524

    Article  MathSciNet  Google Scholar 

  22. Sahmani S, Aghdam MM, Bahrami M (2015) On the free vibration characteristics of postbuckled third-order shear deformable FGM nanobeams including surface effects. Compos Struct 121:377–385

    Article  Google Scholar 

  23. Kiani K (2016) Free dynamic analysis of functionally graded tapered nanorods via a newly developed nonlocal surface energy-based integro-differential model. Compos Struct 139:151–166

    Article  Google Scholar 

  24. Akbarzadeh Khorshidi M, Shariati M, Emam SA (2016) Postbuckling of functionally graded nanobeams based on modified couple stress theory under general beam theory. Int J Mech Sci 110:160–169

    Article  Google Scholar 

  25. Lei J, He Y, Guo S, Li Z, Liu D (2016) Size-dependent vibration of nickel cantilever microbeams: experiment and gradient elasticity. AIP Adv 6:105202

    Article  Google Scholar 

  26. Sahmani S, Aghdam MM, Bahrami M (2017) An efficient size-dependent shear deformable shell model and molecular dynamics simulation for axial instability analysis of silicon nanoshells. J Mol Graph Model 77:263–279

    Article  Google Scholar 

  27. Ziaee S (2017) The steady-state response of size-dependent functionally graded nanobeams to subharmonic excitation. J Eng Math 104:19–39

    Article  MathSciNet  MATH  Google Scholar 

  28. Guo S, He Y, Lei J, Li Z, Liu D (2017) Individual strain gradient effect on torsional strength of electropolished microscale copper wires. Scripta Mater 130:124–127

    Article  Google Scholar 

  29. Sahmani S, Aghdam MM (2017) Size dependency in axial postbuckling behavior of hybrid FGM exponential shear deformable nanoshells based on the nonlocal elasticity theory. Compos Struct 166:104–113

    Article  Google Scholar 

  30. Sahmani S, Aghdam MM (2017) Nonlinear instability of hydrostatic pressurized hybrid FGM exponential shear deformable nanoshells based on nonlocal continuum elasticity. Compos Part B Eng 114:404–417

    Article  Google Scholar 

  31. Sahmani S, Aghdam MM (2017) Temperature-dependent nonlocal instability of hybrid FGM exponential shear deformable nanoshells including imperfection sensitivity. Int J Mech Sci 122:129–142

    Article  Google Scholar 

  32. Li Z, He Y, Lei J, Guo S, Liu D, Wang L (2018) A standard experimental method for determining the material length scale based on modified couple stress theory. Int J Mech Sci 141:198–205

    Article  Google Scholar 

  33. Sarafraz A, Sahmani S, Aghdam MM (2019) Nonlinear secondary resonance of nanobeams under subharmonic and superharmonic excitations including surface free energy effects. Appl Math Model 66:195–226

    Article  MathSciNet  Google Scholar 

  34. Lim CW, Zhang G, Reddy JN (2015) A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. J Mech Phys Solids 78:298–313

    Article  MathSciNet  MATH  Google Scholar 

  35. Li L, Hu Y (2015) Buckling analysis of size-dependent nonlinear beams based on a nonlocal strain gradient theory. Int J Eng Sci 97:84–94

    Article  MathSciNet  MATH  Google Scholar 

  36. Li L, Hu Y (2016) Wave propagation in fluid-conveying viscoelastic carbon nanotubes based on nonlocal strain gradient theory. Comput Mater Sci 112:282–288

    Article  Google Scholar 

  37. Yang WD, Yang FP, Wang X (2016) Coupling influences of nonlocal stress and strain gradients on dynamic pull-in of functionally graded nanotubes reinforced nano-actuator with damping effects. Sens Actuators A Phys 248:10–21

    Article  Google Scholar 

  38. Simsek M (2016) Nonlinear free vibration of a functionally graded nanobeam using nonlocal strain gradient theory and a novel Hamiltonian approach. Int J Eng Sci 105:10–21

    Article  MathSciNet  MATH  Google Scholar 

  39. Farajpour A, Haeri Yazdi MR, Rastgoo A, Mohammadi M (2016) A higher-order nonlocal strain gradient plate model for buckling of orthotropic nanoplates in thermal environment. Acta Mech 227:1849–1867

    Article  MathSciNet  MATH  Google Scholar 

  40. Li L, Li X, Hu Y (2016) Free vibration analysis of nonlocal strain gradient beams made of functionally graded material. Int J Eng Sci 102:77–92

    Article  MATH  Google Scholar 

  41. Sahmani S, Aghdam MM (2017) A nonlocal strain gradient hyperbolic shear deformable shell model for radial postbuckling analysis of functionally graded multilayer GPLRC nanoshells. Compos Struct 178:97–109

    Article  Google Scholar 

  42. Sahmani S, Aghdam MM (2017) Nonlinear instability of axially loaded functionally graded multilayer graphene platelet-reinforced nanoshells based on nonlocal strain gradient elasticity theory. Int J Mech Sci 131:95–106

    Article  Google Scholar 

  43. Sahmani S, Aghdam MM (2018) Nonlocal strain gradient shell model for axial buckling and postbuckling analysis of magneto-electro-elastic composite nanoshells. Compos Part B Eng 132:258–274

    Article  Google Scholar 

  44. Li L, Hu Y (2017) Post-buckling analysis of functionally graded nanobeams incorporating nonlocal stress and microstructure-dependent strain gradient effects. Int J Mech Sci 120:159–170

    Article  Google Scholar 

  45. Sahmani S, Aghdam MM (2017) Size-dependent axial instability of microtubules surrounded by cytoplasm of a living cell based on nonlocal strain gradient elasticity theory. J Theor Biol 422:59–71

    Article  MathSciNet  MATH  Google Scholar 

  46. Zhu X, Li L (2017) Closed form solution for a nonlocal strain gradient rod in tension. Int J Eng Sci 119:16–28

    Article  MathSciNet  MATH  Google Scholar 

  47. Zhu X, Li L (2017) On longitudinal dynamics of nanorods. Int J Eng Sci 120:129–145

    Article  MATH  Google Scholar 

  48. Zeighampour H, Tadi Beni Y, Karimipour I (2017) Wave propagation in double-walled carbon nanotube conveying fluid considering slip boundary condition and shell model based on nonlocal strain gradient theory. Microfluid Nanofluid 21:85

    Article  Google Scholar 

  49. Sahmani S, Fattahi AM (2018) Small scale effects on buckling and postbuckling behaviors of axially loaded FGM nanoshells based on nonlocal strain gradient elasticity theory. Appl Math Mech 39:561–580

    Article  MathSciNet  MATH  Google Scholar 

  50. Li L, Tang H, Hu Y (2018) Size-dependent nonlinear vibration of beam-type porous materials with an initial geometrical curvature. Compos Struct 184:1177–1188

    Article  Google Scholar 

  51. Sahmani S, Aghdam MM (2017) Size-dependent nonlinear bending of micro/nanobeams made of nanoporous biomaterials including a refined truncated cube cell. Phys Lett A 381:3818–3830

    Article  MathSciNet  Google Scholar 

  52. Sahmani S, Aghdam MM (2018) Nonlinear primary resonance of micro/nanobeams made of nanoporous biomaterials incorporating nonlocality and strain gradient size dependency. Results Phys 8:879–892

    Article  Google Scholar 

  53. Li L, Tang H, Hu Y (2018) The effect of thickness on the mechanics of nanobeams. Int J Eng Sci 123:81–91

    Article  MathSciNet  MATH  Google Scholar 

  54. Sahmani S, Aghdam MM, Rabczuk T (2018) Nonlinear bending of functionally graded porous micro/nanobeams reinforced with graphene platelets based upon nonlocal strain gradient theory. Compos Struct 186:68–78

    Article  Google Scholar 

  55. Sahmani S, Aghdam MM, Rabczuk T (2018) A unified nonlocal strain gradient plate model for nonlinear axial instability of functionally graded porous micro/nano-plates reinforced with graphene platelets. Mater Res Express 5:045048

    Article  Google Scholar 

  56. Sahmani S, Aghdam MM, Rabczuk T (2018) Nonlocal strain gradient plate model for nonlinear large-amplitude vibrations of functionally graded porous micro/nano-plates reinforced with GPLs. Compos Struct 198:51–62

    Article  Google Scholar 

  57. Radic N (2018) On buckling of porous double-layered FG nanoplates in the Pasternak elastic foundation based on nonlocal strain gradient elasticity. Compos Part B Eng 153:465–479

    Article  Google Scholar 

  58. Halpin JC, Kardos JL (1976) The Halpin-Tsai equations: a review. Polym Eng Sci 16:344–352

    Article  Google Scholar 

  59. Hejazi SM, Abtahi SM, Safaie F (2016) Investigation of thermal stress distribution in fiber reinforced roller compacted concrete pavements. J Ind Text 45:869–914

    Article  Google Scholar 

  60. Shen H-S, Yang D-Q (2015) Nonlinear vibration of functionally graded fiber reinforced composite laminated beams with piezoelectric fiber reinforced composite actuators in thermal environments. Eng Struct 90:183–192

    Article  Google Scholar 

  61. Shen H-S, Chen X, Huang X-L (2016) Nonlinear bending and thermal postbuckling of functionally graded fiber reinforced composite laminated beams with piezoelectric fiber reinforced composite actuators. Compos Part B Eng 90:326–335

    Article  Google Scholar 

  62. Sahmani S, Aghdam MM (2017) Nonlocal strain gradient beam model for nonlinear vibration of prebuckled and postbuckled multilayer functionally graded GPLRC nanobeams. Compos Struct 179:77–88

    Article  Google Scholar 

  63. Li C, Zhang Y, Tu W, Jun C, Liang H, Yu H (2017) Soft measurement of wood defects based on LDA feature fusion and compressed sensor images. J For Res 28:1285–1292

    Article  Google Scholar 

  64. Sahmani S, Aghdam MM (2017) Nonlinear vibrations of pre- and post-buckled lipid supramolecular micro/nano-tubules via nonlocal strain gradient elasticity theory. J Biomech 65:49–60

    Article  Google Scholar 

  65. Shen H-S, Xiang Y (2018) Postbuckling of functionally graded graphene-reinforced composite laminated cylindrical shells subjected to external pressure in thermal environments. Thin-Walled Struct 124:151–160

    Article  Google Scholar 

  66. Sahmani S, Aghdam MM (2018) Nonlocal strain gradient beam model for postbuckling and associated vibrational response of lipid supramolecular protein micro/nano-tubules. Math Biosci 295:24–35

    Article  MathSciNet  MATH  Google Scholar 

  67. Sahmani S, Khandan A, Saber-Samandari S, Aghdam MM (2018) Nonlinear bending and instability analysis of bioceramics composed with magnetite nanoparticles: fabrication, characterization, and simulation. Ceram Int 44:9540–9549

    Article  Google Scholar 

  68. Sahmani S, Shahali M, Khandan A, Saber-Samandari S, Aghdam MM (2018) Analytical and experimental analyses for mechanical and biological characteristics of novel nanoclay bio-nanocomposite scaffolds fabricated via space holder technique. Appl Clay Sci 165:112–123

    Article  Google Scholar 

  69. Shen H-S, Xiang Y (2018) Postbuckling behavior of functionally graded graphene-reinforced composite laminated cylindrical shells under axial compression in thermal environments. Comput Methods Appl Mech Eng 330:64–82

    Article  MathSciNet  Google Scholar 

  70. Sahmani S, Khandan A, Saber-Samandari S, Aghdam MM (2018) Vibrations of beam-type implants made of 3D printed bredigite-magnetite bio-nanocomposite scaffolds under axial compression: application, communication and simulation. Ceram Int 44:11282–11291

    Article  Google Scholar 

  71. Sahmani S, Saber-Samandari S, Shahali M, Yekta HJ, Aghadavoudi F, Montazeran AH, Aghdam MM, Khandan A (2018) Mechanical and biological performance of axially loaded novel bio-nanocomposite sandwich plate-type implant coated by biological polymer thin film. J Mech Behav Biomed Mater 88:238–250

    Article  Google Scholar 

  72. Wang Z-X, Shen H-S (2018) Nonlinear vibration of sandwich plates with FG-GRC face sheets in thermal environments. Compos Struct 192:642–653

    Article  Google Scholar 

  73. Sahmani S, Saber-Samandari S, Khandan A, Aghdam MM (2019) Nonlinear resonance investigation of nanoclay based bio-nanocomposite scaffolds with enhanced properties for bone substitute applications. J Alloy Compd 773:636–653

    Article  Google Scholar 

  74. Liu F, Ming P, Li J (2007) Ab initio calculation of ideal strength and phonon instability of graphene under tension. Phys Rev B 76:064120

    Article  Google Scholar 

  75. Rafiee MA, Rafiee J, Wang Z, Song H, Yu Z-Z, Koratkar N (2009) Enhanced mechanical properties of nanocomposites at low graphene content. ASC Nano 3:3884–3890

    Article  Google Scholar 

  76. Sahmani S, Ansari R (2012) Small scale effect on vibrational response of single-walled carbon nanotubes with different boundary conditions based on nonlocal beam models. Commun Nonlinear Sci Numer Simul 17:1965–1979

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Rotating Equipment Department, Niroo Research Institute (NRI), Tehran, 14665-517, Iran

    S. Sahmani

  2. Mechanical Engineering Science Department, Faculty of Engineering and Built Environment, University of Johannesburg, Johannesburg, 2006, South Africa

    A. M. Fattahi & N. A. Ahmed

Authors
  1. S. Sahmani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. M. Fattahi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. N. A. Ahmed
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to S. Sahmani or A. M. Fattahi.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sahmani, S., Fattahi, A.M. & Ahmed, N.A. Analytical mathematical solution for vibrational response of postbuckled laminated FG-GPLRC nonlocal strain gradient micro-/nanobeams. Engineering with Computers 35, 1173–1189 (2019). https://doi.org/10.1007/s00366-018-0657-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00366-018-0657-8

Keywords

Search

Navigation