Skip to main content
SpringerLink
Log in

Size-dependent nonlinear forced oscillation of self-assembled nanotubules based on the nonlocal strain gradient beam model

  • Technical Paper
  • Published:
Journal of the Brazilian Society of Mechanical Sciences and Engineering Aims and scope Submit manuscript

Abstract

Self-assembled protein micro/nanotubule has been aroused considerable interest as a self-assembled supramolecular structure with bioactive properties. The prime aim of this study is to predict size dependency in the nonlinear forced oscillation of the self-assembled nanotubules embedded in an elastic biomedium. To accomplish this end, the nonlocal strain gradient elasticity theory including both softening and stiffening features of size effect is applied to the refined hyperbolic shear deformable beam model. By using the principle of Hamilton, unconventional governing differential equations of motion have been extracted. Subsequently, generalized differential quadrature method in conjunction with the Galerkin technique is employed to solve the nonclassical problem numerically. The nonlocal strain gradient frequency response and amplitude response relevant to the primary resonance of the self-assembled nanotubules are obtained corresponding to different types of boundary conditions. It is anticipated that the nonlocal size effect causes to decrease the excitation amplitudes associated with both bifurcation points, but its effect on the first one is more considerable. However, the strain gradient size dependency has an opposite influence and leads to increase them. Furthermore, it is found that by changing the end supports from simply supported to clamped one, the influence of the nonlocality on the excitation amplitude associated with the bifurcation points increases, but the influence of the strain gradient size dependency decreases.

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.

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. Zheng J, Buxbaum RE, Heidemann SR (1993) Investigation of microtubule assembly and organization accompanying tension-induced neurite initiation. J Cell Sci 104:1239–1250

    Google Scholar 

  2. Omelchanko T, Vasiliev JM, Gelfand IM, Feder HH, Bonder EM (2002) Mechanisms of polarization of the shape of fibroblasts and epitheliocytes: separation of the roles of microtubules and Rho-dependent actin–myosin contractility. Proc Natl Acad Sci USA 99:10452–10457

    Article  Google Scholar 

  3. Gupton SL, Salmon WC, Waterman-Storer CM (2002) Converging populations of f-actin promote breakage of associated microtubules to spatially regulate microtubule turnover in migrating cells. Curr Biol 12:1891–1899

    Article  Google Scholar 

  4. Pokorny J, Hasek J, Jelinek F (2005) Electromagnetic field of microtubules: effects on transfer of mass particles and electrons. J Biol Phys 31:501–514

    Article  Google Scholar 

  5. Pokorny J, Hasek J, Vanis J, Jelinek F (2008) Biophysical aspects of cancer-electromagnetic mechanism. Indian J Exp Biol 46:310–321

    Google Scholar 

  6. Atanasov AT (2014) Calculation of vibration modes of mechanical waves on microtubules presented like strings and bars. Am J Mod Phys 3:1–11

    Article  Google Scholar 

  7. Thai H-T, Vo TP (2012) A nonlocal sinusoidal shear deformation beam theory with application to bending, buckling and vibration of nanobeams. Int J Eng Sci 54:58–66

    Article  MathSciNet  Google Scholar 

  8. Wang L, Xu YY, Ni Q (2013) Size-dependent vibration analysis of three-dimensional cylindrical microbeams based on modified couple stress theory: a unified treatment. Int J Eng Sci 68:1–10

    Article  MathSciNet  Google Scholar 

  9. Liu C, Ke LL, Wang YS, Yang J, Kitipornchai S (2014) Buckling and post-buckling of size-dependent piezoelectric Timoshenko nanobeams subject to thermo-electro-mechanical loadings. Int J Struct Stab Dyn 14:1350067

    Article  MathSciNet  Google Scholar 

  10. Shojaeian M, Tadi Beni Y (2015) Size-dependent electromechanical buckling of functionally graded electrostatic nano-bridges. Sens Actuators, A 232:49–52

    Article  Google Scholar 

  11. Sarvestani HY, Ghayoor H (2015) Free vibration analysis of curved nanotube structures. Int J Non Linear Mech 86:167–173

    Article  Google Scholar 

  12. Sahmani S, Aghdam MM, Bahrami M (2015) On the postbuckling behavior of geometrically imperfect cylindrical nanoshells subjected to radial compression including surface stress effects. Compos Struct 131:414–424

    Article  Google Scholar 

  13. Sahmani S, Aghdam MM, Bahrami M (2015) Nonlinear buckling and postbuckling behavior of cylindrical nanoshells subjected to combined axial and radial compressions incorporating surface stress effects. Compos B Eng 79:676–691

    Article  Google Scholar 

  14. Sahmani S, Bahrami M, Aghdam MM (2015) Surface stress effects on the postbuckling behavior of geometrically imperfect cylindrical nanoshells subjected to combined axial and radial compressions. Int J Mech Sci 100:1–22

    Article  Google Scholar 

  15. Nami MR, Janghorban M (2015) Free vibration analysis of rectangular nanoplates based on two-variable refined plate theory using a new strain gradient elasticity theory. J Braz Soc Mech Sci Eng 37:313–324

    Article  Google Scholar 

  16. Shaat M, Abdelkefi A (2016) Size dependent and micromechanical modeling of strain gradient-based nanoparticle composite plates with surface elasticity. Eur J Mech A Solids 58:54–68

    Article  MathSciNet  Google Scholar 

  17. Akbarzadeh Khorshidi M, Shariati M (2016) Free vibration analysis of sigmoid functionally graded nanobeams based on a modified couple stress theory with general shear deformation theory. J Braz Soc Mech Sci Eng 38:2607–2619

    Article  Google Scholar 

  18. Sahmani S, Bahrami M, Aghdam MM (2016) Surface stress effects on the nonlinear postbuckling characteristics of geometrically imperfect cylindrical nanoshells subjected to axial compression. Int J Eng Sci 99:92–106

    Article  MathSciNet  Google Scholar 

  19. Sahmani S, Aghdam MM, Bahrami M (2016) Size-dependent axial buckling and postbuckling characteristics of cylindrical nanoshells in different temperatures. Int J Mech Sci 107:170–179

    Article  Google Scholar 

  20. Mohammadimehr M, Rousta Navi B, Ghorbanpour Arani A (2016) Modified strain gradient Reddy rectangular plate model for biaxial buckling and bending analysis of double-coupled piezoelectric polymeric nanocomposite reinforced by FG-SWNT. Compos B Eng 87:132–148

    Article  Google Scholar 

  21. Zeighampour H, Shojaeian M (2017) Size-dependent vibration of sandwich cylindrical nanoshells with functionally graded material based on the couple stress theory. J Braz Soc Mech Sci Eng 39:2789–2800

    Article  Google Scholar 

  22. Tavakolian F, Farrokhabadi A, Mirzaei M (2017) Pull-in instability of double clamped microbeams under dispersion forces in the presence of thermal and residual stress effects using nonlocal elasticity theory. Microsyst Technol 23:839–848

    Article  Google Scholar 

  23. 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 

  24. Sahmani S, Aghdam MM, Bahrami M (2017) Nonlinear buckling and postbuckling behavior of cylindrical shear deformable nanoshells subjected to radial compression including surface free energy effects. Acta Mech Solida Sin 30:209–222

    Article  Google Scholar 

  25. Lotfi M, Moghimi Zand M, Hosseini II, Baghani M, Dargazany R (2017) Transient behavior and dynamic pull-in instability of electrostatically-actuated fluid-conveying microbeams. Microsyst Technol 23:6015–6023

    Article  Google Scholar 

  26. Farajpour A, Rastgoo A, Mohammadi M (2017) Vibration, buckling and smart control of microtubules using piezoelectric nanoshells under electric voltage in thermal environment. Phys B 509:100–114

    Article  Google Scholar 

  27. Wang Y-G, Song H-F, Lin W-H, Xu L (2017) Large deflection analysis of functionally graded circular microplates with modified couple stress effect. J Braz Soc Mech Sci Eng 39:981–991

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  29. 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 

  30. Ebrahimi F, Haghi P (2017) Wave propagation analysis of rotating thermoelastically-actuated nanobeams based on nonlocal strain gradient theory. Acta Mech Solida Sin 30:647–657

    Article  Google Scholar 

  31. Fathi M, Ghassemi A (2017) The effects of surface stress and nonlocal small scale on the uniaxial and biaxial buckling of the rectangular piezoelectric nanoplate based on the two variable-refined plate theory. J Braz Soc Mech Sci Eng 39:3203–3216

    Article  Google Scholar 

  32. Fattahi AM, Sahmani S (2017) Size dependency in the axial postbuckling behavior of nanopanels made of functionally graded material considering surface elasticity. Arab J Sci Eng 42:4617–4633

    Article  MathSciNet  Google Scholar 

  33. Imani Aria A, Biglari H (2018) Computational vibration and buckling analysis of microtubule bundles based on nonlocal strain gradient theory. Appl Math Comput 321:313–332

    MathSciNet  Google Scholar 

  34. Mehralian F, Tadi Beni Y (2018) Vibration analysis of size-dependent bimorph functionally graded piezoelectric cylindrical shell based on nonlocal strain gradient theory. J Braz Soc Mech Sci Eng 40(27):1–15

    Google Scholar 

  35. Jiang J, Wang L (2018) Analytical solutions for thermal vibration of nanobeams with elastic boundary conditions. Acta Mech 229:2203–2219

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  37. Sahmani S, Aghdam MM (2018) Thermo-electro-radial coupling nonlinear instability of piezoelectric shear deformable nanoshells via nonlocal elasticity theory. Microsyst Technol 24:1333–1346

    Article  Google Scholar 

  38. Sahmani S, Fattahi AM (2018) Development of efficient size-dependent plate models for axial buckling of single-layered graphene nanosheets using molecular dynamics simulation. Microsyst Technol 24:1265–1277

    Article  Google Scholar 

  39. Mohammadi M, Eghtesad M, Mohammadi H (2018) Stochastic analysis of pull-in instability of geometrically nonlinear size-dependent FGM micro beams with random material properties. Compos Struct 200:466–479

    Article  Google Scholar 

  40. Ma LH, Ke LL, Reddy JN, Yang J, Kitipornchai S, Wang YS (2018) Wave propagation characteristics in magneto-electro-elastic nanoshells using nonlocal strain gradient theory. Compos Struct 199:10–23

    Article  Google Scholar 

  41. Sahmani S, Aghdam MM (2018) Small scale effects on the large amplitude nonlinear vibrations of multilayer functionally graded composite nanobeams reinforced with graphene-nanoplatelets. Int J Nanosci Nanotechnol 14:207–227

    Google Scholar 

  42. Sahmani S, Aghdam MM (2018) Boundary layer modeling of nonlinear axial buckling behavior of functionally graded cylindrical nanoshells based on the surface elasticity theory. Iran J Sci Technol Trans Mech Eng 42:229–245

    Article  Google Scholar 

  43. Sahmani S, Fotouhi M, Aghdam MM (2018) Size-dependent nonlinear secondary resonance of micro-/nano-beams made of nano-porous biomaterials including truncated cube cells. Acta Mech. https://doi.org/10.1007/s00707-018-2334-9

    Article  Google Scholar 

  44. Wang J, Shen H, Zhang B, Liu J, Zhang Y (2018) Complex modal analysis of transverse free vibrations for axially moving nanobeams based on the nonlocal strain gradient theory. Physica E 101:85–93

    Article  Google Scholar 

  45. Sahmani S, Fattahi AM, Ahmed NA (2018) Nonlinear torsional buckling and postbuckling analysis of cylindrical silicon nanoshells incorporating surface free energy effects. Microsyst Technol. https://doi.org/10.1007/s00542-018-4246-y

    Article  Google Scholar 

  46. Sahmani S, Fattahi AM, Ahmed NA (2018) Analytical mathematical solution for vibrational response of postbuckled laminated FG-GPLRC nonlocal strain gradient micro-/nanobeams. Eng Comput. https://doi.org/10.1007/s00366-018-0657-8

    Article  Google Scholar 

  47. Sahmani S, Khandan A (2018) Size dependency in nonlinear instability of smart magneto-electro-elastic cylindrical composite nanopanels based upon nonlocal strain gradient elasticity. Microsyst Technol. https://doi.org/10.1007/s00542-018-4072-2

    Article  Google Scholar 

  48. Sahmani S, Aghdam MM (2018) Nonlocal electrothermomechanical instability of temperature-dependent FGM nanopanels with piezoelectric facesheets. Iran J Sci Technol Trans Mech Eng. https://doi.org/10.1007/s40997-018-0180-y

    Article  Google Scholar 

  49. Babu B, Patel BP (2019) Analytical solution for strain gradient elastic Kirchhoff rectangular plates under transverse static loading. Eur J Mech A Solids 73:101–111

    Article  MathSciNet  Google Scholar 

  50. 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 

  51. Gao Y, An L (2010) A nonlocal elastic anisotropic shell model for microtubule buckling behaviors in cytoplasm. Physica E 42:2406–2415

    Article  Google Scholar 

  52. Taj M, Zhang JQ (2012) Analysis of vibrational behaviors of microtubules embedded within elastic medium by Pasternak model. Biochem Biophys Res Commun 424:89–93

    Article  Google Scholar 

  53. Baninajjaryan A, Tadi Beni Y (2015) Theoretical study of the effect of shear deformable shell model, elastic foundation and size dependency on the vibration of protein microtubule. J Theor Biol 382:111–121

    Article  MathSciNet  Google Scholar 

  54. Civalek B, Demir C (2016) A simple mathematical model of microtubules surrounded by an elastic matrix by nonlocal finite element method. Appl Math Comput 289:335–352

    MathSciNet  MATH  Google Scholar 

  55. Deng J, Liu Y, Zhang Z, Liu W (2017) Size-dependent vibration and stability of multi-span viscoelastic functionally graded material nanopipes conveying fluid using a hybrid method. Compos Struct 179:590–600

    Article  Google Scholar 

  56. 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  Google Scholar 

  57. Sahmani S, Aghdam MM (2017) Nonlinear instability of hydrostatic pressurized microtubules surrounded by cytoplasm of a living cell including nonlocality and strain gradient microsize dependency. Acta Mech 229:403–420

    Article  MathSciNet  Google Scholar 

  58. 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 

  59. 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  Google Scholar 

  60. 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  Google Scholar 

  61. 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 

  62. 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 248:10–21

    Article  Google Scholar 

  63. 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  Google Scholar 

  64. 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  Google Scholar 

  65. Tang Y, Liu Y, Zhao D (2017) Wave dispersion in viscoelastic single walled carbon nanotubes based on the nonlocal strain gradient Timoshenko beam model. Physica E 87:301–307

    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  Google Scholar 

  67. Li X, Li L, Hu Y, Ding Z, Deng W (2017) Bending, buckling and vibration of axially functionally graded beams based on nonlocal strain gradient theory. Compos Struct 165:250–265

    Article  Google Scholar 

  68. Lu L, Guo X, Zhao J (2017) Size-dependent vibration analysis of nanobeams based on the nonlocal strain gradient theory. Int J Eng Sci 116:12–24

    Article  MathSciNet  Google Scholar 

  69. 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 

  70. 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 

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

    Article  Google Scholar 

  72. 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 

  73. 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 

  74. Zhen Y-X, Wen S-L, Tang Y (2019) Free vibration analysis of viscoelastic nanotubes under longitudinal magnetic field based on nonlocal strain gradient Timoshenko beam model. Physica E 105:116–124

    Article  Google Scholar 

  75. Sawant MK, Dahake AG (2014) A new hyperbolic shear deformation theory for analysis of thick beam. Int J Innov Res Sci Eng Technol 3:9636–9643

    Google Scholar 

  76. Faghih Shojaei M, Ansari R, Mohammadi V, Rouhi H (2014) Nonlinear forced vibration analysis of postbuckled beams. Arch Appl Mech 84:421–440

    Article  Google Scholar 

  77. Ansari R, Mohammadi V, Faghih Shojaei M, Gholami R, Sahmani S (2014) On the forced vibration analysis of Timoshenko nanobeams based on the surface stress elasticity theory. Compos B Eng 60:158–166

    Article  Google Scholar 

  78. Sahmani S, Bahrami M, Aghdam MM, Ansari R (2014) Surface effects on the nonlinear forced vibration response of third-order shear deformable nanobeams. Compos Struct 118:149–158

    Article  Google Scholar 

  79. Keller BH (1977) Numerical solution of bifurcation and nonlinear eigenvalue problems, applications of bifurcation theory. University of Wisconsin, Madison, New York

    MATH  Google Scholar 

  80. de Pablo PJ, Schaap IAT, Mackintosh FC, Schmidt CF (2003) Deformation and collapse of microtubules on the nanometer scale. Phys Rev Lett 91:098101

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mechanical 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.

Ethics declarations

Conflict of interest

It is confirmed that there is no conflict of interest.

Additional information

Technical Editor: Paulo de Tarso Rocha de Mendonça, Ph.D.

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. Size-dependent nonlinear forced oscillation of self-assembled nanotubules based on the nonlocal strain gradient beam model. J Braz. Soc. Mech. Sci. Eng. 41, 239 (2019). https://doi.org/10.1007/s40430-019-1732-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40430-019-1732-9

Keywords

Search

Navigation