, Volume 51, Issue 6, pp 1481–1489 | Cite as

Linear free vibration in pre/post-buckled states and nonlinear dynamic stability of lipid tubules based on nonlocal beam model

  • Jun Zhong
  • Yiming Fu
  • Chang Tao


In this paper, considering the small scale effect, the linear free vibration in pre/post-buckled states and nonlinear dynamic stability of lipid tubules with in-plane movable ends are studied. The small scale effect is characterized by nonlocal elasticity theory. The vibration in pre/post-buckled regions is solved by the differential quadrature method (DQM), and the nonlinear dynamic stability is solved by incremental harmonic balance method (IHBM). In numerical results, the effects of small scale parameter, types of lipid tubule on vibration in pre/post-buckled states and nonlinear dynamic stability are discussed.


Lipid tubules Nonlocal theory Vibration in pre/post-buckled states Nonlinear dynamic stability 



This study is supported by the National Natural Science Foundation of China under Grant No. 11272117.


  1. 1.
    Fang J (2007) Ordered arrays of self-assembled lipid tubules: fabrication and Applications. J Mater Chem 17:3479–3484ADSCrossRefGoogle Scholar
  2. 2.
    Meilander NJ, Pasumarthy MK, Kowalczyk TH, Cooper MJ, Bellamkonda RV (2003) Sustained release of plasmid DNA using lipid microtubules and agarose hydrogel. J Control Release 88:321–331CrossRefGoogle Scholar
  3. 3.
    Meilander NJ, Yu X, Ziats NP, Bellamkonda RV (2001) Lipid-based microtubular drug delivery vehicles. J Control Release 71:141–152CrossRefGoogle Scholar
  4. 4.
    Kameta N, Masuda M, Minamikawa H, Goutev NV, Rim JA, Jung JH, Shimizu T (2005) Selective construction of supramolecular nanotube hosts with cationic inner surfaces. Adv Mater 17:2732–2736CrossRefGoogle Scholar
  5. 5.
    Kameta N, Masuda M, Mizuno G, Morii N, Shimizu T (2008) Supramolecular nanotube endo sensing for a guest protein. Small 4:561–565CrossRefGoogle Scholar
  6. 6.
    Yamada K, Ihara H, Ide T, Fukumoto T, Hirayama C (1984) Formation of helical super structure from single-walled bilayers by amphiphiles with aligo-l-glutamic acid-head group. Chem Lett 13:1713–1716CrossRefGoogle Scholar
  7. 7.
    Yager P, Schoen PE (1984) Formation of tubules by a polymerizable surfactant. Mol Cryst Liq Cryst 106:371–381CrossRefGoogle Scholar
  8. 8.
    Fujima T, Frusawa H, Minamikawa H, Ito K, Shimizu T (2006) Elastic precursor of the transformation from glycolipid nanotube to vesicle. J Phys: Condens Matter 18:3089ADSGoogle Scholar
  9. 9.
    Rosso R, Virga EG (1998) Exact statics and approximate dynamics of adhering lipid tubules. Contin Mech Thermodyn 10:107–119ADSMathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Rosso R, Virga EG (1998) Adhesion by curvature of lipid tubules. Contin Mech Thermodyn 10:359–367ADSMathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Stepanyants N, Jeffries GD, Orwar O, Jesorka A (2012) Radial sizing of lipid nanotubes using membrane displacement analysis. Nano Lett 12:1372–1378ADSCrossRefGoogle Scholar
  12. 12.
    Zhao Y, Mahajan N, Fang J (2006) Bending and radial deformation of lipid tubules on Self-Assembled Thiol Monolayers. J Phys Chem B 110:22060–22063CrossRefGoogle Scholar
  13. 13.
    Zhao Y, Tamhane K, Zhang X, An L, Fang J (2008) Radial elasticity of self-assembled lipid tubules. ACS Nano 2:1466–1472CrossRefGoogle Scholar
  14. 14.
    Zhao Y, An L, Fang J (2007) Buckling of lipid tubules in shrinking liquid droplets. Nano Lett 7:1360–1363ADSCrossRefGoogle Scholar
  15. 15.
    Zhao Y, An L, Fang J (2009) Buckling instability of lipid tubules with multibilayer walls under local radial indentation. Phys Rev E 80:021911ADSCrossRefGoogle Scholar
  16. 16.
    Zhao Y, Fang J (2008) Zigzag lipid tubules. J Phys Chem B 112:10964–10968CrossRefGoogle Scholar
  17. 17.
    Eringen AC (1983) On differential equations of nonlocal elasticity and solutions of screw dislocation and surface Waves. J Appl Phys 54:4703–4710ADSCrossRefGoogle Scholar
  18. 18.
    Yang F, Chong A, Lam D, Tong P (2002) Couple stress based strain gradient theory for elasticity. Int J Solids Struct 39:2731–2743CrossRefMATHGoogle Scholar
  19. 19.
    Lam D, Yang F, Chong A, Wang J, Tong P (2003) Experiments and theory in strain gradient elasticity. J Mech Phys Solids 51:1477–1508ADSCrossRefMATHGoogle Scholar
  20. 20.
    Shen H-S (2011) Nonlinear analysis of lipid tubules by nonlocal beam model. J Theor Biol 276:50–56MathSciNetCrossRefGoogle Scholar
  21. 21.
    Gao Y, Lei F-M (2009) Small scale effects on the mechanical behaviors of protein microtubules based on the nonlocal elasticity theory. Biochem Biophys Res Commun 387:467–471CrossRefGoogle Scholar
  22. 22.
    Civalek Ö, Demir Ç (2011) Bending analysis of microtubules using nonlocal euler–bernoulli beam theory. Appl Math Model 35:2053–2067MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Shen H-S (2013) A two-step perturbation method in nonlinear analysis of beams, plates and shells. Wiley, SingaporeCrossRefMATHGoogle Scholar
  24. 24.
    Fu Y, Bi R, Zhang P (2009) Nonlinear dynamic instability of double-walled carbon nanotubes under periodic excitation. Acta Mech Solida Sin 22:206–212CrossRefGoogle Scholar
  25. 25.
    Fu Y, Wang J, Mao Y (2012) Nonlinear analysis of buckling, free vibration and dynamic stability for the piezoelectric functionally graded beams in thermal environment. Appl Math Model 36:4324–4340MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Lau S, Cheung Y, Wu S (1982) A variable parameter incrementation method for dynamic instability of linear and nonlinear elastic systems. J Appl Mech 49:849–853CrossRefMATHGoogle Scholar
  27. 27.
    Bert CW, Malik M (1996) Differential quadrature method in computational mechanics: a review. Appl Mech Rev 49:1–28ADSCrossRefGoogle Scholar
  28. 28.
    Shen H-S (2011) A novel technique for nonlinear analysis of beams on two-parameter elastic foundations. Int J Struct Stab Dyn 11:999–1014MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Nayfeh AH, Emam SA (2008) Exact solution and stability of postbuckling configurations of beams. Nonlinear Dyn 54:395–408MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Girish J, Ramachandra L (2005) Thermal postbuckled vibrations of symmetrically laminated composite plates with initial geometric imperfections. J Sound Vib 282:1137–1153ADSCrossRefGoogle Scholar
  31. 31.
    Bolotin VV (1964) The dynamic stability of elastic systems. Holden-Day, San FranciscoMATHGoogle Scholar
  32. 32.
    Wang CM, Zhang YY, He XQ (2007) Vibration of nonlocal Timoshenko beams. Nanotechnology 18(10):105401ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.College of Mechanical and Vehicle EngineeringHunan UniversityChangshaChina

Personalised recommendations