Skip to main content
SpringerLink
Log in

Active vibration control of nanotube structures under a moving nanoparticle based on the nonlocal continuum theories

  • Published:
Meccanica Aims and scope Submit manuscript

Abstract

The active vibration control of nanotubes under action of a moving nanoscale particle is carried out using nonlocal elasticity theory of Eringen. The nanotube is simulated by an equivalent continuum structure under simply supported boundary conditions within the framework of nonlocal Rayleigh as well as Timoshenko beam theory. A Dirac-delta function is applied to model the location of the moving mass along the nanotube as well as its inertial effects. In the following a linear classical optimal control algorithm with a time varying gain matrix and displacement-velocity feedback is applied to vibration suppression in nanotube structure. The effects of the moving nanoparticle velocity, slenderness ratio of nanotube and small scale effect parameter on the dynamic deflection are studied in some detail for both the Rayleigh and Timoshenko theories. Finally the proficiency of the control algorithm in suppressing the response of the nanostructure under the effect of moving nanoparticle with different number of controlled modes and control forces is studied. The results of the present work can be used as a benchmark in future studies of nanomachines and nanosystems.

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
Fig. 10

Similar content being viewed by others

References

  1. Shirai Y, Morin JF, Sasaki T, Guerrero JM, Tour JM (2006) Recent progress on nanovehicles. Chem Soc Rev 35(11):1043–1055. doi:10.1039/b514700j

    Article  Google Scholar 

  2. Lipowsky R, Klumpp S (2005) ‘Life is motion’: multiscale motility of molecular motors. Phys A 352(1):53–112. doi:10.1016/j.physa.2004.12.034

    Article  Google Scholar 

  3. Konstas K, Langford SJ, Latter MJ (2010) Advances towards synthetic machines at the molecular and nanoscale level. Int J Mol Sci 11(6):2453–2472. doi:10.3390/ijms11062453

    Article  Google Scholar 

  4. Porto M, Urbakh M, Klafter J (2000) Atomic scale engines: cars and wheels. Phys Rev Lett 84(26):6058

    Article  ADS  Google Scholar 

  5. Regan B, Aloni S, Ritchie R, Dahmen U, Zettl A (2004) Carbon nanotubes as nanoscale mass conveyors. Nature 428(6986):924–927

    Article  ADS  Google Scholar 

  6. Parmeggiani A, Schmidt C (2004) Micromechanics of molecular motors: experiments and theory. In: Deutsch A, Howard J, Falcke M, Zimmermann W (eds) Function and regulation of cellular systems. Mathematics and biosciences in interaction. Birkhäuser Basel, pp 151–176. doi:10.1007/978-3-0348-7895-1_15

  7. Jia L, Moorjani SG, Jackson TN, Hancock WO (2004) Microscale transport and sorting by kinesin molecular motors. Biomed Microdev 6(1):67–74

    Article  Google Scholar 

  8. Shirai Y, Osgood AJ, Zhao Y, Yao Y, Saudan L, Yang H, Yu-Hung C, Alemany LB, Sasaki T, Morin J-F (2006) Surface-rolling molecules. J Am Chem Soc 128(14):4854–4864

    Article  Google Scholar 

  9. Shirai Y, Osgood AJ, Zhao Y, Kelly KF, Tour JM (2005) Directional control in thermally driven single-molecule nanocars. Nano Lett 5(11):2330–2334

    Article  ADS  Google Scholar 

  10. Hackney DD (2007) Processive motor movement. Science 316(5821):58–59

    Article  Google Scholar 

  11. Marx A, Müller J, Mandelkow E-M, Hoenger A, Mandelkow E (2006) Interaction of kinesin motors, microtubules, and MAPs. J Muscle Res Cell Motil 27(2):125–137

    Article  Google Scholar 

  12. Russell JT, Wang B, Král P (2012) Nanodroplet transport on vibrated nanotubes. J Phys Chem Lett. doi:10.1021/jz201614m

    Google Scholar 

  13. Falvo M, Clary G, Taylor R, Chi V, Brooks F, Washburn S, Superfine R (1997) Bending and buckling of carbon nanotubes under large strain. Nature 389(6651):582–584

    Article  ADS  Google Scholar 

  14. Nardelli MB, Yakobson B, Bernholc J (1998) Mechanism of strain release in carbon nanotubes. Phys Rev B 57(8):R4277

    Article  ADS  Google Scholar 

  15. Sánchez-Portal D, Artacho E, Soler JM, Rubio A, Ordejón P (1999) Ab initio structural, elastic, and vibrational properties of carbon nanotubes. Phys Rev B 59(19):12678

    Article  ADS  Google Scholar 

  16. Ansari R, Mirnezhad M, Sahmani S (2013) An accurate molecular mechanics model for computation of size-dependent elastic properties of armchair and zigzag single-walled carbon nanotubes. Meccanica 48(6):1355–1367. doi:10.1007/s11012-012-9671-x

    Article  MATH  MathSciNet  Google Scholar 

  17. Krishnan A, Dujardin E, Ebbesen T, Yianilos P, Treacy M (1998) Young’s modulus of single-walled nanotubes. Phys Rev B 58(20):14013

    Article  ADS  Google Scholar 

  18. Yakobson BI, Brabec C, Bernholc J (1996) Nanomechanics of carbon tubes: instabilities beyond linear response. Phys Rev Lett 76(14):2511–2514

    Article  ADS  Google Scholar 

  19. Arash B, Wang Q (2012) A review on the application of nonlocal elastic models in modeling of carbon nanotubes and graphenes. Comput Mater Sci 51(1):303–313. doi:10.1016/j.commatsci.2011.07.040

    Article  Google Scholar 

  20. Eringen AC (2002) Nonlocal continuum field theories. Springer, Berlin

    MATH  Google Scholar 

  21. Peddieson J, Buchanan GR, McNitt RP (2003) Application of nonlocal continuum models to nanotechnology. Int J Eng Sci 41(3):305–312

    Article  Google Scholar 

  22. Aksencer T, Aydogdu M (2011) Levy type solution method for vibration and buckling of nanoplates using nonlocal elasticity theory. Phys E 43(4):954–959. doi:10.1016/j.physe.2010.11.024

    Article  Google Scholar 

  23. Ansari R, Shahabodini A, Rouhi H (2013) Prediction of the biaxial buckling and vibration behavior of graphene via a nonlocal atomistic-based plate theory. Compos Struct 95:88–94. doi:10.1016/j.compstruct.2012.06.026

    Article  Google Scholar 

  24. Aydogdu M (2009) A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration. Phys E 41(9):1651–1655. doi:10.1016/j.physe.2009.05.014

    Article  Google Scholar 

  25. 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 

  26. Ghorbanpour Arani A, Shajari AR, Amir S, Loghman A (2012) Electro-thermo-mechanical nonlinear nonlocal vibration and instability of embedded micro-tube reinforced by BNNT, conveying fluid. Phys E 45:109–121. doi:10.1016/j.physe.2012.07.017

    Article  Google Scholar 

  27. Khodami Maraghi Z, Ghorbanpour Arani A, Kolahchi R, Amir S, Bagheri MR (2013) Nonlocal vibration and instability of embedded DWBNNT conveying viscose fluid. Compos B Eng 45(1):423–432. doi:10.1016/j.compositesb.2012.04.066

    Article  Google Scholar 

  28. Lee H-L, Chang W-J (2008) Free transverse vibration of the fluid-conveying single-walled carbon nanotube using nonlocal elastic theory. J Appl Phys 103(2):024302. doi:10.1063/1.2822099

    Article  ADS  Google Scholar 

  29. Narendar S, Roy Mahapatra D, Gopalakrishnan S (2011) Prediction of nonlocal scaling parameter for armchair and zigzag single-walled carbon nanotubes based on molecular structural mechanics, nonlocal elasticity and wave propagation. Int J Eng Sci 49(6):509–522. doi:10.1016/j.ijengsci.2011.01.002

    Article  MATH  MathSciNet  Google Scholar 

  30. Reddy JN (2007) Nonlocal theories for bending, buckling and vibration of beams. Int J Eng Sci 45(2–8):288–307. doi:10.1016/j.ijengsci.2007.04.004

    Article  MATH  Google Scholar 

  31. Roque CMC, Ferreira AJM, Reddy JN (2011) Analysis of Timoshenko nanobeams with a nonlocal formulation and meshless method. Int J Eng Sci 49(9):976–984. doi:10.1016/j.ijengsci.2011.05.010

    Article  MATH  Google Scholar 

  32. Shen HS (2011) Nonlinear analysis of lipid tubules by nonlocal beam model. J Theor Biol 276(1):50–56. doi:10.1016/j.jtbi.2011.02.001

    Article  Google Scholar 

  33. Thai H-T (2012) A nonlocal beam theory for bending, buckling, and vibration of nanobeams. Int J Eng Sci 52:56–64. doi:10.1016/j.ijengsci.2011.11.011

    Article  MathSciNet  Google Scholar 

  34. 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. doi:10.1016/j.ijengsci.2012.01.009

    Article  MathSciNet  Google Scholar 

  35. Ansari R, Ramezannezhad H (2011) Nonlocal Timoshenko beam model for the large-amplitude vibrations of embedded multiwalled carbon nanotubes including thermal effects. Phys E 43(6):1171–1178. doi:10.1016/j.physe.2011.01.024

    Article  Google Scholar 

  36. Ansari R, Rouhi H, Sahmani S (2011) Calibration of the analytical nonlocal shell model for vibrations of double-walled carbon nanotubes with arbitrary boundary conditions using molecular dynamics. Int J Mech Sci 53(9):786–792. doi:10.1016/j.ijmecsci.2011.06.010

    Article  Google Scholar 

  37. Arash B, Ansari R (2010) Evaluation of nonlocal parameter in the vibrations of single-walled carbon nanotubes with initial strain. Physica E 42(8):2058–2064. doi:10.1016/j.physe.2010.03.028

    Article  ADS  Google Scholar 

  38. Aydogdu M, Filiz S (2011) Modeling carbon nanotube-based mass sensors using axial vibration and nonlocal elasticity. Phys E 43(6):1229–1234. doi:10.1016/j.physe.2011.02.006

    Article  Google Scholar 

  39. Claeyssen JR, Tsukazan T, Coppeti RD (2013) Nonlocal effects in modal analysis of forced responses with single carbon nanotubes. Mech Syst Signal Process 38(2):299–311. doi:10.1016/j.ymssp.2013.01.014

    Article  ADS  Google Scholar 

  40. Fang B, Zhen Y-X, Zhang C-P, Tang Y (2013) Nonlinear vibration analysis of double-walled carbon nanotubes based on nonlocal elasticity theory. Appl Math Model 37(3):1096–1107. doi:10.1016/j.apm.2012.03.032

    Article  MathSciNet  Google Scholar 

  41. Fazelzadeh SA, Ghavanloo E (2012) Nonlocal anisotropic elastic shell model for vibrations of single-walled carbon nanotubes with arbitrary chirality. Compos Struct 94(3):1016–1022. doi:10.1016/j.compstruct.2011.10.014

    Article  Google Scholar 

  42. Mehdipour I, Barari A, Kimiaeifar A, Domairry G (2012) Vibrational analysis of curved single-walled carbon nanotube on a Pasternak elastic foundation. Adv Eng Softw 48:1–5

    Article  Google Scholar 

  43. Kiani K (2010) A meshless approach for free transverse vibration of embedded single-walled nanotubes with arbitrary boundary conditions accounting for nonlocal effect. Int J Mech Sci 52(10):1343–1356. doi:10.1016/j.ijmecsci.2010.06.010

    Article  Google Scholar 

  44. Kiani K (2013) Vibration behavior of simply supported inclined single-walled carbon nanotubes conveying viscous fluids flow using nonlocal Rayleigh beam model. Appl Math Model 37(4):1836–1850. doi:10.1016/j.apm.2012.04.027

    Article  MathSciNet  Google Scholar 

  45. Lee H-L, Chang W-J (2009) Vibration analysis of a viscous-fluid-conveying single-walled carbon nanotube embedded in an elastic medium. Phys E 41(4):529–532. doi:10.1016/j.physe.2008.10.002

    Article  Google Scholar 

  46. Lee H-L, Chang W-J (2010) Surface effects on frequency analysis of nanotubes using nonlocal Timoshenko beam theory. J Appl Phys 108(9):093503. doi:10.1063/1.3503853

    Article  ADS  MathSciNet  Google Scholar 

  47. Lim CW, Li C, Yu JL (2012) Free torsional vibration of nanotubes based on nonlocal stress theory. J Sound Vib 331(12):2798–2808. doi:10.1016/j.jsv.2012.01.016

    Article  ADS  Google Scholar 

  48. Lu P, Lee HP, Lu C, Zhang PQ (2007) Application of nonlocal beam models for carbon nanotubes. Int J Solids Struct 44(16):5289–5300. doi:10.1016/j.ijsolstr.2006.12.034

    Article  MATH  Google Scholar 

  49. Murmu T, Adhikari S (2011) Nonlocal vibration of carbon nanotubes with attached buckyballs at tip. Mech Res Commun 38(1):62–67. doi:10.1016/j.mechrescom.2010.11.004

    Article  MATH  Google Scholar 

  50. Narendar S, Gopalakrishnan S (2009) Nonlocal scale effects on wave propagation in multi-walled carbon nanotubes. Comput Mater Sci 47(2):526–538. doi:10.1016/j.commatsci.2009.09.021

    Article  Google Scholar 

  51. Wang B (2012) Dynamic analysis of embedded curved double-walled carbon nanotubes based on nonlocal Euler-Bernoulli Beam theory. Multidiscip Model Mater Struct 8(4):432–453. doi:10.1108/15736101211281470

    Article  ADS  Google Scholar 

  52. Wang CY, Zhang J, Fei YQ, Murmu T (2012) Circumferential nonlocal effect on vibrating nanotubules. Int J Mech Sci 58(1):86–90. doi:10.1016/j.ijmecsci.2012.03.009

    Article  Google Scholar 

  53. Yang J, Ke LL, Kitipornchai S (2010) Nonlinear free vibration of single-walled carbon nanotubes using nonlocal Timoshenko beam theory. Phys E 42(5):1727–1735. doi:10.1016/j.physe.2010.01.035

    Article  Google Scholar 

  54. Wang CM, Zhang YY, He XQ (2007) Vibration of nonlocal Timoshenko beams. Nanotechnology 18(10):105401. doi:10.1088/0957-4484/18/10/105401

    Article  ADS  Google Scholar 

  55. Aydogdu M (2009) Axial vibration of the nanorods with the nonlocal continuum rod model. Phys E 41(5):861–864. doi:10.1016/j.physe.2009.01.007

    Article  Google Scholar 

  56. 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

  57. Rahmani O, Noroozi Moghaddam MH (2014) On the vibrational behavior of piezoelectric nano-beams. Adv Mater Res 829:790–794

  58. Rahmani O, Ghaffari S (2014) Frequency analysis of nano sandwich structure with nonlocal effect. Adv Mater Res 829:231–235

  59. Rahmani O (2014) On the flexural vibration of pre-stressed nanobeams based on a nonlocal theory. Acta Physica Polonica A 125(2)

  60. Pirmohammadi AA, Pourseifi M, Rahmani O, Hoseini SAH (2014) Modeling and active vibration suppression of a single-walled carbon nanotube subjected to a moving harmonic load based on a nonlocal elasticity theory. Appl Phys A 117:1547–1555. doi:10.1007/s00339-014-8592-z

  61. Li C, Lim CW, Yu JL, Zeng QC (2011) Analytical solutions for vibration of simply supported nonlocal nanobeams with an axial force. Int J Struct Stab Dyn 11(02):257–271. doi:10.1142/s0219455411004087

    Article  MATH  MathSciNet  Google Scholar 

  62. Mohammadi B, Ghannadpour SAM (2011) Energy approach vibration analysis of nonlocal Timoshenko beam theory. Proced Eng 10:1766–1771. doi:10.1016/j.proeng.2011.04.294

    Article  Google Scholar 

  63. Eltaher MA, Emam SA, Mahmoud FF (2012) Free vibration analysis of functionally graded size-dependent nanobeams. Appl Math Comput 218(14):7406–7420. doi:10.1016/j.amc.2011.12.090

    Article  MATH  MathSciNet  Google Scholar 

  64. Ke L-L, Wang Y-S (2012) Thermoelectric-mechanical vibration of piezoelectric nanobeams based on the nonlocal theory. Smart Mater Struct 21(2):025018. doi:10.1088/0964-1726/21/2/025018

    Article  ADS  MathSciNet  Google Scholar 

  65. Ke L-L, Wang Y-S, Wang Z-D (2012) Nonlinear vibration of the piezoelectric nanobeams based on the nonlocal theory. Compos Struct 94(6):2038–2047. doi:10.1016/j.compstruct.2012.01.023

    Article  Google Scholar 

  66. Murmu T, Adhikari S (2012) Nonlocal elasticity based vibration of initially pre-stressed coupled nanobeam systems. Eur J Mech A Solids 34:52–62. doi:10.1016/j.euromechsol.2011.11.010

    Article  MathSciNet  Google Scholar 

  67. Şimşek M (2012) Nonlocal effects in the free longitudinal vibration of axially functionally graded tapered nanorods. Comput Mater Sci 61:257–265. doi:10.1016/j.commatsci.2012.04.001

    Article  Google Scholar 

  68. Torabi K, Nafar Dastgerdi J (2012) An analytical method for free vibration analysis of Timoshenko beam theory applied to cracked nanobeams using a nonlocal elasticity model. Thin Solid Films 520(21):6595–6602. doi:10.1016/j.tsf.2012.06.063

    Article  ADS  Google Scholar 

  69. 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 

  70. Behera L, Chakraverty S (2014) Free vibration of nonhomogeneous Timoshenko nanobeams. Meccanica 49(1):51–67. doi:10.1007/s11012-013-9771-2

    Article  MATH  MathSciNet  Google Scholar 

  71. Anjomshoa A (2013) Application of Ritz functions in buckling analysis of embedded orthotropic circular and elliptical micro/nano-plates based on nonlocal elasticity theory. Meccanica 48(6):1337–1353. doi:10.1007/s11012-012-9670-y

    Article  MATH  MathSciNet  Google Scholar 

  72. Babaei H, Shahidi AR (2013) Free vibration analysis of quadrilateral nanoplates based on nonlocal continuum models using the Galerkin method: the effects of small scale. Meccanica 48(4):971–982. doi:10.1007/s11012-012-9646-y

    Article  MATH  MathSciNet  Google Scholar 

  73. Karamooz Ravari MR, Shahidi AR (2013) Axisymmetric buckling of the circular annular nanoplates using finite difference method. Meccanica 48(1):135–144. doi:10.1007/s11012-012-9589-3

    Article  MATH  MathSciNet  Google Scholar 

  74. Karamooz Ravari MR, Talebi S, Shahidi AR (2014) Analysis of the buckling of rectangular nanoplates by use of finite-difference method. Meccanica 49(6):1443–1455. doi:10.1007/s11012-014-9917-x

    Article  MATH  MathSciNet  Google Scholar 

  75. Kiani K (2010) Longitudinal and transverse vibration of a single-walled carbon nanotube subjected to a moving nanoparticle accounting for both nonlocal and inertial effects. Phys E 42(9):2391–2401. doi:10.1016/j.physe.2010.05.021

    Article  Google Scholar 

  76. Kiani K (2010) Application of nonlocal beam models to double-walled carbon nanotubes under a moving nanoparticle. Part I: theoretical formulations. Acta Mechanica 216(1–4):165–195. doi:10.1007/s00707-010-0362-1

    Google Scholar 

  77. Kiani K (2010) Application of nonlocal beam models to double-walled carbon nanotubes under a moving nanoparticle. Part II: parametric study. Acta Mechanica 216(1–4):197–206. doi:10.1007/s00707-010-0363-0

    Google Scholar 

  78. Kiani K, Mehri B (2010) Assessment of nanotube structures under a moving nanoparticle using nonlocal beam theories. J Sound Vib 329(11):2241–2264. doi:10.1016/j.jsv.2009.12.017

    Article  ADS  Google Scholar 

  79. Şimşek M (2010) Dynamic analysis of an embedded microbeam carrying a moving microparticle based on the modified couple stress theory. Int J Eng Sci 48(12):1721–1732. doi:10.1016/j.ijengsci.2010.09.027

    Article  MATH  Google Scholar 

  80. Şimşek M (2010) Vibration analysis of a single-walled carbon nanotube under action of a moving harmonic load based on nonlocal elasticity theory. Phys E 43(1):182–191. doi:10.1016/j.physe.2010.07.003

    Article  Google Scholar 

  81. Kiani K (2011) Small-scale effect on the vibration of thin nanoplates subjected to a moving nanoparticle via nonlocal continuum theory. J Sound Vib 330(20):4896–4914. doi:10.1016/j.jsv.2011.03.033

    Article  ADS  Google Scholar 

  82. Şimşek M (2011) Nonlocal effects in the forced vibration of an elastically connected double-carbon nanotube system under a moving nanoparticle. Comput Mater Sci 50(7):2112–2123. doi:10.1016/j.commatsci.2011.02.017

    Article  Google Scholar 

  83. Ghorbanpour Arani A, Roudbari MA, Amir S (2012) Nonlocal vibration of SWBNNT embedded in bundle of CNTs under a moving nanoparticle. Phys B 407(17):3646–3653. doi:10.1016/j.physb.2012.05.043

    Article  ADS  Google Scholar 

  84. Kiani K, Wang Q (2012) On the interaction of a single-walled carbon nanotube with a moving nanoparticle using nonlocal Rayleigh, Timoshenko, and higher-order beam theories. Eur J Mech A Solids 31(1):179–202. doi:10.1016/j.euromechsol.2011.07.008

    Article  MATH  MathSciNet  Google Scholar 

  85. Burl JB (1998) Linear optimal control: H (2) and H (Infinity) methods. Addison-Wesley Longman Publishing Co., Inc., Redwood City

    Google Scholar 

  86. Kwakernaak H, Sivan R (1972) Linear optimal control systems, vol 172. Wiley-Interscience, New York

    MATH  Google Scholar 

  87. Gupta S, Batra R (2008) Continuum structures equivalent in normal mode vibrations to single-walled carbon nanotubes. Comput Mater Sci 43(4):715–723

    Article  Google Scholar 

  88. Eringen AC (1983) On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 54(9):4703–4710

    Article  ADS  Google Scholar 

  89. Wang L, Hu H (2005) Flexural wave propagation in single-walled carbon nanotubes. Phys Rev B 71(19):195412

    Article  ADS  Google Scholar 

  90. Wang Q, Han Q, Wen B (2008) Estimate of material property of carbon nanotubes via nonlocal elasticity. Adv Theor Appl Mechan 1(1):1–10

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Smart Structures and New Advanced Materials Laboratory, Department of Mechanical Engineering, University of Zanjan, Zanjan, Iran

    M. Pourseifi, O. Rahmani & S. A. H. Hoseini

Authors
  1. M. Pourseifi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. O. Rahmani
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. A. H. Hoseini
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to O. Rahmani.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pourseifi, M., Rahmani, O. & Hoseini, S.A.H. Active vibration control of nanotube structures under a moving nanoparticle based on the nonlocal continuum theories. Meccanica 50, 1351–1369 (2015). https://doi.org/10.1007/s11012-014-0096-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11012-014-0096-6

Keywords

Search

Navigation