Skip to main content
SpringerLink
Log in

Nonlocal second-order strain gradient elasticity model and its application in wave propagating in carbon nanotubes

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

Abstract

Recent experimental studies indicate that Young’s modulus of carbon nanotubes increases steeply with tube diameter decreasing. The consideration of this effect is of great importance for the fabrication and exploitation of nano-electromechanical devices. Nevertheless, the rapid stiffness enhancement effect noticed from experimental observation maybe unable to be predicted by using size-dependent elasticity models available in literatures. It is strongly necessary to further shed light on the size-dependent mechanical mechanism and characterize the rapid strengthening effect of stiffness for nano-sized materials. To achieve this goal, the nonlocal second-order strain gradient elasticity model is established by introducing the second-order strain gradient field with nonlocal effect into the stored energy function of nonlocal first-order strain gradient elasticity theory. With the aids of the laws of thermodynamics, the constitutive relations are obtained. The Hamilton principle is used to derive the governing equations of equilibrium and boundary conditions. The proposed model is applied to investigate the problem of wave propagating in carbon nanotubes. The new dispersion relations derived are presented for evaluating the influences of size-dependent parameters on the characteristics of wave propagation. The results show that present model can predict the rapid increasing effect of carbon nanotubes with the decrease of tube size.

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

Similar content being viewed by others

References

  • Agrawal PM, Sudalayandi BS, Raff LM, Komanduri R (2006) A comparison of different methods of Young’s modulus determination for single-wall carbon nanotubes (SWCNT) using molecular dynamics (MD) simulations. Comput Mater Sci 38:271–281

    Article  Google Scholar 

  • Aifantis EC (1992) On the role of gradients in the localization of deformation and fracture. Int J Eng Sci 30:1279–1299

    Article  MATH  Google Scholar 

  • Akgöz B, Civalek O (2011) Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams. Int J Eng Sci 49:1268–1280

    Article  MathSciNet  MATH  Google Scholar 

  • Arash B, Wang Q (2012) A review on the application of nonlocal elastic models in modeling of carbon nanotubes and graphenes. Comput Mater Sci 51:303–313

    Article  Google Scholar 

  • Bahrami A (2017a) A wave-based computational method for free vibration, wave power transmission and reflection in multi-cracked nanobeams. Compos B Eng 120:168–181

    Article  Google Scholar 

  • Bahrami A (2017b) Free vibration, wave power transmission and reflection in multi-cracked nanorods. Compos B Eng 127:53–62

    Article  MathSciNet  Google Scholar 

  • Bahrami A, Teimourian A (2015) Nonlocal scale effects on buckling, vibration and wave reflection in nanotubes via wave propagation approach. Compos Struct 134:1061–1075

    Article  Google Scholar 

  • Bahrami A, Teimourian A (2016) Study on the effect of small scale on the wave reflection in carbon nanotubes using nonlocal Timoshenko beam theory and wave propagation approach. Compos B Eng 91:492–504

    Article  Google Scholar 

  • Bahrami A, Teimourian A (2017) Small scale effect on vibration and wave power reflection in circular annular nanoplates. Compos B Eng 109:214–226

    Article  MATH  Google Scholar 

  • Behera L, Chakraverty S (2017) Recent researches on nonlocal elasticity theory in the vibration of carbon nanotubes using beam models: a review. Arch Comput Method Eng 24:481–494

    Article  MathSciNet  MATH  Google Scholar 

  • Bi MH, Hao YA, Zhang JM, Lei M, Bi K (2017) Particle size effect of BaTiO3 nanofillers on the energy storage performance of polymer nanocomposites. Nanoscale 9:16386–16395

    Article  Google Scholar 

  • Bouchaala AMS (2018) Size effect of a uniformly distributed added mass on a nanoelectromechanical resonator. Microsyst Technol 24:2765–2774

    Article  Google Scholar 

  • Chen P, Xiao TY, Qian YH, Li SS, Yu SH (2013) A nitrogen-doped graphene/carbon nanotube nanocomposite with synergistically enhanced electrochemical activity. Adv Mater 25:3192–3196

    Article  Google Scholar 

  • Dehrouyeh-Semnani AM, Bahrami A (2016) On size-dependent Timoshenko beam element based on modified couple stress theory. Int J Eng Sci 107:134–148

    Article  MathSciNet  MATH  Google Scholar 

  • Deng SC, Liu JX, Liang NG (2007) Wedge and twist disclinations in second strain gradient elasticity. Int J Solids Struct 44:3646–3665

    Article  MATH  Google Scholar 

  • Dupuis AC (2005) The catalyst in the CCVD of carbon nanotubes—a review. Prog Mater Sci 50:929–961

    Article  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 filed theories. Springer, New York

    MATH  Google Scholar 

  • Eringen AC, Edelen DBG (1972) On nonlocal elasticity. Int J Eng Sci 10:233–248

    Article  MathSciNet  MATH  Google Scholar 

  • Fang TH, Chang WJ, Feng YL (2016) Mechanical characteristics of graphene nanoribbons encapsulated single-walled carbon nanotubes using molecular dynamics simulations. Appl Surf Sci 356:221–225

    Article  Google Scholar 

  • Golmakani ME, Vahabi H (2017) Nonlocal buckling analysis of functionally graded annular nanoplates in an elastic medium with various boundary conditions. Microsyst Technol 23:3613–3628

    Article  Google Scholar 

  • Govindjee S, Sackman JL (1999) On the use of continuum mechanics to estimate the properties of nanotubes. Solid State Commun 110:227–230

    Article  Google Scholar 

  • Goya K, Fuchiwaki Y, Tanaka M, Addinall R, Ooie T, Sugino T, Asaka K (2017) A micropipette system based on low driving voltage carbon nanotube actuator. Microsyst Technol 23:2657–2661

    Article  Google Scholar 

  • Hsieh JY, Lu JM, Huang MY, Hwang CC (2006) Theoretical variations in the Young’s modulus of single-walled carbon nanotubes with tube radius and temperature: a molecular dynamics study. Nanotechnology 17:3920–3924

    Article  Google Scholar 

  • Iijima S (1991) Helical microtubules of graphitic carbon. Nature 354:56–58

    Article  Google Scholar 

  • Ilkhani MR, Bahrami A, Hosseini-Hashemi SH (2016) Free vibration of thin rectangular nano-plates using wave propagation approach. Appl Math Model 40:1287–1299

    Article  MathSciNet  Google Scholar 

  • Karparvarfard SMH, Asghari M, Vatankhah R (2015) A geometrically nonlinear beam model based on the second strain gradient theory. Int J Eng Sci 91:63–75

    Article  MathSciNet  MATH  Google Scholar 

  • Ke LL, Wang YS (2011) Flow-induced vibration and instability of embedded double-walled carbon nanotubes based on a modified couple stress theory. Phys E 43:1031–1039

    Article  Google Scholar 

  • Khodabakhshi P, Reddy JN (2015) A unified integro-differential nonlocal model. Int J Eng Sci 95:60–75

    Article  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 

  • Lazar M, Maugin GA, Aifantis EC (2006) Dislocations in second strain gradient elasticity. Int J Solids Struct 43:1787–1817

    Article  MATH  Google Scholar 

  • Lekawa-Raus A, Patmore J, Kurzepa L, Bulmer J, Koziol K (2014) Electrical properties of carbon nanotube based fibers and their future use in electrical wiring. Adv Funct Mater 24:3661–3682

    Article  Google Scholar 

  • Li L, Hu YJ, Ling L (2016a) Wave propagation in viscoelastic single-walled carbon nanotubes with surface effect under magnetic field based on nonlocal strain gradient theory. Phys E 75:118–124

    Article  Google Scholar 

  • Li CL, Guo HL, Tian XG (2016b) A size-dependent generalized thermoelastic diffusion theory and its application. J Therm Stress 40:603–626

    Article  Google Scholar 

  • Li CL, Guo HL, Tian XG (2017) Shock-induced thermal wave propagation and response analysis of a viscoelastic thin plate under transient heating loads. Wave Random Complex 28:270–286

    Article  Google Scholar 

  • Li CL, Guo HL, Tian XG (2018) Size-dependent effect on thermo-electro-mechanical responses of heated nano-sized piezoelectric plate. Wave Random Complex. https://doi.org/10.1080/17455030.2018.1450539

    Google Scholar 

  • Liew KM, Wang Q (2007) Analysis of wave propagation in carbon nanotubes via elastic shell theories. Int J Eng Sci 45:227–241

    Article  Google Scholar 

  • Liew KM, He XQ, Wong CH (2004) On the study of elastic and plastic properties of multi-walled carbon nanotubes under axial tension using molecular dynamics simulation. Acta Mater 52:2521–2527

    Article  Google Scholar 

  • 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 

  • Lipomi DJ, Vosgueritchian M, Tee BCK, Hellstrom SL, Lee JA, Fox CH, Bao ZN (2011) Skin-like pressure and strain sensors based on transparent elastic films of carbon nanotubes. Nat Nanotechnol 6:788–792

    Article  Google Scholar 

  • Liu X, Yang QS, Liew KM, He XQ (2017) Superstretchability and stability of helical structures of carbon nanotube/polymer composite fibers: coarse-grained molecular dynamics modeling and simulation. Carbon 115:220–228

    Article  Google Scholar 

  • Ma HM, Gao XL, Reddy N (2008) A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. J Mech Phys Solids 56:3379–3391

    Article  MathSciNet  MATH  Google Scholar 

  • Mindlin RD (1964) Micro-structure in linear elasticity. Arch Ration Mech Anal 16:51–78

    Article  MathSciNet  MATH  Google Scholar 

  • Mindlin RD (1965) Second gradient of strain and surface-tension in linear elasticity. Int J Solids Struct 1:414–438

    Google Scholar 

  • Narendara S, Mahapatra DR, 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 49:509–522

    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 

  • Polizzotto C (2014) Surface effects, boundary conditions and evolution laws within second strain gradient plasticity. Int J Plast 60:197–216

    Article  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • Reddy JN (2007) Nonlocal theories for bending, buckling and vibration of beams. Int J Eng Sci 45:288–307

    Article  MATH  Google Scholar 

  • Reddy JN (2011) Microstructure-dependent couple stress theories of functionally graded beams. J Mech Phys Solids 59:2382–2399

    Article  MathSciNet  MATH  Google Scholar 

  • Reddy JN, Srinivasa AR (2014) Non-linear theories of beams and plates accounting for moderate rotations and material length scales. Int J Nonlinear Mech 66:43–53

    Article  Google Scholar 

  • Shodja HM, Ahmadpoor F, Tehranchi A (2012) Calculation of the additional constants for fcc materials in second strain gradient elasticity: behavior of a nano-sized Bernoulli–Euler beam with surface effects. J Appl Mech 79:021008

    Article  Google Scholar 

  • Simsek M, Reddy JN (2013) Bending and vibration of functionally graded microbeams using a new higher order beam theory and the modified couple stress theory. Int J Eng Sci 64:37–53

    Article  MathSciNet  MATH  Google Scholar 

  • Srinivasa AR, Reddy JN (2013) A model for a constrained, finitely deforming, elastic solid with rotation gradient dependent strain energy, and its specialization to von Kármán plates and beams. J Mech Phys Solids 61:873–885

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  • Tang YG, Liu Y, Zhao D (2016) Viscoelastic wave propagation in the viscoelastic single walled carbon nanotubes based on nonlocal strain gradient theory. Phys E 84:202–208

    Article  Google Scholar 

  • Thai HT (2012) A nonlocal beam theory for bending, buckling, and vibration of nanobeams. Int J Eng Sci 52:56–64

    Article  MathSciNet  MATH  Google Scholar 

  • Thostenson ET, Ren ZF, Chou TW (2001) Advances in the science and technology of carbon nanotubes and their composites: a review. Compos Sci Technol 61:1899–1912

    Article  Google Scholar 

  • Tjong SC (2013) Recent progress in the development and properties of novel metal matrix nanocomposites reinforced with carbon nanotubes and graphene nanosheets. Mater Sci Eng R 74:281–350

    Article  Google Scholar 

  • Toupin RA (1963) Elastic materials with couple-stresses. Arch Ration Mech Anal 11:385–414

    Article  MathSciNet  MATH  Google Scholar 

  • Treacy MMJ, Ebbesen TW, Gibson JM (1996) Exceptionally high Young’s modulus observed for individual carbon nanotubes. Nature 381:678–680

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Yao N, Lordi V (1998) Young’s modulus of single-walled carbon nanotubes. J Appl Phys 64:1939–1943

    Article  Google Scholar 

  • Yu CH, Shi L, Yao Z, Li DY, Majumdar A (2005) Thermal conductance and thermopower of an individual single-wall carbon nanotube. Nano Lett 5:1842–1846

    Article  Google Scholar 

Download references

Acknowledgements

This work is supported National Natural Science Foundation of China (11572237, 11732007) and the Fundamental Research Funds for the Central Universities.

Author information

Authors and Affiliations

  1. State Key Laboratory for Strength and Vibration of Mechanical Structures, Shaanxi Engineering Research Center of Nondestructive Testing and Structural Integrity Evaluation, Xi’an Jiaotong University, Xi’an, 710049, People’s Republic of China

    Chenlin Li, Huili Guo & Xiaogeng Tian

Authors
  1. Chenlin Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Huili Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiaogeng Tian
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiaogeng Tian.

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

Li, C., Guo, H. & Tian, X. Nonlocal second-order strain gradient elasticity model and its application in wave propagating in carbon nanotubes. Microsyst Technol 25, 2215–2227 (2019). https://doi.org/10.1007/s00542-018-4085-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00542-018-4085-x

Search

Navigation